Pergamon Prog. Crystal Growth and Charact. Vol. 34, pp. 95-131, 1997

© 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved

0960-8974/97 $32.00

PII: S0960-8974(97)00007-7

ICOSAHEDRAL SYMMETRY IN CLUSTERS

Vijay Kumar

Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, india

ABSTRACT

First principles calculations and simulations based on interatomic potentials together with experimental studies of abundance spectrum suggest icosahedral structures to be common for some magic clusters of

diw~rse systems such as rare gases, metals, covalently bonded systems and water. Close packing models obtained from pair potentials are shown to be good representations of the structure of rare gas clusters. However, the electronic structure is found to play the important role in the atomic structure and re- lated properties of other clusters. Results of recent studies of fullerenes, their derivatives as well as some large icosahedral metal clusters containing several thousand atoms are also presented. Further results on doped icosahedral clusters are discussed which hold promise for the development of new materials and for understanding the occurrence of icosahedral order in several aluminum alloys.

KEYWORDS

Clusters; icosahedra] symmetry; fullerenes; quasicrystals; mas~ spectrum; electronic structure; total energy.

INTRODUCTION

Clusters form an important class of materials because of their technological applications such as in catalysis, photographic films, magnetic tapes, electronic devices, etc. and as models to study complex materials such as glasses, chemisorption and surface reactions , role of impurities in materials, nucle- ation and growth etc. There is also academic interest in understanding properties of aggregates as a function of their size in our quest of designing materials for specific applications. Interest in such studies has grown rapidly in the past decade because of the advances in mass spectrometry experiments and laser evaporation technique which have made the studies of mass selected clusters of a wide variety of materials possible. Also much progress has been made in theoreticM studies of the electronic structure of an aggregate and the methodologies of calculating the total energy and optimization techniques which have helped in understanding the structure, growth, abundance, bonding nature, ionization potential, polarizability and chemisorpt.ion properties etc. of several clusters (for an overview of this field see e.g.

95

96 V. Kumar

Phillips, 1986; Sugano et al., 1987; Jena et al., 1992; Kumar et al., 1993). In this review we focus our attention on the structural aspects and in particular on the icosahedral packings which have been found to occur prominently in clusters of a wide variety of materials.

Unlike crystals where translational periodicity enforces certain symmetries on the structure and inhibits icosahedral long range order, small systems have the freedom to adopt any structure that would lower their free energy. This makes the study of clusters difficult as very little experimental information is available on their structure. Most studies have been on large supported clusters to understand the nucleation and growth behaviour. These have revealed multiple twinning or icosahedral morphology of particles for some metals such as Au, Pd, and Ni (Allpress and Sanders, 1967; Gillet, 1977). Atomic calculations have been done using a Lennard- Jones (L J) type pair potential (Hoare and Pal, 1972, 1975; Hoare, 1979; Farges el al., 1983; 1986)

V(d) = e0[(a/d) -'~ - 2(~/d)-6], (1)

where d = a represents the minimum of the potential with the strength e0. Such calculations have shown icosahedral packing to be energetically most favorable for small clusters and evidence for the same has since then been obtained in free clusters of rare gases (Echt et al., 1981). Besides such clusters, the other most interesting family of clusters which has attracted great attention in recent years and in which icosahedral symmetry plays a very important role, is clusters of carbon in the size range of N>20 (Rohlfing et al., 1984; Kroto et al., 1985; Heath el al., 1985; Kumar et al., 1993). These clusters are now

referred to as fullerenes after Mr. Buckminsterfuller who designed the geodesic dombs with icosahedral symmetry. The most important member of this family is C~0 which has a truncated icosahedron struc- ture. In a subsequent major breakthrough, Kr£tschmer et al. (1990) developed a new phase of carbon, the solid C60, in which C60 clusters act as the superatoms of the structure. Doping of this new phase

with K, Rb, Cs etc. has led to a new family of relatively high temperature superconductors (Hebard et al., 1991; Rosseinsky et al., 1991; Holczer et al., 1991; Tanigaki et al., 1991). These developments have generated considerable interest in the study of other clusters which could be used to form new materials. Such studies have been recently done on doped carbon and aluminum clusters and their

results look promising.

Local arrangement of atoms which is nearly perfect icosahedral, also exists in several other crystals such as AI5 compounds, Frank-Kasper phases, All=Mo type alloys, boron and its compounds etc. (Pearson, 1973; Shoemaker and Shoemaker, 1988; Widom, 1988). Also Frank (1952) has argued that small icosa- hedral clusters should exist in supercooled liquids. In some systems local icosahedral order is so prevalent

that it gives rise to complicated structures having several hundred atoms in their unit cells (Pearson, 1973). When many of these alloys were rapidly solidified, it was shown that even long range icosahedral order could exist (Shechtman et al., 1984, Ramachandra Rao and Shastri, 1985, Chattopadhayay et al., 1985). These so called quasicrystals have orientational order but no translational periodicity. Since this discovery, many more alloys have been found to solidify in the quasicrystalline phase and it has even become possible to produce single quasicrystals, llowever, a proper understanding of the physical reasons for such an ordering is yet to be achieved. While first principles studies of large aggregates of such systems are difficult, certain clusters can be considered as building blocks (superatoms) of these complex materials (Gong and Kumar, 1994: Janot and Boissieu, 1994) and their properties studied from a local point of view. Electronic structure and total energy calculations have been performed in recent )'ears on icosahedral aluminum-transition metal clusters and these results will be discussed here.

Physical and chemical properties such as ionization potentiM, cohesive energies, fragmentation, magne- tization, chemisorption behaviour etc. of clusters are found to exhibit a large variation and in general

Icosahedral Symmetry in Clusters 97

2

r~

-I

- 2

- 1

- 5

ahedron

l c o s ~ h e d ~

I | i I, $3 l

I 20 ~ 40 50 60

Number of atoms in cluster

Figure 1: Binding energy per atom of icosahedral and fcc clusters. After Allpress and Sanders (1970).

an oscillatory behaviour as a function of their size (see e.g. de Heer et al., 1987; Kumar and Car, 1991 and references therein). Finite temperature properties (Beck and Berry, 1988; Honeycutt and Andersen, 1987) of clusters have been found to be quite different from bulk due to a large fraction of atoms lying at the surface. Fast ionic diffusion in clusters could give rise to dynamic structural transformations and some small clusters may behave like a liquid even at very low temperatures as compared to their bulk melting point, Thus treating a cluster as a liquid drop, some general features in the properties of metal clusters have been understood from electronic structure calculations on a simple spherical jellium model (SJM) in which the ionic charge is smeared out into a uniform positive background (de Heer et al., 1987). However, for several systems of interest such as transition metals, semiconductors and multicornponent systems, a jellium model is not quite appropriate and atomic calculations are needed to understand their structure and physico-chemical properties. Similar studies arc'. also required e.g. in understanding catalytic properties as the reactions could be site specific-. Recent progress in the total energy calculations within the density functional theory has made such studies on cluste.rs having upto a few tens of atoms possible and some of these on icosahedral clusters would be discussed here.

In the following sections we first discuss some of the early work on the structure of clusters. Selected experimental and theoretical results are then presented on clusl,ers of different materials which are found to have icosahedral packing.

EARLY WORK ON STRUCTURE OP SMALL AGGREGATES

van Harteveld and Hartog (1969) and Romanowski (1969) studied structure of clusters by considering different polyhedral aggregates obtained from common lattices. The stability of these aggregates was studied on the basis of binding energies obtained by simply counting the number of nearest neighbours. Such a procedure led to nearly spherical clusters to be of lowest energy. Subsequently spherical fcc clus- ters were used for rare gases and metals. A]lpress and Sanders (1970) optimized interplaner distances in fcc clusters and obtained an expansion of the lattice constant at the surfa.ce. More detailed recent calculations (Delley el al., 1983) on met.M clusters, however, in generM predict a coutraction of the hond lengths as compared 1o bulk. An important result of the studies of Allpress and Sanders (1970) was that

98 V. Kumar

a b c d

Figure 2: Transformation of a cuboctahedron into an icosahedron. Vertices of a cuboctahedron (a) lie

on the corners of three mutually perpendicular squares (b). By small deformations the three squares

can be converted into golden rectangles (c) which give rise to a regular icosahedron (d). Center to

vertex and surface bonds are equal in (a) whereas the surface bonds in (d) are about 5% longer than

the center to vertex bonds.

icosahedral clusters were found to be lower in energy than the ['cc clusters (Fig. 1). These calculations

showed particular stability for 13- and 55- atom icosahedral clusters. Hoare and Pal (1972) have done

extensive calculations on LJ clusters and found ieosahedral structures to be lowest in energy. These

studies showed that the use of the crystalline fragments for representing cluster structure might not be

appropriate and gave rise to the study of non-crystalline structures for clusters. Figure 2 shows how

an icosahedron of 13 atoms could be obtained from a cuboctahedron by small distortions of the three

square planes along the three coordinate axes such that. these become golden rectangles (the two sides

are in the ratio of the golden mean, r = ( v ~ + 1)/2) (Samson, 1968). Though in these two packings

the center atom has the same coordination, a cuboctahedron has equilateral triangles and squares a,s

its faces whereas in an icosahedron all the faces are equilateral triangles which represent the densest

packing. But, if we consider atoms to be hard spheres, then in an icosahedron atoms on the surface

do not touch each other because the distance from the center to a vertex is about 5% smaller than the

distance between two vertices. This is the reason why icosahedral packing can not be continued for

very large clusters having only one type of atoms, ttowever, it is still unknown when an icosahedral

packing would transform into a crystalline structure. As we shall discuss later, there are experimental

evidences that icosahedral packing may continue for clusters having several lhousand atoms. Further,

it has been found that the icosahedral frustration could bc eliminated if more than one type of atoms

are used as it is the case of quasicrystals.

