13.8 Multipole expansion 361

subsequently be expanded by the spherical harmonics addition theorem:11

l

(l ’ |m|)! m

Yl (θ, φ)Yl’m (θj φj ),

Pl (cos ±) = (13.138)

(l + |m|)!

m=’l

where

|m|

Ylm (θ, φ) = Pl (cos θ)eimφ (13.139)

|m|

are the spherical harmonic functions and Pl are the associated Legendre

functions.12 For l ¤ 3 these functions are:

0

l = 0 : P0 (cos θ) = 1, (13.140)

0

l = 1 : P1 (cos θ) = cos θ, (13.141)

1

: P1 (cos θ) = sin θ, (13.142)

l = 2 : P2 (cos θ) = 1 (3 cos2 θ ’ 1),

0

(13.143)

2

1

: P2 (cos θ) = 3 sin θ cos θ, (13.144)

: P2 (cos θ) = 3 sin2 θ,

2

(13.145)

cos3 θ ’ 3 cos θ,

0 5

l = 3 : P3 (cos θ) = (13.146)

2 2

sin θ(5 cos2 θ ’ 1),

1 3

: P3 (cos θ) = (13.147)

2

: P3 (cos θ) = 15 sin2 θ cos θ,

2

(13.148)

: P3 (cos θ) = 15 sin3 θ.

3

(13.149)

The result is

∞ l

1 (l ’ |m|)! m m

M Y (θ, φ),

4πµ0 φ(r) = (13.150)

rl+1 (l + |m|)! l l

l=0 m=’l

where Mm are the 2l + 1 components of the l-th spherical multipole:

l

qj rj Yl’m (θj , φj ).

Mm = l

(13.151)

l

j

These spherical harmonic de¬nitions are related to the cartesian tensor def-

initions of (13.132) to (13.135).

11 We use simple non-normalized spherical harmonics. Our de¬nition of the spherical multipole

moments corresponds to Hirschfelder et al. (1965). De¬nitions in the literature may di¬er as

to the normalization factors and sign of the functions for odd m. See, e.g., Weisstein (2005),

Abramowitz and Stegun (1965) and Press et al. (1992).

12 See, e.g., Jahnke and Emde (1945), who list Legendre functions up to l = 6 and associated

functions up to l = 4.

362 Electromagnetism

13.9 Potentials and ¬elds in non-periodic systems

Given a set of charges, the calculation of potentials, ¬elds, energies and forces

by summation of all pairwise interactions is a problem of N 2 complexity that

easily runs out of hand for large systems. The use of a cut-o¬ radius reduces

the problem to order-N , but produces large and often intolerable artefacts

for the fairly long-ranged Coulomb forces. For gravitational forces, lacking

the compensation of sources with opposite sign, cut-o¬s are not allowed at

all. E¬cient methods that include long-range interactions are of two types:

(i) hierarchical and multipole methods, employing a clustering of sources

for interactions at longer distances; and

(ii) grid methods, essentially splitting the interaction into short- and

long-range parts, solving the latter by Poisson™s equation, generally

on a grid.

The second class of methods are the methods of choice for periodic system,

which are treated in detail in the next section. They can in principle also

be used for non-periodic systems “ and still with reasonable e¬ciency “ by

extending the system with periodic images. But also without periodicity the

same method of solution can be used when the Poisson equation is solved

on a grid with given boundary conditions, possibly employing multigrid

methods with spatial resolution adjusted to local densities.

As we emphasize molecular simulations where the long-range problem con-

cerns Coulomb rather than gravitational forces, we shall not further consider

the hierarchical and “fast multipole” methods, which are essential for astro-

physical simulations and are also used in molecular simulation,13 but have

not really survived the competition with methods described in the next

section. Whether the fast multipole methods may play a further role in

molecular simulation, is a matter of debate (Board and Schulten, 2000).

13.10 Potentials and ¬elds in periodic systems of charges

In periodic systems (see Section 6.2.1) the Coulomb energy of the charges

is given by (see (13.42)):

’ φself ), i ∈ unit cell,

1

UC = i qi (φ(ri ) (13.152)

2

13 The basic articles on hierarchical and fast multipole methods are Appel (1985), Barnes and Hut

(1986) and Greengard and Rokhlin (1987). Niedermeier and Tavan (1994) and Figueirido et

al. (1997) describe the use of fast multipole methods in molecular simulations. It is indicated

that these methods, scaling proportional to N , are computationally more e¬cient than lattice

summation techniques for systems with more than about 20 000 particles.

13.10 Potentials and ¬elds in periodic systems of charges 363

with

qj

1

φ(r) =

|r ’ r j ’ n1 a ’ n2 b ’ n3 c|

4πµ0

n1 ,n2 ,n3 ∈Z

j

qj

1

= , (13.153)

|r ’ rj ’ Tn|

4πµ0

n∈Z3

j

where T is the transformation matrix from relative coordinates in the unit

cell to cartesian coordinates (see (6.3) on page 143), i.e., a matrix of which

the columns are the cartesian base vectors of the unit cell a, b, c. The

last line of (13.153) is in matrix notation; the meaning of |x| is (xT x)1/2 .

Note that the displacements can be either subtracted (as shown) or added

in (13.153). The self-energy contains the diverging interaction of qi with

itself, but not with the images of itself; the images are to be considered

as di¬erent particles as in a crystal. The interaction of a charge with its

images produces zero force, as for every image there is another image at

equal distance in the opposite direction; thus the interaction energy of each

charge with its own images is a constant, which diverges with the number

of periodic images considered. In order to avoid the divergence we may

assume that every charge has a homogeneous charge distribution of equal

magnitude but opposite sign associated with it. If the total charge in a unit