90 8 5 8

(1 ’ ) cos κ ’ sin κ + 2 ,

Wκ = (13.193)

κ4 V κ2 κ κ

where κ = krc . The short-range force function (13.161) is

5r 9r3 5r4

1

f (r) = 2 ’ 3 + 5 ’ 6 for r < rc ,

s

r rc rc rc

= 0 for r ≥ rc ,(13.194)

13.10 Potentials and ¬elds in periodic systems of charges 371

3

2.5 •s(r)

2 r ’1

1.5 r ’1 ’ •s(r)

1

0.5 w(r)

0.25 0.5 0.75 1 1.25 1.5

r/rc

Figure 13.7 Functions for the cubic spread function: w(r) is proportional to the

spread function; the potential functions are as in Fig. 13.6, but scaled by 2 to make

them comparable.

and the short-range potential function (13.159) is

5r2 9r4 r5

1 9

’ + 3 ’ 5 + 6 for r < rc ,

r 4rc 2rc 4rc rc

= 0 for r ≥ rc . (13.195)

13.10.5 Net dipolar energy

Special attention needs to be given to the energetic e¬ects of a net non-zero

dipole moment, as has been carefully done by de Leeuw et al. (1980).15 The

problem is that Coulomb lattice sums over unit cells with non-vanishing

total dipole moment converge only conditionally, i.e., the sum depends on

the sequence of terms in the summation. Consider summation over a chunk

of matter containing a (very large, but not in¬nite) number of unit cells. The

total dipole moment of the chunk of matter is proportional to the volume of

the chunk. The Coulomb energy, given by the summed dipolar interactions,

now depends on the shape of the chunk and on its dielectric environment. For

example, in a ¬‚at disc perpendicular to the dipole moment, the interaction is

unfavorable (positive), but in a long cylinder parallel to the dipole moment

the interaction is favorable (negative). In a sphere of radius R with cubic

unit cells the interactions sum to zero, but there will be a reaction ¬eld ERF

15 See also Caillol (1994), Essmann et al. (1995) and Deserno and Holm (1998a).

372 Electromagnetism

0.04

1

0.02

0.8 0

-0.02

7.5 10 12.5 15 17.5

0.6

0.4

0.2

0

2.5 5 7.5 10 12.5 15 17.5 20

krc

Figure 13.8 Fourier transforms of the cubic (solid line) and Gaussian (dashed line)

spread functions. For the Gaussian transform β was set to 2/rc . The inset magni¬es

the tails.

due to the polarizability of the medium in which the sphere is embedded

(see (13.83) on page 347):

μtot 2(µr ’ 1)

ERF = , (13.196)

4πµ0 R3 2µr + 1

where µr is the relative dielectric constant of the medium. The energy per

unit cell ’μtot ERF /(2N ) (where N is the number of unit cells in the sphere)

in the reaction ¬eld can now be written as

μ2 2(µr ’ 1)

=’

URF , (13.197)

6µ0 V 2µr + 1

where μ is the unit cell dipole moment and V the unit cell volume. This

term does not depend on the size of the system since the R3 proportionality

in the volume just cancels the R’3 proportionality of the reaction ¬eld. For

lower multipoles (i.e., for the total charge) the energy diverges, and the

system is therefore required to be electroneutral; for higher multipoles the

lattice sum converges unconditionally so that the problem does not arise.

It is clear that the boundary conditions must be speci¬ed for periodic

systems with non-vanishing total dipole moment. The system behavior,

especially the ¬‚uctuation of the dipole moment, will depend on the chosen

boundary conditions. A special case is the tin-foil or metallic boundary

condition, given by µr = ∞, which is equivalent to a conducting boundary.

13.10 Potentials and ¬elds in periodic systems of charges 373

Applied to a sphere, the RF energy per unit cell then becomes

μ2

=’

URF (spherical tin-foil b.c.). (13.198)

6µ0 V

Since the question of the boundary condition did not come up when solving

for the long-range Coulomb interaction, leading to (13.174), one wonders

whether this equation silently implies a speci¬c boundary condition, and

if so, which one. By expanding exp(±ik · r) in powers of k, we see that

Qk Q’k = (k · μ)2 + O(k 4 ), while Wk = (1/V ) + O(k 2 ). The term (k · μ)2

equals 1 μ2 k 2 when averaged over all orientations of the dipole moment.

3

Thus the energy term k ’2 Qk Q’k Wk /(2µ0 ) equals ’μ2 /(6µ0 V ) + O(k 2 ),

which is exactly the dipolar energy for the tin-foil boundary conditions.

The conclusion is that application of the equations for the Coulomb energy,

as derived here based on a splitting between short- and long-range compo-

nents, and consequently also for the Ewald summation, automatically imply

tin-foil boundary conditions.

If one wishes to exert spherical boundary conditions corresponding to a

dielectric environment with relative dielectric constant µr rather than con-

ducting boundary conditions, an extra term making up the di¬erence be-

tween (13.197) and (13.198) must be added to the computed energy. This

extra term is

μ2 1

Udipole = . (13.199)

2µ0 V (2µr + 1)

This term is always positive, as the tin-foil condition (for which the cor-

rection is zero) provides the most favorable interaction. In a vacuum envi-

ronment (µr = 1) it is more unfavorable to develop a net dipole moment,

and in a dipolar ¬‚uid with ¬‚uctuating net dipole moment, the net dipole

moment is suppressed compared to tin-foil boundary conditions. The most

natural boundary condition for a dipolar ¬‚uid would be a dielectric environ-

ment with a dielectric constant equal to the actual dielectric constant of the

medium.

13.10.6 Particle“mesh methods

The computational e¬ort of the Ewald summation scales as N 2 with the

number of charges N and becomes prohibitive for large systems.16 Fast

Fourier transforms (FFT)17 are computationally attractive although they

16 With optimized truncation of the real and reciprocal sums (Perram et al., 1988) a N 3/2 -scaling

can be accomplished. The computation can also be made considerably faster by using tabulated

functions (see Chapter 19).