  A numerical algorithm based on the probabilistic path integral approach for
solving Schroedinger equation has been devised to treat molecular systems
without Born-Oppenheimer approximation in the non relativistic limit at zero
temperature as an alternative to conventional Variational and perturbation
methods. Using high quality variational trial functions and path integral
method based on Generalized Feynman-Kac method, we have been able to calculate
the non-Born-Oppenheimer energy for hydrogen molecule for the sigma state and
hydrogen molecular ion. Combining these values and the value for ionization
potential for atomic hydrogen, dissociation energy and ionization potential for
hydrogen molecule have been determined to be 36 113.672(3) cm inverse and
124.446.066(10) cm inverse.respectively. Our results favorably compare with
other theoretical and experimental results and thus show the promise of being a
nonperturbative alternative for testing fundamental physical theories.
