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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
. Through this article, the primed coordinates such 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. We also use spherical coordinates throughout, e.g., the vector
has coordinates
where
is the radius,
is the colatitude and
is the azimuthal angle.
Spherical multipole moments of a point charge[]
Figure 1: Definitions for the spherical multipole expansion
The electric potential due to a point charge located at
is given by
![\Phi(\mathbf{r}) =
\frac{q}{4\pi\varepsilon} \frac{1}{R} =
\frac{q}{4\pi\varepsilon}
\frac{1}{\sqrt{r^{2} + r^{\prime 2} - 2 r^{\prime} r \cos \gamma}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6590aeadec4180575ddb5ef07a66ab0bc19449af)
where
is the distance between the charge position and the observation point
and
is the angle between the vectors
and
.
If the radius
of the observation point is greater than the radius
of the charge,
we may factor out 1/r and expand the square root in powers of
using Legendre polynomials
![\Phi(\mathbf{r}) =
\frac{q}{4\pi\varepsilon r} \sum_{l=0}^{\infty}
\left( \frac{r^{\prime}}{r} \right)^{l} P_{l}(\cos \gamma )](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2e193a8eac962020c7c85de1ef67e0b070a40a)
This is exactly analogous to the axial
multipole expansion.
We may express
in terms of the coordinates
of the observation point and charge position using the
spherical law of cosines (Fig. 2)
![\cos \gamma =
\cos \theta \cos \theta^{\prime} +
\sin \theta \sin \theta^{\prime} \cos(\phi - \phi^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1460980d69ffb060a067f6b58d436b003f01e4)
Figure 2: Angles between the unit vectors
![\mathbf {\hat {z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bdb354dce52452a8b65ebca5427d3012427412f)
(the coordinate axis),
![\mathbf {\hat {r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe52dfe80c9a6604b3a46b24d65eb02c92c59e9)
(the observation point) and
![\mathbf{\hat{r}^{\prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b17d2e3dadd97e462e2a0b5eaacd2965bbf4d18)
(the charge position).
Substituting this equation for
into
the Legendre polynomials and factoring the primed and unprimed
coordinates yields the important formula known as the spherical harmonic addition theorem
![P_{l}(\cos \gamma) = \frac{4\pi}{2l + 1} \sum_{m=-l}^{l}
Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffa2b37d329affd120faa7390a5ac46b50e5364)
where the
functions are the spherical harmonics.
Substitution of this formula into the potential yields
![\Phi(\mathbf{r}) =
\frac{q}{4\pi\varepsilon r} \sum_{l=0}^{\infty}
\left( \frac{r^{\prime}}{r} \right)^{l}
\left( \frac{4\pi}{2l+1} \right)
\sum_{m=-l}^{l}
Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/2549ec6d6900baaf74632391c55305ded0a356c3)
which can be written as
![\Phi(\mathbf{r}) =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty} \sum_{m=-l}^{l}
\left( \frac{Q_{lm}}{r^{l+1}} \right)
\sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3671ef9f274ae174e48f6b1c35da6609f78f345e)
where the multipole moments are defined
.
As with axial multipole moments, we may also consider
the case when
the radius
of the observation point is less
than the radius
of the charge.
