Selection Rule Tables

A selection rule tells you when symmetry forces a transition amplitude $\langle f|\hat O|i\rangle$ to vanish. The operator matters: a pair of levels may forbid electric-dipole absorption but permit magnetic-dipole or electric-quadrupole excitation. Passing the rules does not guarantee a strong line; radial overlap, interference, population, and detection geometry still matter.

Use the tables below for a quick comparison. Read Atomic selection rules or Selection rules in spectroscopy for detailed applications. Rules involving $L,S$, or an individual electron’s $l$ require the stated coupling approximation; rules involving exact parity and total angular momentum apply to eigenstates with those symmetries.

Required background. The Wigner–Eckart theorem separates the angular factor from the reduced matrix element.

Universal Tensor Rule

For a spherical tensor component $T_q^{(k)}$, one common Wigner–Eckart convention is

$$\langle \alpha_f J_f M_f| T_q^{(k)} |\alpha_i J_i M_i\rangle = (-1)^{J_f-M_f} \begin{pmatrix} J_f & k & J_i\\ -M_f & q & M_i \end{pmatrix} \langle \alpha_f J_f \|T^{(k)}\| \alpha_i J_i\rangle.$$

The $3j$ symbol gives three universal necessary conditions:

$$|J_f-J_i| \le k \le J_f+J_i,$$ $$M_f=M_i+q, \qquad q=-k,-k+1,\ldots,k,$$

and every $J$ and $M$ triple must be admissible. The reduced matrix element contains parity, internal symmetry, radial, spin, configuration, and body-fixed information. Passing the $3j$ test is therefore not sufficient.

Why triangle conditions are safer

The shorthand

$$\Delta J=0,\pm1,\ldots,\pm k$$

can hide low-$J$ exclusions. For a rank-2 operator, for example, $J_i=0\to J_f=1$ has $\Delta J=1$ but fails

$$|0-1|\le2\le0+1.$$

By contrast, $J_i=0\to J_f=2$ passes the rank-2 triangle condition. When in doubt, use the triangle inequality rather than the change rule.

Multipole Parity Rule

Let $\pi_i,\pi_f=\pm1$ be parity eigenvalues. The leading electromagnetic multipoles have operator parity

$$\pi(Ek)=(-1)^k, \qquad \pi(Mk)=(-1)^{k+1}.$$

A nonzero matrix element requires

$$\pi_f\pi_i=\pi(\hat O).$$

Therefore:

Multipole	Tensor rank	Required state parity
electric dipole E1	$1$	changes
magnetic dipole M1	$1$	unchanged
electric quadrupole E2	$2$	unchanged
magnetic quadrupole M2	$2$	changes
electric octupole E3	$3$	changes

Parity rules are exact only while parity is a symmetry of the Hamiltonian. A static electric field, chiral environment, weak interaction, or other parity-mixing perturbation can make field-dressed states cease to be parity eigenstates.

Atomic Multipole Table

The NIST atomic-spectroscopy reference tabulates the angular and parity rules. The compact table here separates those conditions from approximation-dependent restrictions described below.

Operator	Angular momentum	Parity
E1	rank $1$; $\Delta J=0,\pm1$, excluding $0\leftrightarrow0$	changes
M1	rank $1$; $\Delta J=0,\pm1$, excluding $0\leftrightarrow0$	unchanged
E2	rank $2$; $\lvert J_f-J_i\rvert\le2\le J_f+J_i$	unchanged

For E1 and M1, $\Delta M=0,\pm1$; when $J_f=J_i$, the $M_i=M_f=0$ matrix element vanishes. For E2, $\Delta M=0,\pm1,\pm2$, subject to the full $3j$ symbol, which can impose additional zeros.

For E2, the rank-2 triangle excludes

$$0\not\leftrightarrow0, \qquad 0\not\leftrightarrow1, \qquad \frac12\not\leftrightarrow\frac12.$$

It permits $0\leftrightarrow2$. A table that says only $\Delta J=0,\pm1,\pm2$ misses these distinctions.

