Interactions - Group Theory Applied to Chemistry - page 127

The theorem is based on a perturbation of the Hamiltonian by small displace-
ments of the nuclei. A high-symmetry geometry is chosen as the origin, and the
nuclear displacements are described by normal modes which transform as irreps of
the point group. The nuclear positions are parameters in the electronic Hamiltonian.
One has, to second-order:

H = H 0 + H

∂ ∂ 2

2
Q Γγ

H
∂Q Γγ

1 H

∂Q Γγ ∂Q Γ γ

(6.54)

H = Q Γγ + Q Γγ Q Γ γ

0 0

Q Γγ Q Γ γ

The partial derivatives with respect to the normal modes will affect only the elec-
trostatic V Ne term in the Hamiltonian. These operators are thus electrostatic one-
electron operators. At the coordinate origin, the electronic state is degenerate, and is
described by a set of wavefunctions, , where Γ a is a degenerate irrep. The en-

ergies as functions of the coordinates are obtained by diagonalizing the Hamiltonian
matrix,

| Γ a γ a

H , with elements:

H γ a γ b = Γ a γ a | H | Γ a γ b = E 0 δ γ a γ b + Γ a γ a | H | Γ a γ b (6.55)

H is also called the Jahn-Teller (JT) matrix. The linear terms in this
matrix are of type:
Γ a γ a

Thematrixin

Q Γγ

Γ a γ a ∂

Γ a γ b Γ a γ b

H
∂Q Γγ

H
∂Q Γγ

∂

= Q Γγ (6.56)

0 0

We have used the fact that the integration in this matrix element runs over elec-
tronic coordinates, and does not affect the nuclear coordinates. The Wigner-Eckart
theorem can be applied to derive the selection rules. Since the Hamiltonian is in-
variant under the elements of the symmetry group, the transformation properties of
the operator part in this matrix element will be determined by the partial derivatives,
∂/∂Q Γγ . As we have seen in Sect. 1.3 , a partial derivative in a variable has the same
transformation properties as the variable itself. 3 The operator part is thus given by:

Γ a γ a

Γ a γ b

H
∂Q Γγ

∂

0 =

Γ a Γ Γ a Γ a γ a | ΓγΓ a γ b (6.57)

The coupling coefficient on the right-hand side of Eq. ( 6.57 ) restricts the symmetry
of the nuclear displacements to the direct square of the irrep of the electronic wave-
function. This selection rule is made even more stringent by time-reversal symmetry.
The Hamiltonian is based on displacement of nuclear charges, and not on momenta,
so as an operator it is time-even or real. 4

For spatially-degenerate irreps, which are

3 For complex variables, variable and derivative have complex-conjugate transformation properties.
4 The general time-reversal selection rules are discussed in Sect. 7.6 .