Electronic transitions, which involve an electron moving from one energy level to another, are fundamental to understanding how molecules and atoms interact with light. Not all theoretically possible transitions are observed; instead, specific selection rules dictate which transitions are allowed and which are forbidden based on quantum mechanical principles and the conservation of angular momentum. These rules ultimately determine the intensity of absorption or emission bands in spectroscopy.
The primary selection rules governing transitions between electronic energy levels, including those involving microstates and terms, are:
Key Electronic Transition Selection Rules
1. The Spin Selection Rule ($\Delta S = 0$)
The Spin Selection Rule states that during an electronic transition, the total spin angular momentum of the system must remain unchanged. This means the change in the total spin quantum number, S, must be zero:
- $\Delta S = 0$
For a transition to be allowed, the multiplicity of the initial and final states must be the same. Multiplicity is given by $(2S + 1)$. For example, a transition from a singlet state (S=0, multiplicity=1) to another singlet state is allowed, but a transition from a singlet state to a triplet state (S=1, multiplicity=3) is generally forbidden.
Practical Insight: Violations of this rule often lead to very weak or "spin-forbidden" transitions, which can still occur due to phenomena like spin-orbit coupling, especially in heavier atoms.
2. The Laporte (or Orbital) Selection Rule
The Laporte Selection Rule primarily governs transitions where there is a change in the electron's spatial distribution or angular momentum. It has slightly different interpretations for atoms and centrosymmetric molecules.
-
For Atoms ($\Delta l = \pm 1$)
For a one-electron transition in an atom, the change in the azimuthal (or orbital angular momentum) quantum number, l, must be $\pm 1$:- $\Delta l = \pm 1$
This means an electron must change its orbital type (e.g., from an s orbital ($l=0$) to a p orbital ($l=1$), or from a p orbital to a d orbital ($l=2$)). Transitions like s→s, p→p, or s→d are forbidden by this rule.
-
For Centrosymmetric Molecules ($g \leftrightarrow u$)
In molecules that possess a center of inversion (centrosymmetric molecules), electronic transitions must involve a change in parity. This means the transition must be from a gerade (g) state to an ungerade (u) state, or vice versa:- $g \rightarrow u$ or $u \rightarrow g$
Gerade (g) orbitals are symmetric with respect to inversion through the center of symmetry (e.g., s orbitals, d orbitals), while ungerade (u) orbitals are antisymmetric (e.g., p orbitals, f orbitals). Therefore, transitions like g→g or u→u are forbidden.
Examples:
- In transition metal complexes, d-d transitions are often Laporte forbidden because both the initial and final d orbitals are gerade. However, these transitions frequently appear as weak bands in spectra due to relaxation of the rule through vibronic coupling or deviations from perfect centrosymmetry.
- In diatomic molecules like O$_2$, $\sigma_g \rightarrow \pi_u$ transitions are Laporte allowed, while $\pi_u \rightarrow \pi_u$ are forbidden.
Summary of Key Selection Rules
The table below summarizes the fundamental selection rules for electronic transitions:
| Rule | Description | Condition for Allowed Transition | Forbidden Transition Example |
|---|---|---|---|
| Spin Selection | Total spin angular momentum must not change. | $\Delta S = 0$ (Multiplicity remains same) | Singlet $\leftrightarrow$ Triplet |
| Laporte (Atoms) | Change in azimuthal quantum number. | $\Delta l = \pm 1$ | s $\leftrightarrow$ s, p $\leftrightarrow$ p, s $\leftrightarrow$ d |
| Laporte (Centrosymmetric Molecules) | Change in parity through a center of inversion. | $g \leftrightarrow u$ | g $\leftrightarrow$ g, u $\leftrightarrow$ u |
Origin and Relaxation of Selection Rules
These rules arise from the mathematical properties of the transition moment integral, which describes the probability of a transition occurring. An allowed transition has a non-zero transition moment integral, while a forbidden transition has an integral that is theoretically zero.
While these rules provide a strong guideline, "forbidden" transitions are often observed with low intensity. This is typically due to:
- Spin-Orbit Coupling: Interaction between the electron's spin and orbital angular momenta can slightly mix states of different spin, relaxing the spin selection rule. This is more pronounced in heavier elements.
- Vibronic Coupling: The simultaneous absorption or emission of a photon and a phonon (vibrational quantum) can distort the molecular geometry, momentarily breaking the symmetry required for the Laporte rule, especially in centrosymmetric molecules. This allows for weak Laporte-forbidden transitions to occur.
- External Fields: Strong electric or magnetic fields can also perturb energy levels and relax selection rules.
Understanding these selection rules is crucial for interpreting spectroscopic data, predicting the color of compounds, and designing materials with specific optical properties.
