Group Theory and Electronic Transition Selection Rules

Welcome to the final lesson of our module on symmetries in crystalline solids. In our previous lessons, we built a comprehensive picture of how symmetry governs the behavior of electrons in crystals. We saw how spatial symmetries lead to Bloch's theorem and the classification of electronic states according to the irreducible representations (irreps) of the crystal's point group. We then introduced time-reversal symmetry, a non-spatial symmetry that gives rise to the fundamental constraint of Kramers' degeneracy.

So far, we have focused on the properties of static electronic states—their energies and degeneracies. In this lesson, we will put this knowledge into action to understand a dynamic process: the interaction of light with a crystal. We will use the full power of group theory to determine the selection rules for optical transitions between electronic bands. This will allow us to predict which transitions are "allowed" or "forbidden" and how they depend on the polarization of light, connecting our abstract symmetry analysis directly to measurable experimental quantities like absorption spectra.

The Quantum Mechanical Basis for Optical Transitions

When light shines on a semiconductor, its oscillating electromagnetic field can perturb the electrons, potentially causing an electron to jump from a filled valence band state to an empty conduction band state. The probability of such a transition occurring is governed by quantum mechanics.

According to Fermi's Golden Rule, the transition rate $W$ from an initial state $∣ ψ_{i} ⟩$ to a final state $∣ ψ_{f} ⟩$ is proportional to the square of the matrix element of the interaction Hamiltonian, $H_{in t}$ :

W_{i \to f} \propto ∣ ⟨ ψ_{f} ∣ H_{in t} ∣ ψ_{i} ⟩ ∣^{2}

For optical frequencies, the interaction between light and an electron is well-described by the electric dipole approximation. In this approximation, the interaction Hamiltonian is proportional to the coupling between the electric field of the light $E$ and the electron's dipole moment operator $μ = - e r$ , where $r$ is the position operator.

Therefore, the central task in determining selection rules is to figure out when the transition dipole moment, $⟨ ψ_{f} ∣ r ∣ ψ_{i} ⟩$ , is non-zero. A transition is considered "allowed" if this matrix element is non-zero, and "forbidden" if it is zero.

The Vanishing Integral Theorem: Group Theory's Verdict

This is where group theory provides an elegant and powerful tool. The vanishing integral theorem (which you first encountered in Module 4) states that for an integral of the form $⟨ f ∣ O^∣ i ⟩$ to be non-zero, the direct product of the irreducible representations of the functions involved must contain the totally symmetric irreducible representation of the group ( $Γ_{1}$ ).

The condition is:

Γ (ψ_{f}^{*}) \otimes Γ (O^) \otimes Γ (ψ_{i}) \supset Γ_{1}

where $Γ (ψ_{f}^{*})$ is the irrep of the complex conjugate of the final state, $Γ (O^)$ is the irrep of the operator, and $Γ (ψ_{i})$ is the irrep of the initial state. For unitary representations, which are common in physics, the character of $Γ (ψ_{f}^{*})$ is the complex conjugate of the character of $Γ (ψ_{f})$ . A more intuitive and equivalent way to phrase this is that the decomposition of the direct product $Γ (O^) \otimes Γ (ψ_{i})$ must contain the irrep of the final state, $Γ (ψ_{f})$ .

The following reading explains this theorem and its application to spectroscopy in detail.

Group theory

This document from ETH Zurich provides a rigorous derivation of spectroscopic selection rules using group theory.

Please read the section 4.3.6 "Selection rules with group theory". Focus on the derivation leading up to the vanishing integral theorem (Eq. 4.32) and its application to vibrational and electronic spectroscopy. The logic for electronic spectroscopy, summarized in Eq. 4.36, is precisely what we will be using.

Selection Rules for Direct Interband Transitions

Let's apply this formalism to transitions between electronic bands in a solid.

