Inversion of the Partition Function. II. The Full Steepest-Descent Method

Abstract

The steepest‐descent method for the computation for energy‐level densities, $\mathrm{g(E)}$ , described by Hoare and Ruijgrok [J. Chem. Phys. 52, 113 (1969)] is here extended to higher orders giving an asymptotic expansion of $\mathrm{g(E)}$ in inverse powers of the number of degrees of freedom $N$ . The higher terms take the form of cumulant‐type expressions related to the specific heat of the system. When tested against exactly counted levels for harmonic oscillators, the second‐order term gives a useful improvement on the first‐order prediction, the third‐order term is unimportant, and fourth and higher corrections completely negligible. In all cases examined the series appears to diverge in the higher orders only for uninterestingly small energies.