Representations and Matrices

Introduction

Geometric operations become useful in quantum chemistry when we know how they act on wavefunctions, coordinates or atomic displacements. A representation assigns a matrix to each group operation on a selected basis of functions. Those matrices may be large for many orbitals or as small as one number for a single symmetry species. The matrix viewpoint converts pictures of rotated molecules into algebra that can predict exact zeros, degeneracies and block structure in a quantum calculation.

Core explanation

Choose a basis f₁, f₂, …, fₙ. Applying a symmetry operation R to each basis function produces a linear combination of the same basis functions if the basis space is closed under the group. The coefficients make the columns of an n × n matrix D(R), subject to a declared convention for active versus passive transformations. This matrix describes the operation on the function space, not a change in the number of atoms. For a valid representation, D(E) is the identity matrix and D(RS) = D(R)D(S) under a consistent right-first operation convention.

A simple two-function example is a pair of equivalent ligand orbitals h₁ and h₂. An operation exchanging them sends h₁ to h₂ and h₂ to h₁. In basis (h₁,h₂), its matrix has zero diagonal and ones off diagonal: [[0,1],[1,0]]. Applying it twice gives the identity matrix, matching the geometric fact that two swaps restore the start. The matrix trace is zero. In the transformed basis h₊ = (h₁+h₂)/√2 and h₋ = (h₁−h₂)/√2, the same operation has diagonal matrix diag(1,−1). Its trace remains zero, demonstrating that the character is independent of basis.

Coordinate functions provide familiar matrices. For a 180° rotation about z, x → −x, y → −y and z → z. On basis (x,y,z), D(C₂z) is diag(−1,−1,+1). Reflection in the xz plane maps y → −y while leaving x and z unchanged, giving diag(+1,−1,+1). A 120° rotation around z mixes x and y; its xy block contains cosine and sine of 120°, rather than separate plus and minus signs for x and y. This mixing shows why two-dimensional representations arise in groups such as C₃v.

A representation can often be made block diagonal by changing basis. The exchange matrix for h₁ and h₂ becomes two one-dimensional blocks in the h₊, h₋ basis. A larger set of ligand orbitals may separate into several smaller invariant spaces. Such a representation is reducible. If no common basis change divides the representation into smaller invariant blocks for all operations, the representation is irreducible. One must test all group operations: a matrix that happens to be diagonal for a single operation does not prove the whole representation is reducible in that chosen basis.

The representation dimension equals the number of basis functions, and the character of an operation is the trace of its matrix. At identity, the character is n because D(E) is the n × n identity. For a permutation basis of equivalent local orbitals, the character can often be found by counting basis functions left on themselves, with attention to any sign reversal or orbital orientation. For vector displacements, a fixed atom can contribute the trace of the coordinate transformation rather than simply one. These distinctions matter when building vibrational representations.

The molecular Hamiltonian is invariant under exact molecular symmetry operations for a fixed ideal geometry. When a basis is organised into irreducible symmetry blocks, matrix elements between incompatible blocks vanish. This saves computational work and supplies meaningful orbital labels. However, a basis must be complete enough for its intended calculation and the geometry must retain the assumed symmetry. A distorted structure can couple blocks that were distinct at higher symmetry.

Step-by-step reasoning

Specify basis order and operation convention. Transform the first basis function, write its coefficients as the first matrix column and repeat for each function. Verify D(E) is identity and check a known operation product using matrix multiplication. Compute the trace, then try a physically motivated sum-and-difference basis if the matrix appears reducible.

Visual explanation

Draw two identical ligand circles labelled h₁ and h₂. An arrow exchanging them corresponds to a two-by-two matrix with off-diagonal ones. Below, redraw the basis as a same-sign sum and opposite-sign difference; the exchange now leaves the sum alone and reverses the difference. Put the diagonal matrix beside this second drawing.

Real-world analogy

Changing from east–north coordinates to coordinates along and across a road does not change the underlying movement; it changes the numbers used to describe it. Similarly, changing orbital basis can make a symmetry matrix simpler without changing the physical operation. Its trace remains the same because that number describes the transformation rather than a particular coordinate choice.

Real-world example

In a calculation on water, two H 1s orbitals can be replaced by symmetric and antisymmetric combinations. Hamiltonian and overlap matrices written in the new basis separate compatible interactions with O 2s/2p z from those with O 2p y under the earlier coordinate convention. This avoids computing or interpreting symmetry-forbidden mixing as if it were a small accidental numerical value.

Why?

Why require D(RS) = D(R)D(S)? A representation should preserve the structure of the symmetry group. If two geometric actions compose to a third, their matrices must compose to the matrix of that third action. Without this product rule, arbitrary matrices could be assigned to operation names but would not support valid character tables or selection-rule calculations.

Common misconception

The same geometric operation does not have one universal matrix. Its matrix depends on the chosen basis, its order and the active/passive convention. What is basis-independent is the character within equivalent representations. Also, a trace of zero does not mean the operation has no effect; a swap matrix has trace zero while exchanging two functions completely.

Worked example

Let R exchange h₁ and h₂. Then D(R) = [[0,1],[1,0]], so D(R)² = [[1,0],[0,1]] = D(E), as required for a twofold exchange. The normalised combinations h₊ = (h₁+h₂)/√2 and h₋ = (h₁−h₂)/√2 are eigenfunctions of R with eigenvalues +1 and −1. In that basis D(R) = diag(+1,−1). Both matrices have character 0 and dimension 2.

Quick check

1. What is the character of E for a five-function basis? Answer: Five, because D(E) is the five-by-five identity matrix with trace five. 2. Does changing basis alter a representation's character for a given operation? Answer: No. Equivalent matrices are related by similarity and have the same trace.

Exam focus

Write basis order and operation order before constructing matrices. Distinguish matrix dimension from character. Use the full group to decide reducibility, and verify at least one multiplication identity so sign or column-convention mistakes are caught early.

Advanced insight

For real orthonormal coordinate bases, point-group operation matrices are orthogonal and preserve vector lengths. Orbital bases need not be orthonormal initially, and quantum-chemical calculations often use an overlap matrix. Symmetry still acts linearly, but transformations and block diagonalisation must respect the basis metric when quantitative matrix equations are solved.

Summary

A representation maps each symmetry operation to a matrix acting on a selected function basis while preserving operation products. Basis changes can expose invariant blocks but leave characters unchanged. Matrices turn geometric symmetry into algebraic constraints on orbital mixing, vibrations and quantum calculations, provided the basis and geometry are specified consistently.

Practice questions

1. What matrix represents a C₂ rotation about z on coordinate basis (x,y,z), and what is its trace? Answer: diag(−1,−1,+1), because x and y reverse while z remains unchanged. Its trace is −1. 2. Why is the two-ligand exchange representation reducible? Answer: The h₊ and h₋ combinations each transform within their own one-dimensional spaces under the exchange, so a basis change splits the two-dimensional matrix into two blocks. 3. If a numerical orbital calculation finds weak coupling between two functions declared symmetry-incompatible, what should be checked? Answer: Check the actual geometry, basis orientation, numerical tolerances and symmetry labels. A real distortion may remove the prohibition, while an ideal-symmetry calculation should give zero apart from numerical error.