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,…ϕn, where ϕ0 is the pure vibrational wave function and ϕι 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 ϕι at any points; (ii) the eigenvalues of the eigenvalue equations of the functions ϕ0,ϕ1,ϕ2,…ϕn which are, respectively, the vibrational energy Ev, the rotational constant Bv, and the large order centrifugal distortion constants Dv,Hv,Lv….. 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.

Other Applications of Neural Networks to Quantum Mechanical Problems

The usual method for solving the vibrational Schrödinger equation to obtain molecular vibrational spectra and the associated wave functions generally involves the expansion of the vibrational wave function, ψk(y), in terms of a linear combination of a set of basis functions.