Selection Rule Tables

A selection rule is a statement that a transition matrix element vanishes under specified assumptions. It is not merely a change in a quantum number, and “forbidden” does not mean impossible. A useful rule must name the transition operator, the state labels, the symmetry or coupling limit, and any external fields that define or mix those labels.

These tables are a quick lookup layer. The canonical derivations live in Atomic Selection Rules, Selection Rules in Spectroscopy, and the Wigner–Eckart Theorem. Use those pages when a zero, line strength, or exception must be derived rather than recalled.

Canonical Scope

This page owns compact, qualified tables for:

• electric-dipole, magnetic-dipole, and electric-quadrupole atomic transitions;
• parity, angular-momentum, and polarization conditions;
• hyperfine-resolved transitions;
• pure rotational and rovibrational electric-dipole spectra;
• harmonic vibrational infrared activity;
• vibrational and rotational Raman activity;
• molecular electronic shorthand rules; and
• common mechanisms that open nominally forbidden lines.

It does not own transition-rate formulas, detailed line-strength calculations, or complete molecular-symmetry character tables. A listed rule is normally a necessary condition. A transition satisfying every row can still have a zero reduced matrix element, an accidentally small radial integral, destructive interference, negligible initial population, or an undetectable experimental signal.

How to Read a Rule

Before applying any table, write a five-entry ledger.

Entry	Question
operator	E1, M1, E2, Raman polarizability, collision, or something else?
exact labels	Which quantum numbers commute with the Hamiltonian actually used?
approximation	Is LS coupling, a rigid rotor, harmonic motion, or field-free parity assumed?
geometry	What is the quantization axis, propagation direction, and polarization?
observable	Is the question about a nonzero amplitude, line strength, population, or measured peak?

The amplitude is

$$\mathcal M_{fi} = \langle f|\hat O|i\rangle.$$

A selection rule tests whether symmetry forces $\mathcal M_{fi}=0$. If a small perturbation mixes an allowed state into $|i\rangle$ or $|f\rangle$ with amplitude $\varepsilon$, then a formerly forbidden amplitude is often of order $\varepsilon$, while its intensity is of order $|\varepsilon|^2$.

Universal Tensor Rule

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

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

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 first three rows below reproduce the rigorous angular and parity content used by the NIST atomic-spectroscopy reference. Conditions listed in the final column require additional approximations.

Type	Rigorous $J$ rule	Projection rule	Parity	Common additional rule
E1	$\Delta J=0,\pm1$; $0\not\leftrightarrow0$	$\Delta M=0,\pm1$; for $\Delta J=0$, $M=0\not\leftrightarrow0$	changes	one active electron: $\Delta l=\pm1$; pure LS: $\Delta S=0$, $\Delta L=0,\pm1$ with $L=0\not\leftrightarrow0$
M1	$\Delta J=0,\pm1$; $0\not\leftrightarrow0$	$\Delta M=0,\pm1$; for $\Delta J=0$, $M=0\not\leftrightarrow0$	unchanged	negligible configuration mixing: same configuration; pure LS: $\Delta S=0$, usually $\Delta L=0$
E2	rank-2 triangle; shorthand $\Delta J=0,\pm1,\pm2$	$\Delta M=0,\pm1,\pm2$	unchanged	one active electron: $\Delta l=0,\pm2$; pure LS: $\Delta S=0$ and $

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 refer to the full multipole operator and field-free angular-momentum eigenstates. The $L$, $S$, $l$, and configuration entries are model-dependent:

• $\Delta S=0$ follows because the leading electric operator does not act on spin, but spin–orbit mixing makes intercombination lines weakly allowed.
• One-electron $\Delta l$ rules are not exact statements about a correlated many-electron eigenstate.
• Configuration interaction can transfer E1, M1, or E2 strength between nominal configurations.
• Intermediate coupling replaces pure LS labels by dominant-component labels.

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

Atomic Electric-Dipole Checklist

For a field-free E1 transition between atomic levels, check the rules in this order.

