Density functional study of structural and electronic properties of AlnN (1 ≤ n ≤ 12) clusters

Density Functional Study of Structuraland Electronic Properties ofAlnN (1 � n � 12) Clusters

LING GUO, HAI-SHUN WUSchool of Chemistry and Material Science, Shanxi Normal University, Linfen 041004, People’sRepublic of China

Received 19 August 2005; accepted 12 September 2005Published online 14 November 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20870

ABSTRACT: Low-lying equilibrium geometric structures of AlnN (n � 1–12) clustersobtained by an all-electron linear combination of atomic orbital approach, within spin-polarized density functional theory, are reported. The binding energy, dissociationenergy, and stability of these clusters are studied within the local spin densityapproximation (LSDA) and the three-parameter hybrid generalized gradientapproximation (GGA) due to Becke–Lee–Yang–Parr (B3LYP). Ionization potentials,electron affinities, hardness, and static dipole polarizabilities are calculated for theground-state structures within the GGA. It is observed that symmetric structures withthe nitrogen atom occupying the internal position are lowest-energy geometries.Generalized gradient approximation extends bond lengths as compared with the LSDAlengths. The odd–even oscillations in the dissociation energy, the second differences inenergy, the highest occupied molecular orbital–lowest unoccupied molecular orbital(HOMO–LUMO) gaps, the ionization potential, the electron affinity, and the hardnessare more pronounced within the GGA. The stability analysis based on the energiesclearly shows the Al7N cluster to be endowed with special stability. © 2005 WileyPeriodicals, Inc. Int J Quantum Chem 106: 1250–1257, 2006

Key words: aluminum nitrogen; DFT theory; stability

1. Introduction

S mall clusters composed of aluminum atomhave been the subjects of intensive studies for

the past two decades. A large number of studies ofaluminum clusters, both theoretical as well as ex-

perimental have been reported (see, e.g., the re-views in Refs. [1–4].) One of the main motivationsbehind these studies is to understand the evolutionof physical properties with the size of the cluster.Many properties of aluminum clusters can be un-derstood using the spherical jellium model (SJM), inwhich the ions are smeared out in a uniformlycharged sphere leading to electronic shell closuresfor clusters containing a “magic” number 2, 8, 20,Correspondence to: L. Guo; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 106, 1250–1257 (2006)© 2005 Wiley Periodicals, Inc.

40, 58, 92, 138, . . . of valence electrons. These find-ings were subsequently confirmed by first-princi-ples theoretical calculations in which the ions wererepresented by pseudo-potentials [5]. The questionaddressed in the present study is the effect of dop-ing by a single impurity on the electronic structureand geometry of these clusters. In bulk materials, asmall percentage of impurity is known to affect theproperties significantly. In clusters, the impurityeffect should be even more pronounced and influ-enced by the finite size of the system. Under vac-uum condition, and using the magnetron reactivesputtering technique, the sputtering technique, thesputtered Al atoms can react with N2 to form a newtype of AlN nanofilm, and some AlnN precursorintermediates have been experimentally alreadyobserved [6].

This experimental work triggered an interest insimulations of N doped aluminum clusters. Nayakaet al. [7] reported the equilibrium geometries, bind-ing energy, and electronic structure of AlnN (n �1–6) clusters with BPW91 functional at generalizedgradient approximation (GGA). Leskiw et al. [8]observed and theoretically investigated AlnN (n �1–8) neutral and anionic clusters. Other theoreticalstudies on AlnN have also been published [9].

To provide further insight into AlnN clusters, wehave carried out a detailed systematic study of theequilibrium structure and various electronic struc-ture-related properties of these clusters, employingboth the local spin density approximation (LSDA)and hybrid generalized gradient approximation(GGA) for the exchange-correlation potential. Weinvestigate the relative ordering of these structureswith the N impurity occupying the internal andother position, and show that the ground-statestructures have N taking a internal position. Thecalculations are explicitly carried out, to our knowl-edge for the first time, by considering all electronsin the calculations with no pseudo-potentials (withnonlocal gradient corrections). Furthermore, the all-electron treatment eliminates issues such as core-valence exchange-correlation, which occurs in thepseudo-potential treatment when there is nomarked distinction between the core and valenceregions [10a]. In this work, we study the evolutionof the ionization potential, electron affinity, highestoccupied molecular orbital–lowest unoccupied mo-lecular orbital (HOMO–LUMO) gap, hardness, po-larizability, dissociation energy, and binding en-ergy for AlnN clusters up to n � 12. These physicalquantities are compared with their counterpartscalculated at the same level (all-electron B3LYP/6-

311�G*) for pure aluminum clusters, which to ourknowledge also represent the first all-electron withgradient corrections calculations in these systems[10b].