One way to look at the growth of icosahedral clusters is to consider an icosahedron to be a particle with

20 multiply twinned slightly distorted letrahedra. Regular tetrahedra represent the closest packing of

identical spheres. Howew.'r, no lattice can be generated from packing of regular tetrahedra alone. In

the case of some multicomponent systems such as Frank-Kasper phases (Pearson, 1973), packing of

atoms can be described in terms of slightly distorted tetrahedra. Also packing in metallic glasses and

undercooled liquids has been considered in terms of space filling models of distorted tetrahedra (see

Nelson and Spaepen, 1989). For clusters, packings of tetrahedra have been considered by Hoare and Pal

(1972, 197.5). Extensive studies of the growth models for clusters (N _< 65) with a LJ potential showed

icosahedral structures to be of lowest energy. A 55- atom cluster was shown to have full icosahedral

symmetry (nee below) wheree~s for other clusters, the core of the clusters was found to have icosahedrM

type packing. These non-crystalline structures were referred I.o as polytetrahedrM packings and are

good models for the structure of rare gas clusters and alike. These models have been refined further

Icosahedral Symmetry in Clusters 99

Figure 3: Mackay icosahedra with 13, 55, 147, 309 and 561 atoms.

and will be discussed in the following sections.

The most important family of structures that find support from both experiments as well ms model

calculations, are the Mackay icosahedra (MI). In 1962 Mackay developed dense non-crystallographic

packings of equal spheres. In these packings an icosahedron of 12 atoms around a central atom is

surrounded by another icosahedral shell of 42 atoms and this procedure can be continued to generate

bigger ieosahedra of 147, 309, 561, ... atoms. In general the number of atoms in a Mackay icosahedron

with i shells is given by N ( i ) = 1 + ~ : ( 1 0 s ~ + 2). As shown in Fig. 3, atoms on all the faces of Mackay

icosahedra are on a fragment of a triangular lattice. The density in such an icosahedral packing of

spheres reaches a limiting value of 0.68818 as compared to 0.68017 in bcc and 0.74048 in fcc structures.

Therefore, in the limit of a large aggregate, the icosahedral packing of identical spheres is not the

densest. But for small clusters, most of the atoms lie on the surface and therefore the surface energy

contribution plays a very important role in the determination of the ground state structure. In general one <:an write the total energy of a cluster as,

Ed~,st = Neb,~lk + E~,~. s . . . . (2)

where eb~tk is the energy per atom in the bulk of that material. Esurl~c~ is the deviation of the total

energy from the bulk value and in the limit of a very large aggregate, it would approach to the value of

the surface energy of that material. For small clusters Es,~r]~c ~ dominates and therefore a cluster recon-

structs itself to lower the surface energy, whereas with the increase in size, the binding energy in the

inner part of the cluster would ten(] towards the bulk value and therefore the cluster would tend to have

bulk structure while the surface region may still have deviations similar to reconstruction/relaxation

at surfaces. Fig. 4 shows the fraction, N~/N, (Ns being the number of surface atoms) of atoms lying

on the surface in a particle having different morphologies and structures such as tetrahedral, octahe-

dral and cuboctahedral particles of fcc structure and MI packings. This fraction is the smallest for a

cuboctahedron among the fcc fragments bul it is the least for MI. As the cluster size increa,ses, the

difference between the different values, though decreases, it is o n l y for clusters with about 10 ~ atoms

that this fraction becomes nearly equal for cuboctahedral and icosahedral packings. Also the number of

unsatm'ated bonds, important to determine the surface energy, are more in the case of a cuboctahedron

100 V. Kumar

t ~

O

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

, , e , g u ~ . . O . ~ , , , . . . . . . n - ~ , , . . . . . . . . . . . - ~ - , , , . . . . . . , , , , ,

~ ,,, " n o Tetrahedron '" ~ . , ~ + Octahedron

' ""~ ; CcUb2ht;h;dr°n

1 • i i i i i i i i i L i i i 1 , 11 ~ i 1 , , 1 1 1 1 I A _ L X l I I I I I I L r k L ~ L I ~ i ~ t a

lO 1 10 2 10 3 10 4 lO s 10 6

N

Figure 4: Fraction of atoms lying on the surface of a particle, a) tetrahedral, b) octahedrM and c) cuboctahedral fragments of fee structure and d) Mackay icosahedra.

as compared to an icosahedron. For example in a 13 atom cuboctahedron, each surface atom has 4 nearest neighbour bonds on the surface arid therefore there are 24 surface bonds in all. Whereas in

an icosahedron there are 30 surface bonds with each vertex atom having fiw~ nearest neighbours on the surface. Though the bond lengths in the two cases are slightly different, in a crude sense counting the bonds alone, one finds that the icosahedral packing has less number of unsaturated bonds and is therefore more favorable as compared to the cuboctahedral orle and this is what has also been observed.

In particular within the LJ potential model the total energy of 13- atom icosahedral cluster is - 39.972eo counting only the nearest neighbour bonds whereas for the fcc cluster it is - 36e0. However, the same may not necessarily be true in cases where a pair potential description is not applicable as in metals for which the detailed electronic structure would play an important role in determining the atomic structure.

SELECTED EXPERIMENTAL RESULTS ON [COSAttEDRAL CI,USTERS

'File structure of supported clusters of metals has been studied over tile past thirty years through electron diffraction arid electron microscopy (Ino, 1966; Gillet, 1976; 1977 and references therein) or using the extended x-ray absorption fine structure (EXAFS) (see e.g. Sinfelt, 1979) technique. Gold and silver clusters were observed to have icosahedral morphology (Gillet, 1977; Burton and Garten, 1977 and references therein; Marks, 1984). Figure 5 shows an electron micrograph of pentagonal and ieosahedral gold particles obtained by Giltet (1977). The gold vapor was condensed on an alkali halide

crystal under ul t ra high vacuum conditions. Particles with diameters in the range of 30 - 150 ~t had p¢,utagoual decahedral (or pentagonal bipyramidal) or icosahedral (hexagonal outlines) morphology. In

Icosahedral Symmetry in Clusters t 01

? i !' Q .

Pl • O

L a 5 ° ~ 4

Figure 5: Electron micrograph of pentagonal (P) or icosahedral (I) gold particles, t 'l and P2 refer to the

pentagonal particles with axis inclined on the substrate plane and normal to the substrate respectively. (After Gillet (1977))

such experiments, however, the effects of the substrate on the cluster structure are not known and the

influence of the electron beam may not be negligible. In fact lijima and lchihashi (1986) have reported

structurM transformation of gold particles by electron beam irradiation. Earlier studies on metal smoke

by electron micrography have also indicated tetrahedral, octahedral, pentagonal bipyramidal and icosa-

hedral forms of particles in the range of 10 ~ - l0 s atoms (Ino, [966). The first electron diffraction

experiments on unsuppoted clusters were done by Farges et a1.(1973) on Ar clusters of about 100 A size

obtained from a supersonic beam source. They found evidence for non-crystalline structures in addition

to the fcc packing. These experiments, however, suffered from a drawback that these were not done on

size selected clusters and the size distribution was not, known. Though information on small <:lusters

could not be obtained, these experiments gave support to the prediction of Hoare and Pal regarding

the polytetrahedral packing.

Magic Clusters of Rare Gases

The first observation of mass spectrum of free clusters was done by Echt et al. (1981) for xenon. Sim-

ilar to the work of Farges et al. (1973), clusters were obtained by adiabatic expansion of xenon vapor

through a capillary into ultra high vacuum. The mass spectrum thus obtained (Fig. 6) shows significant

variation in the peak intensities of these clusters. By controlling the nozzle geometry, temperature, To,

as well as the stagnation pressure, P0, the distribution of the cluster size could be. varied. This variation

in the peak intensities could be due to several reasons. It could be attributed to fragmentation of

clusters in the electron- or photo-ionization process which is used for the mass analysis of the beam.