The States: The initial state $∣ ψ_{i} ⟩$ is a Bloch state in a valence band, $∣ ψ_{v, k} ⟩$ , belonging to an irrep $Γ_{v}$ of the little group of $k$ . The final state $∣ ψ_{f} ⟩$ is a Bloch state in a conduction band, $∣ ψ_{c, k^{'}} ⟩$ , belonging to an irrep $Γ_{c}$ of the little group of $k^{'}$ .
Wavevector Conservation: The momentum of a photon in the optical range is very small compared to the size of the Brillouin zone. This leads to the conservation of the crystal momentum of the electron. This first, crucial selection rule is $k^{'} \approx k$ . This means that allowed optical transitions are, to a very good approximation, vertical on an $E - k$ band diagram. These are called direct transitions.
The Operator: The interaction operator is the position operator $r$ , which transforms as a vector. Its components $(x, y, z)$ transform according to the vector representation of the point group, $Γ_{v ec}$ . We can find this representation by looking for how $x, y,$ and $z$ are listed in the rightmost columns of a character table.

Combining these points, the selection rule for a direct transition at wavevector $k$ is that the matrix element $⟨ ψ_{c, k} ∣ r ∣ ψ_{v, k} ⟩$ must be non-zero. The group-theoretical condition is:

Γ_{c}^{*} \otimes Γ_{v ec} \otimes Γ_{v} \supset Γ_{1}

Or, equivalently, the direct product of the operator's representation and the initial state's representation must contain the final state's representation:

Γ_{v ec} \otimes Γ_{v} \supset Γ_{c}

This is the central result. To see if a transition from a valence band of symmetry $Γ_{v}$ to a conduction band of symmetry $Γ_{c}$ is allowed, we compute the direct product $Γ_{v ec} \otimes Γ_{v}$ and check if $Γ_{c}$ is one of the irreps in its decomposition.

The document below provides a concise summary of this principle in the context of semiconductors, where it is often referred to as an application of the Wigner-Eckart theorem.

[PDF] Chapter 4 Optical properties of semiconductors - Digital CSIC

This chapter excerpt from a book on optical properties of semiconductors directly links the band structure to optical transitions.

Please read the two paragraphs in Section 4.3 that begin with this sentence. This section introduces Fermi's Golden Rule for solids and explicitly states the selection rule based on the direct product of representations.

An Illustrative Example: Transitions at the $Γ$ point

Let's make this concrete with an example. We will consider a crystal with $T_{d}$ point group symmetry (like zincblende semiconductors, e.g., GaAs or InP) and look at transitions at the $Γ$ point ( $k = 0$ ). The character table for $T_{d}$ is:

$T_{d}$	E	8 $C_{3}$	3 $C_{2}$	6 $S_{4}$	6 $σ_{d}$	Basis Functions
$A_{1}$	1	1	1	1	1	$x^{2} + y^{2} + z^{2}$
$A_{2}$	1	1	1	-1	-1
$E$	2	-1	2	0	0	$(2 z^{2} - x^{2} - y^{2}, x^{2} - y^{2})$
$T_{1}$	3	0	-1	1	-1	$(R_{x}, R_{y}, R_{z})$
$T_{2}$	3	0	-1	-1	1	$(x, y, z), (x y, x z, y z)$

From the table, we can see two key things:

The vector $(x, y, z)$ transforms as the $T_{2}$ irreducible representation. So, $Γ_{v ec} = T_{2}$ .
The top of the valence band in many zincblende semiconductors, derived from p-like atomic orbitals, has $T_{2}$ symmetry (when neglecting spin-orbit coupling). So, let's take $Γ_{v} = T_{2}$ .
The bottom of the conduction band, derived from s-like atomic orbitals, has $A_{1}$ symmetry. Let's see if a transition from the $T_{2}$ valence band to the $A_{1}$ conduction band is allowed.

Our selection rule is $Γ_{v ec} \otimes Γ_{v} \supset Γ_{c}$ . Here, that's $T_{2} \otimes T_{2} \supset A_{1}$ . We need to compute the direct product $T_{2} \otimes T_{2}$ . We can do this using the characters:

χ (T_{2} \otimes T_{2}) = [χ (T_{2})]^{2} = [3, 0, - 1, - 1, 1]^{2} = [9, 0, 1, 1, 1]

