Variational Formulation of Local Molecular Field Theory

TL;DR

This paper reformulates the Local Molecular Field theory as a free energy extremum problem, deriving a new explicit formula for the ideal free energy term that simplifies solvent modeling using molecular dynamics data.

Contribution

It introduces a variational formulation of LMF theory with a novel explicit ideal free energy expression based on short-range molecular dynamics, enabling more straightforward implementation.

Findings

Derives a new free energy functional for LMF theory.

Shows the functional minimizes at the self-consistent LMF solution.

Provides a practical approach for solvent modeling using all-atom MD data.

Abstract

In this note, we show that the Local Molecular Field theory of Weeks et. al. can be re-derived as an extremum problem for an approximate Helmholtz free energy. Using the resulting free energy as a classical, fluid density functional yields an implicit solvent method identical in form to the Molecular Density Functional theory of Borgis et. al., but with an explicit formula for the 'ideal' free energy term. This new expression for the ideal free energy term can be computed from all-atom molecular dynamics of a solvent with only short-range interactions. The key hypothesis required to make the theory valid is that all smooth (and hence long-range) energy functions obey Gaussian statistics. This is essentially a random phase approximation for perturbations from a short-range only, 'reference,' fluid. This single hypothesis is enough to prove that the self-consistent LMF procedure minimizes a novel density functional whose 'ideal' free energy is the molecular system under a specific, reference Hamiltonian, as opposed to the non-interacting gas of conventional density functionals. Implementation of this new functional into existing software should be straightforward and robust.