Vibrations of excited molecules at combination and difference frequencies

In the context of the theory of molecular vibrations, by applying the direct product operation of matrices, an equation for the vibrations of vibrationally excited molecules at combination and difference frequencies is obtained. From the solution of this equation, elements of vibration patterns and expressions that can be used for the study of changes in the molecular structural parameters and the coefficients of kinematic interactions are determined. The formulas obtained are applied to the calculation of the anharmonic electro-optical parameters of the molecules of water H₂O and its two isotopes H₂ ¹⁷O and H₂ ¹⁸O. These calculations are performed in terms of the semiempirical quantum-chemical CNDO/2 method and by numerical differentiation of dipole moment functions by employing a cubic spline approximation.