Weiner's theory for exactly solvable Schr\"odinger equation with symmetric double well potential

TL;DR

This paper develops a modified Wiener’s theory for exactly solvable Schrödinger equations with symmetric double-well potentials, enabling more accurate proton transfer rate calculations, exemplified by the hydrogen bond in ammonia dimer.

Contribution

The paper introduces a modified Wiener’s theory that eliminates severe approximations, allowing precise analytical calculation of proton transfer rates in symmetric double-well potentials.

Findings

Derived an analytic formula for proton transfer rate calculation.

Applied the method to hydrogen bond in ammonia dimer, matching experimental data.

Showed transition from Arrhenius to quantum tunneling behavior.

Abstract

The Weiner's theory (WT) is developed on the basis of the exactly solvable Schr\"odinger equation with trigonometric double-well potential (TDWP). The symmetric case of TDWP is considered. This modified version of WT (mWT) enables one to eliminate some severe approximations of the original Weiner's approach and to obtain more accurate results. An analytic formula is derived which provides the calculation of the proton transfer rate with the help of elements implemented in {\sl {Mathematica}}. We exemplify the application of mWT by calculating the proton transfer rate constant in the hydrogen bond of the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ (${\rm{H_3N\cdot\cdot\cdot H^{+} \cdot\cdot\cdot NH_3}}$). The parameters of the model for this object are extracted from available literature data on IR spectroscopy and quantum chemical calculations. The approach yields the transition from the Arrhenius-like exponential temperature dependence characteristic of thermal activation to that of quantum tunneling. Besides it is well suited for describing the phenomenon of vibrationally enhanced tunnelling.