Structural consequences of the jellium model for alkali metal clusters

Chemical Physics 137 ( 1989) 15-24 North-Holland, Amsterdam

STRUCTURAL CONSEQUENCES OF THE JELLIUM MODEL FOR ALKALI METAL CLUSTERS D. Michael P. MINGOS and Zhenyang LIN Inorganic Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OXI 3QRS UK Received 4 April I989

Alkali metal clusters with closed-shell electronic configurations according to the jellium model adopt geometries of high symmetry and based on the Td, Oh and I,, point groups. Clusters with incomplete shells adopt either prolate or oblate geometries. The conclusions have been supported by crystal field perturbation theory arguments and molecular orbital calculations. For high nuclearity clusters alternative high symmetry structures can occur and those which are either the most close packed or spherical are predicted to be the most stable. When the jellium closed-shell “magic numbers” coincide with one of these high symmetry structures then the cluster will be particularly stable.

1. Introduction

The success of the jellium model in explaining the “magic numbers” observed in the mass spectra of homo-nuclear alkali metal clusters has attracted a great deal of attention [ l-4 1. This simple free-electron model has successfully accounted for the large peaks observed at N=2,8, 20, 30, 58 and 92 atoms in the mass spectra of sodium and potassium clusters produced in a supersonic expansion with argon carrier gas. These magic numbers are associated with electronic closed-shell structures where the ordering of the electronic energy levels is 1s, 1p, 1d, 2d, 1f, 2p, lg, ... . Relative stability maxima are then predicted for the closed-shell clusters which occur at n=2, 8, 18, 20, 34, 40, 58, .... where n is the number of valence electrons in the cluster. Although the jellium model has successfully explained the magic numbers for alkali metal clusters more sophisticated quantum chemical methods are required to understand the details of the orbital energies, shell structures, ionisation energies and geometries of these clusters. Ab initio calculations have been widely used to study these aspects [ 5-71. Such calculations have been carried out on alkali metal clusters with up to 20 atoms and the results have provided detailed information on the geometries and stabilities of lithium clusters and sodium clusters.

However, limitations on computing facilities have prevented the application of ab initio calculations to larger systems even when only alkali metals are present. In this paper, we describe some general principles governing the shapes of alkali metal clusters and provide a methodology for suggesting geometries for larger alkali metal clusters with closed-shell electronic structures, i.e. for those with magic numbers. Recently, we have developed a structural jellium model which accounts for the electronic structures of clusters using a crystal-field perturbation [ 8 1. The perturbation analysis not only enabled us to understand qualitatively the success of the jellium model for alkali metal clusters and the reasons for failure in other situations but also underlined the geometric consequences for jellium closed-shell clusters. In this paper these geometric conclusions are developed in more detail. The magic numbers in clusters of rare gases have been explained in terms of icosahedral close packed geometries, for which the van der Waals interactions lead to densely packed and nearly spherical geometries corresponding to observed discontinuities at the following numbers of atoms N= 13, 19, 25, 55, 71, 87, and 147, etc. [9]. The magic numbers of mixed rare gases have also been explained by the icosahedral close packing of spheres for two kinds of soft

0301-0104/89/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

16

D.M.P. Mmgos, Z. Lin / The jeifium model for alkali metal clusters

spheres [ lo]. A similar method will be employed this paper.

in

2. The splitting of cluster orbitals The spherical free electron jellium model does not provide any geometric insights, but a crystal field perturbation can be used to consider the effects of the location of the nuclei on the splittings of the P, D, F, ... shells. The crystal field perturbation can be introduced as follows [ 8 ] :

The matrix elements of the individual potential are as follows:

terms in the

(nlml V~~nlm')=F~,(lm~LM~fm')

