NBS Monograph 115: 2.9. Relations Between Matrix Elements

2.9. Relations Between Matrix Elements

Thus far in chapter 2, we have concerned ourselves with a determination of the symmetry properties of molecular wave functions. Symmetry operations can serve another purpose, however. They can be used to obtain a relation between values of two different matrix elements. Their usefulness here arises from the fact that the value of any matrix element is unchanged if the two wave functions and the operator involved in the integral are subjected to a symmetry operation, since such an operation corresponds to a change in variables everywhere in the integral. Sometimes the relation obtained is equivalent to a selection rule. Suppose we have two wave functions Ψ₁, Ψ₂ and an operator L. Suppose further that these quantities obey the following transformation equations under some symmetry operation σ

eq 2.21 (2.21)

By applying the symmetry operation σ to the integrand below, we obtain

(2.22)

The integral given in (2.22) is therefore nonvanishing only if n₁ + n₂ + n₃ is an even integer. We have thus obtained a selection rule.

Equations (2.21) lead to a selection rule because Ψ₁, Ψ₂, and L all transform into some constant times themselves. The quantities Ψ₁, Ψ₂, and L might transform into some constant times Ψ₃, Ψ₄, and L′, say. When this occurs, we do not obtain a selection rule, but obtain rather a relation between two different matrix elements. For example, Hamiltonian , ${\cal H}_{\rm ev}$ , and ${\cal H}_{rm r}$ of eq (1.1) are all invariant under the symmetry operation σ_υ. Thus, by applying σ_υ to both wave functions and to ${\cal H}_{\rm ev}$ in the matrix element $\langle L^\prime\Lambda^\prime S^\prime\Sigma^\prime|{\cal H}_{{\rm e}v}|L\Lambda S\Sigma\rangle$ , and by using the transformation equations (2.11), we obtain the following equality

eq 2.23 (2.23)

where the first and second factors, respectively, come from the transformation properties of the first and second wave functions. Just as in (2.11), it is often possible to obtain consistent results, when some of the quantum numbers in (2.23) do not have definite values, by arbitrarily assigning them values. These values must be used throughout all calculations, however; i.e., the values chosen here must agree with those chosen for eqs (2.11) and (2.16).

As mentioned above, ${\cal H}_{rm r}$ is also invariant under σ_υ. However, the individual operators occurring in ${\cal H}_{rm r}$ can be shown to transform as follows [15,16]:

eq 2.24 (2.24)

Transformation equations for L_±, L_z, L² and J_±, J_z, J² can be obtained by making the obvious substitutions everywhere in (2.24). By combining the transformation properties (2.24) with those of the wave functions given in (2.11), various matrix elements of the Hamiltonian can be related to each other.

2.10. Example: $L_\perp^2 \equiv L^2 - L_z^2$

We now show that the matrix elements of $L_\perp^2$ in the basis set used to label (1.18) are all equal. By applying the symmetry operation σ_υ to the wave functions and operators on the left side of the two equalities below, we obtain

eq 2.25 (2.25)

In addition, if we assume that the matrix elements of the orbital operator $L_\perp^2$ do not depend on the spin quantum numbers, then we see that

(2.26)

Equation (2.26) rests ultimately on the assumption that the nonrotating-molecule basis set functions $|\Lambda S\Sigma\rangle$ can be written as the product |Λ⟩ |SΣ⟩ of an orbital function |&Lambda⟩ and a spin function |SΣ⟩, and on the further assumption that the same orbital function is associated with all 2S + 1 spin functions corresponding to given S. These assumptions will be valid to the extent that Λ, S, and Σ are good quantum numbers. As a consequence of (2.25) and (2.26), the four matrix elements of $L_\perp^2$ in the basis set used to label the matrix (1.18) are all equal. The equalities represented by (2.25) are exact; that represented by (2.26) is only approximate.