A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, ${\displaystyle \mathbb {R} ^{3))$. 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- or complex-valued and is defined either on ${\displaystyle \mathbb {R} ^{3))$, or less often on ${\displaystyle \mathbb {R} ^{n))$ for some other ${\displaystyle n}$.

Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, 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.^[1]

The multipole expansion is expressed as a sum of terms with progressively finer angular features (moments). The first (the zeroth-order) term is called the 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 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).^[2][3][4] A multipole moment usually involves 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 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 ${\displaystyle f(\theta ,\varphi )}$ as the sum

$${\displaystyle f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }\,C_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi )}$$

${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}$

${\displaystyle C_{\ell }^{m))$

${\displaystyle C_{0}^{0))$

${\displaystyle C_{1}^{-1},C_{1}^{0},C_{1}^{1))$

$${\displaystyle f(\theta ,\varphi )=C+C_{i}n^{i}+C_{ij}n^{i}n^{j}+C_{ijk}n^{i}n^{j}n^{k}+C_{ijk\ell }n^{i}n^{j}n^{k}n^{\ell }+\cdots }$$

${\displaystyle n^{i))$

${\displaystyle \theta }$

${\displaystyle \varphi }$

${\displaystyle C}$

${\displaystyle C_{i))$

In the above expansions, the coefficients may be real or 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

$${\displaystyle C_{\ell }^{-m}=(-1)^{m}C_{\ell }^{m\ast }\,.}$$

$${\displaystyle C=C^{\ast };\ C_{i}=C_{i}^{\ast };\ C_{ij}=C_{ij}^{\ast };\ C_{ijk}=C_{ijk}^{\ast };\ \ldots }$$

While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank.^[6] This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves.

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, ${\displaystyle r}$—most frequently, as a Laurent series in powers of ${\displaystyle r}$. For example, to describe the electromagnetic potential, ${\displaystyle V}$, from a source in a small region near the origin, the coefficients may be written as:

$${\displaystyle V(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }C_{\ell }^{m}(r)\,Y_{\ell }^{m}(\theta ,\varphi )=\sum _{j=1}^{\infty }\,\sum _{\ell =0}^{\infty }\,\sum _{m=-\ell }^{\ell }{\frac {D_{\ell ,j}^{m)){r^{j))}\,Y_{\ell }^{m}(\theta ,\varphi ).}$$

Applications

Multipole expansions are widely used in problems involving gravitational fields of systems of masses, electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves. A classic example is the calculation of the exterior multipole moments of 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 Greengard and Rokhlin, a general technique for efficient computation of energies and forces in systems of interacting particles. 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: r_i < 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., |R| > 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 1 / |r − R|, which was done once and for all by Legendre in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion—usually only the first few terms are given followed by an ellipsis.

Expansion in Cartesian coordinates

Assume v(r) = v(−r) for convenience. The Taylor expansion of v(r − R) around the origin r = 0 can be written as

$${\displaystyle v(\mathbf {r} -\mathbf {R} )=v(\mathbf {R} )-\sum _{\alpha =x,y,z}r_{\alpha }v_{\alpha }(\mathbf {R} )+{\frac {1}{2))\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )-\cdots +\cdots }$$

$${\displaystyle v_{\alpha }(\mathbf {R} )\equiv \left({\frac {\partial v(\mathbf {r} -\mathbf {R} )}{\partial r_{\alpha ))}\right)_{\mathbf {r} =\mathbf {0} }\quad {\text{and))\quad v_{\alpha \beta }(\mathbf {R} )\equiv \left({\frac {\partial ^{2}v(\mathbf {r} -\mathbf {R} )}{\partial r_{\alpha }\partial r_{\beta ))}\right)_{\mathbf {r} =\mathbf {0} }.}$$

$${\displaystyle \left(\nabla ^{2}v(\mathbf {r} -\mathbf {R} )\right)_{\mathbf {r} =\mathbf {0} }=\sum _{\alpha =x,y,z}v_{\alpha \alpha }(\mathbf {R} )=0,}$$

$${\displaystyle \sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )={\frac {1}{3))\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}\left(3r_{\alpha }r_{\beta }-\delta _{\alpha \beta }r^{2}\right)v_{\alpha \beta }(\mathbf {R} ),}$$

