Research ArticleThe Electronic Structure of Short Carbon NanotubesThe Effects of Correlation

Vijay Gopal Chilkuri1 Stefano Evangelisti1 Thierry Leininger1 and Antonio Monari23

1Laboratoire de Chimie et Physique Quantiques IRSAMC Universite de Toulouse et CNRS 118 Route de Narbonne31062 Toulouse Cedex France2Universite de Lorraine Nancy Theorie-Modelisation-Simulation SRSMC Boulevard des Aiguillettes31062 Vandœuvre-les-Nancy France3CNRS Theorie-Modelisation-Simulation SRSMC Boulevard des Aiguillettes 31062 Vandœuvre-les-Nancy France

Correspondence should be addressed to Stefano Evangelisti stefanoevangelistiirsamcups-tlsefr

Received 20 April 2015 Revised 16 July 2015 Accepted 5 August 2015

Academic Editor Ashok Chatterjee

Copyright copy 2015 Vijay Gopal Chilkuri et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a tight binding and ab initio study of finite zig-zag nanotubes of various diameters and lengths The verticalenergy spectra of such nanotubes are presented as well as their spin multiplicities The calculations performed using the tightbinding approach show the existence of quasi-degenerate orbitals located around the Fermi level thus suggesting the importanceof high-quality ab initio methods capable of a correct description of the nondynamical correlation Such approaches (CompleteActive Space SCF and Multireference Perturbation Theory calculations) were used in order to get accurate ground and nearestexcited-state energies along with the corresponding spin multiplicities

1 Introduction

Besides the two lightest elements hydrogen and heliumcarbon is one of themost widespread elements in the universeand one of the best known ones Indeed its three-dimensionallotropic forms diamond and graphite are well known sincethe antiquity For this reason the recent discovery of newlow-dimensional allotropic forms such as fullerenes carbonnanotubes and graphene came out rather unexpected [1 2]In a few years a completely new branch of science wasborn whose scientific and technological impact can hardlybe overestimated Indeed these findings had and still havean enormous importance in the discovery of novel andadvanced materials and have been one of the key factorsin the development of nanoscience The peculiar propertiesof graphene (that concern both the ldquoinfiniterdquo ideal sheetand finite nonoislands) raise new challenging theoreticalproblems whose understanding and modeling will bringsignificant insight in our comprehension of the structure ofmatter Nanotubes which can be obtained from a graphenestripe by enrolling it along longitudinal axes present

an even larger spectrum of interesting behaviors It is clearthat a better understanding of these systems would make ourknowledge of the general properties of solid-state physics andchemistry much deeper

The new allotropic forms of carbon share the particu-larity of being all of low dimensionality Indeed the almostspherical fullerene [3] can be considered as a quasi-zero-dimensional (0D) structure while the graphene one-atom-thick surface is a strictly two-dimensional (2D) structureBetween these two extreme cases are located the carbon nan-otubes that can be considered as essentially one-dimensional(1D) materials The presence of carbon 120587-conjugated latticeusually resembling a honeycomb hexagonal lattice is thecommon feature to all these forms and is at the origin oftheir striking properties Indeed the strength of the carbon-carbon sigma bond and the rigidity of the resulting structureconfer graphene and nanotubes a great stability and rigidityand make them appealing to be used as building-blockfibers for materials having to resist considerable stress [4]Another remarkable aspect that should be stressed is thehydrophobicity and the resistance of these materials a fact

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2015 Article ID 475890 14 pageshttpdxdoiorg1011552015475890

2 Advances in Condensed Matter Physics

that makes them ideal candidates for drug delivery andvectorization [5] In addition to these mechanical propertieswhich are already extensively exploited nowadays there arealso peculiar and surprising electronic properties exhibitedby these materials For instance graphene can be consideredas a zero-gap semiconductor showing an infinite electronicmobility at the Fermi level Carbon nanotubes [6] on theother hand which are formally obtained by enrolling agraphene sheet along a helicoidal symmetry axis show a vastvariety of electronic behaviors They exhibit extremely inter-esting properties with a quite complex and rich structure-property correlation pattern In particular depending on theway the original graphene sheet is enrolled around the axisnanotubes can present or not chirality and their conductivityproperties can vary dramatically So different classes ofnanotubes can present a behavior that passes from metallicto low-gap semiconductor to real semiconductor materialsThis peculiar characteristic can efficiently be exploited in thefield of opto- andmolecular-electronics devices the potentialperformance and efficiency of carbon based nanostructurescould potentially induce a revolution in that field comparableto the one produced by silicon based microdevices [1 7]

Because of the scientific and technological importance ofcarbon nanotubes a huge amount of scientific work boththeoretical and experimental has been devoted to the studyof the properties and the electronic structure of these systemsUp to now however the great majority of such studieswere conducted on infinitely extended systems Moreoverin many cases the Hamiltonians used were restricted tosemiempiricalmodels ab initio calculations beingmore oftenperformed at density functional theory level only Howevereven if the study of extended systems deserves a considerableimportance both from a theoretical and a practical pointof view finite-size effects occurring in short finite clusterscan also reveal new and unexpected properties which canstrongly modify the physical behavior with respect to theinfinite system In particular it is known that graphenenanoclusters and open carbon nanotubes can present the so-called edge orbitals that are partially filled orbitals whoseelectron density is mainly concentrated on the border regionsof the system A peculiarity of these orbitals is given by thefact that their energy lies close to the Fermi level that isthey are the frontier orbitals of the system Depending onthe geometry of the cluster the edge orbitals show interestingdegeneracy patterns These features will imply the presenceof different types of open-shell electronic structures possiblyinducing magnetic behaviors characterized by a high-spinground state Moreover there is the possibility of a closingof the Fermi level gap to give rise to a metallic behavior inthe limit of large systems

The presence of an open-shell structure will also inducea strong multiconfigurational nature of the low-lying statesof the system (in particular of those of lower multiplicity)This fact implies that multireference methods are required toproperly take into account the effects of the static correlation[8] Note as previously evoked that the magnetic behavioris strongly dependent on the geometry of the cluster andcan be rationalized by invoking the Ovchinnicov Rule [9 10]and Liebrsquos theorem [11] For instance it has been argued that

triangular-shape clusters are ferromagnetic while hexagonalshapes are expected to be diamagnetic and show a gap at theFermi level Note also that the magnitude of the magneticcoupling between the states can be tuned by varying thesize of the cluster All these important characteristics canbeen exploited by connecting in a network different grapheneclusters characterized by differentmagnetic properties Let usalso cite the fact that the required precise control on the shapeand size of the cluster can nowadays be achieved by bottom-up techniques based on the precise deposition of carbonatoms on different surfaces In fact such techniques havealready allowed to produce very specific graphene clustersand even graphene antidots [12] Finally it is also noteworthyto remind that carbon nanostructures can be used to host andstabilize magnetic linear structures and that the interactionwith conjugated carbon structures can also strongly modifythe magnetic behavior of the linear aggregate

In the present contribution we want to extend our anal-ysis of border effects performed on graphene nanoclusters tothe domain of nanotubes In particular we will concentrateon zig-zag nonchiral nanotubes and we will consider onlyopen structures that present terminal edges saturated withhydrogens Indeed this simple choice allows to keep all thecarbon atoms 1199041199012-hybridized and to limit the edge effectsThis is often done in theoretical studies even if in mostexperimental samples there are no terminal hydrogens

Infinite nanotubes are uniquely characterized (apart fromthe sign of helicity if they are chiral) by a pair of non-negative integer numbers 119899 and 119898 (with 119898 le 119899) andthe corresponding nanotube is denoted as (119899119898) These twonumbers describe how a graphene sheet is enrolled aroundthe helicoidal axes in order to generate the infinite nanotubeAs the two extreme cases we find nonchiral nanotubes thezig-zag nanotubes are of the type (119899 0) while the corre-sponding armchair structures are of type (119899 119899) When 119898 isdifferent from both 0 and 119899 we have chiral nanotubes Iffinite nanotubes are considered the situation is much morecomplex since one must consider not only the length ofthe nanotube itself but also the details of its two bordersThis is particularly true for chiral nanotubes whose helicityprevents the possibility of cutting the extremities in such away to respect the nanotube symmetry along the central axisNonchiral nanotubes however can be truncated in such away that the two edges respect the nanotube symmetry Inzig-zag nanotubes (the systems that are the object of thiswork) the integer number 119899 indicates the number of con-tiguous hexagon units that we find on a circular path aroundthe tube the 119898 value on the other hand is redundant sinceit is identically equal to 0 Finite-length zig-zag nanotubesare characterized by 119899 as the corresponding infinite systemsand by the length of the nanotube We will indicate by 119901

this length and in order to avoid confusion with the well-established notation (119899119898) we indicate by (119899 119901) a fragmentof a (119899 0) (ie zig-zag) nanotube having length 119901

To study the nature of edge orbitals and their influenceon the global electronic structure of the nanostructure aswell as its evolution with the different nanotube structuralparameters we will apply two different computational strate-gies a very simple semiempirical Huckel approach which

Advances in Condensed Matter Physics 3

can treat a large variety of different structures and highlycorrelated multireference wave-function-based approachesable to give quantitative results but whose use is restricted torelatively small systemsThe first crude approximation allowsus to obtain a qualitative overview of the density of statesand in particular to characterize the orbital degeneracies atthe Fermi level Note that this approach will also allow usto consider very large aggregates that is to come close tothe infinite system limit The wave-function-based approachon the other hand takes care of the static and dynamiccorrelation and is able to study the low-lying spectrumof the magnetic states evaluating its evolution with thegeometry and the structural parameters To the best of ourknowledge this is the first contribution applying high levelMultireference Perturbation Theory (MRPT) to the study ofthis kind of systems A similar approach was already appliedto closed polyacenes structures [13] whereas most of previousworks on nanotubes [14ndash22] or nanoribbons [23] principallyrely on density functional theory or tight binding resultsThe main conclusion of studies on There also have beensome studies in the solid-state community on the energyspectrum [24] and properties of single walled nanotubes [25]using sophisticated analytic tools Similar structures as thoseconsidered consist in an antiferromagnetic coupling of twoelectrons localized in the edge orbitals

This paper is organized as follows In Section 2 Theo-retical Methods we briefly describe the methods that weemployed in this study and we justify their use The com-putational details of the calculations are given in Section 3in order to permit the reproduction of our calculations infuture further studies In Section 4 we present the nanotubestructure and symmetry along with the particular terminol-ogy used in this work Results at theHuckel and ab initio levelare presented and discussed in Section 5 Final conclusionsare drawn in Section 6 together with plans for future works

2 Theoretical Methods

In this section we briefly describe the theoretical methodsand the computational approaches used in the present workThe computational details concerning both the ab initio andthe tight binding approaches will be discussed in detail in thenext section in order to facilitate the reproduction of ourresults

Carbon nanotubes as it is the case of most graphene-derived nanostructures show the presence of edge molecularorbitals (MO) that are neither doubly occupied nor totallyempty in the electronic ground state (see eg [12]) Thesimplest Hamiltonian that is able to give a qualitative descrip-tion of the electronic wavefunction of 1199041199012-hybridized carbonstructures is the Huckel Hamiltonian It is a tight bindingHamiltonian that was developed in order to describe smallpolyenes and in general aromatic compounds At Huckellevel only one119901 atomic orbital per carbon atom is considered(the orbital that is locally orthogonal to the 120590 orbitals)and this defines the 120587 system Correspondingly only oneelectron per atom is considered Despite its simplicity theHuckel Hamiltonian is capable of grasping the essence of

the electronic structure of these systems and their low-lyingenergy spectrum obtained at this level is remarkably correctIt is worth noticing that the pioneering 1947 paper on graphite(and graphene) written byWallace describes results obtainedwith this type of approach [26]

As a general strategy that we employed in previousstudies on other 119904119901

2 carbon nanostructures [13 27] weperformed preliminary tight binding calculations by usingthe model Huckel Hamiltonian [28] in order to obtain a firstapproximation of the energy spectrum at one-electron levelThis is very important for subsequent calculations at many-electron level by using an ab initio Hamiltonian in orderto clearly identify the orbitals that form a quasi-degeneratepartly occupied manifold A proper ab initio treatment ofthis group of quasi-degenerate orbitals requires the introduc-tion of the nondynamical correlation [29] and this is mostconveniently done by the use of the Complete Active SpaceSelf-Consistent Field (CAS-SCF) formalism [30] In thisapproach the correlation of the electrons distributed amongthe quasi-degenerate orbitals (the active orbitals) is fullyconsidered at a configuration interaction level At the sametime the shape of all the molecular orbitals (active or not)is fully optimized self-consistently It is known that a CAS-SCF approach is suited for a qualitative description of thelow-lying energy spectrum of open-shell systems (magneticstructures mixed-valence systems) For quantitative resultsthe introduction of the dynamical correlation is also requiredsince the weights of the Slater determinants belonging to thequasi-degenerate manifold can be strongly modified by theinteraction Among the possible multireference approachesthe Quasi-Degenerate Perturbation Theory is the only onethat is capable of treating these relatively large systems Inparticular contracted methods have the advantage of a weakcomputational dependence on the dimension of the activespace and can therefore be used in the present case In thiswork we chose the n-Electron Valence Perturbation Theoryat second-order level (NEVPT2) because of its capability ofdealing with the intruder-state problem without the need ofintroduction of arbitrary parameters [31 32]

In doing a CAS-SCF calculation the choice of theguess starting orbitals plays a key role in the final resultIndeed being a nonlinear optimization algorithm CAS-SCFadmits a huge number of different solutions sometimes notpresenting a particular physical interest It is also frequentbecause of this nonlinear character of the formalism tohave instabilities that lead to symmetry-breaking phenomenain the case of strictly degenerate states as the ones thatcharacterize nanotubes having non-Abelian symmetries Inthe present work we performed High-Spin Restricted Open-Shell Hartree-Fock (ROHF) calculations on the differentsystems by choosing as single-occupied orbitals precisely theorbitals that show a quasi-degenerate character at Huckellevel Once the manifold of quasi-degenerate orbitals hasbeen identified we performed frozen-orbital CAS-SCF (ieCAS-CI) calculations on this active space It is known thatfor states having nonionic character the resulting energyspectrum is very close to the CAS-SCF one at a considerablylower computational cost

4 Advances in Condensed Matter Physics

3 Computational Details

In the next subsections we describe separately the tight bind-ing and ab initio approaches

31 Tight Binding All computations at tight binding level(Huckel calculations in the language of chemists) have beenperformed by using a specific home-made code[33ndash35] TheHuckel Hamiltonian has been constructed starting from theconnectivity of each carbon atom by defining as usual

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119894⟩ = 120572 = 0

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119895⟩ = 120573120574

119894119895

(1)

where 120574119894119895is 1 if sites 119894 and 119895 are connected in the nanostructure

skeleton and 0 otherwise The value of the hopping integral120573 should depend in principle on the distance betweenthe two connected atoms 119894 and 119895 However since the C-C bonds have very similar length in all of our structureswe assumed the same values of 120573 for all the topologicallyconnected pair of atoms In our calculations this parameterwas fixed to the arbitrary value 120573 = minus1 We remind thatminus120573 is sometimes called the hopping integral in the physicsliterature where it is usually indicated as 119905 In the sectionResults and Discussion we report the energy spectrum ofseveral nanotubes computed at the Huckel levelThis is doneas a function of the orbital number In order to facilitate thecomparison between systems having a different number oforbitals the orbital numbers are normalized in the segment[0 1]