Section 2 briefly outlines the computationalmethodology. Section 3 discusses the results. Theconclusions are presented in Section 4.

2. Methodology and ComputationalDetails

The geometry optimization and electronic struc-ture calculation is carried out using a molecularorbital approach within the framework of spin-po-larized density functional theory (DFT) [11, 12]. Anall-electron 6-311�G* basis set was employed [13].We have employed the Kohn–Sham (KS) exchangealong with the Vosko–Wilk–Nussair [14] parame-trization of homogeneous electron gas data attrib-utable to Ceperley and Alder [15] for the correlationpotential. We shall henceforth refer to this specificLSDA approach as SVWN. We have also carriedout calculations that go beyond the LSDA and takeinto account gradient corrections. In this case,we have used Becke’s three-parameter functional(B3LYP) [16], which uses part of the Hartree–Fock(HF) exchange (but calculated with KS orbitals) andBecke’s exchange functional [17] in conjunctionwith the Lee–Yang–Parr [18] functional for correla-tion. The ionic configuration was regarded as opti-mized when the maximum force, the root meansquare (RMS) force, the maximum displacement ofatoms, and the RMS displacement of atoms havemagnitudes of less than 0.0045, 0.0003, 0.0018, and0.0012 a.u., respectively. We carried the calculationsout for spin multiplicities of 2S�1 � 1 and 2S�1 �2 for clusters with even and odd numbers of elec-trons, respectively, except for AlN, which is a spintriplet. All calculations are carried out using theGaussian 98 [16] suite of programs.

3. Results and Discussion

The geometries of all the clusters obtained withinthe LSDA and B3LYP are similar, apart from thelarger bond distances observed in B3LYP, althoughthe order of isomers is reversed in some cases withB3LYP. We therefore present only the structuresobtained within the B3LYP scheme in Figures 1 and2. Geometric parameters of the structures of Figures1 and 2 are listed in Table I.

DFT STUDY OF STRUCTURAL AND ELECTRONIC PROPERTIES OF ALnN (1 � n � 12) CLUSTERS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1251

The bond lengths of the AlN molecule are1.805 Å and 1.778 Å within the B3LYP and SVWNschemes, respectively, which are smaller than thevalue 1.813 Å reported by Langhoff et al. [19]obtained from a MRCI calculation. The better

agreement of the B3LYP bond length with itsMRCI counterpart is due to the effect of the non-local corrections contained in B3LYP, which par-tially corrects the overbinding tendency of theLSDA.

FIGURE 1. Geometries of AlnN structures.

FIGURE 2. Geometries of AlnN structures.

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In the B3LYP scheme, the lowest-energy struc-ture for Al2N is linear, with the N atom occupy-ing the central position [Fig. 1(a)]. A linear struc-

ture in which N takes a terminal position and astructure forming an isosceles triangle with N atapex [Fig. 1(b)] are two low-lying structures at,respectively, 3.86 and 0.04 eV above the moststable structure.

Nayak et al. [7] investigated three different neu-tral isomers using a DFT-GGA calculation. We sup-port their predictions that the energetically mostfavorable isomer is the planar (D3h, 1A1�) structurewith a central N bound to three Al atoms [Fig. 1(a)].The next structure in the B3LYP scheme withoutany imaginary harmonic vibrational frequency is alinear C�v configuration [Fig. 1(b)], which is 2.06 eVabove the ground state.

In the case of Al4N, three low-lying nearly de-generate structures are found, all of which are pla-nar. The one with the lowest energy is a planarstructure with a central N surrounded by four Alatoms forming a square [Fig. 1(a)]. Next, Al4N iso-mer [Fig. 1(b)] in energy ordering is also a planarstructure with C2v (2A1) symmetry. Their energydifferences of 0.14 eV are close to the value of 0.20eV obtained by Nayak et al. [7] at the GGA level oftheory with the 6-311G** basis set. Another planarC2v (2B2) is 0.44 eV less stable, which is a transitionstate with an imaginary frequency at 20i cm�1. Notethat N is threefold coordinated to Al in the twolow-lying structures, and the average AlON bondlengths of 1.846 Å are very close to the value of1.850 Å seen in Al3N. The number of AlON bondsin the ground state is more than those of the twolow-lying isomers, which indicates the ground stateshould correspond to a structure that would maxi-mize the number of AlON bonds and agrees withthe Nayak et al. [7] prediction.

