Standard time-dependent perturbation theory yields an expression for the Einstein B coefficients [see equation (A)]
that is proportional to the squared absolute value of an infinite sum. The first term in this sum is proportional to the quantum-mechanical matrix element ⟨1|d|2⟩, where |1⟩ and |2⟩ are the quantum states of the ion before and after emission, and d is the operator for the ion's electric dipole moment. The two next terms in the series are proportional to ⟨1|q|2⟩ and ⟨1|m|2⟩, where q and m are the ion's electric quadrupole and magnetic dipole moments. On dimensional grounds the relative magnitudes of these quantities are of order 
, where λ is the wavelength of the photon concerned and a₀ is the Bohr radius. Since λ/a₀ ≈ 5000 in the optical band, the first (dipole) contribution to B dominates unless it vanishes by virtue of some exact or near symmetry of the ion. Forbidden transitions are those for which ⟨1|d|2⟩ vanishes. Their forbidden nature is indicated by surrounding the symbols of the ions in which they occur by square brackets thus: [O²⁺]. If ⟨1|d|2⟩ is much smaller than expected on dimensional grounds, the transition is called semi-forbidden and a square bracket is put to the right of the ion involved: N⁺].
A transition is allowed (i.e., not forbidden) if the differences in the quantum numbers of the states |1⟩ and |2⟩ obey the electric-dipole selection rules. The most important quantum numbers characterizing these states are the atom's total angular momentum F, its total electronic angular momentum J, its total electronic spin S and its total orbital angular momentum L. The selection rules for these quantities are
F and J are always good quantum numbers but L and S are good quantum numbers only to the extent that the Russell–Saunders or L – S coupling scheme is applicable. The quantum states of atoms are generally classified under the assumption that L – S coupling applies, although this coupling scheme is never more than an approximation, and one that deteriorates as one proceeds down the periodic table. In reality a state |1⟩ that is classified as having, say, S = 1, may be a linear combination of a predominant state that has S = 1 and a state that has S = 0. This mixed state can decay by emission of electric dipole radiation to a state classified as having S = 0 without violation of the electric-dipole selection rule. Such intercombination transitions are semi-forbidden.
