character table

Every point group comes with a small, fixed grid of numbers that summarizes everything its symmetry implies — the character table. If the point group is the molecule's passport, the character table is the page of stamps and codes that you actually read to get work done. There is one standard table per point group (C2v, C3v, Td, Oh, and so on), and they are printed in the back of every inorganic textbook, ready to be looked up rather than derived from scratch.

The table is laid out like a spreadsheet. Across the top run the symmetry operations of the group, grouped into classes (operations that are equivalent by symmetry, such as the three C2 rotations of a group sharing one column). Down the left side run the irreducible representations — the fundamental symmetry types, given short Mulliken labels like a1, a2, b1, b2 (singly-degenerate), e (doubly-degenerate), or t (triply-degenerate). The body of the table holds the characters: a single number for each representation under each operation, telling you whether something of that symmetry type is left unchanged (+1), flipped in sign (-1), or partly mixed (0 or larger numbers for degenerate sets). Two extra columns on the right list which mathematical functions transform as each row — the linear functions x, y, z (which behave like p orbitals and a dipole) and the quadratic functions like z^2, x^2-y^2, xy (which behave like d orbitals and Raman activity).

The character table is the workhorse that turns symmetry into predictions. Reading off the x, y, z column tells you which vibrations are infrared-active; reading the quadratic column tells you which are Raman-active; matching orbital symmetries lets you build the right bonding combinations for a molecular-orbital or ligand-field diagram. You rarely need to construct a character table — the skill is learning to read one fluently, because almost every quantitative result in this field is obtained by looking up the relevant row or column.