Properties of the spectra of asymmetric molecules: matrix evaluation in bases of spherical harmonics and common generating function

Abstract

The Schrödinger equation for the rotational states of asymmetric molecules is known to be separable in spheroconal coordinates and integrable in terms of Lamé functions. However, the numerical evaluation of the latter has not been developed efficiently, thereby limiting the practical application of such solutions. In this article, the matrix evaluation of the rotational states is formulated and implemented numerically for any asymmetric molecule, using the familiar bases of spherical harmonics. The matrix of the Hamiltonian - in a frame of reference fixed in the molecule and oriented along its principal axes - is constructed in the chosen basis and shown to separate into blocks of $(\ell+1) \times (\ell+1)$ and $\ell \times \ell$, for each value of the angular momentum quantum number $\ell$. The diagonalization of the successive blocks leads to accurate values of eigenenergies and eigenvectors for all values of the asymmetry parameters. The connection between these rotational states and their Lamé function representation is also established, identifying at the same time a common generating function for spherical harmonics and spheroconal harmonics.