A Method of Obtaining the Expanded Secular Equation for the Vibration Frequencies of a Molecule

TLDR

The secular equation for molecular vibration frequencies is expressed as an algebraic equation rather than a determinant. The paper presents a method to derive the expanded secular equation for molecular vibration frequencies. The method derives the expanded secular equation using literal force constants, optionally literal or numeric masses, applies symmetry to factor the equation, and offers several approximate root techniques. An algebraic example of a nonlinear general triatomic molecule demonstrates the method.

Abstract

A method is described for obtaining the secular equation for the vibration frequencies of a molecule directly in expanded form, i.e. as an algebraic rather than a determinantal equation. The force constants occur literally and the masses may occur either literally or numerically, as desired. The symmetry may be employed as usual to factor the secular equation. Several methods of obtaining approximate roots are described for which the expanded form is particularly suitable. Finally, an example, the nonlinear general triatomic molecule, is worked out algebraically.