A Theoretical Study on Vibrational Energies of Molecular Hydrogen and its Isotopes using a Semi-Classical Approximation

This study aims to apply a semi-classical approach using some analytically solvable potential functions to accurately compute the first ten pure vibrational energies of molecular hydrogen (H 2 ) and its isotopes in their ground electronic states. This study also aims at comparing the accuracy of the potential functions within the framework of the semi-classical approximation. The performance of the approximation was investigated as a function of the molecular mass. In this approximation, the nuclei were assumed to move in a classical potential. The Bohr-Sommerfeld quantization rule was then applied to calculate the vibrational energies of the molecules numerically. The results indicated that the first vibrational transition frequencies (v 1 s0 ) of all hydrogen isotopes were consistent with the experimental ones, with a minimum percentage error of 0.02% for ditritium (T 2 ) molecule using the Modified-Rosen-Morse potential. It was also demonstrated that, in general, the Rosen-Morse and the Modified-Rosen-Morse potential functions were better in terms of calculating the vibrational energies of the molecules than Morse potential. Interestingly, the Morse potential was found to be better than the Manning-Rosen potential. Finally, the semi-classical approximation was found to perform better for heavier isotopes for all potentials applied in this study.