A Mathematical Theory for Vibrational Levels Associated with Hydrogen Bonds II: The Non--Symmetric Case

TL;DR

This paper develops a new mathematical approach to model vibrational levels in molecules with non-symmetrical hydrogen bonds, improving accuracy over traditional methods by considering different mass scalings and specialized energy surfaces.

Contribution

It introduces an alternative to the Born--Oppenheimer approximation tailored for non-symmetric hydrogen bonds, with detailed analysis and formulas validated against experimental and numerical data.

Findings

Proves existence of quasimodes and quasienergies to arbitrary order

Provides formulas for quasienergies up to order epsilon cubed

Results compare well with experimental and numerical data

Abstract

We propose an alternative to the usual time--independent Born--Oppenheimer approximation that is specifically designed to describe molecules with non--symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. As a result, the different vibrational modes appear at different orders of approximation. Although we develop a general theory, our analysis is motivated by an examination of the F H Cl- ion. We describe our results for it in detail. We prove the existence of quasimodes and quasienergies for the nuclear vibrational and rotational motion to arbitrary order in the Born--Oppenheimer parameter epsilon. When the electronic motion is also included, we provide simple formulas for the quasienergies up to order epsilon cubed that compare well with experiment and numerical results.