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

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 develops an exact theoretical framework that allows us to clarify the approximations involved in the QM/cDFT formulation. We, therefore, 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, which can be related to a partial Wigner transform. 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. Therefore, 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 material.