Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule

Abstract

The reduced mass of the effective quantum particle for the inversion mode ν2 of phosphine molecule PH3 is known to be a position dependent one. In the present article the inversion spectrum of PH3 is considered with the help of the Schrödinger 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 ν2=E2-E0=992.1cm-1 (2ν2=E4-E0=1972.5cm-1; 3ν2=E6-E0=2940.8cm-1; 4ν2=E8-E0=3895.9cm-1) obtained by Špirko et al. Also we use the value for the barrier height Eb=12300cm-1 that seems to be commonly accepted at present and the hypothetical value for the energy splitting of the 11-th doublet of the 10ν2 band s10=E21-E20≈7.2cm-1 suggested in the literature. Definite predictions are derived for the energy splitting of the 4-th doublet of the 3ν2 band s3=E7-E6 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 {ν2;Eb;s10} yielding s3=6.21·10-12cm-1.