Comparative Test of Approximations for Calculation of Energy-Level Densities

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Abstract
Two molecular models, one representing a “small” molecule and the other a “large” molecule, both with several free rotors, are used to test, by comparison with exact count, seven approximation formulas for the calculation of density of harmonic vibration–rotation states, at energies between 1000 and 20 000 cm−1. The techniques of the classical approach, the Laplace transform, steepest descents, and symmetric functions are represented among the seven formulas. Also shown is the effect of the various approximations on the energy dependence of the computed unimolecular rate constant, and on ion breakdown curves calculated by the statistical theory of mass spectra. It is concluded that the techniques of the Laplace transform [J. Chem. Phys. 46, 3736 (1967)] and steepest descents, as formulated in an Appendix, yield the best all‐around approximation, while a semiclassical‐type formula [J. Chem. Phys. 41, 1883 (1964)] requires least computational effort and still gives excellent results in most cases. In an Appendix, the method of steepest descents is generalized and its connection with the Laplace transform noted.