Statistical rate theory requires the knowledge of the density of allowed states of the decomposing molecule. States may be disallowed if they are unbound, or if they are excluded by angular-momentum conservation requirements. To investigate such exclusion, the generalized method of steepest descents, as recently formulated by the authors [J. Chem. Phys. 51, 3006 (1969)], is used to compute the density and sum of states for a molecular species represented by a system of independent quantum oscillators and independent classical free rotors under the restriction that (1) all oscillators (harmonic or anharmonic) are subject to vibrational cutoff; or (2) one anharmonic oscillator is coupled with a two-dimensional rotor whose rotational energy is restricted by angular-momentum conservation, while its vibrational energy is not allowed to exceed its effective dissociation energy, all other oscillators being subject to Restriction (1). The molecular model used is a “small molecule” where the two restrictions produce an easily noticeable effect. It turns out that Restriction (1) has an appreciable effect on the sum of states and on the unimolecular rate constant when the oscillators are assumed to be harmonic, but the sum of states and rate constant calculated for Morse oscillators does not differ from the strictly harmonic (no cutoff) case by more than an order of magnitude at energies below the total dissociation energy. Restriction (2), applied to the weakest oscillator (bond) in the molecule, has an effect on the rate constant that depends on symmetry. If molecule and complex are of the same symmetry (angular momentum strictly conserved), the rotations are “adiabatic” and the unimolecular rate constant increases with increasing rotational quantum number J. If molecule and complex are of different symmetries (angular momentum not strictly conserved), the rotations are “inactive” and the unimolecular rate constant decreases at first with J and then goes through a minimum, at constant excitation energy in the complex. Experimental data on this point would be most desirable. Such checks as were feasible show the method of calculation to be accurate within better than 10%.

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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.

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