Interactions - Group Theory Applied to Chemistry - page 133

case of a non-degenerate ground state, , which is coupled to an excited mani-

fold. The Hamiltonian is identical to the expression in Eq. ( 6.54 ). Application of the
selection rules shows that the diagonal contribution,

| ψ Σσ

, is limited to the
totally-symmetric operator associated with the harmonic restoring-force, as in Eq.
( 6.66 ). Perturbation theory further provides interactions between the ground and ex-
cited states. These interactions are usually limited to first-order contributions, which
give rise to a quadratic coordinate dependence. Hence one has, to second-order in
the displacements:

ψ Σσ | H | ψ Σσ

2 K Γ
γ

Q 2 Γγ

1

E(Q) = E 0 +

Γ

H
∂Q Γγ |

∂ 2

| ψ Λλ | ψ Σσ 0 |

Q 2 Γγ

+ (6.80)

E 0 − E Λ

Λλ Γγ

where we have used the property that the Hamiltonian matrix is hermitian. The
selection rule in this process resides with the matrix elements in the enumerator
of the bilinear term. The vibronic operator must couple ground and excited states;
hence, it is required that their triple direct product contains the totally-symmetric
irrep:

Γ 0 ∈ Γ Λ ×

×

Γ Σ (6.81)

Applying the Wigner-Eckart theorem to the matrix element yields:
ψ Λλ

ψ Σσ

H
∂Q Γγ

∂

0 =

|

Λλ ΓγΣσ Λ Γ Σ (6.82)

The sum over the λ components of the excited state, transforming as the Λ irrep,
can be simplified by using the orthonormality property of the coupling coefficients
from Eq. ( 6.16 ).

∂ H 2

| ψ Λλ |

∂Q Γγ |

ψ Σσ 0 |

2

| Λλ | ΓγΣσ Λ Γ Σ |

=

E 0 − E Λ E 0 − E Λ

λ ∈ Λ λ ∈ Λ

2

= |

Λ Γ Σ |

(6.83)

E 0 − E Λ

Note that in case of a non-degenerate ground state the product Γ Σ yields only

one irrep, since the norm of the product character string equals the order of the
group.

×

χ Γ × Σ

χ Γ × Σ = χ Γ χ Σ χ Γ χ Σ = χ Γ χ Γ =|

| | | G | (6.84)