32 Ab Initio All the systems have been studied using theminimal STO-3G Gaussian basis [36] for both carbon andhydrogen Although this basis set is certainly too small togive quantitatively reliable results it is known to reproducecorrectly the qualitative behavior of many organic and inor-ganic systems Its reduced size on the other hand permitsto investigate at a correlated ab initio level systems whosesize is relatively large However in order to explore theeffect of a more realistic basis set a limited number ofstructures representative of each class of nanotubes havebeen studied with a larger basis set the cc-pVDZ correlationconsistent basis set of Dunning [37] These are the structurescharacterized by 119899 = 6 7 and 119901 = 4 5 for which a(311990421199011119889) contraction for C and (21199041119901) for H were used Thiscorresponds to a valence double-zeta plus polarization (vdzp)basis set

The systemsrsquo geometries have been optimized at Restrict-ed Open-Shell Hartree-Fock (ROHF) level for the lowest-energy state This means the triplet state for the nanotubescharacterized by an even value of 119899(119890 119901) and the quintet statefor the nanotubes characterized by an odd value of 119899(119900 119901)

During geometry optimization the 119863119899119889

or 119863119899ℎ

sym-metries have been imposed depending on the nanotubesclasses Thus all the coordinates have been relaxed allow-ing the nanotube to lose its cylindrical shape The onlyremaining constraint was the order of the rotation axis Actu-ally due to restrictions of the Molpro quantum chemistry

package [38] optimization has been carried out using theappropriate Abelian subgroups Even if this may resultin symmetry breaking of the electronic wavefunction itnever appeared for the systems considered in this workFor all the nanotubes it appears that only slight distor-tions from the regular ideal cylindrical shape have beenobtained Even the terminal hydrogens just slightly pointtowards the axis of the nanotube All the detailed geometriesare given in the Supplementary Material available onlineat httpdxdoiorg1011552015475890 At these optimizedgeometries the lowest states have been studied at CompleteActive Space Self-Consistent Field (CAS-SCF) level [39]Then the effect of dynamical correlation was introduced atMultireference Perturbation Theory (MR-PT) level usingthe partially contracted version of second-order n-ElectronsValence PerturbationTheory (NEVPT2) formalism [40ndash42]In all the CAS-SCF calculations the active space has beenselected by choosing the orbitals that are strictly degeneratedor quasi-degenerated with the Huckel Hamiltonian at theFermi level This means that our active spaces are eitherCAS(22) (for the (119890 119901) tubes) or CAS(44) (for the (119900 119901)

tubes) The geometries have been optimized at ROHF levelfor the high-spin wavefunctionThe orbitals obtained thereofhave then been kept frozen for the CAS-SCF calculationson the other spin multiplicities In this way our calculationsare actually of CAS-CI [43 44] type which in our case giveenergies close to CAS-SCF onesTheNEVPT2 formalism hasthen been applied to these CAS-CI wavefunctions in order torecover the dynamical correlation

Obviously our geometrical constraints prevent Jahn-Teller distortion which could be expected in the case ofdegenerate orbitals In this preliminary work we decided notto consider this possibility However due to the stiffness ofthe nanotube backbone we believe these distortions to be ofsmall size

4 Electronic Structure ofCarbon 119904119901

2 Nanostructures

The net of carbon atoms forming a graphene layer is theconceptual starting point to produce most of the carbon 1199041199012-hybridized nanostructures A crucial feature to rationalizethe relation between structure and electronic properties andmore generally to study finite-size effects in these structuresis to recognize that the honeycomb graphene skeleton formsa bipartite lattice [12] with two compenetrating triangularsublattices 119860 and 119861 Each carbon atom belonging to agraphene nanostructure is associated with one of these twosublattices and one can speak of119860-type and 119861-type grapheneatoms or centers A lattice is bipartite if each 119860-type atomhas only 119861-type nearest neighbors and vice versa So forinstance zig-zag edges are composed of atoms that all belongto the same sublattices A necessary condition for a latticeto be bipartite is the absence of odd-number carbon cyclesIn order to rationalize the emergence of magnetic propertiesin graphene nanosystems one can recall Liebrsquos theorem[11] This theorem also known as the theorem of itinerantmagnetism and usually applied within the framework of the

Advances in Condensed Matter Physics 5

Hubbard one-orbital model is able to predict the total spinof the ground state in bipartite lattices In particular one cansee that an imbalance on the number of atom in one sublatticeresults in a magnetic ground state with spin

119878 =

1003816100381610038161003816119873119860minus 119873119861

1003816100381610038161003816

2

(2)

where 119873119860

and 119873119861are the numbers of atoms of 119860- and

119861-type respectively It is also possible to show that fora Huckel Hamiltonian |119873

119860minus 119873119861| is also the number of

eigenvalues equal to zero It is important to note that themagnetization originates from localized edge-states that givealso rise to a high density of states at the Fermi levelwhich in turn can determine a spin polarization instabilityMoreover the relation between the unbalanced number ofatoms and the ground state spin implies that two centerswill be ferromagnetically coupled if they belong to the samesublattice and antiferromagnetically coupled if they do not[45]

We consider now the symmetry of the (119899 119901) nanotubesIt is easy to see that these structures fall into two classesaccording to the parity of the length 119901 The (119899 119901) tubes have119863119899ℎ

symmetry if 119901 is odd ((119899 119900) tubes) and 119863119899119889

symmetryfor 119901 even ((119899 119890) tubes) In all these cases (with the partialexception of the case 119899 = 2 which is however too narrow toform a tube) we are in the presence of non-Abelian symmetrygroups We recall that our ab initio calculations have beenperformed by usingAbelian subgroups of the full point groupof the system According to the parity even (119890) or odd (119900) of 119899and 119901 we have the following Abelian subgroups119862

2V for (119890 119890)1198632ℎfor (119890 119900) 119862

2ℎfor (119900 119890) and 119862

2V for (119900 119900)We call edge carbon atoms the carbon atoms that are

saturated with hydrogens The role of these edge atoms iscrucial in order to understand the finite-size effects in carbonnanotubes The edge carbon atoms of the tubes of type (119890 119901)can support molecular orbitals (MO) of 120587-nature havingalternating sign on any pair of consecutive hexagons Let usconsider the MO combinations that are localized on one ofthe tube extremities only AtHuckel level it is straightforwardto verify that both these alternating edge combinations areeigenfunctions of the one-electron Hamiltonian correspond-ing to eigenvalues equal to zeroTherefore the Huckel energyspectrum presents a pair of degenerate orbitals at the Fermilevel for all (119890 119901) tubes regardless of the parity of 119901 Thesituation is different as far as the ab initio Hamiltonian isconsidered since these two localized edge combinations areno longer exact molecular orbitals of the Fock HamiltonianAs a result at ab initio level the exact degeneracy can be lostIn particular it turns out that while the two edge orbitalsare exactly degenerate for (119890 119890) tubes the degeneracy is onlyapproximated in the case of the (119890 119900) structures This pointcan be easily understood by symmetry considerations In factthe two localized edge orbitals are interchanged by some ofthe symmetry operations of the point group of the system(119863119899ℎ

for (119890 119900) and 119863119899119889

for (119890 119890) tubes) For this reasonthey give rise to a representation of the molecule symmetrygroup In the case of (119890 119890) tubes the representation isirreducible and it still corresponds to two exactly degenerate

molecular orbitals of the system (eg see Figure 6 wherethe two degenerate orbitals have 119890

3symmetry) In the case

of the (119890 119900) tubes on the other hand the two combinationsbelong to two different irreducible representations of theAbelian symmetry group of the system The two orbitals areonly approximately degenerate (see Figure 7 where the twoorbitals have 119887

1119906and 1198872119892

symmetries resp) This argumentexplains the remarkable difference between the tight bindingand ab initio energy spectra in the Fermi level region for the(119890 119900) tubes

5 Results and Discussion

51 Tight Binding Even if the tight binding approach repre-sents a crude approximation of a chemical system it can beextremely useful in order to sketch out some general tenden-cies and to elucidate the behavior of the different classes ofcompoundsThis is particularly true in the present case sincea tight binding approach allows the treatment of very largesystems due to its extremely reduced computational costs Itis possible in this way to explore the behavior of virtuallyldquoinfiniterdquo systems

Short zig-zag nanotubes can be divided into four dif-ferent classes characterized by different symmetries andmolecular-orbital patterns depending on the parity of 119899 and119901 The Huckel calculations for these classes permitted tocharacterize the different active spaces which were essentiallyconfirmed at the ab initio level

Nanotubes of the type (119899 119890) have 119863119899119889

symmetry whilethose of type (119899 119900) have 119863

119899ℎsymmetry At the Huckel level

both (119890 119890) and (119890 119900) have two exactly degenerate orbitalsat the Fermi level hosting two electrons This is illustratedin Figures 1 and 2 where the energy spectra of (6 4) and(6 5) structures are reported For the sake of comparison thespectra of long (6 100) or very wide (100 4) and (100 5)structures are also reported in the figures None of thesespectra shows the presence of an energy gap at the Fermi leveland the wide structures have even an infinite density of levelsat the Fermi level Nanotubes of the types (119900 119890) and (119900 119900) onthe other hand have four quasi-degenerate levels (two quasi-degenerate pairs of exactly degenerate levels) near the Fermilevel hosting a total of four electrons This fact is illustratedfor the (7 4) and (7 5) structures in Figures 3 and 4 Fromthese figures it appears that in the case of very long (119900 119901)

nanotubes four degenerate isolated orbitals hosting a total offour electrons are located at the Fermi level As a systematicstudy of large systems at theHuckel level involves a digressionfrom the present topic a general analysis of the four classesof the nanotubes will follow

511 Nanotubes of Type 119890 119890 The Huckel calculation for thenanotubes characterized by an even parity for both 119899 and119901 showed that there were two orbitals at the Fermi levelThe calculations Figure 1 are shown for very wide and verylong nanotubes for the purpose of comparison The abscissa(119896) represents the Huckel orbital number mapped onto theinterval [0 1] It can be seen that there is no gap at the Fermilevel this allows us to characterize such nanotubes as being

6 Advances in Condensed Matter Physics

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100464

Figure 1 Huckel orbital energies for the nanotube (6 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100565

Figure 2 Huckel orbital energies for the nanotube (6 5) where |120573|is the energy unit and 119896 is the normalized orbital number

metallic One can also see in Figure 1 the presence of fourdegenerate levels at the |120573| = 1 level corresponding to therespective value of 119901 (four in this case)

512 Nanotubes of Type 119890 119900 The Huckel calculations arepresented in Figure 2 as one can see there are two degeneratelevels similar to the previous case It is interesting to notealso that as in the previous case there is no gap at the Fermilevel and one can thus characterize the nanotubes as beingmetallicThe figure also shows the presence of five degenerateorbitals at the level |120573| = 1 as a consequence of the value of 119901being five

513 Nanotubes of Type 119900 119890 The case where the nanotubeshave an odd number of rings is characteristically differentfrom the above two cases In the Huckel calculations shown

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100474

Figure 3 Huckel orbital energies for the nanotube (7 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100575

Figure 4 Huckel orbital energies for the nanotube (7 5) where |120573|is the energy unit and 119896 is the normalized orbital number

in Figure 3 one can see the presence of four quasi-degenerateorbitals at the Fermi level On close analysis one can distin-guish pairs or orbitals which are degenerate and there is aslight energy difference between the two pairs making themquasi-degenerate The calculations for very long nanotubesand very wide nanotubes in Figure 3 show that in contrast tothe previous two cases where the number of hexagons (119899) wasa multiple of three there is a significant gap at the Fermi levelbetween the occupied and the virtual orbitals making thenanotubes insulating It is known that the infinite nanotubescharacterized by 119899 and 119901 where 119899 is a multiple of three aremetallic without the presence of a gap at the Fermi level (aband gap)

514 Nanotubes of Type 119900 119900 Figure 4 represents the casecharacterized by an odd value of 119899 and 119898 The similarity to

Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

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2 Advances in Condensed Matter Physics

that makes them ideal candidates for drug delivery andvectorization [5] In addition to these mechanical propertieswhich are already extensively exploited nowadays there arealso peculiar and surprising electronic properties exhibitedby these materials For instance graphene can be consideredas a zero-gap semiconductor showing an infinite electronicmobility at the Fermi level Carbon nanotubes [6] on theother hand which are formally obtained by enrolling agraphene sheet along a helicoidal symmetry axis show a vastvariety of electronic behaviors They exhibit extremely inter-esting properties with a quite complex and rich structure-property correlation pattern In particular depending on theway the original graphene sheet is enrolled around the axisnanotubes can present or not chirality and their conductivityproperties can vary dramatically So different classes ofnanotubes can present a behavior that passes from metallicto low-gap semiconductor to real semiconductor materialsThis peculiar characteristic can efficiently be exploited in thefield of opto- andmolecular-electronics devices the potentialperformance and efficiency of carbon based nanostructurescould potentially induce a revolution in that field comparableto the one produced by silicon based microdevices [1 7]

Because of the scientific and technological importance ofcarbon nanotubes a huge amount of scientific work boththeoretical and experimental has been devoted to the studyof the properties and the electronic structure of these systemsUp to now however the great majority of such studieswere conducted on infinitely extended systems Moreoverin many cases the Hamiltonians used were restricted tosemiempiricalmodels ab initio calculations beingmore oftenperformed at density functional theory level only Howevereven if the study of extended systems deserves a considerableimportance both from a theoretical and a practical pointof view finite-size effects occurring in short finite clusterscan also reveal new and unexpected properties which canstrongly modify the physical behavior with respect to theinfinite system In particular it is known that graphenenanoclusters and open carbon nanotubes can present the so-called edge orbitals that are partially filled orbitals whoseelectron density is mainly concentrated on the border regionsof the system A peculiarity of these orbitals is given by thefact that their energy lies close to the Fermi level that isthey are the frontier orbitals of the system Depending onthe geometry of the cluster the edge orbitals show interestingdegeneracy patterns These features will imply the presenceof different types of open-shell electronic structures possiblyinducing magnetic behaviors characterized by a high-spinground state Moreover there is the possibility of a closingof the Fermi level gap to give rise to a metallic behavior inthe limit of large systems

The presence of an open-shell structure will also inducea strong multiconfigurational nature of the low-lying statesof the system (in particular of those of lower multiplicity)This fact implies that multireference methods are required toproperly take into account the effects of the static correlation[8] Note as previously evoked that the magnetic behavioris strongly dependent on the geometry of the cluster andcan be rationalized by invoking the Ovchinnicov Rule [9 10]and Liebrsquos theorem [11] For instance it has been argued that

triangular-shape clusters are ferromagnetic while hexagonalshapes are expected to be diamagnetic and show a gap at theFermi level Note also that the magnitude of the magneticcoupling between the states can be tuned by varying thesize of the cluster All these important characteristics canbeen exploited by connecting in a network different grapheneclusters characterized by differentmagnetic properties Let usalso cite the fact that the required precise control on the shapeand size of the cluster can nowadays be achieved by bottom-up techniques based on the precise deposition of carbonatoms on different surfaces In fact such techniques havealready allowed to produce very specific graphene clustersand even graphene antidots [12] Finally it is also noteworthyto remind that carbon nanostructures can be used to host andstabilize magnetic linear structures and that the interactionwith conjugated carbon structures can also strongly modifythe magnetic behavior of the linear aggregate