Exact and approximate columns

The $J$, $M$, and parity entries concern field-free eigenstates with those labels. Additional useful limits are:

Approximation	Further restriction
one active electron, E1	$\Delta l=\pm1$
one active electron, E2	$\Delta l=0,\pm2$, with $\lvert l_f-l_i\rvert\le2\le l_f+l_i$
pure LS coupling, E1	$\Delta S=0$; rank-$1$ triangle for $L$
leading nonrelativistic M1 in pure LS coupling	$\Delta S=0$, $\Delta L=0$; orbital/configuration overlap must also be nonzero
pure LS coupling, E2	$\Delta S=0$; rank-$2$ triangle for $L$

Spin–orbit mixing can open weak intercombination E1 lines. Configuration mixing changes strengths and orbital assignments without violating exact parity or total-angular-momentum rules. A one-electron $\Delta l$ rule is not a general restriction on a correlated many-electron state.

Name a line “spin-forbidden E1,” “parity-forbidden E1,” or “E2-allowed,” not simply “forbidden.”

Polarization and Magnetic Sublevels

Choose the quantization axis as $z$. For absorption, label the light component by the angular momentum $q$ it transfers along that axis:

$$\Delta M=M_f-M_i=q.$$

Light component	Spherical index	Absorption rule	Common name
electric field parallel to $z$	$q=0$	$\Delta M=0$	$\pi$
component driving an increase of $M$	$q=+1$	$\Delta M=+1$	$\sigma^+$
component driving a decrease of $M$	$q=-1$	$\Delta M=-1$	$\sigma^-$

Here $q$ labels the tensor component acting on the atom. To connect it to a field convention, with time dependence $e^{-i\omega t}$ a positive-frequency electric vector proportional to $(\hat x+i\hat y)$ drives $q=+1$, up to an irrelevant overall phase. Helicity is defined relative to propagation, whereas $q$ is defined relative to the chosen $z$ axis. Avoid assigning a helicity sign without specifying both directions and absorption versus emission.

The relative angular line strength contains

$$\left| \begin{pmatrix} J_f & 1 & J_i\\ -M_f & q & M_i \end{pmatrix} \right|^2.$$

Equal permission does not imply equal strength. For a beam propagating along the quantization axis, the electric field is transverse and has no $q=0$ component. For propagation perpendicular to the axis, linear polarization can be chosen to produce a $\pi$ component or a superposition of $\sigma^\pm$ components.

Hyperfine-Resolved Rules

Let

$$\mathbf F=\mathbf I+\mathbf J,$$

where $I$ is nuclear spin and $J$ is electronic angular momentum. For a rank-1 transition between hyperfine levels,

$$\Delta F=0,\pm1, \qquad F=0\not\leftrightarrow F'=0,$$

and

$$\Delta m_F=q=0,\pm1.$$

Process	Main hyperfine rule	Projection rule	Important qualifier
optical E1	$\Delta F=0,\pm1$, not $0\leftrightarrow0$	$\Delta m_F=0,\pm1$	parent electronic E1 $J$ and parity rules still apply
microwave M1	rank-1 $F$ triangle, often $\Delta F=0,\pm1$	$\Delta m_F=0,\pm1$	state composition and magnetic moment determine strength
electric quadrupole	rank-2 $F$ triangle	$\Delta m_F=0,\pm1,\pm2$	parity unchanged for E2

For states with unmixed electronic $J$, hyperfine recoupling does not rescue an electronically forbidden $J=0\leftrightarrow0$ E1 transition. The reduced hyperfine matrix element contains the electronic reduced matrix element and a $6j$ coefficient; if the electronic factor is zero, angular recoupling alone cannot make it nonzero. The hyperfine interaction can itself mix different electronic $J$ values; then a nominal $J=0\leftrightarrow0$ transition can acquire weak E1 strength. The exact $F$ and parity conditions still apply.

A “clock transition” often uses $\Delta m_F=0$ states with a first-order magnetic-field-insensitive operating point. That is an experimental choice and Hamiltonian property, not a universal selection rule.

Pure Rotational Electric-Dipole Rules

A pure rotational E1 line requires a nonzero molecule-fixed permanent dipole component and a nonzero rotational matrix element. Molecules without a permanent dipole still possess rotational levels; they simply lack ordinary pure rotational E1 absorption in the ideal field-free model.

Linear closed-shell rotor

For the ideal linear rigid rotor,

$$\Delta J=\pm1.$$

Absorption from a lower rotational level has $\Delta J=+1$; stimulated or spontaneous emission can have $\Delta J=-1$. The $J=0\leftrightarrow0$ transition is absent.

Symmetric and asymmetric tops

Brown and Carrington (2003) develop the molecular angular factors and coupling conventions.

