transitionmoments

Transition moments are quantum-mechanical quantities that quantify the amplitude for a system to undergo a transition between two states when interacting with an external perturbation, such as an electromagnetic field. In spectroscopy, they determine the intensity of transitions between energy levels and thus the strengths of spectral lines.

In general, the transition moment is a matrix element of a transition operator between the initial and final state. For an electric dipole transition, the electric dipole transition moment is M_if = ⟨ψ_f|μ|ψ_i⟩, where μ is the electric dipole operator and ψ_i, ψ_f are the initial and final state wavefunctions. Other types of transitions have corresponding moments, such as the magnetic dipole moment ⟨ψ_f|m|ψ_i⟩ and the electric quadrupole moment, each defined with its appropriate operator.

Physical significance follows from the relation between the transition moment and transition probability. The probability of a transition is proportional to the modulus squared of the transition moment, a connection formalized in Fermi’s golden rule. Selection rules arise from the symmetries of the states and the transition operator, restricting which transitions are allowed.

Computation and applications: Transition moments are computed with quantum-chemical methods, including ab initio techniques and time-dependent density functional theory, to predict absorption intensities, radiative lifetimes, and line strengths. They underpin UV–visible spectroscopy, fluorescence, phosphorescence, and other spectroscopic analyses, and are often compared with experimental line intensities to validate models. Variants include magnetic dipole and electric quadrupole moments, and vibronic effects can modify moments through Herzberg–Teller coupling.

See also: oscillator strength, selection rules, transition density, Einstein A coefficient.