Recent developments in classical density functional theory: Internal energy functional and diagrammatic structure of fundamental measure theory

TL;DR

This paper reviews recent advances in classical density functional theory, focusing on the internal energy functional, diagrammatic structure of fundamental measure theory, and extensions to complex hard sphere mixtures.

Contribution

It introduces a generalized variational approach using Levy’s method to express internal energy as a functional of density and entropy, and details the diagrammatic structure of fundamental measure theory.

Findings

Explicit expression for Helmholtz free energy functional without external potential

Generalization to internal energy as a functional of density and entropy

Diagrammatic structure of Rosenfeld's fundamental measure functional

Abstract

An overview of several recent developments in density functional theory for classical inhomogeneous liquids is given. We show how Levy's constrained search method can be used to derive the variational principle that underlies density functional theory. An advantage of the method is that the Helmholtz free energy as a functional of a trial one-body density is given as an explicit expression, without reference to an external potential as is the case in the standard Mermin-Evans proof by reductio ad absurdum. We show how to generalize the approach in order to express the internal energy as a functional of the one-body density distribution and of the local entropy distribution. Here the local chemical potential and the bulk temperature play the role of Lagrange multipliers in the Euler-Lagrange equations for minimiziation of the functional. As an explicit approximation for the free-energy functional for hard sphere mixtures, the diagrammatic structure of Rosenfeld's fundamental measure density unctional is laid out. Recent extensions, based on the Kierlik-Rosinberg scalar weight functions, to binary and ternary non-additive hard sphere mixtures are described.