Example

Consider now the following form of v(r − R):

$${\displaystyle v(\mathbf {r} -\mathbf {R} )\equiv {\frac {1}{|\mathbf {r} -\mathbf {R} |)).}$$

$${\displaystyle v(\mathbf {R} )={\frac {1}{R)),\quad v_{\alpha }(\mathbf {R} )=-{\frac {R_{\alpha )){R^{3))},\quad {\hbox{and))\quad v_{\alpha \beta }(\mathbf {R} )={\frac {3R_{\alpha }R_{\beta }-\delta _{\alpha \beta }R^{2)){R^{5))}.}$$

$${\displaystyle q_{\mathrm {tot} }\equiv \sum _{i=1}^{N}q_{i},\quad P_{\alpha }\equiv \sum _{i=1}^{N}q_{i}r_{i\alpha },\quad {\text{and))\quad Q_{\alpha \beta }\equiv \sum _{i=1}^{N}q_{i}(3r_{i\alpha }r_{i\beta }-\delta _{\alpha \beta }r_{i}^{2}),}$$

$${\displaystyle {\begin{aligned}4\pi \varepsilon _{0}V(\mathbf {R} )&\equiv \sum _{i=1}^{N}q_{i}v(\mathbf {r} _{i}-\mathbf {R} )\\&={\frac {q_{\mathrm {tot} )){R))+{\frac {1}{R^{3))}\sum _{\alpha =x,y,z}P_{\alpha }R_{\alpha }+{\frac {1}{2R^{5))}\sum _{\alpha ,\beta =x,y,z}Q_{\alpha \beta }R_{\alpha }R_{\beta }+\cdots \end{aligned))}$$

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

$${\displaystyle \sum _{\alpha }v_{\alpha \alpha }=0\quad {\hbox{and))\quad \sum _{\alpha }Q_{\alpha \alpha }=0.}$$

Note: If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance d apart, so that d/R ≫ (d/R)², it is easily shown that the dominant term in the expansion is

$${\displaystyle V(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}R^{3))}(\mathbf {P} \cdot \mathbf {R} ),}$$

Spherical form

The potential V(R) at a point R outside the charge distribution, i.e. |R| > r_max, can be expanded by the Laplace expansion:

$${\displaystyle V(\mathbf {R} )\equiv \sum _{i=1}^{N}{\frac {q_{i)){4\pi \varepsilon _{0}|\mathbf {r} _{i}-\mathbf {R} |))={\frac {1}{4\pi \varepsilon _{0))}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R} )\sum _{i=1}^{N}q_{i}R_{\ell }^{m}(\mathbf {r} _{i}),}$$

${\displaystyle I_{\ell }^{-m}(\mathbf {R} )}$

${\displaystyle R^{\ell +1))$

${\displaystyle R_{\ell }^{m}(\mathbf {r} )}$

$${\displaystyle Q_{\ell }^{m}\equiv \sum _{i=1}^{N}q_{i}R_{\ell }^{m}(\mathbf {r} _{i}),\quad \ -\ell \leq m\leq \ell .}$$

A spherical harmonic depends on the unit vector ${\displaystyle {\hat {R))}$. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as

$${\displaystyle I_{\ell }^{m}(\mathbf {R} )\equiv {\sqrt {\frac {4\pi }{2\ell +1))}{\frac {Y_{\ell }^{m}({\hat {R)))}{R^{\ell +1))))$$

$${\displaystyle {\begin{aligned}V(\mathbf {R} )&={\frac {1}{4\pi \varepsilon _{0))}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R} )Q_{\ell }^{m}\\&={\frac {1}{4\pi \varepsilon _{0))}\sum _{\ell =0}^{\infty }\left[{\frac {4\pi }{2\ell +1))\right]^{1/2}\;{\frac {1}{R^{\ell +1))}\sum _{m=-\ell }^{\ell }(-1)^{m}Y_{\ell }^{-m}({\hat {R)))Q_{\ell }^{m},\qquad R>r_{\mathrm {max} }\end{aligned))}$$

