Symmetry and Fourier analysis of the ab initio-determined torsional variation of structural and Hessian-related quantities for application to vibration–torsion–rotation interactions in CH 3OH

Abstract

The aim of the present paper is to investigate the use of quantum chemistry calculations to obtain the torsional dependence of various structural and vibrational-force-field-related quantities that could help in estimating the vibration–torsion–rotation interaction terms needed to treat perturbations observed in the spectra of methanol-like molecules. We begin by using the Gaussian suite of programs to determine the steepest-descent path from a stationary point at the top of the internal rotation potential barrier in methanol to the equilibrium structure at the bottom of the barrier. This procedure requires determining the gradient ∇ V of the potential (as calculated in mass-weighted Cartesian coordinates) along the internal rotation path. In addition, we use the Gaussian suite to calculate the Hessian ∇∇ V along this path and to generate from these second derivatives the 3 N − 7 small-amplitude vibrational frequencies and the 3 N Cartesian vibrational displacements for each of these vibrations. We then symmetrize the internal coordinates used in presenting the structures, gradients, Hessians and vibrational displacements along the path to take into account the periodic variation of the behavior of the three methyl hydrogen atoms H i as they pass in turn through the C s -plane of the HOC frame. The symmetrized linear combinations of the CH i stretches, of the OCH i bends, and of the HOCH i dihedral angles of the methyl group depend on the internal rotation angle γ and they are determined by considering coordinate transformations from the G 6 permutation-inversion group appropriate for internally rotating methanol. This symmetrization procedure permits us to explore the feasibility of expressing the structures, gradients, Hessians, and vibrational displacement vectors along the internal rotation path as short Fourier series in γ, which is one of the main goals of this paper. In summary, we find that the symmetrized structures, gradients, and Hessians, as well as nine of the 11 projected vibrational frequencies and the vibrational displacement vectors for the three vibrations occurring primarily in the HOC frame can be expressed by short Fourier series expansions to their precision in the Gaussian output, and that these series involve only sin 3 nγ or only cos 3 nγ terms, as required by G 6 symmetry considerations. A preliminary discussion is given of why short Fourier expansions fail for the projected frequencies of the two methyl asymmetric stretches, and for the vibrational displacement vectors of the methyl group vibrational modes. Looking more closely at the symmetrized and projected 3 N × 3 N Hessian, we find algebraically that only elements in the (3 N − 7) × (3 N − 7) small-amplitude-vibrational block of the Hessian are useful for spectroscopic problems. Non-zero elements in the rest of the 3 N × 3 N symmetrized and projected Hessian cannot be converted into quantities needed for perturbation studies.