3.2 Selection rules and transition probabilities

Electronic Transitions: Selection Rules and Probabilities

Selection rules for electronic transitions

Selection rules tell you which electronic transitions can actually happen (and show up in a spectrum) versus which ones are formally forbidden. They come from conservation laws and symmetry requirements in quantum mechanics.

Spin selection rule: The total spin quantum number must be conserved, so $\Delta S = 0$. This prohibits transitions between states of different spin multiplicity. A singlet-to-triplet transition, for example, is spin-forbidden. In practice, spin-orbit coupling can partially relax this rule, especially in molecules containing heavy atoms.

Laporte (parity) selection rule: In centrosymmetric molecules, transitions must involve a change in parity. Only g → u or u → g transitions are allowed. Transitions between states of the same parity (g → g or u → u) are forbidden. This is discussed in more detail below.

Orbital angular momentum selection rule: For atoms, the orbital angular momentum quantum number must change by exactly one: $\Delta L = \pm 1$. This is why an s → p transition is allowed but an s → d transition is not.

Total angular momentum selection rule: The total angular momentum quantum number follows $\Delta J = 0, \pm 1$, with the added restriction that $J = 0 \rightarrow J = 0$ is forbidden.

A transition is formally "allowed" only when it satisfies all applicable selection rules simultaneously. Violating even one rule reduces the transition probability, often by orders of magnitude.

Calculation of transition dipole moments

The transition dipole moment is the central quantity that connects selection rules to measurable absorption intensities. It captures how effectively the molecule's charge distribution is reorganized during a transition.

The transition dipole moment between an initial state $\psi_i$ and a final state $\psi_f$ is defined as:

$\mu_{if} = \int \psi_f^* \hat{\mu} \, \psi_i \, d\tau$

where $\hat{\mu}$ is the electric dipole moment operator. If this integral evaluates to zero, the transition is dipole-forbidden. If it's nonzero, the transition is dipole-allowed.

Transition probability is proportional to the square of the transition dipole moment:

$P_{if} \propto |\mu_{if}|^2$

This is why selection rules matter so concretely: they determine whether $\mu_{if}$ is zero or nonzero, and therefore whether the transition has any measurable intensity.

Oscillator strength provides a dimensionless measure of transition intensity, connecting the quantum mechanical result to a classical oscillator framework:

$f = \frac{4\pi m_e \nu}{3 e^2 \hbar} |\mu_{if}|^2$

Fully allowed transitions typically have $f$ values near 1, while forbidden transitions can have $f$ values of $10^{-3}$ or smaller.

Franck-Condon factors account for the overlap between vibrational wavefunctions of the initial and final electronic states. Even if the electronic part of $\mu_{if}$ is nonzero, the vibrational overlap modulates the intensity of each vibronic band within the transition. This is why electronic absorption bands often show vibrational fine structure, with some vibronic peaks stronger than others depending on the geometry change between the two electronic states.

Laporte rule in centrosymmetric molecules

The Laporte rule applies specifically to molecules with an inversion center (centrosymmetric molecules, such as octahedral complexes). It states that transitions between states of the same parity are forbidden:

This has direct consequences for transition metal chemistry. In octahedral complexes, all d orbitals have gerade (g) symmetry, so d-d transitions are Laporte-forbidden. This is why d-d absorption bands are characteristically weak, with molar absorptivities ($\varepsilon$) typically in the range of 1–100 $\text{M}^{-1}\text{cm}^{-1}$. By contrast, charge-transfer transitions (which are parity-allowed) often have $\varepsilon$ values of $10^3$–$10^4$ $\text{M}^{-1}\text{cm}^{-1}$ or higher.

Despite being formally forbidden, d-d transitions do occur. Several mechanisms partially break the Laporte rule:

• Vibronic coupling: Asymmetric vibrations temporarily distort the molecular geometry, removing the inversion center for brief moments. This is the most common mechanism and is why octahedral transition metal complexes are colored at all.
• Static distortions: Structural effects like the Jahn-Teller distortion permanently remove the inversion center, making the Laporte rule no longer strictly applicable.
• Spin-orbit coupling: This mixes states of different parity, partially relaxing the rule. The effect is stronger for heavier atoms (the heavy atom effect).

Factors affecting transition intensity

Several factors combine to determine how intense a spectral band actually appears:

Electronic factors:

• Transition dipole moment magnitude: Larger $|\mu_{if}|^2$ values produce stronger absorptions. The $\pi \rightarrow \pi^*$ transition, for instance, is typically much more intense than $n \rightarrow \pi^*$ because the orbital overlap contributing to the transition dipole moment is greater for $\pi \rightarrow \pi^*$.
• Selection rule compliance: Transitions that obey all selection rules are orders of magnitude more intense than forbidden ones. A spin-allowed, Laporte-allowed charge-transfer band will dwarf a spin-forbidden d-d band.
• Franck-Condon overlap: The vibrational wavefunction overlap distributes intensity across vibronic sub-bands and determines which vibronic peak is strongest.

Environmental factors:

• Solvent effects: The solvent can shift transition energies and alter intensities through changes in molecular geometry or stabilization of excited states. This phenomenon is called solvatochromism.
• Temperature: Higher temperatures populate higher vibrational levels of the ground state, giving rise to "hot bands" and altering the apparent intensity distribution.

Measurement factors:

• Concentration and path length determine the observed absorbance through the Beer-Lambert law: $A = \varepsilon \, c \, l$. These affect the measured signal but not the intrinsic transition probability.
• Spectral bandwidth of the instrument affects peak shape and resolution but does not change the integrated absorption intensity.

The key distinction to keep straight: transition dipole moments, oscillator strengths, and selection rules govern the intrinsic probability of a transition. Concentration, path length, and instrumental settings affect what you measure but not the underlying physics.