The ionization <:ross section could itself depend upon the <:luster size and the kinetics of the growth of

different clusters may effect their distribution. Also whether the seed is neutral or charged can affect

the growth of clusters. There could also be evaporation of atoms, llowever, reproducibility of the

mass sl)ectrum and the observation of similar sl)ectrnm for some other rare gas elements (Friedman and Beuhler, 1983: Miehle et al., 1989; Lethbridge and State, 1989) suggest this kind of structure in

the mass spectrum to be an intrinsic property of the stability and abundance of clusters. It can 1)e

102 V. Kumar

f -

t ~ l - -

u

4 I G I f - i

g

F-

o

1

L i

13 19

71

X e n

Po = 3 0 0 m b a r

To = 175K 2S

~z,3)l

, I I s s

!IUtlI~IIIII~F'"~ ' ? ; .... ~,;: i ....... ' ,,.

~9 7'6

55 i 87 ! 1811 I

177

Cluster size n

Figure 6: Mass spectrum of xenon clusters produced by adiabatic expansion and ionized by an electron beam. The observed magic numbers are marked in boldface above the curve. Brackets are used for

numbers with less pronounced effects. Numbers below the curve indicate predictions or distinguished

sphere packings. (After Echt et al. (1981))

Icosahedral Symmetry in Clusters

5 10 13 15 20 25 30 35 ~ r : l ; 1 : [ : i : I ; : r ; i I , i ; ] i i - -

103

j i l ' l

, UUUUUUL.~L I

200 400 600 800 1000 MASS NUMBER Ira/z)

Figure 7: Mass spectrum of negatively charged aluminum clusters. (After Nakajima el at. (1991a)).

noted from Fig. 6 that there are some clusters for which the intensity increases to a maximum and

thereafter decreases significantly. These are the clusters which have stronger stability as compared to

the ones with one more atom and have been referred to above as magic clusters. The most prominent

of these peaks occur for rare gas clusters with 13, 19, 25, 55, 71, 87 and 147 atoms. Following Itoare

and Pal (1975) and Mackay (1962), the structure, of these clusters was suggested to he based on the

Mackay icosahedra containing 13, 55 and 147 atoms. The intermediate, clusters could correspond to

partial completion of a shell. Thus 19- atom cluster can be considered as an icosahedron with a cap of

six atoms. As rare gases crystallize in the fee structure, it is also natural to consider the fee packing for

these clusters. However, if the clusters should grow in the fcc structure, then the shell closing would

occur at 13, 19, 55, 79, 87, 135, 141, 177, ... atoms. While clusters with 13, 19, 55, and 87 atoms are

observed, the others have not been. This has been considered to be an evidence for icosahedral packing

(Echt et al., 1981). Mass spectra of clusters of Ar and Kr (Lethbridge and Stace, 1989; Miehle et al.,

1989) in the size range N < 1000 show similar trends at least for clusters with more than 150 atoms.

In the small size range the spectra of Ar and Kr do not show pronounced peaks as compared to Xe.

Some differences in the mass spectrum of Kr are due to abundance of different isotopes whereas for Ar

the presence of charged seeds has been found to improve the growth of clusters (Friedman and Beuhler,

1983; ttarris et al., 1984).

Metal Clusters

Besides rare gas clusters, upsupported metal clusters have also been h)und to possess icosahedral struc-

ture. Figure 7 shows the mass spectrum of negatively charged aluminum clusters (Naka.jima el al., 1991a) in which AI~- a is strongly abundant. Similar results have bc~n obtained for boron doped alu-

minum clusters in which A I ~ B - is strongly abundant (Nakajima et al., 1991b). Yi el al. (1990, 1991)

have studied the relative stability of 13- and 55- atom icosahedral and cuboctahedral clusters using

the ab-initio molecular dynamics method in which tile total energy of the cluster was cah'ulated within

the density fimctional formalism (for more details see below). Symmetry preserving relaxations gave

icosahedron to be of lower energy than a euboetahedron for a 13-atom cluster, t{owever, the same

procedure gave cuboctahedron to be of lower energy for the 55- atom cluster. Annealing tile structures

104 V. Kumar

led to a slightly distorted icosahedron for Alia whereas tor Alss there were significant distortions and

the structure could be assigned to have icosahedral origin. Several structures were found which were

nearly degenerate and had quite similar structure factors. However, the pair- correlation and angular-

distribution functions differed substantially. The stability of Alia cluster can be understood in terms of

the filling of the electronic shells in a jellium model (de Heer et al., 1987) which we shall discuss in more

detail in the following section. There are no other small magic clusters of AI (except for AI + which

may have a pentagonal bipyramid structure) with structural analogy to the magic clusters known for

the rare gases. However, for large clusters having several hundred or thousand atoms, a transition is

found to occur where instead of filling the electronic shells, shells of atmns are completed. Martin and

coworkers (1990, 1991, 1992) have studied the mass spectrum for such large clusters of Na, Mg and A1.

They have suggested an icosahedral or cuboctahedral structure for these large clusters. Figure 8 shows

the ratio of the intensities in the mass spectra obtained from averaging over 500 and 5000 t ime channels

for magnesium clusters. The large peaks (filled) ,corresponding to the magic clusters, are found to

occur at a regular interval when the intensity is plotted as a function of N l/s. 3'his has been associated

with the filling of the icosahedral atomic shells. Considering the Mackay icosahedra it can be seen that

for N > 55 and for successive shells, A ( N l/a) ~ 1.419 which one can notice in the figure. The shell

structure also occurs at approximately equal intervals (with a different period) in the case of electronic

shells when plotted on an (N~) ~/a scale, where N~ is the total number of valence electrons in the cluster.

The reason for the linear behaviour in the two cases can be understood from the fact thai. one can

associate a characteristic length with the filling of a shell. For the atomic shells, it is approximately

the interatomic distance, whereas, for the electronic shells, it is approximately the wavelength, AI, as-

sociated with the highest occupied level. In the free electron model 3,f = (32rc'a/9)l/3r,, where r, is the

radius of a sphere containing an electron and it is related to the valence electron charge density in that

material, l'~or alkali metals, r , equals the Wigner-Seitz radius and considering different clusters to be of

the same density, one can show that Aj is nearly equal to the increment in the electron orbit due to an

increment in the radius of the cluster and hence N~ for snccessiw; electronic shell filling (Bjornholm el

al., 1990). For Na clusters, the transition from the electronic to the atomic shells occurs iil the range

of 1400-2000 atoms and shows up in the form of an abrupt change in the period of appearance of the

features of the abundance spectrum (Martin el al., 1990). As mentioned earlier, small clusters may

behave more like a liquid (electronic shell structure being not very sensitive to the atomic structure),

whereas hn" larger clusters, l;he core may be, come more like a solid and the growth of clusters may then

become layer by layer.

In another experiment Klots et al. (1990) have studied H20 adsorption on hydrogenated Co clusters

and from the abundance spectrum they suggested icosahedral structures for these clusters. A large

number of small clusters of other transition metals have been studied but progress in understanding the

structure has been limited due to difficulties in dealing with the d-electrons.

I'hllere'ne,~

The mass spectrum of carbon clusters as obtained by Rohlfing et al. (1984) is shown in Fig. 9. It is

broadly separated into two parts. Clusters with N < 40 in which 11- and 15- atom clusters are abundant

and the rest which has only tile even number clusters in abundance. Kroto et al. (1985) studied the

mass spectrum of carbon clusters under ditferent exl)erilnental conditions and noted that C60 cluster

wa.s special. They optimized conditions such that predominently ('.~0 clusters were in abundance. The

mass spe(:trum in Fig. 10 clearly shows (:~0 to be the only prominent cluster with a small fraction of

(!to present in the vapour. Kroto e:! al. (1985) suggested truncated icosahedron structure (Fig. 1 la)

Icosahedral Symmetry in Clusters 105

15 E t_

~t

.9 '6 n ,

IO

N = (Mg) !4_2 N

3O9 5 6 1 9 2 3 1415 2 0 5 7 2 8 6 9

r - - I - - I - - - - t

0 1000 2000 3000

Number of Atoms, N'

Figure 8: a) Ratio mass spectrum of MgN clusters .

N = (Mg) N 15 309

- - 56_2 92A3 ~4j_5

8 10 I I

34

9

The filled mass peaks correspond to completely

filled icosahedral shells and are nearly equally spaced on the N t/3 scale shown in (b). The four mass

peaks observed between shell closing in (b) indicate highly stable partial shells. (After Martin el al.

(199l)1.

for (?6o with 12 pentagonal and 20 hexagonal faces. This has been confirmed fwm several experiments

such as NMR (Johnson et al., 1990; Tycko et al., 1991; Yannoni et al., 1991) and scanning tunnelling

microscopy (Li et al., 1991 ). This soccer ball molecule with empty space inside has all the sites identical.

