ON THE USE OF MATRIX FUNCTIONS IN THE THEORY OF VIBRATIONAL-ROTATIONAL SPECTRA

Abstract

A study of the algebraic methods used to simplify vibrational-rotational Hamiltonians led to the realization that many of the canonical or Van Vleck transformations used for this purpose in approximation procedures could be performed exactly. Formulas of the type $e^{iS}ze^{-iS} = e^{k}z$ can be used where S is usually a bilinear or quadratic function of the canonical variables, and K is a square matrix. Any function of a matrix of rank N can be reduced to a sum of N matrices whose coefficients can be represented either as functions of the eignevalues or as expansions in terms of the Cayley-Hamilton coefficients of the matrix, but efficient methods for this reduction had not been available until recently. In the engineering literature methods utilizing Vandermonde matrices have been suggested for this $purpose.^{1}$ We have now formulated substantial further generalizations and simplifications of these $procedures.^{2}$ How these methods can be used to simplify the reduction of vibrational-rotational Hamiltonians will be illustrated in a few examples.