Reading Character Tables

Introduction

A character table is a compact reference, but it is easy to misuse by reading labels without checking axis conventions. Its central grid records how each irreducible symmetry species transforms under every conjugacy class. Adjacent columns often list translations, rotations and quadratic functions that share those species. These entries are not decorations: they connect group theory to molecular orbitals, vibrational selection rules and matrix elements.

Core explanation

The table heading names the point group and its operation classes. A class may be written 2C₃, meaning two operations with the same character. The number before an operation is a multiplicity in group sums; it is not a numerical character. Rows are irreducible representations such as A₁, A₂, B₁, B₂, E or T. The first character in each row, under identity, equals the dimension of that irrep. In a one-dimensional row, characters are typically +1 or −1 for real point groups. A two-dimensional row has character 2 at identity and may have other values such as −1 or 0.

For a concrete C₂v convention, take z as the C₂ axis and list columns E, C₂(z), σ(xz), σ(yz). The four irrep character rows are A₁: (1,1,1,1), A₂: (1,1,−1,−1), B₁: (1,−1,1,−1), and B₂: (1,−1,−1,1). A₁ is totally symmetric. In this convention z belongs to A₁, x to B₁ and y to B₂. A table using a different x/y placement can exchange B₁ and B₂ labels without changing the physics; always read the stated axes.

Rotational functions Rₓ, Rᵧ and R z transform as axial vectors, not ordinary coordinates. Reflections can therefore give different signs from those of x, y and z. In the stated C₂v convention, R z is A₂, Rᵧ is B₁ and Rₓ is B₂. These columns are important when subtracting whole-molecule rotational motions from a 3N displacement representation. Translational motions transform like x, y and z themselves.

Quadratic functions are products of coordinates. For the same C₂v convention, x², y² and z² are A₁; xy is A₂; xz is B₁; yz is B₂. Components of molecular polarizability transform like such quadratic functions, so their table entries help identify Raman-active vibrations. Components of an electric dipole transform like x, y and z, so the linear-coordinate entries help identify infrared activity and electronic electric-dipole transitions. A mode matching a dipole component is symmetry-allowed in IR, but intensity can still be small.

The table can also be read as a parity ledger. Under σ(yz), x changes sign but y and z do not. Thus the x row must have −1 in the σ(yz) column. Under C₂(z), x and y both change sign while z does not. Checking a few coordinate transformations is a powerful defence against copying a table with a different axis convention or misaligning its operation columns.

Character tables do not give every physical answer by inspection. For orbital mixing, compare the full symmetry species of two functions and the Hamiltonian operator. For a transition, combine initial state, operator and final state through a direct product. For vibrations, obtain mode species only after constructing the displacement representation and removing translations and rotations. The coordinate and quadratic labels tell us what activity is possible once the relevant mode species is known.

Step-by-step reasoning

Read the point-group name and operation order first. Confirm the identity-column dimensions and class multiplicities. Identify the totally symmetric row. Check how x, y and z transform under a simple operation to verify the axis convention. Then use the desired function column—linear coordinate for dipole, quadratic function for polarizability, or rotational coordinate for rigid rotation—rather than guessing from the row name alone.

Visual explanation

Draw a small C₂v table with four operation columns and four row labels. Highlight the x entry next to B₁ and shade the C₂(z) and σ(yz) cells where x changes sign. Draw an arrow from x to an electric-dipole component, and another from xy to a polarizability component. This links table entries to measurable quantities.

Real-world analogy

A train timetable is useful only after checking which direction the columns represent and what each symbol means. A character table likewise requires the point group, operation order and coordinate convention. Once oriented, a short row encodes many transformation tests that would otherwise need separate drawings.

Real-world example

Water has C₂v symmetry in a suitable coordinate frame. Its normal modes can be labelled by rows of the C₂v table. If a mode belongs to a species listed under a dipole coordinate, an IR transition can be symmetry-allowed; if it belongs under a quadratic coordinate, Raman scattering can be allowed. Water's actual frequencies and intensities still require measurements or quantitative calculation.

Why?

Why do quadratic functions appear beside irrep rows? Polarizability is a rank-two response linking induced dipole to electric field, and its components transform like products of two coordinates. A vibrational displacement that changes a polarizability component can produce Raman activity. The table's quadratic entries therefore translate abstract symmetry into a concrete spectroscopic selection rule.

Common misconception

The number 2 in 2C₃ is a class size, while a 2 under identity in an E row is an irrep dimension. Mixing these meanings causes errors in reduction formulas. Another mistake is to treat a character-table row as an energy level: symmetry species classify transformation, but do not by themselves order orbital energies.

Worked example

Use the stated C₂v columns E, C₂(z), σ(xz), σ(yz). The x coordinate is unchanged by E, reversed by C₂(z), unchanged by σ(xz) and reversed by σ(yz). Its pattern is (1,−1,1,−1), so x transforms as B₁. The product xz has the same pattern because z is unchanged by every operation; hence xz is also B₁. A B₁ mode can couple to an x-directed dipole derivative and an xz polarizability derivative by symmetry.

Quick check

1. What does the character under E tell you for an irreducible row? Answer: The dimension of that irreducible representation. 2. Which kind of table entries are used for a simple Raman-activity test? Answer: Quadratic-coordinate functions corresponding to polarizability components.

Exam focus

Copy the table's operation order and coordinate convention before calculations. Distinguish class multiplicities from characters and row dimensions. Use linear-coordinate functions for electric-dipole activity, quadratic functions for Raman activity and rotational functions when removing rigid-body motions.

Advanced insight

The table rows are basis independent even though listed example functions are coordinate dependent. A rotation of the laboratory x/y axes changes which explicit function is written under a species but cannot change a molecule's physical predictions. Direct products give the more general rule: an integral can be nonzero only if the integrand's overall representation contains the totally symmetric species.

Summary

Character tables organise operation classes into columns and irreducible symmetry species into rows. Identity characters show dimensions, while coordinate, rotation and quadratic entries identify functions with the same symmetry. Correct axis and class-size reading supports orbital classification and spectral selection rules; the table constrains what is allowed, not the numerical energy or intensity.

Practice questions

1. In the stated C₂v convention, which species contains y and yz? Answer: Both transform as B₂. The z coordinate is A₁, so multiplying y by z leaves y's sign pattern unchanged. 2. A table column is headed 3σv and a row entry under it is −1. What do the two numbers mean? Answer: Three is the number of reflections in that class. The −1 is the character of the representation for each reflection in that class. 3. Does an A₂ C₂v vibrational mode have to be IR active because it is a valid mode? Answer: No. In the stated convention x, y and z belong to B₁, B₂ and A₁, not A₂, so a pure A₂ mode lacks a symmetry-allowed first-order dipole derivative at ideal C₂v geometry.