Solution of the Rovibrational Schrödinger Equation of a Molecule Using the Volterra Integral Equation

By using the Rayleigh-Schrödinger perturbation theory the rovibrational wave function is expanded in terms of the series of functions φ₀,φ₁,φ₂,.φ_n, where φ₀ is the pure vibrational wave function and φ_i are the rotational harmonics. By replacing the Schrödinger differential equation by the Volterra integral equation the two canonical functions α₀ and β₀ are well defined for a given potential function. These functions allow the determination of (i) the values of the functions φ_i at any points; (ii) the eigenvalues of the eigenvalue equations of the functions φ₀,φ₁,φ₂,.φ_n which are, respectively, the vibrational energy E_v, the rotational constant B_v, and the large order centrifugal distortion constants D_v,H_v,L_v. Based on these canonical functions and in the Born-Oppenheimer approximation these constants can be obtained with accurate estimates for the low and high excited electronic state and for any values of the vibrational and rotational quantum numbers v and J even near dissociation. As application, the calculations have been done for the potential energy curves: Morse, Lenard Jones, Reidberg-Klein-Rees (RKR), ab initio, Simon-Parr-Finlin, Kratzer, and Dunhum with a variable step for the empirical potentials. A program is available for these calculations free of charge with the corresponding author.