Linearly scaling computation of ddPCM solvation energy and forces using the fast multipole method

This paper proposes the first linear scaling implementation for the domain decomposition approach of the polarizable continuum model (ddPCM) for the computation of the solvation energy and forces. The ddPCM-equation consists of a (non-local) integral equation on the van der Waals (vdW) or solvent accessible surface (SAS) of the solute’s cavity resulting in a dense solution matrix and, in turn, one matrix-vector multiplication has a quadratic arithmetic complexity with respect to the number of atoms of the solute molecule. The use of spherical harmonics as basis functions makes it natural to employ the fast multipole method (FMM) in order to provide an asymptotically linear scaling method. In the present paper, we employ the FMM in a non-uniform manner with a clusterization based on a recursive inertial bisection. We present some numerical tests illustrating the accuracy and scaling of our implementation.