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

The reduced mass of the effective quantum particle for the inversion mode ${\nu }_2$ of phosphine molecule $\mathrm{P}{\mathrm{H}}_3$ is known to be a position dependent one. In the present article the inversion spectrum of $\mathrm{P}{\mathrm{H}}_3$ 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 ${\nu }_2=E_2-E_0=992.1\mathrm{c}{\mathrm{m}}^{-1}$ ($2\nu_2=E_4-E_0=1972.5
{\rm cm^{-1}}$;$3\nu_2=E_6-E_0=2940.8\ {\rm cm^{-1}}$;$4\nu_2=E_8-E_0=3895.9
{\rm cm^{-1}}$) obtained by \v Spirko et al. Also we use the value for the barrier height $E_b=12300\mathrm{c}{\mathrm{m}}^{-1}$ 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 $s_{10}=E_{21}-E_{20}\approx 7.2\mathrm{c}{\mathrm{m}}^{-1}$ 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$s_3= 6.21\cdot 10^{-12}\ {\rm cm^{-1}}$.