Analytic Energy, Gradient, and Hessian of Electrostatic Embedding QM/MM Based on Electrostatic Potential-Fitted Atomic Charges Scaling Linearly with the MM Subsystem Size.

The electrostatic potential fitting method (ESPF) is a powerful way of defining atomic charges derived from quantum density matrices fitted to reproduce a quantum mechanical charge distribution in the presence of an external electrostatic potential. These can be used in the Hamiltonian to define a robust and efficient electrostatic embedding QM/MM method. The original formulation of ESPF QM/MM was based on two main approximations, namely, neglecting the grid derivatives and nonconserving of the total QM charge. Here, we present a new ESPF atomic charge operator, which overcomes these drawbacks at virtually no extra computational cost. The new charge operators employ atom-centered grids and conserve the total charge when traced with the density matrix. We present an efficient and easy-to-implement analytic form for the energy, gradient, and hessian that scales linearly with the MM subsystem size. We show that grid derivatives and charge conservation are fundamental to preserve the translational invariance properties of energies and their derivatives and exact conditions to be satisfied by the atomic charge derivatives. As proof of concept, we compute the transition state that leads to the formation of hydrogen peroxide during cryptochrome's reoxidation reaction. Last, we show that the construction of the full QM/MM hessian scales linearly with the MM subsystem size.