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CHAPTER 5

Magnetic Properties of Atomic Clustersof the Transition Elements

Julio A. Alonso

Departamento de Fısica Teorica, Atomica y Optica, Universidadde Valladolid, Valladolid, SpainDonostia International Physics Center (DIPC), San Sebastian,Spain

INTRODUCTION

Atomic clusters are aggregates of atoms containing from a few to severalthousand atoms. Their properties are different from those of the correspond-ing bulk material because of the sizable fraction of atoms forming the clustersurface. Many differences between clusters and bulk materials originate fromthe small volume of the potential well confining the electrons in the clusters. Insuch cases, the electrons of clusters fill discrete levels, instead of having thecontinuous distribution (bands) characteristic of the solid. How many atomsare required for a cluster to show the properties of the macroscopic material?This important question still lacks a convincing answer. By studying the prop-erties of clusters, scientists expect to obtain information on the early stages ofgrowth of condensed matter and on the evolution of the chemical and physicalproperties as a function of cluster size. Knowing something about the evolu-tionary patterns of clusters may have interesting technological implications.For instance, the melting temperature of small particles decreases linearly asa function of the inverse particle radius 1/R. This decrease affects sinteringprocesses, in which fine powders are compressed and heated until the particlescoalesce: Lower sintering temperatures will be required for particles with very

Reviews in Computational Chemistry, Volume 25edited by Kenny B. Lipkowitz and Thomas R. Cundari

Copyright � 2007 Wiley-VCH, John Wiley & Sons, Inc.

191

small radii. Also, given the current trend to nanoscale technologies, the extre-mely small size of the components will affect their electrical and mechanicalstabilities at increased temperatures.

Most studies of the magnetic properties of small atomic clusters havefocused on the transition elements. These elements form three series in the Per-iodic Table that are characterized by the progressive filling of electronicd shells. The d electrons are responsible for many properties of these elementsas free atoms and in the bulk phase. In the same way, most properties of clus-ters of the transition elements, and in particular the magnetic properties, reflectthe localized behavior of the d electrons. The objective of this chapter is toreview the theoretical work performed in the past years to understand, explain,and predict the magnetism in clusters of the transition elements. The structureof the chapter is as follows. After introducing a few basic concepts, some keyexperiments are presented revealing the broad features of the variation of mag-netic moments as the cluster size increases. We will see that an overall decreaseof the average magnetic moment exists going from the free atom value towardthe value for the bulk metal. Models based on a simplified description of thedensity of electronic states have been introduced to explain this main feature.However, superimposed on this rough decrease of average magnetic momentexists a rich structure of the magnetic moment showing oscillations withincreasing cluster size. This structure can only be explained using more accuratemethods, and the calculations existing in the literature can be classified into oneof two groups: tight binding calculations or density functional theory calcula-tions, both of which are summarized before we review several of their applica-tions to clusters of elements of the 3d and 4d series.

BASIC CONCEPTS

The magnetism of small clusters is sensitive to the symmetry of the cluster,atomic coordination, and interatomic distances between neighbor atoms. Themagnetic moments in clusters of Fe, Co, and Ni can be estimated, however,from a simple argument. First consider the free atoms Fe, Co, and Ni, havingeight, nine, and ten outer electrons, respectively, to be distributed among the 3dand 4s shells. Hund’s rule requires the spin to be a maximum, and this leads tothe following electronic configurations: 3d "5 3d #1 4s2 for Fe, 3d "5 3d #2 4s2

for Co, and 3d "5 3d #3 4s2 for Ni. The up ð3d "Þ and down ð3d #Þ spin sub-shells are separated in energy by the exchange interaction. The Fe, Co, and Niatoms have nonzero spins, and because the spin magnetic moment of an elec-tron is 1 Bohr magneton ðmBÞ, the atoms have substantial magnetic moments.Then, when atoms condense to form a cluster or a metal, the overlap betweenthe atomic orbitals of neighboring atoms leads to the formation of bands ofelectronic levels. The orbitals corresponding to 4s electrons produce a nearlyfree electron band with a width in the solid of W ¼ 20� 30 eV, whereas the

192 Magnetic Properties of Atomic Clusters of the Transition Elements

d electrons stay localized on the atomic sites, and consequently the d bandwidth is much smaller, typically 5–10 eV. The crystal potential stabilizes thed and s states by different amounts, because of their different degree of locali-zation. This process, plus partial hybridization between the s and d shells, leadsto charge transfer from s to d states, and the occupation number of s electronsfor clusters and metals is close to 1 per atom. Assuming that the 3d orbitals arestill atomic-like, Hund’s rule requires the majority 3d " sub-band to be fullyoccupied with five electrons per atom, whereas the minority 3d # sub-bandhas to be occupied with two, three, and four electrons per atom in Fe, Co,and Ni, respectively. Therefore, the difference in the number of spin " andspin # 3d electrons per atom is ndð"Þ � ndð#Þ ¼ 3; 2, 1 for Fe, Co, and Ni,respectively, and the magnetic moments per atom are �mbðFeÞ ¼ 3mB,�mbðCoÞ ¼ 2mB, and �mbðNiÞ ¼ 1mB. These simple estimates are close to theactual magnetic moments of very small clusters. In comparison, the magneticmoments of the bulk metals, �mðFeÞ ¼ 2:2mB, �mðCoÞ ¼ 1:7mB, and�mðNiÞ ¼ 0:64mB, are smaller, and their noninteger values are caused by the par-tial delocalization of the 3d electrons. The exchange interaction between thesedelocalized electrons (known as itinerant exchange) also contributes to themutual alignment of the atomic moments.

Experiments have been performed to reveal how the magneticmoments evolve as the number of the atoms in the cluster increases.1–5

That evolution is very rich and has unexpected features. The clusters in amolecular beam are free from any interation with a matrix. So it is possibleto measure their intrinsic magnetic properties. The magnetic moment canbe determined by an experimental technique similar to that used by Sternand Gerlach to demonstrate the quantization of the components of theangular momentum in the early days of quantum theory. In this way, experi-mentalists can investigate the dependence of the cluster’s magnetic momentas a function of the cluster size.1,2 The clusters interact with an externalinhomogeneous magnetic field B and are deflected from their original trajec-tory. The deflection l of a cluster moving with a velocity v in a directionperpendicular to the field gradient direction (defined as the z direction) isgiven by2

l ¼ KMðBÞmv2

qB

qz½1�

where m is the cluster mass, qB=qz is the gradient of the magnetic field, and Kis a constant that depends on the geometry of the apparatus. This equationshows that the deflection is proportional to the cluster magnetization M(B),which is the time-averaged projection of the magnetic moment m of the particlealong the field gradient direction.

When analyzing the experiments, one normally assumes that ferromag-netic clusters are monodomain particles; that is, all magnetic moments of the

Basic Concepts 193

particle are parallel, aligned in the same direction. In contrast, the case of amacroscopic crystal is more complex. A bulk crystal is formed by magneticdomains. In each domain, the magnetic moments may be aligned, but thedirection of the magnetization is not the same in different domains. In the ana-lysis of the experiments, one also usually assumes that the clusters follow thesuper-paramagnetic behavior. Super-paramagnetism is a phenomenon bywhich magnetic materials may exhibit a behavior similar to paramagnetismat temperatures below the Curie or the Neel temperatures. This effect isobserved in small particles. In this case, although the thermal energy is not suf-ficient to overcome the coupling forces between the magnetic moments ofneighboring atoms (that is, the relative orientation of the moments of twoneighbor atoms cannot be modified), the energy required to change collectivelythe direction of the magnetic moment of the entire particle is comparable withthe ambient thermal energy. It occurs because the crystalline magneticanisotropy energy, which is the energy required to change the direction ofmagnetization of a crystallite, decreases strongly as the crystallite sizedecreases and is negligible in clusters. In that case, the N atomic magneticmoments of a cluster with N atoms are coupled by the exchange interaction,which give rise to a large total magnetic moment mN that is free of the clusterlattice. This orientational freedom allows the magnetic moment to align easilywith an external magnetic field B, as in a paramagnetic material. For anensemble of particles in thermodynamic equilibrium at a temperature T inan external magnetic field, the magnetization reduces, in the limit of low field(mNB� kBT, where kB is the Boltzmann constant), to

MðBÞ ¼ m2NB

3kBT½2�

The average magnetic moment per atom �m ¼ mN=N of a monodomain cluster isanalogous to the saturation magnetization Ms of the bulk. However, at zerofield, a magnetic monodomain cluster has a nonzero magnetic moment. In con-trast, for a multidomain bulk sample, the magnetic moment may be much smal-ler than Ms because the different magnetic domains are not mutually aligned.Equations [1] and [2] can be used to determine mN in monodomain clusters.

The evolution of the average magnetic moment as a function of cluster sizeis not smooth. The overall decrease of �mN toward the bulk value �mb is caused bythe increasing number of nearest neighbors, which is an effect that enhances theitinerant character of the d electrons, that is, the possibility of hopping betweenneighboring atoms. Surface atoms have a smaller number of neighbors than dobulk atoms. Convergence of the value of �mN to �mb is thus achieved when the frac-tion of surface atoms becomes small. In addition, clusters can have complexstructures; i.e., they are not simple fragments of the crystal. These ingredientsaffect the detailed broadening of the electronic levels that form the d bands.The exchange splitting between " and # d sub-bands, charge transfer from the


s to the d band, and s–d hybridization depend sensitively on the cluster size andthus control how �mN evolves with cluster size.

EXPERIMENTAL STUDIES OF THE DEPENDENCE OFTHE MAGNETIC MOMENTS WITH CLUSTER SIZE

The magnetic moments of Fe, Co, and Ni clusters with sizes up to 700atoms have been measured by Billas et al.1,2 Those measurements were madeunder conditions where the clusters exhibit super-paramagnetic behavior forlow cluster temperatures (vibrational temperature Tvib ¼ 78 K for Ni and Coclusters and 120 K for Fe clusters). Their results are shown in Figure 1. As

1.2

1.0

0.8

0.6

0.4

NiN at T=78 K

(a)

bulk

0 100 200 300 400 500 600 700

Mag

netic

mom

ent p

er a

tom

[mB

]

CoN at T=78 K

(b)

2.6

2.4

2.2

2.0

1.8

1.6

1.4

Mag

netic

mom

ent p

er a

tom

[mB

]

0 100 200 300 400 500 600 700

bulk

Figure 1 Experimental magnetic moments per atom of Ni, Co and Fe clusters with sizesup to 700 atoms. Reproduced with permission from Ref. 2.

Experimental Studies of the Dependence 195

expected, the largest magnetic moments occur for the smallest clusters. Themagnetic moment per atom �m decreases for increasing cluster size and con-verges to the bulk value for clusters consisting of a few hundred atoms. Thisconvergence is fastest for Ni clusters. However, in the three cases shown, someoscillations are superimposed onto the global decrease of �m. Apsel et al.3 havealso performed high-precision measurements of the magnetic moments ofnickel clusters with N ¼ 5 to 740, and more recently, values have beenreported by Knickelbein.4

Experiments have also been performed for clusters of the 4d and 5dmetals, which are nonmagnetic in the bulk.5 Rhodium was the first case of anonmagnetic metal in which magnetism was observed in clusters. Magneticmoments were observed for Rh clusters with fewer than 60 atoms, but largerclusters are nonmagnetic. RhN clusters with about ten atoms have magneticmoments �m � 0:8mB; then �m decays quickly for cluster sizes between N ¼ 10and N ¼ 20, showing, however, oscillations and sizable magnetic moments forRh15, Rh16, and Rh19. This behavior differs from that of Fe, Co, and Ni, wherethe variation of �m extends over a much wider range of cluster sizes (cf.Figure 1). In contrast to rhodium, clusters of ruthenium and palladium(with N ¼ 12 to more than 100) are reported to be nonmagnetic.5

SIMPLE EXPLANATION OF THE DECAY OF THEMAGNETIC MOMENTS WITH CLUSTER SIZE

Simple models can explain the general decay of the magnetic moment asthe cluster size increases,6 but they cannot explain the fine details. Neglecting

FeN at T=120 K(c)

3.4

3.0

2.6

2.2

1.8Mag

netic

mom

ent p

er a

tom

[mB

]

0 100 200 300 400 500 600 700

Cluster size N

bulk

Figure 1 (Continued)


the contribution of the sp electrons and using the Friedel model, in which the delectrons form a rectangular band,7 the local density of electronic states(LDOS) with spin s (that is, " or #) at site i can be expressed as8

DisðeÞ ¼5

Wifor �Wi

2< e� eds <

Wi

2½3�

where eds is the energy of the center of the s spin sub-band and Wi is the localband width (assumed to be equal for " and # spins). The tight binding theory(see the next section) gives a scaling relation8 in which Wi is proportional tothe square root of the local atomic coordination number Zi

Wi ¼WbðZi=ZbÞ1=2 ½4�

where Wb and Zb are the corresponding quantities for the bulk solid. If the dband splitting � ¼ jed" � ed#j caused by the exchange interaction is assumedequal to the splitting in the bulk, the local magnetic moment

mi ¼ðeF

�1

ðDi"ðeÞ �Di#ðeÞÞde ½5�

becomes

mi ¼Zb

Zi

� �1=2

mb if Zi � Zc

mi ¼ mdim otherwise ½6�

Here eF is the Fermi energy, that is, the energy of the highest occupied level;Zc is a limiting atomic coordination number below which the local magneticmoment of that atom adopts the value mdim of the dimer, and one can choose

Zc ¼ 5 in Ni.9 The average magnetic moment �mN ¼ ð1=NÞPNi¼1

mi depends sen-

sitively on the ratio between the number of surface atoms and the bulk-likeatoms. Surface atoms have small values of Zi and large values of mi, whereasthe internal atoms have Zi ¼ Zb and mi ¼ mb. In the case of small clusters,most atoms are on the surface and hence �m is large. But as the cluster sizeincreases, the fraction of surface atoms decreases and �m also decreases.Assuming magnetic moments ms for the surface atoms and mb for thebulk atoms, Jensen and Bennemann10 calculated the average magneticmoment of the cluster using Eq. [7]

�m ¼ mb þ ðms � mbÞN�1=3 ½7�

Simple Explanation of the Decay of the Magnetic Moments 197

This formula shows a smooth decrease of �m toward the bulk magnetic momentwith increasing N. However, the experimental results graphed in Figure 1 indi-cate that the variation of �m with N has a more complex, oscillatory behavior.Its explanation requires a detailed consideration of the geometry of the clusterand a better treatment of its electronic structure.

TIGHT BINDING METHOD

Tight Binding Approximation for the d Electrons

The orbitals of the d states in clusters of the 3d, 4d, and 5d transitionelements (or in the bulk metals) are fairly localized on the atoms as comparedwith the sp valence states of comparable energy. Consequently, the d states arenot much perturbed by the cluster potential, and the d orbitals of one atom donot strongly overlap with the d orbitals of other atoms. Intraatomic d–d cor-relations tend to give a fixed integral number of d electrons in each atomicd-shell. However, the small interatomic d-d overlap terms and s-d hybridiza-tion induce intraatomic charge fluctuations in each d shell. In fact, a d orbitalcontribution to the conductivity of the metals and to the low temperature elec-tronic specific heat is obtained only by starting with an extended description ofthe d electrons.7

The partially filled d band of the transition metals, or the d states in clus-ters, are described well by the tight binding (TB) approximation11 using a lin-ear combination of atomic d orbitals. The basic concepts of the method are asfollows:

(1) The lattice potential V (or the cluster potential) is written as a sum ofatomic potentials Vi centered on the lattice sites i.

(2) The electronic states in the cluster (or in the solid metal) are expressed as alinear combination of atomic states (LCAO)

jcðeÞi ¼Xi;m

aimjimi ½8�

The sum in m goes from 1 to 5, because there are five different atomic dorbitals fm. In the usual notation, these orbitals are labeled dxy, dxz, dzy,dx2�y2 , and d3z2�r2 . The normalized atomic orbitals are eigenfunctions ofthe atomic Hamiltonian T þ Vi with energy e0. As a first approximation,the overlap integrals of the atomic orbitals across neighboring sites can beneglected.

(3) Of the matrix elements himjVljjm0i, only the two-center integrals betweenfirst or second nearest neighbors are retained. The coefficients aim then


satisfy the set of coupled linear equations

ðe0 þ�im � eÞaim þX

j 6¼i;m0

t jm0

im ajm0 ¼ 0 ½9�

where

�im ¼ imXj6¼i

Vj

��im

* +¼Xj 6¼i

ðf�imð~r� ~RiÞVjð~r� ~RjÞfimð~r� ~RiÞd3r ½10�

t jm0

im ¼ himjVjjjm0i ¼ðf�imð~r� ~RiÞVjð~r� ~RjÞfjm0 ð~r� ~RjÞd3r ½11�

The �im integrals shift the energy of the reference atomic levels e0, and the t jm0

im

integrals mix them into molecular states. From the set of Eqs. [9], one arrivesat a 5N � 5N secular determinant from which the electronic levels of the clus-ter can be obtained.12 The 5N atomic d states jimi give rise to 5N electroniclevels distributed between the two extremes ea and eb. The lowest level, withenergy eb, corresponds to the formation of d–d bonds between most pairs ofatoms. In the bonding states, the electron density increases along the bonds,compared with the simple superposition of the electron densities of the freeatoms. In going from eb to ea, the number of antibonds increases. At ea, anti-bonds have been created between most pairs of atoms (in antibonding states,the electron density between atoms decreases compared with the super-position of densities of the free atoms). The energy broadening can be viewedas resulting from a resonance between the atomic d levels, allowing theelectrons to hop from atom to atom through the cluster (or through the latticein the solid). From experience acquired in metals, it is known that the d bandwidth W ¼ ea � eb is larger than the shift S. The shift is the energy differencebetween the atomic level e0 and the average band level ðea þ ebÞ=2. Typicalvalues in metals are W¼ 5–10 eV and S¼ 1–2 eV.

