The steepesthyphen;descent method for the computation for energyhyphen;level densities,glpar;Erpar;, described by Hoare and Ruijgrok J. Chem. Phys.52,113 (1969) is here extended to higher orders giving an asymptotic expansion ofglpar;Erpar;in inverse powers of the number of degrees of freedomN. The higher terms take the form of cumulanthyphen;type expressions related to the specific heat of the system. When tested against exactly counted levels for harmonic oscillators, the secondhyphen;order term gives a useful improvement on the firsthyphen;order prediction, the thirdhyphen;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.
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