Nayak et al. [7] and Leskiw et al. [8] studieddifferent structures of the Al5N cluster. Both con-sidered the three-dimensional (3D) structure [Fig.1(b)] as the ground state of neutral Al5N, whichcould be envisioned as an Al3N with an Al2 dimmerattached to one side. However, our calculations findit a transition state indeed with an imaginary fre-quency at 45i cm�1, and a planar isomer [Fig. 1(a)]with C2v (1A1) symmetry is confirmed to be theground state without imaginary frequency lying0.29 and 0.14 eV below the [Fig. 1(b)] isomer opti-mized with the B3LYP and SVWN methods, respec-tively.

Leskiw et al. [8] gave three lower isomers andfound the C2v (2A1) structure to be the most stablefor the Al6N. The present calculation also concludesthe same conclusion. It is a distorted D2d structure.The two kinds of AlON bond lengths of 1.894 and

TABLE I ______________________________________Geometric parameters of AlnN (n � 1–12) clusters atB3LYP/6-311�G* level.

Molecule Type L/Å Molecule Type L/Å

AlN 1-2 1.805 3-7 2.147Al2N(a) 1-3 1.731 4-5 2.750Al2N(b) 1-2 1.723 4-8 2.742Al3N(a) 1-2 1.850 5-7 1.948Al3N(b) 1-2 2.819 2-6 2.977

2-3 1.695 5-6 2.9923-4 1.779 Al9N 1-5 2.653

Al4N(a) 1-2 1.964 2-4 1.9181-3 2.770 2-5 2.856

Al4N(b) 1-4 1.835 2-9 2.9502-3 2.901 3-4 2.1382-4 1.850 4-6 2.0812-5 2.485 5-6 2.703

Al4N(c) 1-2 1.848 Al10N 1-4 2.8061-3 1.828 1-6 2.7023-4 2.715 2-3 2.764

Al5N(a) 1-2 2.940 2-5 2.1901-3 1.889 2-6 2.8413-4 1.975 2-8 2.718

Al5N(b) 1-2 2.812 2-11 2.8381-6 2.541 3-5 1.9552-3 1.847 Al11N 1-2 2.6653-4 1.831 1-3 2.746

Al6N(a) 1-2 1.894 1-12 2.2041-3 2.032 2-7 2.7112-4 2.725 2-12 2.0543-4 2.700 3-5 2.641

Al6N(b) 1-2 2.869 3-7 2.6871-3 1.917 4-6 2.690

Al7N(a) 1-2 2.122 4-7 2.6691-3 2.108 6-7 2.7871-6 2.133 7-12 2.0682-3 2.649 Al12N 1-2 2.7653-6 2.613 1-13 2.029

Al7N(b) 1-3 2.711 2-3 2.7721-4 2.097 2-9 2.5941-8 2.652 3-8 2.7892-4 2.138 3-9 2.7683-4 2.161 3-10 2.649

Al8N(a) 1-5 2.653 3-13 1.9552-5 2.798 4-13 2.1222-7 2.008 5-13 2.1684-7 2.024 6-13 2.0926-7 1.999 7-9 2.745

Al8N(b) 1-4 2.742 7-12 2.6391-7 2.001 8-10 2.6162-7 2.025



2.032 Å are in good agreement with the results of1.910 and 2.030 Å, respectively, reported by Leskiwet al. [8]. Next, the Al6N isomer in the energy or-dering is a planar D2h (2B2g) structure [Fig. 1(b)]. Ithas an imaginary bending-mode frequency, whichis located 0.52 eV above the ground state.

In good agreement with previous calculations[8], the lowest-energy isomer found for neutralAl7N is a Cs (1A�) form [Fig. 1(a)] with the N atomscentrally located. Next Al7N isomer in the energyordering (only 0.02 eV less stable) possesses C2v

(1A1) geometry [Fig. 1(b)] with an imaginary fre-quency at 39i cm�1. The bending mode frequencyshows a tendency to want to form the ground state.Their energies should be evaluated with SVWNbecause of their near degeneracy. It is found thatFigure 1(a) is 0.02 eV more stable than Figure 1(b),in agreement with the density functional calcula-tion.

