Spherical multipole moments

Spherical multipole moments are the coefficients in a series expansion
of a potential that varies inversely with the distance R to a source, "i.e.", as 1/"R". Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.
For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density formula_1. Through this article, the primed coordinates such as formula_2 
refer to the position of charge(s), whereas the unprimed coordinates such as formula_3 refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector formula_2 has coordinates formula_5 where formula_6 is the radius, formula_7 is the colatitude and formula_8 is the azimuthal angle.
Spherical multipole moments of a point charge.
The electric potential due to a point charge located at formula_2 is given by 
where 
formula_11
is the distance between the charge position and the observation point
and formula_12 is the angle between the vectors formula_3 and formula_2.
If the radius formula_15 of the observation point is greater than the radius formula_6 of the charge, 
we may factor out 1/"r" and expand the square root in powers of formula_17 using Legendre polynomials
This is exactly analogous to the axial
multipole expansion. 
We may express formula_19 in terms of the coordinates
of the observation point and charge position using the 
spherical law of cosines (Fig. 2)
Substituting this equation for formula_19 into
the Legendre polynomials and factoring the primed and unprimed
coordinates yields the important formula known as the spherical harmonic addition theorem
where the formula_23 functions are the spherical harmonics.
Substitution of this formula into the potential yields
which can be written as 
where the multipole moments are defined
As with axial multipole moments, we may also consider 
the case when 
the radius formula_15 of the observation point is less 
than the radius formula_6 of the charge.
In that case, we may write 
which can be written as
where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics
The two cases can be subsumed in a single expression if
formula_32 and formula_33 are defined
to be the lesser and greater, respectively, of the two
radii formula_15 and formula_6; the
potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion
Exterior spherical multipole moments.
It is straightforward to generalize these formulae by replacing the point charge formula_37
with an infinitesimal charge element formula_38 
and integrating. The functional form of the expansion is the same. In the exterior case, where formula_39, the result is:
where the general multipole moments are defined 
Note.
The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to "Y"lm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.
Interior spherical multipole moments.
Similarly, the interior multipole expansion has the same functional form. In the interior case, where formula_42, the result is:
with the interior multipole moments defined as 
Interaction energies of spherical multipoles.
A simple formula for the interaction energy of two non-overlapping
but concentric charge distributions can be derived. Let the 
first charge distribution formula_45 
be centered on the origin and lie entirely within the second charge 
distribution formula_46. The interaction energy between any two static charge distributions is defined by
The potential formula_48 
of the first (central) charge distribution 
may be expanded in exterior multipoles
where formula_50 represents the formula_51
exterior multipole moment of the first charge distribution. 
Substitution of this expansion yields the formula
Since the integral equals the complex conjugate 
of the interior multipole moments formula_53 of the 
second (peripheral) charge distribution, the energy
formula reduces to the simple form
For example, this formula may be used to determine the electrostatic 
interaction energies of the atomic nucleus with its surrounding 
electronic orbitals. Conversely, given the interaction energies
and the interior multipole moments of the electronic orbitals, 
one may find the exterior multipole moments (and, hence, shape)
of the atomic nucleus. 
Special case of axial symmetry.
The spherical multipole expansion takes a simple form if the charge 
distribution is axially symmetric (i.e., is independent of the azimuthal angle formula_8). 
By carrying out the formula_8 integrations that 
define formula_57 and formula_58, it can be shown the
multipole moments are all zero except when formula_59. Using the
mathematical identity
the exterior multipole expansion becomes
where the axially symmetric multipole moments are defined 
In the limit that the charge is confined to the formula_63-axis, 
we recover the exterior axial multipole moments. 
Similarly the interior multipole expansion becomes 
where the axially symmetric interior multipole moments are defined 
In the limit that the charge is confined to the formula_63-axis, 
we recover the interior axial multipole moments. 