Editing Selection rule (section)

== Overview ==
In [[quantum mechanics]] the basis for a spectroscopic selection rule is the value of the ''transition moment integral''<ref>Harris & Bertolucci, p. 130.</ref>
:<math>m_{1,2} = \int \psi_1^* \, \mu \, \psi_2 \, \mathrm{d}\tau,</math>
where <math>\psi_1</math> and <math>\psi_2</math> are the [[wave function]]s of the two states, "state&nbsp;1" and "state&nbsp;2", involved in the transition, and {{mvar|μ}} is the [[Transition dipole moment|transition moment operator]]. This integral represents the [[propagator]] (and thus the probability) of the transition between states&nbsp;1 and 2; if the value of this integral is ''zero'' then the transition is "[[#anchor_forbidden_trans|forbidden]]".

In practice, to determine a selection rule the integral itself does not need to be calculated: It is sufficient to determine the [[symmetry]] of the ''transition moment function'' <math>\psi_1^* \, \mu \, \psi_2.</math> 
If the transition moment function is symmetric over all of the totally symmetric representation of the [[point group]] to which the atom or molecule belongs, then the integral's value is (in general) ''not'' zero and the transition ''is'' allowed. Otherwise, the transition is "[[#anchor forbidden trans|forbidden]]".

The transition moment integral is zero if the ''transition moment function'', <math>\psi_1^* \, \mu \, \psi_2,</math> is anti-symmetric or [[odd functions|odd]], i.e. <math>y(x) = -y(-x)</math> holds. The symmetry of the transition moment function is the [[direct product of groups|direct product]] of the [[Even and odd functions|parities]] of its three components. The symmetry characteristics of each component can be obtained from standard [[character tables]]. Rules for obtaining the symmetries of a direct product can be found in texts on character tables.<ref name=sw/>
{|class="wikitable"
|+ Symmetry characteristics of transition moment operator<ref name=sw>{{cite book |last1=Salthouse |first1=J. A. |last2=Ware |first2=M. J. |year=1972 |title=Point Group Character Tables and Related Data |publisher=[[Cambridge University Press]] |isbn=0-521-08139-4}}</ref>
! Transition type !! {{mvar|μ}} transforms as !! Context
|-
! Electric dipole
| {{mvar|x, y, z}} || Optical spectra
|-
! Electric quadrupole
| {{mvar|x<sup>2</sup>, y<sup>2</sup>, z<sup>2</sup>, xy, xz, yz}} || Constraint {{mvar|x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup>}} = 0
|-
! Electric polarizability
| {{mvar|x<sup>2</sup>, y<sup>2</sup>, z<sup>2</sup>, xy, xz, yz}} || Raman spectra
|-
! Magnetic dipole
| {{mvar|R}}<sub>x</sub>, {{mvar|R}}<sub>y</sub>, {{mvar|R}}<sub>z</sub> || Optical spectra (weak)
|}