Quadratic ζ-sum rules in symmetric top molecules

Abstract

In symmetric top molecules, the summation in quadratic ζ-sum rules Σ r′ ζ rr′ ( α)2 often extends over vibrations of more than one symmetry species. It is shown that in such a case the sum can be further divided into sums in each of which the summation goes over vibrations of only one symmetry species. First, the “irreducible” quadratic sum rules of the components of ℓ-matrix are derived by using an argument similar to that adopted by Boyd and Longuet-Higgins in their derivation of linear ζ-sum rules. Then the linear sum rules of A rr ( αα) and quadratic sum rules of a r ( αβ) and ζ rr′ ( α) are derived. Several examples are given for various types of molecules.