# The standard mean-field treatment of inter-particle attraction in classical DFT is better than one might expect

## TL;DR

This paper demonstrates that the standard mean-field approach in classical DFT for attractive interactions provides more accurate inhomogeneous fluid predictions than expected, surpassing the simple RPA approximation in a one-dimensional model.

## Contribution

The study reveals that mean-field DFT, when used with the test-particle method, outperforms RPA in predicting fluid structure, challenging assumptions about its accuracy based on bulk correlation functions.

## Key findings

- Mean-field DFT yields superior inhomogeneous fluid structure results.
- Results from a 1D model show better accuracy than RPA predictions.
- The quality of DFT should not be judged solely by bulk correlation approximations.

## Abstract

In classical density functional theory (DFT) the part of the Helmholtz free energy functional arising from attractive inter-particle interactions is often treated in a mean-field or van der Waals approximation. On the face of it, this is a somewhat crude treatment as the resulting functional generates the simple random phase approximation (RPA) for the bulk fluid pair direct correlation function. We explain why using standard mean-field DFT to describe inhomogeneous fluid structure and thermodynamics is more accurate than one might expect based on this observation. By considering the pair correlation function $g(x)$ and structure factor $S(k)$ of a one-dimensional model fluid, for which exact results are available, we show that the mean-field DFT, employed within the test-particle procedure, yields results much superior to those from the RPA closure of the bulk Ornstein-Zernike equation. We argue that one should not judge the quality of a DFT based solely on the approximation it generates for the bulk pair direct correlation function.