Electrostatic Forces: Formulas for the First Derivatives of a Polarizable, Anisotropic Electrostatic Potential Energy...

Explicit formulas are derived analytically for the first derivatives of a (i) polarizable, (ii) high-rank multipolar electrostatic potential energy function for (iii) flexible molecules. The potential energy function uses a machine learning method called Kriging to predict the local-frame multipole moments of atoms defined via the Quantum Chemical Topology (QCT) approach. These atomic multipole moments then interact via an interaction tensor based on spherical harmonics. Atom-centered local coordinate frames are used, constructed from the internal geometry of the molecular system. The forces involve derivatives of both this geometric dependence and of the trained kriging models. In the near future, these analytical forces will enable molecular dynamics and geometry optimization calculations as part of the QCT force field.