Multipole expansion

Multipole expansion

A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used 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. For example, the initial term — called the zero-th, or monopole, moment — is a constant, independent of angle. The following term — the first, or dipole, moment — varies once from positive to negative around the sphere. Higher-order terms (like the quadrupole and octupole) vary more quickly with angles.


Expansion in spherical harmonics

Most commonly, the series is written as a sum of spherical harmonics. Thus, we might write a function f(θ,ϕ) as the sum

f(\theta,\phi) = \sum_{l=0}^\infty\, \sum_{m=-l}^{l}\, C^m_l\, Y^m_l(\theta,\phi).

Here, Y^m_l(\theta,\phi) are the standard spherical harmonics, and C^m_l are constant coefficients which depend on the function. The term C^0_0 represents the monopole; C^{-1}_1,C^0_1,C^1_1 represent the dipole; and so on. Equivalently, the series is also frequently written[2] as

f(\theta,\phi) = C + C_i n^i + C_{ij}n^i n^j + C_{ijk}n^i n^j n^k + C_{ijkl}n^i n^j n^k n^l + \cdots.

Here, each ni represents a unit vector in the direction given by the angles θ and ϕ, and indices are implicitly summed. Here, the term C is the monopole; Ci is a set of three numbers representing the dipole; and so on.

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

C_l^m = (-1)^m C^{m\ast}_l\ .

In the multi-vector expansion, each coefficient must be real:

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.[3] 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, r -- most frequently, as a Laurent series in powers of r. For example, to describe the electromagnetic potential, V, from a source in a small region near the origin, the coefficients may be written as:

V(r,\theta,\phi) = \sum_{l=0}^\infty\, \sum_{m=-l}^{l}\, C^m_l(r)\, Y^m_l(\theta,\phi)= \sum_{j=1}^\infty\, \sum_{l=0}^\infty\, \sum_{m=-l}^{l}\, \frac{D^m_{l,j}}{r^j}\, Y^m_l(\theta,\phi) .

Applications of multipole expansions

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 [4] 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., if 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 qi with position vectors ri. We assume the charges to be clustered around the origin, so that for all i: ri < rmax, where rmax has some finite value. The potential V(R), due to the charge distribution, at a point R outside the charge distribution, i.e., |R| > rmax, 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 some dots.

Expansion in Cartesian coordinates

The Taylor expansion of an arbitrary function v(r-R) around the origin r = 0 is,

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})


v_\alpha(\mathbf{R}) \equiv\left( \frac{\partial v(\mathbf{r}-\mathbf{R}) }{\partial r_\alpha}\right)_{\mathbf{r}= \mathbf0}\quad\hbox{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}

If v(r-R) satisfies the Laplace equation

\left(\nabla^2 v(\mathbf{r}- \mathbf{R})\right)_{\mathbf{r}=\mathbf0}  = \sum_{\alpha=x,y,z} v_{\alpha\alpha}(\mathbf{R})  = 0

then the expansion can be rewritten in terms of the components of a traceless Cartesian second rank tensor,

\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} (3r_\alpha  r_\beta - \delta_{\alpha\beta} r^2) v_{\alpha\beta}(\mathbf{R}),

where δαβ is the Kronecker delta and r2 ≡ |r|2. Removing the trace is common, because it takes the rotational invariant r2 out of the second rank tensor.


Consider now the following form of v(r-R),

v(\mathbf{r}- \mathbf{R}) \equiv \frac{1}{|\mathbf{r}- \mathbf{R}|},

then by direct differentiation it follows that

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}.

Define a monopole, dipole and (traceless) quadrupole by, respectively,

q_\mathrm{tot} \equiv \sum_{i=1}^N q_i, \quad P_\alpha \equiv\sum_{i=1}^N q_i r_{i\alpha}, \quad \hbox{and}\quad Q_{\alpha\beta} \equiv \sum_{i=1}^N q_i (3r_{i\alpha}  r_{i\beta} - \delta_{\alpha\beta} r_i^2)

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,

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}{6 R^5}\sum_{\alpha,\beta=x,y,z} Q_{\alpha\beta} (3R_\alpha  R_\beta - \delta_{\alpha\beta} R^2) +\cdots

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 linear dependent quantities, for

\sum_{\alpha} v_{\alpha\alpha} = 0 \quad \hbox{and}\quad \sum_{\alpha} Q_{\alpha\alpha} = 0.