Now we use the reduction formula (which you learned in Module 2) to decompose this reducible representation. For the $A_{1}$ component:

n_{A_{1}} = \frac{1}{h} C \sum N_{C} χ_{r e d} (C) χ_{A_{1}} (C) = \frac{1}{24} [1 \cdot 9 \cdot 1 + 8 \cdot 0 \cdot 1 + 3 \cdot 1 \cdot 1 + 6 \cdot 1 \cdot 1 + 6 \cdot 1 \cdot 1] = \frac{1}{24} [9 + 3 + 6 + 6] = \frac{24}{24} = 1

The decomposition of $T_{2} \otimes T_{2}$ is $A_{1} \oplus E \oplus T_{1} \oplus T_{2}$ . Since this decomposition does contain $A_{1}$ , the transition from a $T_{2}$ valence band to an $A_{1}$ conduction band is allowed. This corresponds to the fundamental band gap absorption in many direct-gap semiconductors.

Polarization Dependence

The power of this method extends further. Notice in the $T_{d}$ table that $(x, y, z)$ together form the basis for $T_{2}$ . In lower symmetry groups, different components can transform according to different irreps. For example, in a tetragonal crystal ( $D_{2 d}$ symmetry), $z$ might transform as one irrep (e.g., $B_{2}$ ) while $(x, y)$ transform as another (e.g., $E$ ).

An annotated character table for the C3v point group. Note how different functions are associated with different irreducible representations (Mulliken labels). The linear functions 'z' and '(x, y)' are associated with A1 and E, respectively. This determines the symmetry of the dipole operator for different light polarizations.

This means that transitions allowed by the $Γ (z)$ operator will be activated by light with its electric field polarized along the crystal's z-axis ( $E ∥ z$ ). Transitions allowed by the $Γ (x, y)$ operator will be activated by light polarized in the xy-plane ( $E ⊥ z$ ). This results in polarization-dependent absorption, a powerful experimental tool for probing the symmetries of electronic states.

The following video, though focused on vibrational spectroscopy (IR and Raman), perfectly illustrates the logic of using a character table to determine which modes are active. The principle is identical for electronic transitions.

Symmetry: IR and Raman Spectroscopy

This video by Melissa Garrett demonstrates how to use a character table to determine spectroscopic activity.

Please watch the final two segments. The first, assigning IR and Raman activity, explains how the presence of x, y, z (for IR/dipole transitions) or quadratic functions (for Raman) next to an irrep determines its activity. The second, summary of rules, provides a concise summary. This is a practical application of the selection rule: if \Gamma_i is the ground state (\Gamma_1), a transition to \Gamma_f is allowed if \Gamma_f itself has the same symmetry as one of the components of \vec{r}.

For a more advanced and concrete application in condensed matter, the resource LINK you looked at earlier shows a band structure diagram for chalcopyrite (Fig 4.9), where allowed transitions for different polarizations ( $E ⊥ c$ and $E ∥ c$ ) are explicitly marked with different arrows, directly demonstrating the outcome of these polarization-dependent selection rules.

Conclusion

In this lesson, we have connected our abstract understanding of crystal symmetry to a concrete, measurable phenomenon: the absorption of light. By combining Fermi's Golden Rule with the vanishing integral theorem, we have derived a powerful group-theoretical tool for predicting the selection rules for electronic transitions.

Key Takeaways:

The probability of an optical transition is governed by the matrix element $⟨ ψ_{c} ∣ r ∣ ψ_{v} ⟩$ .
Group theory dictates that this transition is allowed only if the direct product of representations $Γ_{c}^{*} \otimes Γ_{v ec} \otimes Γ_{v}$ contains the totally symmetric irrep, $Γ_{1}$ .
For direct transitions ( $k^{'} = k$ ), this allows us to use the character table of the point group at $k$ to determine allowed transitions between bands of known symmetry.
Because different components of the position operator $r$ can transform according to different irreps, we can predict which transitions will be active for different polarizations of light.

This lesson concludes our module on symmetries in crystalline solids. You have now seen how the principles of group theory—from discrete translations and point groups to the more abstract time-reversal symmetry—provide a rigid and predictive framework for understanding the electronic and optical properties of materials.

In our next module, we will shift our focus from the discrete symmetries of crystals to the continuous symmetries that underlie the fundamental laws of nature, beginning with the celebrated Noether's theorem.

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