Tunneling splittings in vibrational spectra of non-rigid molecules. X. Reaction path Hamiltonian as zero-order approximation

Abstract

This paper completes the development of the perturbative instanton approach (PIA), presented in previous papers of this series, as a rigorous semiclassical method for the calculation of the tunneling dynamics in multidimensional potential energy surfaces. The precision of the PIA is established here by calculating upper as well as lower bounds for quantities characterizing the tunneling. Two perturbative solutions of the integral equation of instanton motion are derived using the straight-line path (SLP) and minimum energy path (MEP) as zero-order approximations. For arbitrary coupling strengths between the tunneling coordinate, ( X), and small-amplitude coordinates, { Y k }, the multidimensional extreme tunneling trajectory (ETT) is shown lie inside a strip, the boundaries of which are the trajectories ( X 1 (0), { Y k ( X 1 (0))}) and ( X 2 (0), { Y k ( X 2 (0))}). X 1 (0)( t) and X 2 (0)( t)= X 1 (0)( αt), correspond to 1D tunneling motions along the SLP in the bare potential and along the MEP in the adiabatic potential, modified by taking into account that the effective mass is X-dependent. The scaling coefficient, α, is defined by the spectral density of transverse vibrations. Along the boundary trajectories, the X 1 (0)( t) and X 2 (0)( t) motions induce transverse displacements, Y k , proportional to the coupling constants. Since the boundary trajectories are related by a scaling transformation in time, both perturbative series differ only by a renormalization of the transverse frequencies. The method of two perturbative series is also exploited for solving the semiclassical equations. The contributions of each transverse coordinate to the Euclidean action along the ETT, the width of tunneling channel, and the prefactors of the semiclassical wave functions are shown to be limited from below and from above by the corresponding parameters calculated for the boundary trajectories. These “conditions of the strip” establish the accuracy of tunneling splittings calculated within the PIA. The thus completed PIA is well adapted for the study of non-rigid molecules with strong XY-coupling. As an example, the tunneling dynamics of malonaldehyde is recalculated, and lower and upper boundaries for the proton transfer barrier are determined to equal 4.30 and 4.39 kcal/mol.