Midtown splines: An optimal charge assignment for electrostatics calculations

Transferring particle charges to and from a grid plays a central role in the particle-mesh algorithms widely used to evaluate the electrostatic energy in molecular dynamics (MD) simulations. The computational cost of this transfer process represents a substantial part of the overall time required for simulation and is primarily determined by the size of the support (the set of grid nodes at which the transfer function is evaluated). The accuracy of the resulting approximation depends on the form of the transfer function, of which several have been proposed, as well as the size and shape of its support. Here, we show how to derive the transfer function that yields maximal asymptotic accuracy for a given support in the limit of fine grid resolution, finding that all such functions are splines, and we determine these functions (which we refer to as midtown splines) for a variety of choices of support to find optimally efficient transfer functions at accuracy levels relevant to MD simulations. We describe midtown splines that achieve fourth- and sixth-order accuracy in the grid spacing while requiring a support size of 32 and 88 grid nodes, respectively, compared to the 64 and 216 nodes required by the most widely used transfer functions (B-splines). At accuracy levels typically used in MD simulations, the use of midtown splines thus cuts the time required for charge spreading by roughly a factor of two.