Editing Multipole expansion (section)

===Molecular moments===
All atoms and molecules (except [[Azimuthal quantum number|''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''<sub>''i''</sub>. (Electrons have a ''Z''-value of −1, while for nuclei it is the [[atomic number]]). Particle ''i'' has spherical polar coordinates ''r''<sub>''i''</sub>, ''θ''<sub>''i''</sub>, and φ<sub>''i''</sub> and Cartesian coordinates ''x''<sub>''i''</sub>, ''y''<sub>''i''</sub>, and ''z''<sub>''i''</sub>.
The (complex) electrostatic multipole operator is
<math display="block">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 harmonics|solid harmonic]] function in [[Solid harmonics#Racah's normalization|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 value (quantum mechanics)|expectation (expected) value]]:
<math display="block">M^m_\ell \equiv \langle \Psi \mid Q^m_\ell \mid \Psi \rangle.</math>
If the molecule has certain [[Molecular symmetry#Point group|point group symmetry]], then this is reflected in the wave function: Ψ transforms according to a certain [[irreducible representation]] λ of the [[group (mathematics)|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 <math> Q^m_1 </math> vanish for {{math|1=''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 [[Spherical harmonics#Condon–Shortley phase|Condon-Shortley phase]]) give:
<math display="block"> M^0_0 = \sum_{i=1}^N e Z_i, </math>
(the total charge of the molecule). The (complex) dipole components are:
<math display="block"> M^1_1 = - \tfrac{1}{\sqrt 2} \sum_{i=1}^N e Z_i \langle \Psi | x_i+iy_i | \Psi \rangle\quad \hbox{and} \quad
M^{-1}_{1} = \tfrac{1}{\sqrt 2} \sum_{i=1}^N e Z_i \langle \Psi | x_i - iy_i | \Psi \rangle. </math>
<math display="block"> M^0_1 = \sum_{i=1}^N e Z_i \langle \Psi | z_i | \Psi \rangle.</math>

Note that by a simple [[Solid harmonics#Real form|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 display="block">\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\; \left(3z_i^2-r_i^2\right), \\
C^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i x_i, &
S^1_2 &= \sqrt{3}\sum_{i=1}^N eZ_i\; z_i y_i, \\
C^2_2 &= \frac{1}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; \left(x_i^2-y_i^2\right), &
S^2_2 &= \frac{2}{3}\sqrt{3}\sum_{i=1}^N eZ_i\; x_i y_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 [[Spherical multipole moments#General spherical multipole moments|this article]], which follows the definition of the standard textbook on [[classical electrodynamics]] by Jackson,<ref name=Jackson75/>{{rp|137}} except for the normalization. Moreover, in the classical definition of Jackson the equivalent of the ''N''-particle [[quantum mechanics|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<ref>U. Fano and G. Racah, ''Irreducible Tensorial Sets'', Academic Press, New York (1959). p. 31</ref> and Brink and Satchler.<ref>D. M. Brink and G. R. Satchler, ''Angular Momentum'', 2nd edition, Clarendon Press, Oxford, UK (1968). p. 64. See also footnote on p. 90.</ref>