Calculation of the reaction field due to off-center point multipoles

Two general methods for calculating the reaction field generated by a set of off-center point multipoles in a spherical cavity are presented. The methods are a generalization of Kirkwood's original theory for an arbitrary charge distribution. A polytensor formulation, similar to that previously developed for direct multipole interactions, serves to organize the computation and allows straightforward extension to higher derivatives of reaction field energy and gradients of the potential. The computation is reduced to calculation of the Cartesian derivatives of biaxial harmonics. Recursive and explicit formulas are given for the calculation. As an example, the incorporation of reaction field effects in computation of induced dipole moments is discussed. The second procedure, the central multipole method, scales linearly in calculation time with the size of the system. Methods to obtain derivatives analytically based on this method are also described. Our developments allow use of reaction field energy terms with atomic multipole-based empirical potential energy functions. Both methods show particular promise for use in simulation of heterogeneous systems, such as biopolymers, where the remainder of the cavity can be filled explicitly with solvent.