,

where F,,, is a function of geometry and number of atoms in the cluster and I nfm) is a jellium wavefunction with a principal quantum number n and angular momentum quantum numbers 1 and m. Although the spherical harmonic expansion of V( r, 8, 4) is an infinite series, the matrix elements of V( r, f?,6) can be reduced to sums of several terms when only a definite I state is considered. This is because #I

unless m = M+ m’ and L = 0,2,4, .... 21.The requirement that each term in the potential must belong to the totally symmetric representation of the point group provides another restriction on those LA4 which appear. From this analysis we concluded that an alkali metal cluster with a complete jellium closed-shell structure will adopt a high symmetry structure, which most closely approximates to spherical and minimises the splittings of nl shells. Furthermore, we concluded that for clusters with several shells of atoms complementary angular coordinates for atoms on successive layers also minimise the splittings of nl shells. The Vro term in the above expansion [ 81 corresponds to an oblate or prolate distortion of a sphere. ” For crystal field theory see ref. [ II].

These oblate and prolate distortions produce complementary splitting effects, as illustrated in fig. 1 for the P (I = 1) shell. These complementary splittings give rise to specific geometric preferences for clusters with incomplete nl subshells. For example (P )’ and (P)4 configurations lead to prolate and oblate geometries respectively. The same conclusions concerning the relative splittings of nl shells can be derived within a molecular orbital framework [ 12 1. Alkali metal clusters with Td, Oh and Ih symmetries have a degenerate set of P skeletal bonding molecular orbitals whereas oblate and prolate clusters have a split set of P skeletal molecular orbitals. These splittings are shown in fig. 2 which summarises the orbital splittings for some polyhedral alkali metal clusters derived from a molecular orbital analysis. In oblate clusters the P + , set is always more stable than PO, whereas prolate clusters have PO below P + , . The energy of the lower lying S bonding molecular orbital is determined by the connectivity of the skeletal atoms. The higher the connectivities the more stable is S. Consequently, polyhedral geometries with the maximum number of nearest neighbours maximise the stabilisation of S and produce the maximum mean stabilisation energies for the bonding nf shells. The optimum geometries for Li4, Li6 and Lig clusters found in the ab initio calculations and illustrated in fig. 3 support the above arguments. A similar pattern is evident from ab initio calculations [ 13 ] for Na2Mg, Na,Mg and Na6Mg, the results of which are shown in fig. 4.

PO Pt1

P

PiJ

--;-----:I

64 Fig. 1.The splitting clusters.

PO

(b)

of the P shell in (a) oblate and (b) prolate

D.M.P. Mingos, Z. Lin / Thejeliium modelfor alkalimetal clusters

all

orbitals

are

17

o-type

A

S

S

S

Fig. 2. The energies of L” MOs of some polyhedral clusters.

Fig. 5a shows the results of an extended Htlckel calculation on a Na& cluster based on fee close packing arrangement with 0, symmetry. In this point group ‘the D shell is split into tzg and eBcomponents. The tilling of this shell and lower-lying P and S shells leads to a stable configuration for 18 electrons. Prolate and oblate distortions result in the splittings shown in the figure. The complementary nature of the splittings can also be derived from a crystal field analysis. Clearly it can be predicted that clusters with 14 valence electrons prefer oblate geometries, 16 electrons prolate. Fig. 5b shows the splitting of F orbitals from an extended Htickel calculation on Na,,

cluster with fee close packing of O,, symmetry, i.e. a truncated octahedron (see fig. 6d below). The orbitals with lower energies are 1S, IP, lD, 2S and 2P. Therefore it can be predicted from fig. 5b that clusters with 36 valence electrons prefer a prolate geometry, 38 electrons oblate and 40 electrons with closedshell electronic structures prefer an Oh high symmetry geometry. In our discussion of shell splittings within a firstorder perturbation approximation [ 81 it was found that locating atoms of different angular directions on concentric spheres with different radii leads to small shell splittings. Therefore close packed or nearly close

18

D.M.P. Mingos, 2. Lin / The jellium model$or alkali metal clusters

Fig. 3. Calculated geometries for some Li, clusters.