In the present contribution we want to extend our anal-ysis of border effects performed on graphene nanoclusters tothe domain of nanotubes In particular we will concentrateon zig-zag nonchiral nanotubes and we will consider onlyopen structures that present terminal edges saturated withhydrogens Indeed this simple choice allows to keep all thecarbon atoms 1199041199012-hybridized and to limit the edge effectsThis is often done in theoretical studies even if in mostexperimental samples there are no terminal hydrogens

Infinite nanotubes are uniquely characterized (apart fromthe sign of helicity if they are chiral) by a pair of non-negative integer numbers 119899 and 119898 (with 119898 le 119899) andthe corresponding nanotube is denoted as (119899119898) These twonumbers describe how a graphene sheet is enrolled aroundthe helicoidal axes in order to generate the infinite nanotubeAs the two extreme cases we find nonchiral nanotubes thezig-zag nanotubes are of the type (119899 0) while the corre-sponding armchair structures are of type (119899 119899) When 119898 isdifferent from both 0 and 119899 we have chiral nanotubes Iffinite nanotubes are considered the situation is much morecomplex since one must consider not only the length ofthe nanotube itself but also the details of its two bordersThis is particularly true for chiral nanotubes whose helicityprevents the possibility of cutting the extremities in such away to respect the nanotube symmetry along the central axisNonchiral nanotubes however can be truncated in such away that the two edges respect the nanotube symmetry Inzig-zag nanotubes (the systems that are the object of thiswork) the integer number 119899 indicates the number of con-tiguous hexagon units that we find on a circular path aroundthe tube the 119898 value on the other hand is redundant sinceit is identically equal to 0 Finite-length zig-zag nanotubesare characterized by 119899 as the corresponding infinite systemsand by the length of the nanotube We will indicate by 119901

this length and in order to avoid confusion with the well-established notation (119899119898) we indicate by (119899 119901) a fragmentof a (119899 0) (ie zig-zag) nanotube having length 119901

To study the nature of edge orbitals and their influenceon the global electronic structure of the nanostructure aswell as its evolution with the different nanotube structuralparameters we will apply two different computational strate-gies a very simple semiempirical Huckel approach which

Advances in Condensed Matter Physics 3

can treat a large variety of different structures and highlycorrelated multireference wave-function-based approachesable to give quantitative results but whose use is restricted torelatively small systemsThe first crude approximation allowsus to obtain a qualitative overview of the density of statesand in particular to characterize the orbital degeneracies atthe Fermi level Note that this approach will also allow usto consider very large aggregates that is to come close tothe infinite system limit The wave-function-based approachon the other hand takes care of the static and dynamiccorrelation and is able to study the low-lying spectrumof the magnetic states evaluating its evolution with thegeometry and the structural parameters To the best of ourknowledge this is the first contribution applying high levelMultireference Perturbation Theory (MRPT) to the study ofthis kind of systems A similar approach was already appliedto closed polyacenes structures [13] whereas most of previousworks on nanotubes [14ndash22] or nanoribbons [23] principallyrely on density functional theory or tight binding resultsThe main conclusion of studies on There also have beensome studies in the solid-state community on the energyspectrum [24] and properties of single walled nanotubes [25]using sophisticated analytic tools Similar structures as thoseconsidered consist in an antiferromagnetic coupling of twoelectrons localized in the edge orbitals

This paper is organized as follows In Section 2 Theo-retical Methods we briefly describe the methods that weemployed in this study and we justify their use The com-putational details of the calculations are given in Section 3in order to permit the reproduction of our calculations infuture further studies In Section 4 we present the nanotubestructure and symmetry along with the particular terminol-ogy used in this work Results at theHuckel and ab initio levelare presented and discussed in Section 5 Final conclusionsare drawn in Section 6 together with plans for future works

2 Theoretical Methods

In this section we briefly describe the theoretical methodsand the computational approaches used in the present workThe computational details concerning both the ab initio andthe tight binding approaches will be discussed in detail in thenext section in order to facilitate the reproduction of ourresults

Carbon nanotubes as it is the case of most graphene-derived nanostructures show the presence of edge molecularorbitals (MO) that are neither doubly occupied nor totallyempty in the electronic ground state (see eg [12]) Thesimplest Hamiltonian that is able to give a qualitative descrip-tion of the electronic wavefunction of 1199041199012-hybridized carbonstructures is the Huckel Hamiltonian It is a tight bindingHamiltonian that was developed in order to describe smallpolyenes and in general aromatic compounds At Huckellevel only one119901 atomic orbital per carbon atom is considered(the orbital that is locally orthogonal to the 120590 orbitals)and this defines the 120587 system Correspondingly only oneelectron per atom is considered Despite its simplicity theHuckel Hamiltonian is capable of grasping the essence of

the electronic structure of these systems and their low-lyingenergy spectrum obtained at this level is remarkably correctIt is worth noticing that the pioneering 1947 paper on graphite(and graphene) written byWallace describes results obtainedwith this type of approach [26]

As a general strategy that we employed in previousstudies on other 119904119901

2 carbon nanostructures [13 27] weperformed preliminary tight binding calculations by usingthe model Huckel Hamiltonian [28] in order to obtain a firstapproximation of the energy spectrum at one-electron levelThis is very important for subsequent calculations at many-electron level by using an ab initio Hamiltonian in orderto clearly identify the orbitals that form a quasi-degeneratepartly occupied manifold A proper ab initio treatment ofthis group of quasi-degenerate orbitals requires the introduc-tion of the nondynamical correlation [29] and this is mostconveniently done by the use of the Complete Active SpaceSelf-Consistent Field (CAS-SCF) formalism [30] In thisapproach the correlation of the electrons distributed amongthe quasi-degenerate orbitals (the active orbitals) is fullyconsidered at a configuration interaction level At the sametime the shape of all the molecular orbitals (active or not)is fully optimized self-consistently It is known that a CAS-SCF approach is suited for a qualitative description of thelow-lying energy spectrum of open-shell systems (magneticstructures mixed-valence systems) For quantitative resultsthe introduction of the dynamical correlation is also requiredsince the weights of the Slater determinants belonging to thequasi-degenerate manifold can be strongly modified by theinteraction Among the possible multireference approachesthe Quasi-Degenerate Perturbation Theory is the only onethat is capable of treating these relatively large systems Inparticular contracted methods have the advantage of a weakcomputational dependence on the dimension of the activespace and can therefore be used in the present case In thiswork we chose the n-Electron Valence Perturbation Theoryat second-order level (NEVPT2) because of its capability ofdealing with the intruder-state problem without the need ofintroduction of arbitrary parameters [31 32]

In doing a CAS-SCF calculation the choice of theguess starting orbitals plays a key role in the final resultIndeed being a nonlinear optimization algorithm CAS-SCFadmits a huge number of different solutions sometimes notpresenting a particular physical interest It is also frequentbecause of this nonlinear character of the formalism tohave instabilities that lead to symmetry-breaking phenomenain the case of strictly degenerate states as the ones thatcharacterize nanotubes having non-Abelian symmetries Inthe present work we performed High-Spin Restricted Open-Shell Hartree-Fock (ROHF) calculations on the differentsystems by choosing as single-occupied orbitals precisely theorbitals that show a quasi-degenerate character at Huckellevel Once the manifold of quasi-degenerate orbitals hasbeen identified we performed frozen-orbital CAS-SCF (ieCAS-CI) calculations on this active space It is known thatfor states having nonionic character the resulting energyspectrum is very close to the CAS-SCF one at a considerablylower computational cost

4 Advances in Condensed Matter Physics

3 Computational Details

In the next subsections we describe separately the tight bind-ing and ab initio approaches

31 Tight Binding All computations at tight binding level(Huckel calculations in the language of chemists) have beenperformed by using a specific home-made code[33ndash35] TheHuckel Hamiltonian has been constructed starting from theconnectivity of each carbon atom by defining as usual

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119894⟩ = 120572 = 0

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119895⟩ = 120573120574

119894119895

(1)

where 120574119894119895is 1 if sites 119894 and 119895 are connected in the nanostructure

skeleton and 0 otherwise The value of the hopping integral120573 should depend in principle on the distance betweenthe two connected atoms 119894 and 119895 However since the C-C bonds have very similar length in all of our structureswe assumed the same values of 120573 for all the topologicallyconnected pair of atoms In our calculations this parameterwas fixed to the arbitrary value 120573 = minus1 We remind thatminus120573 is sometimes called the hopping integral in the physicsliterature where it is usually indicated as 119905 In the sectionResults and Discussion we report the energy spectrum ofseveral nanotubes computed at the Huckel levelThis is doneas a function of the orbital number In order to facilitate thecomparison between systems having a different number oforbitals the orbital numbers are normalized in the segment[0 1]

32 Ab Initio All the systems have been studied using theminimal STO-3G Gaussian basis [36] for both carbon andhydrogen Although this basis set is certainly too small togive quantitatively reliable results it is known to reproducecorrectly the qualitative behavior of many organic and inor-ganic systems Its reduced size on the other hand permitsto investigate at a correlated ab initio level systems whosesize is relatively large However in order to explore theeffect of a more realistic basis set a limited number ofstructures representative of each class of nanotubes havebeen studied with a larger basis set the cc-pVDZ correlationconsistent basis set of Dunning [37] These are the structurescharacterized by 119899 = 6 7 and 119901 = 4 5 for which a(311990421199011119889) contraction for C and (21199041119901) for H were used Thiscorresponds to a valence double-zeta plus polarization (vdzp)basis set

The systemsrsquo geometries have been optimized at Restrict-ed Open-Shell Hartree-Fock (ROHF) level for the lowest-energy state This means the triplet state for the nanotubescharacterized by an even value of 119899(119890 119901) and the quintet statefor the nanotubes characterized by an odd value of 119899(119900 119901)

During geometry optimization the 119863119899119889

or 119863119899ℎ

sym-metries have been imposed depending on the nanotubesclasses Thus all the coordinates have been relaxed allow-ing the nanotube to lose its cylindrical shape The onlyremaining constraint was the order of the rotation axis Actu-ally due to restrictions of the Molpro quantum chemistry

package [38] optimization has been carried out using theappropriate Abelian subgroups Even if this may resultin symmetry breaking of the electronic wavefunction itnever appeared for the systems considered in this workFor all the nanotubes it appears that only slight distor-tions from the regular ideal cylindrical shape have beenobtained Even the terminal hydrogens just slightly pointtowards the axis of the nanotube All the detailed geometriesare given in the Supplementary Material available onlineat httpdxdoiorg1011552015475890 At these optimizedgeometries the lowest states have been studied at CompleteActive Space Self-Consistent Field (CAS-SCF) level [39]Then the effect of dynamical correlation was introduced atMultireference Perturbation Theory (MR-PT) level usingthe partially contracted version of second-order n-ElectronsValence PerturbationTheory (NEVPT2) formalism [40ndash42]In all the CAS-SCF calculations the active space has beenselected by choosing the orbitals that are strictly degeneratedor quasi-degenerated with the Huckel Hamiltonian at theFermi level This means that our active spaces are eitherCAS(22) (for the (119890 119901) tubes) or CAS(44) (for the (119900 119901)

tubes) The geometries have been optimized at ROHF levelfor the high-spin wavefunctionThe orbitals obtained thereofhave then been kept frozen for the CAS-SCF calculationson the other spin multiplicities In this way our calculationsare actually of CAS-CI [43 44] type which in our case giveenergies close to CAS-SCF onesTheNEVPT2 formalism hasthen been applied to these CAS-CI wavefunctions in order torecover the dynamical correlation

Obviously our geometrical constraints prevent Jahn-Teller distortion which could be expected in the case ofdegenerate orbitals In this preliminary work we decided notto consider this possibility However due to the stiffness ofthe nanotube backbone we believe these distortions to be ofsmall size

4 Electronic Structure ofCarbon 119904119901

2 Nanostructures

The net of carbon atoms forming a graphene layer is theconceptual starting point to produce most of the carbon 1199041199012-hybridized nanostructures A crucial feature to rationalizethe relation between structure and electronic properties andmore generally to study finite-size effects in these structuresis to recognize that the honeycomb graphene skeleton formsa bipartite lattice [12] with two compenetrating triangularsublattices 119860 and 119861 Each carbon atom belonging to agraphene nanostructure is associated with one of these twosublattices and one can speak of119860-type and 119861-type grapheneatoms or centers A lattice is bipartite if each 119860-type atomhas only 119861-type nearest neighbors and vice versa So forinstance zig-zag edges are composed of atoms that all belongto the same sublattices A necessary condition for a latticeto be bipartite is the absence of odd-number carbon cyclesIn order to rationalize the emergence of magnetic propertiesin graphene nanosystems one can recall Liebrsquos theorem[11] This theorem also known as the theorem of itinerantmagnetism and usually applied within the framework of the

Advances in Condensed Matter Physics 5

Hubbard one-orbital model is able to predict the total spinof the ground state in bipartite lattices In particular one cansee that an imbalance on the number of atom in one sublatticeresults in a magnetic ground state with spin

119878 =

1003816100381610038161003816119873119860minus 119873119861

1003816100381610038161003816

2

(2)

where 119873119860

and 119873119861are the numbers of atoms of 119860- and

119861-type respectively It is also possible to show that fora Huckel Hamiltonian |119873

119860minus 119873119861| is also the number of

eigenvalues equal to zero It is important to note that themagnetization originates from localized edge-states that givealso rise to a high density of states at the Fermi levelwhich in turn can determine a spin polarization instabilityMoreover the relation between the unbalanced number ofatoms and the ground state spin implies that two centerswill be ferromagnetically coupled if they belong to the samesublattice and antiferromagnetically coupled if they do not[45]

We consider now the symmetry of the (119899 119901) nanotubesIt is easy to see that these structures fall into two classesaccording to the parity of the length 119901 The (119899 119901) tubes have119863119899ℎ

symmetry if 119901 is odd ((119899 119900) tubes) and 119863119899119889

symmetryfor 119901 even ((119899 119890) tubes) In all these cases (with the partialexception of the case 119899 = 2 which is however too narrow toform a tube) we are in the presence of non-Abelian symmetrygroups We recall that our ab initio calculations have beenperformed by usingAbelian subgroups of the full point groupof the system According to the parity even (119890) or odd (119900) of 119899and 119901 we have the following Abelian subgroups119862

2V for (119890 119890)1198632ℎfor (119890 119900) 119862

2ℎfor (119900 119890) and 119862

2V for (119900 119900)We call edge carbon atoms the carbon atoms that are

saturated with hydrogens The role of these edge atoms iscrucial in order to understand the finite-size effects in carbonnanotubes The edge carbon atoms of the tubes of type (119890 119901)can support molecular orbitals (MO) of 120587-nature havingalternating sign on any pair of consecutive hexagons Let usconsider the MO combinations that are localized on one ofthe tube extremities only AtHuckel level it is straightforwardto verify that both these alternating edge combinations areeigenfunctions of the one-electron Hamiltonian correspond-ing to eigenvalues equal to zeroTherefore the Huckel energyspectrum presents a pair of degenerate orbitals at the Fermilevel for all (119890 119901) tubes regardless of the parity of 119901 Thesituation is different as far as the ab initio Hamiltonian isconsidered since these two localized edge combinations areno longer exact molecular orbitals of the Fock HamiltonianAs a result at ab initio level the exact degeneracy can be lostIn particular it turns out that while the two edge orbitalsare exactly degenerate for (119890 119890) tubes the degeneracy is onlyapproximated in the case of the (119890 119900) structures This pointcan be easily understood by symmetry considerations In factthe two localized edge orbitals are interchanged by some ofthe symmetry operations of the point group of the system(119863119899ℎ

for (119890 119900) and 119863119899119889

for (119890 119890) tubes) For this reasonthey give rise to a representation of the molecule symmetrygroup In the case of (119890 119890) tubes the representation isirreducible and it still corresponds to two exactly degenerate

molecular orbitals of the system (eg see Figure 6 wherethe two degenerate orbitals have 119890