Gate	Test	Status
energy	$E_f-E_i=\hbar\omega$ for absorption	kinematic
total angular momentum	$	J_f-J_i
magnetic projection	$M_f=M_i+q$, $q=0,\pm1$	rigorous once axis and polarization are fixed
parity	$\pi_f=-\pi_i$	rigorous if parity is conserved
total spin	$\Delta S=0$	LS-coupling approximation
total orbital momentum	$\Delta L=0,\pm1$, not $0\leftrightarrow0$	LS-coupling approximation
active-electron orbital	$\Delta l=\pm1$	independent-particle approximation
strength	reduced matrix element and radial overlap	not a selection rule

The first four symmetry gates can all pass while the line remains weak. Conversely, a line failing an approximate LS rule can appear through spin–orbit or configuration mixing without violating the rigorous $J$ and parity rules.

Polarization and Magnetic Sublevels

Choose the quantization axis as $z$. For absorption, define the spherical field component $q$ so that

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

Light component	Spherical index	Absorption rule	Common name
electric field parallel to $z$	$q=0$	$\Delta M=0$	$\pi$
positive-helicity component under the stated convention	$q=+1$	$\Delta M=+1$	$\sigma^+$
negative-helicity component under the stated convention	$q=-1$	$\Delta M=-1$	$\sigma^-$

The labels $\sigma^\pm$ depend on the propagation direction, viewing direction, and spherical-basis convention. A reproducible statement defines the quantization axis and says whether it describes absorption or 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

The hyperfine rule 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.

A “clock transition” often uses $\Delta m_F=0$ states with a first-order magnetic-field-insensitive operating point. That is an engineering 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

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 $0\leftrightarrow0$

“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

Absence of microwave E1 lines is therefore not evidence that a molecule does not rotate.

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

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
Rayleigh component	material state unchanged, often written $\Delta J=0$	elastic scattering

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 mutual-exclusion rule requires an inversion center and the stated leading operators. It is not a universal rule for all molecules, surfaces, solids, resonant processes, or symmetry-broken environments.

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

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

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	strongly enhanced weak Raman 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.

Assignment Checklist

For each candidate transition:

1. Identify the initial and final energy levels.
2. Name the operator and multipole order.
3. Apply the full tensor triangle condition.
4. Apply projection and polarization rules.
5. Apply parity or molecular-symmetry rules.
6. Mark every coupling-dependent rule as approximate.
7. Evaluate or obtain the reduced matrix element.
8. Include lower-state population and degeneracy.
9. Fold in line shape, resolution, and detection geometry.
10. Compare predicted and observed positions and relative strengths.

A selection-rule match is evidence for an assignment, not a unique identification.

Common Mistakes

Using a change rule without the triangle condition

Rank-2 $\Delta J=1$ does not make $0\leftrightarrow1$ an E2 transition. Use $|J_f-J_i|\le k\le J_f+J_i$.

Forgetting the operator

The same pair of levels can be E1-forbidden, M1-allowed, and E2-allowed. “Forbidden” is incomplete without a mechanism.

Treating LS rules as exact

$\Delta S=0$ and $\Delta L$ rules assume a coupling limit. Real eigenstates can contain mixed LS components.

Treating allowed as strong

The reduced matrix element, radial overlap, interference, state population, and experimental geometry control intensity after symmetry permission.

Assigning polarization without an axis convention

$\sigma^+$ means $\Delta M=+1$ here for absorption under the stated convention. Reversing the propagation or viewing convention can reverse a verbal helicity label.

Saying nonpolar molecules have no rotational spectrum

They lack ordinary pure rotational E1 absorption, but rotational Raman, collision-induced, electric-quadrupole, or magnetic-dipole spectra can remain.

Applying the harmonic rule to an anharmonic molecule

$\Delta v=\pm1$ is the leading harmonic, linear-property result. Overtones and combinations are expected at weaker intensity in real molecules.

Applying infrared–Raman mutual exclusion universally

The rule requires inversion symmetry and the leading electric-dipole and polarizability operators.

Exercises

Exercise 1: 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.

Exercise 2: The hidden E2 exclusion

Explain why $J_i=0\to J_f=1$ is not an E2 transition even though $\Delta J=+1$ appears in the shorthand E2 list.

