Boundary Conditions and the Treatment of Long-Range Forces

Boundary Conditions and the Treatment of Long-Range Forces 

Molecular dynamics or Monte Carlo simulations of bulk liquids employ 3-dimensional (3D) periodic boundary conditions (PBC) (using typically a cubic or a truncated octahedron box), which are designed specifically 

The short-range Lennard-Jones interactions may be truncated at a distance (typically done with a continuous switching function with a continuous first derivative ), which can be as small as 2.5 t. One expects an error of only about 1% in the total internal energy for this value of R. However, the error in liquid-surface-related properties is much higher 5% and 20% errors in the liquid and vapor densities, respectively, and 50% error in the surface tension.Thus, longer cutoff distances and corrections must be included to obtain reliable and consistent results. 

In contrast, truncation at a computationally feasible (i.e., small) value of R. produces artifacts when the system is strongly ionic because the potential energy is dominated by the slowly varying 1/r terms.To address this, a popular approach known as Ewald or lattice sum (similar to the one used to calculate the lattice energy of ionic crystals) is used to sum the electrostatic interactions in the simulation box and all of its replicas. This is done by rewriting the sum of the 1/r terms as a sum of a rapidly converged series in real space (so a small cutoff can be used for these terms) and a much more slowly varying smooth function that can be approximated by a few cosine and sine terms in reciprocal (k) space.These are expensive calculations that scale like 

To sum the long-range coulomb interactions for an interfacial system, one can use the 3-dimensional Ewald (3DE) method and its variants mentioned above, provided that the 3D-periodic boundary condition geometry is used (see Eigure la and b). If the system is only periodic in two dimensions (a slab geometry, see Eigure Ic), one must use a 2-dimensional version (2DE), which is computationally expensive. There are ways to approximate the Ewald sum 

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