Character Tables for Degeneracy and Parity

1. The Quantum Mechanical Framework: Symmetry, Degeneracy, and Parity

The connection between symmetry and degeneracy is a cornerstone of quantum mechanics. If a Hamiltonian $H$ is invariant under the operations $\hat{g}$ of a symmetry group $G$, then $[\hat{H},\hat{g}]=0$ for all $g\in G$. From this, a profound consequence follows: the set of degenerate eigenfunctions corresponding to a single energy eigenvalue forms a basis for an irreducible representation of the group $G$.

This means if you know the symmetry group of your system and its character table, you can immediately predict the possible degeneracies of its energy levels—they must be equal to the dimensions of the group's irreps (the character of the identity element, $\chi (E)$).

Parity in Centrosymmetric Systems

A special case arises for systems that possess an inversion center, i.e., whose symmetry group $G$ contains the inversion operator $i$. Such groups are called centrosymmetric. Since $i^2=E$, the eigenvalues of the inversion operator must be $\pm 1$. Eigenfunctions of a centrosymmetric Hamiltonian must also be eigenfunctions of the inversion operator.

• If $\hat{i}\psi (\mathbf{r})=+\psi (\mathbf{r})$, the wavefunction has even parity and is labeled gerade (German for "even"), denoted by a subscript g.
• If $\hat{i}\psi (\mathbf{r})=-\psi (\mathbf{r})$, the wavefunction has odd parity and is labeled ungerade ("odd"), denoted by a subscript u.

As we will see, the character tables for centrosymmetric groups like O_h explicitly classify their irreps with 'g' and 'u' subscripts, directly telling us the parity of any state that transforms according to that irrep.

2. Splitting of Atomic Orbitals: Crystal Field Theory

Let's consider a practical application from your condensed matter physics course: an atom or ion in a crystal. In free space, the Hamiltonian for the atom's electrons is spherically symmetric; its symmetry group is O(3) (or SO(3) if we ignore inversion). This high symmetry leads to the familiar (2l+1)-fold degeneracy of atomic orbitals with angular momentum $l$.

When this atom is placed in a crystal, the surrounding ions create an electrostatic potential, the crystal field, which has the symmetry of the atom's local site in the lattice, not the full O(3) group. This reduction in symmetry partially or fully lifts the orbital degeneracy. Group theory allows us to predict precisely how these levels will split without solving the Schrödinger equation.

The procedure is as follows: The set of (2l+1) degenerate atomic orbitals forms a basis for a (2l+1)-dimensional irreducible representation of O(3). When the symmetry is lowered to a point group $G\subset O(3)$, this representation becomes a (generally) reducible representation of $G$. Our task is to decompose this reducible representation into the irreps of $G$.

a) The Octahedral Field (O_h)

Imagine an ion at the center of an octahedron of negative charges (ligands). This is a very common coordination in crystals. The symmetry group is O_h. Let's determine how the five d-orbitals ($l=2$) split in this environment. The five orbitals form a 5-dimensional representation of O_h, which we can call ${\Gamma }_d$. To decompose it, we would need its characters, which can be derived using formulas for the characters of O(3) representations.

The group theoretical statement is:

Here, $D^{(l=2)}$ is the $l=2$ irrep of O(3), and the arrow signifies restriction to the O_h subgroup. The result tells us that the five d-orbitals split into two distinct energy levels: one that is doubly degenerate (transforming as E_g) and one that is triply degenerate (transforming as T_2g).

The 'g' subscript tells us that both sets of resulting orbitals have even parity, which is correct, since the original d-orbitals have parity $(-1{)}^l=(-1{)}^2=+1$.

A physical understanding of this splitting is invaluable. Let's visualize why these particular orbitals group together.

As the video explains, in an octahedral geometry, the ligands lie along the Cartesian axes.

• The $d_{z^2}$ and $d_{x^2-y^2}$ orbitals have their lobes pointing directly at the negative ligands. The electrostatic repulsion is strong, so these orbitals are raised in energy. They are degenerate by symmetry and form the basis for the E_g irrep.
• The $d_{xy}$, $d_{xz}$, and $d_{yz}$ orbitals have their lobes pointing between the ligands. The repulsion is weaker, so these orbitals are lower in energy relative to the E_g set. They are degenerate by symmetry and form the basis for the T_2g irrep.

b) The Tetrahedral Field (T_d)

Now let's consider a tetrahedral arrangement of ligands, with symmetry group T_d. A key difference from O_h is that T_d does not have a center of inversion.

The decomposition proceeds similarly, but the result is different. For d-orbitals ($l=2$), the decomposition into irreps of T_d is:

Notice the absence of the 'g' and 'u' subscripts. Since T_d is not centrosymmetric, its irreps (and the wavefunctions that transform according to them) do not have a defined parity. The 5-fold degenerate d-level splits into a doubly degenerate E level and a triply degenerate T_2 level.

In a tetrahedral geometry, none of the d-orbitals point directly at the ligands. However, the $d_{xy}$, $d_{xz}$, and $d_{yz}$ orbitals (the T_2 set) are oriented more closely to the ligands than the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals (the E set). Therefore, in a tetrahedral field, the T_2 orbitals are higher in energy than the E orbitals, the reverse of the octahedral case.

3. A General Tool for Level Splitting

This method of decomposing O(3) irreps is completely general. For any atomic level (s, p, d, f, ...) and any crystal symmetry (cubic, tetragonal, trigonal, etc.), we can predict the splitting pattern. The resource LINK contains a comprehensive table summarizing these results.

Let's use this table to answer a new question: How does an atomic f-level ($l=3$) split in a tetrahedral (T_d) field?

Looking at Table 6.6, we find the row for $l=3$ (f-orbitals) and the column for T_d symmetry. The entry is A_2 ⊕ T_1 ⊕ T_2.

• A_2 is a 1-dimensional irrep (singly degenerate).
• T_1 is a 3-dimensional irrep (triply degenerate).
• T_2 is a 3-dimensional irrep (triply degenerate).

The sum of the dimensions is 1 + 3 + 3 = 7, which correctly matches the original degeneracy of the f-orbitals ($2l+1=2(3)+1=7$). Thus, group theory predicts that the 7-fold degenerate f-level will split into three distinct energy levels: a non-degenerate level (A_2 symmetry) and two different triply-degenerate levels (T_1 and T_2 symmetries).

Conclusion

In this lesson, we have moved from interpreting character tables to actively using them to solve a significant physical problem. You have seen how the abstract machinery of group theory provides concrete, quantitative predictions about the quantum states of atoms in crystals, a central topic in condensed matter physics.

Key Takeaways:

• Degenerate eigenfunctions of a Hamiltonian form a basis for an irreducible representation of its symmetry group; the degeneracy equals the irrep's dimension.
• In centrosymmetric groups (like O_h), irreps are labeled with g (gerade/even) or u (ungerade/odd) to indicate their parity under inversion. In non-centrosymmetric groups (like T_d), states do not have a defined parity.
• The splitting of degenerate atomic orbitals in a crystal field can be determined by decomposing the corresponding irrep of O(3) into the irreps of the crystal's point group.
• For d-orbitals ($l=2$), the splitting is $E_g\oplus T_{2g}$ in an O_h field and $E\oplus T_2$ in a T_d field, with the energy ordering being inverted between the two cases.

You are now equipped to predict the degeneracy and parity of energy levels in highly symmetric systems. Our next step will be to consider transitions between these levels. We will see how group theory provides powerful "selection rules" that tell us whether a transition from a state with one symmetry to a state with another symmetry is allowed or forbidden. This will involve the concept of the direct product of representations.