Tunneling splittings in vibrational spectra of non-rigid molecules:: III. Tunneling coordinate-dependent coupling between small amplitude motions

Abstract

The perturbative instanton approach (PIA), developed previously to evaluate tunneling in multidimensional potential energy surfaces, is extended to include tunneling coordinate ( X)-dependent interactions between small amplitude vibrations Y k . The form of the coupling terms considered here depends on the symmetry of the two, coupled, transverse vibrations. When both vibrations have the same symmetry, the coupling is ∼ X 2 Y k Y l , while ∼ XY k Y l coupling exists between coordinates of the different symmetry. When the large amplitude motions is described by an angular coordinate φ with period 2π/ n, the corresponding interaction terms have the form Y k Y l ·cos( nφ) and Y k Y l ·sin( nφ), respectively. These couplings cause a rotation of the transverse coordinate system along the extreme tunneling trajectory (ETT), and the semi-classical action depends on the sign of coupling constants. The first-order analytical expressions for the ETT, the semi-classical action and the prefactors of the wave functions of the ground and first excited states are found from perturbative solutions of the equations of motion, the Hamilton–Jacobi and the transport equations, respectively. When the longitudinal frequency in the well approaches the renormalized frequency of one of the transverse vibration, a resonant increase of the tunneling splitting results for the corresponding vibrational level.