# Interdisciplinary Applied Mathematics

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N

Vbauss = ~3n-R E Vi’    (16.15)

k=1

The medium surrounding the sphere of simulation boxes must also be considered in the calculation, since the sphere can interact with its surroundings (Sadus, 1999). No correction is required if the surrounding medium is a good conductor. However, if the surrounding medium is a vacuum, the following correction applies:

Kco

2tt

3L18

N

2

J2qi ril

i=1

(16.16)

Consequently, the final expression for the total potential energy is

К = Kreal + Krecip + KGauss + Kcorr-    (16T7)

The computational cost of the reciprocal-space energy (equation (16.14)) scales as N3/2 (Sadus, 1999). Thus, the Ewald summation approach can still be very expensive for large systems.

2. Particle-Mesh Ewald (PME). PME is a method proposed by (Darden et al., 1993) and (Essmann et al., 1995) to improve the performance of the reciprocal summation in the Ewald method. Instead of directly summing reciprocal vectors, the charges are assigned to a grid using cardinal B-spline interpolation. This grid is then Fourier transformed with a 3D FFT algorithm, and the reciprocal energy term is calculated by a single sum over the grid in k-space. The potential at the grid points is calculated by inverse transformation, and by using the interpolation factors, the forces on each atom can be calculated. The PME algorithm scales as Nlog(N), and is substantially faster than the ordinary Ewald summation for medium to large systems. For very small systems it might still be better to use Ewald summation to avoid the overhead in setting up grids and performing transformations.

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