3symmetry) In the case

of the (119890 119900) tubes on the other hand the two combinationsbelong to two different irreducible representations of theAbelian symmetry group of the system The two orbitals areonly approximately degenerate (see Figure 7 where the twoorbitals have 119887

1119906and 1198872119892

symmetries resp) This argumentexplains the remarkable difference between the tight bindingand ab initio energy spectra in the Fermi level region for the(119890 119900) tubes

5 Results and Discussion

51 Tight Binding Even if the tight binding approach repre-sents a crude approximation of a chemical system it can beextremely useful in order to sketch out some general tenden-cies and to elucidate the behavior of the different classes ofcompoundsThis is particularly true in the present case sincea tight binding approach allows the treatment of very largesystems due to its extremely reduced computational costs Itis possible in this way to explore the behavior of virtuallyldquoinfiniterdquo systems

Short zig-zag nanotubes can be divided into four dif-ferent classes characterized by different symmetries andmolecular-orbital patterns depending on the parity of 119899 and119901 The Huckel calculations for these classes permitted tocharacterize the different active spaces which were essentiallyconfirmed at the ab initio level

Nanotubes of the type (119899 119890) have 119863119899119889

symmetry whilethose of type (119899 119900) have 119863

119899ℎsymmetry At the Huckel level

both (119890 119890) and (119890 119900) have two exactly degenerate orbitalsat the Fermi level hosting two electrons This is illustratedin Figures 1 and 2 where the energy spectra of (6 4) and(6 5) structures are reported For the sake of comparison thespectra of long (6 100) or very wide (100 4) and (100 5)structures are also reported in the figures None of thesespectra shows the presence of an energy gap at the Fermi leveland the wide structures have even an infinite density of levelsat the Fermi level Nanotubes of the types (119900 119890) and (119900 119900) onthe other hand have four quasi-degenerate levels (two quasi-degenerate pairs of exactly degenerate levels) near the Fermilevel hosting a total of four electrons This fact is illustratedfor the (7 4) and (7 5) structures in Figures 3 and 4 Fromthese figures it appears that in the case of very long (119900 119901)

nanotubes four degenerate isolated orbitals hosting a total offour electrons are located at the Fermi level As a systematicstudy of large systems at theHuckel level involves a digressionfrom the present topic a general analysis of the four classesof the nanotubes will follow

511 Nanotubes of Type 119890 119890 The Huckel calculation for thenanotubes characterized by an even parity for both 119899 and119901 showed that there were two orbitals at the Fermi levelThe calculations Figure 1 are shown for very wide and verylong nanotubes for the purpose of comparison The abscissa(119896) represents the Huckel orbital number mapped onto theinterval [0 1] It can be seen that there is no gap at the Fermilevel this allows us to characterize such nanotubes as being

6 Advances in Condensed Matter Physics

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100464

Figure 1 Huckel orbital energies for the nanotube (6 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100565

Figure 2 Huckel orbital energies for the nanotube (6 5) where |120573|is the energy unit and 119896 is the normalized orbital number

metallic One can also see in Figure 1 the presence of fourdegenerate levels at the |120573| = 1 level corresponding to therespective value of 119901 (four in this case)

512 Nanotubes of Type 119890 119900 The Huckel calculations arepresented in Figure 2 as one can see there are two degeneratelevels similar to the previous case It is interesting to notealso that as in the previous case there is no gap at the Fermilevel and one can thus characterize the nanotubes as beingmetallicThe figure also shows the presence of five degenerateorbitals at the level |120573| = 1 as a consequence of the value of 119901being five

513 Nanotubes of Type 119900 119890 The case where the nanotubeshave an odd number of rings is characteristically differentfrom the above two cases In the Huckel calculations shown

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100474

Figure 3 Huckel orbital energies for the nanotube (7 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100575

Figure 4 Huckel orbital energies for the nanotube (7 5) where |120573|is the energy unit and 119896 is the normalized orbital number

in Figure 3 one can see the presence of four quasi-degenerateorbitals at the Fermi level On close analysis one can distin-guish pairs or orbitals which are degenerate and there is aslight energy difference between the two pairs making themquasi-degenerate The calculations for very long nanotubesand very wide nanotubes in Figure 3 show that in contrast tothe previous two cases where the number of hexagons (119899) wasa multiple of three there is a significant gap at the Fermi levelbetween the occupied and the virtual orbitals making thenanotubes insulating It is known that the infinite nanotubescharacterized by 119899 and 119901 where 119899 is a multiple of three aremetallic without the presence of a gap at the Fermi level (aband gap)

514 Nanotubes of Type 119900 119900 Figure 4 represents the casecharacterized by an odd value of 119899 and 119898 The similarity to

Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Advances in Condensed Matter Physics 3

can treat a large variety of different structures and highlycorrelated multireference wave-function-based approachesable to give quantitative results but whose use is restricted torelatively small systemsThe first crude approximation allowsus to obtain a qualitative overview of the density of statesand in particular to characterize the orbital degeneracies atthe Fermi level Note that this approach will also allow usto consider very large aggregates that is to come close tothe infinite system limit The wave-function-based approachon the other hand takes care of the static and dynamiccorrelation and is able to study the low-lying spectrumof the magnetic states evaluating its evolution with thegeometry and the structural parameters To the best of ourknowledge this is the first contribution applying high levelMultireference Perturbation Theory (MRPT) to the study ofthis kind of systems A similar approach was already appliedto closed polyacenes structures [13] whereas most of previousworks on nanotubes [14ndash22] or nanoribbons [23] principallyrely on density functional theory or tight binding resultsThe main conclusion of studies on There also have beensome studies in the solid-state community on the energyspectrum [24] and properties of single walled nanotubes [25]using sophisticated analytic tools Similar structures as thoseconsidered consist in an antiferromagnetic coupling of twoelectrons localized in the edge orbitals

This paper is organized as follows In Section 2 Theo-retical Methods we briefly describe the methods that weemployed in this study and we justify their use The com-putational details of the calculations are given in Section 3in order to permit the reproduction of our calculations infuture further studies In Section 4 we present the nanotubestructure and symmetry along with the particular terminol-ogy used in this work Results at theHuckel and ab initio levelare presented and discussed in Section 5 Final conclusionsare drawn in Section 6 together with plans for future works

2 Theoretical Methods

In this section we briefly describe the theoretical methodsand the computational approaches used in the present workThe computational details concerning both the ab initio andthe tight binding approaches will be discussed in detail in thenext section in order to facilitate the reproduction of ourresults

Carbon nanotubes as it is the case of most graphene-derived nanostructures show the presence of edge molecularorbitals (MO) that are neither doubly occupied nor totallyempty in the electronic ground state (see eg [12]) Thesimplest Hamiltonian that is able to give a qualitative descrip-tion of the electronic wavefunction of 1199041199012-hybridized carbonstructures is the Huckel Hamiltonian It is a tight bindingHamiltonian that was developed in order to describe smallpolyenes and in general aromatic compounds At Huckellevel only one119901 atomic orbital per carbon atom is considered(the orbital that is locally orthogonal to the 120590 orbitals)and this defines the 120587 system Correspondingly only oneelectron per atom is considered Despite its simplicity theHuckel Hamiltonian is capable of grasping the essence of

the electronic structure of these systems and their low-lyingenergy spectrum obtained at this level is remarkably correctIt is worth noticing that the pioneering 1947 paper on graphite(and graphene) written byWallace describes results obtainedwith this type of approach [26]

As a general strategy that we employed in previousstudies on other 119904119901

2 carbon nanostructures [13 27] weperformed preliminary tight binding calculations by usingthe model Huckel Hamiltonian [28] in order to obtain a firstapproximation of the energy spectrum at one-electron levelThis is very important for subsequent calculations at many-electron level by using an ab initio Hamiltonian in orderto clearly identify the orbitals that form a quasi-degeneratepartly occupied manifold A proper ab initio treatment ofthis group of quasi-degenerate orbitals requires the introduc-tion of the nondynamical correlation [29] and this is mostconveniently done by the use of the Complete Active SpaceSelf-Consistent Field (CAS-SCF) formalism [30] In thisapproach the correlation of the electrons distributed amongthe quasi-degenerate orbitals (the active orbitals) is fullyconsidered at a configuration interaction level At the sametime the shape of all the molecular orbitals (active or not)is fully optimized self-consistently It is known that a CAS-SCF approach is suited for a qualitative description of thelow-lying energy spectrum of open-shell systems (magneticstructures mixed-valence systems) For quantitative resultsthe introduction of the dynamical correlation is also requiredsince the weights of the Slater determinants belonging to thequasi-degenerate manifold can be strongly modified by theinteraction Among the possible multireference approachesthe Quasi-Degenerate Perturbation Theory is the only onethat is capable of treating these relatively large systems Inparticular contracted methods have the advantage of a weakcomputational dependence on the dimension of the activespace and can therefore be used in the present case In thiswork we chose the n-Electron Valence Perturbation Theoryat second-order level (NEVPT2) because of its capability ofdealing with the intruder-state problem without the need ofintroduction of arbitrary parameters [31 32]

In doing a CAS-SCF calculation the choice of theguess starting orbitals plays a key role in the final resultIndeed being a nonlinear optimization algorithm CAS-SCFadmits a huge number of different solutions sometimes notpresenting a particular physical interest It is also frequentbecause of this nonlinear character of the formalism tohave instabilities that lead to symmetry-breaking phenomenain the case of strictly degenerate states as the ones thatcharacterize nanotubes having non-Abelian symmetries Inthe present work we performed High-Spin Restricted Open-Shell Hartree-Fock (ROHF) calculations on the differentsystems by choosing as single-occupied orbitals precisely theorbitals that show a quasi-degenerate character at Huckellevel Once the manifold of quasi-degenerate orbitals hasbeen identified we performed frozen-orbital CAS-SCF (ieCAS-CI) calculations on this active space It is known thatfor states having nonionic character the resulting energyspectrum is very close to the CAS-SCF one at a considerablylower computational cost

4 Advances in Condensed Matter Physics

3 Computational Details

In the next subsections we describe separately the tight bind-ing and ab initio approaches

31 Tight Binding All computations at tight binding level(Huckel calculations in the language of chemists) have beenperformed by using a specific home-made code[33ndash35] TheHuckel Hamiltonian has been constructed starting from theconnectivity of each carbon atom by defining as usual

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119894⟩ = 120572 = 0

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119895⟩ = 120573120574

119894119895

(1)

where 120574119894119895is 1 if sites 119894 and 119895 are connected in the nanostructure

skeleton and 0 otherwise The value of the hopping integral120573 should depend in principle on the distance betweenthe two connected atoms 119894 and 119895 However since the C-C bonds have very similar length in all of our structureswe assumed the same values of 120573 for all the topologicallyconnected pair of atoms In our calculations this parameterwas fixed to the arbitrary value 120573 = minus1 We remind thatminus120573 is sometimes called the hopping integral in the physicsliterature where it is usually indicated as 119905 In the sectionResults and Discussion we report the energy spectrum ofseveral nanotubes computed at the Huckel levelThis is doneas a function of the orbital number In order to facilitate thecomparison between systems having a different number oforbitals the orbital numbers are normalized in the segment[0 1]

32 Ab Initio All the systems have been studied using theminimal STO-3G Gaussian basis [36] for both carbon andhydrogen Although this basis set is certainly too small togive quantitatively reliable results it is known to reproducecorrectly the qualitative behavior of many organic and inor-ganic systems Its reduced size on the other hand permitsto investigate at a correlated ab initio level systems whosesize is relatively large However in order to explore theeffect of a more realistic basis set a limited number ofstructures representative of each class of nanotubes havebeen studied with a larger basis set the cc-pVDZ correlationconsistent basis set of Dunning [37] These are the structurescharacterized by 119899 = 6 7 and 119901 = 4 5 for which a(311990421199011119889) contraction for C and (21199041119901) for H were used Thiscorresponds to a valence double-zeta plus polarization (vdzp)basis set

The systemsrsquo geometries have been optimized at Restrict-ed Open-Shell Hartree-Fock (ROHF) level for the lowest-energy state This means the triplet state for the nanotubescharacterized by an even value of 119899(119890 119901) and the quintet statefor the nanotubes characterized by an odd value of 119899(119900 119901)

During geometry optimization the 119863119899119889

or 119863119899ℎ

sym-metries have been imposed depending on the nanotubesclasses Thus all the coordinates have been relaxed allow-ing the nanotube to lose its cylindrical shape The onlyremaining constraint was the order of the rotation axis Actu-ally due to restrictions of the Molpro quantum chemistry

package [38] optimization has been carried out using theappropriate Abelian subgroups Even if this may resultin symmetry breaking of the electronic wavefunction itnever appeared for the systems considered in this workFor all the nanotubes it appears that only slight distor-tions from the regular ideal cylindrical shape have beenobtained Even the terminal hydrogens just slightly pointtowards the axis of the nanotube All the detailed geometriesare given in the Supplementary Material available onlineat httpdxdoiorg1011552015475890 At these optimizedgeometries the lowest states have been studied at CompleteActive Space Self-Consistent Field (CAS-SCF) level [39]Then the effect of dynamical correlation was introduced atMultireference Perturbation Theory (MR-PT) level usingthe partially contracted version of second-order n-ElectronsValence PerturbationTheory (NEVPT2) formalism [40ndash42]In all the CAS-SCF calculations the active space has beenselected by choosing the orbitals that are strictly degeneratedor quasi-degenerated with the Huckel Hamiltonian at theFermi level This means that our active spaces are eitherCAS(22) (for the (119890 119901) tubes) or CAS(44) (for the (119900 119901)

tubes) The geometries have been optimized at ROHF levelfor the high-spin wavefunctionThe orbitals obtained thereofhave then been kept frozen for the CAS-SCF calculationson the other spin multiplicities In this way our calculationsare actually of CAS-CI [43 44] type which in our case giveenergies close to CAS-SCF onesTheNEVPT2 formalism hasthen been applied to these CAS-CI wavefunctions in order torecover the dynamical correlation

Obviously our geometrical constraints prevent Jahn-Teller distortion which could be expected in the case ofdegenerate orbitals In this preliminary work we decided notto consider this possibility However due to the stiffness ofthe nanotube backbone we believe these distortions to be ofsmall size

4 Electronic Structure ofCarbon 119904119901

2 Nanostructures

The net of carbon atoms forming a graphene layer is theconceptual starting point to produce most of the carbon 1199041199012-hybridized nanostructures A crucial feature to rationalizethe relation between structure and electronic properties andmore generally to study finite-size effects in these structuresis to recognize that the honeycomb graphene skeleton formsa bipartite lattice [12] with two compenetrating triangularsublattices 119860 and 119861 Each carbon atom belonging to agraphene nanostructure is associated with one of these twosublattices and one can speak of119860-type and 119861-type grapheneatoms or centers A lattice is bipartite if each 119860-type atomhas only 119861-type nearest neighbors and vice versa So forinstance zig-zag edges are composed of atoms that all belongto the same sublattices A necessary condition for a latticeto be bipartite is the absence of odd-number carbon cyclesIn order to rationalize the emergence of magnetic propertiesin graphene nanosystems one can recall Liebrsquos theorem[11] This theorem also known as the theorem of itinerantmagnetism and usually applied within the framework of the

