Tunneling splittings in vibrational spectra of non-rigid molecules: VI. Asymmetric double-well potentials Original Research Article

The perturbative instanton approach (PIA) developed in the previous papers of this series [V.A. Benderskii, E.V. Vetoshkin, S.Yu. Grebenshchikov, L. von Laue, H.P. Trommsdorff, Chem. Phys. 219 (1997) 119; V.A. Benderskii, E.V. Vetoshkin, L. von Laue, H.P. Trommsdorff, Chem. Phys. 219 (1997) 143; V.A. Benderskii, E.V. Vetoshkin, H.P. Trommsdorff, Chem. Phys. 234 (1998) 153; V.A. Benderskii, E.V. Vetoshkin, Chem. Phys. 234 (1998) 173] has been extended to asymmetric double-well potentials, characterized by an energy difference, D, between the two potential wells. For multidimensional double-well potentials with an asymmetry comparable to or smaller than the fundamental frequency, Ω, of the tunneling mode in the wells, the semi-classical wave functions and tunneling splittings in the ground and low-lying excited vibrational states are calculated. The semi-classical equations are written in a form, in which the energy asymmetry, D, and eigenvalues, as terms of first-order of ℏ, are replaced from the Hamilton–Jacobi equation (HJE) into the transport equation. This replacement, symmetrizing the HJE, makes it possible to avoid the construction of a tunneling trajectory between inequivalent wells and to use the extreme classical trajectory in the upside-down symmetric potential. Comparison of the results of the quantum and PIA calculations shows that the latter provide an evaluation of the tunneling matrix element with an accuracy of better than 10%, provided that D/ℏΩ≤3 and the semi-classical parameter γ>5. A resonant increase of the tunneling splitting is predicted, if one of the transverse frequencies or the difference between longitudinal and transverse frequencies become equal to D. The relation between these resonances and the Flynn–Stoneham and Skinner–Trommsdorff mechanisms of vibrationally assisted incoherent tunneling is discussed.