Editing Selection rule (section)

== Examples ==

=== Electronic spectra ===
The [[Laporte rule]] is a selection rule formally stated as follows: In a [[centrosymmetric]] environment, transitions between like [[atomic orbitals]] such as ''s''–''s'', ''p''–''p'', ''d''–''d'', or ''f''–''f'', transitions are forbidden. The Laporte rule (law) applies to [[electric dipole transition]]s, so the operator has ''u'' symmetry (meaning {{lang|de|ungerade}}, odd).<ref>Anything with ''u'' ({{langx|de|ungerade}}) symmetry is antisymmetric with respect to the centre of symmetry. ''g'' ({{langx|de|gerade}}) signifies symmetric with respect to the centre of symmetry. If the transition moment function has ''u'' symmetry, the positive and negative parts will be equal to each other, so the integral has a value of zero.</ref> ''p'' orbitals also have ''u'' symmetry, so the symmetry of the transition moment function is given by the product (formally, the product is taken in the [[group (mathematics)|group]])
''u''&times;''u''&times;''u'', which has ''u'' symmetry. The transitions are therefore forbidden. Likewise, ''d'' orbitals have ''g'' symmetry (meaning {{lang|de|gerade}}, even), so the triple product ''g''&times;''u''&times;''g'' also has ''u'' symmetry and the transition is forbidden.<ref>Harris & Berolucci, p. 330.</ref>

The wave function of a single electron is the product of a space-dependent wave function and a [[Spin (physics)|spin]] wave function. Spin is directional and can be said to have odd [[Parity (physics)|parity]]. It follows that transitions in which the spin "direction" changes are forbidden. In formal terms, only states with the same total [[spin quantum number]] are "spin-allowed".<ref>Harris & Berolucci, p. 336.</ref> In [[crystal field theory]], ''d''–''d'' transitions that are spin-forbidden are much weaker than spin-allowed transitions. Both can be observed, in spite of the Laporte rule, because the actual transitions are coupled to vibrations that are anti-symmetric and have the same symmetry as the dipole moment operator.<ref>Cotton Section 9.6, Selection rules and polarizations.</ref>

=== Vibrational spectra ===
{{Main|infrared spectroscopy|Raman spectroscopy}}
In vibrational spectroscopy, transitions are observed between different [[molecular vibration|vibrational states]]. In a fundamental vibration, the molecule is excited from its [[ground state]] (''v'' = 0) to the first excited state (''v'' = 1). The symmetry of the ground-state wave function is the same as that of the molecule. It is, therefore, a basis for the totally symmetric representation in the [[point group]] of the molecule. It follows that, for a vibrational transition to be allowed, the symmetry of the excited state wave function must be the same as the symmetry of the transition moment operator.<ref>Cotton, Section 10.6 Selection rules for fundamental vibrational transitions.</ref>

In [[infrared spectroscopy]], the transition moment operator transforms as either ''x'' and/or ''y'' and/or ''z''. The excited state wave function must also transform as at least one of these vectors. In [[Raman spectroscopy]], the operator transforms as one of the second-order terms in the right-most column of the [[character theory|character]] table, below.<ref name=sw/>

{| class="wikitable" style="text-align:center"
|+ Character table for the ''T<sub>d</sub>'' point group
  ! || E || 8 ''C''<sub>3</sub>|| 3 ''C''<sub>2</sub>|| 6 ''S''<sub>4</sub>|| 6 ''σ<sub>d</sub>''|| ||
  |-
  ! A<sub>1</sub> 
  | 1 || 1 || 1 || 1 || 1 || 
  | ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  ! A<sub>2</sub> 
  | 1 || 1 || 1 || −1 || −1 ||  || 
  |-
  !  E  
  | 2 || −1 || 2 || 0 || 0 || 
  | (2 ''z''<sup>2</sup> − ''x''<sup>2</sup> − ''y''<sup>2</sup>, ''x''<sup>2</sup> − ''y''<sup>2</sup>)
  |-
  ! T<sub>1</sub> 
  | 3 || 0 || −1 || 1 || −1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>'') || 
  |-
  ! T<sub>2</sub> 
  | 3 || 0 || −1 || −1 || 1
  | (''x'', ''y'', ''z'') || (''xy'', ''xz'', ''yz'')
  |-
|}
The molecule methane, CH<sub>4</sub>, may be used as an example to illustrate the application of these principles.  The molecule is [[tetrahedral]] and has ''T<sub>d</sub>'' symmetry. The vibrations of methane span the representations A<sub>1</sub> + E +  2T<sub>2</sub>.<ref>Cotton, Chapter 10 Molecular Vibrations.</ref> Examination of the character table shows that all four vibrations are Raman-active, but only the T<sub>2</sub> vibrations can be seen in the infrared spectrum.<ref>Cotton p. 327.</ref>

