Simple formulas for the vibrational and rotational eigenvalues of the lennard-jones 12-6 potential

Abstract

Dunham expressed the term values of a generalized rotating vibrator as a double power series in the vibrational and rotational quantum numbers. Eighteen coefficients in this series have been evaluated for the Lennard-Jones (12-6) potential. The accuracy of the resulting term-value formula has been tested by direct numerical solution of the radial equation and was found to be very good. The addition of a few empirical corrections for states lying close to the dissociation limit provides an eigenvalue predictor which is accurate to better than a part per million in the most favorable cases and never worse than a part per thousand. The range covered is from systems with only a single bound state to ones with more than ten thousand, e.g., the iodine dimer. When used to provide trial eigenvalues for Cooley's procedure for solving the radial equation, the predictor successfully discriminates between states separated from each other and from the dissociation limit by less than one part in 10 5. Simple formulas have also been derived for the zero-point energy, the number of pure vibrational states, and the total number of bound states. Finally, it is shown that even fourth-order WKB terms can make significant contributions for very light systems and that there is no indication that Dunham's series for this potential model fails to converge even for term values lying arbitrarily close to the dissociation limit.