The Vibration-Rotation Energies of the Nonlinear TriatomicXY2Type of Molecule

The problem of the vibration-rotation energies of the nonlinear triatomic molecule of the $X{Y}_2$ type and their dependence upon the constants occurring in the potential energy function has been considered in the quantum mechanics. The quantum-mechanical equation has been derived in a perfectly general manner and contains all interaction terms between rotation and oscillation and all anharmonic potential energy terms to a second degree of approximation. The approximate eigenvalues of the Schrödinger equation have been calculated by means of the nondegenerate perturbation theory of quantum mechanics and it is found that the energy may be expressed essentially as that of the molecule oscillating anharmonically and that of a semi-rigid rotator where the moments of inertia are no longer strictly speaking constants, but quantities depending to some extent also upon the vibration quantum numbers and the potential energy constants. Expressions for the effective moments of inertia and the coefficients of the centrifugal expansion terms are evaluated and given in explicit form.