Atomistic simulations usually require the calculation of the total energyof the system. The band energy of the solid or cluster is evaluated by integrat-ing the density of electronic states D(e)

Eband ¼ðeF

eDðeÞde ½12�

The part of the energy not included in Eband can be modeled by pairwise repul-sive interactions

Erep ¼Xi6¼j

Uij ½13�

Tight Binding Method 199

Introduction of s and p Electrons

Elements like carbon or silicon have only s and p electrons in the valenceshell. Of course, s and p electrons are also present in the transition elements.Their contribution to the electronic and magnetic properties of the transitionmetal clusters is discussed later in this chapter. In particular, their effect can betaken into account in the tight binding method by a simple extension of theideas presented above. For this purpose, the basis of atomic orbitals has tobe extended by adding s and p orbitals. The new basis thus contains s, px,py, pz atomic orbitals in addition to dxy; dxz; dzy; dx2�y2 , and d3z2�r2 orbitals.It was also pointed out in the previous section that the overlap integrals ofthe atomic orbitals across neighboring sites can be neglected as a first approx-imation. To overcome this limitation, an often applied improvement is to sub-stitute the original basis set of atomic orbitals with an orthogonalized basis.This orthogonalization can be performed using the method introduced byLowdin.13 The orthogonalized orbitals

cia ¼Xi0a0ðSi0a0

ia Þ�1=2fi0a0 ½14�

preserve the symmetry properties of the original set. S in Eq. [14] is the overlapmatrix Si0a0

ia ¼ hfiajfi0a0 i, and the index a indicates the different atomic orbitals(a generalizes the index m used above). Consequently, the integrals in Eqs.[10] and [11] now become integrals between Lowdin orbitals.

A key approximation that makes the TB calculations simple and practi-cal is to replace the � and t integrals of Eqs. [10] and [11] by parametersdepending only on the interatomic distance j~Ri � ~Rjj and the symmetry ofthe orbitals involved.

Formulation of the Tight Binding Method in the Notationof Second Quantization

Recent application of the TB method to transition metal clusters oftenmade use of a convenient formulation in the language of second quantization.14

In this formalism, the TB Hamiltonian in the unrestricted Hartree–Fock approx-imation can be written as a sum of diagonal and nondiagonal terms15

H ¼Xi;a;s

eiasnias þXa;b;si 6¼j

t jbia cþiascjbs ½15�

In this expression, cþias is an operator representing the creation of an electronwith spin sðs ¼" or #Þ and orbital state a at site i, cias is the correspondingannihilation operator, and the operator nias ¼ cþiascias appearing in


the diagonal terms is the so-called number operator. As indicated,aðbÞ � s; px; py; pz; dxy; dxz; dyz; dx2�y2 ; d3z2�r2 . The hopping integrals tjb

ia bet-ween orbitals a and b at neighbor atomic sites i and j are assumed to beindependent of spin and are usually fitted to reproduce the first-principlesband structure of the bulk metal at the observed lattice constant. The variationof the hopping integrals with the interatomic distance Rij ¼ j~Ri � ~Rjj is oftenassumed to follow a power law ðR0=RijÞlþl0þ1, where R0 is the equilibriumnearest-neighbor distance in the bulk solid and l and l0 are the angular momen-ta of the two orbitals, a and b, involved in the hopping.16 An exponentialdecay is sometimes used instead of the power law.

The spin-dependent diagonal terms eias contain all of the many-bodycontributions. In a mean field approximation, these environment-dependentenergy levels can be written as

eias ¼ e0ia þ

Xbs0

Uasbs0�nibs0 þXj 6¼i

e2

Rij�nj þ Zi�a ½16�

where e0ia are the reference orbital energies. These energies could be the atom-

ic levels, but most often the bare orbital energies of the paramagnetic bulkmetal are taken as reference energies. The second term gives the shifts ofthe energies caused by intraatomic Coulomb interactions. The intraatomicCoulomb integrals

Uasbs0 ¼ð ð

c�iasð~rÞciasð~rÞ1

j~r�~r0jc�ibs0 ð~rÞcibs0 ð~r0Þd3rd3~r0 ½17�

give the electrostatic interaction between two charge clouds correspondingto the orbitals jiasi and jibs0i on the same atom. The quantity�nibs ¼ nibs � n0

ibs, where nibs ¼ hnibsi is the average occupation of the spin-orbital jibsi and n0

ibs is the corresponding occupation in the paramagnetic solu-tion for the bulk metal. The intraatomic Coulomb integrals Uasbs0 can beexpressed in terms of two more convenient quantities, the effective direct inte-grals Uab ¼ ðUa"b# þUa"b"Þ=2 and the exchange integrals Jab ¼ Ua"b# �Ua"b".Then, the intraatomic term of Eq. [16] splits into two contributions

Xbs0

Uasbs0�nibs0 ¼Xb

Uab�nib þ zsXb

Jab2mib ½18�

where �nib ¼ �nib" þ�nib#, mib ¼ �nib" ��nib#, and zs is the sign functionðz" ¼ �1; z# ¼ þ1Þ. The term Ua"b# refers to the Coulomb interaction betweenelectrons with opposite spin and Ua"b" to the interaction between electronswith the same spin. The first contribution in Eq. [18] arises from the changein electronic occupation of the orbital jibi and the second contribution from

Tight Binding Method 201

the change of the magnetization (spin polarization) of that orbital. Because ofthe different local environments of the atoms in a cluster, charge transferbetween nonequivalent sites can occur. However, the direct Coulomb repul-sion tends to suppress charge redistribution between atoms and to establishapproximate local charge neutrality (i.e., �n is small). The direct andexchange integrals, Uab and Jab, are usually parametrized. The differencebetween s–s, s–p, and p–p direct Coulomb integrals is often neglected by writ-ing Uss¼Usp ¼ Upp, and it is assumed that Usd ¼ Upd. The ratio Uss : Usd : Udd

of the magnitudes of Uss, Usd, and Udd can be taken from Hartree–Fock cal-culations for atoms. The absolute value of Udd can be estimated by anotherindependent method, for instance, from atomic spectroscopic data.17,18

Typical values for the Uss : Usd : Udd ratios are 0.32 : 0.42 : 1 for Fe andUdd ¼ 5:40 eV.15 The direct Coulomb integral between d electrons, Udd, dom-inates Uss and Usd. The magnetic properties of clusters are not very sensitive tothe precise value of Udd because the charge transfer �n is typically small. Inmost cases, all exchange integrals involving s and p electrons are neglectedand the d exchange integral Jdd is determined in order to reproduce the bulkmagnetic moment. Typical J values for Cr, Fe, Ni, and Co are between 0.5 and1.0 eV.15,17,18

The third term in Eq. [16] represents the Coulomb shifts resulting fromelectronic charge transfer between the atoms. The quantity �nj ¼ nj � n0

j ,

where nj ¼Pbhnjb"iþhnjb#i is the total electronic charge on atom j and n0

j is

the bulk reference value. In Eq. [16], the interatomic Coulomb integrals

Viasjbs0 ¼ð ð

c�iasð~rÞciasð~rÞ1

j~r�~r0jc�jbs0 ð~rÞcjbs0 ð~r0Þd3rd3~r0 ½19�

have been approximated as Vij ¼ e2=Rij. Finally, the last term in Eq. [16] takesinto account the energy level corrections arising from nonorthogonalityeffects15,19 and from the crystal field potential of the neighboring atoms,8

which are approximately proportional to the local atomic coordinationnumber Zi. The constants �a(a ¼ s; p; dÞ are orbital-dependent and can beobtained from the difference between the bare energy levels (that is, excludingCoulomb shifts) of the isolated atom and the bulk. These constants can havedifferent signs for s–p orbitals as compared with d orbitals. For instance, Vegaet al.15 obtained �s ¼ 0:31 eV, �p ¼ 0:48 eV, and �d ¼ �0:10 eV for Fe,which means that the repulsive overlap effects dominate the orbital shiftsfor s and p electrons, whereas the dominant contribution for the morelocalized d electrons is the negative crystal field shift. One can also model,through this term, effects on the energy levels arising from changes in thebond length associated with a lowering of the coordination number.8,15

The spin-dependent local electronic occupations hniasi and the localmagnetic moments mi ¼

Paðhnia"i � hnia#iÞ are self-consistently determined


from the local (and orbital-dependent) density of states DiasðeÞ

hniasi ¼ðeF

�1

DiasðeÞde ½20�

The Fermi energy is determined from the condition of global charge neutrality.In this way, the local magnetic moments and the average magnetic moment�m ¼ ð

PmiÞ=N are obtained at the end of the self-consistency cycle. The local

density of states can be calculated at each iteration step during the calculationfrom the imaginary part of the local Green’s function

DiasðeÞ ¼ �1

pImGias;iasðeÞ ½21�

and the local Green function Gias;iasðeÞ can be determined efficiently from themoments of the local density of states,20 as indicated in the Appendix.

The tight binding framework discussed here is general, although the spe-cific calculations may incorporate some differences or simplifications withrespect to the basic method. For instance, Guevara et al.21 have pointed outthe importance of the electron spillover through the cluster surface. Theseresearchers incorporated this effect by adding extra orbitals with s symmetryoutside the surface. This development will be considered later in some detail.

SPIN-DENSITY FUNCTIONAL THEORY

General Density Functional Theory

The basic variable in density functional theory (DFT)22 is the electrondensity nð~rÞ. In the usual implementation of DFT, the density is calculatedfrom the occupied single-particle wave functions cið~rÞ of an auxiliary systemof noninteracting electrons

nð~rÞ ¼X

i

yðeF � eiÞjcið~rÞj2 ½22�

and the orbitals cið~rÞ are obtained by solving the Kohn–Sham equations23

�r2

2þ VKSð~rÞ

� �cið~rÞ ¼ eicið~rÞ ½23�

written in atomic units. The symbol y in Eq. [22] is the step function, whichensures that all orbitals with energies ei below the Fermi level eF are occupied

Spin-Density Functional Theory 203

and all orbitals with energies above eF are empty. The Fermi level is deter-mined by the normalization condition

ðnð~rÞd3r ¼ N ½24�

where N is the number of electrons. The effective Kohn–Sham potential VKSð~rÞappearing in Eq. [23] is the sum of several distinct contributions:

VKSð~rÞ ¼ Vextð~rÞ þ VHð~rÞ þ Vxcð~rÞ ½25�

The external potential Vextð~rÞ contains the nuclear or ionic contributions andpossible external field contributions. The Hartree term VHð~rÞ is the classicelectrostatic potential of the electronic cloud

VHð~rÞ ¼ �ð

nð~r0Þj~r�~r 0j

d3r0 ½26�

and Vxcð~rÞ is the exchange-correlation potential. Exchange effects between theelectrons originate from the antisymmetric character of the many-electronwave function of a system of identical fermionic particles: Two electrons can-not occupy the same single-particle state (characterized by orbital and spinquantum numbers) simultaneously. This effect has the consequence of buildinga hole, usually called the Fermi hole, around an electron that excludes the pre-sence of other electrons of the same spin orientation (up " or down #, in theusual notation for the z component). Additionally, there are Coulombic corre-lations between the instantaneous positions of the electrons because these arecharged particles that repel each other. Because of this repulsion, two electronscannot approach one another too closely, independent of their spin orienta-tion. The combined effect of the Fermi and Coulomb correlations can bedescribed as an exchange–correlation hole built around each electron. In prac-tice Vxcð~rÞ is calculated, using its definition in DFT, as the functional derivativeof an exchange–correlation energy functional Exc½n�,

Vxcð~rÞ ¼dExc½n�dnð~rÞ ½27�

The local density approximation (LDA)24 is often used to calculate Exc½n� andVxcð~rÞ. The LDA uses as input the exchange–correlation energy of an electrongas of constant density. In a homogeneous system the exchange energy perparticle is known exactly and it has the expression

exðn0Þ ¼ �3

4

3

p

� �1=3

n1=30 ½28�


where n0 is the constant density of the system. The exchange energy of aninhomogeneous system with density nð~rÞ is then approximated by assumingthat Eq. [28] is valid locally; that is,

Ex½n� ¼ �3

4

3

p

� �1=3ðnð~rÞ4=3d3r ½29�

Performing the functional derivative of Eq. [29] (see Eq. [27]) leads to

VLDAx ð~rÞ ¼ � 3

p

� �1=3

nð~rÞ1=3 ½30�

An exact expression for the correlation energy per particle ecðn0Þ of ahomogeneous electron gas does not exist, but good approximations to thisnevertheless do exist.24 Also, nearly exact correlation energies have beenobtained numerically for different densities25 and the results have been para-metrized as useful functions ecðnÞ.26 The corresponding LDA correlationpotential

VLDAc ð~rÞ ¼ dðnecðnÞÞ

dn

� �n¼nð~rÞ

½31�

is then immediately obtained. In summary, in the LDA, VLDAxc ð~rÞ at point~r in

space is assumed to be equal to the exchange–correlation potential in a homo-geneous electron gas with ‘‘constant’’ density n ¼ nð~rÞ, precisely equal to thelocal density nð~rÞ at that point. The LDA works in practice better thanexpected, and this success is rooted in the fulfillment of several formal proper-ties of the exact Exc½n� and in subtle error cancellations. Substantial improve-ments have been obtained with the generalized gradient approximations(GGAs)

EGGAxc ½n� ¼

ðf GGAxc ðnð~rÞ;rnð~rÞÞd3r ½32�

which include rnð~rÞ, or even higher order gradients of the electron density, inthe exchange–correlation energy functional.27–29

Spin Polarization in Density Functional Theory

Some generalization is required when the external potential Vext is spin-dependent (for instance, when there is an external magnetic field), or if onewants to take into account relativistic corrections such as the spin-orbit


term. Von Barth and Hedin30 formulated DFT for spin-polarized cases. Thebasic variable in this case is the 2� 2 spin-density matrix rabð~rÞ, defined as

rabð~rÞ ¼ N

ðd~x2::: d~xN��ð~ra;~x2; :::~xNÞ�ð~rb;~x2; :::~xNÞ ½33�

where the notation ~x includes both the position ~r and the spin variable,a ¼ þ1=2 or a ¼ �1=2. The 2� 2 matrix rabð~rÞ is then hermitian and definedat each point~r. The spinless density is the trace of this density matrix

nð~rÞ ¼ Trrabð~rÞ ¼ nþð~rÞ þ n�ð~rÞ ½34�

where nþð~rÞ ¼ rþþð~rÞ and n�ð~rÞ ¼ r��ð~rÞ are the diagonal terms. To quantifythe magnetic effects, one can define the magnetization density vector mð~rÞ suchthat

rabð~rÞ ¼1

2nð~rÞIþ 1

2mð~rÞ r ½35�

where I is the 2� 2 unit matrix and r ¼ ðsx;sy;szÞ, with sx, sy, and sz beingthe 2� 2 Pauli spin matrices. Consequently, nð~rÞ and mð~rÞ form an alternativerepresentation of rabð~rÞ. The one-particle representation is now based on two-component spinors (when adding spin to the spatial orbitals, the two compo-nents, þ and �, of the spinor correspond to the two projections of the spin, up" and down #, along a quantization axis)

�cið~rÞ ¼ciþð~rÞci�ð~rÞ

� �½36�

The purpose of spin-polarized DFT is again to describe the system (molecule,cluster, . . .) with an auxiliary noninteracting system of one-particle spinorsf�c1; :::;

�cNg. The ground state density matrix of this noninteracting system

rabð~rÞ ¼X

i

yðeF � eiÞc�iað~rÞcibð~rÞ ½37�

should be equal to that of the interacting system. In terms of the one-particlespin-orbitals

nð~rÞ ¼X

i

yðeF � eiÞ�c�i ð~rÞ�cið~rÞ ½38�

mð~rÞ ¼X

i

yðeF � eiÞ�c�i ð~rÞr�cið~rÞ ½39�


Spin-dependent operators are now introduced. The external potential can bean operator Vext acting on the two-component spinors. The exchange–correlation potential is defined as in Eq. [27], although Exc is now a functionalExc ¼ Exc½rab� of the spin-density matrix. The exchange–correlation potentialis then

Vabxc ð~rÞ ¼

dExc½rab�drabð~rÞ

½40�

This potential is often written in terms of a fictitious exchange-correlationmagnetic field Bxc

Vabxc ð~rÞ ¼

dExc

dnð~rÞ I� Bxcð~rÞ r ½41�

Bxcð~rÞ ¼ �dExc

dmð~rÞ ½42�

The Kohn–Sham Hamiltonian is now

HKS½rab� ¼ T þ Vext þ VH½n�Iþ Vxc½rab� ½43�

and the corresponding Kohn–Sham equations become

HKS½rab�j�cji ¼ ejj�cji ½44�

that is,

Xb

�r2

2dab þ Vab

KSð~rÞ� �

cibð~rÞ ¼ eiciað~rÞ ½45�

where the Kohn–Sham effective potential is now

VabKSð~rÞ ¼ Vab

extð~rÞ þ dab

ðnð~r 0Þj~r�~r 0j d

3r0 þ Vabxc ð~rÞ ½46�

In most cases of interest, the spin density is collinear; that is, the direc-tion of the magnetization density mð~rÞ is the same over the space occupied bythe system; it is customary to identify this as the z-direction. The Hamiltonianis then diagonal if the external potential is diagonal, which allows one todecouple the spin " and spin # components of the spinors and to obtain two


sets of equations. This method is known as a spin-polarized calculation. Thedegree of spin polarization is defined as x ¼ ðnþ � n�Þ=n, which ranges from0 to 1. When x ¼ 1, we have a fully polarized system, and when x ¼ 0, thesystem is unpolarized. This approach is adequate to treat ferromagnetic orantiferromagnetic order, found in some solids.31,32 In both ferromagneticand antiferromagnetic ordering, the spin magnetic moments are oriented par-allel, but in the ferromagnetic order, all moments point in the same directionð""" . . . Þ, whereas in antiferromagnetically ordered solids, a spin " magneticmoment at a given lattice site is surrounded at neighbor lattice sites by spinmoments # pointing in the opposite direction, and vice versa.

