Chemistry Reference
In-Depth Information
under the axis and antisymmetric with respect to the vertical
σ
v
planes, which
invert the sense of rotation. For the tetrahedron, the face circulations transform as
A
2
+
ˆ
T
2
, as shown in Fig.
6.8
.
Γ
=
Γ(a
2
C
3
v
↑
T
d
)
=
A
2
+
T
1
(6.131)
The following theorem [
22
] applies:
Theorem 16
The alternating sum of induced representations of the vertex nodes
,
edge arrows
,
and face rotations
,
is equal to the sum of the totally-symmetric repre-
sentation
,
Γ
0
,
and the pseudo-scalar representation
,
Γ
.
The latter representation
is symmetric under proper symmetry elements and antisymmetric under improper
symmetry elements
.
Γ
σ
(v)
−
Γ
(e)
+
Γ
(f)
=
Γ
0
+
Γ
(6.132)
The Euler theorem may be considered as the dimensional form of this theorem,
which states that the alternating sum of the characters of the induced representations
under the unit element,
E
, is equal to 2, but the present theorem extends this char-
acter equality to all the operations of the group. The theorem silently implies that
irreps can be added and subtracted. In the example of the tetrahedron, the theorem
is expressed as:
Γ
σ
(v)
−
Γ
(e)
+
Γ
(f)
=
(A
1
+
T
2
)
−
(T
1
+
T
2
)
+
(A
2
+
T
2
)
=
A
1
+
A
2
(6.133)
A straightforward interpretation of the theorem is possible in terms of fluid flow
on the surface of a polyhedron.
10
Suppose observers are positioned on the vertices,
edge centres and face centres, and register the local fluid flow. When the incoming
and outgoing currents at a node are not in balance, the observers located on these
nodes will report piling up or depletion of the local fluid level. This is the scalar
property represented by the vertex term. The corresponding connection between
edge flow and vertex density is expressed by the
boundary operation
, indicated by
δ
in Fig.
6.8
. Taking the boundary of an edge arrow means replacing the arrow by
the difference of two vertex-localized scalars: a positive one (indicated by a white
circle in the figure) at the node to which the arrow's head is pointing, and a negative
one (indicated by a black circle) at the node facing the arrow's tail. This projection
from edge to vertex will not change the symmetry. Hence, in this way, the boundary
of the
T
2
edge irrep is the
T
2
vertex SALC, as illustrated in the figure. Similarly,
observers in face centres will notice the net current that is circulating around the
face. Such a circular current through the edges does not give rise to changes at the
nodes (indeed the incoming flow at a node is also leaving again), but is observable
from the centre of the face around which the current is circulating. The boundaries
10
This flow description provides a simple pictorial illustration of the abstract homology theory.
The standard reference is: [
23
].
