Routes to the density profile and structural inconsistency

Classical density functional theory (DFT) is the primary method for investigations of inhomogeneous fluids in external fields. It requires the excess Helmholtz free energy functional as input to an Euler-Lagrange equation for the one-body density. A variant of this methodology, the force-DFT, uses instead the Yvon-Born-Green equation to generate density profiles. It is known that the latter are consistent with the virial route to the thermodynamics, while DFT is consistent with the compressibility route. In this work we will show an alternative DFT scheme using the Lovett-Mou-Buff-Wertheim (LMBW) equation to obtain density profiles, that are shown to be also consistent with the compressibility route. However, force-DFT and LMBW DFT can both be implemented using a closure relation on the level of the two-body correlation functions. This is proven to be an advantageous feature, opening the possibility of an optimisation scheme in which the structural inconsistency between different routes to the density profile is minimized. (Structural inconsistency is a generalization of the notion of thermodynamic inconsistency, familiar from bulk integral equation studies.) Numerical results are given for the density profiles of two-dimensional systems of hard-core Yukawa particles with a repulsive or an attractive tail, in planar geometry.

💡 Research Summary

This paper addresses a fundamental inconsistency that arises in classical density functional theory (DFT) when describing inhomogeneous fluids under external fields. Traditional DFT determines the equilibrium one‑body density ρ(r) by minimizing a grand‑potential functional that contains the excess Helmholtz free‑energy functional F_exc. While this variational route yields results that are consistent with the compressibility route to thermodynamics, an alternative “force‑DFT” based on the Yvon‑Born‑Green (YBG) equation produces density profiles that obey the virial route. Because the two routes are generally not equivalent when approximate two‑body correlations are used, a structural inconsistency appears – a generalisation of the well‑known thermodynamic inconsistency in bulk integral‑equation theories.

The authors introduce a new DFT scheme that employs the Lovett‑Mou‑Buff‑Wertheim (LMBW) sum‑rule instead of the YBG equation. By inserting the inhomogeneous Ornstein‑Zernike (OZ) relation together with a chosen closure (e.g., the Verlet or Modified Verlet closure) into the LMBW equation, they obtain an expression for the gradient of the density that involves a convolution of the direct correlation function c(r₁,r₂) with the density gradient. They analytically demonstrate that the resulting density profiles satisfy the compressibility route, i.e., the pressure obtained from the wall‑contact theorem matches the compressibility pressure derived from the bulk direct correlation function at zero wave‑vector. Thus, LMBW‑DFT provides a compressibility‑consistent alternative to standard variational DFT.

A key insight of the work is that both the YBG and LMBW sum‑rules can be implemented with the same closure relation. The Modified Verlet closure introduces a tunable parameter α_V that can be adjusted to minimise the difference between the YBG‑derived and LMBW‑derived density profiles. This optimisation constitutes a novel “structural consistency” scheme: by varying α_V the authors enforce that the two routes to the density profile produce the same result, thereby eliminating the structural inconsistency.

To validate the theory, the authors consider two‑dimensional fluids of hard‑core Yukawa particles confined by a planar wall. The pair potential consists of an impenetrable hard core of diameter σ plus an exponential tail ϕ_tail(r)=ε exp