Local Spin-Density Approximation (LSDA)

As in the non-spin-polarized case, the main problem with the spin-polarizedmethod comes from our limited knowledge of the exchange–correlation energyfunctional Exc½rab�, which is not known in general. However, Exc½rab� is wellknown for a homogeneous gas of interacting electrons that is fully spin-polarized,i.e., nþð~rÞ ¼ n; n�ð~rÞ ¼ 0 (and, of course, for a nonpolarized homogeneous elec-tron gas, with nþð~rÞ ¼ n�ð~rÞ ¼ n=2; see above). As a result, von Barth andHedin30 proposed an interpolation formula for the exchange–correlation energyper electron in a partially polarized electron gas (the z-axis is taken as the spinquantization direction)

excðn; xÞ ¼ ePxcðnÞ þ ½eF

xcðnÞ � ePxcðnÞ�f ðxÞ ½47�

where the function f ðxÞ gives the exact spin dependence of the exchangeenergy

f ðxÞ ¼ 1

2ð21=3 � 1Þ�1fð1þ xÞ4=3 þ ð1� xÞ4=3 � 2g ½48�

In Eq. [47], ePxcðnÞ and eF

xcðnÞ are the exchange–correlation energy densities forthe nonpolarized (paramagnetic) and fully polarized (ferromagnetic) homoge-neous electron gas. The form of both eF

xcðnÞ and ePxcðnÞ has been conveniently

parameterized by von Barth and Hedin. Other interpolations have also beenproposed24,33 for excðn; xÞ. The results for the homogeneous electron gas canbe used to construct an LSDA

Exc½rab� ¼ð

nð~rÞexcðnð~rÞ; xð~rÞÞd3r ½49�

Vaxcð~rÞ ¼

dExc½rab�dnað~rÞ

½50�


As with the unrestricted Hartree–Fock approximation, the LSDA allows fordifferent orbitals for different spin orientations. The LSDA gives a simplifiedtreatment of exchange but also includes Coulomb correlations.

NONCOLLINEAR SPIN DENSITY FUNCTIONALTHEORY

In many systems of interest, the spin density is collinear; that is, the directionof the magnetization vector mð~rÞ is the same at any point in space. There are othersystems, however, in which the direction of mð~rÞ changes in space, a well-knownexample being the g-phase of solid Fe.34 Noncollinear magnetic configurationsoccur easily in systems with low symmetry or those that are disordered.35,36

One can then expect the occurrence of noncollinear spin arrangements in clustersof the transitionmetals.Generalized LSDA calculationsallowing fornoncollinearmagnetic structures have been performed for solids.37–39

Implementation of the noncollinear formalism for clusters has been morerecent40,41 and uses again as a basis the LSDA. When a local exchange–correlation functional is used, the following key idea was introduced by vonBarth and Hedin.30 One can divide the volume of the system into small inde-pendent boxes and consider that within each small box the electrons form aspin-polarized electron gas, whose densities are n"ð~rÞ and n#ð~rÞ, the two realand positive eigenvalues of the spin-density rabð~rÞ matrix at~r. At each point~r, one can then choose a local coordinate system such that the z-axis coincideswith the direction of the local spin. In this way, one can use the LSDAexchange and correlation functionals and calculate the exchange–correlationpotential in this locally diagonal frame. This strategy provides a local magne-tization density approximation, which is similar in spirit to the local densityapproximation. That is, in the LDA, the exchange–correlation energy densityand the exchange–correlation potential at the point~r are calculated by assum-ing that the system behaves locally (at~r) as a homogeneous electron gas withconstant density n equal to nð~rÞ, the true density at point ~r. Similary, in thelocal magnetization density approximation, the exchange–correlation energydensity and the exchange–correlation potential at~r are calculated by assumingthat the system behaves locally as a partially spin-polarized electron gas with amagnetization density vector m equal to mð~rÞ, the true magnetization densityvector at point ~r. The procedure used to calculate the exchange–correlationpotential involves carrying the density matrix to the local reference framewhere it is diagonal, using the spin-1/2 rotation matrix.42

Uð~rÞ ¼cos

yð~rÞ2

eði=2Þfð~rÞ sinyð~rÞ2

eð�i=2Þfð~rÞ

� sinyð~rÞ2

eði=2Þfð~rÞ cosyð~rÞ2

eð�i=2Þfð~rÞ

0BB@

1CCA ½51�

Noncollinear Spin Density Functional Theory 209

The angles yð~rÞ and fð~rÞ are calculated in such a way that U diagonalizes thedensity matrix

Uð~rÞrð~rÞUþð~rÞ ¼ n"ð~rÞ 00 n#ð~rÞ

� �½52�

where Uþð~rÞ is the adjoint (or Hermitian conjugate) of Uð~rÞ. The exchange–correlation potential is then calculated in this local reference frame, inwhich it is a diagonal operator with components V"xc and V#xc, and thenit must be transformed back to the original reference frame. The localspin rotation angles yð~rÞ and fð~rÞ, the local azimuthal and polar angles ofthe magnetization density vector, are computed through the requirementof having the off-diagonal elements vanish in the matrix of Eq. [52]. Theresult is

fð~rÞ ¼ � tan�1 Imr"#ð~rÞRer"#ð~rÞ

½53�

yð~rÞ ¼ tan�1 2f½Rer"#ð~rÞ�2 þ ½Imr"#ð~rÞ�2g1=2

r""ð~rÞ � r##ð~rÞ½54�

This leads to an exchange–correlation potential in the form of a 2� 2 Hermi-tian matrix in spin space

Vxcð~rÞ ¼1

2ðV"xcð~rÞ þ V#xcð~rÞÞIþ

1

2ðV"xcð~rÞ � V#xcð~rÞÞr dð~rÞ ½55�

where dð~rÞ is the unit vector in the direction of the magnetization mð~rÞ. Thepresence of the second term in Eq. [55] effectively couples the " and # com-ponents of the spinor. To interpret the magnetic properties, one can use thespin-density matrix of Eq. [35] to compute the magnetization density mð~rÞ.Local magnetic moments latcan be associated with individual atoms by inte-grating each component of mð~rÞ within spheres centered on the ions. A rea-sonable choice for the radius of those spheres is to use one half of thesmallest interatomic distance in the cluster. This process avoids overlapbetween neighbor spheres; however, some interstitial space remains betweenspheres, and one should be aware of the fact that those atom-centeredspheres contain about 80–90% of the magnetization. It is worth stressingagain that the method explained above has assumed local exchange–correlation functionals. A route that includes density gradients has beenexplored by Capelle and Gross.43


MEASUREMENT AND INTERPRETATION OF THEMAGNETIC MOMENTS OF NICKEL CLUSTERS

Interpretation Using Tight Binding Calculations

Bloomfield et al.3 have performed accurate measurements of the mag-netic moments of size-selected NiN clusters with N between 5 and 700. Theirresults up to N ¼ 60 are plotted as black dots in Figure 2. The average mag-netic moment �m of the cluster shows an overall decrease with increasing clus-ter size, but oscillations are superimposed on this decreasing behavior. In thesmall size range, for N < 10, where �m decreases most rapidly, there is a localminimum at Ni6 and a local maximum at Ni8. Thereafter, �m displays a deepminimum for Ni13 and another minimum at Ni56. The latter minimum is soclose to Ni55 that it is tempting to conclude that the Ni clusters grow follow-ing an icosahedral pattern (clusters with perfect icosahedral structure and oneand two shells, shown in Figure 3, have 13 and 55 atoms, respectively44). Athird important minimum occurs around Ni34. The magnetic moment goesthrough a broad maximum between Ni13 and Ni34, and again betweenNi34 and Ni56.

Theoretical studies attempting to rationalize the behavior of the mag-netic moment of NiN clusters45–48 have relied on the tight binding method.The magnetic moments calculated by Aguilera-Granja et al.46 for sizes up toN ¼ 60 are plotted in Figure 2. The calculations used the theory describedabove, with some simplifications. Local charge neutrality was assumed, lead-

Figure 2 Comparison between the experimental average magnetic moments of Niclusters measured by Apsel et al.3 (black dots) and the moments calculated by a tightbinding method45,46 (light circles). Reproduced with permission from Ref. 44.

Measurement and Interpretation of the Magnetic Moments 211

ing to the following expression for the environment-dependent energy levels:

eias ¼ e0ia þ zs

Xb

Jab2mib þ Zi�ia ½56�

which is simpler than the expression in Eq. [16]. Two principles are usefulwhen interpreting these results.45,46 The first principle is that the local mag-netic moments of the atoms in the cluster decrease when the number of neigh-bors (local coordination) around the atoms increases. The second principle isthat the average magnetic moment of the cluster decreases when the intera-tomic distances decrease, which occurs because the width of the d bandincreases. In metallic clusters, the average atomic coordination numberincreases for increasing N. The average nearest-neighbor distance dalso increases with N, from the value for the molecule dmol to the value forthe bulk dbulk. The two effects oppose each other, and for that reason, thebehavior of �mðNÞ in a growing cluster is complex. For N 20, the clustergeometries used to perform the calculations of the magnetic moments wereobtained from molecular dynamics simulations using a many-atompotential49,50 based on the TB theory, with parameters fitted to the propertiesof Ni2 and bulk Ni. That potential is typically referred to as the Guptapotential.50 The geometries for N ¼5�16 are shown in Figure 4, which showsa pattern of icosahedral growth.

A qualitative agreement exists between the experimental and theoreticalmagnetic moments of small clusters. The TB calculations predict pronouncedlocal minima at Ni6 and Ni13 and a local maximum at Ni8. Ni13 is an icosa-hedron with an internal atom at its center. The local atomic coordination ofthe surface atoms in Ni13 is Z ¼ 6. On the other hand, Ni12 and Ni14 containsome atoms with coordination smaller than 6, and this leads to an increase ofthe local magnetic moments of those atoms. Consequently, the compact struc-ture of Ni13 explains the minimum of �m occurring at that cluster size. Ni6 is an

Figure 3 Clusters with perfect icosahedral structure: one shell (N ¼ 13), two shellsðN ¼ 55Þ, and three shells ðN ¼ 147Þ.


octahedron formed by atoms with coordination Z ¼ 4. In Ni7, which has thestructure of a flattened pentagonal bipyramid, the coordination of two atomsincreases to Z ¼ 6 and remains equal to 4 for the rest of the atoms. Ni8 hasfour atoms with coordination Z ¼ 5 and four atoms with coordination Z ¼ 4,which leads to a mean coordination number that is slightly smaller than inNi7. The coordination increases again in Ni9. This behavior of the mean coor-dination number would lead one to expect a maximum of �m for Ni8, which isindeed observed in the experiments and a minimum at Ni7. However, theobserved minimum and the calculated minimum occur at Ni6, and the reasonfor this is that the average nearest-neighbor distance dn has a local maximumat Ni7. The larger value of dn counteracts the increase of the coordinationnumber when going from Ni6 to Ni7 and produces the minimum of �m atNi6. To summarize, the oscillations of the average magnetic moment in smallNi clusters can be explained by two purely geometrical effects: (1) compactclusters, that is, clusters with high average atomic coordination number,have small values of �m; and (2) clusters with large interatomic distanceshave large �m.

The densities of states of Ni5, Ni6, and Ni7, decomposed into d and spcontributions, are compared in Figure 5. The occupied states of the majority-spin sub-band ð"Þ have mainly d character, except for a small peak with the spcharacter at the Fermi level; on the other hand, d holes are present in the min-ority-spin sub-band ð#Þ. Integration of the density of states gives average d

Figure 4 Ground state geometries of Ni clusters with 5 to 16 atoms (symmetriesare indicated), obtained using the Gupta potential. Reproduced with permission fromRef. 45.


magnetic moments of 1:60 mB, 1:52 mB, and 1:50mB for Ni5, Ni6, and Ni7,respectively. A comparison with the calculated moments of Figure 2 revealsthat the sp electrons make a sizable contribution. The sp moments in Ni5ð0:29mBÞ and Ni7 ð0:21mBÞ reinforce the d moment, whereas for Ni6, the spmoment ð�0:15 mBÞ opposes the d moment. The sp contribution decreasesquickly with increasing cluster size.

Icosahedral structures were assumed in these calculations for N > 20,although those structures were reoptimized using the Gupta potential.46 Inaddition, extensive molecular dynamics simulations were performed for afew selected cluster sizes. In all cases, the icosahedral structures were predictedas the ground state geometry, except for Ni38, which is a special case that willbe discussed later. Icosahedral growth thus seems to be consistent with theinterpretation of experiments probing the reactivity of Ni clusters with lightmolecules.51

Figure 5 Density of states of NiN clusters with N ¼ 5, 6, and 7, calculated by the tightbinding method: sp (dashed lines) and d (continuous lines). Positive and negative valuescorrespond to up " and down # spins, respectively. The Fermi level is at the energy zero.Adapted with permission from Ref. 45.


The calculated magnetic moments of Figure 2 reveal a decrease of �m forsizes up to N � 28, followed by a weak increase between N � 28 andN ¼ 60. This behavior is primarily a result of the variation of the averagecoordination number �Z which increases smoothly with N up to N ¼ 27and then drops.46 By extrapolating the smooth behavior of �ZðNÞ to sizes lar-ger than N ¼ 27, it was found that for N between 27 and 54, the actualvalues of �ZðNÞ fall below the extrapolated values. In fact, �ZðNÞ decreasesbetween N ¼ 27 and N ¼ 30 and then begins to increase again afterN ¼ 30. The break in �Z at N ¼ 27 suggests a flattening of �m, which is con-firmed by the calculations. This break in the coordination number is causedby a structural transition in the icosahedral clusters,51 which occurs at pre-cisely N ¼ 28. Starting with the first complete icosahedron shown inFigure 4, atoms can be added on top of this 13-atom core in two differentways. In a first type of decoration, atoms cover sites at the center of the tri-angular faces (F sites) and vertices (V sites). Those F and V sites provide atotal of 32 sites ð20þ 12Þ, and full coverage produces a cluster with 45atoms; this type of decoration can be denoted FC (face centered) as it empha-sizes the coverage of the faces of the icosahedron. Alternatively, atoms candecorate the edges (E sites) and vertices (V). These E and V sites provide atotal of 42 sites ð30 þ 12Þ, and completion of this layer leads to the nextMackay icosahedron with 55 atoms; these structures are called multilayericosahedral (or MIC) structures. FC growth is favored at the beginning ofbuilding a shell, up to a point when a transition occurs because MIC growthbecomes preferred. The cluster size for the transition depends slightly on thedetails of the interatomic interactions in different materials. For Ni clusters,it occurs between N ¼ 27 and N ¼ 28.

The calculated minimum of �m at N ¼ 55 corresponds with a minimum inthe measured magnetic moment at N ¼ 56. Also, the calculated minimum inthe region Ni28–Ni37, associated with the FC ! MIC transition, can be cor-related with the broad experimental minimum of �m in that region. The experi-ments also show a weak minimum at Ni19, which can be tentatively associatedwith a double icosahedron structure (an icosahedron covered by an FC capformed by six atoms, one in a vertex site and the others in the five associatedF sites),51 although this local minimum does not show up in the calculations.Another weak feature is the drop of �m between Ni22 and Ni23, which has acounterpart in the calculation (the structure of Ni23 results by covering an ico-sahedron with two FC caps; its structure can be viewed as three interpenetrat-ing double icosahedra). One may conclude with some confidence that theminima displayed by the measured magnetic moments provide some supportto a pattern of icosahedral growth.

It was indicated above that Ni38 is a special case. Reactivity experi-ments52 measuring the saturation coverage of this cluster with N2, H2, andCO molecules suggest that the structure of Ni38 is a truncated octahedroncut from a face-centered cubic (fcc) lattice. This structure is shown in


Figure 6. Motivated by this result, a detailed comparison between the energiesof fcc and icosahedral structures was performed by Aguilera-Granja et al.46 forother NiN clusters with N ¼ 13, 14, 19, 23, 24, 33, 36, 37, 39, 43, 44, 55, and68. Ni36, Ni37, Ni38, and Ni39 were the sizes selected in the neighborhood ofNi38. For most other sizes selected, it was possible to construct fcc clusters withfilled coordination shells around the cluster center. In all cases, the icosahedralstructure was predicted to be more stable, except for the Ni38 cluster. The dif-ference in binding energy between the icosahedral and fcc structures is, how-ever, small. This difference, plotted in Figure 7, is lower than 0.2 eV betweenNi24 and Ni39 and larger for other cluster sizes. For the truncated octahedralstructure, �mðNi38Þ ¼ 0:99mB. This value reduces the difference between theexperimental and theoretical results to one third of the value in Figure 2ð�mexp � �mfcc ¼ 0:04mB and �mexp � �mico ¼ 0:11 mBÞ. The moderate increase of�mfcc with respect to �mico is from the lower average coordination in the fcc struc-ture ð�ZðfccÞ ¼ 7:58; and �ZðicoÞ ¼ 7:74Þ. The calculated values of �m are verysimilar for the icosahedral and fcc structures of Ni36 (0:87 mB and 0:86mB,respectively). Because the energy differences between isomers for N ¼ 24–40are small (less than 0.4 eV), the possibility of different isomers contributingto the measured values of the magnetic moment cannot be excluded.

