Inversion of the Partition Function: The First-Order Steepest-Descent Method

This paper explains and advocates the use of a Laplace‐transform steepest‐descent method for the computation of energy‐level densities $\mathrm{g(E)}$ by inversion of the partition function. The earlier Lin–Eyring approach, based on the Darwin–Fowler method, is shown to be unnecessarily obscure and in part redundant By use of a much‐simpler formulation due to Kubo, it is shown that the first‐order steepest‐descent approximation to the inversion integral follows straightforwardly from a knowledge of the relevant partition function, its energy, and specific‐heat relations. In this way several explicit equations for $\mathrm{g(E)}$ and its integral $\mathrm{W(E)}$ are derived, including two for anharmonic vibration and internal vibration–rotation. The relationship of the S–D method to Tolman's classical formula is discussed and some thermodynamic implications pointed out. Numerical results agree closely with those obtained by the direct counting of energy levels and, with certain exceptions, the indirect computations of other authors.