A variational formulation of the free energy of mixed quantum-classical systems: coupling classical and electronic density functional theories

TL;DR

The paper develops an exact, variational density-functional framework for mixed quantum-classical (QM/MM) systems in the canonical ensemble, clarifying ambiguities in QM/MM and QM/cDFT couplings. Starting from the adiabatic QM/MM density matrix, it derives a Helmholtz free-energy functional $F_0$ that is minimized over the full QM/MM densities and then recast as a universal functional of the one-body QM and MM densities via a Levy-Lieb constrained search, yielding $F_0 = \min_{(\rho,n)} \{ (v^{ext}_{qm}|\rho) + (V^{ext}_{mm}|n) + \mathcal{F}[\rho,n] \}$. The functional is explicitly decomposed into a purely quantum part $\mathcal{F}_{qm}[\rho]$, a purely classical part $\mathcal{F}_{mm}[n]$, and QM/MM coupling terms including a mean-field contribution $\varepsilon_{qm}^{mm}[\rho,n]$ and a correlation functional $\delta\mathcal{F}_{qm}^{mm}[\rho,n]$, linking electronic-structure theory with classical density theory for QM/MM. The formulation is illustrated in the context of solvation with a mean-field approximation and is complemented by supplementary material extending the theory to the semi-grand canonical ensemble. This framework provides a rigorous basis to couple eDFT and cDFT at finite temperature and offers a path to leverage existing DFT strategies for correlated QM/MM problems.

Abstract

Combining classical density functional theory (cDFT) with quantum mechanics (QM) methods offers a computationally efficient alternative to traditional QM/molecular mechanics (MM) approaches for modeling mixed quantum-classical systems at finite temperatures. However, both QM/MM and QM/cDFT rely on somewhat ambiguous approximations, the two major ones being: i) the definition of the QM and MM regions as well as the description of their coupling, and ii) the choice of the methods and levels of approximation made to describe each region. This paper addresses the second point and develop an exact theoretical framework that allows us to clarify the approximations involved in the QM/cDFT formulation. We establish a comprehensive density functional theory (DFT) framework for mixed quantum-classical systems within the canonical ensemble. We start by recalling the expression of the adiabatic equilibrium density matrix for a mixed system made of Nqm quantum and Nmm classical particles. Then, we propose a variational formulation of the Helmholtz free energy in terms of the full, non-equilibrium, QM/MM density matrix. Taking advantage of permutational symmetry and thanks to constrained-search methods, we reformulate the computation of the Helmholtz free energy using only the quantum and classical one-body densities.This paper generalizes both cDFT and electronic DFT (eDFT) to QM/MM systems. We then reformulate the functional to make the standard eDFT and cDFT Levy-Lieb functionals explicitly appear, together with a new universal correlation functional for QM/MM systems. A mean-field approximation is finally introduced in the context of solvation problems and we discuss its connection with several existing mixed cDFT-eDFT schemes. An extension to the semi-grand canonical ensemble, where the number of classical particles is allowed to fluctuate, is provided in the supplementary materials.