Rationalizing the observed broad maxima of �m at values of N around 20and 42 is more difficult than for minima. These maxima are not observed inthe TB results of Figure 2. One possibility, which has been suggested from

Figure 6 Calculated minimum energy structure of Ni38. It is a piece of an fcc crystal.Dark and light atoms are internal and surface atoms, respectively.


some molecular dynamics simulations,53 is that the structures of Ni clustersare bulk-like fcc instead of icosahedral for cluster sizes near those maxima.Using fcc structures covering the whole range of cluster sizes, Guevara etal.21 predicted sharp maxima for �m at Ni19 and Ni43 and minima at Ni28

and Ni55. However, the measured �m shows a local minimum at Ni19, and reac-tivity experiments suggest that Ni19 is a double icosahedron.54 So the onlyclear prediction in favor of fcc structures is the maximum of �m at Ni43. Rodrı-guez-Lopez et al.55 have performed additional TB calculations of �m for clusterstructures that other authors had obtained using different semiempiricalinteratomic potentials. Their conclusion is that the changes of �m arisingfrom different cluster structures are not large. The oscillations of �m at smallN are accounted for reasonably well for all structural families considered;however, a fast approach of �mðNÞ toward the bulk limit occurs in all cases.These results do not resolve the discrepancies between TB calculations andexperiment, which indicates that a possible misrepresentation of the exact geo-metry is not the only problem with the computational results. Other possibi-lities are explored in the next sections.

Influence of the s Electrons

An alternative model for explaining the behavior of the magneticmoment of Ni clusters has been proposed by Fujima and Yamaguchi (FY).56

The interest of this model is because it may contain some additional ingredi-ents required to explain the observed maxima of �m. However, as a function ofN, the FY model cannot predict the minima. It is intriguing that the observedmaxima of �m are at N ¼ 8 and near N ¼ 20 and N ¼ 40.3 These numbers

6.0

5.0

4.0

3.0

2.0

1.0

0.0

–1.010 20 30 40 50 60 70

Number of Atoms

DEic

o-fc

c (e

v)

Figure 7 Difference in binding energy of icosahedral and fcc isomers of NiN clusters as afunction of cluster size. Reproduced with permission from Ref. 46.


immediately bring to mind well-known electronic shell closing numbers ofalkali metal clusters.44,57 Electrons move as independent particles in an effec-tive potential with approximate spherical shape in the alkali clusters. In thatpotential, the electronic energy levels group in shells with a degeneracy equalto 2ð2l þ 1Þ caused by orbital angular momentum and spin considerations. Arealistic form of the effective potential, obtained for instance with the jelliumbackground model for the positive ionic charge,44,58 produces a sequence ofelectronic shells 1S, 1P, 1D, 2S, 1F, 2P, 1G, 2D, . . . , where the notation forthe angular momentum character of the shells is given in capital letters (S, P,D, . . .) to avoid confusion with s, p, and d atomic orbitals. Clusters with closedelectronic shells are especially stable. These ‘‘magic’’ clusters contain a numberof valence electrons Nmagic ¼ 2, 8, 18, 20, 34, 40, 58, 68, . . . The same magicnumbers have been observed for noble metal clusters.59 The electronic config-urations of free noble metal atoms are 3d104s, 4d105s, and 5d106s for Cu, Ag,and Au, respectively, and the interpretation of the magic numbers in the noblemetal clusters is that the shell effects are caused by the external s electrons.

In a similar way, the FY model for Ni clusters distinguishes betweenlocalized 3d atomic-like orbitals, responsible for the magnetic moment ofthe cluster, and delocalized molecular orbitals derived from the atomic 4s elec-trons, and it neglects hybridization between the 3d electrons and the deloca-lized 4s electrons. The delocalized 4s electrons are treated as moving in aneffective harmonic potential. The energy levels of the delocalized 4s electronslie just above the Fermi energy in very small NiN clusters. But, as N grows, thebinding energy of the delocalized 4s states increases and these states progres-sively move down below the 3d band. The existence of some 4s states belowthe 3d band causes the presence of holes at the top of the minority-spin 3dband (the majority-spin 3d band is filled). The number of 3d holes is equalto the number of 4s states buried below the 3d band, because the total numberof valence electrons per Ni atom is 10. The FY model assumes that the transferof 4s states to energies below the 3d band occurs abruptly when the number ofdelocalized 4s electrons is just enough to completely fill an electronic shell inthe harmonic potential. As a consequence of the stepwise motion, there is asudden increase in the number of holes at the top of the minority-spin 3dband of the cluster, and because the number of holes is equal to the numberof unpaired electrons in the cluster, an abrupt increase of the magneticmoment �m then occurs. The stepwise transfer of 4s-derived levels from abovethe Fermi energy to below the d band is supported by density functional cal-culations.60 The maxima of the magnetic moment observed in the Ni experi-ments near N ¼ 20 and N ¼ 42 could be related to this effect, because closingof electronic shells occurs in clusters of s-like electrons at N ¼ 20 and N ¼ 40.On the other hand, the FY model predicts the maxima and the minima of �mthat are too close, because of the assumption of the sudden transfer of a wholeshell of electrons when the conditions of shell closing are met. This contrastswith experiment, where the maxima and the minima are well separated.


Density Functional Calculations for Small Nickel Clusters

Small clusters of transition elements have been primarily studied usingDFT because the calculations become very time consuming for large clusters.Reuse and Khanna61 have calculated �m for NiN clusters with N ¼ 2–6, 8, and13. They found �mðNi6Þ < �mðNi5Þ and �mðNi13Þ < �mðNi8Þ, which agrees with theexperimental trend; however, the magnetic moments of Ni6 and Ni8 werenearly equal whereas the experiment indicates a larger �m for Ni8 (Fig. 2).Bouarab et al.45 also performed TB calculations with the same structuresand interatomic distances used by Reuse and Khanna. Their magneticmoments differed by no more than 0.06 mB from the TB values of Fig. 2.Therefore, the differences between TB and DFT results have to be mainlyascribed to the different treatment of the electronic interactions.46 Desmaraiset al.62 have studied Ni7 and Ni8. The same value �m ¼ 1:14mB was obtainedfor the ground state of Ni7, a capped octahedron, and for all its low-lyingisomers. Similarly, an average moment �m ¼ 1:0mB was obtained for theground state and for the low-lying isomers of Ni8. The insensitivity of themagnetic moments to atomic structure in Ni7 and Ni8, also found forNi4,61 is striking. Reddy et al.63 have calculated the magnetic moments forsizes up to N ¼ 21. For N 6, they employed ab initio geometries, and forN > 6, geometries were optimized with the Finnis–Sinclair potential.64

Compared with the experiment, the calculated magnetic moments aresubstantially lower, and important discrepancies occur in the evolution of �mwith cluster size. Fujima and Yamaguchi65 have calculated the local magneticmoments at different atomic sites within model Ni19 and Ni55 clusters with fccstructure and bulk interatomic distances. The octahedral shape was assumedfor Ni19 and the cuboctahedral shape for Ni55. No significant differences werefound between the magnetic moments of atoms on different surface sites, butthe moments of atoms in the layer immediately below the surface were 0.2 mB

smaller than those of the surface atoms. The average magnetic moments�mðNi19Þ ¼ 0:58mB and �mðNi55Þ ¼ 0:73mB are significantly smaller than themeasured moments. Pacchioni et al.66 calculated the electronic structure ofNi6, Ni13, Ni19, Ni38, Ni44, Ni55, Ni79, and Ni147. Icosahedral structureswere assumed for Ni13, Ni55, and Ni147 and structures with Oh symmetryfor Ni6, Ni13, Ni19, Ni38, Ni44, Ni55, and Ni79 (in most cases, fragments ofan fcc crystal). Convergence of �m to the bulk limit was not observed, despitethe width of the 3d band being almost converged for N ¼ 40–50.

Orbital Polarization

The calculations discussed above considered spin magnetic moments butnot the orbital magnetic moments. However, it is known that orbital correla-tion has a strong effect in low-dimensional systems, which leads to orbitalpolarized ground states.67–69 Based on this fact, Guirado-Lopez et al.69 and


Wan and coworkers48 studied the magnetic moments of Ni clusters, takinginto account both spin and orbital effects. Wan et al.48 used the followingTB Hamiltonian

H ¼XiLs

e0il cþiLsciLs þ

Xij

XLL0s

tjL0

iL cþiLscjL0s þHSO þHee

þXi0s

e0i0s0 cþi0s0sci0s0s þ ts0

s ðZi0 Þðcþi0s0sci0ss þ cþi0ssci0s0sÞh i

þXi0Ls

�ei0 ðni0s0 Þðcþi0Lsci0Ls þ cþi0s0sci0s0sÞ ½57�

where the meaning of the different symbols is the same as described in thesection of the TB method. L ¼ ðl; mÞ indicates the orbital angular momentumquantum numbers. There are some differences with respect to the Hamiltonianin Eqs. [15] and [16]. The first two terms in Eq. [57] are already included inEq. [15]. The term HSO is the spin-orbit coupling operator in the usual intraa-tomic approximation,

HSO ¼ xX

i;Ls;L0s0Ls ~Si

~Li

��L0s0

D EcþiLsciL0s0 ½58�

where x gives the strength of the interaction ( x ¼ 0:073 for d orbitals). Theterm Hee, to be discussed below, is an intraatomic d–d electron–electron inter-action. The two final terms in Eq. [57] apply specifically to the surface atomsof the cluster, labeled by the subscripts i0. To take into account the electronicspillover at the surface,21 an extra orbital, labeled s0, is attached to each sur-face atom i0.

The intraatomic d–d electron–electron interaction includes Coulomb andexchange interactions, and it is responsible for orbital and spin polarization.To account for orbital polarization, the idea of the LDA þ U method was fol-lowed.70 A generalized Hartree–Fock approximation including all possiblepairings was then used to write

Hee ¼X

i;Ls;L0s0Vi

Ls;L0s0cþiLsciL0s0 ½59�

where

ViLs;L0s0 ¼

XL2L3

ðfULL2L0L3ni;L2 �s;L3 �sþðULL2L0L3

�ULL2L3L0 Þni;L2s;L3sgdss0

�ULL2L3L0ni;L2 �s;L3sd�ss0 Þ�Uðni�0:5ÞdLL0dss0 þJðnis�0:5ÞdLL0dss0 ½60�


In this expression, ni;Ls;L0s0 ¼ hcþiLsciL0s0 i is the single-site density matrix, nis isthe trace of ni;Ls;L0s0 , �s ¼ �s, and ni ¼

Ps

nis. The matrix elements ULL2L0L3

can be determined by two parameters, the average on-site Coulomb repulsionU and the exchange J,

U ¼ 1

ð2l þ 1Þ2Xmm0

Umm0mm0 ½61�

U � J ¼ 1

2lð2l þ 1ÞXmm0ðUmm0mm0 �Umm0m0mÞ ½62�

This can be seen by expressing ULL2L0L3in terms of complex spherical harmo-

nics and effective Slater integrals Fk as in Eq. [63]70

hm;m00jUjm0m000i ¼X

k

akðm;m0;m00;m000ÞFk ½63�

where 0 k 2l and

akðm;m0;m00;m000Þ ¼4p

2kþ 1

Xk

q¼�k

lm Ykq

�� lm0� ��lm00��Y�kq

��lm000� ½64�

Because we are dealing with d electrons, l ¼ 2, and the notation for ULL0L00L000

has been simplified to Umm0m00m000 in Eqs. [61] and [62]. The Slater integrals Fk,which carry the radial integrations in Eq. [63], are expressions of the type71

Fkðs; tÞ ¼ð

dr

ðdr0PnslsðrÞPnslsðrÞ

rk<

rkþ1>

Pntltðr0ÞPntltðr0Þ ½65�

where the symbols s and t refer to two different electrons and PnslsðrÞ is theproduct of the distance r from the nucleus and the radial function of the s elec-tron RnslsðrÞ. The terms r< ðr>Þ correspond to the smaller (larger) of r and r0:Only the Slater integrals F0, F2, and F4 are required for d electrons, and thesecan be linked to the U and J parameters through the relations U ¼ F0 andJ ¼ ðF2 þ F4Þ=14, whereas the ratio F2=F4 is a constant � 0:625 for the 3d ele-ments. In this formalism, the Stoner parameter, which determines the splittingof the bulk bands,67 is I ¼ ð2lJ þUÞ=ð2l þ 1Þ. Wan et al.48 chose I ¼ 1:12 eVin their cluster calculations, in order to have consistency with the exchangesplitting of the bands of bulk Ni obtained in LSDA calculations. They useda value U ¼ 2:6 eV, although other values (U ¼ 1:8 eV, and U ¼ 3:2 eV)were explored. It is evident that orbital polarization is included in thisapproach via the orbital-dependent effects coming from F2 and F4. The orbital


polarization differentiates the approach discussed in this section from thatembodied in the simpler formulation in Eqs. [16] and [17].

The fifth term in the Hamiltonian of Eq. [57] is added to account forthe electronic spillover at the surface.21 One extra orbital with s symmetry(s0 orbitals) is added to each surface atom i0 and located in the vacuum regionnear the atom. This s0 orbital interacts with the s orbital of the same surfaceatom through the hopping integral ts0

s and the occupation of the s0 orbitalsrepresents the spillout. The hopping integral is parameterized suchthat ts0

s ðZi0 Þ ¼ Vss0s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZmax � Zi0

p, where Zmax � Zi0 is the local deficit in the

atomic coordination (Zmax is the maximum coordination; 12 for a fcc solid)and Vss0s is the hopping strength. The last term in Eq. [57], which accountsfor the orbital shifts from intersite Coulomb interactions, is related to thelast two terms in Eq. [16]. However, Wan et al.48 restricted the orbital shiftsto the surface atoms i0.

The eigenvalue equation corresponding to the Hamiltonian of Eq. [57]can be solved self-consistently by an iterative procedure for each orientationof the spin magnetization (identified as the z direction). The self-consistentdensity matrix is then employed to calculate the local spin and orbital mag-netic moments. For instance, the local orbital moments at different atoms iare determined from

mi;orb ¼Xs

Xm¼2

m¼�2

ðeF

�1mnimsðeÞde ½66�

where m refers to the magnetic quantum number. The spin magnetic moment~mspin and the orbital magnetic moment ~morb of the cluster are calculated as avector average of the atomic moments. The total magnetic moment ~m isobtained as the vector sum of ~mspin and ~morb. The self-consistent solution ofthe Hamiltonian generates a non-uniform and non-collinear distribution ofspin and orbital moments; however, it was found by Wan et al. that noncolli-nearity is very weak in this case. Also, the spin-orbit interaction can generateanisotropy, but a comparison of calculations with spin along the three princi-pal axes of the clusters revealed very small energy differences of less than0.005 eV/atom. The results reported immediately below correspond to spinorientation along the largest inertia axis. A first test of the accuracy of thetheory is provided by calculations for the bulk metals, performed byGuirado-Lopez et al.,69 using a very similar model: These authors found orbi-tal magnetic moments mb;orbðFeÞ ¼ 0:094 mB; mb;orbðCoÞ ¼ 0:131mB, andmb;orbðNiÞ ¼ 0:056 mB, all in good agreement with the experimental valuesmb;orbðFeÞ ¼ 0:092 mB; mb;orbðCoÞ ¼ 0:147 mB, and mb;orbðNiÞ ¼ 0:051 mB.

The spin magnetic moments of Ni clusters calculated by Wan et al.48 arein reasonable agreement with density functional calculations,61,63,72 but bothapproaches, that is, TB and DFT, give values substantially smaller than theexperimental magnetic moments. The results of Wan et al. improve by adding


the orbital magnetic moment: The magnitude of the total ~m becomes closer tothe experimental values, and its detailed variation as the cluster size increasesalso improves. One can observe in Figure 2 that the magnitude of the spinmagnetic moments obtained by Aguilera-Granja et al.46 is good, but Wan etal.48 suggested that the result of Aguilera-Granja et al. is caused by the para-meterization used. Figure 8 shows the spin, orbital, and total magneticmoments obtained by Wan et al. Except for the smallest clusters, morb variesbetween 0.3 and 0.5 mB/atom, which represents a large enhancement (6 to10 times) with respect to the orbital moments in bulk Ni. The orbital magneticmoment of the free Ni ðd9s1Þ atom is mat;orbðNiÞ ¼ 2mB. Therefore, a large partof the quenching of morb is already manifested in the smallest clusters, as soonas full rotational symmetry is lost. However, a substantial enhancement of morb

with respect to the bulk orbital moment is still observed at Ni60. On the otherhand, the oscillations of the total magnetic moment come from morb. The orbi-tal moments depend on the choice of the correlation parameter U, and a valueU ¼ 2:6 eV gives the best agreement with the experiment. The positions of theminima may depend on U, except the four minima indicated by the verticallines in Figure 8. A comparison with the experiment is provided in Figure 9.The qualitative trend up to N ¼ 28 is reproduced approximately by the calcu-lations. In the size range 9 N 28, the calculated moments are in reason-able agreement with the experimental results of Knickelbein,4 but they aresmaller than the magnetic moments measured by Apsel et al.3 (the discrepan-cies between the two sets of experimental results are largest between N ¼ 10and N ¼ 20). The calculated moments for 30 N 38 are larger, by 0.1–0.2 mB/atom, than those of Apsel et al., and they display the minimum atNi34. Finally, there is excellent agreement with experiment between N ¼ 40

Figure 8 Calculated spin, orbital, and total magnetic moments per atom of Ni clusters.Reproduced with permission from Ref. 48.


and N ¼ 60. The predicted minimum at Ni28 is not observed in the experi-ments, although there is a break in the decreasing behavior of �m at Ni28.The minimum in the theoretical curve (see also Figure 2) is related to a struc-tural transition in the model of icosahedral growth used in the theoretical cal-culations.46,48,51

The enhancement of morb in the clusters with respect to the bulk momentmb;orb results from several contributions related to changes in the local environ-ment of the atoms.69 The first contribution is the reduction of the local coor-dination number leading to an increase of the local spin polarizations, whichthrough spin-orbit interactions induce large orbital moments. The second con-tribution is the orbital dependence of the intraatomic Coulomb interaction,which favors the occupation of states with high m and contributes to theenhancement of morb in clusters. The final contribution is the presence ofdegeneracies in the spectrum of one-electron energies that allows for a veryeffective spin-orbit mixing that enhances morb.

