The Rotation-Vibration Coupling in Diatomic Molecules

A solution of the wave equation for the nuclear motion of a diatomic molecule with a Morse potential function and the rotational term included is given. The wave functions are found to have the same form as the functions obtained when the rotational term is neglected. The constants ${D}_{e}$ and ${\ensuremath{\alpha}}_{e}$ in the equations ${B}_{v}={B}_{e}\ensuremath{-}{\ensuremath{\alpha}}_{e}(v+\frac{1}{2}),$ ${D}_{v}={D}_{e}+{\ensuremath{\beta}}_{e}(v+\frac{1}{2}),$ are found to be given by the relations ${D}_{e}=\ensuremath{-}\frac{4{{B}_{e}}^{3}}{{{\ensuremath{\omega}}_{e}}^{2}}$ ${\ensuremath{\alpha}}_{e}=2{x}_{e}{B}_{e}(3{[\frac{{B}_{e}}{{x}_{e}{\ensuremath{\omega}}_{e}}]}^{\frac{1}{2}}\ensuremath{-}\frac{3{B}_{e}}{{x}_{e}{\ensuremath{\omega}}_{e}}),$ a result which can also be derived from Dunham's formulas. Eq. (2) differs from the corresponding relation in Kratzer's formula by the term in parenthesis. This term is fairly constant for a number of molecules and has an average value of 0.7\ifmmode\pm\else\textpm\fi{}0.1 as was found empirically by Birge. The values of ${\ensuremath{\alpha}}_{e}$ given by (2) show satisfactory agreement with experimental values for many molecules.