Page 263 - Academic Press Encyclopedia of Physical Science and Technology 3rd InOrganic Chemistry

P1: GNH Final Pages
Encyclopedia of Physical Science and Technology EN009M-428 July 18, 2001 1:6

520 Metal Particles and Cluster Compounds

Hartree–Fock results, but it can be seen that substantial
disagreement is still prevalent with theorists when elec-
tronic structures are involved.
Due to computational limitations, ab initio calculations
(more basic, fewer approximations) have been limited to
very small particles such as dimers or trimers. Lithium
has been treated the most extensively. The assumption
that the core electrons can be treated as a pseudopoten-
tial (the same for each metal) allows the extension of
these calculations to other metals. For example, particles
up to ten atoms have been treated theoretically for Ni,
Pd, Cu, and Ag. These calculations become so complex
FIGURE 5 Icosahedran (a) and cuboctahedran (b) show two dif-
even for high-speed computers, that fixed-particle geom-
ferent 13-atom clusters.
etry is often assumed. Thus, no structural information can
be derived although various electronic energies can be
estimated. Further calculations on rare gas atom clusters, which are
Calculations of the binding energy indicate a signifi- believed to model many metal atom clusters, have shown
cant increase in binding energy per atom with increase in that tetrahedral groupings of atoms are usually preferred
cluster size for certain metals. Lithium and copper clusters over octahedral groupings.
up to 13 atoms show a nearly linear relationship between The 13-atom case is most interesting because it repre-
number of atoms and binding energy. It appears that the sents the smallest structure that can have an internal atom,
dimer has a binding energy of one-fourth that of the bulk i.e., one which is not on the surface. Two of the most im-
value and the 13-atom particle has roughly two-thirds that portant 13-atom structures are the cuboctahedran and the
of the bulk value. icosahedran (Fig. 5). The cuboctahedran is derived from
Calculation of ionization potentials provides a good test the face-centered cubic (fcc, closest packed) structure and
of theoretical models. In general, there is a decrease by may be pictured as a central atom in a cube surrounded by
roughly a factor of two from the ionization potential of 12 equivalent atoms at the centers of each edge. This figure
the atoms to the work function of the bulk metal. The de- has eight triangular faces and six square faces. The icosa-
crease is not monatomic, but depends very much on parti- hedran consists of a central atom surrounded by layers of
cle geometry. There is also an odd–even alternation with five atoms each above and below. Each of these layers has
the odd-atom clusters having a lower ion-ization poten- in turn a central atom capping the figure. All twenty faces
tial, presumably because they are odd-electron systems as are triangular and all twelve vertices have fivefold sym-
compared to closed-shell structures for the even-electron metry. This leads to a more closely packed surface for the
systems. These results agree fairly well with the few ex- icosahedran than for the cuboctahedran. The icosahedran
perimental results available for comparison. can be constructed from twenty tetrahedral figures packed
Theoretical calculations of particle geometries have so that they each share three faces with only minimal dis-
◦
also been carried out by several methods. These calcu- tortion (the dihedral angle of a tetrahedron is 70.53 as
◦
lations have spanned the complete range of bond types compared to 72 for the pentagonal angles). The icosa-
including van der Waals clusters of rare gas atoms, ionic hedral structure is dynamically the most stable 13-atom
clusters of salts, and metal clusters. Somewhat surpris- cluster. Inclusion of small three-center forces does not
ingly the rare gases and metals often are predicted to change this conclusion.
have similar shaped clusters perhaps reflecting the nondi- There exist some 988 distinct minimal 13-atom struc-
rectionality of the binding forces. A common goal of tures in these dynamic calculations. Much amorphous
these calculations is to try and predict the bulk three- and/or fluctional character to these small particles is ex-
dimensional crystal structure. In general, this goal has not pected. One might even wish to ask whether they are solid
been met as the small particles often have quite different or liquid.
structure from the bulk, and the transition to bulk geome- Various experiments involving the condensation of
tries occurs only gradually and for rather large particles gases show the presence of “magic numbers,” i.e., clusters
(>500 atoms). As an example of the difficulty, most cal- with certain numbers of atoms that are significantly more
culations for rare gases (and metals) predict clusters with prevalent than others. Often these magic numbers corre-
fivefold symmetry while it is commonly known that no spond in size to the nearly spherical Mackay icosahedra
extended three-dimensional structure may have fivefold (1, 13, 55, 147, 309, 561, 932, ... ). Continuous deforma-
symmetry. tions of these structures can lead to fcc cuboctrahedral