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as a whole. The united atom approach developed above is not limited to the
condensation of pairs of clusters, but may also be extended to collections of
three, four or more atoms [
125
]. It may also be used to interpret the structures of
heteronuclear clusters [
128
], where the site preferences provide an interesting
additional theoretical problem which has been analysed in general terms.
The construction of high nuclearity cluster from vertex, edge and face sharing of
smaller polyhedral units is not limited to gold clusters, but is a characteristic of
metal carbonyl clusters of the later transition metals. The bonding models devel-
oped for metal carbonyl clusters were extended to their condensed clusters by
Mingos [
71
-
73
], and the pecs of condensed clusters were related to those of the
parent polyhedron via simple relationships. It was also noted that as the clusters
became larger, radial bonding takes on a more important role than tangential
bonding and the closed shell electronic structures may be conveniently described
in terms of the simple formula 12
n
S
+
ʔ
i
, where
n
S
is the number of surface atoms
and
ʔ
i
reflects the closed shell requirements of the interstitial group of atoms, i.e. 34
for M
2
, 48 for M
3
, 60 for a tetrahedron, 86 for octahedron and 162 for a centred
icosahedron or cuboctahedron. The 12
n
S
arises from a filled d shell and a single
terminal metal ligand bond.
For gold clusters a similar formula may be proposed and related to the sec which
has been introduced above. In the absence of bridging carbonyls, the thiolato- or
phosphine ligands do not form exclusively ML fragments and ML
2
fragments are
also present (see Fig.
7
). Their metal-ligand requirements lead to 12
a
0
ML
and
14
a
0
ML
2
components to the electron count. The
a
interstitial atoms and those surface
gold atoms which are not able to bind to ligands because of the concavity of the
surface contribute 10 electrons. The electron counts for organothiolato-clusters are
summarised in Table
4
. The pec of the clusters is given by 11(
a
+
a
0
)+3
(
b
+
c
+
d
) since each of the staple ligands donates 3 electrons to the gold kernel.
It is noteworthy that the electron counts in the final two columns of the table agree
for all these examples. The sec is indicated in bold in column 3. It needs to be
emphasised that if formula 4 (
a
0
¼ 2
b
+2
c
) which relates the number of Au-S
bonds to the number of gold atoms capable of forming dative bonds is not valid,
then the total polyhedral electron count will not prove to be a good indicator of the
total pec. The calculation of pec needs to be corrected for the number of 14 electron
centres. These compounds are shown in bold in column 1 of Table
4
. The table
illustrates the good agreement between observed and calculated pecs when this
correction is made.
Acknowledgments I would like to thank Professors Roy Johnston, Zhenyang Lin, Rongchao Jin,
Larry Dahl and Dr Evgueni Mednikov for reading and providing helpful comments on the drafts of
this chapter. I also like to thank Man Sing Cheung and Zhenyang Lin for their assistance with the
molecular orbital calculations.