Dlh Oblate

oh

Ikh

PlQhte

Id

(8)

Fig. 5. (a) The splitting of the D shell for Na,, clusters based on fee arrangement. (b) The splitting of the F shell for NaJ8 clusters based on fee arrangement.

not necessarily minimise of the nl shells.

the extent of the splittings

Fig. 4. Calculated geometries for some Na,Mg mixed clusters.

3. Orbital splittings for the most close packed and packed structures are expected for the alkali metal clusters with closed-shell electronic structures. In summary, ab initio calculations, the crystal field perturbation method and the molecular orbital tensor surface harmonic method are all consistent with the following generalisation: closed-shell alkali metal

clusters, i.e. neutral clusters with 2, 8, 20, 34, ,.. atoms will adopt high symmetry structures with Tti, Oh and I,, symmetries and close packed or nearly close packed geometries. Clusters with incomplete shells adopt lower symmetry oblate and prolate structures, belonging to Dnh, Dnd and their subgroups. The relative stabilities of the alternative structures are determined by the sign of the splittings of P, D, ... shells. Satisfying these geometric requirements often leads to several structural possibilities, since the most close packed structure is not necessarily the most spherical structure. The most close packed structure maximises the number of next neighbour resonance integrals and consequently the average energy of nl molecular orbitals is lowest. However, such structures do

spherical structures The close packed structures maximise the number of inter-atom resonance integrals, but can result in larger splittings of the nl shell than spherical structures. In the crystal field approximation, the nonspherical perturbation potential on a D shell for clusters with tetrahedral or octahedral point group is as follows [ 8 ] :

where Y;l (0, @) are the spherical harmonics and r the distance between the electron and the origin. C depends on the geometry considered and is given by C= (4n/%X

& Y:(6,

@,) ,

where (0, (bi) are the angular coordinates of the ith atom on the surface of sphere with a radius of r,. The smaller C is, the smaller the splitting of the D shell is

D.hf. P. Mngos, Z. Lin / The jellium model for alkali metal clusters

resulted. Calculations of C for a most close packed (fig. 6a) and a more spherical (fig. 7) structure with 20 alkali metal atoms, which will be discussed later, were carried out. The contribution of the interstitial tetrahedral moiety for both structures is the same and

19

is thus not relevant to the comparision. The contribution of the four atoms located on CS axes is defined as C,,, for both structures, and is - 1.04~‘/~/r:~~, where r,=,is the distance of the four atoms to the centre of the cluster. The contributions from the remaining 12 atoms located on the mirrors for the close packed (c.p. ) (fig. 6a) and the more spherical (fig. 7 ) structures are C,,(c.p.)

= 1.00n”2/r~mcc.p.~

and C,,(Vh)

= l.34n”2/rL(sph)

*

It can be seen that the effect of the remaining 12 at-

b

d Fig. 6. Some specific examples of tetrahedral and octahedral arrangements based on fee close packing.

Fig. 7. A spherical structure with Td point group for MZOcluster.

oms opposes that from the tetrahedral atoms for both structures. For the close packed structure (fig. 6a), rtettc.p.) = 1.17~ and rrem(c.p.)= 1.84a (a is the bond distance between two adjacent atoms). The inequality of r tet(c.p.) and rrem(c.p.) causes a big difference in the contributions from the two sets of atoms. The contribution of the tetrahedral set is only about 10% of that from the remaining set because of the factor of 1/r; and Ctet(c.p.j is negligible compared to CEm(c.p.). Therefore a larger splitting of D shell is expected for the close packed structure. In the more spherical structure (fig. 7 ), the cancellation leads to a small C (0.307P/f& where r()=rtei@pph) =rrem+ph)). The spherical structure is thus predicted to produce a smaller splitting of the D shell. This prediction is confirmed by the ab initio calculations on the two structures [ 7 1. The same tendency is found from calculations of M &,clusters (see fig. 6). When C,, is defined as the contribution from the six atoms located on the Cq axes and C,,, from the remaining 12 atoms located on the C2 axes, C,,,, =2.331~‘/~/& and C,,= - 1.17n”2/r5 Te,,,.The cancellation between the two sets of atoms is maximised when r,, = l.l5r,, . In the close packed structure (fig. 6c), r, = 1.4 14r,, . A maximum of cancellation can be achieved by moving the atoms located on the C, axes towards the centre and moving those on C2 axes away from the centre along the axes. Therefore a smaller splitting of the D shell is predicted for a more spherical structure in the M :9 cluster than for the centred octahedron. In summary the splitting of electronic shell structures is

