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{{Short description|Mathematical series}}
A '''multipole expansion''' is a [[Series (mathematics)|mathematical series]] representing a [[Function (mathematics)|function]] that depends on [[angle]]s—usually the two angles used in the [[spherical coordinate system]] (the polar and [[Azimuth|azimuthal]] angles) for three-dimensional [[Euclidean space]], <math>\R^3</math>. Similarly to [[Taylor series]], multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be [[Real number|real]]- or [[complex number|complex]]-valued and is defined either on <math>\R^3</math>, or less often on <math>\R^n</math> for some other {{nowrap|<math>n</math>.}}

Multipole expansions are used frequently in the study of [[electromagnetic field|electromagnetic]] and [[gravitational field]]s, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in [[radius]]. Such a combination gives an expansion describing a function throughout three-dimensional space.<ref name=Edmonds>{{cite book | last = Edmonds | first = A. R. | title = Angular Momentum in Quantum Mechanics | year = 1960 | url = https://archive.org/details/angularmomentumi0000edmo | url-access = registration | publisher = Princeton University Press| isbn = 9780691079127 }}</ref>

The multipole expansion is expressed as a sum of terms with progressively finer angular features ([[Moment (mathematics)|moments]]). The first (the zeroth-order) term is called the [[Monopole (mathematics)|monopole]] moment, the second (the first-order) term is called the [[dipole]] moment, the third (the second-order) the [[quadrupole]] moment, the fourth (third-order) term is called the octupole moment, and so on. Given the limitation of [[Greek numerals|Greek numeral prefixes]], terms of higher order are conventionally named by adding "-pole" to the number of poles—e.g., 32-pole (rarely dotriacontapole or triacontadipole) and 64-pole (rarely tetrahexacontapole or hexacontatetrapole).<ref>{{cite book|last1=Auzinsh|first1=Marcis| last2=Budker|first2=Dmitry|last3=Rochester|first3=Simon|title=Optically polarized atoms : understanding light-atom interactions| date=2010|publisher=New York|location=Oxford|isbn=9780199565122|page=100}}</ref><ref>{{cite journal|last1=Okumura|first1=Mitchio| last2=Chan|first2=Man-Chor|last3=Oka|first3=Takeshi|title=High-resolution infrared spectroscopy of solid hydrogen: The tetrahexacontapole-induced transitions|journal=Physical Review Letters|date=2 January 1989|volume=62|issue=1| pages=32–35| doi=10.1103/PhysRevLett.62.32|pmid=10039541|bibcode=1989PhRvL..62...32O|url=https://authors.library.caltech.edu/5428/1/OKUprl89.pdf }}</ref><ref>{{cite journal|last1=Ikeda|first1=Hiroaki|last2=Suzuki|first2=Michi-To|last3=Arita|first3=Ryotaro| last4=Takimoto|first4=Tetsuya|last5=Shibauchi|first5=Takasada|last6=Matsuda|first6=Yuji|title=Emergent rank-5 nematic order in URu2Si2| journal=Nature Physics|date=3 June 2012|volume=8|issue=7|pages=528–533| doi=10.1038/nphys2330| arxiv=1204.4016| bibcode=2012NatPh...8..528I|s2cid=119108102 }}</ref> A multipole moment usually involves [[Exponentiation|powers]] (or inverse powers) of the distance to the origin, as well as some angular dependence.

In principle, a multipole expansion provides an exact description of the potential, and generally [[Convergent series|converges]] under two conditions: (1) if the sources (e.g. charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called ''exterior multipole moments'' or simply ''multipole moments'' whereas, in the second case, they are called ''interior multipole moments''.

==Expansion in spherical harmonics==
Most commonly, the series is written as a sum of [[spherical harmonics]]. Thus, we might write a function <math>f(\theta,\varphi)</math> as the sum
<math display="block">f(\theta,\varphi) = \sum_{\ell=0}^\infty\, \sum_{m=-\ell}^\ell\, C^m_\ell\, Y^m_\ell(\theta,\varphi)</math>
where <math>Y^m_\ell(\theta,\varphi)</math> are the standard spherical harmonics, and <math>C^m_\ell</math> are constant coefficients which depend on the function. The term <math>C^0_0</math> represents the monopole; <math>C^{-1}_1,C^0_1,C^1_1</math> represent the dipole; and so on. Equivalently, the series is also frequently written<ref>{{cite book | last=Thompson | first=William J. | title=Angular Momentum | publisher=John Wiley & Sons, Inc.}}</ref> as
<math display="block">f(\theta,\varphi) = C + C_i n^i + C_{ij}n^i n^j + C_{ijk}n^i n^j n^k + C_{ijk\ell}n^i n^j n^k n^\ell + \cdots</math>
where the <math>n^i</math> represent the components of a [[unit vector]] in the direction given by the angles <math>\theta</math> and <math>\varphi</math>, and indices are [[Einstein summation convention|implicitly summed]]. Here, the term <math>C</math> is the monopole; <math>C_i</math> is a set of three numbers representing the dipole; and so on.

