Selection Rules: A Phenomenological Overview

Point a microwave beam at HCl gas and it absorbs strongly. Point the same beam at N₂ and nothing happens, even though both are diatomic molecules with rotational energy levels. This split between "spectroscopically active" and "spectroscopically silent" molecules is not an accident of intensity. It is a strict yes-or-no rule written into molecular symmetry and charge distribution. Selection rules tell you, before you ever run an experiment, which transitions can appear in a spectrum and which are forbidden. They matter because they turn a spectrum from a bewildering forest of lines into a decodable message about bond lengths, force constants, and electronic structure. Without them you cannot assign a single peak. This article catalogs the gross and specific selection rules for rotational, vibrational, and electronic transitions. You will see the physical reason behind each rule. You will see how it follows from the transition dipole moment. Worked examples show the rules in action, and later spectroscopy articles build directly on the results collected here.

Key takeaways

A transition is allowed only if the transition dipole moment integral is nonzero; selection rules are shortcuts for spotting this without doing the integral.
Gross selection rules ask whether a mechanism exists at all (e.g., does the molecule have a permanent dipole); specific selection rules restrict $\Delta$ of quantum numbers within that mechanism.
Pure rotational spectra require a permanent dipole moment, with $\Delta J = \pm1$ .
Vibrational (IR) spectra require the dipole moment to change during the vibration, with $\Delta v = \pm1$ in the harmonic approximation.
Electronic spectra follow rules on spin, symmetry (parity, orbital angular momentum) and are refined by the Franck–Condon principle for vibrational structure.
Key idea: a transition appears in a spectrum only if the transition dipole moment ⟨f|μ|i⟩ is nonzero, and selection rules are the symmetry/mechanism shortcuts that predict this sign without integration.

The core insight

Every selection rule, no matter which spectroscopy it governs, is a disguised statement about the transition dipole moment: a transition is spectroscopically allowed if and only if the integral connecting initial and final states through the dipole operator is nonzero, and gross versus specific selection rules simply separate "does the required charge redistribution exist at all" from "which quantum-number jumps that mechanism permits."

The physical picture

A spectroscopic transition is a competition between two things: an oscillating electric field trying to push charge around, and the molecule's own charge distribution deciding whether it can respond. The transition dipole moment,

\boldsymbol{\mu}_{fi} = \int \psi_f^{\,*}\,\hat{\boldsymbol{\mu}}\,\psi_i \, d\tau ,

quantifies that response (see Transition Intensities and the Transition Dipole Moment). When this integral vanishes by symmetry, the transition is forbidden and no photon of that energy is absorbed or emitted, regardless of how strong the incident light is. When it is nonzero, the transition is allowed, and its magnitude sets the line intensity. Selection rules split naturally into two tiers. A gross selection rule asks a yes/no question about the molecule as a whole: does it possess the physical feature (a permanent dipole, a changing dipole during vibration, an allowed change in polarizability) that lets the oscillating field couple to it at all? If the answer is no, every transition of that type is forbidden, full stop — this is why homonuclear diatomics like N₂, O₂, and H₂ show no pure rotational or vibrational infrared spectrum. A specific selection rule then operates within an allowed mechanism, restricting which changes in quantum number ( $\Delta J$ , $\Delta v$ , $\Delta L$ , $\Delta S$ ...) are permitted, because angular momentum carried by the photon (one unit of $\hbar$ ) must be conserved and because the vibrational wavefunctions have definite parity. Rotational, vibrational, and electronic transitions each have their own gross and specific rules, summarized in the figure below, but the underlying logic — nonzero transition dipole, angular momentum conservation, symmetry of the wavefunctions — is identical throughout.

Figure

Gross selection rules: which mechanisms can act

The second useful picture is how the specific selection rule looks on an actual spectrum: it produces a regularly spaced ladder of lines rather than a continuum, because $\Delta J = \pm1$ (or $\Delta v = \pm1$ ) fixes the spacing between adjacent allowed transitions.

Figure

Rotational transitions obey ΔJ = ±1, giving evenly spaced lines

The core relations

\boldsymbol{\mu}_{fi} = \int \psi_f^{\,*}\,\hat{\boldsymbol{\mu}}\,\psi_i\,d\tau \neq 0 \quad \text{(condition for any allowed transition)}

where: $\psi_i,\psi_f$ are initial/final stationary-state wavefunctions and $\hat{\boldsymbol\mu}$ is the electric dipole operator.

\Delta J = \pm 1 \qquad \text{(pure rotational and rotational-vibrational transitions)}

where: $J$ is the rotational quantum number; the photon supplies one unit of angular momentum.

