cancels and need not be invoked.

The direct sum of Coulomb terms is only conditionally convergent (i.e.,

the convergence depends on the sequence of terms in the summation) and

converges very slowly. For an e¬cient evaluation of the lattice energies and

forces it is necessary to split the Coulomb potential into a short-range part

that can be directly evaluated by summation in real space, and a long-range

part that can be e¬ciently computed by solving Poisson™s equation. The

easiest way to accomplish this is to consider each (point) charge as a sum

of two charge distributions (see Fig. 13.5):

qi δ(r ’ r i ) = qi [δ(r ’ r i ) ’ w(r ’ r i )] + qi w(r ’ r i ), (13.154)

where w(r) = w(r) is an isotropic spread function which decreases smoothly

and rapidly with distance and integrates to 1 over space:

∞

w(r)4πr2 dr = 1. (13.155)

0

For the time being we do not specify the spread function and derive the

equations in a general form. Subsequently two speci¬c spread functions will

be considered.

364 Electromagnetism

δ(x)

δ(x)-w(x) w(x)

rc

= +

x

Figure 13.5 A point charge with δ-function distribution (left) is split up into a

distribution with short-range potential (middle) and a distribution with long-range

potential (right) by a smooth charge-spread function w(r).

The total Coulomb energy is split into two contributions:

1 1

qi φl , i ∈ unit cell.

s l

qi φs +

UC = UC + UC = (13.156)

i i

2 2

i i

Note that we did not split the energy into the sum of energies of the two

charge distributions and their interactions, which would require four terms.

Each of the contributions should include the self-energy correction. In ad-

dition there is a contribution to the energy as a result of the net dipole

moment of the unit cell, treated in Section 13.10.5.

13.10.1 Short-range contribution

s

For the short-range contributions UC to the energy we can write:

i ∈ unit cell.

s 1 s

UC = i qi φi , (13.157)

2

1

φs = qj •s (rijn ), (13.158)

i

4πµ0 n

j

def

where the prime in the sum means exclusion of j = i for n = 0, r ijn =

ri ’ rj ’ Tn and •s is a potential function related to the spread function:

∞ ∞

1

def

s

dr 4πr 2 w(r ).

• (r) = dr 2 (13.159)

r

r r

Fs

The force on particle i due to the short-range potential equals the charge

i

qi times the electric ¬eld E s (r i ) = ’(∇φs (r))r i :

qi r ijn

F s = ’qi (∇φs (r))r i = qj f s (rijn ) , (13.160)

i

4πµ0 rijn

n∈Z3

j

where f s is a force function related to the spread function:

∞

d•s (r) 1

def

f (r) = ’

s

w(r ) 4πr 2 dr .

=2 (13.161)

dr r r

13.10 Potentials and ¬elds in periodic systems of charges 365

One may also evaluate the force on particle i from taking minus the gradient

of the total short-range energy (13.158). Although the expression for the

energy contains a factor 1 , particle number i occurs twice in the summa-

2

tion, and one obtains the same equation (13.160) as above. Note that, by

omitting j = i for n = 0 from the sum, the short-range terms are corrected

for the short-range part of the self-energy. Similarly, Coulomb interactions

between speci¬ed pairs can be omitted from the short-range evaluation, if

so prescribed by the force ¬eld. Usually, Coulomb interactions are omitted

between atoms that are ¬rst or second (and often modi¬ed for third) neigh-

bors in a covalent structure because other bond and bond-angle terms take

care of the interaction.

When the spread function is such that the potentials and forces are neg-

ligible beyond a cut-o¬ distance rc , which does not exceed half the smallest

box size, the sums contain only one nearest image of each particle pair,

which can best be evaluated using a pair list that also contains a code for

the proper displacement to obtain the nearest image for each pair.

13.10.2 Long-range contribution

The long-range contribution expressed as an explicit particle sum

1 1

’ •s (rijn )

φl (r i ) = qj (13.162)

4πµ0 rijn

n

j

converges very slowly because of the 1/r nature of the function. The long-

range potential can be e¬ciently evaluated by solving Poisson™s equation

(see (13.47)):

’µ0 ∇2 φl (r) = ρl (r) = qi w(r ’ ri ’ Tn). (13.163)

n∈Z3

i

The solution is equivalent to (13.162), except that no restrictions, such as

j = i for n = 0 or any other speci¬ed pairs, can be included. The Poisson

solution therefore contains a self-energy part (which is a constant, given

the unit cell base vectors and spread function), that must be subtracted

separately. If Coulomb interactions between speci¬ed pairs must be omitted,

their contribution included in the long-range interaction must be subtracted.

The charge distribution is periodic, and so must be the solution of this

equation. The solution is determined up to any additive periodic function

satisfying the Laplace equation ∇2 φ = 0, which can only be a constant if

continuity at the cell boundaries is required. The constant is irrelevant.

There are several ways to solve the Poisson equation for periodic systems,

366 Electromagnetism

including iterative relaxation on a lattice (see Press et al., 1992), but the

obvious solution can be obtained in reciprocal space, because the Laplace

operator then transforms to a simple multiplication. We now proceed to

formulate this Fourier solution.