D.M.P. Mingos, Z. Lin / The jellium model for alkali metal clusters

20

small for those structures with a more spherical geometry and high symmetry.

4. Close packed and spherical high symmetry structures We focus on T,,, 0, and I,, symmetry structures for alkali metal clusters with closed-shell jellium electronic configurations. Tables 1, 2 and 3 summarise the symmetries and the number of atoms in the general and special positions in the Td, Oh and I,, point Table I The symmetry and number of the special and general positions in the tetrahedral point group Symmetry of special position

Number of atoms generated

Location of atoms on

Td C 3” Cl” CS C,

1 4 6 12 24

centre C3 axes C2 axes mirrors general positions

Table 2 The symmetry and number of the special and general positions in the octahedral point group Symmetry of special position

Number of atoms generated

Location of atoms on

Oh C4” C 3” C2” CS C,

1 6 8 12 24 48

centre C, axes C, axes Cz axes mirrors general positions

Table 3 The symmetry and number of the special and general positions in the icosahedral point group Symmetry of special position

Number of atoms generated

Location of atoms on

lh CS” C,” CZ” C, C,

1 12 20 30 60 120

centre C4 axes C1 axes C2 axes mirrors general positions

groups. The magic numbers associated with the jellium model must coincide with permutations of atoms among these positions. For example, an M20 cluster restricted to the Td, Oh or I,, point groups must result from one of the following permutations: T,:

4(C,“)+4(G)+

0,:

6(C,“)+6(C,“)+8(C,“)

12(C)

12(C,,)+8(C,,) I,:

20(&).

Furthermore, these geometries represent composite structures based on tetrahedra ( 4C3,,), octahedra (6C,,), cubes (8&), truncated tetrahedra ( 12C,) and cuboctahedra ( 12CzV). The most close packed structure represents the structure with the maximum numbers of nearest atoms and has successive layers of atoms in complementary positions. At the other extreme the most spherical structure is that which has the maximum numbers of atoms on the outer layer, and evenly distributed. Therefore it is apparent that even for high nuclearity clusters which have closed-shell electronic structures they will have either spherical high symmetrygeometries or close packed or nearly closepacked structures. These two geometric requirements restrict the permutational possibilities. For example, for the four possible permutations above for an MzOcluster, the first choice is the most preferable since it satisfies the requirements of complementary angular coordinates for atoms on successive layers and generates a close packed or nearly close packed structure. For the second permutation, locations of atoms on two sets of CJVspecial positions lead to a structure in which two atoms are placed above each other, i.e., in the same radial direction. Structures derived from the third and fourth permutations cannot lead to nearly close packed arrangements because either many atoms (20 or 12) are located on one shell without centre or a cube (8) is taken as an interstitial moiety. We now look in turn at the close packed and spherical structural possibilities for clusters with closed-shell electronic structures.

D.M.P. Mingos, Z. Lin / The jellium modelfor alkalimetal clusters

5. Close packed structures

5.2. Geometries derived from the fee close packing arrangements

Td, 0, and I,, symmetry structures can be only derived from fee, bee and icp (icosahedral ) close packed arrangements. Those structures derived from the most close packed arrangement (fee) will be the most preferable since they maximise the number of interatom interactions. Therefore, we will focus on the structures derived from the fee arrangement in the following discussion. However, bee and icp will also be considered for those structures which cannot be derived from an fee arrangement.

