Interactions - Group Theory Applied to Chemistry - page 151

of circular currents around face centres are thus chains of arrows on the edges,
which again conserve the symmetry. In Fig. 6.8 the boundary of the T 1 face term is
thus the T 1 edge term. Clearly, the sum of the vertex and face observations should
account for all currents going through the edges, except for two additional terms
which escape edge observations. These are the two Euler invariants: the totally-
symmetric Γ 0 component corresponds to a uniform change of fluid amplitude at all
vertex basins. This does not give rise to edge currents, since it creates no gradients
over the edges. The other is the Γ component. It corresponds to a simultaneous
rotation around all faces in the same sense. Again, such rotor flows do not create net
flows through the edges, because two opposite currents are flowing through every
edge. The Euler invariant thus points to two invariant characteristic modes of the
sphere. They are not boundaries of a mode at a higher level, nor are they bounded
by a mode at a lower level. The phenomena, that these two terms describe, might
also be referred to in a topological context as the electric and magnetic monopoles .
Because of this connection to density and current, this theorem may be applied in
various ways to describe chemical bonding, frontier orbital structure, and vibrational
properties. The applications of this theorem can be greatly extended by introducing
fibre representations, as is shown below.

Taking the Dual To take the dual of a polyhedron is to replace vertices by faces
and vice-versa, as was already mentioned in Sect. 3.7 in relation to the Platonic
solids. The dual has the same number of edges as the original, but every edge is
rotated 90 ◦ . Hence the relations between v D ,e D ,f D for the dual and v,e,f for the
original are:

v D = f

e D = e (6.134)

f D = v

As a result the Euler formula is invariant under the dual operation.

v D e D f D v − e + f = − + = 2 (6.135)

A similar invariance holds for the symmetry extension, but in this case “to take the
dual” corresponds to multiplying all terms by the pseudo-scalar irrep Γ .Theterms
are then changed as follows:

Γ f D
Γ (e) × Γ = Γ ⊥ (e) = Γ e D

Γ σ (v) × Γ = Γ (v) =

(6.136)

Γ σ v D Γ (f) × Γ = Γ σ (f) =

(Γ 0 + Γ ) × Γ = Γ 0 + Γ

Hence, if the theorem holds for the original, it also holds for the dual.
Γ σ (v)

Γ (f) × Γ σ v D − Γ e D + Γ f D =

− Γ (e) + Γ = Γ 0 + Γ (6.137)