Multiplicity of the Solution of the Vibrational Force-Constant Problem

Abstract

When the secular equation in the form | H — Eλ | =0, where H is the matrix product GF or FG, is expanded into a polynomial equation of degree n in λ, a set of n algebraic equations in the n unknowns Ftt′ is obtained. The degrees of these equations are 1, 2, 3, ···, n, respectively. The number of solutions (sets of force constants) is n!, the number of possible sets of frequency assignments is also n!, and one set of force constants corresponds to each set of frequency assignments. When the frequencies are all different, the sets of force constants are all different, and in a diagonal F matrix all the force constants are real and positive. As an illustration, the six sets of force constants for the valence force treatment of a triatomic model of the ethyl chloride skeleton are computed, and the corresponding frequency assignments are determined.