In the [[Quantum harmonic oscillator|harmonic approximation]], it can be shown that [[Overtone band|overtone]]s are forbidden in both infrared and Raman spectra. However, when [[anharmonicity]] is taken into account, the transitions are weakly allowed.<ref>{{cite book |last=Califano |first=S. |title=Vibrational states |publisher=Wiley |date=1976 |isbn=0-471-12996-8 |chapter=Chapter 9: Anharmonicity}}</ref>

In Raman and infrared spectroscopy, the selection rules predict certain vibrational modes to have zero intensities in the Raman and/or the IR.<ref>{{cite journal | doi=10.1366/000370271779948600 | title=Infrared and Raman Selection Rules for Lattice Vibrations: The Correlation Method | date=1971 | last1=Fateley | first1=W. G. | last2=McDevitt | first2=Neil T. | last3=Bentley | first3=Freeman F. | journal=Applied Spectroscopy | volume=25 | issue=2 | pages=155–173 | bibcode=1971ApSpe..25..155F }}</ref> Displacements from the ideal structure can result in relaxation of the selection rules and appearance of these unexpected phonon modes in the spectra. Therefore, the appearance of new modes in the spectra can be a useful indicator of symmetry breakdown.<ref>{{cite journal | doi=10.1103/PhysRevB.82.214302 | title=Raman study of phonon modes in bismuth pyrochlores | date=2010 | last1=Arenas | first1=D. J. | last2=Gasparov | first2=L. V. | last3=Qiu | first3=Wei | last4=Nino | first4=J. C. | last5=Patterson | first5=Charles H. | last6=Tanner | first6=D. B. | journal=Physical Review B | volume=82 | issue=21 | page=214302 | bibcode=2010PhRvB..82u4302A | hdl=2262/72900 | hdl-access=free }}</ref><ref>{{cite journal | doi=10.1103/PhysRevB.84.205330 | title=Phonons in Bi<sub>2</sub>S<sub>3</sub> nanostructures: Raman scattering and first-principles studies | date=2011 | last1=Zhao | first1=Yanyuan | last2=Chua | first2=Kun Ting Eddie | last3=Gan | first3=Chee Kwan | last4=Zhang | first4=Jun | last5=Peng | first5=Bo | last6=Peng | first6=Zeping | last7=Xiong | first7=Qihua | journal=Physical Review B | volume=84 | issue=20 | page=205330 | bibcode=2011PhRvB..84t5330Z }}</ref>

=== Rotational spectra ===
{{Main|Rigid rotor}}
The [[rigid rotor#Selection rules|selection rule]] for rotational transitions, derived from the symmetries of the rotational wave functions in a rigid rotor, is Δ''J'' = ±1, where ''J'' is a rotational quantum number.<ref>{{cite book |last=Kroto |first=H. W. |title=Molecular Rotation Spectra |publisher=Dover |location=New York |date=1992 |isbn=0-486-49540-X }}</ref>

=== Coupled transitions ===
{{Coupling in molecules}}
[[File:HCl rotiational spectrum.jpg|thumb|left|The infrared spectrum of [[HCl]] gas]]
There are many types of coupled transition such as are observed in [[rotational–vibrational coupling|vibration–rotation]] spectra. The excited-state wave function is the product of two wave functions such as vibrational and rotational. The general principle is that the symmetry of the excited state is obtained as the direct product of the symmetries of the component wave functions.<ref>Harris & Berolucci, p. 339.</ref> In [[rovibronic coupling|rovibronic]] transitions, the excited states involve three wave functions.

