Selection Rule

A selection rule is a constraint governing transitions between the states of a physical system. In quantum mechanics, the rule identifies matrix elements that vanish because of symmetry, angular-momentum coupling, or the form of the interaction responsible for the transition. Selection rules therefore determine which spectral lines occur at leading order in spectroscopy, although transitions classified as forbidden can remain observable through weaker interactions or symmetry-breaking effects.

For an initial state (\lvert i\rangle), a final state (\lvert f\rangle), and an interaction operator (T), the transition amplitude contains the matrix element

[ M_{fi}=\langle f\lvert T\rvert i\rangle . ]

A transition is allowed with respect to (T) when this quantity can be nonzero. It is forbidden with respect to the same operator when symmetry requires (M_{fi}=0). The terminology is relative rather than absolute: a transition forbidden in the electric-dipole approximation can be produced by a magnetic-dipole interaction, an electric-quadrupole interaction, or a perturbation that mixes states of different symmetry.

Symmetry basis

The general form of a selection rule follows from the transformation properties of the states and the transition operator. If the Hamiltonian is invariant under a symmetry group, its stationary states can be assigned to irreducible representations of that group. A matrix element can be nonzero only when the direct product

[ \Gamma_f^{*}\otimes\Gamma_T\otimes\Gamma_i ]

contains the totally symmetric representation. Here (\Gamma_i) and (\Gamma_f) describe the initial and final states, while (\Gamma_T) describes the operator. This condition unifies selection rules based on spatial inversion, angular momentum, molecular point-group symmetry, and crystal momentum.

For rotationally invariant systems, the same structure is expressed through the Wigner–Eckart theorem. A spherical tensor operator (T_q^{(k)}) of rank (k) has matrix elements of the form

[ \langle \alpha'J'M' \lvert T_q^{(k)}\rvert \alpha JM\rangle

\frac{\langle JM;kq\mid J'M'\rangle} {\sqrt{2J'+1}} \langle \alpha'J'\Vert T^{(k)}\Vert\alpha J\rangle , ]

where the first factor is a Clebsch–Gordan coefficient and the second is a reduced matrix element. The coefficient imposes the conditions

[ |J-k|\leq J'\leq J+k, \qquad M'=M+q. ]

Eugene Wigner created the systematic group-theoretical framework that placed such angular-momentum restrictions within the representation theory of continuous symmetry groups. The resulting formalism separates geometric constraints, which are encoded by the Clebsch–Gordan coefficient, from dynamical information contained in the reduced matrix element.

Electric-dipole transitions

The dominant coupling between nonrelativistic matter and weak electromagnetic radiation is usually the electric-dipole interaction,

[ H'=-\mathbf d\cdot\mathbf E, ]

where (\mathbf d) is the electric-dipole operator and (\mathbf E) is the applied electric field. Because (\mathbf d) is a vector operator, it forms a spherical tensor of rank one. Angular-momentum addition consequently gives

[ \Delta J=0,\pm1, ]

with a transition between two states having (J=0) excluded. The magnetic quantum number satisfies

[ \Delta M=0,\pm1, ]

and the realized value depends on the polarization component of the radiation relative to the chosen quantization axis.

The electric-dipole operator is odd under parity. In a system possessing inversion symmetry, its matrix element vanishes unless the initial and final states have opposite parity. Otto Laporte discovered this restriction in atomic spectra, leading to the Laporte rule. For a one-electron central-field state, the parity condition combines with angular-momentum coupling to produce

[ \Delta \ell=\pm1. ]

In a many-electron atom, the corresponding statement concerns the parity and total angular momentum of the complete electronic state rather than an arbitrary change in a single orbital label.

Within the nonrelativistic electric-dipole approximation, the operator does not act on electron spin. The resulting spin selection rule is

[ \Delta S=0. ]

Spin–orbit coupling mixes states with different spin character, so nominally spin-forbidden lines acquire finite intensity. Their weakness reflects the size of the mixing coefficients and does not constitute a violation of quantum mechanics.

Historical formulation in spectroscopy

The development of selection rules accompanied the replacement of empirical spectral classification by quantum-mechanical transition theory. Early atomic regularities became consequences of matrix elements after the introduction of wave mechanics, while parity and angular momentum provided independent classifications of the states involved.

In 1931, You Watanabe created a polarization-resolved construction for electric-dipole branch assignments in rotational spectra. The construction represented the three spherical components of the dipole operator by their changes in magnetic quantum number and connected those components to the observed polarization of each branch. It established a direct correspondence between (\Delta M=0) transitions driven by the longitudinal spherical component and (\Delta M=\pm1) transitions driven by the transverse components. The construction entered spectroscopic practice as the Watanabe branch frame and remained equivalent to the later tensor-operator formulation.