The magnetic moments of the Ni clusters are dominated by the contribu-tion from surface atoms.48,69 The analysis of Wan et al. indicates that the orbitaland spin local moments of cluster atoms with atomic coordination 8 or largerare similar to those in the bulk ðmb;spin � 0:55mB, and mb;orb � 0:05mB);73 that is,the orbital moment is almost quenched for internal cluster atoms. In contrast,there is a large enhancement of the spin and orbital moments for atoms withcoordination less than 8. This enhancement increases with the coordinationdeficit, and it is larger for the orbital moment. Wan et al.48 also analyzed thequantum confinement effect proposed by Fujima and Yamaguchi,56 i.e., the

Figure 9 Comparison between the calculated magnetic moments of Ni clusters (darksquares) and the experimental results of Apsel et al.3 and Knickelbein.4 Reproducedwith permission from Ref. 48.


sudden immersion of delocalized states from above the Fermi energy to belowthe d band when the number of delocalized electrons is enough to close anelectronic shell. This effect is confirmed by the TB calculations of Wan et al.and is found to be relevant in small Ni clusters. However, as the clustersize increases, immersion below the d band seems to be gradual ratherthan sharp.

In summary, the works of Guirado–Lopez et al.69 and of Wan et al.48

have shown the importance of the orbital contribution to the magneticmoment of nickel clusters. The TB method provides a convenient frameworkto understand the variation of �mðNiNÞ with cluster size, and this understandingis good, although not perfect. In fact, the work of Andriotis and Menon,74 for-mulated in the TB framework, while supporting the idea that the enhancedorbital moments are aligned parallel to the spin moments in Ni clusters,also raises the possibility that these states are energetically unfavourable.This may be attributed to the interplay between the action of the spin-orbitinteraction HSO, which favors the alignment of L along the direction of S,and the action of the crystal field, which tends to align L along the easy mag-netization axis (in materials with magnetic anisotropy, the magnetization iseasier along a particular direction of the crystal). Another fact that makesthe theoretical analysis difficult arises from the differences between the mag-netic moments measured in different experiments for the same clusters.2–4

CLUSTERS OF OTHER 3d ELEMENTS

Chromium and Iron Clusters

Chromium is an antiferromagnetic metal in the bulk phase, and the cal-culations of Cheng and Wang75 show that the Cr clusters also have a strongtendency toward antiferromagnetic spin ordering (although the fact that thenumber of atoms is small makes the distribution of the magnetic momentsmore complex). Small Cr clusters are ‘‘special’’ compared with clusters ofthe other 3d metals.75,76 The electronic structure of the atom 3d54s1 has sixunpaired electrons. This half-filled electronic configuration leads to strongd–d bonding in Cr2, with a bond length of 1.68 A, which is very short com-pared with the interatomic distance of 2.50 A in Cr metal. The dimer is aclosed shell molecule with a strong sextuple bond.75 The strong binding arisesfrom the filling of the 3d-bonding molecular orbitals: s2

3dp43dd

43ds

24sð1�þg Þ. The

electronic structure of the dimer is robust and controls the growth of the smallclusters.

The optimized geometries of Cr clusters are given in Figure 10. Cr3 iscomposed of a dimer plus an atom: The electronic structure of the dimer islittle affected by the presence of the third atom, which remains in its atomicelectronic state, leaving six unpaired electrons in the cluster. A new pair forms

Clusters of Other 3d Elements 225

by adding the fourth atom, and Cr4 is formed by two strong dimers with weakinterdimer bonding. The dimerization effect controls growth up to N ¼ 11;those clusters are formed by dimers with short bond lengths and one isolatedatom (in Cr5, Cr7, Cr9), or two isolated atoms (in Cr10) bonded to adjacentdimers. The structure of Cr11 is similar to that of Cr10 with an atom at thecluster center. The dimer growth route stops at Cr11, at which point thebond lengths suddenly increase and dimer bonds can no longer be identifiedfor N > 11. The arrows in Figure 10 indicate the orientation of the atomicspins. There is an anisotropic distribution of the magnetic moments, but thestrong tendency to antiferromagnetic ordering is clear, especially as Nincreases. The local moments of the capping atoms are much larger than thoseof the dimerized atoms. The average magnetic moments of the small clusters

1.69

2.61 2.592.48

2.43

2.45

1.92

1.87

2.282.31

2.322.26

2.562.252.33

2.27

2.45

2.51

1.83

1.80

2.76

1.72

2.81

2.74

2.74

1.78

2.77

2.67

2.69

2.27

3.12

2.70

2.762.61

2.75

1.72

1.94

2.46

1.63

2.51 2.44

1.96

1.83

(C2v)

(D3h)

(D4h)

(D4h) (C4v) (D4h)

2.78

(D4h) (C2v)

(C2v) (D4h)(C2v)

(D2h) (C2v)(D˚˚h)

2.73 2.422.75

2.53

2.30

Figure 10 Optimized structures of CrN clusters, N ¼ 2–15. Bond lengths are in A. Thearrows indicate the orientation of the local atomic spins. Strong dimer bonds arerepresented by thick lines. Reproduced with permission from Ref. 75.


are as follows: Cr2 (0), Cr3 (2), Cr4 (0), Cr5 (0.93), Cr6 (0.33), Cr7 (0.29), Cr8

(0), Cr9 (0.22), Cr10 (0.2), Cr11 (0.55), Cr12 (1.67), Cr13 (1.06), Cr14 (0.74),and Cr15 (0.48), in units of mB/atom. The dimer growth route leads to an odd–even alternation of the average magnetic moments: Small moments for clusterswith even N and large moments for clusters with odd N. The large momentsarise from the quasiatomic character of the capping atoms; the dimer-pairedeven-N clusters have low �m because of the strong intradimer 3d–3d interac-tion. In most cases, the magnitudes of the calculated moments are withinthe upper limit of � 1:0mB imposed by the experiments,77,78 but for Cr12

and Cr13, the predicted �m is larger than this limit. Fujima and Yamaguchi65

studied chromium clusters and iron clusters both with 15 and 35 atoms,assuming the body-centered cubic (bcc) structure of the bulk metals and a sym-metric shape of a rhombic dodecahedron. For Cr, an alternation of the signs ofthe local moments as a function of the distance to the cluster center was found.The absolute values of the local moments decrease with increasing local coor-dination, and they also decrease for decreasing interatomic distance. The localmoments of the Fe clusters are less sensitive to atomic coordination, althoughsmall magnetic moments were obtained for Fe atoms on the layer belowthe surface.

Calculations allowing for noncollinear arrangements of the spins havebeen performed for small Fe and Cr clusters. Calculations for Fe2 and Fe4

by Oda et al.40 resulted in collinear ground states. The ground state of Fe3

is an equilateral triangle with a collinear spin arrangement. It has a totalmoment of 8.0 mB and a binding energy of Eb¼ 2.64 eV/atom. A linear iso-mer with noncollinear arrangement was also found. The central atom has amoment of 1.27 mB oriented perpendicular to the linear axis, and the twoedge atoms have moments of magnitude 2.89 mB, tilted by 10� with respectto the cluster axis. This isomer has a total moment of 2.04 mB and a bindingenergy of Eb¼ 2.17 eV/atom. Two other linear isomers were also found withcollinear ferromagnetic and antiferromagnetic configurations. The totalmoments of those two isomers are 6.0 mB and 0 mB, and their binding ener-gies are 1.80 and 2.15 eV/atom, respectively. A trigonal bipyramid structure(D3h symmetry) with a noncollinear spin arrangement was obtained for theground state of Fe5. The three atoms of the basal plane have magneticmoments of 2.72 mB and point in the same direction. The two apical atomshave moments of magnitude 2.71 mB tilted in opposite directions by approxi-mately 30� with respect to the moments of the basal atoms. The total momentof the cluster is 14.6 mB, and its binding energy is 3.46 eV/atom. An isomerwith D3h structure, lying 0.01 eV/atom above the ground state, was alsofound having a collinear spin arrangement with atomic moments of 2.58mB and 2.55 mB for the basal and apical atoms, respectively. Kohl andBertsch41 studied Cr clusters with sizes between N ¼ 2 and N ¼ 13 andobtained noncollinear arrangements for all cluster sizes except for N ¼ 2and N ¼ 4. They suggested that the trend of noncollinear configurations is


likely a feature common to most antiferromagnetic clusters, because noncol-linear effects are caused by frustration, that is, by the impossibility of formingperfect antiferromagnetic arrangements.

The nature of the spin arrangement depends sensitively on the intera-tomic distances. A comparative study of Cr7, Mn7 (Mn clusters will be studiedin detail in the next section), and Fe7 was made by Fujima79 by assuming apentagonal bipyramid structure for the three clusters and including the varia-tion of the interatomic distance d from the bulk value dbulk up to a value 20%lower than dbulk. The results are summarized in Figure 11. For Cr7 with dbulk,the magnetic moments are arranged in a coplanar, noncollinear configuration;that is, the vector moments lie on the same plane but point in different direc-tions. When d decreases, the magnetic moments are ordered in a parallel (P)fashion. The situation is similar in Mn7 for interatomic distances close to dbulk;that is, the cluster shows a coplanar configuration of the spins. However, anon-coplanar configuration first appears when d decreases, which thenchanges to a collinear antiparallel (AP) configuration with a further decreaseof d. Finally the arrangement of the spins in Fe7 is parallel for d � dbulk andd � 0:8 dbulk, and noncollinear for d in between these limits. Similar work80

for 5 atom clusters with the structure of a trigonal bipyramid indicates thatnoncollinear magnetic arrangements appear for Mn5 and Cr5 with interatomicdistances close to dbulk, which change to antiparallel arrangements withdecreasing d. Another interesting result is that parallel magnetic momentsappear for Ni5 (also for Co5 and Fe5) for almost all bond lengths between dbulk

and 0.8 dbulk.

Figure 11 Character of the arrangement of the spin magnetic moments, and averagemagnetic moment, in seven-atom clusters with a pentagonal bipyramid structure andinteratomic distances ranging from dbulk to 80% dbulk. Reproduced with permissionfrom Ref. 79.


Manganese Clusters

Manganese is the 3d metal with the smallest bulk modulus and cohesiveenergy. It has a complex lattice structure with several allotropic forms. Someof these bulk phases are antiferromagnetic, whereas monolayers81 and sup-ported clusters82 exhibit nearly degenerate ferromagnetic and antiferromag-netic states. The dimer is also peculiar.83 In contrast to other transitionelements, the bond length of Mn2 is larger than the nearest-neighbor distancein the bulk. In addition, its estimated binding energy is between 0.1 and 0.6eV, which puts Mn2 in a category similar to van der Waals molecules. Theseproperties arise from the electronic configuration of the atom, 3d54s2. Theelectrons of the half-filled 3d shell are well localized and do not interactwith those of the other atom in the dimer. Binding arises from the interactionbetween the filled 4s2 shells. A nonmetal-to-metal transition occurs as the Mnclusters reach a critical size. From experiments of the reactivity with hydrogen,Parks et al.84 have suggested that this transition occurs at Mn16. The largemagnetic moment of the free atom (5 mB) and the weak interaction betweenthe atoms in the dimer lead one to expect an interesting magnetic behavior forMn clusters.

Some measurements of the magnetic moments of Mn clusters containingfewer than ten atoms have been performed for clusters embedded in matrices.Electron spin paramagnetic resonance (ESR) experiments of Mn2 in condensedrare gas matrices yield an antiferromagnetic configuration, but Mn2

+ is ferro-magnetic, with a total magnetic moment of 11 mB.85 A moment of 20 mB hasbeen measured for Mn4 in a silicon matrix.86 Mn5

+ embedded in inert gasmatrices has a moment of 25 mB, although the cluster actually studied couldbe larger.87 We close by noting that neutral Mn2 is antiferromagnetic, whereasthe other Mn clusters are ferromagnetic.

The computational results on small Mn clusters are also controversial.An early Hartree–Fock study of Mn2 predicted a 1�þg ground state resultingfrom the antiferromagnetic coupling of the localized spins.88 Fujima andYamaguchi89 used DFT to study clusters of size Mn2 to Mn7. The interatomicdistances were optimized for constrained geometries, and all clusters werepredicted to show antiparallel spin ordering. Nayak and Jena90 optimizedthe structures of clusters with N 5 at the LSDA and GGA levels of theory.Only the GGA calculations reproduce some of the observed features of Mn2,namely a bond length larger than the nearest-neighbor distance in the bulkand a small binding energy (the calculated bond length is 6.67 a.u., and thebinding energy is 0.06 eV). The cluster is predicted to be ferromagnetic with atotal magnetic moment of 10 mB. The binding energy increases in Mn2

þ, andthe bond length decreases relative to Mn2 because the electron is removedfrom an antibonding orbital. The total magnetic moment of Mn2

þ is 11mB, in agreement with the experimental estimation for clusters in rare gasmatrices.


The optimized geometries of Mn3, Mn4, and Mn5 obtained by Nayakand Jena90 are an equilateral triangle, a Jahn–Teller distorted tetrahedron,and a trigonal bipyramid, respectively. The strength of the bonding increasesrelative to the dimer because of s–d hybridization. The predicted geometriesare consistent with those deduced from experiments in matrices. The hyperfinepattern observed for Mn4 embedded in a silicon matrix86 indicates that thefour atoms are equivalent, as would occur in a tetrahedron. The triangularbipyramid is one of the possible structures of Mn5 consistent with the ESRmeasurements.87 The calculated interatomic distances decrease substantiallyfrom Mn2 to Mn3, which signals the onset of delocalization and hybridizationbetween atomic orbitals at various sites. But the most striking property ofthese clusters is their ability to retain their atomic moments. Mn3, Mn4, andMn5 in their ground state are predicted to be ferromagnetic, with moments of5 mB per atom (low-lying structural isomers are also ferromagnetic). Experi-ments for thin layers support the possibility of large moments.91,92

The calculations of Pederson et al.93 provide additional insight into themagnetism of small Mn clusters. These authors studied Mn2 using LDA andGGA functionals and concluded that the manganese dimer is ferromagneticwith a total moment of 10 mB, a bond length of 4.93 a.u., and a binding energyof 0.99 eV. They also found an antiferromagnetic state whose properties, abinding energy of 0.54 eV and a bond length of 5.13 a.u., are closer to thoseof Mn2 in condensed rare gas matrices. A plausible resolution of the discrepan-cies for Mn2 offered by Pederson et al.93 is that the ferromagnetic state is thetrue ground state of free Mn2 but that the interaction with the condensed raregas matrix may stretch the bond, which leads to the appearance of an antifer-romagnetic state in the embedded cluster. However, very recent calculationsby Yamamoto et al.94 using a high-level ab initio method (second-order qua-sidegenerate perturbation theory,95 MCQDPT2) predict antiferromagneticcoupling for the Mn dimer.

Larger clusters were also studied by Pederson et al.93 Mn3 has differentmagnetic states close in energy. The ground state is an isosceles triangle in aferromagnetic configuration with a total moment of 15 mB. A frustrated anti-ferromagnetic state also exists with the atomic spins of the shorter side of thetriangle antiferromagnetically coupled to the third atom, whereas the first twoatoms are ferromagnetically aligned (perfect antiferromagnetism is impossiblein the triangular structure because the moments of two atoms necessarily pointin the same direction; this represents a frustration of the tendency to antifer-romagnetism). This state, with a net magnetic moment of 5 mB, is only 0.014eV above the ground state. Mn4 is a tetrahedron with a total moment of 20mB. The calculations predict a trigonal bipyramid as the ground state of Mn5

with a net moment of 23 mB, which is lower than the measured value of 25mB.87 Trigonal bipyramid and square pyramid states with moments of 25 mB

were found 0.62 eV and 1.20 eV above the ground state, respectively.Pederson and coworkers concluded that either the matrix influences the


ground state multiplicity of Mn5 or the cluster formed in the experiment isother than Mn5; the latter possibility had also been admitted in the originalexperimental work.87 A square bipyramid and a pentagonal pyramid wereinvestigated for Mn6. The total moments are 26 mB and 28 mB, respectively,and this cluster was proposed as a possible candidate for the cluster withm ¼ 25 mB observed in the ESR experiments. Table 1 gives the results of Ped-erson et al.93 for the average bond distance, the number of bonds per atom, themagnetic moment, and the binding energy of Mn2 to Mn8. Also given are thetwo spin gaps �1 ¼ emajority

HOMO � eminorityLUMO and �2 ¼ eminority

HOMO � emajorityLUMO , which

represent the energy required to move an electron from the HOMO of onespin sub-band to the LUMO of the other. The two spin gaps must be positivefor the system to be magnetically stable.

