Flexible multipole moments in smooth particle mesh Ewald

The smooth particle mesh Ewald sum is extended with additional force terms that arise from the so-called flexible multipole moments. These are multipole moments (of any rank) that depend explicitly on atomic positions in some local environment that can be made arbitrarily large. By introducing explicit dependence on atomic positions, flexible multipole moments are polarized by their local environment, allowing both intramolecular and intermolecular polarizations to be captured. Multipolar torques are discussed in detail, and it is shown that they arise naturally in the presented framework. Furthermore, we give details of how we validated our implementation of the flexible smooth particle mesh Ewald sum by considering two mathematical limits of the smooth particle mesh Ewald summation.