# Interdisciplinary Applied Mathematics

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the N — 1 particles, but also with an infinite number of images of these particles.

1. The Ewald Summation. The Ewald summation is a popular technique to compute electrostatic interactions (Frenkel and Smit, 2002). The total electrostatic energy of N particles with charges qi in a cubic simulation box (box length: L) and their periodic images is given by

V

to N N

l2ZZZ

n=0 i=1 j=1

QiQj

I ru + П г

(16.12)

where the summation over n is taken over all periodic images, n = (nxL, nyL,    nzL)    with    nx,    ny,    nz    integers.    The    prime indicates    the

omission of i = j for n = 0. Note that the prefactor 1/4^e0 is omitted for simplicity. Direct implementation of equation (16.12) is difficult because the summation is conditionally convergent, and converges very slowly. In the Ewald summation, equation (16.12) is converted into two series terms each of which converges more rapidly (De Leeuw et al., 1980; Heyes, 1981). The first step is to impose a neutralizing charge distribution (typically a Gaussian distribution) of equal magnitude but of opposite sign to each charge. Then, the summation over point charges becomes a summation of the interaction between charges plus the neutralizing distributions. The new summation is often referred to as the “real space” summation. The real space energy is given by 17 18

“reciprocal space” energy,

К

recip

1

2

N N

EEE

k=0i=1j = 1

Anqiqj

k18L19

exp

4k18

cos(k • rij),

(16.14)

where k= 2nn/L18 are the reciprocal vectors. The summation of Gaussian functions in real space also includes the interaction of each Gaussian with itself. This interaction energy is given by

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