Chemistry Reference
In-Depth Information
In this LCAO space several irreps occur multiple times, but they can all be distin-
guished by the specific direct product from which they originated.
6.10 Application: Bonding Schemes for Polyhedra
Leonhard Euler dominated the mathematics of the 18th century. One of his famous
discoveries was the polyhedral theorem, which marks the beginning of topology.
A polyhedron has three structural elements: vertices, edges, and faces.
9
The num-
bers of these will be represented as
v
,
e
, and
f
, respectively. Then, for a polyhedron,
the following theorem holds:
Theorem 15
In a convex polyhedron the alternating sum of the numbers of vertices
,
edges
,
and faces is always equal to
2.
v
−
e
+
f
=
2
(6.128)
2.
The 2 in the right-hand side of Eq. (
6.128
) is called the Euler invariant. It is a
topo-
logical
characteristic. Topology draws attention to properties of surfaces, which are
not affected when surfaces are stretched or deformed, as one can do with objects
made of rubber or clay. Topology is thus not concerned with regular shapes, and in
this sense seems to be completely outside our subject of symmetry; yet, as we intend
to show in this section, there is in fact a deep connection, which also carries over to
molecular properties. The surface to which the 2 in the theorem refers is the surface
of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded
or mapped on the surface of a sphere. Group theory, and in particular the induction
of representations, provides the tools to understand this invariant. To this end, each
of the terms in the Euler equation is replaced by an induced representation, which
is based on the particular nature of the corresponding structural element. In Fig.
6.8
we illustrate the results for the case of the tetrahedron.
As an example, in a cube one has
v
=
8
,e
=
12
,f
=
6, and hence 8
−
12
+
6
=
•
The vertices, being zero-dimensional points, form a set of nodes,
, which are
permuted under the symmetry operations of the polyhedron. The representation
of this set is the positional representation,
Γ
σ
(v)
.The
σ
here refers to the fact that
the sites themselves transform as totally-symmetric objects in the site group. If
the cluster contains several orbits, the induced representation is of course the sum
of the individual positional representations. In Fig.
6.8
the vertex representation is
A
1
+
T
2
. In Sect.
4.7
we have already encountered these irreps, when discussing
the
sp
3
{
u
}
hybridization of carbon.
Γ
σ
(v)
=
Γ(a
1
C
3
v
↑
T
d
)
=
A
1
+
T
2
(6.129)
9
In geometry a
vertex
is a point were two or more lines meet.
