The use of pair force laws to predict liquid structure is exemplified by the calculations of Swamy (1986) on liquid Na and Al. Using the hypernetted chain ((5.5) with E → c), and also the method of Machin-Woodhead and Chihara (MWC): see Chihara (1984; and other references there), Swamy has explored the results of using various oscillatory potentials to assess the applicability of these integral equations for such force laws. The results thereby obtained have been compared with molecular-dynamics simulation. His studies indicate that the HNC equation underestimates the main peak in S(k). The MWC method seems to give good results for S(k) according to Swamy's studies for shorter-range potentials, but when oscillatory effects are included, it is also deficient near the first peak of S(k).

While difficulties remain with integral-equation treatments of liquid structure for such direct calculations of S(k) from a given potential, it has been known for a long time that the inverse problem of extracting φ(r) from an experimentally determined S(k) (Johnson and March, 1963) is a much more stringent probe of an approximate integral equation. This point has been made quite clear in the work of Levesque, Weis and Reatto (1985; see also Reatto, 1988), which is summarized in Section 5.4.

In this appendix, procedures will be described by which the inversion of liquid structure data can yield pair potentials when these are constrained by requirements imposed by the form of pseudopotential theory, already discussed in Chapters 3 and 4.