Multipole expansions are useful because, similar to Taylor series, oftentimes only the first few terms are needed to provide a good approximation of the original function.
[1] The multipole expansion is expressed as a sum of terms with progressively finer angular features (moments).
If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties.
[6] This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves.
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.
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 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.
The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required.
and the expansion can be rewritten in terms of the components of a traceless Cartesian second rank tensor:
Removing the trace is common, because it takes the rotationally invariant r2 out of the second rank tensor.
This expansion of the potential of a discrete charge distribution is very similar to the one in real solid harmonics given below.
The potential V(R) at a point R outside the charge distribution, i.e. |R| > rmax, can be expanded by the Laplace expansion:
In order to derive this multipole expansion, we write rXY = rY − rX, which is a vector pointing from X towards Y.
This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance RAB apart.
Because it is in complex form it has as the further advantage that it is easier to manipulate in calculations than its real counterpart.
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
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").
A well-known example of this is the fact that molecules with an inversion center do not carry a dipole (the expectation values of
The lowest explicit forms of the regular solid harmonics (with the Condon-Shortley phase) give:
Note that by a simple linear combination one can transform the complex multipole operators to real ones.
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.
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.
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.