\Delta v = \pm 1 \qquad \text{(vibrational fundamental, harmonic approximation)}

where: $v$ is the vibrational quantum number; overtones ( $\Delta v = \pm2,\pm3,\dots$ ) appear only through anharmonicity.

\tilde{\nu}_{J\to J+1} = 2\tilde{B}(J+1), \qquad J = 0,1,2,\dots

where: $\tilde B = h/(8\pi^2 c I)$ is the rotational constant and $I$ the moment of inertia; consecutive lines are spaced by $2\tilde B$ .

\Delta S = 0, \qquad \Delta \Lambda = 0,\pm1, \qquad g \leftrightarrow u \ (\text{Laporte rule})

where: $S$ is total spin, $\Lambda$ the projection of orbital angular momentum on the molecular axis (diatomics), and $g/u$ label parity under inversion in centrosymmetric molecules.

Derivation

Derivation AFrom the transition dipole integral to "allowed" or "forbidden"

Goal. Show how the single requirement $\boldsymbol{\mu}_{fi}\neq0$ generates the distinct gross and specific rules for rotation, vibration, and electronic transitions, without carrying out the full quantum-mechanical integration.

Step 1 — Split the dipole operator by degree of freedom. For a diatomic (or any molecule with separable rotation, vibration, electronic motion under the Born–Oppenheimer approximation), the total dipole moment is a function of nuclear coordinates that can be Taylor-expanded about the equilibrium geometry $Q=0$ :

\mu(Q) = \mu_0 + \left(\frac{\partial \mu}{\partial Q}\right)_0 Q + \tfrac12\left(\frac{\partial^2 \mu}{\partial Q^2}\right)_0 Q^2 + \cdots

The zeroth-order term $\mu_0$ is the permanent dipole moment; it governs pure rotational transitions. The first-order term, proportional to the derivative of the dipole with respect to the normal coordinate $Q$ , governs vibrational transitions.

Step 2 — Rotational transitions: isolate the $\mu_0$ term. For pure rotation, the relevant matrix element is $\mu_0\int Y_{J'M'}^*(\theta,\phi)\,(\text{direction cosine})\,Y_{JM}(\theta,\phi)\,d\Omega$ . This integral is zero identically unless $\mu_0 \neq 0$ — the gross selection rule: only molecules with a permanent dipole show a pure rotational spectrum. Symmetric molecules (N₂, CO₂, CH₄) automatically fail this test. Evaluating the angular integral (properties of spherical harmonics) further requires

\Delta J = \pm1, \qquad \Delta M_J = 0,\pm1,

the specific selection rule, which is simply conservation of the one unit of angular momentum carried by the absorbed or emitted photon.

Step 3 — Vibrational transitions: isolate the $(\partial\mu/\partial Q)_0$ term. Substituting the expansion of $\mu(Q)$ into the transition moment and keeping the linear term gives, for a transition between harmonic-oscillator states $v$ and $v'$ ,

\mu_{fi} \propto \left(\frac{\partial \mu}{\partial Q}\right)_0 \int \psi_{v'}^{*}\, Q\, \psi_v \, dQ.

If $(\partial\mu/\partial Q)_0 = 0$ — the dipole does not change as the nuclei move along that normal mode — the transition is forbidden regardless of what the integral over $Q$ gives. This is the gross selection rule for IR activity: a vibration is infrared-active only if it changes the molecule's dipole moment. The symmetric stretch of CO₂, which leaves the dipole at zero throughout, is IR-silent for exactly this reason; the antisymmetric stretch and bend, which do change $\mu$ , are IR-active.

Step 4 — The harmonic-oscillator integral fixes Δv. The integral $\int \psi_{v'}^{*}\,Q\,\psi_v\,dQ$ is a standard result from raising/lowering (ladder) operator algebra applied to harmonic-oscillator wavefunctions; it vanishes unless

\Delta v = v' - v = \pm 1.

This is the specific selection rule for the vibrational fundamental. Overtones ( $\Delta v = \pm 2, \pm3,\dots$ ) become weakly allowed only when the true (anharmonic) potential mixes harmonic-oscillator states of different $v$ , explaining why overtone bands are always much weaker than the fundamental.

Step 5 — Electronic transitions: symmetry and spin of the full wavefunction. The electronic transition moment $\int \psi_{el,f}^{*}\,\hat{\boldsymbol\mu}\,\psi_{el,i}\,d\tau_{el}$ is evaluated using group-theoretic (symmetry) arguments rather than a simple radial integral. Because the dipole operator does not act on spin, the integral vanishes unless the initial and final electronic states have the same total spin multiplicity:

\Delta S = 0.

For centrosymmetric molecules, the dipole operator is odd under inversion, so the integral is nonzero only between states of opposite parity:

g \leftrightarrow u \quad (\text{Laporte rule}), \qquad g \leftrightarrow g \text{ and } u \leftrightarrow u \text{ forbidden}.