In the above expansions, the coefficients may be [[real number|real]] or [[complex number|complex]]. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have
<math display="block">C_\ell^{-m} = (-1)^m C^{m\ast}_\ell \, .</math>
In the multi-vector expansion, each coefficient must be real:
<math display="block">C = C^\ast;\ C_i = C_i^\ast;\ C_{ij} = C_{ij}^\ast;\ C_{ijk} = C_{ijk}^\ast;\ \ldots</math>

While expansions of [[scalar (mathematics)|scalar]] functions are by far the most common application of multipole expansions, they may also be generalized to describe [[tensor]]s of arbitrary rank.<ref>{{cite journal | last=Thorne | first=Kip S. | journal=Reviews of Modern Physics | title=Multipole Expansions of Gravitational Radiation |date=April 1980 | volume=52 | issue=2 | pages=299–339 | doi=10.1103/RevModPhys.52.299 | bibcode=1980RvMP...52..299T| url=https://authors.library.caltech.edu/11159/1/THOrmp80a.pdf }}</ref> This finds use in multipole expansions of the [[vector potential]] in electromagnetism, or the metric perturbation in the description of [[gravitational wave]]s.

For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, <math>r</math>—most frequently, as a [[Laurent series]] in powers of <math>r</math>. For example, to describe the electromagnetic potential, <math>V</math>, from a source in a small region near the origin, the coefficients may be written as:
<math display="block">V(r,\theta,\varphi) = \sum_{\ell=0}^\infty\, \sum_{m=-\ell}^\ell C^m_\ell(r)\, Y^m_\ell(\theta,\varphi)= \sum_{j=1}^\infty\, \sum_{\ell=0}^\infty\, \sum_{m=-\ell}^\ell \frac{D^m_{\ell,j}}{r^j}\, Y^m_\ell(\theta,\varphi) .</math>

==Applications==
Multipole expansions are widely used in problems involving [[gravitational field]]s of systems of [[mass]]es, [[electric field|electric]] and [[magnetic field]]s of charge and current distributions, and the propagation of [[electromagnetic wave]]s. A classic example is the calculation of the ''exterior'' multipole moments of [[atomic nucleus|atomic nuclei]] from their interaction energies with the ''interior'' multipoles of the electronic orbitals. The multipole moments of the nuclei report on the distribution of charges within the nucleus and, thus, on the shape of the nucleus. Truncation of the multipole expansion to its first non-zero term is often useful for theoretical calculations.

Multipole expansions are also useful in numerical simulations, and form the basis of the [[fast multipole method]] of [[Leslie Greengard|Greengard]] and [[Vladimir Rokhlin (American scientist)|Rokhlin]], a general technique for efficient computation of energies and forces in systems of interacting [[particle]]s. The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments. The efficiency of the fast multipole method is generally similar to that of [[Ewald summation]], but is superior if the particles are clustered, i.e. the system has large density fluctuations.

==Multipole expansion of a potential outside an electrostatic charge distribution==
Consider a discrete charge distribution consisting of ''N'' point charges ''q''''i'' with position vectors '''r'''''i''. We assume the charges to be clustered around the origin, so that for all ''i'': {{math|''r''''i'' &lt; ''r''max}}, where ''r''max has some finite value. The potential ''V''('''R'''), due to the charge distribution, at a point '''R''' outside the charge distribution, i.e., {{math|{{abs|'''R'''}} &gt; ''r''max}}, can be expanded in powers of 1/''R''. Two ways of making this expansion can be found in the literature: The first is a [[Taylor series]] in the [[Cartesian coordinates]] ''x'', ''y'', and ''z'', while the second is in terms of [[spherical harmonics]] which depend on [[spherical polar coordinates]]. The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. Its disadvantage is that the derivations are fairly cumbersome (in fact a large part of it is the implicit rederivation of the Legendre expansion of {{math|1 / {{abs|'''r''' − '''R'''}}}}, which was done once and for all by [[Adrien-Marie Legendre|Legendre]] in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion&mdash;usually only the first few terms are given followed by an ellipsis.