5.2. I. Neutral metal clusters

5.1. Polyhedra derivedfrom close packed arrangements The dependence of the number of atoms (NT) in the fragments of fee crystalline-like based on tetrahedral, octahedral, and cubic geometries on the number of atoms lying on an equivalent edge (k) (comer atoms included) was formulated in the 1960’s in the discussion of the statistics of surface atoms and surface sites on metal crystals in catalytic chemistry [ 141. Table 4 lists the total numbers of atoms for some value of k and the general expression of NT as a function of k. Fig. 6 shows some specific examples. It can be seen from table 3 and fig. 6 that the simple tetraha dron and octahedron series are not responsible for the geometries of the alkali metal clusters with magic numbers although they possess high symmetries. However, the geometries of clusters with different numbers of atoms with the same symmetries ( Td and 0,) can be derived by capping or truncating these polyhedra. This will be discussed below. The corresponding cubic, octahedral and rhombic dodecahedral polyhedra based on bee packing are summarised in table 5.

21

The magic numbers for alkali metal clusters observed in experiments are 8,20,40,58 and 92 atoms. Possible geometries based on close packed arrangements can be deduced from the above discussion. M,: The tetra-capped tetrahedron with Td symmetry represents the geometry of Mg. Ab initio calculations on L& have confirmed that the tetra-capped tetrahedron is the most stable geometry [ 5 1. Mzo: It can be seen from table 4 and fig. 6 that a simple tetrahedron with k=4 is a possible geometry for this cluster. An ab initio calculation has indicated that this structure has the lowest Hartree-Fock energy [71. MdO:The total number of atoms based on a tetrahedron with k= 6 is 56 (see table 3). When a fouratom tetrahedron is taken away from each comer of this tetrahedron a truncated tetrahedron with 40 atomsisderived (56-4~4). MS8:Taking a vertex away from each comer of a tetrahedron with k= 6 and capping each edge of this tetrahedron with one atom, generates a geometry of 58atomswithTdsymmetry (56-4+6). Mgz: A tetrahedron with k=7 has 84 atoms. Taking one atom away from each vertex of this tetrahedron and capping each face with 3 atoms, leaves a Mg2cluster with Td symmetry. Those clusters with closed-shell electronic structures which are not observed as large peaks in the mass spectra, e.g. Mi8, MS4, Mh8 and MT0 clusters, could have low intensities either because no high symmetry structures (e.g., ML8, MS4 and MT0clusters) can be generated or accidental degeneracies of the energy levels do not lead to a clear definition of the shell structures. The latter factor can be understood from the following example. In the original discussion of

Table 4 The dependence of the total number of atoms (N) on the number of atoms (k) on each edge of a tetrahedron, octahedron and cube based on face centred cubic close packing (fee)

Ntetrahedron) N (octahedron) N (cube)

k=2

k=3

k=4

k=5

General expression

4 6 14

10 19 63

20 44 172

35 85 465

ik(k+l)(k+2) fk(2kxk+l) k(4kxk-6k+3)

22

D.M.P. Mngos, Z. Lin / The jellium model for alkali metal clusters

Table 5 The dependence of the total number of atoms (N) on the number of atoms (k) on each edge of a cube, octahedron and rhombic dodecahedron based on body centred cubic (bee)

N (cube) N (octahedron ) N (rhombic dodecahedron)

k=2

k=3

k=4

k=5

General expression

9 15 15

35 57 65

12 143 175

189 289 369

k3+(k-1)) +(2k2+1)(4k-3) (2k-1)(2k2-2k+l)

the jellium model [ 11, the difference in the total electronic energy between MN and MN+, clusters, E(N) -E(N+ 1 ), was defined as d(N). The change in this quantity, d (N+ 1) -d(N), was plotted against N. Peaks result when A( N+ 1) increases in a discontinuous fashion, as an energy level is just tilled at certain N and the next orbital which is much less stable starts to be occupied in the cluster with N+ 1 atoms. When If orbitals are close to 2p orbitals the peak of N= 34 cannot be observed because A( N+ 1) -A(N) is too small to cause a discontinuity.

