Isolated effective Hamiltonians for two nearly degenerate modes coupled by Coriolis and centrifugal terms

Abstract

A quantum mechanical canonical transformation due to Bogoliubov and Tyablikov (BT), applied previously to eliminate a single Coriolis term coupling two nearly degenerate vibrational degrees of freedom of a polyatomic molecule [J. Chem. Phys. 94, 461 (1991); 95, 1884 (1991)], is generalized. First, we show how to use an angular momentum dependent BT transformation to effectively decouple two vibrational degrees of freedom Coriolis coupled by two components of the angular momentum. This is accomplished by choosing the molecule-fixed axes so that, in the rotated frame, there is only one Coriolis term. Redefining the orientation of the molecule-fixed axes at equilibrium and using the BT transformation enables us to move large off-block matrix elements into vibrational blocks so that we can use second order perturbation theory to treat strong Coriolis coupling. Second, we develop a mixed BT transformation-perturbation theory method to calculate energy levels for molecules for which both Coriolis and centrifugal coupling are present. The method is tested on a two-mode model of formaldehyde using an ab initio Hamiltonian. Third, we combine our BT transformation method and conventional contact transformation theory to derive effective Hamiltonians for nearly degenerate vibrational states coupled by Coriolis and centrifugal terms without resorting to a reduced dimensional model. It is very difficult to use perturbation theory after having applied a BT transformation. We circumvent this problem by first using standard perturbation theory to decouple modes whose zeroth-order energies are well-separated and then applying a BT transformation for the strongly coupled modes. The theory is applied to an experimentally determined Hamiltonian for formic acid.