===Expansion in Cartesian coordinates===
Assume {{math|1=''v''('''r''') = ''v''(−'''r''')}} for convenience. The [[Taylor series|Taylor expansion]] of {{math|1=''v''('''r''' − '''R''')}} around the origin {{math|1='''r''' = '''0'''}} can be written as
<math display="block">v(\mathbf{r}- \mathbf{R}) = v(\mathbf{R}) - \sum_{\alpha=x,y,z} r_\alpha v_\alpha(\mathbf{R}) +\frac{1}{2} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R})
- \cdots + \cdots</math>
with Taylor coefficients
<math display="block">v_\alpha(\mathbf{R}) \equiv\left( \frac{\partial v(\mathbf{r}-\mathbf{R}) }{\partial r_\alpha}\right)_{\mathbf{r} = \mathbf 0} \quad\text{and} \quad
v_{\alpha\beta}(\mathbf{R}) \equiv\left( \frac{\partial^2 v(\mathbf{r}-\mathbf{R}) }{\partial r_{\alpha}\partial r_{\beta}}\right)_{\mathbf{r}= \mathbf0} .</math>
If {{math|''v''('''r''' − '''R''')}} satisfies the [[Laplace equation]], then by the above expansion we have
<math display="block">\left(\nabla^2 v(\mathbf{r}- \mathbf{R})\right)_{\mathbf{r}=\mathbf0} = \sum_{\alpha=x,y,z} v_{\alpha\alpha}(\mathbf{R}) = 0,</math>
and the expansion can be rewritten in terms of the components of a traceless Cartesian second rank [[tensor]]:
<math display="block">\sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} r_\alpha r_\beta v_{\alpha\beta}(\mathbf{R})
= \frac{1}{3} \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z} \left(3r_\alpha r_\beta - \delta_{\alpha\beta} r^2\right) v_{\alpha\beta}(\mathbf{R}) ,</math>
where {{math|''δ''''αβ''}} is the [[Kronecker delta]] and {{math|''r''2 ≡ {{abs|'''r'''}}2}}. Removing the trace is common, because it takes the rotationally invariant {{math|''r''2}} out of the second rank tensor.

====Example====

Consider now the following form of {{math|''v''('''r''' − '''R''')}}:
<math display="block">v(\mathbf{r}- \mathbf{R}) \equiv \frac{1}{|\mathbf{r}- \mathbf{R}|} .</math>
Then by direct [[differentiation (mathematics)|differentiation]] it follows that
<math display="block">v(\mathbf{R}) = \frac{1}{R},\quad v_\alpha(\mathbf{R})= -\frac{R_\alpha}{R^3},\quad \hbox{and}\quad v_{\alpha\beta}(\mathbf{R}) = \frac{3R_\alpha R_\beta- \delta_{\alpha\beta}R^2}{R^5} .</math>
Define a monopole, dipole, and (traceless) quadrupole by, respectively,
<math display="block">q_\mathrm{tot} \equiv \sum_{i=1}^N q_i , \quad P_\alpha \equiv\sum_{i=1}^N q_i r_{i\alpha} , \quad \text{and}\quad Q_{\alpha\beta} \equiv \sum_{i=1}^N q_i (3r_{i\alpha} r_{i\beta} - \delta_{\alpha\beta} r_i^2) ,</math>
and we obtain finally the first few terms of the '''multipole expansion''' of the total potential, which is the sum of the Coulomb potentials of the separate charges:<ref name="Jackson75">{{cite book| last1=Jackson|first1=John David| title=Classical electrodynamics|date=1975| publisher=Wiley|location=New York| isbn=047143132X|edition=2d|url-access=registration| url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|pages=137–138}}
<math display="block">\begin{align}
4\pi\varepsilon_0 V(\mathbf{R}) &\equiv \sum_{i=1}^N q_i v(\mathbf{r}_i-\mathbf{R}) \\
&= \frac{q_\mathrm{tot}}{R} + \frac{1}{R^3}\sum_{\alpha=x,y,z} P_\alpha R_\alpha +
\frac{1}{2 R^5}\sum_{\alpha,\beta=x,y,z} Q_{\alpha\beta} R_\alpha R_\beta + \cdots
\end{align}</math>

