Spherical multipole moments

In physics, 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 ⁠

refer to the position of charge(s), whereas the unprimed coordinates such as

refer to the point at which the potential is being observed.

The electric potential due to a point charge located at

is the distance between the charge position and the observation point and

of the charge, we may factor out 1/r and expand the square root in powers of

in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig.

into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem

Substitution of this formula into the potential yields

As with axial multipole moments, we may also consider the case when the radius

where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics

; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion

It is straightforward to generalize these formulae by replacing the point charge

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 Yℓm, not to its complex conjugate.

Similarly, the interior multipole expansion has the same functional form.

A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived.

be centered on the origin and lie entirely within the second charge distribution

The interaction energy between any two static charge distributions is defined by

of the first (central) charge distribution may be expanded in exterior multipoles

exterior multipole moment of the first charge distribution.

Since the integral equals the complex conjugate of the interior multipole moments

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.

The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle

where the axially symmetric multipole moments are defined

-axis, we recover the exterior axial multipole moments.

where the axially symmetric interior multipole moments are defined

-axis, we recover the interior axial multipole moments.