Inphysics andchemistry, aselection rule, ortransition rule, formally constrains the possible transitions of a system from onequantum state to another. Selection rules have been derived forelectromagnetic transitions inmolecules, inatoms, inatomic 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 inchemical reactions, where some are formallyspin-forbidden reactions, that is, reactions where the spin state changes at least once fromreactants toproducts.

In the following, mainly atomic and molecular transitions are considered.

Inquantum mechanics the basis for a spectroscopic selection rule is the value of thetransition moment integral^[1]

${\displaystyle m_{1,2}=\int {\psi }_1^{\ast }\mu {\psi }_2\mathrm{d}\tau ,}$

where${\displaystyle {\psi }_1}$ and${\displaystyle {\psi }_2}$ are thewave functions of the two states, "state 1" and "state 2", involved in the transition, andμ is thetransition moment operator. This integral represents thepropagator (and thus the probability) of the transition between states 1 and 2; if the value of this integral iszero then the transition is "forbidden".

In practice, to determine a selection rule the integral itself does not need to be calculated: It is sufficient to determine thesymmetry of thetransition moment function${\displaystyle {\psi }_1^{\ast }\mu {\psi }_2.}$ If the transition moment function is symmetric over all of the totally symmetric representation of thepoint group to which the atom or molecule belongs, then the integral's value is (in general)not zero and the transitionis allowed. Otherwise, the transition is "forbidden".

The transition moment integral is zero if thetransition moment function,${\displaystyle {\psi }_1^{\ast }\mu {\psi }_2,}$ is anti-symmetric orodd, i.e.${\displaystyle y(x)=-y(-x)}$ holds. The symmetry of the transition moment function is thedirect product of theparities of its three components. The symmetry characteristics of each component can be obtained from standardcharacter tables. Rules for obtaining the symmetries of a direct product can be found in texts on character tables.^[2]

TheLaporte rule is a selection rule formally stated as follows: In acentrosymmetric environment, transitions between likeatomic orbitals such ass–s,p–p,d–d, orf–f, transitions are forbidden. The Laporte rule (law) applies toelectric dipole transitions, so the operator hasu symmetry (meaningungerade, odd).^[3]p orbitals also haveu symmetry, so the symmetry of the transition moment function is given by the product (formally, the product is taken in thegroup)u×u×u, which hasu symmetry. The transitions are therefore forbidden. Likewise,d orbitals haveg symmetry (meaninggerade, even), so the triple productg×u×g also hasu symmetry and the transition is forbidden.^[4]

The wave function of a single electron is the product of a space-dependent wave function and aspin wave function. Spin is directional and can be said to have oddparity. It follows that transitions in which the spin "direction" changes are forbidden. In formal terms, only states with the same totalspin quantum number are "spin-allowed".^[5] Incrystal 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.^[6]

In vibrational spectroscopy, transitions are observed between differentvibrational states. In a fundamental vibration, the molecule is excited from itsground 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 thepoint 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]

Ininfrared spectroscopy, the transition moment operator transforms as eitherx and/ory and/orz. The excited state wave function must also transform as at least one of these vectors. InRaman spectroscopy, the operator transforms as one of the second-order terms in the right-most column of thecharacter table, below.^[2]

The molecule methane, CH₄, may be used as an example to illustrate the application of these principles. The molecule istetrahedral and hasT_d symmetry. The vibrations of methane span the representations A₁ + E + 2T₂.^[8] Examination of the character table shows that all four vibrations are Raman-active, but only the T₂ vibrations can be seen in the infrared spectrum.^[9]

In theharmonic approximation, it can be shown thatovertones are forbidden in both infrared and Raman spectra. However, whenanharmonicity 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]

Theselection rule for rotational transitions, derived from the symmetries of the rotational wave functions in a rigid rotor, is ΔJ = ±1, whereJ is a rotational quantum number.^[14]

There are many types of coupled transition such as are observed invibration–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.^[15] Inrovibronic transitions, the excited states involve three wave functions.