because the edge capped atom lies directly above an edge atom of the tetrahedron. M&: A face capped octahedron with k= 5 has a total number of 93 atoms (85 + 8 ). Magic numbers, n = 19, 21, 35 and 41 for alkali metal cluster cations generated from the liquid metal ion source (LMIS ) , have been observed [ 15 1. These magic numbers and the existence of high symmetry geometries to account for their structures contribute to the success of the jellium model.

5.2.2. The alkali metal cluster cations The magic numbers for cluster cations are those with closed-shell electronic structures. The following M,+ clusters can be derived using the same argument as above. M,+: Body centred cubic geometry (see table 5 ) has 0, symmetry. An ab initio calculation found that the centred square antiprism was the most stable structure. This is because a larger stabilisation energy is gained by the 1s orbital in spite of the destabilisation resulting from the splitting of lp orbitals during the conversion from body centred cube to centred square antiprism [ 5 1. MT,: A simple octahedron with k= 3 (see table 4 and fig. 6c) has a total number of 19 atoms. MT, can be also derived from icp close packing (a form of double icosahedron) but it has D5,, rather than I,, symmetry. M& : Taking one atom away from each vertex of an octahedron with k= 3 and capping one atom to each face of the octahedron, a geometry with 21 atoms (0,) can bederived (19-6+8). M&: A simple tetrahedron with k= 5 (see fig. 6b) has a total of 35 atoms (Td). M& : Capping one atom on each edge of the above tetrahedron (k= 5), a cluster with 41 atoms is generated ( 3 5 + 6 ) . However, this structure is unfavored

6. Spherical geometries In our previous paper, we concluded that when a cluster has a complete jellium closed-shell structure it will adopt a polyhedral geometry based on a high symmetry point group. The adoption of a geometric arrangement which most closely approximates to spherical will minimise the splittings of the nl shells. Consequently alternative possible geometries which are not only high symmetry but also of spherical arrangement can be proposed, which have approximately close packed arrangements of atoms. These spherical geometries can be obtained from those discussed above by assuming that the interstitial atoms are fixed and the surface atoms are allowed to move on the surface of a defined sphere in such a way that the symmetry of the cluster is maintained in the process. Maintaining the symmetry of the cluster simplifies the search for an optimum structure since those atoms lying on special positions on C, (n 3 2) axes are fixed (as long as the radius of the sphere is defined) because the atoms located on special positions on C, axes are not allowed to move away from the rotation axes. Therefore only those atoms lying on the general positions and the special positions on a mirror are allowed to move. The optimum structure is defined as that where all the surface atoms are most

