Chemistry Reference
In-Depth Information
Fig. 6.8
Face, edge and vertex SALCs for a tetrahedron. The
δ
symbol denotes taking the bound-
ary, from faces to edges, and from edges to vertices (see text). The two topological invariants are
the
A
2
face term and the
A
1
vertex term
•
The edges are one-dimensional lines. They form a set of ordered pairs,
.
Each of these can be thought of as an arrow, directed along the edge. The sym-
metry operations will interchange these arrows, but may also change their sense.
The corresponding representation is labelled as
Γ
(e)
. This symbol indicates that
the basic objects on the edge sites are not symmetric points but directed arrows.
The site group through the centre of an edge has maximal symmetry
C
2
v
and in
this site group the arrows transform as the
b
1
irrep, which is symmetric under
reflection in a plane containing the edge and antisymmetric under the symme-
try plane perpendicular to the edge. For a tetrahedron there are six edge vectors,
transforming as
T
1
+
T
2
.
{
u,v
}
Γ
(e)
=
Γ(b
1
C
2
v
↑
T
d
)
=
T
1
+
T
2
(6.130)
•
The faces may be represented as closed chains of nodes, which are bordering
a polyhedral face,
. The sequence forms a circulation around the
face, in a particular sense (going from
{
u,v,w,...
}
, etc.). The set of face
rotations forms the basis for the face representation, which is denoted as
Γ
(f)
.
In a polyhedron the maximal site group of a face is
C
nv
, and in this site group
the face rotation transforms as the rotation around the
C
n
axis, i.e. it is symmetric
u
to
w
over
v