The most stable geometry of Al8N [Fig. 1(a)] isthe result of the addition of the Al atom to thelow-lying geometry of Al7N cluster, which caps thedifferent side. The structure [Fig. 1(b)] is onlyhigher by 0.07 eV. The lowest-energy state of AlnNstructures for n � 8 are results of further capping ofAl8N, with the N impurity getting trapped withinthe cage of Al atoms (Fig. 2).

Thus, our all-electron spin-polarized resultsshow that the N impurity prefers an internal posi-tion, and they validate results reported earlier [7, 8].In general, the N impurity in the most stable struc-tures of AlnN clusters can be looked upon as asubstitutional impurity in pure Aln�1 clusters.

We now discuss the relative stability of theseclusters by computing the energetics that are indic-ative of the stability. We compute the atomizationor binding energy (BE) per atom, the dissociation

energy (DE), and the second differences of energyas, respectively,

Eb�AlnN� � nE�Al� � E�N� � �AlnN�/n � 1, (1)

�E�AlnN� � E�AlnN� � E�Aln�1N� � E�Al�, (2)

�2E�AlnN� � �Aln�1N� � E�Aln�1N� � 2E�AlnN�. (3)

The calculated binding energies are shown in Fig-ure 3. The binding energy increases rapidly from2.69 eV (2.55 eV with LSDA) for AlN to 2.87 eV (3.41eV with LSDA) for Al5N, with a small peak at Al3N,and beyond it tends to saturate. The LSDA bindingenergies are larger than the B3LYP binding ener-gies. This trend is consistent with the generallyobserved overbinding tendency within LSDA. Theodd–even oscillations in the BE curve are not pro-nounced with both B3LYP and LSDA (cf. Fig. 3).The BE curve of pure aluminum clusters calculatedat the B3LYP/6-311�G* level of theory is alsoshown in the same figure. Its comparison with theBE curve for AlnN clusters show that the smallclusters of AlnN are strongly bound. As the clustergrows in size, the difference between the BE curvesof AlnN clusters and pure aluminum clusters steaddiminishes, indicating that the bonding in dopedclusters is essentially similar to that in pure clusters.

The calculated values of the energy required fordissociation of AlnN into Aln-1N and Al are pre-sented in Figure 4. The LSDA values of dissociationenergy are higher than their B3LYP counterpart,again indicating its overbinding nature. The curveshows odd–even oscillations with a peak for clus-ters with an even number of electrons. The peak atAl7N within the LSDA is especially prominent. The20 valence–electron Al5N cluster has a dissociationenergy comparable to its neighbor within LSDA,

FIGURE 3. Binding energy in electron volt. Solid line,B3LYP; dashed line, SVWN; dotted dashed line, Alnwith the B3LYP.

FIGURE 4. Dissociation energy in electron volt. Solidline: B3LYP. Dotted line: SVWN.

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while it is clearly more stable than its neighbor withthe B3LYP scheme. This is consistent with the ob-servation from the BE curve.

The values of second difference of cluster energy,Eq.(3), are plotted in Figure 5. This curve againshows odd–even oscillation within B3LYP, withpeaks (dips) at an odd (even) number of aluminumatoms, whose origin is the full occupation of the lastorbital for the even-electron system and incompleteoccupation for the odd-electron system. While thepeak at Al7N is conspicuous in both the LSDA andB3LYP schemes, the peak for the 20 valence–elec-tron Al5N is prominent only in B3LYP. These ob-servations are consistent with those from the bind-ing and dissociation energies, indicating that theB3LYP, the 20 valence–electron system is more sta-ble.

We have also calculated the adsorption energy ofN, i.e., the energy released upon adsorption of N bya pure aluminum cluster, according to

Ead � E�AlnN� � E�Aln� � E�N�. (4)

The calculated values of Ead for the clusters up toAl12 ranges between 5.39 and 10.87 eV (Table II).The minimum value (5.39 eV) occurs for AlN, whileit takes the maximum value (10.87 eV) for Al3N.

The HOMO–LUMO gap is a useful quantity forexamining the stability of clusters. It is found thatsystems with larger HOMO–LUMO gaps are, in

general, less reactive. In the case of an odd-electronsystem, we calculate the HOMO–LUMO gap asthe smallest spin-up-spin-down gap. The HOMO–LUMO gaps as thus calculated are presented inFigure 6. The clusters with an even number ofelectrons, with the exception of Al5N, have peaksindicating their enhanced stability with respect totheir neighbors

Experimentally, the electronic structure isprobed via measurements of ionization potentials,electron affinities, polarizabilities, etc. Therefore,we also study these quantities to understand theirevolution with size. These quantities are deter-mined within B3LYP for the lowest-energy struc-tures obtained within the same scheme.