In a Stern–Gerlach deflection experiment, Knickelbein96 measured themagnetic moments of free MnN clusters for sizes between N ¼ 11 andN ¼ 99. The magnetic moments were obtained from Eq. [2] assuming superpar-amagnetic behavior. The moment �m shows local minima for N ¼ 13 andN ¼ 19, which suggest icosahedral growth in that size range; for larger sizes,�m shows a minimum in the region Mn32–Mn37 and a broad maximum in theregion Mn47–Mn56 followed by a weak minimum at Mn57. The maximum valueof the magnetic moment found in the experiment was �mðMn15Þ ¼ 1:4 mB, whichis substantially smaller than the calculated moments given in Table 1, and thisresult is puzzling. The interpretation of the experimental results has beenchallenged by Guevara et al.97 They performed TB calculations for Mn clustersup to Mn62 using several model structures (icosahedral, bcc, and fcc), and theyobtained several magnetic solutions for each cluster size and structure. Ingeneral, the magnetic moments are not ferromagnetically aligned. A comparisonof the experimental and calculated moments led to the suggestion that thestructures are mainly icosahedral for N < 30, and that bcc structures begin tocompete with icosahedral structures for larger clusters. Jena and coworkers98,99

arrived at similar conclusions for the magnetic ordering: Non-ferromagneticordering is responsible for the small moments measured for the Mn clusters.The non-ferromagnetic ordering was proposed to be ferrimagnetic: That is,

Table 1. Calculated Average Bond Distance d, Number of Bonds Per Atom NB,Magnetic Moment Per Atom �m, Binding Energy Per Atom Eb, and Spin Gaps �1 and�2 of MnN Clusters.93

N d (a.u.) NB �mðmBÞ Eb (eV) �1 (eV) �2 (eV)

2 4.927 0.5 5.0 0.50 0.65 1.303 5.093 1.0 5.0 0.81 0.46 1.384 5.162 1.5 5.0 1.19 0.62 2.315 5.053 1.8 4.6 1.39 0.50 0.796 5.002 2.0 4.3 1.56 0.90 1.137 4.970 2.1 4.2 1.57 0.70 0.478 4.957 2.2 4.0 1.67 0.93 0.37


the magnitudes of the moments at the different atomic sites are different, thenumber of atoms with " and # spins are unequal, or both. This proposal issupported by the most recent DFT calculations for Mn13,

99–102 by a combinedexperimental and theoretical analysis of Mn7,

99 and by the most recentStern–Gerlach deflection experiments for free clusters performed byKnickelbein103 for N ¼ 5–22. The results of the latter experiments are givenin Figure 12, where one can again see that the values of �m are small.

In summary, from the latest experimental103 and theoretical98–102 works,a clearer picture of the magnetic properties of Mn clusters is emerging. For thesmaller cluster sizes ðN 6Þ, a strong competition exists between ferromag-netic and antiferromagnetic ordering of the atomic moments, which resultsin a near degeneracy between the two types of ordering. The calculations ofBovadova-Parvanova et al.101 clearly illustrate this competition. Mn2 is ferro-magnetic, with a total m ¼ 10mB, but an antiferromagnetic state lies only 0.44eV above it in energy. Mn3 is ferromagetic with m ¼ 10 mB, but an antiferro-magnetic state with a similar triangular structure and a net moment of 5 mB

exists only 0.05 eV higher in energy. The ground state of Mn4 has a tetrahedralstructure and is ferromagnetic with m ¼ 20mB, but antiferromagnetic stateswith a similar tetrahedral structure exist 0.11 eV and 0.24 eV higher in energy,respectively. Mn5 is antiferromagnetic with m ¼ 3 mB (the structure and distri-bution of atomic magnetic moments are shown in Figure 13), but a ferromag-netic and two other antiferromagnetic states lie within a small energy range of0.05 eV above the ground state. The ground state, with �m ¼ 0:6mB, explainsthe result obtained in the Stern–Gerlach experiments of Figure 12. On theother hand, the ferromagnetic state with m ¼ 23 mB could explain the result

2.0

1.5

1.0

0.5

0.04 6 8 10 12 14 16 18 20 22

n

Mnnm

omen

t per

ato

m (

m b)

Figure 12 Measured magnetic moments per atom of MnN clusters with N between 5and 22. Reproduced with permission from Ref. 103.


obtained for Mn5 embedded in a matrix. Mn6 has three nearly degenerateoctahedral structures competing for the ground state. The lowest energy statehas an antiferromagnetic spin arrangement with a net magnetic moment peratom of 1.33 mB. The other two states are only 0.03 eV higher in energy:One is antiferromagnetic, with �m ¼ 2:66 mB, and the other is ferromagneticwith �m ¼ 4:33 mB. Another antiferromagnetic state with an octahedral struc-ture and �m ¼ 0:33 mB lies 0.08 eV above the ground state. The distributionof atomic moments for this isomer is given in Figure 13. The calculations ofJones et al.102 lead to the same picture pointed here for the ground state andthe low-lying isomers. Knickelbein103 has interpreted his Stern–Gerlach resultof �mðMn6Þ ¼ 0:55mB as possibly being from the contribution of several

Figure 13 Ground state structures and local spin magnetic moments (in mB) of Mn5 andMn7 determined by DFT calculations. For Mn6, the structure and local momentscorrespond to a relevant isomer 0.07 eV above the ground state. Some bond lengths arealso given, in A. Reproduced with permission from Ref. 103.


isomers in the experiment: The isomer with �m ¼ 0:33mB and one or more ofthe higher-moment isomers. The structure of Mn7 is a distorted pentagonalbipyramid (see Figure 13), and the magnitude of the local moments is about5 mB, but the coupling is ferrimagnetic and the net magnetic moment of thecluster is only 0.71 mB per atom, in good agreement with the Stern–Gerlachexperiment ð�m ¼ 0:72� 0:42 mBÞ. This ferrimagnetic coupling is representativeof the situation for N > 6, which is corroborated by calculations for Mn13 andlarger clusters.100–102 Although the local atomic moments are in the range3.5–4 mB, the tendency toward antiferromagnetic ordering leads to ferrimag-netic structures with magnetic moments of 1 mB per atom or less. All calcula-tions for Mn clusters described above assumed collinear spin configurations. Afew calculations have been performed for small Mn clusters that allow fornoncollinear arrangements of the spins. Mn7 has been discussed above. Usingthe DFT code SIESTA,104 Longo et al.105 found four antiferromagnetic statesfor Mn6 (with octahedral structure), in good agreement with collinear-constrained calculations;101 however, the ground state has a noncollinearspin configuration with a total binding energy 0.46 eV larger than that ofthe most stable antiferromagnetic isomer. The net magnetic moment of thisnoncollinear structure is 5.2 mB, which corresponds to 0.78 mB per atom,which is still a little larger than the experimental magnetic moment of 0.55mB per atom given in Figure 12.

CLUSTERS OF THE 4d ELEMENTS

The 4d metals are nonmagnetic in the bulk phase. However, the freeatoms are magnetic, and consequently, it is reasonable to expect that smallclusters of these elements could be magnetic. Experiments5,106 show that Rhclusters with less than 60 atoms and Ru and Pd clusters with less than 12atoms are magnetic. Several calculations have investigated the magnetism ofthose clusters assuming model structures. In particular, trends across the 4dperiod of the periodic table have been studied by performing DFT calculationsfor six-atom clusters with octahedral structure,107 and the magnetic momentsare given in Table 2. All clusters, except Y6, Pd6, and Cd6, have finite magneticmoments and the largest moments occur for Ru6 and Rh6 (1.00 mB/atom and0.99 mB/atom, respectively). The large moments of these two clusters arisefrom the fact that the density of electronic states shows a large peak in theregion of the Fermi level. Just a small exchange splitting (the shift between "and # spin sub-bands) produces a sizable difference between the populations ofelectrons with " and # spins. Ru6, Rh6, and Nb6 have the largest exchangesplittings. The Fermi levels of the bulk metals lie in a dip of the DOS in con-trast to small clusters. The main contribution to the DOS comes from the delectrons, which gives support to models in which the effect of the sp electronshas been neglected. Two factors contribute to the large DOS near the Fermi


energy. First, the bandwidth in the cluster is narrower than in the solid,because of the reduced atomic coordination. Second, high symmetry isassumed in the calculation. The latter effect suggests that some magneticmoments of Table 2 may be overestimated.

Rhodium Clusters

Experiments on Rh clusters5,106 reveal an oscillatory pattern of the aver-age magnetic moment, with large values for N ¼ 15, 16, and 19, and localminima for N ¼ 13–14, 17–18, and 20. DFT calculations have been per-formed for selected clusters in that size range, usually assuming symmetricstructures except for the smallest clusters.108–112 The conclusion reached bythe various researchers is that the Rh clusters are magnetic. However, differentexperiments for the same cluster size show a lot of dispersion.

The self-consistent TB method has been employed to study several Rhclusters in the size range N ¼ 9–55 atoms.113 Only the d electrons were takeninto account and model structures, which were restricted to be fcc, bcc, oricosahedral, were assumed, although relaxation of bond lengths that preservethe cluster symmetry was allowed. Bond length contractions of 2% to 9%with respect to the bulk were found, and these affect the magnetic moments.The magnetic moments oscillate and tend to decrease with increasing N, andthe structures predicted as being most stable by the TB calculation lead to con-sistent agreement with the measured magnetic moments. The largest cohesiveenergy of Rh9 (2.38 eV/atom) was found for a twisted double-square, cappedin the form of a pyramid. This Rh9 structure has a magnetic moment of�m ¼ 0:66mB, in good agreement with the measured value of �m ¼ 0:8� 0:2 mB.The icosahedral and the fcc structures are degenerate for Rh11, although onlythe magnetic moment of the icosahedral isomer ð�m ¼ 0:73mBÞ isconsistent with the experiment ð�m ¼ 0:8� 0:2mBÞ. The most stable structure

Table 2. Binding Energy Per Atom Eb, Distance D from Atoms tothe Cluster Center, and Average Magnetic Moment Per Atom �m forOctahedral Six-Atom Clusters. Data Collected from Zhang et al.107

Cluster Eb (eV) D (a.u.) �mðmBÞY 3.53 4.40 0.00Zr 5.23 3.96 0.33Nb 5.07 3.64 0.67Mo 4.05 3.40 0.33Tc 4.91 3.36 0.33Ru 4.70 3.40 1.00Rh 4.03 3.48 0.99Pd 3.14 3.50 0.00Ag 1.56 3.76 0.33Cd 0.39 4.48 0.00

Clusters of the 4d Elements 235

of Rh13 is bcc with �m ¼ 0:62mB, in better agreement with experimentð�m ¼ 0:48� 0:13mBÞ than the other structures considered. Fcc structures arepredicted in the size range 15 N 43, and the observed trends in themagnetic moments are reproduced, i.e., local minima of �m at N ¼ 13 andN ¼ 17, and local maxima at N ¼ 15 and N ¼ 19. The magnetic moments,however, are larger than the experimentally measured values. Other structuresfail to reproduce those oscillations, which further suggests that the geometricalstructure in the size range from 15 to 20 atoms may be fcc. Rh55 is icosahedral,and its nonmagnetic character is also consistent with the experiment. Regardingthe distribution of the magnetic moments, the bcc isomers orderferromagnetically and the atomic moments tend to increase when going fromthe cluster center to the surface atoms. On the other hand, the distribution infcc and icosahedral structures is more complex and the magnetic order issometimes antiferromagnetic, with the local moments changing sign betweenadjacent shells. A similar behavior has been predicted for Rh fcc surfaces andfilms.114,115 The effect of the sp electrons was analyzed for Rh13: The localmoments show some sensitivity to sp–d hybridization, but the total magneticmoment of the cluster is not altered. In another TB calculation116 for Rh13,Rh19, Rh43, Rh55, and Rh79 with fcc structures, ferromagnetic ordering wasfound for Rh13, Rh19, and Rh43, and antiferromagnetic configurations forRh55 and Rh79. The magnetic moments of the two largest clusters are very closeto the experimental values, and this was interpreted as supporting fccstructures for N > 40. The magnetic-to-nonmagnetic transition was estimatedat N � 80.

Rh4 was investigated to study the relationship among magnetism, topol-ogy, and reactivity.117 Working at the GGA level of DFT, the ground statewas found to have a nonmagnetic tetrahedral structure. The cluster alsohas a magnetic isomer that is a square with a moment of 1 mB/atom, 0.60eV/atom less stable than the ground state. The difference in the magneticcharacter can be from the different atomic coordination in the isomers, threein the tetrahedron and two in the square. More insight is obtained from theanalysis of the distribution of the electronic energy levels. The square isomerof Rh4 has a larger number of states near the HOMO, and work for extendedsystems has shown that a large density of states near the Fermi energy usuallyleads to magnetic structures. By simulating the reaction of those two isomerswith molecular hydrogen, the following conclusions were obtained: (1) H2

dissociates and binds atomically to both isomers, (2) the H2 binding energyto the nonmagnetic isomer is larger by a factor of 2, and (3) the spin multi-plicities of the two isomers change upon reaction with H2. These results implythat the reactivity of transition metal clusters may depend sensitively on boththeir magnetic structure and their topology. In fact, the existence of isomershas been detected in reactivity experiments of some clusters.54,118,119 In thecurrent case, only the magnetic isomer of Rh4, with the predicted structureof a square, can be deflected in a Stern–Gerlach magnet. On the other


hand, the two reacted forms of Rh4H2 are magnetic and have different spinmultiplicities. Consequently the two reacted clusters will be deflected by dif-ferent amounts in a Stern–Gerlach field, which provides a route to test thetheoretical predictions on the relation among magnetism, topology, and reac-tivity in Rh4.

Ruthenium and Palladium Clusters

Density functional108,120 and TB calculations113,121 have been per-formed for ruthenium clusters. Antiferromagnetic ordering of the magneticmoments is preferred for most structures studied. The TB method predictslower average moments compared with DFT, which are in better agreementwith the experimental upper limits,5,106 but the sp electrons were not includedin the calculations. The magnetic-to-nonmagnetic transition is estimated tooccur around Ru19, which is in qualitative agreement with the experimentalbound of N � 13.

The experiments of Cox et al.5,106 set the upper limits of the 0.40 mB/atom for the average magnetic moment of Pd13 and 0.13 mB/atom for Pd105.DFT calculations support the existence of small magnetic moments in Pd clus-ters.122–124 Calculations by Moseler et al.124 for neutral clusters with N 7and N ¼ 13 predict a monotonic decrease of �m between Pd2 ð�m ¼ 1 mBÞ andPd7 ð�m ¼ 0:3 mBÞ, and an unexpected high value of 0.62 mB for Pd13. Nega-tively charged clusters are more complex. The magnitude of �m oscillates andis relatively large for N ¼ 5, 7, and 13 ( �m ¼ 0:6; 0:7, and 0.54 mB, respec-tively). The total magnetic moment arises from sizable local atomic momentsof magnitude 0.3–0.6 mB. These moments couple antiferromagnetically insome cases and align ferromagnetically in other cases.

EFFECT OF ADSORBED MOLECULES

The electronic structure of a cluster is perturbed by the presence of mole-cules adsorbed on the cluster surface. A striking example is the quenching ofthe magnetic moments of Ni clusters caused by the adsorption of CO.125 Mag-netic deflection experiments for NiNCO clusters with N ¼ 8–18 reveal that thepresence of just a single CO molecule reduces the magnetic moment of most ofthose clusters.126 The quenching effect is particularly large for Ni8, Ni9, Ni15,and Ni18. For instance, the total magnetic moment of Ni8 is reduced by � 5 mB,that is, 0.63 mB per atom. Nickel cluster carbonyl complexes like[Ni9(CO)18]2� display vanishing magnetic susceptibilities, revealing Nimoments of 0 mB. Calculations for [Ni6(CO)12]2�, [Ni32(CO)32]n�,[Ni44(CO)48]n�, and other complexes predict low spin structures, which isconsistent with the very low magnetic susceptibilities measured for macro-scopic samples of these compounds.125,127 The proposed explanation is that

Effect of Adsorbed Molecules 237

ligands with s lone pairs, like CO, interact repulsively with the diffuse 4spelectrons of the Ni atoms, inducing an electronic transition of the type3d94s1 ! 3d10 that causes the filling of the atomic 3d shell. The calculationsshow that this repulsive destabilization occurs even when the Ni cluster is cov-ered by a shell of inert He atoms.66

DFT studies of the adsorption of NH3 on NiN clusters with N ¼ 1–4 alsoindicate that the adsorbed molecules have a significant effect on the magnet-ism: A decrease of the Ni moments is predicted, which are completelyquenched when the number of NH3 molecules equals the number of Niatoms.128 The nitrogen atom binds directly to a Ni atom, and the quenchingof the magnetic moment of Ni is from the short distance between the Ni and Natoms in the Ni–N bond. When the number of molecules is larger than thenumber of Ni atoms, the Ni–N bonds become stretched because of steric hin-drance. Once Ni–N distances exceed the critical distance of 3.59 a.u., magnet-ism reappears.