This expansion of the potential of a discrete charge distribution is very similar to the one in real solid harmonics given below. The main difference is that the present one is in terms of linearly dependent quantities, for
<math display="block">\sum_{\alpha} v_{\alpha\alpha} = 0 \quad \hbox{and} \quad \sum_{\alpha} Q_{\alpha\alpha} = 0 .</math>

'''Note:'''
If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance {{mvar|d}} apart, so that {{math|''d''/''R'' ≫ (''d''/''R'')2}}, it is easily shown that the dominant term in the expansion is
<math display="block">V(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0 R^3} (\mathbf{P}\cdot\mathbf{R}) ,</math>
the electric [[Dipole#Field from an electric dipole|dipolar potential field]].

===Spherical form===
The potential {{math|''V''('''R''')}} at a point {{math|'''R'''}} outside the charge distribution, i.e. {{math|{{abs|'''R'''}} > ''r''max}}, can be expanded by the [[Laplace expansion (potential)|Laplace expansion]]:
<math display="block">V(\mathbf{R}) \equiv \sum_{i=1}^N \frac{q_i}{4\pi \varepsilon_0 |\mathbf{r}_i - \mathbf{R}|}
=\frac{1}{4\pi \varepsilon_0} \sum_{\ell=0}^\infty \sum_{m=-\ell}^{\ell}
(-1)^m I^{-m}_\ell(\mathbf{R}) \sum_{i=1}^N q_i R^m_\ell(\mathbf{r}_i),</math>
where <math>I^{-m}_{\ell}(\mathbf{R})</math> is an irregular [[solid harmonic]] (defined below as a [[spherical harmonic]] function divided by <math>R^{\ell+1}</math>) and <math>R^m_{\ell}(\mathbf{r})</math> is a regular solid harmonic (a spherical harmonic times {{math|r''ℓ''}}). We define the ''spherical multipole moment'' of the charge distribution as follows
<math display="block">Q^m_\ell \equiv \sum_{i=1}^N q_i R^m_\ell(\mathbf{r}_i),\quad\ -\ell \le m \le \ell.</math>
Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the ''N'' charges).

A [[spherical harmonic]] depends on the unit vector <math>\hat{R}</math>. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as
<math display="block">I^m_{\ell}(\mathbf{R}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \frac{Y^m_{\ell}(\hat{R})}{R^{\ell+1}}</math>
so that the ''multipole expansion'' of the field {{math|''V''('''R''')}} at the point {{math|'''R'''}} outside the charge distribution is given by

<math display="block">\begin{align}
V(\mathbf{R})
& = \frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}
\sum_{m=-\ell}^{\ell}(-1)^{m} I^{-m}_{\ell}(\mathbf{R}) Q^{m}_{\ell}\\
& = \frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}\left[\frac{4\pi}{2\ell + 1}\right]^{1/2}\;\frac{1}{R^{\ell + 1}}
\sum_{m=-\ell}^{\ell}(-1)^{m} Y^{-m}_{\ell}(\hat{R}) Q^{m}_{\ell}, \qquad R > r_{\mathrm{max}}
\end{align}</math>

This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the [[spherical multipole moments]] appear as coefficients in the {{math|1/''R''}} expansion of the potential.

It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks.
Since the summand of the ''m'' summation is invariant under a unitary transformation of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a [[Solid harmonics#Real form|unitary transformation]], we can simply substitute real irregular solid harmonics and real multipole moments. The {{math|1=''ℓ'' =&thinsp;0}} term becomes
<math display="block">V_{\ell=0}(\mathbf{R}) =
\frac{q_\mathrm{tot}}{4\pi \varepsilon_0 R} \quad\hbox{with}\quad q_\mathrm{tot}\equiv\sum_{i=1}^N q_i.</math>
This is in fact [[Coulomb's law]] again. For the {{math|1=''ℓ'' =&thinsp;1}} term we introduce
<math display="block">\mathbf{R} = (R_x, R_y, R_z),\quad \mathbf{P} = (P_x, P_y, P_z)\quad
\hbox{with}\quad P_\alpha \equiv \sum_{i=1}^N q_i r_{i\alpha}, \quad \alpha=x,y,z.</math>
Then
<math display="block">V_{\ell=1}(\mathbf{R}) =
\frac{1}{4\pi \varepsilon_0 R^3} (R_x P_x +R_y P_y + R_z P_z) = \frac{\mathbf{R} \cdot \mathbf{P} }{4\pi \varepsilon_0 R^3} =
\frac{\hat\mathbf{R} \cdot \mathbf{P} }{4\pi \varepsilon_0 R^2}.</math>
This term is identical to the one found in Cartesian form.

