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.