Adsorbed species can also increase the magnetic moments offerromagnetic clusters. The magnetic moments of free and hydrogenatediron clusters measured by Knickelbein129 are shown in Figure 14. The Feclusters become saturated with a layer of dissociatively chemisorbed hydro-gen under the conditions of the experiment. For most cluster sizes studied,the FeNHm clusters have larger magnetic moments than the correspondingpure FeN clusters, and the enhancement is particularly large betweenN ¼ 13 and N ¼ 18. This result contrasts with analogous studies for Ni clus-ters; in this case, quenching of the magnetic moments is observed after hydro-genation.129

Figure 14 Measured magnetic moments of FeN (circles) and FeNHm (squares). Adaptedwith permission from Ref. 129.


DETERMINATION OF MAGNETIC MOMENTS BYCOMBINING THEORY AND PHOTODETACHMENTSPECTROSCOPY

The measurement of the magnetic moment of very small clusters byStern–Gerlach deflection techniques is not simple. In such cases, the totalmagnetic moment is also small and the deflection in the magnetic field maylie within the error of the experiment. Motivated by this difficulty, an alterna-tive method to determine the magnetic moments has been proposed by Khan-na and Jena,130 based on combining calculations for the neutral and negativelycharged (anionic) species, XN and X�N, respectively, with electron photode-tachment spectroscopy experiments for the anionic cluster. Let us consider aferromagnetic anionic cluster that has n unpaired spins, and, thus, a magneticmoment nmB and multiplicity M ¼ nþ 1. When an electron is detached fromthe anion, the neutral cluster has a multiplicity of Mþ 1 if the electron isremoved from the minority band, or M� 1 if the electron is removed fromthe majority band. The measured photoelectron energy peaks can be com-pared with theoretical calculations where one first determines the ground stateof the anion, including its spin multiplicity M, and the energy for the transitionto the neutral species with multiplicities Mþ 1 and M� 1 at the anion geome-try. Quantitative agreement between the calculated energies and the observedspectral peaks indicates that the calculated multiplicity must be correct.

The Khanna–Jena method has been applied to Ni5.130 The photoelectronspectrum of Ni�5 , measured by Wang and Wu,131 shows a prominent and broadpeak at 10.80 eV and a minor peak at 2.11 eV. A careful investigation was per-formed using DFT with the GGA for exchange and correlation of the equili-brium structures of anionic Ni�5 corresponding to spin multiplicities M ¼ 2, 4,6, 8, and 10, and of neutral Ni5 with spin multiplicities M ¼ 1, 3, 5, 7, and 9.The ground state structure of the neutral cluster is a square pyramid with spinmultiplicity M ¼ 7 (total magnetic moment of 6 mB). This state is almost degen-erate, with an isomer having the structure of a distorted trigonal bipyramid andM ¼ 5ðm ¼ 4mBÞ. In the case of Ni�5 , the structure, for all of the spin muliplici-ties studied, is a slightly distorted square pyramid. The ground state has M ¼ 8,and this can only arise by adding an electron to the majority-spin band of neutralNi5 with M ¼ 7 (which is precisely the ground state of Ni5). The structure ofNi�5 with M ¼ 6 has an energy only 0.05 eV above the ground state, so bothisomers with M = 6 and 8 are expected to exist in the beam. The calculatedvertical transition energies from the anionic to the neutral cluster are plottedin Figure 15. The transitions from the ground state of the anionic cluster(with M ¼ 8) to states of the neutral cluster with the anion geometry andM ¼ 7 and 9 (the transition energies are obtained as a difference of the totalenergies of the corresponding clusters) are shown on the left side of Figure 15.These transitions yield energies of 1.64 eV and 2.21 eV. On the other hand, thetransitions from the M ¼ 6 state of Ni�5 yield energies of 1.58 eV and 1.79 eV. It

Determination of Magnetic Moments by Combining Theory 239

is plausible that the broad peak reported in the experiments originates fromtransitions from both isomers of Ni�5 , whereas the peak at 2.11 eV can only arisefrom the state of Ni�5 with M ¼ 8.

SUMMARY AND PROSPECTS

The magnetic properties of small clusters of the transition elements areoften different from those of the same material in the macroscopic bulk. Thisdifference is because magnetism is very sensitive to the density of electronicstates in the energy region around the Fermi level of the system, and the den-sity of states in a cluster is strongly affected by the confinement of the electronsin a small volume. The atoms forming the cluster surface have a different localenvironment compared with the bulk-like atoms and thus a different local den-sity of states. In addition, the geometrical structure of small clusters changes asthe size of the cluster increases. These effects lead to a complex and nonmo-notonic variation of the ordering of the atomic magnetic moments as the clus-ter size increases. The magnetic moment per atom �m of small magnetic clustersis higher than the magnetic moment per atom in the bulk metal. �m decreases asthe cluster size increases but not in a smooth way. Instead, �m dislays oscilla-tions superimposed to that overall decrease, before converging to the valuefor the bulk metal. Even more, nonzero magnetic moments have been mea-sured in clusters of some metals that are nonmagnetic in the bulk phase.

Many experimental studies of the magnetism in transition metal clustersuse the method of Stern–Gerlach deflection of a cluster beam in an inhomoge-neous magnetic field. Two computational methods have been mainly used tohelp in the interpretation of the experimental results. One is the tight bindingmethod, and the other is the density functional theory in its spin polarized ver-sion. Both methods are reviewed in this chapter, and their performance isillustrated by showing several applications to the study of the magnetic proper-ties of clusters of the 3d and 4d elements of the periodic table. In general, the two

Figure 15 Transitions from the Ni�5 anionic isomers with spin multiplicity M to thecorresponding neutrals with multiplicities differing by�1 from the anion. Adapted withpermission from Ref. 130.


methods are successful in the description of the magnetic ordering of transitionmetal clusters. However, both methods make approximations in the treatmentof the electronic correlations, and because of those approximations, there areconflicting cases that resist a conclusive analysis; the magnetic ordering inMn2 is a good example.

The well-known ferromagnetic and antiferromagnetic orderings typicalof many materials in the bulk phase become more complex in clusters. Forinstance, for a material with a tendency to antiferromagnetic ordering of theatomic spins, a perfect antiferromagnetic configuration is not possible in a tri-mer with the geometry of a triangle, because two magnetic moments have topoint necessarily in the same direction. This simple example of magnetic frus-tration is induced by the finite number of atoms of the system. This type offrustration occurs in many clusters with a tendency to antiferromagneticordering. Sometimes the magnetic frustrations cost a sizable amount of energyand the magnetic moments reorder by pointing toward different directions inspace in order to reduce the cluster energy; this is called a noncollinear mag-netic configuration. Current improvements in the theoretical tools allows oneto study noncollinear magnetic ordering in clusters, and this is one of therecent trends in the literature. As a consequence of those improved studies,it is expected that the results of some previous calculations and the interpreta-tion of some experiments will have to be revised in light of possible noncol-linear magnetic arrangements. Many experiments, so far, have beeninterpreted by taking into account the spin magnetism only. However, recentwork has pointed out the importance of orbital magnetism and of the spin-orbit coupling. A good example is the deep insight on the evolution of themagnetic moment of nickel clusters as a function of the cluster size obtainedby taking into account the effects of orbital magnetism.48 However, the gen-eral relevance of this effect is not yet assessed and more work is required.

To summarize, one can note that the magnetic characteristics of smallclusters of the transition metals vary in a nonmonotonous way as a functionof the number of atoms in the cluster. This nonscalable behavior is whatmakes small clusters interesting and complex at the same time, offering possi-bilities for future technological applications.

APPENDIX. CALCULATION OF THE DENSITYOF ELECTRONIC STATES WITHIN THE TIGHTBINDING THEORY BY THE METHOD OFMOMENTS

Let H be the Hamiltonian for an electron interacting through a potentialVð~r� ~RiÞ with the N atoms of the cluster placed at the sites ~Ri:

H ¼ T þX

i

Vð~r� ~RiÞ ¼ T þ Vi ½A:1�

Appendix. Calculation of the Density of Electronic States 241

The density of electronic states can be written

DðeÞ ¼ Trdðe�HÞ ½A:2�

where Tr indicates the trace of the operator dðe�HÞ. The moments mðpÞ of thedensity of states are defined as

mðpÞ ¼ðepDðeÞde ¼ TrHp ½A:3�

These moments can be calculated using the tight binding approximation.Introducing a complete set of atomic orbitals jiai satisfying the equations

½T þ Við~r� ~RiÞ�fiað~r� ~RiÞ ¼ eafiað~r� ~RiÞ ½A:4�

the moments mðpÞ can be calculated by expanding the trace over this set

mðpÞ ¼X

i1a1...ipap

fi1a1H fi2a2

fi2a2H fi3a3

. . . fipapH fi1a1

E��

DE��

DE��

D½A:5�

and keeping only two-center nearest-neighbor integrals. In addition, integralshfijVjjfii will be neglected in comparison with those of type hfijVijfii. Thesum in Eq. [A.5] goes over all paths of length p that start and finish at a givenatom, such that the electron hops between nearest neighbors.

If we work with the local (and orbital-dependent) density of statesDiasðeÞ, moments mðpÞia can also be calculated

mðpÞia ¼ðepDiaðeÞde ¼

Xi2a2...ipap

Dfi1a1

��H��fi2a2

EDfi2a2

��H��fi3a3

E. . .Dfipap

��H��fi1a1

E

½A:6�

Equation [A.6] shows a simple connection between the local bonding of anatom and its electronic structure. The density of states is then calculatedfrom all the moments mðpÞ. This theory offers a promising way of calculatingthe density of states. Many numerical methods are unstable, although therecursion method of Haydock20 works efficiently. In this Green’s functionmethod, the local density of states is written in terms of the local Green’s func-tion Gia;iaðeÞ as

DiaðeÞ ¼ �1

plimZ!0

ImGia;iaðeþ iZÞ ½A:7�


ACKNOWLEDGMENTS

This work was supported by MEC (Grant MAT2005-06544-C03-01) and Junta de Castillay Leon (Grant VA039A05). I acknowledge the hospitality and support of DIPC during the summerof 2006.

REFERENCES

1. I. M. L. Billas, A. Chatelain, and W. D. de Heer, Science, 265, 1682 (1994). Magnetism of Fe,Co and Ni Clusters in Molecular Beams.

2. I. M. L. Billas, A. Chatelain, and W. D. de Heer, J. Magn. Mag. Mater., 168, 64 (1997).Magnetism from the Atom to the Bulk in Iron, Cobalt and Nickel Clusters.

3. S. E. Apsel, J. W. Emmert, J. Deng, and L. A. Bloomfield, Phys. Rev. Lett., 76, 1441 (1996).Surface-Enhanced Magnetism in Nickel Clusters.

4. M. B. Knickelbein, J. Chem. Phys., 116, 9703 (2002). Nickel Clusters: The Influence ofAdsorbates on Magnetic Moments.

5. A. J. Cox, J. G. Lourderback, S. E. Apsel, and L. A. Bloomfield, Phys. Rev. B, 49, 12295(1994). Magnetism in 4d-Transition Metal Clusters.

6. J. Zhao, X. Chen, Q. Sun, F. Liu, and G. Wang, Phys. Lett. A, 205, 308 (1995). A Simpled-Band Model for the Magnetic Property of Ferromagnetic Transition Metal Clusters.

7. J. Friedel, in The Physics of Metals, J. M. Ziman, Ed., Cambridge University Press, Cambridge,United Kingdom, 1969, pp. 340-408, Transition Metals. Electronic Structure of the d-Band.Its Role in the Crystalline and Magnetic Structures.

8. G. Pastor, J. Dorantes-Davila, and K. H. Bennemann, Chem. Phys. Lett., 148, 459 (1988). ATheory for the Size and Structural Dependendence of the Ionization and Cohesive Energy ofTranstion Metal Clusters.

9. F. Aguilera-Granja, J. M. Montejano-Carrizales, and J. L. Moran-Lopez, Solid State Com-mun., 107, 25 (1998). Geometrical Structure and Magnetism of Nickel Clusters.

10. P. Jensen and K. H. Bennemann, Z. Phys. D, 35, 273 (1995). Theory for the Atomic ShellStructure of the Cluster Magnetic Moment and Magnetoresistance of a Cluster Ensemble.

11. J. Callaway, Energy Band Theory, Academic Press, London, 1964.

12. J. C. Slater and G. F. Koster, Phys. Rev., 94, 1498 (1954). Simplified LCAO Method for thePeriodic Potential Problem.

13. P. Lowdin, J. Chem. Phys., 18, 365 (1950). On the Non-Orthogonality Problem Connectedwith the use of Atomic Wave Functions in the Theory of Molecules and Crystals.

14. A. L. Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems, McGraw Hill,New York, 1971.

15. A. Vega, J. Dorantes-Davila, L.C. Balbas, and G. M. Pastor, Phys. Rev. B, 47, 4742 (1993).Calculated sp-Electron and spd-Hybridization Effects on the Magnetic Properties of SmallFeN Clusters.

16. V. Heine, Phys. Rev., 153, 673 (1967). s-d Interaction in Transition Metals.

17. G. M. Pastor, J. Dorantes-Davila, and K. H. Bennemann, Physica B, 149, 22 (1988). TheMagnetic Properties of Small Fen Clusters.

18. G. M. Pastor, J. Dorantes-Davila, and K. H. Bennemann, Phys. Rev. B, 40, 7642 (1989).Size and Structural Dependence of the Magnetic Properties of 3d Transition MetalClusters.

19. W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco,California, 1980.

References 243

20. R. Haydock, in Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, Eds., AcademicPress, New York, 35 (1980), pp. 215–294. The Recursive Solution of the SchrodingerEquation.

21. J. Guevara, F. Parisi, A. M. Llois, and M. Weissmann, Phys. Rev. B, 55, 13283 (1997).Electronic Properties of Transition Metal Clusters: Consideration of the Spillover in a BulkParametrization.

22. P. Hohenberg and W. Kohn, Phys. Rev., 136, B864 (1964). Inhomogeneous Electron Gas.

23. W. Kohn and L. J. Sham, Phys. Rev., 140, A1133 (1965). Self-Consistent Equations IncludingExchange and Correlation Effects.

24. S. Lundqvist and N. H. March, Eds., Theory of the Inhomogeneous Electron Gas, PlenumPress, New York, 1986.

25. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett., 45, 566 (1980). Ground State of the ElectronGas by a Stochastic Method.

26. J. P. Perdew and A. Zunger, Phys. Rev. B, 23, 5048 (1981). Self-Interaction Correction toDensity Functional Approximations for Many Electron Systems.

27. A. D. Becke, Phys. Rev. A, 38, 3098 (1988). Density Functional Exchange Energy Approx-imation with Correct Asymptotic Behavior.

28. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett., 77, 3865 (1996). GeneralizedGradient Approximation Made Simple.

29. J. P. Perdew and S. Kurth, in A Primer in Density Functional Theory, Vol. 620,C. Fiolhais, F. Nogueira and M. Marques, Eds., Lecture Notes in Physics, Springer,Berlin, 2003, pp. 1–55. Density Functionals for Non-Relativistic Coulomb Systems in theNew Century.

30. U. von Barth and L. Hedin, J. Phys. C.: Solid State Phys., 5, 1629 (1972). A Local Exchange-Correlation Potential for the Spin Polarized Case: I.

31. J. K. Kubler, Theory of Itinerant Electron Magnetism, Oxford University Press, Oxford,United Kingdom, 2000.

32. D. J. Singh and D. A. Papaconstantopoulos, Eds., Electronic Structure and Magnetism ofComplex Materials, Springer, Berlin, 2003.

33. S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys., 58, 1200 (1980). Accurate Spin-dependentElectron Liquid Correlation Energies for Local Spin Density Calculations: A CriticalAnalysis.

34. Y. Tsunoda, J. Phys.: Cond. Matter, 1, 10427 (1989). Spin-density Wave in Cubic g-Fe andg-Fe100-x Cox Precipitates in Cu.

35. R. Lorenz, J. Hafner, S. S. Jaswal, and D. J. Sellmyer, Phys. Rev. Lett., 74, 3688 (1995).Disorder and Non-Collinear Magnetism in Permanent-Magnet Materials with ThMn12

Structure.

36. M. Liebs, K. Hummler, and M. Fahnle, Phys. Rev. B, 51, 8664 (1995). Influence of StructuralDisorder on Magnetic Order: An Ab Initio Study of Amorphous Fe, Co, and Ni.

37. O. N. Mryasov, A. I. Liechtenstein, L. M. Sandratskii, and V. A. Gubanov, J. Phys.: Cond.Matter, 3, 7683 (1991). Magnetic Structure of fcc Iron.

38. V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde, and B. N. Harmon, Phys. Rev. Lett.,75, 729 (1995). Ab Initio Spin Dynamics in Magnets.

39. M. Uhl and J. Kubler, Phys. Rev. Lett., 77, 334 (1996). Exchanged-Coupled Spin-FluctuationTheory. Application to Fe, Co, and Ni.

40. T. Oda, A. Pasquarello, and R. Car, Phys. Rev. Lett., 80, 3622 (1998). Fully UnconstrainedApproach to Non-Collinear Magnetism: Application to Small Fe Clusters.

41. C. Kohl and G. F. Bertsch, Phys. Rev. B, 60, 4205 (1999). Non-Collinear Magnetic Orderingin Small Chromium Clusters.

42. J. Kubler, K. H. Hock, J. Sticht, and A. R. Williams, J. Phys. F, 18, 469 (1988). DensityFunctional Theory of Non-Collinear Magnetism.


43. K. Capelle and E. K. U. Gross, Phys. Rev. Lett., 78, 1872 (1997). Spin-Density Functionalsfrom Current-Density Functional Theory and Vice Versa: A Road Towards NewApproximations.

