Irreducible Representations

Introduction

A large orbital or displacement basis usually contains several patterns of symmetry behaviour. The smallest blocks that cannot be split further while respecting every operation are irreducible representations, or irreps. Their labels are the alphabet of character tables and molecular-orbital diagrams. Understanding what the labels mean prevents a common mistake: treating A, B, E and T as arbitrary names rather than indicators of transformation dimension and symmetry relations.

Core explanation

A function space is invariant under a group when applying any group operation to a function in the space produces another function in the same space. A representation is reducible if it can be expressed as a direct sum of smaller invariant spaces using one common basis change for all its matrices. An irrep has no such proper nonzero subspace. A one-dimensional representation is necessarily irreducible. A two-dimensional representation can be irreducible when operations mix its two functions so that no single one-dimensional direction remains invariant under the entire group.

For C₃v, the x and y functions form an E irrep. A C₃ rotation mixes them, and a vertical reflection acts differently on components aligned and perpendicular to its plane. No single real direction in the xy plane is preserved as a one-dimensional representation by all C₃v operations. By contrast, z is A₁ and stands alone. A ligand basis of three H 1s functions in ammonia reduces to A₁ ⊕ E: its three-dimensional representation is not irreducible, but the two-dimensional E part is.

Character-table labels follow common conventions. A and B are one-dimensional types, E is two-dimensional and T is three-dimensional in many molecular point groups. Subscripts distinguish species by behaviour under particular operations; for example A₁ is usually totally symmetric, with character +1 for every operation. In centrosymmetric groups, g and u mark even and odd inversion parity. In groups with a horizontal mirror plane, primes mark even or odd behaviour under σh. These symbols must be interpreted from the actual table, because coordinate conventions and group structure matter.

The number of irreps equals the number of conjugacy classes. Their dimensions obey Σdᵢ² = h, where h is the number of group operations. For C₃v, h = 6 and dimensions 1, 1 and 2 satisfy 1 + 1 + 4 = 6. This both checks the table and shows why an E species is needed. For C₂v, four singleton classes and four one-dimensional irreps give 1 + 1 + 1 + 1 = 4. A proposed extra independent symmetry species would violate these constraints.

Symmetry can enforce degeneracy. If a Hamiltonian retains the full point group, states belonging to a multidimensional irrep can form degenerate partner functions, such as the E pair in a suitable molecular orbital or vibration. This is a consequence of symmetry and the Hamiltonian's invariance, not a mere coincidence of calculated numbers. However, two states from different one-dimensional irreps may occasionally have equal energies by accident, and a perturbation lowering symmetry can split an E level. Group labels describe transformation, not a guaranteed universal energy order.

In practical chemistry, reducing a representation tells us how many symmetry-adapted combinations of each type must exist. If four ligand functions reduce to A₁ + B₁ + E, dimensions 1 + 1 + 2 account for four functions. An atomic orbital or normal coordinate can then be assigned the matching symmetry species and compared. Mixing requires the same species for a symmetry-preserving Hamiltonian; spectral intensity requires a more specific direct-product test with the transition operator.

Step-by-step reasoning

Define the basis and calculate its full class-character set. Consult the group character table and decompose it into irreducible rows with nonnegative integer coefficients. Confirm that the sum of dimensions equals the original basis dimension at E. Then use symmetry species to organise interactions, while checking actual energy and overlap before claiming a strong bond or transition.

Visual explanation

Draw a three-dimensional block representing the three NH₃ hydrogen functions. Split it into a one-dimensional line labelled A₁ and a two-dimensional plane labelled E. Show C₃ leaving the line unchanged but rotating vectors within the plane. No further split of that plane remains valid under all operations.

Real-world analogy

A three-note musical chord can be decomposed into one uniform component and a two-component contrast pattern. The uniform part behaves independently under rearrangements; the contrast pair rotates into itself. This resembles A₁ plus E, although irreps are defined by exact transformation matrices rather than an auditory impression.

Real-world example

In ideal octahedral coordination chemistry, metal d orbitals are organised into an E g pair and a T₂g triple under Oh symmetry. This tells us their symmetry-enforced dimensions before ligand-field energies are calculated. Ligands determine the energy splitting between those sets, but the two-versus-three grouping follows from the point-group representation.

Why?

Why not classify each orbital by its appearance alone? An operation may mix several orbitals, so one real orbital may not be a complete invariant object. Irrep analysis finds the smallest closed function spaces and prevents assigning incompatible one-dimensional labels to partner functions that must be treated together.

Common misconception

An E irrep is not the identity operation E. One is a two-dimensional symmetry species, the other a group element. Also, two-dimensional symmetry does not mean exactly two electrons occupy the orbital set; representation dimension counts independent partner functions. Energy degeneracy follows only when the Hamiltonian preserves the relevant symmetry and no symmetry-breaking field intervenes.

Worked example

Suppose a group has six operations and three conjugacy classes. The C₃v character table has irreducible dimensions 1, 1 and 2. Their squared dimensions sum to 1² + 1² + 2² = 6, matching the group order. The ammonia H 1s representation has dimension three and reduces to A₁ + E, whose dimensions 1 + 2 = 3. The checks address different statements: squared dimensions test the complete table; ordinary dimensions test one basis decomposition.

Quick check

1. Can an E irrep be split into two one-dimensional irreps while keeping all C₃v operations represented correctly? Answer: No. By definition its two-function space is irreducible under the full group. 2. How many irreducible rows should a group with five conjugacy classes have? Answer: Five irreducible representations, one per class.

Exam focus

Distinguish representation dimension, group order and number of classes. Apply both dimension checks in their proper contexts. Explain irreducibility using a common invariant subspace for all operations, rather than saying merely that one displayed matrix cannot be diagonalised.

Advanced insight

Schur's lemma explains why a symmetry-preserving operator acts in a constrained way within irreducible spaces and has zero couplings between inequivalent irreps. For a Hamiltonian, this yields block structure and symmetry-enforced partner degeneracy under suitable conditions. The lemma is the algebraic foundation behind many qualitative molecular-orbital selection rules.

Summary

Irreps are the smallest function spaces closed under every point-group operation. Their dimensions and characters classify orbitals, vibrations and states. A reducible basis decomposes into a direct sum of irreps, whose dimensions account for the full basis. Class count and squared-dimension relations check a character table, while symmetry labels constrain mixing and degeneracy.

Practice questions

1. A six-dimensional ligand basis reduces to A₁ + A₂ + 2E in C₃v. Is its dimension consistent? Answer: Yes. The dimensions add as 1 + 1 + 2(2) = 6. This alone does not prove the decomposition, but it passes the identity-character check. 2. Why may two apparently separate x- and y-like orbitals need one E label? Answer: C₃v rotations mix x and y into linear combinations. The pair is a single two-dimensional invariant space that cannot be decomposed consistently under all operations. 3. If a distortion removes C₃ rotation from ammonia, must an E vibrational pair remain exactly degenerate? Answer: No. The original symmetry protection is lost, so the two components may split in frequency under the lower-symmetry geometry.