Chemistry Reference
In-Depth Information
identify the more favourable structures. For example, the sequence in
b
suggests a
threefold axis and the total number of gold atoms in the kernel (
a
+
a
0
) suggests one
or two interstitial gold atoms. The methodology developed above may be extended
to other organothiolato-clusters and the results are summarised in Table
2
.
3.9 Spherical and Close-Packed Arrangements
The discussion of high nuclearity clusters above has stressed the electronic factors
which favour specific nuclearities and geometries and the purely geometric aspects,
which become increasingly important as the clusters increase in size, have not been
discussed in detail. The terms spherical and close packed have been mentioned, but
not defined very precisely. Ignoring possible ligand affects, a cluster which is
spherical in shape, has a close-packed arrangement of atoms and the requisite
number of valence electrons to complete an electronic shell is very likely to be
very stable. However, if it is not possible to satisfy these criteria simultaneously and
none of the criteria predominates, then a balance has to be struck. Ab initio, crystal
field perturbation analyses and the TSHMs all suggest that stable alkali metal and
gold clusters are associated with closed electronic shells with 2, 8, 18, 20, 34,
...
electrons, i.e. the closed shells which H¨kkinen has emphasised in his superatom
analyses [
84
,
97
,
99
-
105
]. There have also been a number of theoretical studies on
non-spherical clusters and their occurrence when the shells are partially filled
[
98
,
113
-
118
]. The sphericity of a metal cluster may be defined by doing a moment
of inertia analysis or using the definition originally proposed by Wadell in 1935
[
117
,
118
]. He defined sphericity
as the ratio of surface area of a sphere with the
same volume as the given polyhedron to the surface area of the particle:
ʨ
3
6
V
p
2
3
1
ʨ
¼
ˀ
,
A
p
where
V
p
is volume of the particle and
A
p
is the surface area of the particle. The
sphericity of a sphere is 1, and by the
isoperimetric inequality
, any particle which is
not a sphere will have sphericity less than 1. Table
3
summarises the sphericities of
some common high symmetry polyhedra which are relevant to the current review
and emphasises that for clusters with up to 13 atoms the icosahedron is the most and
the tetrahedron the least spherical. As the sphericities of the polyhedra approach
1, their description using particle in a sphere free electron models become more
appropriate. It is also noteworthy that by this criterion the icosahedron is signifi-
cantly more spherical than the cuboctahedron although they have the same number
of vertices. Condensed polyhedra become more spherical as they progress from
vertex and edge sharing to face sharing, and this transformation underlines the
united atom approach which has been discussed above. Since structures with
fivefold symmetry are incompatible with translational symmetry, they cannot
