# DETERMINATION OF THE POTENTIAL AND WAVEFUNCTIONS OF THE HYDROGEN MOLECULE BY THE USE OF THE GEL'FAND-LEVITAN EQUATION

#### Abstract

The purpose of this thesis is to calculate the potential and the wavefunctions associated with the ground state of the hydrogen molecule. There are other methods to obtain the potential, but most allow only one minimum, are generated by a single convergent power series in (v = 1/2), where v is the quantum vibrational number, have a specific formula which involves derived molecular constants and require a specific model, as for example treating the molecule as an harmonic or anharmonic oscillator. For a number of excited electronic states, there are definite signs of an abrupt change in the rate of convergence of the vibrational energy levels, which are then best represented by two power series in (v + 1/2). When there are strong perturbations, it may be quite impracticable to use a modification of the power series method. It is then desirable to make some use of the experimental determined energy levels themselves rather than some derived formulae for these levels. The potential and wavefunctions in this thesis are derived from the integral equation of Gel'fand-Levitan and the only data needed to generate the G-L potential and wavefunctions are the vibrational levels of the molecule. Both the G-L potential and wavefunctions involve a matrix which consists of a row and column for each vibrational level. An attempt is made to simplify the G-L potential for small and large internuclear separations. Computer calculations follow Cooley's method for eigenvalue corrections in solving Schrodinger's equation for central fields. By substituting the G-L potential for each of the 15 levels of hydrogen into Schrodinger's equation, the correct energy eigenvalues are obtained. The G-L potential for the ground state of the hydrogen molecules which contain 15 vibrational levels is correct, since it reproduces the correct eigenvalues and agrees with the RKR potential, which is the accepted potential.

#### Subject Area

Molecules

#### Recommended Citation

JENNINGS, THOMAS ROGER, "DETERMINATION OF THE POTENTIAL AND WAVEFUNCTIONS OF THE HYDROGEN MOLECULE BY THE USE OF THE GEL'FAND-LEVITAN EQUATION" (1983). *ETD Collection for Fordham University*. AAI8323532.

https://fordham.bepress.com/dissertations/AAI8323532