D.M.P. Mingos, Z. Lin / The jeilium model for alkali metal clusters

evenly distributed on the surface of the defined sphere. The most evenly distributed structure is defined by maximising the sum of all distances between surface atoms. Under this condition, the least repulsion between surface atoms on the sphere is expected and the most evenly “spherical” structure is found. A Fortran program has been written to generate these optimum structures with magic numbers. One chooses the relative orientation of the outer “sphere” and the interstitial moiety such that no two atoms are placed in the same radical direction. This restricts a random choice in the permutation of the numbers from tables 1, 2 and 3 to get a magic number. The individual geometries are discussed below. M20: Assuming an interstitial four-atom tetrahedral moiety made of the four face centre atoms of the tetrahedron with k= 4 (see fig. 6a) and allowing the remaining 16 atoms, which include 12 special positions on mirror and 4 on C3 axes, to move, we find a Td optimum geometry which is shown in fig. 7. When the bond distances between the interstitial atoms are a and the radius of the outer sphere is 1.467a, the bond distances between the surface atoms are found to be about 1.40~. This indicates that there is less interaction between the surface atoms and at the Hartree-Fock level this structure is less stable than the close packed alternative. Calculations including correlation appear to invert the relative energies of the two possibilities [ 71. M& : An I,, symmetry structure for this cluster can be generated when 20 atoms are located on the C3 axes of I,, point group (see table 3 ) and one atom is placed on the centre. M& : It can be seen from fig. 6b that the interstitial moiety is a centred cuboctahedron. The remaining 22 surface atoms include 12 special positions on mirrors, 4 on C) axes and 6 on C2 axes. The optimum geometry with Td point group of the 22 surface atoms is shown in fig. 8 (for clarity, the interstitial cuboctahedron has been omitted from the figure). The radius of the sphere is chosen as 1.73a and the distance between surface atoms is about 1.33~ These ratios are similar to those for the MZ,,cluster. M4,,: When a tetrahedral moiety is taken as interstitial, the 36 surface atoms are then located on the 12 special mirror positions and 24 on the general positions. The optimum geometry is shown in fig. 9, which can be visualised as the superposition of figs.

23

Fig. 8. The optimum geometry with Td point group for M$ cluster (the interstitial cuboctahedron is omitted for reason ofclarity).

Fig. 9. The optimum geometry with Td point group for Ma0 cluster (the interstitial tetrahedron is omitted for reason of clarity).

Fig. 10. The two sets of atoms for the optimum structure of MN cluster (seefig. 9) (a) general positions and (b) special mirror positions.

10a and lob. The radius of the polyhedron is 1.8a and the distances between surface atoms are found to be about the same as those between the interstitial atoms. The interstitial four-atom tetrahedron has been omitted from this figure for reasons for clarity.

24

D.M. P. Mingos. Z. Lin / The jellium model for alkali metal clusters

The atoms in the interstitial tetrahedron are located in the radial directions to the centres of the four hexagonal planes (see fig. 1Oa) . M4: : A centred cuboctahedron ( 13 atoms) is the interstitial moiety and the remaining 28 atoms are located on the general positions (24) and the special positions on C3 axes (4). A Td molecule is obtained for M4: alkali metal cluster cation. MSs: A ten-atom tetrahedron with k= 3 (see table 4) can be defined as the interstitial moiety for this cluster. The remaining 48 atoms are located on the general positions of the octahedral point group. The combination of the interstitial moiety with Td symmetry and the surface geometry with 0, point group produces a structure with Td symmetry. Mgz: A truncated octahedron with k=4 (44- 6 = 38) (see fig. 6d) is taken as the interstitial moiety and the remaining 54 atoms are placed on the general positions (48) and the special positions (6) on C, axes. An Oh symmetry is generated for this cluster. M&: When a face-capped centred icosahedron ( 1 + 12 + 20~ 33 ) is taken as the interstitial moiety (see table 3) and the remaining atoms are located on the 60 mirror positions, an I,, symmetry structure is obtained for this cluster. For other high nuclearity alkali metal clusters with magic numbers, a similar method can be used to generate the possible high symmetry geometries. It depends on the choice of the interstitial moiety and the atoms on the outer sphere. When more atoms come to be considered, more inner spheres can be chosen. This means that more possible structures with high symmetry have to be considered.

7. Conclusion This paper provides a general method for generating the geometries of high nuclearity alkali metal clusters according to the jellium model with magic numbers. A theoretical justification for the adoption of high symmetry structures has been given within the framework of molecular orbital and crystal field the-

ories. Clusters which satisfy the magic numbers and can adopt a high symmetry close packed or spherical geometry are predicted to be particularly stable. Clusters with incomplete jellium closed-shell electronic structures adopt lower symmetry oblate and prolate structures.

Acknowledgement The SERC and the Chinese Academy of Sciences are thanked for their financial support and Dr. Tom Slee for helpful discussions.

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