We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all electrostatic parameters for all possible geometrical perturbations
in order to obtain the energy gradient required for performing molecular dynamics simulations. By making use of a Lagrange formalism, however, this computational demanding task can be replaced by solving a single equation similar to that for determining the polarization energy itself. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic parameters from the energy expression depending on these parameters without a large computational penalty provides flexibility in the design of new force fields.
nonvariational multipoles supplementary