This is perhaps the reason for its special stability as the strain due to curling of graphite sheet is equally

distributed. In this molecule there are two types of bonds with lengths 1.40 and 1.45 Jt. The bonds

sharing two hexagons are short (bouble bonds) whereas the bonds sharing a pentagon and a hexagon

a.re long (single bonds). Larger fullerenes have also been identified in the mass spectrum. Among these

C70 has been isolated. It has an egg shaped structure (Fig. l l b ) which can be obtained from C60 by

inserting a ring of hexagons. In fact from Euler's network closure requirement, we have

12 = 3rl.3 + 2n4 + 17z5 + On~ - lnr - 2ns - ..., (3)

where, n,, is the number of m-gons in the network. It can be seen that there is no restriction on the

number of hexagons and therefore larger fi, llerer, es can be generated by increasing their number while

keeping the number of pentagons 12. Ugarte (1993) has also observed multishell fullerene structures.

These are referred to as bucky onions but their properties are yet to be properly understood.

Doped F'ldlerene.s

Several studies have been made on associating atoms or molecules inside and outside the fullerene cage.

Also substitution of C by B ((luo et al., 1991) or N (Pradeep et aL. 1991) has been studied. Boron

doped carbon clusters have been demonstrated to be fullerenes and are found to act as Lewis acids,

readily chenlisorbing one ammonia molecule per surface boron atom (Gno et al. 1991). Heath et al.

(1985) produced some lanthanum complexes of carbon clusters which could have been the first studies

of endohedral fullerenes. Lat.er C h a i e t al (1991) studied doping of fullerenes with La, K, and B. It

was shown that the metal atoms were encapsulated inside the cage. LaCs~ was found to be uniquely

stable. Evidence for coalescence of such fullerenes at high temperatures was Mso obtained which led

to formatioll of larger fullerenes with as many as three lanthanum atoms inside. In an independent

sl udy Ya.nnoni el al. (1992) have reporled encapsulation of scandium clusters in fullerene cages. These

106 V. Kumar

II" SO -

! - 40 3

20 40 :

0 0 20 4 0 60

x l O

70

80

l _ _ A _ _ 80 100 120

C r a t e r Stze (Atoms)

Figure 9: Time of flight mass spectrum of carbon dusters obtained with 40mJ doubled Nd:YAC va-

porizing laser energy and with 1.6 mJ unfocused ArF (193 nm) ionizing laser energy. This spectrum is a combination of two different spectra. For C~, 1 < N <_ 30, the vertical deflection plate voltage used 300 V which optimizes C2 + collection while for C+N, 20 _< N _< 50, 600 V was used, optimizing

, + Cl0 o. The gain in the latter instance was also increased by a factor of 10. The signal intensities of the two distributions cannot be compared directly since the small clusters are two-photon ionized while the

large clusters are single photon ionized. (After Rohlfing et al., 198/)

Icosahedral Symmetry in Clusters 107

Q

No. of carbon atoms per cluster

Figure 10: Time of flight mass spectra of carbon clusters obtained by laser vaporization of graphite and cooled in the supersonic beam. The three spectra shown differ in the extent of helium collisions occuring in the supersonic nozzle. In e, the effective helium density over the graphite target was less than l0 torr while in b roughly 760 torr was present. The enhancement of C6o and Cr0 clusters was believed to be due to gas phase reactions. The spectrum in a was obtained by maximizing these cluster

thermalization and cluster-cluster reactions. The main feature of this spectrum is the occurence of C~ as the main specie with a small fraction of Cro. After Kroto et al. 1985.

Figure 11: a) Truncated icosahedron structure ot C,;0 and b) rugby structure of Cro.

108 V. Kumar

endohedral fullerenes, represented by (Chai et al., 1991; Smalley, 1992) the fornmla M(~CN (M = en- capsulated atom), have created much excitement as these could lead to the development of some novel materials but unlike pure Ce, o, isolation of such fullerenes has been difficult.

Some of the initial efforts to associate atoms/molecules outside the C60 cage were with hydrogen to saturate the valence of carbon atoms. Hydrogenated fullerenes are also of interest in understanding the unidentified infrared features observed in astronomical objects in which intersteller or circumsteller dust is illuminated by ultraviolet radiation from a star (Webster, 1991). Haufler et al. (1990) partially hydrogenated C~o using Birch reduction to obtain CsoH36. They proposed a tetrahedral structure for this molecule such that the 24 unsaturated carbon atoms get paired to form 12 double bonds which are bisected four at a time by three mutually perpendicular planes. Dunlap et al. (1991) have shown that the CsoH60 molecule is somewhat less stable than the Cs0H36 molecule which could be the reason for partial hydrogenation of C6o.

In a computer experiment Kohanoff et al. (1992) studied 12 Li atoms outside the C60 fullerene to form

a stable Li12C60 cluster with one Li atom on top of each pentagon. The high stability of this cluster was interpreted to be due to complete filling of the tl~ and tlg unoccupied states of C6o (see the following section). The C6o cluster was reported to be in a very high charged state in which all the valence electrons of Li atoms get transfered to Cs0. When a smaller atom such as Ba, Ca, or Sr was taken, then

Zimmermann et al. (1994) obtained C60M~ and CToM~ fullerene complexes with M = Ca, St, and Ba. The mass spectrum (Fig. 12) shows marked stability of C60Ba32 cluster. Similar results were obtained for Ca. There are 32 pentagonal and hexagonal rings in C60 and therefore it has been interpreted that each alkaline earth (AE) metal atom occupied the pentagonal or hexagonal ring position. In the case of Cz0, addition of 37 AE atoms has been found to show marked stability. This corresponds to 37 rings in CTu. It has, therefore, been suggested to be a possible probe to study the number of rings in [ullerene molecules. It is also found that further shells of metal atoms can be added onto these (:lusters. Thus

C60Ca104, C70Ca114, C60Ca236 and Cr, oCa44s have been found to be abundant. Though, confirmation of the structure of these large clusters would need detailed ab initio calculations, it is expected that the metal atoms would make only a little change in the strongly stable C60 cage.

In another significant development, Guo et al. (1992a, b) have studied transition metal doped carbon clusters and found strong abundance of MsCI~ (M = V, Zr, ttf, and Ti) molecular clusters. These metallo-carbohedrenes were suggested to have a structure similar to the one of C20 (believed to be a dodecahedron with 12 pentagonal faces) in which each metal atom has three carbon atoms as near-

est neighbours, the metal-carbon bond length being different, from carbon-carbon bond length. These studies have opened up new directions for making other stable molecules which could possibly be used to form new materials.

W a t e r Clusters

Bonding in water clusters is quite different as compared to systems described above. Yet the obser- vation (Fig. 13) of (H~O)20H +, (H20)21H + and related complexes in the abundance spectrum (Yang and Castleman, Jr., 1989; Wei et al., 1991) has been explained in terms of the formation of pentagonal dodecahedral clathrate structure of (H20)2o shown in Fig. 14. There are 30 self-hydrogen bonded hy- drogen atoms along the 12 pentagonal rings and 10 hydrogen atoms sticking out of the oxygen atoms. For the (H20)21H + cluster, the H30 + ion is encaged inside the clathrate whereas for the (H~O)20H + there is a strong evidence that there are l l hydrogen atoms extending out from the (H20)~0H + ion

Icosahedral Symmetry in Clusters 109

s 0 0 [ - - - , - ,

I C6oBax x=32

X = 3 2

C , m B a . +

3S ,13

0 i • ; - , - , . . . . . . , -

1 0 0 0 3 0 0 0 500 ,

I t lass [ a to l l ]

4

CauBa~ +

4

35 ' ~'

i 3. i;

7 0 0 0

l'igure 12: Mass spectrum of photoionized C60Ba~ clusters containing both singly and doubly ionized species. The solid line connects peaks of singly ionized clusters. There is a sharp edge at the cluster with 32 metal atoms. (After Zimmermann et al., 1994).

suggesting a clathrate structure with a proton incorporated in the lattice. This proton is believed to be mobile. The 10 hydrogen-bonding sites in these clusters are found to form a fully solvated hydrogen- bonding structure (Wei et al., 1991) such as (tt20)2t.(TMA)10.H + (TMA = trimethylamine).

