On the connections and differences among three mean-field approximations: a stringent test

This letter attempts to clarify the meaning of three closely related mean-field approximations: random phase approximation (RPA), local molecular field (LMF) approximation, and symmetry-preserving mean-field (SPMF) approximation, and their use of reliability and validity in the field of theory and simulation of liquids when the long-ranged component of the intermolecular interaction plays an important role in determining density fluctuations and correlations. The RPA in the framework of classical density functional theory (DFT) neglects the higher order correlations in the bulk and directly applies the long-ranged part of the potential to correct the pair direct correlation function of the shortranged system while the LMF approach introduces a nonuniform mimic system under a reconstructed static external potential that accounts for the average effect arising from the long-ranged component of the interaction. Furthermore, the SPMF approximation takes the viewpoint of LMF but instead instantaneously averages the long-ranged component of the potential over the degrees of freedom in the direction with preserved symmetry. The formal connections and the particular differences of the viewpoint among the three approximations are explained and their performances in producing structural properties of liquids are stringently tested using an exactly solvable model. We demonstrate that the RPA treatment often yields uncontrolled poor results for pair distribution functions of the bulk system. On the other hand, the LMF theory produces quite reasonably structural correlations when the pair distribution in the bulk is converted to the singlet particle distribution in the nonuniform system. It turns out that the SPMF approach outperforms the other two at all densities and under extreme conditions where the long-ranged component significantly contributes to the structural correlations.