Rotor or band type	Body-fixed rule	Space-fixed rule
symmetric-top parallel	$\Delta K=0$	$\Delta J=0,\pm1$, not $0\leftrightarrow0$
symmetric-top perpendicular	$\Delta K=\pm1$	$\Delta J=0,\pm1$, not $0\leftrightarrow0$
asymmetric-top $a$ type	$\Delta K_a$ even, $\Delta K_c$ odd	$\Delta J=0,\pm1$, not $0\leftrightarrow0$
asymmetric-top $b$ type	$\Delta K_a$ odd, $\Delta K_c$ odd	$\Delta J=0,\pm1$, not $0\leftrightarrow0$
asymmetric-top $c$ type	$\Delta K_a$ odd, $\Delta K_c$ even	$\Delta J=0,\pm1$, not $0\leftrightarrow0$

The symmetric-top rows give angular matrix-element conditions and also apply to rovibrational bands. A parallel pure-rotational transition within the ideal rigid-top spectrum, with $\Delta J=\Delta K=0$, has zero energy change and therefore gives no finite-frequency absorption line. The $K=0$ parallel band also has a vanishing $\Delta J=0$ matrix element.

“Even” includes $0,\pm2,\ldots$ and “odd” includes $\pm1,\pm3,\ldots$. These asymmetric-top rules identify the dipole component that can carry a line in the rigid-rotor limit. Nuclear-spin statistics, torsion, large-amplitude motion, electronic spin, hyperfine structure, and state mixing add labels and restrictions.

Quick molecular comparison

Molecule	Permanent dipole?	Pure rotational E1?	Rotational Raman?
HCl	yes	yes, chiefly $\Delta J=\pm1$	yes if polarizability is anisotropic
CO	yes	yes	yes
N$_2$	no	no in the ideal E1 model	yes, chiefly $\Delta J=\pm2$
O$_2$	no	no ordinary E1; magnetic-dipole structure can occur	yes

A molecule can therefore rotate without having microwave E1 absorption.

Vibrational Infrared Rules

Expand the electric dipole along a normal coordinate $Q_k$:

$$\boldsymbol\mu(\mathbf Q) = \boldsymbol\mu_0 + \sum_k \left( \frac{\partial\boldsymbol\mu} {\partial Q_k} \right)_0 Q_k + \cdots.$$

A harmonic fundamental is infrared active when

$$\left( \frac{\partial\boldsymbol\mu} {\partial Q_k} \right)_0 \ne0$$

in at least one laboratory component. In point-group language,

$$\Gamma(Q_k) \subset \Gamma(\mu_x,\mu_y,\mu_z).$$

Model	Main rule	What opens additional bands?
one-dimensional harmonic oscillator with linear dipole	$\Delta v=\pm1$	higher dipole derivatives or mechanical anharmonicity
polyatomic harmonic normal modes	one quantum in an IR-active mode	mode mixing, anharmonicity, resonance
hot band	commonly $v\to v+1$ from $v>0$	finite initial thermal population
overtone	$\Delta v=2,3,\ldots$	electrical or mechanical anharmonicity
combination band	changes in more than one mode	mixed dipole derivatives, anharmonic coupling

A molecule need not have a permanent dipole to possess an infrared-active vibration. The criterion is the derivative of the dipole along the mode. Conversely, a polar molecule can have an individual normal mode with zero dipole derivative.

Rovibrational Branches

For a rank-1 rovibrational E1 transition, the space-fixed rotational shorthand is

$$\Delta J=-1,0,+1, \qquad J=0\not\leftrightarrow J'=0.$$

Branch	$\Delta J$	Typical label
P	$-1$	lower rotational angular momentum in the final state
Q	$0$	same $J$
R	$+1$	higher rotational angular momentum in the final state

Not every band has all three branches. For a simple linear $\Sigma\leftrightarrow\Sigma$ electric-dipole band, the Q branch is absent. Perpendicular bands, electronic angular momentum, vibrational angular momentum, spin, parity, and molecular symmetry can permit or suppress different branch structures.

For symmetric tops:

• a parallel transition moment gives $\Delta K=0$;
• a perpendicular transition moment gives $\Delta K=\pm1$.

Branch letters describe $\Delta J$, not the complete selection rule.