TIIEORETICAL STUDIES OF ICOSAIIEDRAL CLUSTERS

Pair lTzteraction Model and Cluster.s of Rare Gases

The most extensive theoretical studies of the structure and growth behaviour of clusters have been using the simplest Lennard-.lones type pair potentials as also discussed in the beginning. This model serves as a good example for understanding properties of rare gas clusters because 1) experimental results are available and 2) pair potential model is a good representation of the interaction between rare gas atoms. tlowever, even with this simple pair interaction model it, is difficult to find the ground state of clusters

as the number of possible configurations increases very rapidly with the size. Hoare and Pal estimated 988 configurations for a 13- atom cluster. It is obvious that one can not explore all such configurations. Therefore progress has largely been made by starting with some judicious guesses which are then relaxed to obtain the minimum energy structure. Hoare and Pal started with a tetrahedron or an octahedron

seed to find the structure of larger clusters by adding atoms to such seeds and then optimized the slructure. As mentioned earlier, their extensive calculations indicated noncrystallographic structures with local or global icosahedral symmetry to be of lowest energy. Similar conclusions were also obtained by Farges and coworkers (1983, 1986) from their molecular dynamics calculations on Ar clusters using a L,I potential. In particular they obtained pentagonal bipyramid, icosahedron, double icosahedron (an icosahedron with a pentagonal cap of 6 atoms on one of its faces), dodecahedron, 55+ atom Mackay icosahedron and polyicosahedron (interpenetrating icosahedra) structures for intermediate clusters from cooling a liquid drop. For larger clusters of Ar Farges et al. (1986) suggested multilayer icosahedral structures which are drived from M1 and a transition to bulk structure in the region of N = 750. Itoare and Pal also found Mackay icosahedron for N = 55 for l~.l and Morse potentials. Barker and Hoare

(see" ltoare, 1979) have designed ot.her complicated stru('t.tlrcs with icosahedral symmetry particularly

110 V. Kumar

e~

(a} ( b} "" 1

Cluster Size (Atoms) Flight Time (microsecond)

Figure 13: a) Mass spectrum of protonated water clusters (H20)NH + (N = 4 - 45) at 119 K and 0.3 Torr He. (After Yang and Castleman, Jr., 1989) b) A portion of the time of flight mass spectrum of mixed water-trimethylamine cluster ions, (H20),(TMA)qH + with different combinations of (p,q). The mixed cluster ions (tt20)21(TMA)t0H + is clearly the most abundant species and an abrupt intensity drop at (21,11) is evident. All the peaks shown are fully assignable, e.g. the mixed cluster series (p,7) are labeled as 'A'. (After Wei et al.,

1991)

I !

/

" -x /T\ / I

I t

Figure 14: Clathrate structure of (H20)20 comprising 12.5-nmmber rings. Tile 30 self-hydrogen-bonded hydrogen atoms in the five-member rings are not drawn. (After Wei c/ al., 1991)

Icosahedral Symmetnj in Clusters 111

Figure 15: Rhombicosadodecahedral s t ructure for a 115 a tom cluster obtained from Lennard-Jones

potentiM. (After Hoare, 1979).

in between the numbers corresponding to M1. The mosl interesting of these is the 115-atom cluster (see

also Farges et al., 1986) which has the rhombicosidodecahedral s t ructure (Fig. 15) . It is a 55 Mackay

icosahedron with a further shell of a toms added at stacking fault positions with respect to the surface

planes. Another LJ min imum is a rhombic t r icontrahedron of 471 atoms (Fig. 16) whose binding

energy is comparable to a s t ructure based upon MI. These studies therefore suggest tha t the closed

shell icosahedral s t ructures const i tu te particularly stable configurations for L.J clusters. D)r nonclosed

shell configurations, the clusters tend to have a largest, possible closed shell unit at the center with the

remaining a toms dis t r ibuted on the surface in one or lnore complete layers.

Looking into the growth of clusters by adding atoms to an existing closed shell icosahedral cluster, one

finds tha t an a tom carl occupy either the te t rahedral sile or the edge site with two nearest neighbour

a toms below or the vertex site. Northby (1!187) considered a lattice of such sites and developed a search

and opt imizat ion procedure to lind minimum energy s t ructures on art icosahedrally derived lattice.

Struclures generated frorn lhese initial configurations were optimized using a I,J pair potent ial as well

as a nearest neighbour radial square potential w e l l

+100 i f d/do < 0.8

V(d) 1 i f 0.8 < d/do < 1.3,

0 i./" d/do > 1.3.

(4)

to obtain local energy minimum. [lere d is the mteratomic distance and do is tile dis tance at the po-

tentiM minimum. The result ing s t ructures were found to be lower or equal in energy to the s t ructures

proposed earlier by Iioare and Pal. Also these were claimed to be the most t ightly bound multilayer

icosahedral s t ructures arid were suggested to br reasonable candidates for the lowest energy clusters of

rare gases and alike, in the size range of I:~ to [ t7 atoms, l:ol N _< 21 (except 17) and N = 25. 26, 29

and 55 these s t ructures are [lie same as oblained by lloa,'c and PM.

112 V. Kumar

1 \

Figure 16: Rhombic tricontrahedron of 471 atoms. (After tloare, 1979)

Electronic Structure and Stability of Clusters

Besides rare gases, clusters of alkali metals and those of sirnple metals such as AI, Mg and Be have

been studied in great detail. Bonding in such metals is of nearly free electron type and therefore the

electronic structure and related properties of these materials can be understood in terms of a simple

jelliurn model. As the structure of a cluster is not known a priori, in the first approximation one can

treat it to be spherical. Therefore, a spherical jellium model has b ~ n extensively used to study the

stability of metal clusters. In this model the ionic charges are replaced by a uniform positive spherical background such that

no i f r < R n+(r) = 0 otherwise . (5)

Here R is the radius of the cluster which is related to the number of atoms, N in tile cluster,

~ r R a = N~, (6)

f~, being the volume per atom in the macroscopic m e t a l The constant positive density no is related to

[~ and the number of valence electrons Z by Z = n0fL The electronic structure and the total energy for

such a potential can then be obtained by solving the Kohn-Sham equations (Kohn and Sham, 1965),

[ - ~ V 2 + V~,(r) + VH(r) + V~(r)]~/(r) = ~iWi(r), (7)

where V~ , Vn and V~ are the external (electron - ion), Hartree and exchange-correlation potentials

respectively (atomic units e = h = m~ = 1 are used.). The electron-ion potential can be an all electron

or a pseudopotential or the uniform background. The other two contributions to the potential are

$'}/(r) = ./dr 'n~(r ') / lr - r'l (8)

a n d

Icosahedral Symmetry in Clusters 113

gE~. ~';~(r)- drip(r)' (9)

where the electronic charge density n¢(r) is given by,

o c c

n~(r) = ~/( i) l~p~(r)l 2, (10) i

f(i), being the occupation number for the state i. The wave function ~b,(r) is expressed as a linear combination of either plane waves or suitably chosen orbitals. The ground state total energy E for a given configuration of the ions {R~} is

o c t

1 [drdr, n<(r)n,(r') 1 ~ Z,Z, (11)

Here ZI is the charge on the I ~h ion at the position Hi. I£~¢[n~] is the exchange-correlation energy. '['his is calculated within the local density approximation (LDA) or the local spin density approximation (LSD) in most of the calculations,

E...[,~.] = f dr~°(r),~.[~dr)], (12)

where e~¢[n¢(r)] is the exchange-correlation energy per particle for a homogeneous electron gas with density n~(r). In the case of spin polarized calculation, this energy should correspond to the den- sity with the same spin polarization as in the actual system. There are several approximate forms available for e=~ and therefore E,~ can be calculated once the density is known. This approximation yields reliable binding energy trends and equilibrium structures for a variety of molecules (Jones and (]unnarsson, 1989) and solids (Moruzzi et al., 1978). However, recently there have been interesting developments where non-local exchange-correlation functionals have been successfully used which have provided improved agreement with experiments for several systems (see von Barth, 1994). But since

most applications have been done within the LDA/LSD we shall restrict our discussion to LDA/LSD only.

Metal Clusters. Similar to the shell model of nuclei, in tile SJM also, solution of Eq. (7) leads to electronic shells is, lp, ld, 2s, lf, 2p, lg, 2d, ... with principle quantum number n and tile angular

momentum quantum number l. These shells Call be completely occupied successively by 2, 8, 18, 20, 34, 40, 58, 68, ... valence electrons. Alkali and other simple metal clusters with these number of valence electrons have been found to be magic (Kumar, 199,t). Detailed calculations have been performed for small clusters of Na, K, Be, Mg, and AI in which the structure has been optimized e.g. using the ab irfitio molecular dynamics method developed by Car and Parrinello (1985). In this method the total energy

(11) of a system is calculated using the Kohn-Sham equations (7) as compared to some pair potentials used in the standard molecular dynamics approach. For metals a pair potential description is difficult to justify and also for systems such as clusters, it may not be possible to develop good pair potentials as the structure is not known a priori. The Car-Parrinello method has been the most interesting development in this direction and it has been successfidly applied to clusters o[' several systems using the simulated annealiug proeedure to obtain the lowest energy structures and to study tire dynamical behavionr of clusters. The reader is referred to a recent review by l(umar (199,l) on applications of this method to various clusters. Of our interest are tire results on Mg4 (i{umar and (?at, 1991) and Be4 (Kawai and Weare, 1!)91 ) which are tet, rahedra similar to the tetramers of rare gases. These have 8 valence electrons

114 V. Kumar

2L i i i I i I I J I j l i _

r-

O 5 1• N

2 . 5

2.0

Ot.N

Figure 17: The chemisorption energy of Mg on MgN clusters shows a lower value for magic clusters and

is correlated with the large HOMO-LUMO gap. (After Kumar and Car, 1991)

which correspond to a shell closing in the jellium model. Alkali metal clusters with 8 atoms are also

magic. Nas has been found to have a dodecahedron structure (R6thlisberger and Andreoni, 1991) with

local pentagonal environment of some atoms. This structure is also known as Bernal structure and cor-

responds to the optimum arrangement of spherical balls. For LJ potential, the lowest energy structure

is a capped pentagonal bipyramid and it is close in energy to the Bernal structure. Mg7 and Be7 have

a pentagonal bipyramid structure similar to the 7- atom clusters of rare gases. These clusters have 14

valence electrons which correspond to the filling of a subshell that arise when the spherical potential

is modified to have an ellipsoidal symmetry (de Heer, 1987). Beryllium and magnesium have a closed

shell atomic electronic configuration and therefore their dimers have weak van der Waals type bonding.

