Closure for the Ornstein-Zernike equation with pressure and free energy consistency

Abstract

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions, for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volume-like term, similar to excess chemical potential correction terms recently developed. Using our new bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ρ * < 1 and temperatures of T * = 1.5, 2.74, and 5. Our new closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and Verlet-modified closure only exhibit consistency between a few relations. Accuracy of the new closure is comparable to Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation

“…New closures requiring pressure and free energy consistency have been proposed recently and could be tried also 99 . Reassembling those ideas in a consistent way and validating them for the systems studied in this paper will be the focus of a forthcoming work.…”

Evolutionary optimization of the Verlet closure relation for the hard-sphere and square-well fluids

“…Most closure relations that enforce thermodynamic consistency do not have a unique set of parameters that solve the optimization problem 22 . A solution to this problem would be TABLE I. Compressibility factor Z = β P/ρ of the hard-sphere fluid for different values of the reduced number density, ρσ 3 .…”

“…This is the so-called partial thermodynamic consistency, which is simple to implement and use, but it has the disadvantage that other state variables might still give different results depending on the route used to calculate them 1,20 . For more robust consistency requirements, free energy must also yield the same result through different routes, and other proposals that include this information have been studied for the case of the hard-sphere fluid 17,21,22 .…”

Data-driven approximations to the bridge function yield improved closures for the Ornstein–Zernike equation

“…This suggests that taking h(r) and c(r) together as input features should be more informative than g(r) alone. 43 An additional feature w(r) can be constructed from the fluctuations of the pair distribution,…”