128
6 Interactions
Combination of Eqs. (6.50) and (6.51) then yields the Wigner–Eckart theorem of
Eq. (6.46), where the total interaction is the product of a scalar interaction constant
and a CG coupling coefficient. The former refers to the interaction itself, the latter
extracts the transformation properties. In case of product multiplicity, there will be
one reduced matrix element for every coupling channel, and the matrix element is
decomposed into a sum over the channels. The Wigner–Eckart theorem or matrixelement theorem is at the heart of most chemical applications of group theory. It
provides an elegant method for separating interactions into an intrinsic part and a
part that depends only on the symmetry of the problem under consideration.
An important consequence of the matrix element theorem concerns the definition
of selection rules. An interaction will be forbidden if the corresponding coupling
coefficient in the Wigner–Eckart theorem is zero. The conditions that control the
zero values of the coupling coefficients are called triangular conditions, since they
involve the combination of three irreps. Two kinds of triangular conditions must be
taken into account:
1. Selectivity on the representations: an interaction element is forbidden if the coupling of the three irreps involved is zero, i.e. if the direct product of the operator
and ket parts does not include the irrep of the bra.
Ω/ ∈ Λ × Γ
(6.52)
The triad of the three irreps may also be seen as a triple direct product,
¯
Ω × Λ × Γ , where the bra irrep appears in its complex-conjugate form. Equation (6.8) can now also be read as the character overlap between the totallysymmetric irrep and the triple product. Accordingly, the selection rule of Eq.
(6.52) can also be reformulated as: an interaction will be forbidden if the triple
product of the irreps does not contain the totally-symmetric irrep.
Γ 0 /
∈ ¯
Ω × Λ × Γ
(6.53)
2. Selectivity on the subrepresentations: subrepresentations that are defined in a
splitting field must obey the triangular conditions for the subduced irreps in the
corresponding subgroup.
6.6 Application: The Jahn–Teller Effect
In 1937 Jahn and Teller made the claim that degenerate states of molecules are
intrinsically unstable [6, 7].
Theorem 13 Non-linear molecules in a spatially-degenerate electronic state are
subject to spontaneous symmetry-breaking forces that distort the molecule to a geometry of lower symmetry, where the degeneracy is removed.
6 Interactions
Combination of Eqs. (6.50) and (6.51) then yields the Wigner–Eckart theorem of
Eq. (6.46), where the total interaction is the product of a scalar interaction constant
and a CG coupling coefficient. The former refers to the interaction itself, the latter
extracts the transformation properties. In case of product multiplicity, there will be
one reduced matrix element for every coupling channel, and the matrix element is
decomposed into a sum over the channels. The Wigner–Eckart theorem or matrixelement theorem is at the heart of most chemical applications of group theory. It
provides an elegant method for separating interactions into an intrinsic part and a
part that depends only on the symmetry of the problem under consideration.
An important consequence of the matrix element theorem concerns the definition
of selection rules. An interaction will be forbidden if the corresponding coupling
coefficient in the Wigner–Eckart theorem is zero. The conditions that control the
zero values of the coupling coefficients are called triangular conditions, since they
involve the combination of three irreps. Two kinds of triangular conditions must be
taken into account:
1. Selectivity on the representations: an interaction element is forbidden if the coupling of the three irreps involved is zero, i.e. if the direct product of the operator
and ket parts does not include the irrep of the bra.
Ω/ ∈ Λ × Γ
(6.52)
The triad of the three irreps may also be seen as a triple direct product,
¯
Ω × Λ × Γ , where the bra irrep appears in its complex-conjugate form. Equation (6.8) can now also be read as the character overlap between the totallysymmetric irrep and the triple product. Accordingly, the selection rule of Eq.
(6.52) can also be reformulated as: an interaction will be forbidden if the triple
product of the irreps does not contain the totally-symmetric irrep.
Γ 0 /
∈ ¯
Ω × Λ × Γ
(6.53)
2. Selectivity on the subrepresentations: subrepresentations that are defined in a
splitting field must obey the triangular conditions for the subduced irreps in the
corresponding subgroup.
6.6 Application: The Jahn–Teller Effect
In 1937 Jahn and Teller made the claim that degenerate states of molecules are
intrinsically unstable [6, 7].
Theorem 13 Non-linear molecules in a spatially-degenerate electronic state are
subject to spontaneous symmetry-breaking forces that distort the molecule to a geometry of lower symmetry, where the degeneracy is removed.