However, as the cluster size increases, there occurs a non-metal - metal transition because in the bulk

both are good metals. While the size range in which a transition to metallic nature takes place is not

known, Kumar and Car (1991) have shown that the metallization starts in some bonds preferentially

and one can say that for small clusters there may be some features corresponding to weak interaction

while for larger clusters there is a gradual change from weak chemical to covalent to metallic bonding.

This is clear from the fact that a 13-atom cluster of these elements is not an icosahedron (Kumar and

Car, 1991). This has 26 valence electrons which do not correspond to a shell closing in the SJM witich

could have been a very good representation for a highly symmetric icosahedral structure. However, All3

has been found to be a nearly perfect icosahedron. It has 39 valence electrons which is just one short

of the completion of an electronic shell in the SJM and is a strongly abundant cluster as it, has been

mentioned earlier. On the other hand A15s does not correspond to electronic shell closing in the SJM

and consequently there are large Jahn-Teller distortions from Mackay icosahedral structure as discussed above.

'Fhe above results suggest that though metal clusters have different interatornic interaction as compared

to rare gases, magic clusters with the same number of atoms in tile two cases tend to have the same

closed packed structures. This indicates a strong effect of the surface energy in the determination of tile (:luster structure.

Icosahedral Symmetry in Clusters 115

-28

-30

-32

"-" - 3 4 . > v

- 3 e

¢: :~ - 3 8 .,~ m

-40

- 4 2

- 4 4

4 . 8

' I%. " ' I ' I ' I ' 1

AhOts" ~ /

/Z

AlltC_

AI~sB ~ "

I I , I , 1 , I z I 4,8 5.0 5.2 5.4 5.8

(a.u.)

Figure 18: Binding energies of some 40 valence electron A112X icosahedral clusters as a function of the

cente, r to vertex distance, R~. X represents the dopant at the (:enter of the icosahedron. (After Gong

and Kumar, 1993)

It ha~s been observed that magic clusters such as AI~- 3 do not interact with e.g. oxygen and behave more

like inert gases (Leuchtner el al., 1989). This has also been demonstrated from ab initio calculations of

chemisorption of Mg on MgN clusters (Kulnar and Car, 1991) which give smaller chemisorption energy

for magic clusters as shown in Fig. 17. it also shows that the magic: clusters have a large highest

occupied-lowest unoccupied molecular orbital (HOMO-LUMO) energy gap and therefore behave like

inert g a s a t o m s (Kumar and Car, 1991). Interaction between magic clusters should therefore be of

van der Waals type. This has heen substantiated from the results of Khanna and Jena (1!193) who

studied interaction between two tetrahedra of Mg and of Kumar (1993) who studied the lowest energy

structure of Sbs cluster which can be described as two weakly interacting tetrahedra. Both of these

calculations suggest van der Waals type weak interaction between magic clusters which can be treated

as superatoms for making materials as it has been the case of solid Ct~0.

Doped metal clusters. Recent[y efforts have also been made to dope All3 cluster by replacing the central

AI atom by a tetravalent atom such as Si, Ge, ... to make it a 40 valence electron cluster (Khanna

and .lena, 1993; Gong and Kumar, 1993). Gong and Kumar (1993) calculated the total energy of

AIj2X icosahedral clusters within the local spin density and frozen core approximations using a. linear

combination of atomic orbitals for X = B, AI, Ga, C, Si. (ie. As and Ti. The struclure was optimized

116 V. Kumar

by keeping the ieosahedral symmetry of the cluster. Figure 18 shows the binding energy of various 40

valence electron clusters as a function of the center to vertex distance. It can be seen that the doping

can enhance the stability of this cluster by a few eV. As mentioned earlier, in an icosahedron the vertex

to vertex distance is about 5% longer than the center to vertex. Therefore substitution of a smaller

atom at the center could provide energetically more favorable binding between the Al atoms. Gong and

Kumar (1993) have shown that the substitution of Si, Ge, C and B at the center leads to a large gain

in energy due to chemical bonding between the center atom and the 12 vertex A1 atoms whereas the

gain due to a contraction in the center to vertex bond length is small. Among the neutral clusters the

gain is the largest in the case of All2C but C is not optimally bonded and this cluster has been found

(Kumar and Sundararajan, unpublished) to reconstruct. In the case of Si and Ge doping.the cluster is

very compact and stable. The electronic energy spectrum of the 40 valence electron clusters is shown in

Fig. 19. The general features of this spectrum are similar to those expected in a SJM. It can be noted

that all the states except those of A 9 and T1~ (near the Fermi level) symmetries occur approximately

at the same energy. This is likely due to the fact that the states with 4- and 5- fold symmetries will not

interact with the orbitals of the central atom. The potential of the central atom has an effect on the

ordering of some states such as in Al12As + and All2C and therefore it is possible that the abundance of

clusters in such systems may be different than predicted by the S.]M. All the clusters with 40 valence

clectrons have a HOMO-LUMO gap of about 2 eV. However, in the ca.se of Ti substitution at the

center, the highest occupied sp-d hybridized state is only partially occupied, indicating the structure to

be unstable. Therefore, the SJM may not be applicable to systems involving d-electrons. It was also

pointed out that in addition to the contraction in the center to vertex bond length, the large gain in the

binding energy of icosahedral A[ clusters with Si substitution could be responsible for the stabilization

of some quasicrystals such as AI-Mn. Seitsonen et al. (1992) have studied the possibility of forming a

fcc solid out of Alt~Si clusters. Similar to Mg, AI~2Si has been found to be metallic in the form of a solid.

Gong and Kumar (1994) have studied the electronic structure and relative stability of AI12'I'M (TM =

transition metal) icosahedral clusters within the local spin density functional theory to understand the

formation of quasicrystals and some other phases in which local icosahedral order is strongly prevalent.

The binding energy of the icosahedrally relaxed clusters is shown in Fig. 20. [t is maximum in the

middle of a d-series as all the bonding d-states are occupied. It, should be noted that for several of the

transition metal elements lying in the middle of a d-series (Mn, Cr, Mo, Tc, W, Re), there exists AII2MO

phase in which Ala2Mo icosahedra are distributed on a bcc lattice (Pearson, 1973). The electronic en-

ergy spectrum of these clusters shows an interesting variation across a d-series and can be analysed in

terms of the interaction between the energy levels of the empty centered A112 icosahedron and the TM

atom (Fig. 21). The eigenvalue spectrum of the empty center A1 icosahedron has a H.q level in the

occupied part of the spectrum and a ltg level above the Fermi level. The d-level of the transition metal

atom remains 5-fold degenerate in the icosahedral symmetry and interacts with the H a levels of tile A112

cluster. With the substitution of successive elements in a d-series, the hybridized d-level of the TM

atom becomes progressively more strongly bound and the hybridization increases with the occupied lit

level of AIj2. For Mn, the d electrons are nearly fully polarized. This leads to a large magnetic moment

on the Mn atom. In AI-Mn quasicrystals, experiments (Youngquist et al., 1!)86) reveal existance of large

local moments on the Mn sites which is not the case for some crystalline phases of A1-Mn. McHenry et al., (1988) have shown the importance of symmetry on the magnetic moments. The icosahedra] envi-

ronment reduces the sp-d.hybridization as compared to a cubic environment and it results in a larger

magnetic moment. Beyond Mn this sp-d hybridized level starts rising up again and the hybridiza.tion

with the occupied H~l level of Ala2 decrea,ses and in the ease of Ni it decreases significantly. However,

for ('.u, the hybridized d-level ['~lls down again and lhe hybridization increases while in the case of Zn,

the d-level lies deep below the spectrum of All2 and the hybridiza.tion is nearly zero. Similar results

Icosahedral Symmetry in Clusters 117

;> v

r-,

c~

r

4

2

0 -

- 2

- 4

-6

--8

- l (

- 1 2

JU-t. ~I:.Si AI.,G. A I ~ ' A I ~ .tlmB" Al.,r.,i'~

T,,. G., Ti. Tll,_._

T~" ' - T~ T~ Tb

A I w

A,.__ X , ~ H - - . , - - ~ . - - X__ X . - - ~ ~ A ' -

TL~'-'~ T t i ' ~ T l ~ T l ¢ ~ TIl"-'~ Tl l " - - T t I ' ~

A , - -

Figure 19: The electronic energy spectrum of some 40 valence electron AII2X optimized icosahedral clusters. (After Gong and Kumar, 1993)

118 V. Kumar

34

A

,~ 36

1.4

= 38

.-. 4 0 en

4 2

Zr Nb Mo Tc Ru Rh Pd AI. q i | I i i 1 i 1

- - ?

o 4d series l l?/...~......