The infrared spectrum ofhydrogen 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-calledP andR branches. TheQ branch, located at the vibration frequency, is absent.Symmetric top molecules display theQ branch. This follows from the application of selection rules.^[16]

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.^[17] In spite of appearances, the selection rules are the same as in Raman spectroscopy.^[18]

In general, electric (charge) radiation or magnetic (current, magnetic moment) radiation can be classified intomultipoles Eλ (electric) or Mλ (magnetic) of order 2^λ, e.g., E1 for electricdipole, E2 forquadrupole, 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.^[19]

The emitted particle carries away angular momentum, with quantum numberλ, which for the photon must be at least 1, since it is a vector particle (i.e., it hasJ^P = 1⁻). Thus, there is no radiation from E0 (electric monopoles) or M0 (magnetic monopoles, which do not seem to exist).

Since the total angular momentum has to be conserved during the transition, we have that

${\displaystyle {\mathbf{J}}_{\text{i}}={\mathbf{J}}_{\text{f}}+\mathit{\lambda },}$

where${\displaystyle \Vert \mathit{\lambda }\Vert =\sqrt{\lambda (\lambda +1)}\hslash ,}$ and itsz projection is given by${\displaystyle {\lambda }_z=\mu \hslash ;}$ and where${\displaystyle {\mathbf{J}}_{\text{i}}}$ and${\displaystyle {\mathbf{J}}_{\text{f}}}$ are, respectively, the initial and final angular momenta of the atom. The corresponding quantum numbersλ andμ (z-axis angular momentum) must satisfy

${\displaystyle |J_{\text{i}}-J_{\text{f}}|\le \lambda \le J_{\text{i}}+J_{\text{f}}}$

and

${\displaystyle \mu =M_{\text{i}}-M_{\text{f}}.}$

Parity is also preserved. For electric multipole transitions

${\displaystyle \pi (\mathrm{E}\lambda )={\pi }_{\text{i}}{\pi }_{\text{f}}=(-1{)}^{\lambda },}$

while for magnetic multipoles

${\displaystyle \pi (\mathrm{M}\lambda )={\pi }_{\text{i}}{\pi }_{\text{f}}=(-1{)}^{\lambda +1}.}$

Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles.

These considerations generate different sets of transitions rules depending on the multipole order and type. The expressionforbidden transitions is often used, but this does not mean that these transitionscannot occur, only that they areelectric-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.^[20]

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 ofLS coupling.

