Degeneracy and Symmetry

Introduction

In a one-dimensional box every energy level belongs to exactly one wavefunction. Move to a square or a cube, and something new appears: different wavefunctions can have exactly the same energy. This is degeneracy , and it is everywhere in chemistry. The three 2p orbitals of an atom, the five d orbitals of a free metal ion and the paired π orbitals of benzene are all degenerate. The particle in a box shows, in the simplest possible way, that degeneracy is a direct consequence of symmetry.

Core explanation

Degeneracy in a square box. For a square of side L the energy is

E = (h²/8mL²)(nₓ² + n y²)

The states (1, 2) and (2, 1) both have nₓ² + n y² = 5, so they have the same energy: this level is doubly degenerate. The wavefunctions are different — one has a nodal line parallel to y, the other a nodal line parallel to x — but one is simply the other rotated by 90°. The lowest levels of a square box, in units of h²/8mL², are:

Energy States Degeneracy --- --- --- 2 (1, 1) 1 5 (1, 2), (2, 1) 2 8 (2, 2) 1 10 (1, 3), (3, 1) 2 13 (2, 3), (3, 2) 2

Degeneracy in a cubic box. For a cube, E = (h²/8mL²)(nₓ² + n y² + n z²):

Energy States Degeneracy --- --- --- 3 (1, 1, 1) 1 6 (2, 1, 1) and permutations 3 9 (2, 2, 1) and permutations 3 11 (3, 1, 1) and permutations 3 12 (2, 2, 2) 1 14 (3, 2, 1) and permutations 6

When all three quantum numbers differ, there are 3! = 6 arrangements; when two are equal, 3; when all are equal, 1.

Symmetry as the cause. Rotating a cube by 90° about any axis, or reflecting it, leaves it unchanged. Such an operation turns one wavefunction into another with the x, y and z labels interchanged. Because the Hamiltonian is unchanged by the operation, the transformed wavefunction must have the same energy. The higher the symmetry, the more operations there are and the larger the possible degeneracies.

Lifting degeneracy. Stretch the cube slightly along z so that c > a = b. Now

E = (h²/8m)[(nₓ² + n y²)/a² + n z²/c²]

The three states of the E = 6 level split: (1, 1, 2) drops in energy because the long z direction costs less, while (2, 1, 1) and (1, 2, 1) stay together as a pair, since x and y are still equivalent. Lowering symmetry partially removes degeneracy, exactly as a distorting ligand field splits d orbitals in a metal complex.

Accidental degeneracy. Some equalities have no geometric origin. In a cube, (3, 3, 3) and (5, 1, 1) both give 27, so these four states are degenerate even though no rotation of the cube converts (3, 3, 3) into (5, 1, 1). In a square box, (1, 7) and (5, 5) both give 50. Such degeneracies arise from number theory rather than from spatial symmetry.

Formulae

Square box: E = (h²/8mL²)(nₓ² + n y²). Cubic box: E = (h²/8mL²)(nₓ² + n y² + n z²). Degeneracy of a cube level with three different quantum numbers = 6; with two equal = 3; with all equal = 1.

Step-by-step reasoning

To find the degeneracy of a cubic-box level:

1. Compute the sum nₓ² + n y² + n z² for the level. 2. List every set of positive integers that gives this sum. 3. For each set, count the distinct permutations (6, 3 or 1). 4. Add the permutations for all sets to obtain the total degeneracy. 5. Note whether any sets are unrelated by permutation, signalling accidental degeneracy.

Visual explanation

Draw the (2, 1, 1), (1, 2, 1) and (1, 1, 2) wavefunctions as cubes cut in half by a single nodal plane perpendicular to x, y and z respectively. Each drawing is the others turned on its side. Since turning a cube does not change it, the three states must share the same energy.

Real-world analogy

A square table with four identical chairs offers the same view from each chair after a quarter turn. For a rectangular table the long-side chairs and end chairs differ. Symmetry makes positions equivalent; reducing symmetry makes them distinct.

Real-world example

In an octahedral transition-metal complex, the five d orbitals split into a triply degenerate t₂g set and a doubly degenerate e g set. If the octahedron is elongated along one axis, as in the Jahn–Teller effect seen in many copper(II) complexes, these sets split further, just like the stretched cube.

Why?

Why do degenerate states matter chemically? Degeneracy determines how many electrons can occupy a level, the magnetic properties of molecules and the populations of states at thermal equilibrium, since a level of degeneracy g contributes g times the Boltzmann factor to the partition function.

Common misconception

"Degenerate states are the same state." Degenerate wavefunctions have the same energy but different shapes and nodal patterns; they are linearly independent and can each hold electrons. Any linear combination of them is also a valid eigenfunction with the same energy.

Worked example

Question: Find the degeneracy of the level with nₓ² + n y² + n z² = 27 in a cubic box.

Reasoning: Sets of positive integers whose squares sum to 27: (3, 3, 3) gives 9 + 9 + 9 = 27, with 1 arrangement; (5, 1, 1) gives 25 + 1 + 1 = 27, with 3 arrangements. No other sets work.

Answer: Degeneracy = 1 + 3 = 4, of which the grouping of (3, 3, 3) with (5, 1, 1) is accidental.

Quick check

1. What is the degeneracy of the level with energy 14h²/8mL² in a cubic box? Answer: Six, from the six permutations of the quantum numbers (3, 2, 1).

Exam focus

Tables of cubic-box levels and degeneracies are standard. Show your permutation counting and state clearly that degeneracy arises from symmetry. Be ready to explain what happens when a box is distorted, and link this to orbital splitting.

Advanced insight

Group theory makes the link between symmetry and degeneracy exact: the degeneracy of symmetry-required levels equals the dimension of an irreducible representation of the point group. The cube belongs to the group Oₕ, whose irreducible representations have dimensions 1, 2 and 3, so symmetry alone never forces more than threefold degeneracy. The sixfold (3, 2, 1) level must therefore be built from more than one irreducible representation: part of its degeneracy comes from the special separable form of the box potential rather than from the geometry of the cube alone.

Summary

Degeneracy is the number of independent wavefunctions sharing an energy. Square and cubic boxes show degeneracy because rotations and reflections interchange coordinates without changing the Hamiltonian. Distorting the box lowers symmetry and splits degenerate levels. Some equalities, such as (3, 3, 3) and (5, 1, 1), are accidental. These ideas carry directly over to atomic and molecular orbitals.

Practice questions

1. List the degenerate states of a square box with energy 10h²/8mL². Answer: (1, 3) and (3, 1), a doubly degenerate level. 2. A cube is compressed along z so that c < a = b. Which member of the (2, 1, 1) set rises in energy? Answer: (1, 1, 2), because the excitation lies along the shortened z direction, which now costs more energy. 3. Give an example of accidental degeneracy in a square box. Answer: (1, 7), (7, 1) and (5, 5), all with nₓ² + n y² = 50. 4. Why does a rectangular box with Lₓ ≠ L y show no symmetry-required degeneracy? Answer: No rotation or reflection of the rectangle interchanges x and y, so no operation relates states such as (1, 2) and (2, 1), and their energies differ.