1 l I I 1 I I 1 I Ti V Cr Mn Fe Co Ni Cu Zn

Figure 20: Binding energies of AIt2TM optimized icosahedral clusters. (After Gong and Kumar, 1994)

have been obtained for the 4d series also. Of particular interest are the results ['or Ru, Rh and Pd all of

which show strong contraction in the bond length and signiticant sp d hybridization. Gong and Kumar

(1994) have suggested three factors to be responsible for the icosahedral phase formation: 1) gain in

binding energy due to substitution of AI by a TM atom, 2) contraction of the center to vertex bond

length which makes icosahedral packing more favorable and 3) the sp d hybridization which leads to

the indirect interaction between the TM atoms via the AI atoms a.nd which would be responsible for

the ordering in the system. Thus these calculations suggested Mn, Fe, Co, Cu, Ru, Rh and Pd to be

good candidates for icosahedral ordering. Interestingly except for Rh, all others haw~" been found to

form binary or ternary quasicrystals. Pd and Cu have been predicted Io behave similarly except thai,

the number of valence electrons in the two cases are different. This could be important in counting

the number of valence electrons while using t tume Rothery type arguments for stability and for the

relative concentration of the species for the formation of stable quasicryst, als. According to the results

of these (:alculations Zn is predicted to be distributed randomly as it is in the. case of (AIZn)4sMg32

Mloys (Pearson, 197;I). A significant contraction in the center to vertex bond length and the large sp-d

hybridization were suggested to play a particularly important role iu lh(, stabilization of the icosahedral

phase because the binding energy in the case of Pd and Cu substitution at the center is the same as for

All:~. The significant corltraction in the center to vertex distance for llu and (',u is in agreement with

the I';XAI"S measurements on AIl{uCu quasicrys(als (Sadoc ('1 a/.. 1~93).

Igdlerenes. Electronic structure ariel stability of fullerenes have been investigated extensively during

the past few years. First principles determination of the structure of carbon clusters frorn molecular

dynamics calculations arid simulated annealing is very expensiw' as the atomic orbitals of carbon are

more localized when compared to systems such as AI and Si and therefore a large number of plane waves

are required to expattd the wave functiom [Iowew'r, in a few cases 1lie structure has been det, ermined

experimentally which can be used to sludy their properties. Also, cah:ulations using simpler methods

such as a tight binding molecular dynamics approach (Wang el. al, 1993: Tomanek and Schhlter, 1991)

Icosahedral Symmetry in Clusters 119

v

- 5

- 1 0

Gi _ - -HI - -HI _HA

HI _ l i l t - - - -HI Jig - - 118 - -

,-..4111 H I - - - - H I - - H I _HI

--T1u --T1u - -T lu H I Tit ] - - T l u m l l j

• T I ~ . . . . . ; i . t u ~ . . . . HII ~ . . . . . . . . . . T I u -4~u "--Gu TI u--_. .G u TIu - -

Cu - - Cu ~.-Jq l Gu - - Ou - - - G u

- - ~ u - --' i~u u - ' - T 2 u HI -- '~T2u T2 - -T,u ~ u -'~ u ~J ~'~ T2u-- U-- - -AI

A¢ - - t l - - rl E - -

I l l - - _ i 4 1 - -HI -- t ' , l - -H! - - I l l HII - - HI + | Ig - -

t lg - -

--TLU - -T lu --TIu - - T l u

T I U _ - T l u T l u - - T l u - - T I u - - T | u - -

All - -

- - H I HI - - - - H I l t l - - - - H I H I - - }tll - - - - H a HB - - --Hx

T t u H TIV "l 'lu - - T l u - - T l u T l u . . . . . . . . . . I . _ - ~ . . . . . . . . . . . . Y ~ " ~ i ~ , - - + T l u - - H'i T t u - -Gu H I - - - - H I

- - C u Gu - - Gu _ ' - - G u .--Gu Cu . - - G u

O u - - T2u-- -T2u T 2 u - - ' ~ u T~I- - - -T2u T~u~Gu ~ --T2u T2u-- ~T'~u

Itll All - - - - ' q All - - - -A! --All - - Art

H I - - --H8

H| -- --H|

- -HI Hit -- - -H| HI - - - - t t l --utg H* I - -

H I - -

- - T I u ~ T I u - - T I u - - T I u T I u ~ T I u - -

- - T I u T t u - - T I u - -

T I U - - Hg - - - - H |

--A|

- - ~ AS ~ ~ A g - i t t l - - - t l

A i - "tit -

Aft= AlltTi AlnV AlnCr AlmMn AlmPe A118Co AlnNi AlteCu AlleZn

Figure 21: Eigen value spectrum for empty center icosahedral All2 and doped <:lusters. (After Gong and t(urnar, 1994)

0.2

0.4

0.6

0.8 Y. it .4

o 1.0

1.2 20

' l " ' l l r ' , l , , , , i , , , , i , , , , i , , , , l ~ + , , i , , , i

6O 70

30 40 50 60 70 80 90 100 Cluster S i z e X

I"i~ure 22: Binding en('rg+> per atom a.s a function of size in carl)ou clust('zs. (After \Van~ fl M., 1993)

120 V. Kumar

> ~J

:>+

(a) i n _ .

I

2 5 3 l

Degeneracy I ~ X W L I E X

/

1 {b}

. . . . . . . . . , \ I T / \

+

I /

~%'/ 4 2 ̧ ',,

A X W I I )~ X

Figure 23: (a)Electronic eigen value speelrum for (;60 cluster (lefl pane.l) and the band structure, of the

fcc (J+;o crystal (right panel), fcc C60 is found to be a semiconductor with a direct gap at the X point. In

the hgure the valence [)and top at 1he X point is defined as the zero of energy. The lowest six optically

allowed excitations have also been shown hy arrows. (b) Band strucDire of [)c (?{10 around the energy

gap. (After Saito and Oshiyama, 1991 )

predict fullerene slructure of carbon clusters iIl the studied l'angc of 20 to 100 atoms. The binding

energy per atom increases with the size of the clus*er as showll ill Fig. 92 and would reach towards

its value in graphite. IIowever, C60 and C70 show locally particular stability. In I"ig. 23 we show the

electronic eigenvalue spectrnin of Q0, obtained by Saito and Oshiyama (19!)1) within the local density

and frozen core approxirnations using the norln-conserving pseudopolenlials (Iiachelel el al., 1982) and

a (;aussian orbital basis set. The occupied spectrum has a width of aboul 20 eV. The deeper levels arise

from the o" bonded states whereas the corresponding antihonding states are unoccupied. States near

th{, highest occupied level arise mainly from the 7r bonded p~ orbitals. The. I lOMO is of h= symmetry

whereas IDle LUMO has the t l , symmetry. The ]tOMO-LITMO gap is about 1.9(!\". Considering C6u

a perfect spherical ball, its energy levels can be characterized by angular molnentum l. The HOMO

and I,UMO belong to the l ~- 5 state. The second and the third highest occupied slates with ~Ja and

hq symmetries correspond to the l - 4 stale. In a spherically symmetric potential, the allowed optical

transitions are | k o m l t o l + 1 states (Al - 1). Fk}r (:60 lhe lowest six optically allowed excitations,

h,, -+ /.lq, h.v --+ tl,,, h,, --4 hg, .q, -+ Z2~,, h9 ~ &~,, and h,+ ~ 9:; are show|3 in Fig. 2;I. The calculated

excilatiorl energies, 2.87, 3.07..1.06, 5.09. 5. I7, and 5.87 eV agree well with the experimental photoab-

sorption spectrurn which exhibits peaks at 3.0& :1.76, 4.82, and 5.85 eV (Aiie eZ al., 1990). Also shown

in the figure is the band struct.ure of k c (?++0 which is found l.o be a semiconductor with a. direct gap

at the X point. Due to weak interaction hetwee.n lhe (?6o molecules, lhe bands are w~ry narrow and

il has been considered that the electron correlations couht be imporl,anl irl lhese malerials but l,heir

eiders oft the electronic structure of doped fullerites is yet Io he properly understood. Doping the solid

(~,o with metal atoms leads to the occupalion of the conduclion hand and a charge transfer from metal

atom,s to the fullerenes. The bandwidth and 1he densily o1" sl.a~c,s at lh¢, l:¢'rmi h'vPI gets significantly

atfected by changing the intermoJecular distance e.g. hy ,lol)ing or I)y" al>l)lyiug pressure and Ibis leads

1o a variation in the superconduclivity Iransilio]~ Wu/I)erature.