The vertical ionization potential (VIP) is calcu-lated as the self-consistent energy difference be-tween the cluster and its positive ion with the samegeometry. The VIP is plotted in Figure 7 as a func-tion of cluster size. The VIP decreases as the clustersize increases, and shows oscillations with peaksfor clusters with an even number of electrons. Thepeak occurring at Al7N is especially prominent,with large drops for the following clusters. Alsoshown in Figure 7 are the VIPs of pure aluminumclusters. These have also been calculated at theB3LYP/6-311�G* level of theory, with structuresoptimized at the same level of theory. The compar-ison of the two curves shows that odd–even oscil-

FIGURE 5. Second difference in energy. Solid line,B3LYP; dotted line, SVWN.

TABLE II ______________________________________________________________________________________________Adsorption energies (in eV) (see text for full details) calculated within B3LYP with (6-311�G*) basis set.

n 1 2 3 4 5 6 7 8 9 10 11 12

Ead 5.39 9.72 10.87 10.52 10.02 10.26 9.36 8.84 9.46 8.52 8.04 7.81

FIGURE 6. HOMO-LUMO gap in electron volt. Solidline: B3LYP; dotted line, SVWN.



lations are observed in pure Al clusters right fromthe beginning, while in AlnN clusters, the VIP de-creases monotonically up to Al4N, beyond which itexhibits odd–even pattern. It is also interesting tonote that replacing one Al in an Aln cluster with N,to give Aln�1N, results in smaller VIPs for evenvalues of n, and larger VIPs for odd values of n.

Clusters (see Fig. 8) by assuming the geometryfor the charged cluster to be the same as for theneutral one. The VEA also exhibits an odd–evenpattern. This is again a consequence of the electronpairing effect. In the case of clusters with an evennumber of valence electrons, the extra electron hasto go into the next orbital, which costs energy,resulting in a lower value of VEA. A comparison ofthe VEAs of AlnN clusters and pure aluminumclusters shows that the odd–even pattern is shiftedby one unit upon doping. This observation is notconsistent with the observations from VIPs.

Another useful quantity is the chemical hardness[20], which can be approximated as

� � 1/ 2I � A, (5)

where A and I are the electron affinity and ioniza-tion potential, respectively. Clusters with large val-ues of hardness are, in general, less reactive andmore stable. The hardness of AlnN clusters, calcu-lated according to Eq. (5), using VIP for the ioniza-tion potential and VEA for the electron affinity, isshown in Figure 9. The hardness decreases with thesize of the clusters and exhibits peaks for evenvalence–electron clusters. The observed structureof the hardness curve is consistent with the findingsfrom the VIP, VEA, and stability criteria, in thateven valence–electron clusters are more stable thanodd valence–electron clusters.

Table III presents the polarizabilities for the low-est-energy structures calculated within the B3LYPscheme. The polarizabilities per atom of AlnN clus-ters decreases from 64.6 a.u. for Al2N to 41.7 a.u. forAl12N, with the lowest value (38.3 a.u.) for Al7N.

FIGURE 7. Ionization potential in electron volt calcu-lated at B3LYP/6-311�G* level for AlnN and Aln clus-ters. Solid line, AlnN clusters; dotted line, Aln clusters.

FIGURE 8. Electron affinity in electron volt calculatedat B3LYP/6-311�G* level. Solid line, AlnN clusters; dot-ted line, Aln clusters.

FIGURE 9. Hardness of AlnN clusters in electron voltscalculated within the B3LYP.

TABLE III _____________________________________Polarizabilities (in atomic units) calculated within theB3LYP with (6-311�G*) basis set.

System �xx �yy �zz � �/n

Al2N 66.5 66.5 254.9 129.3 64.6Al3N 136.9 136.9 106.9 126.9 42.3Al4N 211.4 211.4 122.9 181.9 45.5Al5N 144.1 238.1 350.6 244.3 48.9Al6N 243.0 330.0 271.1 281.4 46.9Al7N 269.1 269.2 267.0 268.4 38.3Al8N 254.1 445.9 388.4 362.8 45.4Al9N 395.5 287.2 472.7 385.1 42.8Al10N 358.9 385.7 542.1 428.9 42.9Al11N 512.5 522.4 357.2 464.0 42.2Al12N 566.9 526.6 408.3 500.6 41.7

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The lowest value of polarizability per atom occursfor Al7N, which could be due to a combined effectof the compactness of structure and the electronicshell closure that occurs for this cluster. The closed-shell electronic configuration of Al7N would resultin the low response of the electrons to the appliedelectric field, resulting, thereby, in lower value ofpolarizability. It is also evident from Table III thatthe odd–even oscillations that were present in thebinding energy, dissociation energy, VIP, VEA, andhardness are not seen here. The polarizability peratom seems to saturate beyond Al7N.

