As s¹ metal clusters become larger the stability of symmetrically packed ion cores should begin to dominate over Jahn-Teller distorted structures that stabilize the valence electrons in smaller clusters. The icosahedral structure of inert gas clusters has been known and investigated for quite some time.¹⁴⁶ Metal clusters with cuboctahedral or icosahedral packings may be present in molecular beams but their properties are difficult to distinguish from those of other approximately spherical structures. Nonetheless there is some experimental evidence for icosahedral structure in metal clusters. It has been shown that icosahedral topologies are responsible for some of the periodicities in the IP's of large Mg¹⁴⁷ clusters. Gas phase reactivity measurements on Cu clusters have suggested icosahedral structures for clusters with more than 70 atoms¹⁴⁸ and there is similar indirect evidence for icosahedral structure in Co^{149 150} and Ni^{150 151} clusters. Icosahedral structure has also been observed in Ag particles deposited on a surface using the apparatus described previously.¹⁵² Knowing that icosahedral structure could be present, it is important to distinguish between the characteristics of this geometry and the spherical or ellipsoidal structures discussed previously. Tables with the numbers of atoms in various packing geometries such as icosahedral are available,¹⁴⁷ but less is known of the electronic structure of these shapes and, in the case of metal clusters, the effects of electronic structure are likely to be apparent in the properties of the cluster.

Figure 6.1. Examples of compact cluster structures. The atoms could also represent integration points to fill such a volume. Shown first is the 13 atom icosahedral cluster that is a stable rare-gas and Lennard-Jones structure. Next shown is a 1415 atom icoshedral cluster whose 'eigenmodes' have been studied here. Finally, an example is given of a nearly spherical cluster that is face-centered-cubic (FCC) packed. Large clusters may begin to approach the bulk FCC structure in geometries such as this. The symmetrically oriented faces of this configuration would affect the 'eigenvalue' spectrum, although this has not yet been studied.

In s¹ metal clusters the valence electrons are nearly free and the cluster can be approximated as a potential box containing electrons. There are a number of possible approaches to integrate the wave function over this volume but here a hybrid method that is related to both the Hückel method and to a numerical grid integration is chosen. Similar techniques could also be used to study FCC-pieces or random packings¹⁵³ as might occur for some metal clusters. Note that this approach is still compatible with ellipsoidal distortions and that these could be applied to the icosahedral packing.

To evaluate the energy level spacings in a bounding icosahedral volume, the volume can be filled with 'atoms,' in analogy to the cluster itself, and the common Hückel model can be employed. In this model the energy levels are eigenvalues of the Hamiltonian matrix defined by:

(6.1)

where i and j are atom indices, a is a constant energy shift (the Coulomb integral), and b gives the energy spectrum its scale (the resonance integral).

If the atoms are closely packed then the Hückel model is very similar to the free-electron model solved on a discrete mesh of points located at the atomic nuclei. For example on a one dimensional grid of points labeled i with spacing a, the Hamiltonian at one point is,

(6.2)

The matrix then has terms Hii=1/a2 and Hij=-1/(2a2) and is analogous to the Hückel matrix. This type of model has been used by Manninen et. al. and the following builds on that research. ¹⁵⁴ The model is only an accurate expression of the free electron model if (1) the number of grid points (or atoms) is large enough to accurately integrate the volume, and (2) the b values are chosen to accurately represent a discretization of the Laplacian, Ѳ.

This type of Hückel or discrete integration model can be used to conveniently investigate the effects of cluster geometry on electronic shell structure. In the calculations that follow no explicit use is made of the high symmetry of the icosahedron since in the future intermediate growth sequences that correspond to the addition of atoms or integration points onto the icosahedron will be investigated. In the computations, eigenvalues of the rather large Hamiltonian matrices (up to 3000 by 3000 banded) are computed using banded matrix eigenvalue routines of EISPACK or Lanczos diagonalization routines developed specifically for this purpose.¹⁵⁵ To compare the eigenvalue spectrum to the electronic density of states in the metal cluster, each energy level is filled with two electrons and broadened with a Lorenzian function. The energy levels for a 13-atom Hückel cluster are shown in the figure that follows.

