Editing Dipole (section)

== Atomic dipoles ==
<!-- This section is linked from [[Intermolecular force]] -->
A non-degenerate (''S''-state) atom can have only a zero permanent dipole. This fact follows quantum mechanically from the inversion symmetry of atoms. All 3 components of the dipole operator are antisymmetric under [[Inversion in a point|inversion]] with respect to the nucleus,
: <math> \mathfrak{I} \;\mathfrak{p}\;  \mathfrak{I}^{-1} = -\mathfrak{p}, </math>
where <math>\mathfrak{p}</math> is the dipole operator  and <math>\mathfrak{I}</math> is the inversion operator.

The permanent dipole moment  of an atom in a non-degenerate state (see [[degenerate energy level]]) is given as the expectation (average) value of the dipole operator,
: <math>\left\langle \mathfrak{p} \right\rangle = \left\langle\, S\, | \mathfrak{p} |\, S \,\right\rangle,</math>
where <math> |\, S\, \rangle </math> is an ''S''-state, non-degenerate, wavefunction, which is symmetric or antisymmetric under inversion: <math> \mathfrak{I}\, |\, S\, \rangle = \pm|\, S\, \rangle</math>. Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse,
: <math>
   \left\langle \mathfrak{p} \right\rangle =
   \left\langle\, \mathfrak{I}^{-1}\, S\, | \mathfrak{p} |\, \mathfrak{I}^{-1}\, S\, \right\rangle =
   \left\langle\, S\, | \mathfrak{I}\, \mathfrak{p}\, \mathfrak{I}^{-1} |\, S\, \right\rangle =
  -\left\langle \mathfrak{p} \right\rangle
</math>
it follows that the expectation value changes sign under inversion.  We used here the fact that <math> \mathfrak{I}</math>, being a symmetry operator, is [[unitary operator|unitary]]: <math> \mathfrak{I}^{-1} =  \mathfrak{I}^{*}\,</math> and [[Hermitian adjoint#Definition for bounded operators between Hilbert spaces|by definition]] the Hermitian adjoint <math> \mathfrak{I}^*\,</math> may be moved from bra to ket and then becomes <math> \mathfrak{I}^{**} = \mathfrak{I}\,</math>. Since the only quantity that is equal to minus itself is the zero, the expectation  value vanishes,
: <math>\left\langle \mathfrak{p} \right\rangle = 0.</math>

In the case of open-shell atoms with degenerate  energy levels, one could define a dipole moment by the aid of the first-order [[Stark effect]]. This gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) only if some of the wavefunctions belonging to the degenerate energies have opposite [[parity (physics)|parity]]; i.e., have different behavior under inversion. This is a rare occurrence, but happens for the excited H-atom, where 2s and 2p states are "accidentally" degenerate (see article [[Laplace–Runge–Lenz vector#Quantum mechanics of the hydrogen atom|Laplace–Runge–Lenz vector]] for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).