Exact solution for a class of molecular rotation-inversion spectra by infinite matrix diagonalization

Abstract

Molecules which consist of a rigid frame of C _2v symmetry and an invertor moving around an internal axis fixed in the frame may be considered as a specific case of semi-rigid molecules with one finite internal degree of freedom. Representative examples are ethylene imine, phosphirane, ammonia-d ₁, cyanoamine. For this class of molecules the solution to the quantum mechanical rotation-inversion problem is given. Starting from the classical rotation-inversion hamiltonian, the symmetry group of the problem is shown to be isomorphous to O ⁺(3) × V ₄. The kinetic energy matrix is then calculated analytically in the direct product of the bases of the symmetric rotor and the trigonometric functions. The calculation of the matrix elements, though quite complicated, is possible in closed form by complex integration. This in turn leads to a new aspect of the problem, which consists in a discussion of the zeros of the kinetic determinant |g _mn | = g. The roots of g lying within the unit circle in the complex plane are shown to determine directly the asymptotic behaviour of the matrix elements. Using an appropriate function for the analytical representation of the two minima potential energy it is shown that a finite number of eigenvalues may be calculated to a precision sufficient for high-resolution microwave spectroscopy. Furthermore, selection rules and analytical expressions for electric dipole transition matrix elements are given, based on a model in which the electric dipole moment depends in a simple manner on the inversion angle. An application of the theory to the microwave spectrum of ethylene imine and cyanoamine is briefly discussed.