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 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 unitary transformation, we can simply substitute real irregular solid harmonics and real multipole moments. The ℓ = 0 term becomes

$${\displaystyle V_{\ell =0}(\mathbf {R} )={\frac {q_{\mathrm {tot} )){4\pi \varepsilon _{0}R))\quad {\hbox{with))\quad q_{\mathrm {tot} }\equiv \sum _{i=1}^{N}q_{i}.}$$

$${\displaystyle \mathbf {R} =(R_{x},R_{y},R_{z}),\quad \mathbf {P} =(P_{x},P_{y},P_{z})\quad {\hbox{with))\quad P_{\alpha }\equiv \sum _{i=1}^{N}q_{i}r_{i\alpha },\quad \alpha =x,y,z.}$$

$${\displaystyle V_{\ell =1}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}R^{3))}(R_{x}P_{x}+R_{y}P_{y}+R_{z}P_{z})={\frac {\mathbf {R} \cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{3))}={\frac ((\hat {\mathbf {R} ))\cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{2))}.}$$

In order to write the ℓ = 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

$${\displaystyle Q_{z^{2))\equiv \sum _{i=1}^{N}q_{i}\;{\frac {1}{2))(3z_{i}^{2}-r_{i}^{2}),}$$

Interaction of two non-overlapping charge distributions

Consider two sets of point charges, one set {q_i} clustered around a point A and one set {q_j} clustered around a point B. Think for example of two molecules, and recall that a molecule by definition consists of electrons (negative point charges) and nuclei (positive point charges). The total electrostatic interaction energy U_AB between the two distributions is

$${\displaystyle U_{AB}=\sum _{i\in A}\sum _{j\in B}{\frac {q_{i}q_{j)){4\pi \varepsilon _{0}r_{ij))}.}$$

In order to derive this multipole expansion, we write r_XY = r_Y − r_X, which is a vector pointing from X towards Y. Note that

$${\displaystyle \mathbf {R} _{AB}+\mathbf {r} _{Bj}+\mathbf {r} _{ji}+\mathbf {r} _{iA}=0\quad \iff \quad \mathbf {r} _{ij}=\mathbf {R} _{AB}-\mathbf {r} _{Ai}+\mathbf {r} _{Bj}.}$$

$${\displaystyle |\mathbf {R} _{AB}|>|\mathbf {r} _{Bj}-\mathbf {r} _{Ai}|{\text{ for all ))i,j.}$$

$${\displaystyle {\frac {1}{|\mathbf {r} _{j}-\mathbf {r} _{i}|))={\frac {1}{|\mathbf {R} _{AB}-(\mathbf {r} _{Ai}-\mathbf {r} _{Bj})|))=\sum _{L=0}^{\infty }\sum _{M=-L}^{L}\,(-1)^{M}I_{L}^{-M}(\mathbf {R} _{AB})\;R_{L}^{M}(\mathbf {r} _{Ai}-\mathbf {r} _{Bj}),}$$

${\displaystyle I_{L}^{M))$

${\displaystyle R_{L}^{M))$

$${\displaystyle R_{L}^{M}(\mathbf {r} _{Ai}-\mathbf {r} _{Bj})=\sum _{\ell _{A}=0}^{L}(-1)^{L-\ell _{A)){\binom {2L}{2\ell _{A))}^{1/2}\times \sum _{m_{A}=-\ell _{A))^{\ell _{A))R_{\ell _{A))^{m_{A))(\mathbf {r} _{Ai})R_{L-\ell _{A))^{M-m_{A))(\mathbf {r} _{Bj})\;\langle \ell _{A},m_{A};L-\ell _{A},M-m_{A}\mid LM\rangle ,}$$

$${\displaystyle R_{\ell }^{m}(-\mathbf {r} )=(-1)^{\ell }R_{\ell }^{m}(\mathbf {r} ).}$$

$${\displaystyle {\begin{aligned}U_{AB}={}&{\frac {1}{4\pi \varepsilon _{0))}\sum _{\ell _{A}=0}^{\infty }\sum _{\ell _{B}=0}^{\infty }(-1)^{\ell _{B)){\binom {2\ell _{A}+2\ell _{B)){2\ell _{A))}^{1/2}\\[5pt]&\times \sum _{m_{A}=-\ell _{A))^{\ell _{A))\sum _{m_{B}=-\ell _{B))^{\ell _{B))(-1)^{m_{A}+m_{B))I_{\ell _{A}+\ell _{B))^{-m_{A}-m_{B))(\mathbf {R} _{AB})\;Q_{\ell _{A))^{m_{A))Q_{\ell _{B))^{m_{B))\;\langle \ell _{A},m_{A};\ell _{B},m_{B}\mid \ell _{A}+\ell _{B},m_{A}+m_{B}\rangle .\end{aligned))}$$