For diatomics, the projection of orbital angular momentum along the internuclear axis, $\Lambda$ , obeys $\Delta\Lambda = 0,\pm1$ , the electronic analogue of the photon's one unit of angular momentum. Vibrational fine structure riding on the allowed electronic transition is then governed not by a strict $\Delta v$ rule but by the Franck–Condon principle: intensities are weighted by the square of the vibrational wavefunction overlap, $|\langle v'|v\rangle|^2$ , because electronic transitions are much faster than nuclear motion.

Boxed result.

\boxed{ \begin{aligned} &\text{Rotation: } \mu_0 \neq 0 \ (\text{gross}); \ \Delta J = \pm1 \ (\text{specific})\\ &\text{Vibration: } (\partial\mu/\partial Q)_0 \neq 0 \ (\text{gross}); \ \Delta v = \pm1 \ (\text{specific, harmonic})\\ &\text{Electronic: } \Delta S = 0,\ g\leftrightarrow u,\ \Delta\Lambda = 0,\pm1 \ (\text{specific}); \text{ intensity pattern set by Franck–Condon overlap} \end{aligned}}

Worked examples

Example 1 — Foundational: is CO microwave-active, and where does its first line fall? Carbon monoxide has a permanent dipole moment ( $\mu \approx 0.122$ D), so it passes the gross selection rule immediately: pure rotational absorption is allowed. Its rotational constant is $\tilde B = 1.9313\ \text{cm}^{-1}$ (a standard spectroscopic value, obtained from microwave data). The specific selection rule $\Delta J = \pm1$ means the lowest-energy absorption is $J=0 \to J=1$ :

\tilde\nu_{0\to1} = 2\tilde B (J+1) = 2 \times 1.9313\ \text{cm}^{-1} \times (0+1) = 3.8626\ \text{cm}^{-1}.

Converting to frequency:

\nu = \tilde\nu \, c = (3.8626\ \text{cm}^{-1})(2.998\times10^{10}\ \text{cm s}^{-1}) = 1.1583\times10^{11}\ \text{Hz} \approx 115.8\ \text{GHz}.

Sanity check: this falls squarely in the microwave/millimeter-wave region, consistent with CO's famous use as an interstellar-medium tracer, whose $J=1\to0$ line at 115.27 GHz is one of the most-observed lines in radio astronomy — our calculated value matches to within rounding of $\tilde B$ . The next line, $J=1\to2$ , appears at $2\tilde B(2) = 7.7252\ \text{cm}^{-1}$ , i.e. exactly twice the first spacing — confirming the predicted $2\tilde B$ ladder spacing from the specific selection rule.

Example 2 — Applied: which vibrational modes of CO₂ are IR-active, and what does that predict for its spectrum? Linear CO₂ (O=C=O, point group $D_{\infty h}$ ) has three fundamental vibrational modes: the symmetric stretch ( $\nu_1$ ), the doubly degenerate bend ( $\nu_2$ ), and the antisymmetric stretch ( $\nu_3$ ). Symmetric stretch ( $\nu_1$ , $\tilde\nu_1 \approx 1388$ cm⁻¹): both C–O bonds stretch and contract in phase. By the molecule's center of symmetry, the dipole moment is zero at every point along this coordinate — extension on one side is mirrored by extension on the other, and the two bond dipoles always cancel. Therefore $(\partial\mu/\partial Q_1)_0 = 0$ : the gross selection rule fails, and $\nu_1$ is IR-inactive (though, as later Raman articles will show, it is Raman-active — a preview of the mutual exclusion rule for centrosymmetric molecules). Antisymmetric stretch ( $\nu_3$ , $\tilde\nu_3 \approx 2349$ cm⁻¹): one bond lengthens while the other shortens, so the molecule develops a net oscillating dipole along the axis. $(\partial\mu/\partial Q_3)_0 \neq 0$ : IR-active, and this is in fact the strong absorption responsible for CO₂'s major role as a greenhouse gas near 2349 cm⁻¹ (4.26 μm). Bend ( $\nu_2$ , doubly degenerate, $\tilde\nu_2 \approx 667$ cm⁻¹): bending the O=C=O angle away from 180° creates a transient dipole perpendicular to the original axis. $(\partial\mu/\partial Q_2)_0 \neq 0$ : IR-active — this is the second major CO₂ absorption band, near 667 cm⁻¹ (15 μm), which is the specific band responsible for trapping outgoing terrestrial infrared radiation in the atmosphere. Sanity check: two IR-active fundamentals ( $\nu_2$ , $\nu_3$ ) plus one IR-inactive, Raman-active fundamental ( $\nu_1$ ) is exactly the pattern required by the mutual exclusion rule for any molecule with a center of symmetry — no fundamental can be both IR- and Raman-active. This consistency check confirms the mode assignment is physically sound before any spectrum is even measured.