Icosahedral Symmetry in Clusters 121

When an electron is added (removed) to (from) C60, the highest occupied state becomes partially occu-

pied in the icosahedral symmetry of C60. Consequently some dahn-Teller distortions can be expected. Saito (1991) studied the electronic structure of such charged clusters without taking into account the distortions. The energy level sequences and the spacing between the levels in the vicinity of the HOMO were found to be similar to those obtained in the case of C60. The nearly rigid shift of the states from

C6o to C6 + (C~o) is about -3.3 eV (3,4 eV). The extra hole (electron) was found to be distributed over all the 60 ions. Such studies are significant for developing derivatives of fullerenes and for an understanding of superconductivity in alkali doped solid Ca0. Similar to the graphite intercalation compounds with alkali-metal atoms where the valence electrons of the metal atoms are known to get transferred to the graphite sheets, in KaC60 and other related doped fullerites which are superconducting, the LUMO

derived bands are half filled and the C60 molecule is nearly triply charged with the K becoming ionized. Adding further K leads to K~C60 bcc phase which is insulating. First principles calculations within LDA show C6o in an unusually high charge state with six additional electrons and K atoms almost fully ionized (Erwin and Pederson, 1991).

Endohedral fullerenes. Observation of endohedral fullerenes has led to several studies of their electronic structure. Chang el al. (1991) studied several electropositive and electronegative elements such as O, F, K, Ca, Mn, Cs, Ba, La, Eu and U at the center of the C~o cage and their cations. The metal atoms were found to lose zero, one or two of their outer s-shell electrons to the cage tt~, state or the metal d state or both, depending upon the extent of the s orbital radius. In particular, the alkali atoms were found to transfer their outermost electron to the cage. On the other hand O, F, or CI (see also Saito, 1991) atom at the center interacts little with the cage atoms, in this case also, the highest occupied orbital is of tl~ symmetry but corresponds to the p state of the central atom. The LUMO (h~) and the next highest occupied state are derived from the cage orbitals. Ionization of the metal complexes was found to occur

from the cage tl~ orbital, if occupied. This results in most complexes to have nearly the same ionization potential. Cioslowski and Fleischmann (1991) have studied endohedral complexes of Cs0 with closed shell 10 electron atoms or ions such as, F- , Ne, Na +, Mg 2+ and A] a+. They found Ne to be stable

at the center with very little interaction with the cage. On the other hand complexes with ions are strongly stabilized and the ions, screened. The negatively (positively) charged ions decrease (increase) the cage radius. The ions at the center have only a local minimum. The energy minimum corresponds to the guest ion displaced from the cage center with a negligible dependence on the angular position of the cage. A simple model in which the guest specie interacts with a polarizahle double-layered Cs0 cage with positively charged nuclei inside and negatively charged electron cloud outside was shown to account for most of the properties of endohedral fullerenes.

CLUSTERING, FUSION AND FRAGMENTATION OF CLUSTERS

Some clusters can be used as building blocks (superatoms) for developing new molecules and materi- als. The simplest example would be a dimer of such clusters. As discussed earlier, the closed shell magic clusters such as Mg4 and Sb 4 interact weakly (van der Waals type interaction). On the other hand a dinaer of Na19 is stable and has strong interaction as in the case of monovalent atoms (Saito and Ohnishi, 1987), Nals being of a closed shell configuration. When the two clusters are brought still nearer to each other, a fusion process occurs which leads to a Naas cluster with a gain in energy as compared to the dimer. Such studies of interactions between icosahedral clusters of alulninum may be helpful in understanding the atomic structure of quasicrystals and other complex materials. Depending upon the electrons per cluster, there could be a preference for face. edge or vertex sharing or two icosahedra

cotdd be interacting face to face or edge to edge as in the Al12Mo phase.

122 V. Kumar

E

=2

v .=_ c~

(C 6 " .V=55 O)N

' 1 - , t , i ,

20000

+3

+t31+3 +3 1,1 ~÷31+a +3 J

I +~ +3

4 0 0 O 0 60000 Mass (amu)

Figure 2.4: Mass spectrum of C60 clusters. Notice the peaks at N = 13, 19 and 55.

third peak has high intensity. (After Martin et al., 1993)

Thereafter every

Martin and coworkers (1993) have studied clusters of C~;~). As interaction between C~0 fullerenes is weak,

magi(: clusters with 13, 19, 55, ... C~0 fullerenes }lave been obtaiue(l as in the case of rare gas clusters

(Fig. 24). [towever, there are deviations from the magic numt)ers observed in the case of rare gas

clusters. The oscillatory pattern seen after N - 55 might be due to three moh~(:tlles filling a triangular

face of the icosahedron. It would appear tllat at least nine such faces could be cow~red without adding

ally vertex molecules. Similar r£'.sults have been obtained lot Cr0 also but some of the magic clusters

are different from C~0. A determination of the detailed sl.ruclure and the etfects of weak electrostatic

interactions on the orientational or(ter are yet to be studied to our knowledge in these (:lusters.

Whetteu and Yeretzian (1!~93) have reported collision studies between fidlerenes. Cs0 fullerenes have

been found to coalesce t~J torm larger fullerenes which are multiples of the original smaller fullerenes.

The other process that occurs in the collision is the fragmentation of fullerenes. Very similar to the

nuclear physics experiments, such studies are likely to give clues regarding the formation of fullerenes,

One of the products in the fragmentation is a dimer. Successive photofragrnentation of dimers has also

been done from different fullerenes in order to understand their stability (O'Brien et al., 1988); Guo

el, al., 1991). Such studies have indicated Cae to be the smallest stable fullerene beyond which the

cage structure gets fragmented in the process. In the case of endohedral fullerenes this shrink- wrap

mechanism has been helpful in proving that the guest atom is captured inside the cage. Similar studies

of collision and fragmentations on clusters of other materials would be helpflfi in understanding growth

of clusters.

SUMMARY

Results on clusters of rare gases, metals, carbon and water, as discussed above, suggest that clusters of

systems having very different bonding characteristics exist in icosahedral structures. While the surface

energy considerations are very important in the formation of icosahedral clusters and for rare ga~ses

a pail interaction model provides a very satisfactory description of the structure, the details of the

electronic states distribution play a crucial role l\)r other materials as one finds front tile example of

Icosahedral Symmetry in Clusters 123

Mgl~ cluster for which a relaxed icosahedral structure lies highest in energy as compared to relaxed fcc, hcp and the lowest energy structure obtained from simulated annealing calculations (Kumar and (Jar, 1991). Properties such as magnetic moments and excitation energies of clusters could depend sensitively on the structure as the eigenvalue spectrum can change very significantly depending upon the symmetry. However, properties derived from the shell structure have been shown to be less sensitive to the structure (R6thlishberger and Andreoni, 1991).

Atomic calculations and structural optimization of clusters with more than a few tens of atoms as it would occur in ieosahedral packings are difficult in cases other than the rare gases. The jellium model,

originally developed for studying some bulk properties of simple metals, gives a reasonable description of the electrons in metal clusters as these move freely inside the cluster but are bounded by the surface which leads to the quantization of the electronic states and the shell effects in the properties of metal clusters. In the bulk, however, the lattice determines the bulk electronic wave functions due to the periodic boundary conditions and therefore as the cluster size increases, there should be a transition from the electronic shells to a bulk type behaviour. How big a cluster may be needed for the transition is though not clear. Electronic shell effects have been noted for clusters having of the order of 1000 atoms (Bjornholm et al., 1990). For larger clusters a transition has been noted to atomic shells (super- shells) with icosahedral or cuboctahedrat structure. Simplified approaches such as the effectiw." medium theory (Chetty et al., 1992) arid perhaps the electron diffraction experiments a,s done by Farges et al.

(1!)73) on argon clusters could provide more information about the structure of such clusters. Further, it is hoped that first principles calculations of relatively large clusters will become possible with the availability of better computational resources and the improvements in methodologies (Galli and Par- rinello, 1993; Mauri and Galli. 1994). Such developments would also make it possible to study clusters of technologically more importal,t materials having transition metal atoms and first row elements and do computer experinaents for developing new clusters/molecules in the near future.

One of the exciting developments in research on clusters has been the discovery of fullerenes in which icosahedral, symmetry plays an important role, A variety of their derivatives have further been discov-

ered and efforts are being made to find stable clusters/molecules of other materials which could lead to the development of new molecular solids. As compared to atoms the atomic structure of clusters is likely to force new arrangements and phase transitions in solids similar to solid C~0 (David el al.,

1991). Further, materials with complex structures such as quasicrystals arid their approximants could be viewed in terms of assembly of clusters. Such etforts are currently actively persued. Similar to the

studies of the nucleus structure, recent collision experiments giving rise to fusion and fragmentation of clusters are likely to give further information on the growth and structure of clusters.

The field of clusters is very large and multidisciplinary and we could not include many of the complex

molecules that may also have icosahedral symmetry. It was not possible to be exaustive but I hope that the reader would get some glimpses of the recent developments where icosahedral clusters have been playing an important role.

ACKNOWLED(]EMENTS

l would like to thank my collaborators 1{. (!ar, V. Sundararajan and in particular X.C,. (long for severM fruitful discussions. I aln grateful to the editors for their patience and to Prof. S. Ranganathan for his ellCOU l 'agenlel]t .

124 V. Kumar

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