Methodologies to Solve the Vibrational Schrödinger Equation

Abstract

The solution of the vibrational Schrödinger equation is of fundamental importance in the interpretation of infrared spectra and the information about the fundamental physics encodedtherein. Unfortunately, due to the high dimensionality of vibrational problems, a direct solution rarely presents itself. Therefore, the development of models and methods to provide approximate solutions is crucial to the effective use of vibrational spectroscopies. In this work, various methods to solve the Schrödinger equation and models built off of such methods are presented, applied, and analyzed. First, a broad overview of the key methods in constructing representations of the molecular Hamiltonian is provided. Next, highly-accurate, low-dimensional approaches are applied to understand an interesting correlation between the frequencies of hydrogen bonds and OH bond lengths as well as to the interpretation of the vibrational spectrum of the H5+ molecular ion. After that, vibrational perturbation theory, an approximate, high-dimensional method is introduced and our recent developments in extending the method are discussed. Finally, recent work on using vibrational perturbation theory to identify resonances is introduced and a brief discussion of possible ways to combine vibrational perturbation theory with low-dimensional approaches is provided.