Exactly solvable double-well potential in Schr\"odinger equation for inversion mode of phosphine molecule

TL;DR

This paper presents an exactly solvable Schrödinger equation model with position-dependent mass and a modified double-well potential to analyze the inversion spectrum of phosphine, providing precise predictions for energy splittings.

Contribution

It introduces an exactly solvable model for the inversion mode of phosphine using a position-dependent mass Schrödinger equation with a modified double-well potential, aligning theoretical results with experimental data.

Findings

Accurate energy level predictions for phosphine inversion spectrum

Derived precise estimates for energy splittings in specific vibrational bands

Validated the model against experimental data and provided testable predictions.

Abstract

The reduced mass of the effective quantum particle for the inversion mode $\nu_2$ of phosphine molecule ${\rm{PH_3}}$ is known to be a position dependent one. In the present article the inversion spectrum of ${\rm{PH_3}}$ is considered with the help of the Schr\"odinger equation (SE) with position dependent mass and corresponding modified double-well potential. The SE is shown to be exactly solvable. The results are used for the analysis of the pertinent experimental data available in literature. We are based on the reliable value $\nu_2=E_2-E_0=992.1\ {\rm cm^{-1}}$ ($2\nu_2=E_4-E_0=1972.5\ {\rm cm^{-1}}$; $3\nu_2=E_6-E_0=2940.8\ {\rm cm^{-1}}$; $4\nu_2=E_8-E_0=3895.9\ {\rm cm^{-1}}$) obtained by \v Spirko et al. Also we use the value for the barrier height $E_b=12300 \ {\rm cm^{-1}}$ that seems to be commonly accepted at present and the hypothetical value for the energy splitting of the 11-th doublet of the $10\nu_2$ band $s_{10}=E_{21}-E_{20}\approx 7.2\ {\rm cm^{-1}}$ suggested in the literature. Definite predictions are derived for the energy splitting of the 4-th doublet of the $3\nu_2$ band $s_3=E_{7}-E_{6}$ that is a test one for the observation in the experiment of Okuda et al. SE with position dependent mass provides self-consistently the required values of $\{\nu_2; E_b;s_{10}\}$ yielding $s_3= 6.21\cdot 10^{-12}\ {\rm cm^{-1}}$.