Selection rule
In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic nuclei, and so on. The selection rules may differ according to the technique used to observe the transition. The selection rule also plays a role in chemical reactions, where some are formally spin forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products.
In the following, mainly atomic and molecular transitions are considered.
Overview
In quantum mechanics the basis for a spectroscopic selection rule is the value of the transition moment integral[1]
 ,
where and are the wave functions of the two states involved in the transition and µ is the transition moment operator. This integral represents the propagator (and thus the probability) of the transition between states; therefore if the value of this integral is zero the transition is forbidden. In practice, the integral itself does not need to be calculated to determine a selection rule. It is sufficient to determine the symmetry of transition moment function, If the symmetry of this function spans the totally symmetric representation of the point group to which the atom or molecule belongs then its value is (in general) not zero and the transition is allowed. Otherwise, the transition is forbidden.
The transition moment integral is zero if the transition moment function, , is antisymmetric or odd, i.e. holds. The symmetry of the transition moment function is the direct product of the 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. [2]
Transition type  µ transforms as  note 

Electric dipole  x, y, z  Optical spectra 
Electric quadrupole  x^{2}, y^{2}, z^{2}, xy, xz, yz  Constraint x^{2} + y^{2} + z^{2} = 0 
Electric polarizability  x^{2}, y^{2}, z^{2}, xy, xz, yz  Raman spectra 
Magnetic dipole  R_{x}, R_{y}, R_{z}  Optical spectra (weak) 
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 ss, pp, dd, or ff, transitions are forbidden. The Laporte rule applies to electric dipole transitions, so the operator has u symmetry (meaning ungerade, odd).[3] p orbitals also have u symmetry, so the symmetry of the transition moment function is given by the triple product u×u×u, which has u symmetry. The transitions are therefore forbidden. Likewise, d orbitals have g symmetry (meaning gerade, even), so the triple product g×u×g also has u symmetry and the transition is forbidden.[4]
The wave function of a single electron is the product of a spacedependent wave function and a spin wave function. Spin is directional and can be said to have odd 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 "spinallowed".[5] In crystal field theory, dd transitions that are spinforbidden are much weaker than spinallowed transitions. Both can be observed, in spite of the Laporte rule, because the actual transitions are coupled to vibrations that are antisymmetric and have the same symmetry as the dipole moment operator.[6]
Vibrational spectra
In vibrational spectroscopy, transitions are observed between different 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 groundstate 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.[7]
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 secondorder terms in the rightmost column of the character table, below.[2]
E  8 C_{3}  3 C_{2}  6 S_{4}  6 σ_{d}  

A_{1}  1  1  1  1  1  x^{2} + y^{2} + z^{2}  
A_{2}  1  1  1  1  1  
E  2  1  2  0  0  (2 z^{2}  x^{2}  y^{2},x^{2}  y^{2})  
T_{1}  3  0  1  1  1  (R_{x}, R_{y}, R_{z})  
T_{2}  3  0  1  1  1  (x, y, z)  (xy, xz, yz) 
The molecule methane, CH_{4}, may be used as an example to illustrate the application of these principles. The molecule is tetrahedral and has T_{d} symmetry. The vibrations of methane span the representations A_{1} + E + 2T_{2}.[8] Examination of the character table shows that all four vibrations are Ramanactive, but only the T_{2} vibrations can be seen in the infrared spectrum.[9]
In the harmonic approximation, it can be shown that overtones are forbidden in both infrared and Raman spectra. However, when anharmonicity is taken into account, the transitions are weakly allowed.[10]
In Raman and infrared spectroscopy, the selection rules predict certain vibrational modes to have zero intensities in the Raman and/or the IR.[11] 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.[12][13]
Rotational spectra
The selection rule for rotational transitions, derived from the symmetries of the rotational wave functions in a rigid rotor, is ΔJ = ±1, where J is a rotational quantum number.[14]
Coupled transitions
Coupling in science 