Figure 6.2. Density of State versus energy for a 13 atom 'Hückel' cluster. Energy levels broadened by convolution with a Lorenzian function. <ico13-0.law>

For more relevant larger clusters a direct application of the Hückel model does not result in an accurate integration of the volume. To improve on this, the values of b (or equivalently the integration weights) which more accurately integrate the volume represented by each point were determined. In contrast to an FCC or cubic packing, the atom sites in an icosahedron are not all identical. There are 4 different sites or atoms, as illustrated in the following figure, and each site requires a unique set of integration weights.

Figure 6.3. The four types of points or atoms within the icosahedral cluster. In the upper left is the center point in the cluster and has perfect icosahedral symmetry. In the upper right is a representative 'corner' point of which each icosahedral cluster has twelve. The perspective is rotated – the center atom is hidden – to reveal the lower symmetry. The atoms on top are nearer the cluster center and more closely packed. In the lower left is a representative 'edge' point oriented so that the cluster center is below this figure. Finally, in the lower right is a 'face' point viewed along the cluster surface. Note that these 'types' also occur within the cluster and not only on the exterior as the names edge, corner, and face might imply. As atoms in a Hückel calculation or points for numerical integration each of these types requires its own set of b coefficients or integration weightings. The number of symmetry unique atoms is 2 (center), 5 (corner), 6 (edge), 3 (face). <ico_cen.geo><ico_cor.geo><ico_edge.geo><ico_face.geo>

Although there is not an exact set of integration weights, a nearly optimal set was constructed by minimizing the error in the evaluation of the Laplacian over a set of polynomial functions. This was done for each site type. Briefly, the set of weights wi was determined to minimize the error of the evaluation of Ñ 2 f when f can be any of the polynomial functions: 1, x, y, z, x2, y2, z2, xy, xz, etc.¹⁵⁶ There is also an additional constraint that b ij=b ji to preserve the Hermitian nature of the Hamiltonian. This resulted in a set of (possibly overdetermined) linear equations that were solved in a least squares sense. The set of weights for the center type point has only two unique values: -1.6584 for the center point itself and 0.1382 for the other 12 points. The number of other unique points is noted in the previous figure. The other 3 sets of weights and further details can be found in the program source code.

The following figure compares the density of states (DOS) for a 923 atom cluster based on a direct Hückel model and on the improved integration formula. The detailed structure is changed by the improved integration formula. Since this detailed structure is often used to identify icoshedral structure in metal clusters, this is a significant improvement. The remaining figures give the DOS structure for a 1415 atom structure using the improved integration formula. There are some similarities between the patterns noted here and those observed in the feature of mass spectra of silver clusters but a more definitive conclusion awaits a more detailed comparison.

Figure 6.4. Comparison of density of states spectrum for 923 atom icosahedral cluster with the original Hückel integration formula¹⁵⁴ (dotted line) and the improved integration (solid line) developed here. In this model the unit of energy is arbitrary. Energy levels have been smoothed by convolution with a Lorenzian function, width 0.15, for visibility. Labels indicate the number of electrons required to fill the orbitals. Accurate integration is required to correctly evaluate the DOS in regions between major electronic 'shell closings'.<comp923.xls>

Figure 6.5. Density of states spectrum for 1415 atom icosahedral cluster as a function of orbital energy. Many of the 'closed shell' species are also those of the metal cluster spherical shell model. The regions between closed shells differ as do the values obtained for large numbers of electrons. In this model energy units are arbitrary. Energy levels have been smoothed by convolution with a Lorenzian function, width 0.10, for visibility.

Figure 6.6. Density of states spectrum for 1415 atom icosahedral cluster as a function of the number of atoms or electrons in the cluster. Again, energy levels have been smoothed by convolution with a Lorenzian function, width 0.10, for visibility. Note the damped oscillations that occur in the vicinity of 400 electrons. This may be indicative of 'supershell' structure.^{158 158} <ico1415.xls>