An asymptotic method in asymmetric rotor theory

Abstract

An asymptotic method developed in the theory of the Mathieu equation is adapted to the treatment of asymmetric rotor problems in the domain of high J and low K, i.e., in the domain characterized by the condition δJ(J + 1) n 2 ⪢ 1 , where δ = (B - C) (A - C) is a parameter of asymmetry, and, depending upon the symmetry type, n = K or K + 1. The method is essentially based on the observation that a differential equation of infinite order can be found which generates a matrix of infinite rank possessing elements identical with the elements of the finite asymmetric rotor matrix within the domain in which this matrix is defined. Methods developed in the theory of differential equations can thus be adapted to asymmetric rotor theory. The asymptotic method leads to expansions in terms of descending powers of [δJ(J + 1)] 1 2 and J( J + 1), and to explicit formulas for level energies, transition frequencies, etc. These formulas serve a twofold purpose: (1) They give a more direct insight into features characteristic for the high J-low K domain than numerical results alone. (2) They make possible a straightforward derivation of approximate numerical results in a domain where tabulated values are not available, and where conventional methods become rather laborious and time-consuming. For J = 40, for instance, 110, 96, and 112 of 820 tabulated values for the asymmetric rotor eigenvalues are reproduced by these formulas with errors less than 0.00005, 0.00050, 0.00500, etc. The degree of accuracy attained increases rapidly with increasing J. Substantial further improvement of the asymptotic formalism appears feasible.