Raman Rules

Long (2002) develops the Raman tensor and polarization dependence. Ordinary nonresonant Raman scattering is governed by the molecular polarizability tensor rather than a permanent electric dipole. Expand

$$\alpha_{ab}(\mathbf Q) = \alpha_{ab}^{(0)} + \sum_k \left( \frac{\partial\alpha_{ab}} {\partial Q_k} \right)_0 Q_k + \cdots.$$

A harmonic vibrational mode is Raman active when at least one component satisfies

$$\left( \frac{\partial\alpha_{ab}} {\partial Q_k} \right)_0 \ne0.$$

Equivalently, the mode symmetry must occur in the symmetry representation of the quadratic functions associated with the symmetric polarizability tensor.

Vibrational and rotational table

Raman process	Leading ideal-model rule	Additional requirement
vibrational Stokes	$\Delta v=+1$	nonzero polarizability derivative
vibrational anti-Stokes	$\Delta v=-1$	populated excited vibrational state
vibrational overtone or combination	larger or multimode changes	higher polarizability derivatives, anharmonicity, or resonance
linear-rotor rotational Raman	shifted lines have $\Delta J=\pm2$	anisotropic polarizability
unshifted component	no molecular energy change	the state can change within a degenerate subspace, for example between degenerate $M$ levels

For rotational Raman spectra, branch notation often extends to

Branch	$\Delta J$
O	$-2$
Q	$0$
S	$+2$

The shifted pure rotational Raman lines are the O and S branches. Polarization selection depends on the Raman tensor and the incident and detected polarization vectors; “Raman active” alone does not determine the observed intensity in a chosen geometry.

Centrosymmetric mutual exclusion

For a molecule with inversion symmetry, an ideal normal mode cannot be both electric-dipole IR active and ordinary Raman active:

• ungerade modes can transform like the dipole and be IR active;
• gerade modes can transform like the polarizability and be Raman active.

This parity distinction remains valid for electric-dipole–electric-dipole Raman scattering on resonance when inversion is exact: the product of two odd dipole operators is even. Resonance changes intermediate-state weights, not that parity. Vibronic mixing that preserves inversion also cannot make an odd fundamental E1–E1 Raman active. Broken inversion, higher multipoles, or a different process such as hyper-Raman scattering requires different rules.

Molecular Electronic Shorthand

For an electric-dipole transition in a diatomic or linear molecule, commonly used approximate rules include:

Label	E1 shorthand	Qualification
total spin	$\Delta S=0$	leading spin-independent electric dipole; spin–orbit mixing opens weak bands
orbital projection	$\Delta\Lambda=0,\pm1$	body-fixed rank-1 condition
total angular momentum	$\Delta J=0,\pm1$, not $0\leftrightarrow0$	full angular condition
space-fixed projection	$\Delta M=0,\pm1$	polarization dependent
parity and reflection	must match operator symmetry	notation depends on electronic state and molecular symmetry

The most general symmetry test is

$$\Gamma_f^* \otimes \Gamma(\hat O) \otimes \Gamma_i \supset \Gamma_{\mathrm{tot.sym.}}.$$

The complex conjugate $\Gamma_f^*$ is essential for complex irreducible representations because the final state appears as a bra. For real representations it can be omitted. This representation criterion is safer than memorizing a label change when vibronic coupling, degenerate electronic states, or non-Abelian molecular symmetry is involved.

When Forbidden Lines Appear

Mechanism	What it mixes or changes	Typical borrowed line
spin–orbit coupling	different spin character with compatible total symmetry	intercombination E1
configuration interaction	configurations with different radial or angular strength	weak nominally configuration-forbidden line
static electric field	opposite parity	Stark-induced E1
magnetic field	magnetic sublevels and sometimes angular-momentum labels	field-enabled or redistributed Zeeman components
hyperfine interaction	electronic angular-momentum character through nuclear coupling	hyperfine-induced transition
vibronic coupling	electronic and vibrational symmetries	Herzberg–Teller intensity
anharmonicity	harmonic vibrational basis states	overtones and combination bands
collisions or environment	isolated-system symmetry and coherence	collision-induced or matrix-enabled absorption
resonant Raman denominator	intermediate-state weighting, while exact symmetries remain	enhancement of an already symmetry-permitted channel

An observed weak line should be assigned only after alternatives such as an isotopologue, hot band, impurity, blend, multiple scattering, detector artifact, or calibration error have been checked.