Advances in Condensed Matter Physics 5

Hubbard one-orbital model is able to predict the total spinof the ground state in bipartite lattices In particular one cansee that an imbalance on the number of atom in one sublatticeresults in a magnetic ground state with spin

119878 =

1003816100381610038161003816119873119860minus 119873119861

1003816100381610038161003816

2

(2)

where 119873119860

and 119873119861are the numbers of atoms of 119860- and

119861-type respectively It is also possible to show that fora Huckel Hamiltonian |119873

119860minus 119873119861| is also the number of

eigenvalues equal to zero It is important to note that themagnetization originates from localized edge-states that givealso rise to a high density of states at the Fermi levelwhich in turn can determine a spin polarization instabilityMoreover the relation between the unbalanced number ofatoms and the ground state spin implies that two centerswill be ferromagnetically coupled if they belong to the samesublattice and antiferromagnetically coupled if they do not[45]

We consider now the symmetry of the (119899 119901) nanotubesIt is easy to see that these structures fall into two classesaccording to the parity of the length 119901 The (119899 119901) tubes have119863119899ℎ

symmetry if 119901 is odd ((119899 119900) tubes) and 119863119899119889

symmetryfor 119901 even ((119899 119890) tubes) In all these cases (with the partialexception of the case 119899 = 2 which is however too narrow toform a tube) we are in the presence of non-Abelian symmetrygroups We recall that our ab initio calculations have beenperformed by usingAbelian subgroups of the full point groupof the system According to the parity even (119890) or odd (119900) of 119899and 119901 we have the following Abelian subgroups119862

2V for (119890 119890)1198632ℎfor (119890 119900) 119862

2ℎfor (119900 119890) and 119862

2V for (119900 119900)We call edge carbon atoms the carbon atoms that are

saturated with hydrogens The role of these edge atoms iscrucial in order to understand the finite-size effects in carbonnanotubes The edge carbon atoms of the tubes of type (119890 119901)can support molecular orbitals (MO) of 120587-nature havingalternating sign on any pair of consecutive hexagons Let usconsider the MO combinations that are localized on one ofthe tube extremities only AtHuckel level it is straightforwardto verify that both these alternating edge combinations areeigenfunctions of the one-electron Hamiltonian correspond-ing to eigenvalues equal to zeroTherefore the Huckel energyspectrum presents a pair of degenerate orbitals at the Fermilevel for all (119890 119901) tubes regardless of the parity of 119901 Thesituation is different as far as the ab initio Hamiltonian isconsidered since these two localized edge combinations areno longer exact molecular orbitals of the Fock HamiltonianAs a result at ab initio level the exact degeneracy can be lostIn particular it turns out that while the two edge orbitalsare exactly degenerate for (119890 119890) tubes the degeneracy is onlyapproximated in the case of the (119890 119900) structures This pointcan be easily understood by symmetry considerations In factthe two localized edge orbitals are interchanged by some ofthe symmetry operations of the point group of the system(119863119899ℎ

for (119890 119900) and 119863119899119889

for (119890 119890) tubes) For this reasonthey give rise to a representation of the molecule symmetrygroup In the case of (119890 119890) tubes the representation isirreducible and it still corresponds to two exactly degenerate

molecular orbitals of the system (eg see Figure 6 wherethe two degenerate orbitals have 119890

3symmetry) In the case

of the (119890 119900) tubes on the other hand the two combinationsbelong to two different irreducible representations of theAbelian symmetry group of the system The two orbitals areonly approximately degenerate (see Figure 7 where the twoorbitals have 119887

1119906and 1198872119892

symmetries resp) This argumentexplains the remarkable difference between the tight bindingand ab initio energy spectra in the Fermi level region for the(119890 119900) tubes

5 Results and Discussion

51 Tight Binding Even if the tight binding approach repre-sents a crude approximation of a chemical system it can beextremely useful in order to sketch out some general tenden-cies and to elucidate the behavior of the different classes ofcompoundsThis is particularly true in the present case sincea tight binding approach allows the treatment of very largesystems due to its extremely reduced computational costs Itis possible in this way to explore the behavior of virtuallyldquoinfiniterdquo systems

Short zig-zag nanotubes can be divided into four dif-ferent classes characterized by different symmetries andmolecular-orbital patterns depending on the parity of 119899 and119901 The Huckel calculations for these classes permitted tocharacterize the different active spaces which were essentiallyconfirmed at the ab initio level

Nanotubes of the type (119899 119890) have 119863119899119889

symmetry whilethose of type (119899 119900) have 119863

119899ℎsymmetry At the Huckel level

both (119890 119890) and (119890 119900) have two exactly degenerate orbitalsat the Fermi level hosting two electrons This is illustratedin Figures 1 and 2 where the energy spectra of (6 4) and(6 5) structures are reported For the sake of comparison thespectra of long (6 100) or very wide (100 4) and (100 5)structures are also reported in the figures None of thesespectra shows the presence of an energy gap at the Fermi leveland the wide structures have even an infinite density of levelsat the Fermi level Nanotubes of the types (119900 119890) and (119900 119900) onthe other hand have four quasi-degenerate levels (two quasi-degenerate pairs of exactly degenerate levels) near the Fermilevel hosting a total of four electrons This fact is illustratedfor the (7 4) and (7 5) structures in Figures 3 and 4 Fromthese figures it appears that in the case of very long (119900 119901)

nanotubes four degenerate isolated orbitals hosting a total offour electrons are located at the Fermi level As a systematicstudy of large systems at theHuckel level involves a digressionfrom the present topic a general analysis of the four classesof the nanotubes will follow

511 Nanotubes of Type 119890 119890 The Huckel calculation for thenanotubes characterized by an even parity for both 119899 and119901 showed that there were two orbitals at the Fermi levelThe calculations Figure 1 are shown for very wide and verylong nanotubes for the purpose of comparison The abscissa(119896) represents the Huckel orbital number mapped onto theinterval [0 1] It can be seen that there is no gap at the Fermilevel this allows us to characterize such nanotubes as being

6 Advances in Condensed Matter Physics

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100464

Figure 1 Huckel orbital energies for the nanotube (6 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100565

Figure 2 Huckel orbital energies for the nanotube (6 5) where |120573|is the energy unit and 119896 is the normalized orbital number

metallic One can also see in Figure 1 the presence of fourdegenerate levels at the |120573| = 1 level corresponding to therespective value of 119901 (four in this case)

512 Nanotubes of Type 119890 119900 The Huckel calculations arepresented in Figure 2 as one can see there are two degeneratelevels similar to the previous case It is interesting to notealso that as in the previous case there is no gap at the Fermilevel and one can thus characterize the nanotubes as beingmetallicThe figure also shows the presence of five degenerateorbitals at the level |120573| = 1 as a consequence of the value of 119901being five

513 Nanotubes of Type 119900 119890 The case where the nanotubeshave an odd number of rings is characteristically differentfrom the above two cases In the Huckel calculations shown

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100474

Figure 3 Huckel orbital energies for the nanotube (7 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100575

Figure 4 Huckel orbital energies for the nanotube (7 5) where |120573|is the energy unit and 119896 is the normalized orbital number

in Figure 3 one can see the presence of four quasi-degenerateorbitals at the Fermi level On close analysis one can distin-guish pairs or orbitals which are degenerate and there is aslight energy difference between the two pairs making themquasi-degenerate The calculations for very long nanotubesand very wide nanotubes in Figure 3 show that in contrast tothe previous two cases where the number of hexagons (119899) wasa multiple of three there is a significant gap at the Fermi levelbetween the occupied and the virtual orbitals making thenanotubes insulating It is known that the infinite nanotubescharacterized by 119899 and 119901 where 119899 is a multiple of three aremetallic without the presence of a gap at the Fermi level (aband gap)

514 Nanotubes of Type 119900 119900 Figure 4 represents the casecharacterized by an odd value of 119899 and 119898 The similarity to

Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in Condensed Matter Physics

3 Computational Details

In the next subsections we describe separately the tight bind-ing and ab initio approaches

31 Tight Binding All computations at tight binding level(Huckel calculations in the language of chemists) have beenperformed by using a specific home-made code[33ndash35] TheHuckel Hamiltonian has been constructed starting from theconnectivity of each carbon atom by defining as usual

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119894⟩ = 120572 = 0

⟨119894

10038161003816100381610038161003816

10038161003816100381610038161003816119895⟩ = 120573120574

119894119895

(1)

where 120574119894119895is 1 if sites 119894 and 119895 are connected in the nanostructure

skeleton and 0 otherwise The value of the hopping integral120573 should depend in principle on the distance betweenthe two connected atoms 119894 and 119895 However since the C-C bonds have very similar length in all of our structureswe assumed the same values of 120573 for all the topologicallyconnected pair of atoms In our calculations this parameterwas fixed to the arbitrary value 120573 = minus1 We remind thatminus120573 is sometimes called the hopping integral in the physicsliterature where it is usually indicated as 119905 In the sectionResults and Discussion we report the energy spectrum ofseveral nanotubes computed at the Huckel levelThis is doneas a function of the orbital number In order to facilitate thecomparison between systems having a different number oforbitals the orbital numbers are normalized in the segment[0 1]

32 Ab Initio All the systems have been studied using theminimal STO-3G Gaussian basis [36] for both carbon andhydrogen Although this basis set is certainly too small togive quantitatively reliable results it is known to reproducecorrectly the qualitative behavior of many organic and inor-ganic systems Its reduced size on the other hand permitsto investigate at a correlated ab initio level systems whosesize is relatively large However in order to explore theeffect of a more realistic basis set a limited number ofstructures representative of each class of nanotubes havebeen studied with a larger basis set the cc-pVDZ correlationconsistent basis set of Dunning [37] These are the structurescharacterized by 119899 = 6 7 and 119901 = 4 5 for which a(311990421199011119889) contraction for C and (21199041119901) for H were used Thiscorresponds to a valence double-zeta plus polarization (vdzp)basis set

The systemsrsquo geometries have been optimized at Restrict-ed Open-Shell Hartree-Fock (ROHF) level for the lowest-energy state This means the triplet state for the nanotubescharacterized by an even value of 119899(119890 119901) and the quintet statefor the nanotubes characterized by an odd value of 119899(119900 119901)

During geometry optimization the 119863119899119889

or 119863119899ℎ

sym-metries have been imposed depending on the nanotubesclasses Thus all the coordinates have been relaxed allow-ing the nanotube to lose its cylindrical shape The onlyremaining constraint was the order of the rotation axis Actu-ally due to restrictions of the Molpro quantum chemistry

package [38] optimization has been carried out using theappropriate Abelian subgroups Even if this may resultin symmetry breaking of the electronic wavefunction itnever appeared for the systems considered in this workFor all the nanotubes it appears that only slight distor-tions from the regular ideal cylindrical shape have beenobtained Even the terminal hydrogens just slightly pointtowards the axis of the nanotube All the detailed geometriesare given in the Supplementary Material available onlineat httpdxdoiorg1011552015475890 At these optimizedgeometries the lowest states have been studied at CompleteActive Space Self-Consistent Field (CAS-SCF) level [39]Then the effect of dynamical correlation was introduced atMultireference Perturbation Theory (MR-PT) level usingthe partially contracted version of second-order n-ElectronsValence PerturbationTheory (NEVPT2) formalism [40ndash42]In all the CAS-SCF calculations the active space has beenselected by choosing the orbitals that are strictly degeneratedor quasi-degenerated with the Huckel Hamiltonian at theFermi level This means that our active spaces are eitherCAS(22) (for the (119890 119901) tubes) or CAS(44) (for the (119900 119901)

tubes) The geometries have been optimized at ROHF levelfor the high-spin wavefunctionThe orbitals obtained thereofhave then been kept frozen for the CAS-SCF calculationson the other spin multiplicities In this way our calculationsare actually of CAS-CI [43 44] type which in our case giveenergies close to CAS-SCF onesTheNEVPT2 formalism hasthen been applied to these CAS-CI wavefunctions in order torecover the dynamical correlation

Obviously our geometrical constraints prevent Jahn-Teller distortion which could be expected in the case ofdegenerate orbitals In this preliminary work we decided notto consider this possibility However due to the stiffness ofthe nanotube backbone we believe these distortions to be ofsmall size

4 Electronic Structure ofCarbon 119904119901

2 Nanostructures

The net of carbon atoms forming a graphene layer is theconceptual starting point to produce most of the carbon 1199041199012-hybridized nanostructures A crucial feature to rationalizethe relation between structure and electronic properties andmore generally to study finite-size effects in these structuresis to recognize that the honeycomb graphene skeleton formsa bipartite lattice [12] with two compenetrating triangularsublattices 119860 and 119861 Each carbon atom belonging to agraphene nanostructure is associated with one of these twosublattices and one can speak of119860-type and 119861-type grapheneatoms or centers A lattice is bipartite if each 119860-type atomhas only 119861-type nearest neighbors and vice versa So forinstance zig-zag edges are composed of atoms that all belongto the same sublattices A necessary condition for a latticeto be bipartite is the absence of odd-number carbon cyclesIn order to rationalize the emergence of magnetic propertiesin graphene nanosystems one can recall Liebrsquos theorem[11] This theorem also known as the theorem of itinerantmagnetism and usually applied within the framework of the

Advances in Condensed Matter Physics 5

Hubbard one-orbital model is able to predict the total spinof the ground state in bipartite lattices In particular one cansee that an imbalance on the number of atom in one sublatticeresults in a magnetic ground state with spin

119878 =

1003816100381610038161003816119873119860minus 119873119861

1003816100381610038161003816

2

(2)

where 119873119860

and 119873119861are the numbers of atoms of 119860- and

119861-type respectively It is also possible to show that fora Huckel Hamiltonian |119873

119860minus 119873119861| is also the number of

eigenvalues equal to zero It is important to note that themagnetization originates from localized edge-states that givealso rise to a high density of states at the Fermi levelwhich in turn can determine a spin polarization instabilityMoreover the relation between the unbalanced number ofatoms and the ground state spin implies that two centerswill be ferromagnetically coupled if they belong to the samesublattice and antiferromagnetically coupled if they do not[45]

We consider now the symmetry of the (119899 119901) nanotubesIt is easy to see that these structures fall into two classesaccording to the parity of the length 119901 The (119899 119901) tubes have119863119899ℎ

symmetry if 119901 is odd ((119899 119900) tubes) and 119863119899119889

symmetryfor 119901 even ((119899 119890) tubes) In all these cases (with the partialexception of the case 119899 = 2 which is however too narrow toform a tube) we are in the presence of non-Abelian symmetrygroups We recall that our ab initio calculations have beenperformed by usingAbelian subgroups of the full point groupof the system According to the parity even (119890) or odd (119900) of 119899and 119901 we have the following Abelian subgroups119862

