Some Mathematical Methods for the Study of Molecular Vibrations

TLDR

The paper presents methods to simplify the calculation of vibrational frequencies for complex molecules. It introduces a vectorial scheme and algebraic rules to compute the kinetic‑energy reciprocal matrix and symmetry‑coordinate transformations, handles redundant coordinates, reduces the secular equation by separating high frequencies, and provides isotope‑product rules and normal‑coordinate determination. The methods are illustrated by applying them to the vibrational analysis of CH₃Cl.

Abstract

Developments which reduce the labor of calculating the vibration frequencies of complex molecules are described. In particular a vectorial scheme is given for obtaining the reciprocal of the matrix of the kinetic energy in terms of valence-type coordinates. A general rule for writing down the coefficients of the transformation to symmetry coordinates is derived together with a method of obtaining the kinetic energy reciprocal matrix (G) in terms of symmetry coordinates with a minimum of algebra. A treatment of redundant coordinates is developed. In addition, reduction of the secular equation by the splitting out of high frequencies, a new type of isotope product rule, and the determination of normal coordinates are discussed. The molecule CH3Cl is worked out as an illustration.