Character tables and reducible representations

Reading a character table

For C_{3v} (ammonia):

• Columns are classes of operations; g_c is the number in each class, and the order is h = \sum g_c = 6.
• Rows are irreducible representations. The character under E is the degeneracy: 1 for A/B, 2 for E, 3 for T.
• Mulliken labels: A is symmetric and B antisymmetric under the principal rotation; subscripts g/u show behaviour under i; primes show behaviour under \sigma_h.
• The right-hand columns list which irreducible representation the Cartesian axes, rotations and quadratic functions belong to; this is what links tables to IR and Raman activity.

Building a reducible representation

Choose a basis (bond vectors, orbitals, atomic displacements) and, for one operation per class, ask how much of each basis member ends up on itself: +1 unchanged, −1 reversed, 0 moved elsewhere. Sum these to get the character \chi_\Gamma.

The reduction formula

Worked example: N–H stretches. The three bond vectors give \Gamma = (3, 0, 1): all stay under E, all move under C_3, one stays under each \sigma_v.

• n(A_1) = \frac{1}{6}(3\cdot1 + 2\cdot0\cdot1 + 3\cdot1\cdot1) = 1
• n(A_2) = \frac{1}{6}(3 + 0 - 3) = 0
• n(E) = \frac{1}{6}(3\cdot2 + 0 + 0) = 1

So \Gamma = A_1 + E: one symmetric stretch and a degenerate pair, all IR and Raman active.

For all motions of a molecule, use unmoved atoms multiplied by a contribution per atom (3 for E, −1 for C_2, 1 for \sigma, −3 for i) and subtract translations and rotations to leave the vibrations. For water this gives 2A_1 + B_2.

Projection operators

To find the actual combinations, apply

to one basis function, summing over every operation R. For ammonia, projecting r_1 gives the A_1 mode \tfrac{1}{\sqrt3}(r_1+r_2+r_3) and an E component \tfrac{1}{\sqrt6}(2r_1-r_2-r_3); its orthogonal partner is \tfrac{1}{\sqrt2}(r_2-r_3). The same procedure yields ligand SALCs for MO diagrams.

Behind the reduction formula lies the great orthogonality theorem: irreducible representations behave like orthogonal vectors in a space whose dimension is the number of classes. This is why decomposition is unique and why the sum of the squared dimensions equals the order of the group. Researchers rarely reduce representations by hand, but understanding the structure matters when software output must be interpreted or when a problem falls outside standard tables.

Direct products decide whether integrals vanish: \langle\psi_f|\hat{O}|\psi_i\rangle can be non-zero only if \Gamma_f\otimes\Gamma_O\otimes\Gamma_i contains the totally symmetric representation. This single rule generates electric-dipole, Raman, magnetic-dipole and two-photon selection rules and the symmetry conditions for vibronic coupling. The symmetric part of the direct product of a degenerate state with itself determines which distortions are Jahn–Teller active.

Correlation tables follow representations as symmetry is lowered, from O_h to D_{4h} on tetragonal elongation, or from a molecular group to a crystal site group. They are used to predict band splittings, to assign spectra of distorted complexes and to map orbital labels from computational subgroups back to the true point group.

For heavy-atom and open-shell systems, double groups extend character tables to spinor states, essential for spin–orbit-coupled excited states, lanthanide crystal-field levels and magnetic anisotropy. Magnetic (Shubnikov) groups and space-group representations, with labels at each wavevector in the Brillouin zone, play the equivalent role for phonons, band structures and topological materials, where symmetry indicators computed from characters can flag non-trivial band topology.