2V for (119890 119890)1198632ℎfor (119890 119900) 119862

2ℎfor (119900 119890) and 119862

2V for (119900 119900)We call edge carbon atoms the carbon atoms that are

saturated with hydrogens The role of these edge atoms iscrucial in order to understand the finite-size effects in carbonnanotubes The edge carbon atoms of the tubes of type (119890 119901)can support molecular orbitals (MO) of 120587-nature havingalternating sign on any pair of consecutive hexagons Let usconsider the MO combinations that are localized on one ofthe tube extremities only AtHuckel level it is straightforwardto verify that both these alternating edge combinations areeigenfunctions of the one-electron Hamiltonian correspond-ing to eigenvalues equal to zeroTherefore the Huckel energyspectrum presents a pair of degenerate orbitals at the Fermilevel for all (119890 119901) tubes regardless of the parity of 119901 Thesituation is different as far as the ab initio Hamiltonian isconsidered since these two localized edge combinations areno longer exact molecular orbitals of the Fock HamiltonianAs a result at ab initio level the exact degeneracy can be lostIn particular it turns out that while the two edge orbitalsare exactly degenerate for (119890 119890) tubes the degeneracy is onlyapproximated in the case of the (119890 119900) structures This pointcan be easily understood by symmetry considerations In factthe two localized edge orbitals are interchanged by some ofthe symmetry operations of the point group of the system(119863119899ℎ

for (119890 119900) and 119863119899119889

for (119890 119890) tubes) For this reasonthey give rise to a representation of the molecule symmetrygroup In the case of (119890 119890) tubes the representation isirreducible and it still corresponds to two exactly degenerate

molecular orbitals of the system (eg see Figure 6 wherethe two degenerate orbitals have 119890

3symmetry) In the case

of the (119890 119900) tubes on the other hand the two combinationsbelong to two different irreducible representations of theAbelian symmetry group of the system The two orbitals areonly approximately degenerate (see Figure 7 where the twoorbitals have 119887

1119906and 1198872119892

symmetries resp) This argumentexplains the remarkable difference between the tight bindingand ab initio energy spectra in the Fermi level region for the(119890 119900) tubes

5 Results and Discussion

51 Tight Binding Even if the tight binding approach repre-sents a crude approximation of a chemical system it can beextremely useful in order to sketch out some general tenden-cies and to elucidate the behavior of the different classes ofcompoundsThis is particularly true in the present case sincea tight binding approach allows the treatment of very largesystems due to its extremely reduced computational costs Itis possible in this way to explore the behavior of virtuallyldquoinfiniterdquo systems

Short zig-zag nanotubes can be divided into four dif-ferent classes characterized by different symmetries andmolecular-orbital patterns depending on the parity of 119899 and119901 The Huckel calculations for these classes permitted tocharacterize the different active spaces which were essentiallyconfirmed at the ab initio level

Nanotubes of the type (119899 119890) have 119863119899119889

symmetry whilethose of type (119899 119900) have 119863

119899ℎsymmetry At the Huckel level

both (119890 119890) and (119890 119900) have two exactly degenerate orbitalsat the Fermi level hosting two electrons This is illustratedin Figures 1 and 2 where the energy spectra of (6 4) and(6 5) structures are reported For the sake of comparison thespectra of long (6 100) or very wide (100 4) and (100 5)structures are also reported in the figures None of thesespectra shows the presence of an energy gap at the Fermi leveland the wide structures have even an infinite density of levelsat the Fermi level Nanotubes of the types (119900 119890) and (119900 119900) onthe other hand have four quasi-degenerate levels (two quasi-degenerate pairs of exactly degenerate levels) near the Fermilevel hosting a total of four electrons This fact is illustratedfor the (7 4) and (7 5) structures in Figures 3 and 4 Fromthese figures it appears that in the case of very long (119900 119901)

nanotubes four degenerate isolated orbitals hosting a total offour electrons are located at the Fermi level As a systematicstudy of large systems at theHuckel level involves a digressionfrom the present topic a general analysis of the four classesof the nanotubes will follow

511 Nanotubes of Type 119890 119890 The Huckel calculation for thenanotubes characterized by an even parity for both 119899 and119901 showed that there were two orbitals at the Fermi levelThe calculations Figure 1 are shown for very wide and verylong nanotubes for the purpose of comparison The abscissa(119896) represents the Huckel orbital number mapped onto theinterval [0 1] It can be seen that there is no gap at the Fermilevel this allows us to characterize such nanotubes as being

6 Advances in Condensed Matter Physics

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100464

Figure 1 Huckel orbital energies for the nanotube (6 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100565

Figure 2 Huckel orbital energies for the nanotube (6 5) where |120573|is the energy unit and 119896 is the normalized orbital number

metallic One can also see in Figure 1 the presence of fourdegenerate levels at the |120573| = 1 level corresponding to therespective value of 119901 (four in this case)

512 Nanotubes of Type 119890 119900 The Huckel calculations arepresented in Figure 2 as one can see there are two degeneratelevels similar to the previous case It is interesting to notealso that as in the previous case there is no gap at the Fermilevel and one can thus characterize the nanotubes as beingmetallicThe figure also shows the presence of five degenerateorbitals at the level |120573| = 1 as a consequence of the value of 119901being five

513 Nanotubes of Type 119900 119890 The case where the nanotubeshave an odd number of rings is characteristically differentfrom the above two cases In the Huckel calculations shown

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100474

Figure 3 Huckel orbital energies for the nanotube (7 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100575

Figure 4 Huckel orbital energies for the nanotube (7 5) where |120573|is the energy unit and 119896 is the normalized orbital number

in Figure 3 one can see the presence of four quasi-degenerateorbitals at the Fermi level On close analysis one can distin-guish pairs or orbitals which are degenerate and there is aslight energy difference between the two pairs making themquasi-degenerate The calculations for very long nanotubesand very wide nanotubes in Figure 3 show that in contrast tothe previous two cases where the number of hexagons (119899) wasa multiple of three there is a significant gap at the Fermi levelbetween the occupied and the virtual orbitals making thenanotubes insulating It is known that the infinite nanotubescharacterized by 119899 and 119901 where 119899 is a multiple of three aremetallic without the presence of a gap at the Fermi level (aband gap)

514 Nanotubes of Type 119900 119900 Figure 4 represents the casecharacterized by an odd value of 119899 and 119898 The similarity to

Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in Condensed Matter Physics 5

Hubbard one-orbital model is able to predict the total spinof the ground state in bipartite lattices In particular one cansee that an imbalance on the number of atom in one sublatticeresults in a magnetic ground state with spin

119878 =

1003816100381610038161003816119873119860minus 119873119861

1003816100381610038161003816

2

(2)

where 119873119860

and 119873119861are the numbers of atoms of 119860- and

119861-type respectively It is also possible to show that fora Huckel Hamiltonian |119873

119860minus 119873119861| is also the number of

eigenvalues equal to zero It is important to note that themagnetization originates from localized edge-states that givealso rise to a high density of states at the Fermi levelwhich in turn can determine a spin polarization instabilityMoreover the relation between the unbalanced number ofatoms and the ground state spin implies that two centerswill be ferromagnetically coupled if they belong to the samesublattice and antiferromagnetically coupled if they do not[45]

We consider now the symmetry of the (119899 119901) nanotubesIt is easy to see that these structures fall into two classesaccording to the parity of the length 119901 The (119899 119901) tubes have119863119899ℎ

symmetry if 119901 is odd ((119899 119900) tubes) and 119863119899119889

symmetryfor 119901 even ((119899 119890) tubes) In all these cases (with the partialexception of the case 119899 = 2 which is however too narrow toform a tube) we are in the presence of non-Abelian symmetrygroups We recall that our ab initio calculations have beenperformed by usingAbelian subgroups of the full point groupof the system According to the parity even (119890) or odd (119900) of 119899and 119901 we have the following Abelian subgroups119862

2V for (119890 119890)1198632ℎfor (119890 119900) 119862

2ℎfor (119900 119890) and 119862

2V for (119900 119900)We call edge carbon atoms the carbon atoms that are

saturated with hydrogens The role of these edge atoms iscrucial in order to understand the finite-size effects in carbonnanotubes The edge carbon atoms of the tubes of type (119890 119901)can support molecular orbitals (MO) of 120587-nature havingalternating sign on any pair of consecutive hexagons Let usconsider the MO combinations that are localized on one ofthe tube extremities only AtHuckel level it is straightforwardto verify that both these alternating edge combinations areeigenfunctions of the one-electron Hamiltonian correspond-ing to eigenvalues equal to zeroTherefore the Huckel energyspectrum presents a pair of degenerate orbitals at the Fermilevel for all (119890 119901) tubes regardless of the parity of 119901 Thesituation is different as far as the ab initio Hamiltonian isconsidered since these two localized edge combinations areno longer exact molecular orbitals of the Fock HamiltonianAs a result at ab initio level the exact degeneracy can be lostIn particular it turns out that while the two edge orbitalsare exactly degenerate for (119890 119890) tubes the degeneracy is onlyapproximated in the case of the (119890 119900) structures This pointcan be easily understood by symmetry considerations In factthe two localized edge orbitals are interchanged by some ofthe symmetry operations of the point group of the system(119863119899ℎ

for (119890 119900) and 119863119899119889

for (119890 119890) tubes) For this reasonthey give rise to a representation of the molecule symmetrygroup In the case of (119890 119890) tubes the representation isirreducible and it still corresponds to two exactly degenerate

molecular orbitals of the system (eg see Figure 6 wherethe two degenerate orbitals have 119890

3symmetry) In the case

of the (119890 119900) tubes on the other hand the two combinationsbelong to two different irreducible representations of theAbelian symmetry group of the system The two orbitals areonly approximately degenerate (see Figure 7 where the twoorbitals have 119887

1119906and 1198872119892

symmetries resp) This argumentexplains the remarkable difference between the tight bindingand ab initio energy spectra in the Fermi level region for the(119890 119900) tubes

5 Results and Discussion

51 Tight Binding Even if the tight binding approach repre-sents a crude approximation of a chemical system it can beextremely useful in order to sketch out some general tenden-cies and to elucidate the behavior of the different classes ofcompoundsThis is particularly true in the present case sincea tight binding approach allows the treatment of very largesystems due to its extremely reduced computational costs Itis possible in this way to explore the behavior of virtuallyldquoinfiniterdquo systems

Short zig-zag nanotubes can be divided into four dif-ferent classes characterized by different symmetries andmolecular-orbital patterns depending on the parity of 119899 and119901 The Huckel calculations for these classes permitted tocharacterize the different active spaces which were essentiallyconfirmed at the ab initio level

Nanotubes of the type (119899 119890) have 119863119899119889

symmetry whilethose of type (119899 119900) have 119863

119899ℎsymmetry At the Huckel level

both (119890 119890) and (119890 119900) have two exactly degenerate orbitalsat the Fermi level hosting two electrons This is illustratedin Figures 1 and 2 where the energy spectra of (6 4) and(6 5) structures are reported For the sake of comparison thespectra of long (6 100) or very wide (100 4) and (100 5)structures are also reported in the figures None of thesespectra shows the presence of an energy gap at the Fermi leveland the wide structures have even an infinite density of levelsat the Fermi level Nanotubes of the types (119900 119890) and (119900 119900) onthe other hand have four quasi-degenerate levels (two quasi-degenerate pairs of exactly degenerate levels) near the Fermilevel hosting a total of four electrons This fact is illustratedfor the (7 4) and (7 5) structures in Figures 3 and 4 Fromthese figures it appears that in the case of very long (119900 119901)

nanotubes four degenerate isolated orbitals hosting a total offour electrons are located at the Fermi level As a systematicstudy of large systems at theHuckel level involves a digressionfrom the present topic a general analysis of the four classesof the nanotubes will follow

511 Nanotubes of Type 119890 119890 The Huckel calculation for thenanotubes characterized by an even parity for both 119899 and119901 showed that there were two orbitals at the Fermi levelThe calculations Figure 1 are shown for very wide and verylong nanotubes for the purpose of comparison The abscissa(119896) represents the Huckel orbital number mapped onto theinterval [0 1] It can be seen that there is no gap at the Fermilevel this allows us to characterize such nanotubes as being

6 Advances in Condensed Matter Physics

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100464

Figure 1 Huckel orbital energies for the nanotube (6 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100565

Figure 2 Huckel orbital energies for the nanotube (6 5) where |120573|is the energy unit and 119896 is the normalized orbital number

metallic One can also see in Figure 1 the presence of fourdegenerate levels at the |120573| = 1 level corresponding to therespective value of 119901 (four in this case)

512 Nanotubes of Type 119890 119900 The Huckel calculations arepresented in Figure 2 as one can see there are two degeneratelevels similar to the previous case It is interesting to notealso that as in the previous case there is no gap at the Fermilevel and one can thus characterize the nanotubes as beingmetallicThe figure also shows the presence of five degenerateorbitals at the level |120573| = 1 as a consequence of the value of 119901being five

513 Nanotubes of Type 119900 119890 The case where the nanotubeshave an odd number of rings is characteristically differentfrom the above two cases In the Huckel calculations shown

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100474

Figure 3 Huckel orbital energies for the nanotube (7 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100575

Figure 4 Huckel orbital energies for the nanotube (7 5) where |120573|is the energy unit and 119896 is the normalized orbital number

in Figure 3 one can see the presence of four quasi-degenerateorbitals at the Fermi level On close analysis one can distin-guish pairs or orbitals which are degenerate and there is aslight energy difference between the two pairs making themquasi-degenerate The calculations for very long nanotubesand very wide nanotubes in Figure 3 show that in contrast tothe previous two cases where the number of hexagons (119899) wasa multiple of three there is a significant gap at the Fermi levelbetween the occupied and the virtual orbitals making thenanotubes insulating It is known that the infinite nanotubescharacterized by 119899 and 119901 where 119899 is a multiple of three aremetallic without the presence of a gap at the Fermi level (aband gap)

514 Nanotubes of Type 119900 119900 Figure 4 represents the casecharacterized by an odd value of 119899 and 119898 The similarity to

Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Physics Research International

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Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in Condensed Matter Physics

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100464

Figure 1 Huckel orbital energies for the nanotube (6 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level6100

100565

Figure 2 Huckel orbital energies for the nanotube (6 5) where |120573|is the energy unit and 119896 is the normalized orbital number

metallic One can also see in Figure 1 the presence of fourdegenerate levels at the |120573| = 1 level corresponding to therespective value of 119901 (four in this case)

512 Nanotubes of Type 119890 119900 The Huckel calculations arepresented in Figure 2 as one can see there are two degeneratelevels similar to the previous case It is interesting to notealso that as in the previous case there is no gap at the Fermilevel and one can thus characterize the nanotubes as beingmetallicThe figure also shows the presence of five degenerateorbitals at the level |120573| = 1 as a consequence of the value of 119901being five

513 Nanotubes of Type 119900 119890 The case where the nanotubeshave an odd number of rings is characteristically differentfrom the above two cases In the Huckel calculations shown

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100474

Figure 3 Huckel orbital energies for the nanotube (7 4) where |120573|is the energy unit and 119896 is the normalized orbital number

k

Ener

gy (|120573|)

1080604020

3

2

1

0

minus1

minus2

minus3

Fermi level7100

100575

Figure 4 Huckel orbital energies for the nanotube (7 5) where |120573|is the energy unit and 119896 is the normalized orbital number

in Figure 3 one can see the presence of four quasi-degenerateorbitals at the Fermi level On close analysis one can distin-guish pairs or orbitals which are degenerate and there is aslight energy difference between the two pairs making themquasi-degenerate The calculations for very long nanotubesand very wide nanotubes in Figure 3 show that in contrast tothe previous two cases where the number of hexagons (119899) wasa multiple of three there is a significant gap at the Fermi levelbetween the occupied and the virtual orbitals making thenanotubes insulating It is known that the infinite nanotubescharacterized by 119899 and 119901 where 119899 is a multiple of three aremetallic without the presence of a gap at the Fermi level (aband gap)

514 Nanotubes of Type 119900 119900 Figure 4 represents the casecharacterized by an odd value of 119899 and 119898 The similarity to

Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

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Advances in Condensed Matter Physics

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Advances in Condensed Matter Physics 7

