Calculation of the VIB-rotational state density of polyatomic molecules by adiabatic switching

Abstract

The problem of estimating the vibrational and vib-rotor state density (both local and cumulative) for polyatomic molecules is approached using the Hertz invariant of classical Hamiltonian dynamics. Such state densities are key input to statistical theoreis of reaction dynamics such as that of RRKM and transition-state theory (TST), and have recently found to be seriously under-estimated by standard Harmonic and direct count Morse oscillator methods. In the semi-classical limit (which is quite appropriate for the very high state densities of interest here) the cumulative state density at energy E is simply the volume of the energy shell, V( E), divided by h N , h being Planck's constant, and N the number of degrees of freedom. The local state density is then h −N d V( E)/d E. Use may thus be made of the fact, noted by Hertz, that the energy shell is an adiabatic invariant for ergodic Hamiltonian systems. This invariance, combined with the time-dependent version of Liouville's theorem allows rigorous estimation of the phase volume of ergodic Hamiltonian systems which are adiabatically connected to other ergodic systems of known phase volume. As such ergodic reference systems with known phase volume are not easily found, we show that, when taken as an averaging method, the Hertz invariant gives excellent results when the reference Hamiltonian is not only non-ergodic, but in fact is separable. Thus, using an N degree of freedom isotropic oscillator reference, classical adiabatic switching is shown to give cumulative and local state densities for a model system with numbers of degrees of vibrational freedom ranging from 4, 6, 8…20… and finally up to 1000. In the regime beyond 8 degrees of freedom primitive Monte-Carlo sampling is inapplicable in a practical sense.