Analytic calculation of ground state splitting in symmetric double well potential

TL;DR

This paper derives an exact analytical solution for the ground state splitting in symmetric double-well potentials using spheroidal functions, and compares it with semiclassical WKB approximations to assess their accuracy.

Contribution

It provides a new exact solution method for ground state splitting in symmetric double-well potentials and evaluates the accuracy of existing WKB approximations.

Findings

Exact solution via spheroidal functions matches well with certain WKB methods.

WKB approximations are accurate when the tunneling doublet is well below the barrier.

WKB methods are inaccurate when the doublet is near the barrier top.

Abstract

The exact solution of the one-dimensional Schr\"odinger equation with symmetric trigonometric double-well potential (DWP) is obtained via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of ring-puckering vibration in the 1,3-dioxole (as an example of the case when the ground state tunneling doublet is well below the potential barrier top) and 2,3-dihydrofuran (as an example of the case when the ground state tunneling doublet is close to the potential barrier top) are compared with several variants of approximate semiclassical (WKB) ones. This enables us to verify the accuracy of various WKB formulas suggested in the literature: 1. ordinary WKB, i.e., the formula from the Landau and Lifshitz textbook; 2. Garg's formula; 3. instanton approach. We show that for the former case all three variants of WKB provide good accuracy while for the latter one they are very inaccurate. The results obtained provide a new theoretical tool for describing relevant experimental data on IR spectroscopy of ring-puckering vibrations.