${\displaystyle J=L+S}$ is the total angular momentum,${\displaystyle L}$ is theazimuthal quantum number,${\displaystyle S}$ is thespin quantum number, and${\displaystyle M_J}$ is thesecondary total angular momentum quantum number.Which transitions are allowed is based on thehydrogen-like atom. The symbol${\displaystyle \nleftrightarrow }$ 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)	${\displaystyle \begin{array}{c}\mathrm{\Delta }J=0,\pm 1\\ (J=0\nleftrightarrow 0)\end{array}}$		${\displaystyle \begin{array}{c}\mathrm{\Delta }J=0,\pm 1,\pm 2\\ (J=0\nleftrightarrow 0,1;\begin{array}{c}\frac{1}{2}\end{array}\nleftrightarrow \begin{array}{c}\frac{1}{2}\end{array})\end{array}}$		${\displaystyle \begin{array}{c}\mathrm{\Delta }J=0,\pm 1,\pm 2,\pm 3\\ (0\nleftrightarrow 0,1,2;\begin{array}{c}\frac{1}{2}\end{array}\nleftrightarrow \begin{array}{c}\frac{1}{2}\end{array},\begin{array}{c}\frac{3}{2}\end{array};1\nleftrightarrow 1)\end{array}}$
	(2)	${\displaystyle \mathrm{\Delta }{M}_J=0,\pm 1(M_J=0\nleftrightarrow 0}$ if${\displaystyle \mathrm{\Delta }J=0)}$		${\displaystyle \mathrm{\Delta }{M}_J=0,\pm 1,\pm 2}$		${\displaystyle \mathrm{\Delta }{M}_J=0,\pm 1,\pm 2,\pm 3}$
	(3)	${\displaystyle {\pi }_{\text{f}}=-{\pi }_{\text{i}}}$	${\displaystyle {\pi }_{\text{f}}={\pi }_{\text{i}}}$		${\displaystyle {\pi }_{\text{f}}=-{\pi }_{\text{i}}}$		${\displaystyle {\pi }_{\text{f}}={\pi }_{\text{i}}}$
LS coupling	(4)	One-electron jump ${\displaystyle \mathrm{\Delta }L=\pm 1}$	No electron jump ${\displaystyle \mathrm{\Delta }L=0}$, ${\displaystyle \mathrm{\Delta }n=0}$	None or one-electron jump ${\displaystyle \mathrm{\Delta }L=0,\pm 2}$	One-electron jump ${\displaystyle \mathrm{\Delta }L=\pm 1}$	One-electron jump ${\displaystyle \mathrm{\Delta }L=\pm 1,\pm 3}$	One-electron jump ${\displaystyle \mathrm{\Delta }L=0,\pm 2}$
	(5)	If${\displaystyle \mathrm{\Delta }S=0}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1\\ (L=0\nleftrightarrow 0)\end{array}}$	If${\displaystyle \mathrm{\Delta }S=0}$: ${\displaystyle \mathrm{\Delta }L=0}$	If${\displaystyle \mathrm{\Delta }S=0}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1,\pm 2\\ (L=0\nleftrightarrow 0,1)\end{array}}$		If${\displaystyle \mathrm{\Delta }S=0}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1,\pm 2,\pm 3\\ (L=0\nleftrightarrow 0,1,2;1\nleftrightarrow 1)\end{array}}$
Intermediate coupling	(6)	If${\displaystyle \mathrm{\Delta }S=\pm 1}$: ${\displaystyle \mathrm{\Delta }L=0,\pm 1,\pm 2}$		If${\displaystyle \mathrm{\Delta }S=\pm 1}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1,\\ \pm 2,\pm 3\\ (L=0\nleftrightarrow 0)\end{array}}$	If${\displaystyle \mathrm{\Delta }S=\pm 1}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1\\ (L=0\nleftrightarrow 0)\end{array}}$	If${\displaystyle \mathrm{\Delta }S=\pm 1}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1,\\ \pm 2,\pm 3,\pm 4\\ (L=0\nleftrightarrow 0,1)\end{array}}$	If${\displaystyle \mathrm{\Delta }S=\pm 1}$: ${\displaystyle \begin{array}{c}\mathrm{\Delta }L=0,\pm 1,\\ \pm 2\\ (L=0\nleftrightarrow 0)\end{array}}$

Inhyperfine structure, the total angular momentum of the atom is${\displaystyle F=I+J,}$ where${\displaystyle I}$ is thenuclear spin angular momentum and${\displaystyle J}$ is the total angular momentum of the electron(s). Since${\displaystyle F=I+J}$ has a similar mathematical form as${\displaystyle J=L+S,}$ it obeys a selection rule table similar to the table above.

Insurface vibrational spectroscopy, thesurface selection rule is applied to identify the peaks observed in vibrational spectra. When amolecule isadsorbed on a substrate, the molecule induces opposite image charges in the substrate. Thedipole 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.

• Superselection rule
• Spin-forbidden reactions
• Singlet fission

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• Cotton, F. A. (1990).Chemical Applications of Group Theory (3rd ed.). Wiley.ISBN 978-0-471-51094-9.

• Stanton, L. (1973). "Selection rules for pure rotation and vibration-rotation hyper-Raman 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). "Section 4.1.5: Selection rules for Raman activity".The vibrational spectroscopy of polymers. Cambridge University Press.ISBN 0-521-24633-4.
• Sherwood, P. M. A. (1972). "Chapter 4: The interaction of radiation with a crystal".Vibrational Spectroscopy of Solids. Cambridge University Press.ISBN 0-521-08482-2.

• National Institute of Standards and Technology
• Lecture notes from The University of Sheffield

• Quantum mechanics
• Spectroscopy
• Nuclear magnetic resonance

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