‚3 x± xβ xγ

1 3

= ’15 + 5 (x± δβγ + xβ δγ± + xγ δ±β ),(13.128)

r7

‚x± ‚xβ ‚xγ r r

‚4 x± xβ xγ xδ

1 15

’ 7 (x± xβ δγδ + xβ xγ δδ±

= 105

‚x± · · · ‚xδ r r9 r

+xγ xδ δ±β + xδ x± δβγ + x± xγ δβδ + xβ xδ δγ± )

3

+ 5 (δ±β δγδ + δβγ δδ± + δ±γ δβδ ). (13.129)

r

These derivatives are in vector notation: ∇ 1 , ∇∇ 1 , ∇∇∇ 1 , ∇∇∇∇ 1 .

r r r r

Expanding φ(r) in inverse powers of r is now a matter of substituting these

derivatives into the Taylor expansion of |r’r j |’1 . Using the same de¬nitions

for the multipoles as in (13.114) to (13.117), we obtain:

1 (2) 3x± xβ δ±β

(0) 1 (1) x±

’3

4πµ0 φ(r) = M + M± + M ±β

r3 r5

r 2! r

x± xβ xγ

1 (3) 3

’ 5 (x± δβγ + xβ δγ± + xγ δ±β )

+ M ±βγ 15

r7

3! r

+···. (13.130)

The terms in this sum are of increasing order in r’n , and represent the

potentials of monopole, dipole, quadrupole and octupole, respectively.

(l)

Instead of M we can also use the traceless de¬nitions, because the

trace do not contribute to the potential. For example, the trace part of

the quadrupole (which is a constant times the unit matrix) leads to a con-

tribution 3 r’3 (3x2 ’ r2 ) = 0. Therefore instead of (13.130) we can

±

±=1

write:

3x± xβ δ±β

1 (0) 1 (1) 1 (2)

’3

4πµ0 φ(r) = M + 3 M± x± + M±β

r5

r r 6 r

13.8 Multipole expansion 359

5x± xβ xγ

1 (3) 1

’ 5 (x± δβγ + xβ δγ± + xγ δ±β )

+ M

10 ±βγ r7 r

+..., (13.131)

with

M (0) = qj = q (monopole), (13.132)

j

(1)

M± = qj xj± = μ (dipole), (13.133)

j

(2)

qj (3xj± xjβ ’ rj δ±β) = Q (quadrupole),

2

M±β = (13.134)

j

(3)

qj [5xj± xjβ xjγ ’ rj (xj± δβγ + xjβ δγ± + xjγ δ±β )]

2

M±βγ =

j

= O (octupole). (13.135)

Expressed in terms of the derivative tensors, the potential reads:

q 11 1 1 1

’ μ · ∇ + Q:∇∇ ’ O:∇∇∇ + · · · .

4πµ0 φ(r) = (13.136)

r r6 r 30 r

The multipole de¬nitions obviously also apply to continuous charge distri-

butions, when the summations are replaced by integration over space and

the charges qj are replaced by a charge density. These are the de¬nitions (in

cartesian coordinates) that we shall adopt for the multipole moments. The

reader should be aware that there is no consensus on the proper de¬nition

of multipole moments and di¬erent de¬nitions are used in the literature.10

Not only the de¬nitions may di¬er, but also the choice of center is impor-

tant for all multipoles beyond the lowest non-zero multipole. If the total

charge (monopole) is non-zero, the dipole moment depends on the choice of

origin; the dipole moment will vanish if the center of charge i qi r i / i qi

is chosen as the center of the expansion. Likewise the quadrupole moment

depends on the choice of origin for dipolar molecules, etc.

Another elegant and popular expansion of the source term is in terms of

spherical harmonics Ylm (θ, φ). These are functions expressed in polar and

azimuthal angles; for use in simulations they are often less suitable than

their cartesian equivalents. For higher multipoles they have the advantage

of being restricted to the minimum number of elements while the cartesian

10 Our de¬nition corresponds to the one used by Hirschfelder et al. (1954) and to the one in

general use for the de¬nition of nuclear electric quadrupole moments in NMR spectroscopy

(see, e.g., Schlichter, 1963). In molecular physics the quadrupole is often de¬ned with an

extra factor 1/2, corresponding to the Legendre polynomials with l = 2, as in the reported

quadrupole moment of the water molecule by Verhoeven and Dymanus (1970). The de¬nition

is not always properly reported and the reader should carefully check the context.

360 Electromagnetism

z

θ θj

±

r

rj

y

•j

x •

Figure 13.4 The source is at r j = (rj , θj , φj ) and the potential is determined at

r = (r, θ, φ). The angle between these two vectors is ±.

tensors contain super¬‚uous elements (as 27 cartesian tensor components

against the minimum of seven irreducible components for the octupole). On

the other hand, for numerical computations it is generally advisable not

to use higher multipoles on a small number of centers at all, but rather

use lower multipoles (even only monopoles) on a larger number of centers,

in order to avoid complex expressions. For example, the computation of

a force resulting from dipole“dipole interaction requires the gradient of a

dipole ¬eld, which involves a rank-3 tensor already; this is generally as far

as one is prepared to go. Instead of including quadrupoles, one may choose

a larger number of centers instead, without loss of accuracy.

The expansion in spherical harmonics is based on the fact that the in-

verse distance 1/|r ’ r j | is a generating function for Legendre polynomials

Pl (cos ±), where ± is the angle between r and r j (see Fig. 13.4):

∞

rj rj rj

2 l

)’1/2 =

(1 ’ 2 cos ± + Pl (cos ±), (13.137)

r r r

l=0

where the ¬rst four Legendre polynomials are given by Pl0 in (13.140) to

(13.149) below.

These Legendre polynomials of the cosine of an angle ± between two