Efficient Free Energies from a Simplified Electrostatic Embedding QM/MM Approach Based on Electrostatic Potential Fitted Operators

Periodic boundary condition (PBC)-adapted formulations of quantum mechanics/molecular mechanics (QM/MM) methods allow for the accurate computation of free energies, provided there is sufficient phase-space sampling. In this work, we develop a robust and efficient QM/MM approach based on electrostatic potential fitted (ESPF) charge operators. The method combines smooth particle-mesh Ewald summation to describe QM-MM electrostatic interactions and the Ewald pair potential to treat long-range QM-QM interactions. It is fully compatible with both <i>ab initio</i> DFT and semiempirical DFTB QM/MM frameworks under PBC. We demonstrate the efficiency of the approach by implementing a thermodynamic integration (TI) scheme to compute solvation free energies and redox potentials in condensed-phase systems. For solvation energies, we introduce two separate coupling parameters to independently decouple electrostatic and van der Waals interactions. For redox potentials, a coupling parameter is introduced directly into the density matrix, interpolating between the N- and N ± 1-electron states via fractional occupations. We apply this framework to compute solvation free energies and redox potentials in water for a set of amino acid analogues and aromatic ketones, obtaining results in qualitative agreement with experimental data. This work paves the way for routine free energy calculations using electrostatic embedding QM/MM methodologies.