This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance R_AB apart. Since

$${\displaystyle I_{\ell _{A}+\ell _{B))^{-(m_{A}+m_{B})}(\mathbf {R} _{AB})\equiv \left[{\frac {4\pi }{2\ell _{A}+2\ell _{B}+1))\right]^{1/2}\;{\frac {Y_{\ell _{A}+\ell _{B))^{-(m_{A}+m_{B})}\left({\widehat {\mathbf {R} ))_{AB}\right)}{R_{AB}^{\ell _{A}+\ell _{B}+1))},}$$

Molecular moments

All atoms and molecules (except 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

$${\displaystyle Q_{\ell }^{m}\equiv \sum _{i=1}^{N}eZ_{i}\;R_{\ell }^{m}(\mathbf {r} _{i}),}$$

${\displaystyle R_{\ell }^{m}(\mathbf {r} _{i})}$

${\displaystyle \ell }$

$${\displaystyle M_{\ell }^{m}\equiv \langle \Psi \mid Q_{\ell }^{m}\mid \Psi \rangle .}$$

${\displaystyle Q_{1}^{m))$

The lowest explicit forms of the regular solid harmonics (with the Condon-Shortley phase) give:

$${\displaystyle M_{0}^{0}=\sum _{i=1}^{N}eZ_{i},}$$

$${\displaystyle M_{1}^{1}=-{\tfrac {1}{\sqrt {2))}\sum _{i=1}^{N}eZ_{i}\langle \Psi |x_{i}+iy_{i}|\Psi \rangle \quad {\hbox{and))\quad M_{1}^{-1}={\tfrac {1}{\sqrt {2))}\sum _{i=1}^{N}eZ_{i}\langle \Psi |x_{i}-iy_{i}|\Psi \rangle .}$$

$${\displaystyle M_{1}^{0}=\sum _{i=1}^{N}eZ_{i}\langle \Psi |z_{i}|\Psi \rangle .}$$

Note that by a simple linear combination one can transform the complex multipole operators to real ones. The real multipole operators are of cosine type ${\displaystyle C_{\ell }^{m))$ or sine type ${\displaystyle S_{\ell }^{m))$. A few of the lowest ones are:

$${\displaystyle {\begin{aligned}C_{1}^{0}&=\sum _{i=1}^{N}eZ_{i}\;z_{i}\\C_{1}^{1}&=\sum _{i=1}^{N}eZ_{i}\;x_{i}\\S_{1}^{1}&=\sum _{i=1}^{N}eZ_{i}\;y_{i}\\C_{2}^{0}&={\frac {1}{2))\sum _{i=1}^{N}eZ_{i}\;(3z_{i}^{2}-r_{i}^{2})\\C_{2}^{1}&={\sqrt {3))\sum _{i=1}^{N}eZ_{i}\;z_{i}x_{i}\\C_{2}^{2}&={\frac {1}{3)){\sqrt {3))\sum _{i=1}^{N}eZ_{i}\;(x_{i}^{2}-y_{i}^{2})\\S_{2}^{1}&={\sqrt {3))\sum _{i=1}^{N}eZ_{i}\;z_{i}y_{i}\\S_{2}^{2}&={\frac {2}{3)){\sqrt {3))\sum _{i=1}^{N}eZ_{i}\;x_{i}y_{i}\end{aligned))}$$

Note on conventions

The definition of the complex molecular multipole moment given above is the complex conjugate of the definition given in this article, which follows the definition of the standard textbook on classical electrodynamics by Jackson,^[7]: 137 except for the normalization. Moreover, in the classical definition of Jackson the equivalent of the N-particle 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^[8] and Brink and Satchler.^[9]

Examples

There are many types of multipole moments, since there are many types of potentials and many ways of approximating a potential by a series expansion, depending on the coordinates and the symmetry of the charge distribution. The most common expansions include:

• Axial multipole moments of a 1/R potential;
• Spherical multipole moments of a 1/R potential; and
• Cylindrical multipole moments of a ln R potential

Examples of 1/R potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. An example of a ln 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 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 theory, as directional derivatives.

Multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of 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 and dipole terms. Problems solved once for a given order of multipole moment may be linearly combined to create a final approximate solution for a given source.