Structure of nonuniform hard sphere fluids from shifted linear truncations of functional expansions

Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.