The test-particle induced inhomogeneous direct correlation functions and extensions of Widom's theorem: impacts on...

We develop the potential distributions of several test particles to obtain a hierarchy of the nonuniform singlet direct correlation functions (s-DCFs). These correlation functions are interpreted as the segmental chemical potentials or works of insertion of successive test particles in a classical fluid. The development has several interesting consequences: (i) it extends the Widom particle insertion formula to higher-order theorems, the first member gives the chemical potential as in the original theorem, the second member gives the incremental energy for dimer formation, with higher members giving the energies for forming trimers, tetramers, etc. (ii) The second and third order s-DCFs can be related to the cavity distribution functions y((2)) and y((3)) in the liquid-state theory. Thus we can express the triplet cavity function y((3)) in terms of these s-DCFs in an exact form. This enables us to calculate, as an illustration of the above theoretical developments, the numerical values of the s-DCFs via Monte Carlo (MC) simulation data on hard spheres. We use these data to critically analyze the commonly used approximations, the Kirkwood superposition (KSA) and the linear approximation (LA) for triplet correlation functions. An improved rule over KSA and LA is proposed for triplet hard spheres in the rolling-contact configurations. (iii) The s-DCFs are naturally suited for analyzing the chain-incremental Ansatz or hypothesis in the calculation of the chemical potentials of polymeric chain molecules. The first few segments of a polymer chain have been shown from extensive Monte Carlo simulations to not obey this Ansatz. By examining the insertion energies of successive segments through the s-DCFs, we are able to quantitatively decipher the decay of the segmental chemical potentials for at least the first three segments. Comparison with MC data on 4-mer and 8-mer hard-sphere fluids shows commensurate behavior with the s-DCFs. In addition, an analytical density functional theory is derived, through the potential distribution theorem, for obtaining these nonuniform direct correlation functions.