Multilevel Summation for Dispersion: A Linear-Time Algorithm for r⁻⁶ Potentials

We have extended the multilevel summation (MLS) method, originally developed to evaluate long-range Coulombic interactions in molecular dynamics (MD) simulations [Skeel et al., J. Comput. Chem., 23, 673 (2002)], to handle dispersion interactions. While dispersion potentials are formally short-ranged, accurate calculation of forces and energies in interfacial and inhomogeneous systems require long-range methods. The MLS method offers some significant advantages compared to the particle-particle particle-mesh and smooth particle mesh Ewald methods. Unlike mesh-based Ewald methods, MLS does not use fast Fourier transforms and is thus not limited by communication and bandwidth concerns. In addition, it scales linearly in the number of particles, as compared with the $\mathcal{O}(N \log N)$ complexity of the mesh-based Ewald methods. While the structure of the MLS method is invariant for different potentials, every algorithmic step had to be adapted to accommodate the $r^{-6}$ form of the dispersion interactions. In addition, we have derived error bounds, similar to those obtained by Hardy for the electrostatic MLS [Hardy, Ph.D. thesis, University of Illinois at Urbana-Champaign (2006)]. Using a prototype implementation, we have demonstrated the linear scaling of the MLS method for dispersion, and present results establishing the accuracy and efficiency of the method.