Limiting cases & common traps

Situation	What actually happens	Common mistake
Homonuclear diatomic (N₂, O₂, H₂)	No pure rotational or vibrational IR spectrum (μ₀ = 0 and ∂μ/∂Q = 0)	Assuming "no dipole moment" also means no Raman activity — it does not
CO₂ symmetric stretch	IR-inactive, but Raman-active	Assuming a vibration invisible in IR is invisible everywhere
Overtone bands (Δv = ±2, ±3...)	Weakly allowed via anharmonicity	Treating Δv = ±1 as an absolute law rather than the harmonic-approximation result
Centrosymmetric electronic transitions (e.g., d–d transitions in octahedral complexes)	Formally Laporte-forbidden, observed only weakly via vibronic coupling	Concluding the transition "cannot occur" instead of "occurs only weakly"
Spin-forbidden transitions (e.g., phosphorescence)	Weakly allowed via spin–orbit coupling	Treating ΔS = 0 as absolute rather than a strong propensity rule

The chemist behind it

The systematic use of selection rules to interpret molecular spectra owes an enormous debt to Gerhard Herzberg, whose multi-volume treatises on atomic and molecular spectra (published from the 1930s through the 1960s) turned spectroscopic selection rules from scattered quantum-mechanical results into a working toolkit for chemists identifying real molecules — including, famously, free radicals detected in flames, discharges, and interstellar space.

Origins · 1 minGerhard Herzberg and how this was discovered

Gerhard Herzberg

1904–1999 · Canadian physical chemist and spectroscopist whose Nobel-winning work systematized selection rules for diatomic and polyatomic molecules

Ask the tutor: Want to see how the ΔJ = ±1 rule combines with centrifugal distortion to predict real rotational line positions in microwave spectroscopy?

What is the difference between a gross and a specific selection rule?

A gross selection rule asks whether a physical mechanism for coupling to radiation exists at all — for example, whether the molecule has a permanent dipole (rotation) or whether the dipole changes during a vibration. If the answer is no, every transition of that type is forbidden. A specific selection rule then operates within an allowed mechanism, restricting which quantum-number jumps (ΔJ, Δv, ΔS, and so on) are permitted, usually because the photon must carry away or supply a fixed amount of angular momentum, or because of the parity of the wavefunctions involved.

Why is the symmetric stretch of CO₂ invisible in the infrared?

As the two C=O bonds stretch and compress in phase, the dipole contributions from each bond remain equal and opposite by the molecule's center of symmetry, so the total dipole moment stays exactly zero throughout the vibration. Since the gross selection rule for IR activity requires the dipole moment to change with the vibrational coordinate, and here it never does, the transition dipole moment integral vanishes identically. The mode is, however, Raman-active, illustrating the mutual exclusion rule for centrosymmetric molecules.

Are overtone bands in IR spectra truly forbidden?

Not absolutely. In the strict harmonic-oscillator approximation, Δv = ±1 is the only allowed vibrational transition, making overtones (Δv = ±2, ±3, ...) formally forbidden. Real molecular potentials are anharmonic, however, which mixes harmonic-oscillator wavefunctions of different v and gives overtone transitions a small but nonzero transition dipole moment. This is why overtone bands are routinely observed in real IR spectra, just at intensities roughly one to two orders of magnitude weaker than the fundamental.

Why do Laporte-forbidden d–d transitions still give transition-metal complexes their color?

The Laporte rule forbids g ↔ g electronic transitions in centrosymmetric complexes, and d–d transitions within an octahedral complex are exactly of this type. Strict adherence would make these complexes colorless. Molecular vibrations, however, momentarily distort the complex away from perfect centrosymmetry, mixing in small amounts of odd-parity character through vibronic coupling. This relaxes the rule just enough to give weak but chemically important absorption, which is why many transition-metal complexes are pastel-colored rather than intensely colored like fully allowed charge-transfer transitions.

How do these rules connect to what comes next in molecular spectroscopy?

Everything here is phenomenology: what is allowed and what is forbidden, stated without derivation from the underlying integrals. The next articles put these rules to work. Rotational microwave spectroscopy uses the ΔJ = ±1 rule and permanent-dipole requirement to extract bond lengths from line spacings. Vibrational IR spectroscopy uses the dipole-derivative rule to predict which bonds show up in a fingerprint spectrum. Raman spectroscopy introduces the complementary polarizability-based selection rules, explaining why CO₂'s symmetric stretch reappears there.