  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.