In order to write the {{math|1=''ℓ'' =&thinsp;2}} term, we have to introduce shorthand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type
<math display="block">Q_{z^2} \equiv \sum_{i=1}^N q_i\; \frac{1}{2}(3z_i^2 - r_i^2),</math>
can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.

==Interaction of two non-overlapping charge distributions==
Consider two sets of point charges, one set {{math|{''q''''i''}<nowiki/>}} clustered around a point {{mvar|A}} and one set {{math|{''q''''j''}<nowiki/>}} clustered around a point {{mvar|B}}. Think for example of two [[molecule]]s, and recall that a molecule by definition consists of [[electron]]s (negative point charges) and [[atomic nucleus|nuclei]] (positive point charges). The total electrostatic interaction energy {{math|''U''''AB''}} between the two distributions is
<math display="block">U_{AB} = \sum_{i\in A} \sum_{j\in B} \frac{q_i q_j}{4\pi\varepsilon_0 r_{ij}}.</math>
This energy can be expanded in a power series in the inverse distance of {{mvar|A}} and {{mvar|B}}.
This expansion is known as the '''multipole expansion''' of ''U''''AB''.

In order to derive this multipole expansion, we write {{math|1='''r'''XY = '''r'''''Y'' − '''r'''''X''}}, which is a vector pointing from {{mvar|X}} towards {{mvar|Y}}. Note that
<math display="block">\mathbf{R}_{AB}+\mathbf{r}_{Bj}+\mathbf{r}_{ji}+\mathbf{r}_{iA} = 0
\quad \iff \quad
\mathbf{r}_{ij} = \mathbf{R}_{AB}-\mathbf{r}_{Ai}+\mathbf{r}_{Bj} .</math>
We assume that the two distributions do not overlap:
<math display="block"> |\mathbf{R}_{AB}| > |\mathbf{r}_{Bj}-\mathbf{r}_{Ai}| \text{ for all } i,j.</math>
Under this condition we may apply the [[Laplace expansion (potential)|Laplace expansion]] in the following form
<math display="block">\frac{1}{|\mathbf{r}_{j}-\mathbf{r}_i|} = \frac{1}{|\mathbf{R}_{AB} - (\mathbf{r}_{Ai}- \mathbf{r}_{Bj})| } =
\sum_{L=0}^\infty \sum_{M=-L}^L \, (-1)^M I_L^{-M}(\mathbf{R}_{AB})\;
R^M_L( \mathbf{r}_{Ai} - \mathbf{r}_{Bj}),</math>
where <math>I^M_L</math> and <math>R^M_L</math> are irregular and regular [[solid harmonics]], respectively. The [[Solid harmonics#Addition theorems|translation of the regular solid harmonic]] gives a finite expansion,
<math display="block">R^M_L(\mathbf{r}_{Ai}-\mathbf{r}_{Bj}) = \sum_{\ell_A=0}^L (-1)^{L-\ell_A} \binom{2L}{2\ell_A}^{1/2}
\times \sum_{m_A=-\ell_A}^{\ell_A} R^{m_A}_{\ell_A}(\mathbf{r}_{Ai})
R^{M-m_A}_{L-\ell_A}(\mathbf{r}_{Bj})\;
\langle \ell_A, m_A; L-\ell_A, M-m_A\mid L M \rangle,
</math>
where the quantity between pointed brackets is a [[Clebsch–Gordan coefficient]]. Further we used
<math display="block">R^{m}_{\ell}(-\mathbf{r}) = (-1)^{\ell} R^{m}_{\ell}(\mathbf{r}) .</math>
Use of the definition of [[spherical multipole moments#General spherical multipole moments|spherical multipoles]] {{math|''Q''{{supsub|''m''|''ℓ''}}}} and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of {{mvar|L}}) gives finally

