Multipole moments - Citizendia

Multipole moments are the coefficients of a series expansion of a potential due to continuous or discrete sources (e. In Mathematics, a coefficient is a Constant multiplicative factor of a certain object In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical 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. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a terms are denoted as quadrupole, octupole, etc. A quadrupole or quadrapole is one of a sequence of configurations of — for example — electric charge or current or gravitational mass that can exist in ideal form but it moments.

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

1 Examples of multipoles
2 General mathematical properties
3 Molecular electrostatic multipole moments
- 3.1 Note on conventions
  - 3.1.1 References
4 See also

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 Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The most common expansions include:

Axial multipole moments of the $\frac{1}{R}$ potential,
Spherical multipole moments of the $\frac{1}{R}$ potential, and
Cylindrical multipole moments of the $\ln \ R^{ }$ potential

Examples of $\frac{1}{R}$ potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. Axial multipole moments are a Series expansion of the Electric potential of acharge distribution localized close to the origin along one Cartesian axis Spherical multipole moments are the coefficients in a Series expansion of a Potential that varies inversely with the distance R to a source i Cylindrical multipole moments are the coefficients in a Series expansion of a Potential that varies logarithmically with the distance to a source i At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. Potential energy can be thought of as Energy stored within a physical system An example of a $\ln \ R^{ }$ potential is the electric potential of an infinite line charge. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. In Mathematics, an orthonormal basis of an Inner product space V (i In Physics, a field is a Physical quantity associated to each point of Spacetime. These can be thought of as arranged in various geometrical shapes, or, in the sense of distribution theory, as directional derivatives. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the

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. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a Problems solved once for a given order of multipole moment may be linearly combined to create a final approximate solution for a given source. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics

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). See also List of elements by atomic number In Chemistry and Physics, the atomic number (also known as the proton 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

$Q^m_\ell \equiv \sum_{i=1}^N e Z_i \; R^m_{\ell}(\mathbf{r}_i),$

where $R^m_{\ell}(\mathbf{r}_i)$ is a regular solid harmonic function in Racah's normalization (also known as Schmidt's semi-normalization). In Physics and Mathematics, the solid harmonics are solutions of the Laplace equation in Spherical polar coordinates. If the molecule has total normalized wave function Ψ (depending on the coordinates of electrons and nuclei), then the multipole moment of order $\ell$ of the molecule is given by the expectation (expected) value

$M^m_\ell \equiv \langle \Psi | Q^m_\ell | \Psi \rangle.$

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"). Molecular symmetry in Chemistry describes the Symmetry present in Molecules and the classification of molecules according to their symmetry 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. In Physics and Chemistry, especially in the context of Quantum mechanics, a selection rule is a condition constraining the physical properties of the initial A well-known example of this is the fact that molecules with an inversion center do not carry a dipole (the expectation values of $Q^m_1$ 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:

$M^0_0 = \sum_{i=1}^N e Z_i,$

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

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

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

Note that by a simple linear combination one can transform the complex multipole operators to real ones. In Physics and Mathematics, the solid harmonics are solutions of the Laplace equation in Spherical polar coordinates. The real multipole operators are of cosine type $C^m_\ell$ or sine type $S^m_\ell$ . A few of the lowest ones are:

$\begin{align} C^0_1 &= \sum_{i=1}^N eZ_i \; z_i \\ C^1_1 &= \sum_{i=1}^N eZ_i \;x_i \\ S^1_1 &= \sum_{i=1}^N eZ_i \;y_i \\ C^0_2 &= \frac{1}{2}\sum_{i=1}^N eZ_i\; (3z_i^2-r_i^2)\\ C^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i x_i \\ C^2_2 &= \frac{1}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; (x_i^2-y_i^2) \\ S^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i y_i \\ S^2_2 &= \frac{2}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; x_iy_i \\ \end{align}$

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. Spherical multipole moments are the coefficients in a Series expansion of a Potential that varies inversely with the distance R to a source i 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.

Contents

Examples of multipoles

General mathematical properties

Molecular electrostatic multipole moments

Note on conventions

References

See also