The evaluation of some integrals that arise in the perturbation theory of liquids

ABSTRACT

The venerable, but still very useful, perturbation theory of liquids of Barker and Henderson involves some integrals that must be obtained numerically. There is no avoiding this for potentials such as the Lennard–Jones potential. However, to obtain the pressure, Barker and Henderson also obtained the necessary density derivatives numerically. This is cumbersome and care must be used to avoid excessive error. In this paper, it is noted that the earlier and less successful perturbation theory of Frisch et al. employed an alternative method, all but forgotten, for the evaluation of similar integrals. Their method still requires numerical integration. However, it allows an investigator to obtain the integrals for the pressure by analytical differentiation. The resultant integrals must still be evaluated numerically, but the extra step and annoyance of numerical differentiation is avoided. It is noted here that this method of using density derivatives can be incorporated into the Barker–Henderson theory.

KEYWORDS:

• Perturbation theory
• equation of state
• density derivatives
• Laplace transform

Disclosure statement

No potential conflict of interest was reported by the author.