<math display="block">\begin{align}
U_{AB} = {} & \frac{1}{4\pi\varepsilon_0} \sum_{\ell_A=0}^\infty \sum_{\ell_B=0}^\infty (-1)^{\ell_B} \binom{2\ell_A+2\ell_B}{2\ell_A}^{1/2} \\[5pt]
& \times \sum_{m_A=-\ell_A}^{\ell_A} \sum_{m_B=-\ell_B}^{\ell_B}(-1)^{m_A+m_B} I_{\ell_A+\ell_B}^{-m_A-m_B}(\mathbf{R}_{AB})\;
Q^{m_A}_{\ell_A} Q^{m_B}_{\ell_B}\;
\langle \ell_A, m_A; \ell_B, m_B\mid \ell_A+\ell_B, m_A+m_B \rangle.
\end{align}</math>

This is the '''multipole expansion''' of the interaction energy of two non-overlapping charge distributions which are a distance ''R''''AB'' apart. Since
<math display="block">I_{\ell_A+\ell_B}^{-(m_A+m_B)}(\mathbf{R}_{AB}) \equiv \left[\frac{4\pi}{2\ell_A+2\ell_B+1}\right]^{1/2}\;
\frac{Y^{-(m_A+m_B)}_{\ell_A+\ell_B}\left(\widehat{\mathbf{R}}_{AB}\right)}{R^{\ell_A+\ell_B+1}_{AB}},</math>
this expansion is manifestly in powers of {{math|1 / ''RAB''}}. The function {{math|''Y''''m''''l''}} is a normalized [[spherical harmonic]].

===Molecular moments===
All atoms and molecules (except [[Azimuthal quantum number|''S''-state]] atoms) have one or more non-vanishing permanent multipole moments. Different definitions can be found in the literature, but the following definition in spherical form has the advantage that it is contained in one general equation. Because it is in complex form it has as the further advantage that it is easier to manipulate in calculations than its real counterpart.

We consider a molecule consisting of ''N'' particles (electrons and nuclei) with charges ''eZ''''i''. (Electrons have a ''Z''-value of −1, while for nuclei it is the [[atomic number]]). Particle ''i'' has spherical polar coordinates ''r''''i'', ''θ''''i'', and φ''i'' and Cartesian coordinates ''x''''i'', ''y''''i'', and ''z''''i''.
The (complex) electrostatic multipole operator is
<math display="block">Q^m_\ell \equiv \sum_{i=1}^N e Z_i \; R^m_{\ell}(\mathbf{r}_i),</math>
where <math>R^m_{\ell}(\mathbf{r}_i)</math> is a regular [[Solid harmonics|solid harmonic]] function in [[Solid harmonics#Racah's normalization|Racah's normalization]] (also known as Schmidt's semi-normalization).
If the molecule has total normalized wave function Ψ (depending on the coordinates of electrons and nuclei), then the multipole moment of order <math>\ell</math> of the molecule is given by the [[expectation value (quantum mechanics)|expectation (expected) value]]:
<math display="block">M^m_\ell \equiv \langle \Psi \mid Q^m_\ell \mid \Psi \rangle.</math>
If the molecule has certain [[Molecular symmetry#Point group|point group symmetry]], then this is reflected in the wave function: Ψ transforms according to a certain [[irreducible representation]] λ of the [[group (mathematics)|group]] ("Ψ has symmetry type λ"). This has the consequence that [[selection rule]]s hold for the expectation value of the multipole operator, or in other words, that the expectation value may vanish because of symmetry. A well-known example of this is the fact that molecules with an inversion center do not carry a dipole (the expectation values of <math> Q^m_1 </math> vanish for {{math|1=''m'' = −1, 0, 1)}}. For a molecule without symmetry, no selection rules are operative and such a molecule will have non-vanishing multipoles of any order (it will carry a dipole and simultaneously a quadrupole, octupole, hexadecapole, etc.).

The lowest explicit forms of the regular solid harmonics (with the [[Spherical harmonics#Condon–Shortley phase|Condon-Shortley phase]]) give:
<math display="block"> M^0_0 = \sum_{i=1}^N e Z_i, </math>
(the total charge of the molecule). The (complex) dipole components are:
<math display="block"> M^1_1 = - \tfrac{1}{\sqrt 2} \sum_{i=1}^N e Z_i \langle \Psi | x_i+iy_i | \Psi \rangle\quad \hbox{and} \quad
M^{-1}_{1} = \tfrac{1}{\sqrt 2} \sum_{i=1}^N e Z_i \langle \Psi | x_i - iy_i | \Psi \rangle. </math>
<math display="block"> M^0_1 = \sum_{i=1}^N e Z_i \langle \Psi | z_i | \Psi \rangle.</math>

