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 compu-tationally efficient alternative to traditional QM/molecular mechanics (MM) approaches for modeling mixedquantum-classical systems at finite temperatures. However, both QM/MM and QM/cDFT rely on somewhatambiguous approximations, the two major ones being: i) the definition of the QM and MM regions togetherwith their coupling, and ii) the choice of the methods and levels of approximation made todescribe each region.This paper addresses the second point and develop an exact theoretical framework that allows us to clar-ify the approximations involved in the QM/cDFT formulation. We therefore establish a density functionaltheory (DFT) framework for mixed QM/MM systems within the canonical ensemble. We start by recallingthe expression of the equilibrium density matrix for a mixed system made of N qm quantum and N mm clas-sical particles. Then, we propose a variational formulation of the Helmholtz free energy in terms of the fullQM/MM density matrix. Thanks to constrained-search methods, we reformulate the problem using only thequantum and classical one-body densities. Therefore, this paper generalizes both cDFT and electronic DFT(eDFT) to QM/MM systems.We then make appear the eDFT and cDFT Levy-Lieb functionals, together with a newuniversal correlation functional for QM/MM systems. A mean-field approximation is introduced in the con-text of solvation problems and we discuss its connection with several existing cDFT-eDFT schemes.An extension to the semi-grand canonical ensemble, where the number of classical particles can fluctuate, is provided in the supplementary materials.