Multipole moments
Encyclopedia
In mathematics, especially as applied to physics, multipole moments are the coefficient
s 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
within a charge distribution can often be computed by combining interior and exterior multipoles.
s 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 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.
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 derivative
s.
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.
We consider a molecule consisting of N particles (electrons and nuclei) with charges eZi. (Electrons have the Z-value unity, for nuclei it is the atomic number
). Particle i has spherical polar coordinates ri, θi, and φi and cartesian coordinates xi, yi, and zi.
The (complex) electrostatic multipole operator is
where
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
of the molecule is given by the expectation (expected) value
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 rule
s 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
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:
(the total charge of the molecule). The (complex) dipole components are:
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
or sine type 
. A few of the lowest ones are:
The definition in this article agrees with, among others, the one of Fano and Racah and
Brink and Satchler.
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s of a series expansion
Series expansion
In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators . The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function...
of a potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
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
Monopole
Monopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
moment, the first-order term is denoted as the dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
moment, and the third, fourth, etc. terms are denoted as quadrupole
Quadrupole
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 is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
, octupole, etc. moments.
The potential at a position
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 potentialPotential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
s and many ways of approximating a potential by a series expansion
Series expansion
In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators . The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function...
, depending on the coordinates and the symmetry
Symmetry
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 perfection...
of the charge distribution. The most common expansions include:
- Axial multipole momentsAxial multipole momentsAxial multipole moments are a series expansionof the electric potential of acharge distribution localized close tothe origin along oneCartesian axis,denoted here as the z-axis...
of the 1/R potential, - Spherical multipole momentsSpherical multipole momentsSpherical multipole moments are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as 1/R...
of the 1/R potential, and - Cylindrical multipole momentsCylindrical multipole momentsCylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R...
of the ln R potential
Examples of 1/R potentials include the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
, the magnetic potential
Magnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
and the gravitational potential of point sources. An example of a ln R potential is the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
of an infinite line charge.
General mathematical properties
Multipole moments in mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
form an orthogonal basis
Orthogonal basis
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal...
for the decomposition of a function, based on the response of a field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
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
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
, as directional derivative
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
s.
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
Monopole
Monopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
and
dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
terms. Problems solved once for a given order of multipole moment may be linearly combined
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
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 eZi. (Electrons have the Z-value unity, for nuclei it is the atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
). Particle i has spherical polar coordinates ri, θi, and φi and cartesian coordinates xi, yi, and zi.
The (complex) electrostatic multipole operator is
where
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
of the molecule is given by the expectation (expected) valueIf 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 rule
Selection rule
In physics and chemistry a selection rule, or transition rule, formally constrains the possible transitions of a system from one state to another. Selection rules have been derived for electronic, vibrational, and rotational transitions...
s 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
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:
(the total charge of the molecule). The (complex) dipole components are:
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
or sine type . A few of the lowest ones are: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, 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 and
Brink and Satchler.
See also
- MonopoleMonopoleMonopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
- DipoleDipoleIn physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
- QuadrupoleQuadrupoleA 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 is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
- Multipole expansionMultipole expansionA 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...
- Barnes–Hut simulation
- Solid harmonics
- Axial multipole momentsAxial multipole momentsAxial multipole moments are a series expansionof the electric potential of acharge distribution localized close tothe origin along oneCartesian axis,denoted here as the z-axis...
- Cylindrical multipole momentsCylindrical multipole momentsCylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R...
- Spherical multipole momentsSpherical multipole momentsSpherical multipole moments are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as 1/R...
- Laplace expansionLaplace expansion (potential)In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.The Laplace...
- Legendre polynomials