The multilevel summation method (MLS) was developed to evaluate long-range interactions in molecular dynamics (MD) simulations. Previously MLS was investigated in detail for the electrostatic potential by Hardy et al., and we have applied this new method to dispersion interactions. While dispersion interactions are formally short-ranged, long-range methods have to be used to calculate accurately dispersion forces in certain situations, such as in interfacial systems. Because long-range solvers tend to dominate the time needed to perform a step in MD calculations, increasing their performance is of vital importance.
The multilevel summation method offers some significant advantages when compared to mesh-based Ewald methods like the particle-particle particle-mesh and particle mesh Ewald methods. Because, unlike mesh-based Ewald methods, the multilevel summation method does not use fast Fourier transforms, they are not limited by communication and bandwidth concerns. In addition, it scales linearly in the number of particles, as compared to the O(N log N) complexity of the mesh-based methods.
While the structure of the Multilevel Summation is invariant for different potentials, every algorithmic step had to be adapted to accommodate the 1/r^6 form of the dispersion interactions. In addition, we have derived strict error bounds, similar to those obtained by Hardy for the electrostatic multilevel summation method.
Using an unoptimized implementation in C++, we can already demonstrate the linear scaling of the multilevel summation method for dispersion, and present results that suggest it will be competitive in terms of accuracy and performance with mesh-based methods.