Effective pair potentials in fluids in the presence of three-body forces

The physical properties of a fluid in which there is only a two-body potential, u_αβ , can be expressed in terms of the total correlation function, h ₁₂, which is a sum of all connected graphs with root-points on molecules 1 and 2, whose links are f bonds, where f_αβ = exp (-u_αβ/κT) - 1. It is shown that the total correlation function in the presence of a weak three-body potential, u_αβγ , is h ₁₂*, where h ₁₂* is the sum of all two-body connected graphs in which each f bond is replaced in turn by an f* bond, where

and where ? is a sub-set of the elementary graphs each of which contains one f_αβγ link. We call this sub-set the line-irreducible graphs, and its leading term is a graph discussed by Rushbrooke and Silbert.

The three-body potential is set equal to the dipole-dipole-dipole potential of Axilrod and Teller, and the analytic properties and numerical values of the first term ?₁ examined in detail.

Other effective potentials have been defined and the relations between them are elucidated. In particular it is shown that the first term in ? cannot be used to obtain the effective link f* at liquid densities, but that it can be compared with the dependence on density of the effective potential u* obtained by Mikolaj and Pings from the x-ray diffraction of compressed argon.