Note that by a simple [[Solid harmonics#Real form|linear combination]] one can transform the complex multipole operators to real ones. The real multipole operators are of cosine type
<math> C^m_\ell</math> or sine type <math>S^m_\ell</math>. A few of the lowest ones are:
<math display="block">\begin{align}
C^0_1 &= \sum_{i=1}^N eZ_i \; z_i \\
C^1_1 &= \sum_{i=1}^N eZ_i \;x_i \\
S^1_1 &= \sum_{i=1}^N eZ_i \;y_i \\
C^0_2 &= \frac{1}{2}\sum_{i=1}^N eZ_i\; (3z_i^2-r_i^2)\\
C^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i x_i \\
C^2_2 &= \frac{1}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; (x_i^2-y_i^2) \\
S^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i y_i \\
S^2_2 &= \frac{2}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; x_iy_i
\end{align}</math>

====Note on conventions====
The definition of the complex molecular multipole moment given above is the [[complex conjugate]] of the definition given in [[Spherical multipole moments#General spherical multipole moments|this article]], which follows the definition of the standard textbook on [[classical electrodynamics]] by Jackson,<ref name=Jackson75/>{{rp|137}} except for the normalization. Moreover, in the classical definition of Jackson the equivalent of the ''N''-particle [[quantum mechanics|quantum mechanical]] expectation value is an [[integral]] over a one-particle charge distribution. Remember that in the case of a one-particle quantum mechanical system the expectation value is nothing but an integral over the charge distribution (modulus of wavefunction squared), so that the definition of this article is a quantum mechanical ''N''-particle generalization of Jackson's definition.

The definition in this article agrees with, among others, the one of Fano and Racah<ref>U. Fano and G. Racah, ''Irreducible Tensorial Sets'', Academic Press, New York (1959). p. 31</ref> and Brink and Satchler.<ref>D. M. Brink and G. R. Satchler, ''Angular Momentum'', 2nd edition, Clarendon Press, Oxford, UK (1968). p. 64. See also footnote on p. 90.</ref>

==Examples==
There are many types of multipole moments, since there are many types of [[potential]]s and many ways of approximating a potential by a [[series expansion]], depending on the [[coordinate system|coordinates]] and the [[symmetry]] of the charge distribution. The most common expansions include:

* [[Axial multipole moments]] of a {{math|1/''R''}} potential;
* [[Spherical multipole moments]] of a {{math|1/''R''}} potential; and
* [[Cylindrical multipole moments]] of a {{math|ln&thinsp;''R''}} potential

Examples of {{math|1/''R''}} potentials include the [[electric potential]], the [[magnetic scalar potential|magnetic potential]] and the [[gravitational potential]] of point sources. An example of a {{math|ln&thinsp;''R''}} potential is the [[electric potential]] of an infinite line charge.

==General mathematical properties==
Multipole moments in [[mathematics]] and [[mathematical physics]] form an [[orthogonal basis]] for the decomposition of a function, based on the response of a [[field (physics)|field]] to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of [[Distribution (mathematics)|distribution theory]], as [[directional derivative]]s.

Multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated [[differential equation]]s. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of [[group representation|irreducible representations]] of the rotational [[symmetry group]], which leads to spherical harmonics and related sets of [[orthogonal]] functions. One uses the technique of [[separation of variables]] to extract the corresponding solutions for the radial dependencies.

In practice, many fields can be well approximated with a finite number of multipole moments (although an infinite number may be required to reconstruct a field exactly). A typical application is to approximate the field of a localized charge distribution by its [[Monopole (mathematics)|monopole]] and [[dipole]] terms. Problems solved once for a given order of multipole moment may be [[linear combination|linearly combined]] to create a final approximate solution for a given source.

==See also==
* [[Barnes–Hut simulation]]
* [[Fast multipole method]]
* [[Laplace expansion (potential)|Laplace expansion]]
* [[Legendre polynomials]]
* [[Quadrupole magnet]]s are used in [[particle accelerator]]s
* [[Solid harmonics]]
* [[Toroidal moment]]
* [[Dynamic toroidal dipole]]

==References==
<references />

[[Category:Potential theory]]
[[Category:Vector calculus]]
[[Category:Moment (physics)]]

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