In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic 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 in chemical reactions, where some are formally spin-forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products.

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

In quantum mechanics the basis for a spectroscopic selection rule is the value of the transition moment integral

where ${\displaystyle \psi _{1}}$ and ${\displaystyle \psi _{2}}$ are the wave functions of the two states, "state 1" and "state 2", involved in the transition, and μ is the transition moment operator. This integral represents the propagator (and thus the probability) of the transition between states 1 and 2; if the value of this integral is zero 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 the symmetry of the transition moment function ${\displaystyle \psi _{1}^{*}\,\mu \,\psi _{2}.}$ If the transition moment function is symmetric over all of the totally symmetric representation of the point group to which the atom or molecule belongs, then the integral's value is (in general) not zero and the transition is allowed. Otherwise, the transition is "forbidden".

The transition moment integral is zero if the transition moment function, ${\displaystyle \psi _{1}^{*}\,\mu \,\psi _{2},}$ is anti-symmetric or odd, i.e. ${\displaystyle y(x)=-y(-x)}$ holds. The symmetry of the transition moment function is the direct product of the parities of its three components. The symmetry characteristics of each component can be obtained from standard character tables. Rules for obtaining the symmetries of a direct product can be found in texts on character tables.

The Laporte rule is a selection rule formally stated as follows: In a centrosymmetric environment, transitions between like atomic orbitals such as s–s, p–p, d–d, or f–f, transitions are forbidden. The Laporte rule (law) applies to electric dipole transitions, so the operator has u symmetry (meaning ungerade, odd). p orbitals also have u symmetry, so the symmetry of the transition moment function is given by the product (formally, the product is taken in the group) u×u×u, which has u symmetry. The transitions are therefore forbidden. Likewise, d orbitals have g symmetry (meaning gerade, even), so the triple product g×u×g also has u symmetry and the transition is forbidden.

The wave function of a single electron is the product of a space-dependent wave function and a spin wave function. Spin is directional and can be said to have odd parity. It follows that transitions in which the spin "direction" changes are forbidden. In formal terms, only states with the same total spin quantum number are "spin-allowed". In crystal 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.