In that case, we may write
![\Phi(\mathbf{r}) =
\frac{q}{4\pi\varepsilon r^{\prime}} \sum_{l=0}^{\infty}
\left( \frac{r}{r^{\prime}} \right)^{l}
\left( \frac{4\pi}{2l+1} \right)
\sum_{m=-l}^{l}
Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/66f52ae654e4600fb008d9e1188ab64d031124fa)
which can be written as
![\Phi(\mathbf{r}) =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} I_{lm} r^{l}
\sqrt{\frac{4\pi }{2l+1}}
Y_{lm}(\theta, \phi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a334832e9a150708ac7eba534b4d8e9d6238dc5a)
where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics
![I_{lm} \ \stackrel{\mathrm{def}}{=}\ \frac{q}{\left( r^{\prime} \right)^{l+1}}
\sqrt{\frac{4\pi }{2l+1}}
Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a1a6d6495f19e5a39694957085f72f5d5ec77a)
The two cases can be subsumed in a single expression if
and
are defined
to be the lesser and greater, respectively, of the two
radii
and
; the
potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion
![\Phi(\mathbf{r}) =
\frac{q}{4\pi\varepsilon} \sum_{l=0}^{\infty}
\frac{r_<^{l}}{r_>^{l+1}}
\left( \frac{4\pi}{2l+1} \right)
\sum_{m=-l}^{l}
Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fecfbff09056023566f94ce535463c9b0bbaeb1)
Exterior spherical multipole moments[]
It is straightforward to generalize these formulae by replacing the point charge
with an infinitesimal charge element
and integrating. The functional form of the expansion is the same. In the exterior case, where
, the result is:
![{\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\left({\frac {Q_{lm}}{r^{l+1}}}\right){\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}(\theta ,\phi )\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e0915175d9bd9afd9a923c78fa7a3c01f0e613)
where the general multipole moments are defined
![{\displaystyle Q_{lm}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} ^{\prime }\rho (\mathbf {r} ^{\prime })\left(r^{\prime }\right)^{l}{\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}^{*}(\theta ^{\prime },\phi ^{\prime }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6b70f66857b224c022c2e0614bd347f04d0aa9)
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 Ylm, 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
, the result is:
![{\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{l=0}^{\infty }\sum _{m=-l}^{l}I_{lm}r^{l}{\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}(\theta ,\phi ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb644660e1e984ea3a1ad13e5f0e147e94d49d7)
with the interior multipole moments defined as
![{\displaystyle I_{lm}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} ^{\prime }{\frac {\rho (\mathbf {r} ^{\prime })}{\left(r^{\prime }\right)^{l+1}}}{\sqrt {\frac {4\pi }{2l+1}}}Y_{lm}^{*}(\theta ^{\prime },\phi ^{\prime }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c67f23aead14d01cb2cf2b13217e466602891686)
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
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
![U \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}
\rho_{2}(\mathbf{r}) \Phi_{1}(\mathbf{r})](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0e621642b8a345fe8c983f1945b95ff12a5652)
The potential
of the first (central) charge distribution
may be expanded in exterior multipoles
![\Phi(\mathbf{r}) =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm}
\left( \frac{1}{r^{l+1}} \right)
\sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a42d87cb100d13c0a6939e512daf537cfe6610b0)
where
represents the
exterior multipole moment of the first charge distribution.
Substitution of this expansion yields the formula
![U =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm}
\int d\mathbf{r} \
\rho_{2}(\mathbf{r})
\left( \frac{1}{r^{l+1}} \right)
\sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9906c26eb445e34a824f3bf493531f4ba1cfe4e5)
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
![U =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} I_{2lm}^{*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfc494bf4118eeeedb54377149482571e43b86f)
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
).
By carrying out the
integrations that
define
and
, it can be shown the
multipole moments are all zero except when
. Using the
mathematical identity
![P_{l}(\cos \theta) \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{4\pi}{2l+1}} Y_{l0}(\theta, \phi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/262980f2578a5e263ece858b022616d9727f1e81)
the exterior multipole expansion becomes
![\Phi(\mathbf{r}) =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty}
\left( \frac{Q_{l}}{r^{l+1}} \right)
P_{l}(\cos \theta)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5313c741758236f0acbd3aea9bb1f4a3a017b92)
where the axially symmetric multipole moments are defined
![Q_{l} \ \stackrel{\mathrm{def}}{=}\
\int d\mathbf{r}^{\prime} \rho(\mathbf{r}^{\prime})
\left( r^{\prime} \right)^{l} P_{l}(\cos \theta^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/773a05a11c9512cddaaba5abfe35165deaf5ea9e)
In the limit that the charge is confined to the
-axis,
we recover the exterior axial multipole moments.
Similarly the interior multipole expansion becomes
![\Phi(\mathbf{r}) =
\frac{1}{4\pi\varepsilon}
\sum_{l=0}^{\infty} I_{l} r^{l} P_{l}(\cos \theta)](https://wikimedia.org/api/rest_v1/media/math/render/svg/78f957c24464721eebae8a15cca034437363cb89)
where the axially symmetric interior multipole moments are defined
![I_{l} \ \stackrel{\mathrm{def}}{=}\
\int d\mathbf{r}^{\prime}
\frac{\rho(\mathbf{r}^{\prime})}{\left( r^{\prime} \right)^{l+1}}
P_{l}(\cos \theta^{\prime})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dfe56c77ffe0bb5b1b77582a833ce623a8f06eb)
In the limit that the charge is confined to the
-axis,
we recover the interior axial multipole moments.
See also[]