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6 Interactions
6.1 Overlap Integrals
Operations and representations are merely theoretical constructs. What is actually
observed are the interactions. In quantum mechanics, interactions are expressed as
matrix elements of operators in a function space. When the operator is the unit
operator, the matrix elements are just overlap integrals. These are the simplest form
of interactions.
We start our analysis by examining symmetry selection rules for overlap integrals. Consider the overlap integral between the ith component of a function space
which transforms according to the irrep Γ , and the kth component of another function space transforming as Γ ′ . The overlap integral, S ki , is a scalar quantity and
thus must be invariant under the action of linear symmetry operators acting on the
functions.
S ki =
φ
Γ ′
k
ψ
Γ
i
= ˆ
R
φ
Γ ′
k
ψ
Γ
i
(6.1)
An integral being an infinite sum, the operator can be brought inside the bracket and
then transform the bra and ket parts directly.
ˆ
R
φ
Γ ′
k
ψ
Γ
i
=
ˆ
Rφ
Γ ′
k
ˆ
Rψ
Γ
i
=
jl
¯
D
Γ ′
lk (R)D
Γ
ji (R)
φ
Γ ′
l
ψ
Γ
j
(6.2)
By summing over all ˆ
R ∈ G and dividing by the group order one obtains a form to
which the GOT can be applied.
φ
Γ ′
k
ψ
Γ
i
=
1
|G|
R∈G
ˆ
R
φ
Γ ′
k
ψ
Γ
i
=
1
|G|
jl
R
¯
D
Γ ′
lk (R)D
Γ
ji (R)
φ
Γ ′
l
ψ
Γ
j
= δ Γ ′ Γ δ ik
1
dim(Γ )
j
φ
Γ ′
j
ψ
Γ
j
(6.3)
We now rewrite this result in terms of elements of the overlap matrix S:
S ki = δ Γ ′ Γ δ ik
1
dim(Γ )
j
S jj = δ Γ ′ Γ δ ik
1
dim(Γ )
Tr(S)
(6.4)
This simple derivation yields three important results:
1. Overlap integrals between functions which transform according to different irreps are zero.
2. Overlap integrals between functions which belong to different components of the
same irrep are zero.
3. Overlap integrals between functions with the same symmetry properties, i.e.
transforming as the same component of the same irrep, are independent of the
component choice provided that both components are normalized.