Table 1 The various ground state energy levels and their occupation numbers (STO-3G basis set) for the four different classes of nanotubesalong with their true symmetries It is interesting to note the abelian groups of each class class I (6 4) = 119862

2V class II (6 5) = 1198632ℎ class III

(7 4) = 1198622ℎ class IV (7 5) = 119862

2V

Nanotube Symmetry groupof the nanotube

Orbitalssymmetry

High-spin ROHF orbital energy Electronic ground state CAS-CI occupation number120598119894in au HF CAS-CI 119899

119894

6 4 1198636119889

(i) 1198903

minus01180 31198602

31198602

10000(ii) 1198903

6 5 1198636ℎ

(i) 1198871119906

minus01176 31198602119906

11198601119892

09985(ii) 1198872119892

minus01178 10015

7 4 1198637119889

(i) 1198903119892

minus0135951198601119892

11198601119892

09517(ii) 1198903119892

(iii) 1198903119906

minus01431 10483(iv) 1198903119906

7 5

(i) 11989010158401 minus01384

51198601015840

1

11198601015840

1

09684(ii) 11989010158401

(iii) 119890101584010158401 minus01417 10316

(iv) 119890101584010158401

the previous case is shown by the presence of two pairs ofquasi-degenerate orbitals at the Fermi level and the presenceof a gap between the occupied and the virtual orbitals Onecan see here the presence of a number of degenerate orbitalsat the |120573| = 1 level

52 Ab Initio From the Huckel results it is clear that aclosed-shell determinant is never a good zero-order descrip-tion of the electronic ground state of short zig-zag nanotubesTherefore as discussed in Computational Details we per-formed ROHF calculations to optimize the geometry for thespin multiplicity that gives the minimum energy for eachsystem In the range of the studied values of 119899 and 119901 thismeans triplets for the (119890 119901) nanotubes and quintets for the(119900 119901) ones The orbitals that are obtained from these ROHFcalculations are then used for the subsequent CAS-CI [39]and NEVPT2 [40ndash42] calculations

We will discuss the behavior of the four classes ofnanotubes of zig-zag type As a general remark we notice thatthe frontier orbitals of all four classes are localized at the edgesof the tubes This is exactly what happens at the Huckel level

The most remarkable difference between Huckel and abinitio results concerns the nanotube of the type (119890 119900) having119863119899ℎ

symmetry In fact the 119906 and 119892 combinations of borderorbitals are obviously not degenerated at ab initio level sincethey belong to different irreducible representations of thesystem point group

However at Huckel level it is straightforward to checkthat an alternating orbital located on a single edge of thenanotube (see schematic drawings of Figure 5) is an eigen-function of the Hamiltonian corresponding to a zero valueof the energy This implies that any combination of the twotop and bottom border orbitals will be an eigenfunction withvanishing energy so the 119892 and 119906 combinations are strictlydegenerate Moreover they are completely localized on thenanotube edges

Figure 5 Schemeof the twoopposite alternantHuckel edge orbitals

521 Nanotubes of Type 119890 119890 The nanotubes of this type havea symmetry 119863

119899119889where 119899 is the number of hexagons in a

ring The highest Abelian subgroups are 1198622V as a result of

the absence of the 120590ℎplane contrary to the case where 119901 is

odd see Figure 6These nanotubes for which the active spaceconsisted of two orbitals have two exactly degenerate orbitalsof symmetry 119890

3at the Fermi level (see Table 1) strongly

localized on each edge The geometry was optimized for thetriplet state followed by CAS-CI calculations for the tripletof symmetry 3119860

2and the two singlets 1119860

1and 1119864

2(see

Table 2)It appears that at CAS-CI level the energy difference

between the triplet and the singlet states is almost zero (ofthe order of 10minus5 kJmolminus1 for the studied systems) Thisalmost perfect degeneracy is removed when the effect ofthe dynamical correlation is introduced at NEVPT2 levelHowever the ground state remains a singlet electronic stateresulting from an antiferromagnetic coupling of the two

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in Condensed Matter Physics

Table 2 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 even All possible states for a CASof two electrons in two orbitals have been calculated

np State 2 4 6 8CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601

5 sdot 10minus4 minus195 3 sdot 10minus5 minus59 6 sdot 10minus7

minus24 3 sdot 10minus8 minus1131198602

00 00 00 00 00 00 00 0011198642

4198 1508 5119 1592 5532 1799 5790 2517

8

11198601

3 sdot 10minus3 minus347 1 sdot 10minus4 minus223 8 sdot 10minus6

minus209 1 sdot 10minus6 minus24331198602

00 00 00 00 00 00 00 0011198642

3163 1108 3955 749 4396 minus1150 4569 minus5538

10

11198601

7 sdot 10minus3

minus569 4 sdot 10minus4

minus635 4 sdot 10minus5

minus981 mdash mdash31198602

00 00 00 00 00 00 mdash mdash11198642

2517 849 3199 minus601 3547 minus3908 mdash mdash

e3 e3

Figure 6 Active orbitals of nanotube (6 4) as an example of nanotubes with both 119899 and 119901 evenThe orbitals are of identical size the apparentdifference in size is due to the perspective

(a) 1198871119906 (b) 1198872119892

Figure 7 Active orbitals of nanotube 6 5 as an example of nanotubes with 119899 even and 119901 odd

electrons localized on each edge of the nanotube whateverthe length of the nanotube is

This extremely small singlet-triplet gap at the CAS-CIincreases when the effect of dynamic correlation is intro-duced by the NEVPT2 approach It can also be noted thatfor the case where 119899 is six the singlet-triplet gap decreasesas the length of the nanotube is increased going from about20 to 1 kJmolminus1 as shown in Table 2 The rest of the casesshow a trend where the 1119860

1singlet seems to be replaced

by the 11198642singlet as the length is increased Actually state-

specific CASSCF calculations virtually do not change thesinglet-triplet gap as can be seen in the corresponding tablesof the Supplementary Material This remains true for all theclasses of nanotubes However the difference between thetriplet and the ionic state 1119864

2(see wavefunction composition

in the Supplementary Material) is significantly lowered byabout 261 kJsdotmolminus1 This shows a strong modification ofthe orbitals which can not be overcome by the NEVPT2

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in Condensed Matter Physics 9

Table 3 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 even and 119901 odd All possible states for a CASof two electrons in two orbitals have been calculated

np State 1 3 5 7CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

6

11198601119892

minus03 minus430 minus1 sdot 10minus2 minus101 minus3 sdot 10minus4 minus37 minus2 sdot 10minus5

minus1631198602119906

00 00 00 00 00 00 00 0011198641119892

3193 1624 4764 1519 5539 1693 5661 1901

8

11198601119892

minus37 minus431 minus01 minus259 minus7 sdot 10minus3

minus207 minus9 sdot 10minus4

minus21931198602119906

00 00 00 00 00 00 00 0011198641119892

2375 1389 3641 793 4175 minus37 4458 minus1271

10

11198601119892

minus129 minus347 minus06 minus555 minus01 minus769 mdash mdash31198602119906

00 00 00 00 00 00 mdash mdash11198641119892

1882 1251 2923 271 3398 minus2004 mdash mdash

approach Furthermore it has been shown that for ionicstates NEVPT2 method is inadequate if plain CASSCForbitals are used [46 47] Thus NEVPT2 excitation energiesto ionic states for the other classes are documented in thetables below but will not be discussed

The occupation numbers of Table 1 and the CI vectorsgiven in the Supplementary Material clearly show the mul-tideterminantal character of the wavefunction Indeed foreach of the states considered here the active electrons areevenly distributed among the two active orbitals It can alsobe seen that the effect of the basis set size on the wavefunctionnature is completely negligible in that case

522 Nanotubes of Type 119890 119900 The orbitals for this case arepresented in Figure 7 The nanotube has the symmetry 119863

119899ℎ

as a result of the presence of a 120590ℎplane The two orbitals are

not localized at one edge or the other but are equally presentat the two edges at the same time with gerade and ungeradesymmetry as a consequence the two orbitals are quasi-degenerate The two orbitals are of 119887

1119906and 119887

2119892symmetry

see Table 1 the gerade is more stable due to the antibondinginteractions between the two edge orbitals located at the endsof the nanotube

In this case also the HF calculations and the geom-etry optimization were done for the triplet state and theresulting strongly localized orbitals were used for the CAS-CINEVPT2 calculationsThese calculations were carried outfor the triplet state of symmetry 3119860

2119906followed by the two

singlet states 11198601119892and 1119864

1119892as shown in Table 3 One can see

that only the short nanotubes have 11198601119892as the ground state at

both CAS-CI and NEVPT2 level Again this probably arisesfrom a strong revision of the orbitals for the higher stateswhich is not taken into account here At the CAS-CI level thelowest singlet-triplet gap is very small (except for the (10 1)

case which corresponds to a ring rather than a tube) anddecreases with the length of the nanotube as was observedfor the 119890 119890 case At the NEVPT2 results the trend is oppositeand is even enhanced for the widest nanotubes As in theprevious case 1119860

1119892remains the ground state independently

of the length at CAS-CI level This is also true at NEVPT2

level for the smallest nanotubes while for the larger one weobserve the same limitation as for the 119890 119890 tubes

As for the previous class of nanotubes the wavefunctionof the ground states shows a strong multideterminantalcharacter with a equal mixing of the two configurations|11988711199061198871119906⟩ and |119887

21198921198872119892⟩ due to the quasi-degeneracy of the two

orbitals and leading to an antiferromagnetic coupling of thetwo electrons in the edge orbitals

523 Nanotubes of Type 119900 119890 The orbitals are shown inFigure 8 According to the predictions of the Huckel calcu-lations we find four quasi-degenerate edge orbitals The nan-otube has 119863

119899119889symmetry with the largest Abelian subgroup

being 1198622ℎ The orbitals located at the edges form two pairs of

degenerate orbitals see Table 1 and on taking linear combi-nations of the degenerate orbitals one can construct orbitalssimilar to those of the type 119890 119890 that is two orbitals localizedat one of the edges The pairs of orbitals are separated by anenergy gap due to the anti-interaction between the two edgeorbitals with the ungerade orbitals beingmore stable than thegerade

As for the other classes the geometry was optimizedfor the high-spin quintet state at the ROHF level Theorbitals of the ROHF calculation were used in the CAS-CINEVPT2 calculations These calculations were done forthe quintet state and the lowest lying singlet and tripletstates in each of the four symmetries of the correspondinglargest Abelian group The results are shown in Table 4 Thesinglet open shell of symmetry 1119860

1119892is the ground state for

the short nanotubes and is overtaken by the 11198601119906

state asthe length of the nanotube increases Once again we noticethat the CAS-CI energies are significantly relaxed after theNEVPT2 calculations and that there is a strong competitionbetween several singlet states for the ground state at this levelAgain we believe the low-lying spectrum to be correct andsome more elaborate calculations (state-specific CASSCF +NEVPT2 or multireference configuration interaction) usinglarger basis sets should confirm the nature of the CAS-CIground state

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in Condensed Matter Physics

(a) 1198903119892

(b) 1198903119906

Figure 8 Active orbitals of nanotube (7 4) as an example of nanotubes with 119899 odd and 119901 even

(a) 11989010158403

(b) 119890101584010158403

Figure 9 Active orbitals for nanotube 7 5 as an example of nanotubes with 119899 odd and 119901 odd

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in Condensed Matter Physics 11

Table 4 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 even All possible states for a CASof four electrons in four orbitals have been calculated

np State 2 4 6CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601119892

minus231 minus565 minus09 minus97 minus4 sdot 10minus2

minus2331198602119906

minus148 minus378 minus06 minus65 minus2 sdot 10minus2

minus1651198601119892

00 00 00 00 00 0011198601119906

3464 minus492 3969 minus45 3987 minus1731198641119906

1780 minus171 1985 06 1993 2731198641119892

2031 minus77 1994 13 1994 2711198641119892

4060 minus74 3988 17 3988 2411198641119906

4466 277 5693 459 5997 972

9

11198601119892

minus133 minus515 minus05 minus159 minus2 sdot 10minus2

minus7531198602119906

minus86 minus348 minus03 minus106 minus1 sdot 10minus2 minus5051198601119892

00 00 00 00 00 00011198601119906

3337 minus474 3878 minus379 3824 minus34211198641119892

3845 minus335 3801 minus293 3825 minus32431198641119892

1923 minus219 1900 minus153 1912 minus15531198641119906

1765 minus180 1895 minus155 1912 minus15511198641119906

4099 522 5040 415 5478 minus377

11

11198601119892

minus101 minus616 minus04 minus358 minus3 sdot 10minus2 minus27831198602119906

minus66 minus416 minus03 minus239 minus2 sdot 10minus2

minus18551198601119892

00 00 00 00 00 0011198601119906

3091 minus523 3608 minus889 3629 minus87631198641119892

1814 minus386 1811 minus394 1815 minus43631198641119906

1686 minus263 1807 minus392 1814 minus43711198641119906

3705 513 4498 28 4858 minus55911198641119892

3603 514 3623 minus708 3629 minus821

524 Nanotubes of Type 119900 119900 The orbitals are shown inFigure 9 As in the previous case there are four valenceorbitals located at the edges The nanotube is characterizedby the 119863

119899ℎsymmetry the largest Abelian subgroup being

1198622V The two edge orbitals located on one side of 120590V are

exactly degenerate with the two located at the opposite sideof the 120590V plane see Table 1 The two pairs of orbitals arequasi-degenerate as a result of the syn- and anti-interactionbetween the edge orbitals located at the two edges with theanti-interaction being the more stable

Again the HF calculations and geometry optimizationswere carried out on the quintet state with four orbitals in theopen-shell valence spaceThese calculations were followed bythe CAS-CINEVPT2 calculations with the same orbitals asactive space The lowest lying singlet and triplet state of eachsymmetry in the Abelian subgroup were calculated alongwith the quintet state The results are presented in Table 5It is seen that similar to the previous case there are twodegenerate triplet states and 11198601015840

1remains the lowest lying

state for the nanotubes

525 Basis Set Effects One can see from Tables 6 and 7 thatthe general trend of the lowest electronic states and the orderof magnitude of the CAS-CI energies do not change for the

calculations with the cc-pVDZ basis set This also holds forthe lowest states at the NEVPT2 level An increase of the basisset size involves a relative stabilization of all the levels but theordering of the levels remains the same

6 Conclusion

We presented in this paper a tight binding and ab initio (CAS-CI and NEVPT2) study to evaluate the orbital and verticalelectronic state spectrum of short finite zig-zag nanotubes(119899 0)The calculations were carried out on zig-zag nanotubes(119899 119901) of different size where 119899 indicates the number ofhexagons around the tube while 119901 is the number of hexagonsin the direction of the tube axis The geometry was relaxedand optimized by taking into account the full symmetry ofthe nanotubes It was found that there are four broad classesof systems depending on the parity of the parameters 119899 and119901 There is a close similarity between the tight binding andab initio results with respect to the orbitals pattern close tothe Fermi levelThis confirms the fact often seen in several 120587systems particularly in the largest ones that the basic physicsof these systems is caught already at this very crude level ofapproximation However a significant qualitative differenceis found in the case of (119890 119900) tubes in this case two exactly

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Advances in Condensed Matter Physics

