We construct the contact value approximation (CVA) for the pair distribution function,
g(²)(r₁, r₂), for an inhomogeneous hard sphere fluid. The CVA is an average of two radial
distribution functions, which each take as input the distance between the particles, |r₂ −r₁|,
and the average value of the radial distribution function at contact, gσ(r) at the locations
of each of the particles. In a recently published paper, an accurate function for gσ(r) was
developed, and it is made use of here. We then make a separable approximation to the
radial distribution function, gS(r), which we use to construct the separable contact value
approximation (CVA-S) to the pair distribution function.
We compare the CVA and CVA-S to Monte Carlo simulations that we have developed and
run as well as to two prior approximations to the pair distribution function. This comparison
is done in three main cases: When one particle is near a hard wall; when there is an external
particle the size of a sphere of the fluid; and for various integrals that illustrate typical use-cases
of the pair distribution function. We show reasonable quantitative agreement between
the CVA-S and simulation data, similar to that of the prior approximations. However, due
to its separable nature, the CVA-S can be efficiently used in density functional theory, which
is not the case of the prior approximations.