# Multipole moments

**Multipole moments** are the coefficients of a series expansion of a potential due to continuous or discrete sources (e.g., an electric charge distribution). 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 (e.g., charges)
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**. The zeroth-order term
in the expansion is called
the monopole moment, the first-order term is denoted as the dipole moment, and
the third, fourth, etc. terms are denoted as quadrupole, octupole, etc. moments.

The potential at a position <math>\mathbf{r}</math> *within* a charge distribution can
often be computed by combining interior and exterior multipoles.

## Examples of multipoles

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 the <math>\frac{1}{R}</math> potential,
- Spherical multipole moments of the <math>\frac{1}{R}</math> potential, and
- Cylindrical multipole moments of the <math>\ln \ R^{ }</math> potential

Examples of <math>\frac{1}{R}</math> potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. An example of a <math>\ln \ R^{ }</math> 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.

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.

## Molecular electrostatic multipole 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 the *Z*-value unity, 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>

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 harmonic function in 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 (expected) value

- <math>

M^m_\ell \equiv \langle \Psi | Q^m_\ell | \Psi \rangle.</math>
If the molecule has certain point group symmetry, then this is reflected in the wave function: Ψ transforms according to a certain irreducible representation λ of the group ("Ψ has symmetry type λ"). This has the consequence that selection rules 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 *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 Condon-Shortley phase) give:

- <math> M^0_0 = \sum_{i=1}^N e Z_i, </math>

(the total charge of the molecule). The (complex) dipole components are:

- <math> M^1_1 = - \sqrt{\tfrac{1}{2}} \sum_{i=1}^N e Z_i \langle \Psi | x_i+iy_i | \Psi \rangle\quad \hbox{and} \quad

M^{-1}_{1} = \sqrt{\tfrac{1}{2}} \sum_{i=1}^N e Z_i \langle \Psi | x_i - iy_i | \Psi \rangle. </math>

- <math> M^0_1 = \sum_{i=1}^N e Z_i \langle \Psi | z_i | \Psi \rangle.

</math>

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 <math> C^m_\ell</math> or sine type <math>S^m_\ell</math>. A few of the lowest ones are:

- <math>

\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 this article, which follows the definition of the standard textbook on classical electrodynamics by Jackson^{[1]}, 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^{[2]} and
Brink and Satchler.^{[3]}

##### References

- ↑ J. D. Jackson,
*Classical Electrodynamics*, 2nd edition, Wiley, New York, (1975). p. 137 - ↑ U. Fano and G. Racah,
*Irreducible Tensorial Sets*, Academic Press, New York (1959). p. 31 - ↑ D. M. Brink and G. R. Satchler,
*Angular Momentum*, 2nd edition, Clarendon Press, Oxford, UK (1968). p. 64. See also footnote on p. 90.