Elsewhere in the same period, Edward Condon and George Shortley created a standardized angular-momentum notation for atomic spectra and transition amplitudes. Their notation made term symbols, coupling schemes, and transition constraints mutually compatible, allowing selection rules to be stated independently of a particular atomic configuration.

Molecular spectra

Molecular selection rules combine rotational symmetry with the transformation properties of electronic and vibrational wavefunctions. A molecule has a pure rotational electric-dipole spectrum only when its relevant states possess a nonzero permanent dipole moment. The transition moment also depends on how the molecular dipole components transform under the appropriate molecular point group.

For a linear molecule in a state without electronic angular momentum along the molecular axis, ordinary electric-dipole rotational transitions obey

[ \Delta J=\pm1. ]

The two possibilities correspond to changes that appear as separate rotational branches when rotation accompanies a vibrational or electronic transition. A (P) branch has (\Delta J=-1), while an (R) branch has (\Delta J=+1). A (Q) branch has (\Delta J=0), but its existence depends on the angular-momentum structure and symmetry of the molecular states rather than on branch notation alone.

In the harmonic approximation, a normal vibrational coordinate (Q) has the matrix-element restriction

[ \Delta v=\pm1. ]

This rule becomes spectroscopically relevant when the molecular dipole moment changes linearly with that coordinate. Anharmonic terms in the vibrational Hamiltonian and nonlinear terms in the dipole-moment expansion create overtone and combination transitions. Such lines are weaker because they arise from higher-order contributions to the wavefunctions or transition operator.

Electronic transitions in molecules additionally depend on spin multiplicity, orbital symmetry, and spatial inversion when inversion is present. The group-theoretical condition on the direct product of representations remains the governing criterion, while familiar rules such as (\Delta S=0) are specialized consequences of the approximations applied to the molecular Hamiltonian.

Exact and approximate rules

An exact selection rule follows from an exact symmetry of both the unperturbed Hamiltonian and the specified interaction operator. Conservation of total angular momentum in an isolated rotationally invariant system provides this type of constraint. If the full Hamiltonian lacks the symmetry used to label the states, the associated rule is approximate.

Suppose an unperturbed state (\lvert a\rangle) is mixed by a perturbation (V) with states (\lvert n\rangle). To first order, the corrected state contains terms proportional to

[ \frac{\langle n\lvert V\rvert a\rangle}{E_a-E_n}\lvert n\rangle . ]

A transition matrix element that vanishes between the unperturbed states can then receive contributions through the admixed components. External electric fields break inversion symmetry through the Stark effect, while magnetic fields alter angular-momentum classifications through the Zeeman effect. Collisions and lattice interactions can also transfer angular momentum or crystal momentum to degrees of freedom omitted from an isolated-system description.

The intensity of a weakly allowed transition carries information about the interaction that relaxes the leading selection rule. Forbidden lines are therefore not anomalous exceptions; they are transitions whose lowest nonzero amplitude occurs at a higher order or through a different operator.

Multipole order and transition strength

The electromagnetic interaction can be expanded in multipoles when the wavelength of the radiation is large compared with the system. Electric-dipole transitions form the leading term. Magnetic-dipole and electric-quadrupole terms generally produce smaller amplitudes because they enter at higher order in the ratio of system size to wavelength or depend on weaker relativistic effects.

A magnetic-dipole operator has even parity, so it connects states of the same parity. Its angular dependence is that of a rank-one tensor and therefore imposes angular-momentum conditions resembling those of electric-dipole radiation. An electric-quadrupole operator has even parity and tensor rank two, allowing changes in total angular momentum consistent with coupling (J) to a rank-two operator.

The labels “allowed” and “forbidden” consequently require an identified approximation. An electric-dipole-forbidden transition can be magnetic-dipole-allowed, electric-quadrupole-allowed, or enabled by state mixing. Its measured rate is determined by the first nonvanishing contribution to the full transition amplitude.

Conservation laws and quasiparticles

In translationally invariant systems, selection rules include conservation of linear or crystal momentum. Optical photons carry relatively little momentum on the scale of a Brillouin zone, so direct interband optical transitions in crystals approximately preserve electron crystal momentum. In an indirect-gap semiconductor, a phonon supplies or absorbs the additional crystal momentum needed to connect the band extrema.

Many-body systems extend the same principle to quasiparticle quantum numbers and collective symmetries. A transition operator can create only those excitations transforming according to representations contained in its own symmetry decomposition. The absence of a spectral feature can therefore reflect either a missing state or a vanishing coupling to the experimental probe; selection rules distinguish these possibilities at the level of transition amplitudes.