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

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. This paper aims to develop an exact theoretical framework that allows us to clarify the approximations involved in the QM/cDFT formulation and serves as a starting point for further development. We therefore establish a comprehensive density functional theory (DFT) framework for mixed quantum-classical systems within the canonical ensemble. We start with the adiabatic equilibrium density matrix for a system comprising N qm quantum and N mm classical particles and, based on this, we propose a variational formulation of the Helmholtz free energy in terms of the full 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, thus generalizing both cDFT and electronic DFT (eDFT) for QM/MM systems. We then reformulate the functional in order to explicitly make appear the eDFT and cDFT Levy-Lieb functionals, 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.