Table 5 CAS-CI and the NEVPT2 energies in kJmolminus1 for the class of nanotubes 119899 119901 with 119899 odd and 119901 odd All possible states for a CAS offour electrons in four orbitals have been calculated

np State 1 3 5CAS-CI NEVPT2 CAS-CI NEVPT2 CAS-CI NEVPT2

7

11198601015840

1minus1229 minus1722 minus45 minus222 minus02 minus45

311986010158401015840

2minus700 minus1110 minus30 minus147 minus01 minus30

51198601015840

100 00 00 00 00 00

311986410158401015840

11087 minus655 1952 minus20 1992 22

31198641015840

12164 minus379 2000 minus28 1994 20

111986010158401015840

12551 minus107 3894 168 3984 36

11198641015840

14229 20 3999 minus01 3987 22

111986410158401015840

13083 403 5184 434 5900 903

9

11198601015840

1minus871 minus1245 minus24 minus277 minus01 minus104

311986010158401015840

2minus505 minus831 minus16 minus185 minus01 minus70

51198601015840

100 00 00 00 00 00

31198641015840

11994 minus456 1911 minus191 1903 minus153

311986410158401015840

11197 minus360 1885 minus158 1902 minus150

11198641015840

13691 63 3822 minus302 3807 minus306

111986010158401015840

12606 387 3753 36 3805 minus281

111986410158401015840

12835 923 4659 546 5264 450

11

11198601015840

1minus722 minus1006 minus20 minus447 minus01 minus307

311986010158401015840

2minus421 minus695 minus13 minus298 minus01 minus204

51198601015840

100 00 00 00 00 00

31198641015840

11855 minus555 1810 minus392 1813 minus417

311986410158401015840

11189 minus195 1787 minus345 1812 minus413

11198641015840

13298 minus44 3619 minus645 3626 minus767

111986010158401015840

12540 739 3550 minus59 3624 minus702

111986410158401015840

12436 948 4091 384 4661 minus181

Table 6 Basis set effects for the nanotubes (6 4) and (6 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

6 4

11198601

2 sdot 10minus5

minus24 4 sdot 10minus4

minus3331198602

00 00 000 00011198642

5119 1651 4281 minus117

6 5

11198601119892

3 sdot 10minus4 minus37 minus1 sdot 10minus5 minus3831198602119906

00 00 00 0011198642119906

5359 1693 4523 minus20

degenerate energy levels are found at the Fermi level in thetight binding description on the other hand the energydegeneracy is only approximated at ab initio level

In fact at tight binding level both (119890 119890) and (119890 119900) tubeshave two exactly degenerate energy levels at the Fermi levelHowever only the degeneracy in the case of (119890 119890) tubes isa direct consequence of the space point-group symmetry ofthe tube and is therefore conserved also at ab initio level The(119890 119900) tubes on the other hand have two energy levels thatare only approximately degenerate at ab initio level althoughthe split between the levels goes to zero very quickly as afunction of the tube lengthThe nanotubes with an odd parityof 119899 have four quasi-degenerate orbitals making up the Fermi

Table 7 Basis set effects for the nanotubes (7 4) and (7 5)

119899 119901 State STO-3G VDZCAS-CI NEVPT2 CAS-CI NEVPT2

7 4

11198601119892

minus09 minus97 minus01 minus6031198602119906

minus06 minus65 minus4 sdot 10minus2

minus4051198601119892

00 00 00 0011198601119906

3969 minus45 2998 minus68431198641119906

1985 06 1499 minus33431198641119892

1994 13 1450 minus33511198641119892

3988 17 2999 minus66911198641119906

5693 459 4537 minus586

7 5

11198601015840

1minus02 minus45 minus1 sdot 10minus2 minus31

311986010158401015840

2minus01 minus30 minus7 sdot 10

minus3minus20

51198601015840

100 00 00 00

31198641015840

11994 20 1501 minus329

11198641015840

13987 22 3002 minus668

311986410158401015840

11992 22 1501 minus328

111986010158401015840

13984 36 3002 minus669

111986410158401015840

15990 903 4471 minus450

level They are composed of two pairs of strictly degeneratelevels one slightly below and the other one slightly above

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in Condensed Matter Physics 13

the Fermi level This behavior is qualitatively similar at tightbinding and ab initio level The vertical energy calculationsshow that in all the cases the multiplicity of the ground stateis a singlet in systematical accordance with the Ovchinnikovrule [9 10] The singlet ground state in all the four caseswas found to be of a highly open-shell nature thus requiringa multireference approach for the calculation of the energyspectrum As expected other low-lying states of differentspin multiplicities are found at a very small energy above thesinglet ground state indicating the potential interest of thesemolecular structures in the field of nanotechnology

In future works we are planning to address the problemof the mobility of the electrons within these structures Tothis purpose it is possible to investigate the response of thesystem to the action of electric fields along a direction parallelto the tube axis A further possibility is given by the use ofthe formalism of the so-called Localization Tensor [48ndash52]or Second Moment Cumulant of the Position operator forwhich an implementation at CAS-SCF level has recently beenachieved in our group [53]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the French ldquoCentreNational de la Recherche Scientifiquerdquo (CNRS also underthe PICS action 4263) and ANR under the ANRDFG JointProject MITLOW (ANR-11-INTB-1009) the HPC resourcesof CALMIP under the allocation 2011-[p1048] and the ItalianMinistry of University and Research (MUR) The authorswould also like to thank the Erasmus Mundus Programfunded by the European Commission and in particular itsproject ldquoTCCM Euromaster on Theoretical Chemistry andComputational Modellingrdquo for partial support of this workthrough a generous grant to one of the present authors (VijayGopal Chilkuri)

References

[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004

[2] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[3] W Andreoni The Physics of Fullerene-Based and Fullerene-Related Materials Springer Berlin Germany 2000

[4] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008

[5] J Chen S Chen X Zhao L V Kuznetsova S S Wong andI Ojima ldquoFunctionalized single-walled carbon nanotubes asrationally designed vehicles for tumor-targeted drug deliveryrdquoJournal of the American Chemical Society vol 130 no 49 pp16778ndash16785 2008

[6] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[7] B Trauzettel D V Bulaev D Loss andG Burkard ldquoSpin qubitsin graphene quantumdotsrdquoNature Physics vol 3 no 3 pp 192ndash196 2007

[8] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009

[9] I A Misurkin and A A Ovchinnikov ldquoThe electronic struc-tures and properties of polymeric molecules with conjugatedbondsrdquo Russian Chemical Reviews vol 46 no 10 pp 967ndash9871977

[10] A A Ovchinnikov ldquoMultiplicity of the ground state of largealternant organic molecules with conjugated bondsrdquoTheoreticaChimica Acta vol 47 no 4 pp 297ndash304 1978

[11] E H Lieb ldquoTwo theorems on the Hubbard modelrdquo PhysicalReview Letters vol 62 no 10 pp 1201ndash1204 1989

[12] J F Rossier and J J Palacios ldquoMagnetism in graphene nanois-landsrdquo Physical Review Letters vol 99 Article ID 177204 2007

[13] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoA theoretical study of closed polyacene structuresrdquo PhysicalChemistry Chemical Physics vol 14 no 45 pp 15666ndash156762012

[14] A Rochefort D R Salahub and P Avouris ldquoEffects of finitelength on the electronic structure of carbon nanotubesrdquo Journalof Physical Chemistry B vol 103 no 4 pp 641ndash646 1999

[15] O Hod and G E Scuseria ldquoHalf-metallic zigzag carbonnanotube dotsrdquo ACS Nano vol 2 no 11 pp 2243ndash2249 2008

[16] C M Ruiz and S D Dalosto ldquoElectronic and magneticchanges in a finite-sized single-walled zigzag carbon nanotubeembedded in waterrdquo The Journal of Physical Chemistry C vol117 no 1 pp 633ndash638 2013

[17] S Tang and Z Cao ldquoElectronic and magnetic propertiesof nitrogen-doped finite-size and open-ended zigzag carbonnanotubesrdquo Computational Materials Science vol 50 no 6 pp1917ndash1924 2011

[18] J Wu and F Hagelberg ldquoMagnetism in finite-sized single-walled carbon nanotubes of the zigzag typerdquo Physical Review Bvol 79 Article ID 115436 2009

[19] M Marganska M del Valle S H Jhang C Strunk and MGrifoni ldquoLocalization induced by magnetic fields in carbonnanotubesrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 83 no 19 Article ID 193407 2011

[20] W Chun-Shian and C Jeng-Da ldquoElectronic properties ofzigzag graphene nanoribbons studied by TAO-DFTrdquo Journal ofChemical Theory and Computation vol 11 no 5 pp 2003ndash20112015

[21] R Sundaram S Scheiner A K Roy and T Kar ldquoB=N bondcleavage and BN ring expansion at the surface of boron nitridenanotubes by iminoboranerdquo Journal of Physical Chemistry Cvol 119 no 6 pp 3253ndash3259 2015

[22] Y Wang G Fang J Ma and Y Jiang ldquoFacial dissociations ofwater molecules on the outside and inside of armchair single-walled silicon nanotubes theoretical predictions from mul-tilayer quantum chemical calculationsrdquo Theoretical ChemistryAccounts vol 130 no 2 pp 463ndash473 2015

[23] W Mizukami Y Kurashige and T Yanai ldquoMore 120587 electronsmake a difference emergence of many radicals on graphenenanoribbons studied by Ab initio DMRG theoryrdquo Journal ofChemical Theory and Computation vol 9 no 1 pp 401ndash4072013

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

14 Advances in Condensed Matter Physics

[24] A Donarini A Yar and M Grifoni ldquoSpectrum and Franck-Condon factors of interacting suspended single-wall carbonnanotubesrdquo New Journal of Physics vol 14 no 2 pp 23045ndash23067 2012

[25] A Dirnaichner M Grifoni A Pruing D Steininger A KHuttel and C Strunk ldquoTransport across a carbon nanotubequantum dot contacted with ferromagnetic leads experimentand non-perturbative modelingrdquo Physical Review B vol 91 no19 Article ID 195402 2015

[26] P R Wallace ldquoThe band theory of graphiterdquo Physical Reviewvol 71 pp 622ndash634 1947

[27] M El Khatib S Evangelisti T Leininger and G L BendazzolildquoPartly saturated polyacene structures a theoretical studyrdquoJournal of Molecular Modeling vol 20 no 7 article 2284 2014

[28] R Hoffmann ldquoAn extended Huckel theory I HydrocarbonsrdquoThe Journal of Chemical Physics vol 39 no 6 article 1397 1963

[29] T Helgaker P Joslashrgensen and J Olsen Molecular Electronic-StructureTheory JohnWileyamp Sons NewYorkNYUSA 2000

[30] B O Roos ldquoThe complete active space self-consistent fieldmethod and its applications in electronic structure calcula-tionsrdquo in Advances in Chemical Physics Ab Initio Methods inQuantum Chemistry K P Lawle Ed vol 69 part 2 pp 399ndash445 John Wiley amp Sons New York NY USA 2007

[31] B O Roos and K Andersson ldquoMulticonfigurational pertur-bation theory with level shiftmdashthe Cr

2potential revisitedrdquo

Chemical Physics Letters vol 245 no 3 pp 215ndash223 1995[32] G Ghigo B O Roos and P A Malmqvist ldquoA modified defi-

nition of the zeroth-order Hamiltonian in multiconfigurationalperturbation theory (CASPT2)rdquo Chemical Physics Letters vol396 no 1ndash3 pp 142ndash149 2004

[33] S Evangelisti G L Bendazzoli and A Monari ldquoCharge stateof metal atoms on oxide supports a systematic study based onsimulated infrared spectroscopy and density functional theoryrdquoTheoretical Chemistry Accounts vol 126 no 3 pp 265ndash2732010

[34] G L Bendazzoli S Evangelisti A Monari and R RestaldquoKohnrsquos localization in the insulating state one-dimensionallattices crystalline versus disorderedrdquo The Journal of ChemicalPhysics vol 133 no 6 Article ID 064703 2010

[35] A Monari and S Evangelisti ldquoFinite-size effects in graphenenanostructuresrdquo in Physics and Applications of Graphene The-ory SMikhailov Ed chapter 14 p 303 InTech Rijeka Croatia2011

[36] W J Hehre R F Stewart and J A Pople ldquoSelf-consistentmolecular-orbital methods I Use of gaussian expansions ofslater-type atomic orbitalsrdquoThe Journal of Chemical Physics vol51 no 6 pp 2657ndash2664 1969

[37] T H Dunning Jr ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquoThe Journal of Chemical Physics vol 90 article 10071989

[38] P Knowles and H J Werner ldquoMolpro quantum chemistrypackagerdquo httpwwwmolpronet

[39] B Roos P R Taylor and P E M Siegbahn ldquoA complete activespace SCFmethod (CASSCF) using a densitymatrix formulatedsuper-CI approachrdquoChemical Physics vol 48 no 2 pp 157ndash1731980

[40] C Angeli R Cimiraglia S Evangelisti T Leininger and J-PMalrieu ldquoIntroduction of n-electron valence states formultiref-erence perturbation theoryrdquo Journal of Chemical Physics vol114 no 23 Article ID 10252 2001

[41] C Angeli R Cimiraglia and J-P Malrieu ldquon-electron valencestate perturbation theory a spinless formulation and an efficientimplementation of the strongly contracted and of the partiallycontracted variantsrdquo Journal of Chemical Physics vol 117 no 20pp 9138ndash9153 2002

[42] C Angeli M Pastore and R Cimiraglia ldquoNew perspectives inmultireference perturbation theory the n-electron valence stateapproachrdquo Theoretical Chemistry Accounts vol 117 no 5-6 pp743ndash754 2007

[43] H-J Werner and P J Knowles ldquoA second order multiconfigu-ration SCF procedure with optimum convergencerdquoThe Journalof Chemical Physics vol 82 no 11 pp 5053ndash5063 1985

[44] P J Knowles and H-J Werner ldquoAn efficient second-orderMC SCF method for long configuration expansionsrdquo ChemicalPhysics Letters vol 115 no 3 pp 259ndash267 1985

[45] D Yu E M Lupton H J Gao C Zhang and F Liu ldquoA unifiedgeometric rule for designing nanomagnetism in graphenerdquoNano Research vol 1 no 6 pp 497ndash501 2008

[46] C Angeli ldquoOn the nature of the 120587 rarr 120587 ionic excited states

the V state of ethene as a prototyperdquo Journal of ComputationalChemistry vol 30 no 8 pp 1319ndash1333 2009

[47] C Angeli and C J Calzado ldquoThe role of the magnetic orbitalsin the calculation of the magnetic coupling constants frommultireference perturbation theory methodsrdquo The Journal ofChemical Physics vol 137 no 3 Article ID 034104 2012

[48] R Resta and S Sorella ldquoElectron localization in the insulatingstaterdquo Physical Review Letters vol 82 no 2 pp 370ndash373 1999

[49] C SgiarovelloM Peressi andR Resta ldquoElectron localization inthe insulating state application to crystalline semiconductorsrdquoPhysical Review B vol 64 no 11 Article ID 115202 2001

[50] R Resta ldquoWhy are insulators insulating and metals conduct-ingrdquo Journal of Physics Condensed Matter vol 14 no 20 ppR625ndashR656 2002

[51] R Resta ldquoElectron localization in the quantum hall regimerdquoPhysical Review Letters vol 95 Article ID 196805 2005

[52] R Resta ldquoThe insulating state of matter a geometrical theoryrdquoThe European Physical Journal B vol 79 no 2 pp 121ndash137 2011

[53] M El Khatib T Leininger G L Bendazzoli and S Evange-listi ldquoComputing the Position-Spread tensor in the CAS-SCFformalismrdquo Chemical Physics Letters vol 591 pp 58ndash63 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of