DFT-based QM/MM with particle-mesh Ewald for direct, long-range electrostatic embedding

We present a density functional theory (DFT)-based, quantum mechanics/molecular mechanics (QM/MM) implementation with long-range electrostatic embedding achieved by direct real-space integration of the particle-mesh Ewald (PME) computed electrostatic potential. The key transformation is the interpolation of the electrostatic potential from the PME grid to the DFT quadrature grid from which integrals are easily evaluated utilizing standard DFT machinery. We provide benchmarks of the numerical accuracy with choice of grid size and real-space corrections and demonstrate that good convergence is achieved while introducing nominal computational overhead. Furthermore, the approach requires only small modification to existing software packages as is demonstrated with our implementation in the OpenMM and Psi4 software. After presenting convergence benchmarks, we evaluate the importance of long-range electrostatic embedding in three solute/solvent systems modeled with QM/MM. Water and 1-butyl-3-methylimidazolium tetrafluoroborate (BMIM/BF₄) ionic liquid were considered as "simple" and "complex" solvents, respectively, with water and p-phenylenediamine (PPD) solute molecules treated at the QM level of theory. While electrostatic embedding with standard real-space truncation may introduce negligible errors for simple systems such as water solute in water solvent, errors become more significant when QM/MM is applied to complex solvents such as ionic liquids. An extreme example is the electrostatic embedding energy for oxidized PPD in BMIM/BF₄ for which real-space truncation produces severe errors even at 2-3 nm cutoff distances. This latter example illustrates that utilization of QM/MM to compute redox potentials within concentrated electrolytes/ionic media requires carefully chosen long-range electrostatic embedding algorithms with our presented algorithm providing a general and robust approach.