The infrared spectrum of [[hydrogen chloride]] gas shows rotational fine structure superimposed on the vibrational spectrum. This is typical of the infrared spectra of heteronuclear diatomic molecules. It shows the so-called ''P'' and ''R'' branches. The ''Q'' branch, located at the vibration frequency, is absent. [[Rotational spectroscopy|Symmetric top]] molecules display the ''Q'' branch. This follows from the application of selection rules.<ref>Harris & Berolucci, p. 123.</ref>

[[Resonance Raman spectroscopy]] involves a kind of vibronic coupling. It results in much-increased intensity of fundamental and overtone transitions as the vibrations "steal" intensity from an allowed electronic transition.<ref>{{cite book |last=Long |first=D. A. |title=The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules |publisher=Wiley |date=2001 |isbn=0-471-49028-8 |chapter=Chapter 7: Vibrational Resonance Raman Scattering}}</ref> In spite of appearances, the selection rules are the same as in Raman spectroscopy.<ref>Harris & Berolucci, p. 198.</ref>

=== Angular momentum ===
{{See also|angular momentum coupling}}

In general, electric (charge) radiation or magnetic (current, magnetic moment) radiation can be classified into [[multipole moments|multipoles]] E{{mvar|λ}} (electric) or M{{mvar|λ}} (magnetic) of order 2<sup>{{mvar|λ}}</sup>, e.g., E1 for electric [[dipole]], E2 for [[quadrupole]], or E3 for octupole. In transitions where the change in angular momentum between the initial and final states makes several multipole radiations possible, usually the lowest-order multipoles are overwhelmingly more likely, and dominate the transition.<ref>{{cite book |last=Softley |first=T. P. |title=Atomic Spectra |publisher=[[Oxford University Press]] |location=Oxford, UK |date=1994 |isbn=0-19-855688-8}}</ref>

The emitted particle carries away angular momentum, with quantum number {{mvar|λ}}, which for the photon must be at least 1, since it is a vector particle (i.e., it has [[total angular momentum quantum number|{{mvar|J}}]]<sup>[[Parity (physics)|{{mvar|P}}]]</sup> = 1<sup>−</sup>). Thus, there is no radiation from  E0 (electric monopoles) or  M0  ([[magnetic monopole]]s, which do not seem to exist).

Since the total angular momentum has to be conserved during the transition, we have that
: <math>\mathbf J_\text{i} = \mathbf{J}_\text{f} + \boldsymbol{\lambda},</math>
where <math>\|\boldsymbol{\lambda}\| = \sqrt{\lambda(\lambda + 1)} \, \hbar,</math> and its {{mvar|z}} projection is given by <math>\lambda_z = \mu \hbar;</math> and where <math>\mathbf J_\text{i}</math> and <math>\mathbf J_\text{f}</math> are, respectively, the initial and final angular momenta of the atom. 
The corresponding quantum numbers {{mvar|λ}} and {{mvar|μ}} ({{mvar|z}}-axis angular momentum) must satisfy
: <math>|J_\text{i} - J_\text{f}| \le \lambda \le J_\text{i} + J_\text{f}</math>
and
: <math>\mu = M_\text{i} - M_\text{f}.</math>

Parity is also preserved. For electric multipole transitions 
: <math>\pi(\mathrm{E}\lambda) = \pi_\text{i} \pi_\text{f} = (-1)^{\lambda},</math>
while for magnetic multipoles
: <math>\pi(\mathrm{M}\lambda) = \pi_\text{i} \pi_\text{f} = (-1)^{\lambda+1}.</math>
Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles.

{{anchor|anchor_forbidden_trans}}These considerations generate different sets of transitions rules depending on the multipole order and type. The expression ''[[forbidden transition]]s'' is often used, but this does not mean that these transitions ''cannot'' occur, only that they are ''electric-dipole-forbidden''. These transitions are perfectly possible; they merely occur at a lower rate. If the rate for an E1 transition is non-zero, the transition is said to be permitted; if it is zero, then M1, E2, etc. transitions can still produce radiation, albeit with much lower transitions rates. The transition rate decreases by a factor of about 1000 from one multipole to the next one, so the lowest multipole transitions are most likely to occur.<ref>{{cite book |last1=Condon |first1=E. V. |last2=Shortley |first2=G. H. |orig-year=1935 |title=The Theory of Atomic Spectra |url=https://archive.org/details/in.ernet.dli.2015.212979 |publisher=Cambridge University Press |isbn=0-521-09209-4<!-- reprint --> |year=1999<!-- reprint -->}}</ref>

Semi-forbidden transitions (resulting in so-called intercombination lines) are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated. This is a result of the failure of [[LS coupling]].