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 only non-vanishing term in the expansion is

V(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0 R^3} (\mathbf{P}\cdot\mathbf{R}) 

the electric dipolar potential field.

Spherical form

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

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),

where I^{-m}_{\ell}(\mathbf{R}) is an irregular solid harmonic (defined below as a spherical harmonic function divided by Rl+1) and R^m_{\ell}(\mathbf{r}) is a regular solid harmonic (a spherical harmonic times rl). We define the spherical multipole moment of the charge distribution as follows

Q^m_\ell \equiv \sum_{i=1}^N q_i R^{m}_\ell(\mathbf{r}_i),\qquad -\ell \le m \le \ell.

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

I^m_{\ell}(\mathbf{R}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \frac{Y^m_{\ell}(\hat{R})}{R^{\ell+1}}

so that the multipole expansion of the field V(R) at the point R outside the charge distribution is given by

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}}.

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

V_{\ell=0}(\mathbf{R}) = 
\frac{q_\mathrm{tot}}{4\pi \varepsilon_0 R}\qquad\hbox{with}\quad q_\mathrm{tot}\equiv\sum_{i=1}^N q_i.

This is in fact Coulomb's law again. For the l = 1 term we introduce

\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.


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{R}\cdot\mathbf{P} }{4\pi \varepsilon_0 R^2}.

This term is identical to the one found in Cartesian form.

In order to write the l=2 term, we have to introduce short-hand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type

Q_{z^2} \equiv \sum_{i=1}^N q_i\; \frac{1}{2}(3z_i^2 - r_i^2),

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 {qi } clustered around a point A and one set {qj } 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 UAB between the two distributions is

U_{AB} = \sum_{i\in A} \sum_{j\in B}  \frac{q_i q_j}{4\pi\varepsilon_0 r_{ij}}.

This energy can be expanded in a power series in the inverse distance of A and B. This expansion is known as the multipole expansion of UAB.

In order to derive this multipole expansion, we write rXY = rY-rX, which is a vector pointing from X towards Y. Note that

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

We assume that the two distributions do not overlap:

 |\mathbf{R}_{AB}| > |\mathbf{r}_{Bj}-\mathbf{r}_{Ai}| \quad\hbox{for all}\quad i,j.

Under this condition we may apply the Laplace expansion in the following form

\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}),

where I^M_L and R^M_L are irregular and regular solid harmonics, respectively. The translation of the regular solid harmonic gives a finite expansion,

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}) 
\langle \ell_A, m_A; L-\ell_A, M-m_A| L M \rangle,

where the quantity between pointed brackets is a Clebsch-Gordan coefficient. Further we used

R^{m}_{\ell}(-\mathbf{r}) = (-1)^{\ell} R^{m}_{\ell}(\mathbf{r}) .

Use of the definition of spherical multipoles Qml and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of L) gives finally

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} \,

\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| \ell_A+\ell_B, m_A+m_B \rangle.

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

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}\;

this expansion is manifestly in powers of 1/RAB. The function Yml is a normalized spherical harmonic.

Examples of multipole expansions

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:

Examples of \frac{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

Mathematically, 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.

See also


  1. ^ Edmonds, A. R.. Angular Momentum in Quantum Mechanics. Princeton University Press. 
  2. ^ Thompson, William J.. Angular Momentum. John Wiley & Sons, Inc.. 
  3. ^ Thorne, Kip S. (April 1980). "Multipole Expansions of Gravitational Radiation". Reviews of Modern Physics 52 (2): 299. Bibcode 1980RvMP...52..299T. doi:10.1103/RevModPhys.52.299. 
  4. ^ Ross D. Adamson (January 21, 1999). "The Fast Multipole Method". Retrieved December 10, 2010. 

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