Solution

An E2 operator has rank $k=2$. The exact triangle condition is

$$|J_f-J_i|\le2\le J_f+J_i.$$

For $0\to1$ this becomes

$$1\le2\le1,$$

whose upper inequality fails. The shorthand change rule is necessary but not sufficient at low $J$.

Exercise 3: 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.

Exercise 4: Hyperfine branches

For an optical E1 transition with lower hyperfine level $F_i=1$, list the candidate values of $F_f$. Which candidate would survive if $F_i=0$ instead? Why must the parent electronic transition still be checked?

Solution

The rank-1 hyperfine rule gives

$$\Delta F=0,\pm1.$$

From $F_i=1$, the candidates are

$$F_f=0,1,2.$$

From $F_i=0$, the rank-1 triangle allows only $F_f=1$ because $0\leftrightarrow0$ is forbidden.

The hyperfine reduced matrix element contains the parent electronic reduced matrix element. If the electronic E1 transition fails its $J$ or parity rule, recoupling $J$ with $I$ does not by itself create E1 strength.

Exercise 5: HCl and nitrogen

Compare ideal field-free HCl and N$_2$.

1. Which has pure rotational E1 absorption?
2. What is the leading linear-rotor E1 rule?
3. Which can show pure rotational Raman lines, and with what leading shifted rule?

Solution

HCl has a permanent electric dipole, so it has pure rotational E1 absorption with

$$\Delta J=\pm1.$$

N$_2$ is homonuclear and has no permanent electric dipole, so ordinary pure rotational E1 absorption is absent.

Both molecules have anisotropic polarizability and can show rotational Raman scattering. For a linear rotor, the leading shifted Raman lines obey

$$\Delta J=\pm2.$$

Exercise 6: 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.

The pattern can be relaxed by loss of inversion symmetry, a surface or matrix environment, vibronic mixing, defects, external fields, resonant processes, or assignment of a different species or combination band.

Exercise 7: Audit a weak 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.

Cross-Links

• Term Symbol Reference decodes the atomic and molecular labels on which these rule tables operate.
• Reference and Data is the task-oriented gateway to AMO lookup pages and source-provenance rules.
• Atomic Selection Rules derives E1, M1, E2, hyperfine, and polarization conditions.
• Selection Rules in Spectroscopy develops the operator ledger across atomic and molecular spectra.
• Wigner–Eckart Theorem is the canonical derivation of tensor-rank and projection rules.
• Parity Selection Rules gives the symmetry proof and perturbative failure modes.
• Dipole Transitions develops vector-operator angular factors.
• Rotational Spectroscopy connects rules to line patterns and structure inference.
• Vibrational Spectroscopy explains dipole derivatives, anharmonic bands, and normal-mode assignment.
• Raman Spectroscopy derives polarizability-tensor activity and polarization dependence.
• Hyperfine Structure defines $F$, $m_F$, and the coupling regimes behind resolved components.
• Constants and Conversions translates line positions among frequency, wavelength, wavenumber, energy, and temperature units.

References

1. W. C. Martin and W. L. Wiese, “Atomic Spectroscopy,” in Atomic, Molecular, and Optical Physics Handbook, edited by G. W. F. Drake, AIP Press (1996), NIST web version 2.1 (2007).
2. A. Kramida, Yu. Ralchenko, J. Reader, and the NIST ASD Team, NIST Atomic Spectra Database, version 5.12 (2024), National Institute of Standards and Technology.
3. R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley (1988).
4. I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer (1992), doi:10.1007/978-3-642-76907-8.
5. J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press (2003), doi:10.1017/CBO9780511814808.
6. P. F. Bernath, Spectra of Atoms and Molecules, 4th ed., Oxford University Press (2020).
7. G. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules, 2nd ed., Van Nostrand (1950).
8. D. A. Long, The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules, Wiley (2002), doi:10.1002/0470845767.
9. J. T. Hougen, “Strategies for advanced applications of permutation-inversion groups to the microwave spectra of molecules with large amplitude motions,” Journal of Molecular Spectroscopy 256, 170–185 (2009), doi:10.1016/j.jms.2009.04.006.