==== Summary table ====
<math>J = L + S</math> is the total angular momentum, 
<math>L</math> is the [[azimuthal quantum number]], 
<math>S</math> is the [[spin quantum number]], and
<math>M_J</math> is the [[Total angular momentum quantum number|secondary total angular momentum quantum number]].
Which transitions are allowed is based on the [[hydrogen-like atom]]. The symbol <math>\not\leftrightarrow</math> is used to indicate a forbidden transition.
{| class="wikitable" style="text-align:center"
|-
! colspan=2 | Allowed transitions
! Electric dipole (E1)
! Magnetic dipole (M1)
! Electric quadrupole (E2)
! Magnetic quadrupole (M2)
! Electric octupole (E3)
! Magnetic octupole (M3)
|-
! rowspan=3 | Rigorous rules 	
! (1)
| colspan=2 | <math>\begin{matrix} \Delta J = 0, \pm 1 \\ (J = 0 \not \leftrightarrow 0)\end{matrix}</math>
| colspan=2 | <math>\begin{matrix} \Delta J = 0, \pm 1, \pm 2 \\ (J = 0 \not \leftrightarrow 0, 1;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2}\end{matrix})\end{matrix}</math>
| colspan=2 | <math>\begin{matrix}\Delta J = 0, \pm1, \pm2, \pm 3 \\ (0 \not \leftrightarrow 0, 1, 2;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2} \end{matrix}, \begin{matrix}{3 \over 2}\end{matrix};\ 1 \not \leftrightarrow 1) \end{matrix}</math>
|-
! (2)
| colspan=2 | <math>\Delta M_J = 0, \pm 1 \  (M_J = 0 \not \leftrightarrow 0</math> if <math>\Delta J=0)</math>
| colspan=2 | <math>\Delta M_J = 0, \pm 1, \pm2</math>
| colspan=2 | <math>\Delta M_J = 0, \pm 1, \pm2, \pm 3</math>
|-
! (3)
| <math>\pi_\text{f} = -\pi_\text{i}</math>
| colspan=2 | <math>\pi_\text{f} = \pi_\text{i}</math>	
| colspan=2 | <math>\pi_\text{f} = -\pi_\text{i}</math>
| <math>\pi_\text{f} = \pi_\text{i}</math>
|-
! rowspan=2 | LS coupling
! (4)
| One-electron jump<br/><math>\Delta L = \pm 1</math>
| No electron jump<br/><math>\Delta L = 0</math>,<br><math>\Delta n = 0</math>
| None or one-electron jump<br/><math>\Delta L = 0, \pm 2</math>
| One-electron jump<br/><math>\Delta L = \pm 1</math>
| One-electron jump<br/><math>\Delta L = \pm 1, \pm 3</math>
| One-electron jump<br/><math>\Delta L = 0, \pm 2</math>
|-	
! (5)
| If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math>
| If <math>\Delta S = 0</math>:<br/><math>\Delta L = 0\,</math>
| colspan=2 | If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \pm 2 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}</math>
| colspan=2 | If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \pm 2, \pm 3 \\ (L=0 \not \leftrightarrow 0, 1, 2;\ 1 \not \leftrightarrow 1)\end{matrix}</math>
|-
! Intermediate coupling
! (6)
| colspan=2 | If <math>\Delta S = \pm 1</math>:<br/><math>\Delta L = 0, \pm 1, \pm 2\,</math>
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, 
\pm 3 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math>
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math> 	
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, \pm 3, \pm 4 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}</math> 
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math> 
|}
In [[hyperfine structure]], the total angular momentum of the atom is <math>F = I + J,</math> where <math>I</math> is the [[Quantum number#Nuclear angular momentum quantum numbers|nuclear spin angular momentum]] and <math>J</math> is the total angular momentum of the electron(s). Since <math>F = I + J</math> has a similar mathematical form as <math>J = L + S,</math> it obeys a selection rule table similar to the table above.

=== Surface ===
In [[vibrational spectroscopy|surface vibrational spectroscopy]], the ''surface selection rule'' is applied to identify the peaks observed in vibrational spectra. When a [[molecule]] is [[adsorption|adsorbed]] on a substrate, the molecule induces opposite image charges in the substrate. The [[dipole|dipole moment]] of the molecule and the image charges perpendicular to the surface reinforce each other. In contrast, the dipole moments of the molecule and the image charges parallel to the surface cancel out. Therefore, only molecular vibrational peaks giving rise to a dynamic dipole moment perpendicular to the surface will be observed in the vibrational spectrum.