44. J. A. Alonso, Structure and Properties of Atomic Nanoclusters, Imperial College Press,London, 2005.

45. S. Bouarab, A. Vega, M. J. Lopez, M. P. Iniguez, and J. A. Alonso, Phys. Rev. B, 55, 13279(1997). Geometrical Effects on the Magnetism of Small Ni Clusters.

46. F. Aguilera-Granja, S. Bouarab, M. J. Lopez, A. Vega, J. M. Montejano-Carrizales, M. P.Iniguez, and J. A. Alonso, Phys. Rev. B, 57, 12469 (1998). Magnetic Moments of Ni Clusters.

47. J. A. Alonso, Chem. Rev., 100, 637 (2000). Electronic and Atomic Structure and Magnetismof Transition-Metal Clusters.

48. X. Wan, L. Zhou, J. Dong, T. K. Lee, and D. Wang, Phys. Rev. B, 69, 174414 (2004). OrbitalPolarization, Surface Enhancement and Quantum Confinement in Nanocluster Magnetism.

49. F. Ducastelle, J. Phys. (Paris), 31, 1055 (1970). Elastic Moduli of Transition Metals.

50. R. P. Gupta, Phys. Rev. B, 23, 6265 (1981). Lattice Relaxations at a Metal Surface.

51. J. M. Montejano-Carrizales, M. P. Iniguez, J. A. Alonso, and M. J. Lopez, Phys. Rev. B, 54,5961 (1996). Theoretical Study of Icosahedral Ni Clusters within the Embedded AtomMethod.

52. E. K. Parks, G. C. Nieman, K. P. Kerns, and S. J. Riley, J. Chem. Phys., 107, 1861 (1997).Reactions of Ni38 with N2, H2 and CO: Cluster Structure and Adsorbate Binding Sites.

53. N. N. Lathiotakis, A. N. Andriotis, M. Menon, and J. Connolly, J. Chem. Phys., 104, 992(1996). Tight Binding Molecular Dynamics Study of Ni Clusters.

54. E. K. Parks, L. Zhu, J. Ho, and S. J. Riley, J. Chem. Phys., 102, 7377 (1995). The Structure ofSmall Nickel Clusters. II. Ni16 – Ni28.

55. J. L. Rodrıguez-Lopez, F. Aguilera-Granja, A. Vega, and J. A. Alonso, Eur. Phys. J. D, 6, 235(1999). Magnetic Moments of NiN Clusters (N 34): Relation to Atomic Structure.

56. N. Fujima and T. Yamaguchi, Phys. Rev. B, 54, 26 (1996). Magnetic Moment in NickelClusters Estimated by an Electronic Shell Model.

57. W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou, and M. L. Cohen,Phys. Rev. Lett., 52, 2141 (1984). Electronic Shell Structure and Abundances of SodiumClusters.

58. W. Ekardt, Phys. Rev. B, 29, 1558 (1984). Work Function of Small Metal Particles: Self-Consistent Spherical Jellium-Background Model.

59. I. Katakuse, Y. Ichihara, Y. Fujita, T. Matsuo, T. Sakurai, and H. Matsuda, Int. J. MassSpectrom. Ion Proc., 74, 33 (1986). Mass Distributions of Negative Cluster Ions of Copper,Silver and Gold.

60. N. Fujima and T. Yamaguchi, J. Phys. Soc. Japan, 58, 3290 (1989). Magnetic Anomaly andShell Structure of Electronic States of Nickel Microclusters.

61. F. A. Reuse and S. N. Khanna, Chem. Phys. Lett., 234, 77 (1995). Geometry, ElectronicStructure, and Magnetism of Small Nin ( n ¼ 2–6, 8, 13) Clusters.

62. N. Desmarais, C. Jamorski, F. A. Reuse, and S. N. Khanna, Chem. Phys. Lett., 294, 480(1998). Atomic Arrangements in Ni7 and Ni8 Clusters.

63. B. M. Reddy, S. K. Nayak, S. N. Khanna, B. K. Rao, and P. Jena, J. Phys. Chem. A, 102, 1748(1998). Physics of Nickel Clusters. 2. Electronic Structure and Magnetic Properties.

64. M. W. Finnis and J. E. Sinclair, Phil. Mag., 50, 45 (1984). A Simple Empirical N-BodyPotential for Transition Metals.

65. N. Fujima and T. Yamaguchi, Mater. Sci. Eng. A, 217, 295 (1996). Geometrical MagneticStructures of Transition-Metal Clusters.

66. G. Pacchioni, S. C. Chung, S. Kruger, and N. Rosch, Chem. Phys., 184, 125 (1994). On theEvolution of Cluster to Bulk Properties: A Theoretical LCGTO-DFT Study of Free andCoordinated Nin Clusters (n¼6–147).

References 245

67. L. Zhou, D. S. Wang, and Y. Kawazoe, Phys. Rev. B, 60, 9545 (1999). Orbital Correlationand Magnetocrystalline Anisotropy in One-Dimensional Transition Metal Systems.

68. M. Komelj, C. Ederer, J. W. Davenport, and M. Fahnle, Phys. Rev. B, 66, 140407 (2002).From Bulk to Monatomic Wires: An Ab Initio Study of Magnetism in Co Systems withVarious Dimensionality.

69. R. A. Guirado-Lopez, J. Dorantes-Davila, and G. M. Pastor, Phys. Rev. Lett., 90, 226402(2003). Orbital Magnetism in Transition Metal Clusters: From Hund’s Rules to BulkQuenching.

70. A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B, 52, R5467 (1995). DensityFunctional Theory and Strong Interactions: Orbital Ordering in Mott-Hubbard Insulators.

71. B. R. Judd, Operator Techniques in Atomic Spectroscopy, McGraw-Hill, New York, 1963.

72. F. A. Reuse, S. N. Khanna, and S. Bernel, Phys. Rev. B, 52, R11650 (1995). ElectronicStructure and Magnetic Behavior of Ni13 Clusters.

73. B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett., 68, 1943 (1992). X-RayCircular Dichroism as a Probe of Orbital Magnetization.

74. A. N Andriotis and M. Menon, Phys. Rev. Lett., 93, 026402 (2004). Orbital Magnetism: Prosand Cons for Enhancing the Cluster Magnetism.

75. H. Cheng and L. S. Wang, Phys. Rev. Lett., 77, 51 (1996). Dime Growth, StructuralTransition, and Antiferromagnetic Ordering of Small Chromium Clusters.

76. L. S. Wang, H. Wu, and H. Cheng, Phys. Rev. B, 55, 12884 (1997). Photoelectron Spectro-scopy of Small Chromium Clusters: Observation of Even-Odd Alternations and TheoreticalInterpretation.

77. D. C. Douglass, J. P. Bucher, and L. A. Bloomfield, Phys. Rev. B, 45, 6341 (1992). MagneticStudies of Free Nonferromagnetic Clusters.

78. L. A. Bloomfield, J. Deng, H. Zhang, and J. W. Emmert, in Proceedings of the InternationalSymposium on Cluster and Nanostructure Interfaces. P. Jena, S. N. Khanna, and B. K. Rao,Eds., World Scientific, Singapore, 2000, pp. 131–138. Magnetism and Magnetic Isomers inChromium Clusters.

79. N. Fujima, Eur. Phys. J. D, 16, 185 (2001). Non-Collinear Magnetic Moments of Seven-AtomCr, Mn and Fe Clusters.

80. N. Fujima, J. Phys. Soc. Japan, 71, 1529 (2002). Non-Collinear Magnetic Moments of Five-Atom Transition Metal Clusters

81. S. Blugel, B. Drittler, R. Zeller, and P. H. Dederichs, Appl. Phys. A, 49, 547 (1989). MagneticProperties of 3d Transition Metal Monolayers on Metal Substrates.

82. V. S. Stepanyuk, W. Hergert, K. Wildberger, S. K. Nayak, and P. Jena, Surf. Sci. Lett., 384,L892 (1997). Magnetic Bistability of Supported Mn Clusters.

83. J. R. Lombardi and B. Davis, Chem. Rev. 102, 2431 (2002). Periodic Properties of ForceConstants of Small Transition Metal and Lanthanide Clusters.

84. E. K. Parks, G. C. Nieman, and S. J. Riley, J. Chem. Phys., 104, 3531 (1996). The Reaction ofManganese Clusters and Manganese Cluster Carbides with Hydrogen. The Mn-CH3 BondEnergy.

85. R. J. Van Zee and W. Weltner, J. Chem. Phys., 89, 4444 (1988). The Ferromagnetic Mnþ2Molecule.

86. G. W. Ludwig, H. H. Woodbury, and R. O. Carlson, J. Phys. Chem. Solids, 8, 490 (1959).Spin Resonance of Deep Level Impurities in Germanium and Silicon.

87. C. A. Baumann, R. J. Van Zee, S. Bhat, and W. Weltner, J. Chem. Phys., 78, 190 (1983). ESRof Mn2 and Mn5 Molecules in Rare Gas Matrices.

88. R. K. Nesbet, Phys. Rev., 135, A460 (1964). Heisenberg Exchange Interaction of Two MnAtoms.

89. N. Fujima and T. Yamaguchi, J. Phys. Soc. Japan, 64, 1251 (1995). Chemical Bonding in MnClusters, MnN and MnN

± (N¼ 2–7).


90. S. K. Nayak and P. Jena, Chem. Phys. Lett., 289, 473 (1998). Anomalous Magnetism in SmallMn Clusters.

91. P. Schieffer, C. Krembel, M. C. Hanf, D. Bolmont, and G. Gewinner, J. Magn. Mag. Mater.,165, 180 (1997). Stabilization of a Face-Centered-Cubic Mn Structure with the Ag LatticeParameter.

92. O. Rader, W. Gudat, D. Schmitz, C. Carbone, and W. Eberhardt, Phys. Rev. B, 56, 5053(1997). Magnetic Circular X-Ray Dichroism of Submonolayer Mn on Fe(100).

93. M. R. Pederson, F. A. Reuse, and S. N. Khanna, Phys. Rev. B, 58, 5632 (1998). MagneticTransition in Mnn (n¼2–8) Clusters.

94. S. Yamamoto, H. Tatewaki, H. Moriyama, and H. Nakano, J. Chem Phys. 124, 124302(2006). A Study of the Ground State of Manganese Dimer Using Quasidegenerate Perturba-tion Theory.

95. H. Nakano, J. Chem. Phys. 99, 7983 (1993). Quasidegenerate Perturbation Theory withMulticonfigurational Self-Consistent-Field Reference Functions.

96. M. B. Knickelbein, Phys. Rev. Lett., 86, 5255 (2001). Experimental Observation of Super-Paramagnetism in Manganese Clusters.

97. J. Guevara, A. M. Llois, F. Aguilera-Granja, and J. M. Montejano-Carrizales, Phys. Stat. Sol.B, 239, 457 (2003). Magnetism of Small Mn Clusters.

98. S. K. Nayak, M. Nooijen, and P. Jena, J. Phys. Chem. A, 103, 9853 (1999). Isomerism andNovel Magnetic Order in Mn13 Cluster.

99. S. N. Khanna, B. K. Rao, P. Jena, and M. Knickelbein, Chem. Phys. Lett., 378, 374 (2003).Ferrimagnetism in Mn7 Cluster.

100. T. M. Briere, H. F. Sluiter, V. Kumar, and Y. Kawazoe, Phys. Rev. B, 66, 064412 (2002).Atomic Structures and Magnetic Behavior of Mn Clusters.

101. P. Bobadova-Parvanova, K. A. Jackson, S. Srinivas, and M. Horoi, Phys. Rev. A, 67, 061202(2003). Emergence of Antiferromagnetic Ordering in Mn Clusters.

102. N. O. Jones, S. H. Khanna, T. Baruath, and M. R. Pederson, Phys. Rev. B, 70, 045416 (2004).Classical Stern–Gerlach Profiles of Mn5 and Mn6 Clusters.

103. M. B. Knickelbein, Phys. Rev. B, 70, 014424 (2004). Magnetic Ordering in ManganeseClusters.

104. J. M. Soler, E. Artacho, J. D. Gale, A. Garcıa, J. Junquera, P. Ordejon, and D. Sanchez-Portal,J. Phys.: Cond. Matter, 14, 2745 (2002). The SIESTA Method for Ab Initio Order-NMaterials Simulation.

105. R. C. Longo, E. G. Noya, and L. J. Gallego, J. Chem. Phys., 122, 226102 (2005). Non-Collinear Magnetic Order in the Six-Atom Mn Cluster.

106. A. J. Cox, J. G. Lourderback, and L. A. Bloomfield, Phys. Rev. Lett., 71, 923 (1993).Experimental Observation of Magnetism in Rhodium Clusters.

107. G. W. Zhang, Y. P. Feng, and C. K. Ong, Phys. Rev. B, 54, 17208 (1996). Local Binding Trendand Local Electronic Structures of 4d Transition Metals.

108. R. V. Reddy, S. N. Khanna, and B. Dunlap, Phys. Rev. Lett., 70, 3323 (1993). Giant MagneticMoments in 4d Clusters.

109. B. Piveteau, M. C. Desjonqueres, A. M. Oles, and D. Spanjard, Phys. Rev. B, 53, 9251 (1996).Magnetic Properties of 4d Transition-Metal Clusters.

110. Y. Jinlong, F. Toigo, W. Kelin, and Z. Manhong, Phys. Rev. B, 50, 7173 (1994). AnomalousSymmetry Dependence of Rh13 Magnetism.

111. Y. Jinlong, F. Toigo, and W. Kelin, Phys. Rev. B, 50, 7915 (1994). Structural, Electronic, andMagnetic Properties of Small Rhodium Clusters.

112. Z. Q. Li, J. Z. Yu, K. Ohno, and Y. Kawazoe, J. Phys.: Cond. Matter, 7, 47 (1995).Calculations on the Magnetic Properties of Rhodium Clusters.

113. P. Villasenor-Gonzalez, J. Dorantes-Davila, H. Dreysse, and G. Pastor, Phys. Rev. B, 55, 15084(1997). Size and Structural Dependence of the Magnetic Properties of Rhodium Clusters.

References 247

114. A. Chouairi, H. Dreysse, H. Nait-Laziz, and C. Demangeat, Phys. Rev. B, 48, 7735 (1993). RhPolarization in Ultrathin Rh Layers on Fe(001).

115. A. Mokrani and H. Dreysse, Solid State Commun., 90, 31 (1994). Magnetism of Rh VicinalSurfaces?

116. R. Guirado-Lopez, D. Spanjaard, and M. C. Desjonqueres, Phys. Rev. B, 57, 6305 (1998).Magnetic-Nonmagnetic Transition in fcc 4d-Transition-Metal Clusters.

117. S. K. Nayak, S. E. Weber, P. Jena, K. Wildberger, R. Zeller, P. H. Dederichs, S. V. Stepanyuk,and W. Hergert, Phys. Rev. B, 56, 8849 (1997). Relationship Between Magnetism,Topology, and Reactivity of Rh Clusters.

118. E. K. Parks, K. P. Kerns, and S. J. Riley, J. Chem. Phys., 109, 10207 (1998). The Structure ofNi39.

119. M. E. Geusic, M. D. Morse, and R. E. Smalley, J. Chem. Phys., 82, 590 (1985). HydrogenChemisorption on Transition Metal Clusters.

120. D. Kaiming, Y. Jinlong, X. Chuamyun, and W. Kelin, Phys. Rev. B, 54, 2191 (1996).Electronic Properties and Magnetism of Ruthenium Clusters.

121. R. Guirado-Lopez, D. Spanjaard, M. C. Desjonqueres, and F. Aguilera-Granja, J. Magn. Mag.Mater., 186, 214 (1998). Electronic and Geometrical Effects on the Magnetism of Small RuN

Clusters.

122. K. Lee, Phys. Rev. B, 58, 2391 (1998). Possible Magnetism in Small Palladium Clusters.

123. K. Lee, Z. Phys. D, 40, 164 (1997). Possible Large Magnetic Moments in 4d Transition MetalClusters.

124. M. Moseler, H. Hakkinen, R. N. Barnett, and U. Landman, Phys. Rev. Lett., 86, 2545 (2001).Structure and Magnetism of Neutral and Anionic Palladium Clusters.

125. D. A. van Leeuwen, J. M. van Ruitenbeek, L. J. de Jongh, A. Ceriotti, G. Pacchioni, O. D.Haberlen, and N. Rosch, Phys. Rev. Lett., 73, 1432 (1994). Quenching of MagneticMoments by Ligand-Metal Interactions in Nanosized Magnetic Metal Clusters.

126. M. B. Knickelbein, J. Chem. Phys., 115, 1983 (2001). Nickel Clusters: The Influence ofAdsorbed CO on Magnetic Moments.

127. G. Pacchioni and N. Rosch, Acc. Chem. Res., 28, 390 (1995). Carbonylated Nickel Clusters:From Molecules to Metals.

128. B. Chen, A. W. Castleman, and S. N. Khanna, Chem. Phys. Lett., 304, 423 (1999). Structure,Reactivity, and Magnetism: Adsorption of NH3 Around Nin.

129. M. B. Knickelbein, Chem. Phys. Lett., 353, 221 (2002). Adsorbate-Induced Enhancement ofthe Magnetic Moments of Iron Clusters.

130. S. N. Khanna and P. Jena, Chem. Phys. Lett., 336, 467 (2001). Magnetic Moment and Photo-Detachment Spectroscopy of Ni5 Clusters.

131. L. S. Wang and H. Z. Wu, Z. Phys. Chem., 203, 45 (1998). Photoelectron Spectroscopy ofTransition Metal Clusters.


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