Tunneling splitting in vibrational spectra of non-rigid molecules. I. Perturbative instanton approach

Abstract

A perturbative approach to calculating tunneling splittings in multidimensional potential energy surfaces (PES) is developed. In two-dimensional (2D) model PES, represented by a quartic X 4 potential and a harmonic oscillator Y with frequency ω, both coupled by linear ( CXY) or gated ( CX 2 Y) terms, the extreme tunneling trajectories (ETT) of zero energy are determined by solving the classical equations of motion in the inverted potentia, − V( X, Y), in the form of rapidly converging Taylor series of C/ ω < 1. The series for X( t) and Y( t) contain only even or odd powers of C/ ω, respectively. The semiclassical action on the ETT expands into a series of ( C/ ω 2). When C/ ω < 0.5), second order action reproduces the exact value with an accuracy of better than 5%. On multidimensional PES with one saddle point, the contributions to the action of mutually uncoupled transverse vibrations are additive, which enables us to introduce their spectral density, characterizing the tunneling dynamics. The semiclassical wave function of the ground state is found within the approximation of small fluctuation about the ETT. From this wave function, tunneling splittings are calculated, using the Lifshitz-Herring formula. The values obtained are in satisfactory agreement with the results of numerical diagonalization of the Hamiltonian matrix. Hydrogen transfers in malonaldehyde and in formic acid dimers are treated as examples for the application of this approach.