Classical coupling 
Quantum coupling 

There are many types of coupled transition such as are observed in vibrationrotation spectra. The excitedstate 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.[15] In 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 socalled P and R branches. The Q branch, located at the vibration frequency, is absent. Symmetric top molecules display the Q branch. This follows from the application of selection rules.[16]
Resonance Raman spectroscopy involves a kind of vibronic coupling. It results in muchincreased intensity of fundamental and overtone transitions as the vibrations "steal" intensity from an allowed electronic transition.[17] In spite of appearances, the selection rules are the same as in Raman spectroscopy.[18]
Angular momentum
 See also angular momentum coupling
In general, electric (charge) radiation or magnetic (current, magnetic moment) radiation can be classified into multipoles Eλ (electric) or Mλ (magnetic) of order 2^{λ}, 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 lowestorder multipoles are overwhelmingly more likely, and dominate the transition.[19]
The emitted particle carries away an angular momentum λ, which for the photon must be at least 1, since it is a vector particle (i.e., it has J^{P} = 1^{−}). Thus, there is no E0 (electric monopoles) or M0 (magnetic monopoles, which do not seem to exist) radiation.
Since the total angular momentum has to be conserved during the transition, we have that
where , and its zprojection is given by ; and are, respectively, the initial and final angular momenta of the atom. The corresponding quantum numbers λ and μ (zaxis angular momentum) must satisfy
and
Parity is also preserved. For electric multipole transitions
while for magnetic multipoles
Thus, parity does not change for Eeven or Modd multipoles, while it changes for Eodd or Meven multipoles.
These considerations generate different sets of transitions rules depending on the multipole order and type. The expression forbidden transitions is often used; this does not mean that these transitions cannot occur, only that they are electricdipoleforbidden. These transitions are perfectly possible; they merely occur at a lower rate. If the rate for an E1 transition is nonzero, 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. These are the socalled forbidden transitions. 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.[20]
Semiforbidden transitions (resulting in socalled 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
is the total angular momentum, is the Azimuthal quantum number, is the Spin quantum number, and is the secondary total angular momentum quantum number. Which transitions are allowed is based on the Hydrogenlike atom. The symbol is used to indicate a forbidden transition.
Allowed transitions  Electric dipole (E1)  Magnetic dipole (M1)  Electric quadrupole (E2)  Magnetic quadrupole (M2)  Electric octupole (E3)  Magnetic octupole (M3)  

Rigorous rules  (1)  
(2)  
(3)  
LS coupling  (4)  One electron jump 
No electron jump , 
None or one electron jump 
One electron jump 
One electron jump 
One electron jump 
(5)  If 
If 
If 
If  
Intermediate coupling  (6)  If 
If 
If 
If 
If 
Surface
In surface vibrational spectroscopy, the surface selection rule is applied to identify the peaks observed in vibrational spectra. When a molecule is adsorbed on a substrate, the molecule induces opposite image charges in the substrate. The 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.
Notes
 Harris & Bertolucci, p. 130
 Salthouse, J.A.; Ware, M.J. (1972). Point group character tables and related data. Cambridge University Press. ISBN 0521081394.
 Anything with u (German ungerade) symmetry is antisymmetric with respect to the centre of symmetry. g (German 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.
 Harris & Berolucci, p. 330
 Harris & Berolucci, p. 336
 Cotton Section 9.6, Selection rules and polarizations
 Cotton, Section 10.6 Selection rules for fundamental vibrational transitions
 Cotton, Chapter 10 Molecular Vibrations
 Cotton p. 327
 Califano, S. (1976). Vibrational states. Wiley. ISBN 0471129968. Chapter 9, Anharmonicity
 Fateley, W. G., Neil T. McDevitt, and Freeman F. Bentley. "Infrared and Raman selection rules for lattice vibrations: the correlation method." Applied Spectroscopy 25.2 (1971): 155173.
 Arenas, D. J., et al. "Raman study of phonon modes in bismuth pyrochlores." Physical Review B 82.21 (2010): 214302.  DOI:https://doi.org/10.1103/PhysRevB.82.214302
 Zhao, Yanyuan, et al. "Phonons in Bi 2 S 3 nanostructures: Raman scattering and firstprinciples studies." Physical Review B 84.20 (2011): 205330.  DOI:https://doi.org/10.1103/PhysRevB.84.205330
 Kroto, H.W. (1992). Molecular Rotation Spectra. new York: Dover. ISBN 048649540X.
 Harris & Berolucci, p. 339
 Harris & Berolucci, p. 123
 Long, D.A. (2001). The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules. Wiley. ISBN 0471490288. Chapter 7, Vibrational Resonance Raman Scattering
 Harris & Berolucci, p. 198
 Softley, T.P. (1994). Atomic Spectra. Oxford: Oxford University Press. ISBN 0198556888.
 Condon, E.V.; Shortley, G.H. (1953). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0521092094.
References
Harris, D.C.; Bertolucci, M.D. (1978). Symmetry and Spectroscopy. Oxford University Press. ISBN 0198551525.
Cotton, F.A. (1990). Chemical Applications of Group Theory (3rd ed.). Wiley. ISBN 9780471510949.
Further reading
 Stanton, L. (1973). "Selection rules for pure rotation and vibrationrotation hyperRaman spectra". Journal of Raman Spectroscopy. 1 (1): 53–70. Bibcode:1973JRSp....1...53S. doi:10.1002/jrs.1250010105.
 Bower, D.I; Maddams, W.F. (1989). The vibrational spectroscopy of polymers. Cambridge University Press. ISBN 0521246334. Section 4.1.5: Selection rules for Raman activity.
 Sherwood, P.M.A. (1972). Vibrational Spectroscopy of Solids. Cambridge University Press. ISBN 0521084822. Chapter 4: The interaction of radiation with a crystal.