Common pitfalls

Permitted is not strong. After symmetry permits a matrix element, its radial factor or interference between contributions can still suppress it. Compare relative strengths and line positions before assigning a measured peak.

A missing line is not a missing degree of freedom. A nonpolar molecule still has rotational states. Raman, magnetic-dipole, quadrupole, or collision-induced spectra may reveal them.

Resonance does not remove exact symmetry. A small energy denominator can enhance a permitted Raman amplitude. It cannot change the parity of an E1–E1 operator in a system with exact inversion.

Exercises

E1, M1, or E2?

Classify the angular-momentum and parity permission of these field-free transitions:

1. $J_i^{\pi_i}=0^+\to J_f^{\pi_f}=1^-$;
2. $0^+\to0^-$;
3. $0^+\to2^+$;
4. $1^+\to1^+$.

Consider E1, M1, and E2 only. Do not infer line strength.

Solution

1. $0^+\to1^-$ passes the rank-1 triangle and changes parity, so it is E1-allowed by $J$ and parity. M1 and E2 require unchanged parity and fail.
2. $0^+\to0^-$ has the parity change required by E1 but fails the rank-1 $0\leftrightarrow0$ rule. M1 and E2 fail parity. It is forbidden for all three mechanisms under these assumptions.
3. $0^+\to2^+$ passes the rank-2 triangle and preserves parity, so it is E2-allowed by $J$ and parity. E1 fails both rank and parity; M1 fails rank.
4. $1^+\to1^+$ preserves parity. M1 passes the rank-1 triangle; E2 passes the rank-2 triangle. E1 fails parity. Reduced matrix elements can still remove either permitted multipole.

Polarization from a scalar state

An atom begins in $J_i=0$, $M_i=0$ and is excited by E1 light to $J_f=1$. Which final magnetic sublevel is reached by $q=0,+1,-1$ absorption? What component is absent for a plane wave propagating along the quantization axis?

Solution

The projection rule is

$$M_f=M_i+q=q.$$

Therefore:

• $q=0$ reaches $M_f=0$;
• $q=+1$ reaches $M_f=+1$;
• $q=-1$ reaches $M_f=-1$.

For propagation along the quantization axis, the electric field is transverse, so there is no field component parallel to that axis and no $\pi$ or $q=0$ component.

Infrared and Raman activity

A centrosymmetric molecule has one gerade normal mode and one ungerade normal mode. Under the leading electric-dipole IR and ordinary polarizability Raman operators, which mode can be IR active and which can be Raman active? Name one way the ideal mutual-exclusion pattern can fail experimentally.

Solution

The electric dipole is ungerade, so an ungerade normal mode can be IR active. The polarizability is gerade, so a gerade normal mode can be Raman active. Thus the leading operators give mutual exclusion.

A field or an environment that breaks inversion can relax the exclusion. Resonance or parity-preserving vibronic mixing alone cannot: an E1–E1 amplitude remains even. A combination band can have a different total symmetry from either constituent fundamental, and should be analyzed as its own transition.

Explain a weak intercombination line

An observed line violates $\Delta S=0$ but satisfies the exact $J$ and parity rules for E1. Give a minimal, testable explanation and predict how its intensity scales with a small mixing amplitude $\varepsilon$.

Solution

Spin–orbit coupling can mix a small component of an E1-allowed spin state into one of the levels:

$$|\widetilde f\rangle = |f_0\rangle + \varepsilon|f_{\mathrm{allowed}}\rangle.$$

If

$$\langle f_0|\hat d|i\rangle=0$$

but

$$\langle f_{\mathrm{allowed}}|\hat d|i\rangle\ne0,$$

then

$$\mathcal M_{fi} \sim \varepsilon \langle f_{\mathrm{allowed}}|\hat d|i\rangle.$$

The borrowed intensity scales as

$$I_{\mathrm{borrowed}} \propto|\varepsilon|^2.$$

A test should compare the line strength across a series where spin–orbit mixing changes, or calculate the mixed-state composition and branching ratios.

References

• Brown, J. M., and A. Carrington. Rotational Spectroscopy of Diatomic Molecules. Cambridge University Press, 2003. DOI.
• Long, D. A. The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules. Wiley, 2002. DOI.
• Martin, W. C., and W. L. Wiese. “Atomic Spectroscopy,” in G. W. F. Drake, ed., Atomic, Molecular, and Optical Physics Handbook. AIP Press, 1996. NIST online selection-rule and line-strength reference.