4. Summary and Conclusions

Aluminum clusters doped with a single N impu-rity atom has been studied by an all-electron linearcombination of atomic orbital approach, withinspin-polarized DFT, using both the LSDA and hy-brid GGA schemes for the exchange-correlation.The Al impurity is found to occupy an internalposition. The stability of the lowest-energy struc-tures is investigated by analyzing energies. Odd–even oscillations are observed in most of the phys-ical properties investigated, suggesting that clusterswith an even number of electrons are more stablethan their odd-electron neighboring clusters. Thestability analysis and the various electronic struc-ture properties indicate an even valence–electronAl7N cluster to be the most stable cluster amongthose studied.

References

1. Huang, R. B.; Li, H. D.; Liu, Z. Y.; Yang, S. H. J Phys Chem1995, 99, 1418.

2. Haser, M.; Schneide, U.; Ahlrichs, R. J Am Chem Soc 1992,114, 9551.

3. Jones, R. O.; Gantefor, G.; Hunsicker, S.; Pieperhoff, P.J Chem Phys 1995, 103, 9549.

4. Ballone, P.; Jones, R. O. J Chem Phys 1994, 100, 4941.5. Rao, B. K.; Jena, P. J Chem Phys 1999, 111, 1890.6. (a) Buc, D.; Hotovy, I.; Hascik, S.; Cerven, I. Vacuum 1998,

50, 121; (b) Lee, H. C.; Lee, Y. J. J Mater Sci-Mater El 1997, 18,1229; (c) Muhl, S.; Zapien, J. A.; Mendez, J. M.; Andrade, E.J Phys D Appl Phys 1997, 30, 2147.

7. Nayak, S. K.; Khanna, S. N.; Jena, P. Phys Rev B 1998, 57,3787.

8. Leskiw, B. D.; Castleman, A. W., Jr.; Ashman, C.; Khanna,S. N. J Chem Phys 2001, 114, 1165.

9. Jiang, Z. Y.; Ma, W. J.; Wu, H. S.; Jin, Z. H. J Mol Struct(Theochem) 2004, 678, 123.

10. (a) Porezag, D.; Pederson, M. R.; Liu, A. Y. Phys Rev B 1999,60, 14132; (b) Martinez, A.; Vela, A.; Salahub, D. R. J ChemPhys 1994, 101, 10677.

11. Parr, R. G.; Yang, W. Density Functional Theory of Atomsand Molecules; Oxford University Press: New York, 1989.

12. Jones, R. O.; Gunnarsson, O. Rev Mod Phys 1989, 61, 689.13. (a) Francl, M. M.; Petro, W. J.; Hehre, W. J.; Binkley, J. S.;

Gordon, M. S.; DeFrees, D. J.; Pole, J. A. J Chem Phys 1982,77, 3654; (b) Hariharan, P. C.; Pople, J. A. Theor Chim Acta1973, 28, 213.

14. Vosko, S. H.; Wilk, L.; Nusair, M. Can J Phys 1980, 58, 1200.15. Ceperley, D. M.; Alder, B. J. Phys Rev Lett 1980, 45, 566.16. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.;

Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgom-ery, J. A., Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.;Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.;Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.;Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski,J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.;Malick, D. K.; Rabuck, A. D.; Raghava-chari, K.; Foresman,J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.;Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Mar-tin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.;Nanayakkara, A.; Gonzalez, G.; Challacombe, M.; Gill,P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.;Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A.Gaussian 98, Revision A.7; Gaussian: Pittsburgh, PA, 1999.

17. Becke, A. D. Phys Rev A 1988, 38, 3098.18. Lee, C.; Yang, W.; Parr, R. G. Phys Rev B 1988, 37, 785.19. Langhoff, S. R.; Bauschlicher, C. W.; Pettersson, L. G. M.

J Chem Phys 1988, 89, 7354.20. Parr, R. G.; Pearson, R. G. J Am Chem Soc 1983, 105, 7512.



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Density functional study of structural and electronic properties of AlnN (1 ≤ n ≤ 12) clusters