Character Tables and What Symmetry Predicts

You have assigned point groups from symmetry operations. A character table belongs to one point group. Each irreducible representation has a row. Character columns identify classes of operations. A character is the trace, the sum of diagonal entries, of a transformation matrix. Separate function columns can list coordinates, rotations or quadratic functions; those columns are not operation classes.

For C2v, choose z along C2 and name the mirror planes xz and yz. Use operation order E, C2, σv(xz), σv(yz). Each class contains one operation. The table below gives the four irreducible rows in that order. Every row here is one-dimensional, so a copy of any row contributes one to a representation's dimension.

Reduction finds each irreducible row's multiplicity in Γ. For real C2v characters, multiply matching characters, add the four products and divide by 4. For other finite groups, weight each class's product by its size, use the complex conjugate of the irreducible character, sum and divide by the number of operations. This finite-sum recipe does not apply directly to Dinfh, which has infinitely many operations. Its CO2 mode labels are supplied later, not derived here.

N atoms provide 3N Cartesian displacement coordinates. A nonlinear molecule has three translations and three rotations, leaving 3N - 6 vibrational modes. A linear molecule has only two independent rotations, leaving 3N - 5 modes. These formulas apply to molecules with at least two atoms. A nonlinear five-atom molecule has 15 - 6 = 9 modes. Count degenerate modes separately even when their frequencies match.

To construct a character of Γ3N, count atoms left at their own positions and multiply by the trace of their three-coordinate transformation. Exchanged atoms contribute zero. A fixed atom contributes 3 for E, -1 for a C2 turn and +1 for a mirror reflection. For example, E fixes all four atoms of ammonia, so its Γ3N character is 4 x 3 = 12.

Reduce Γ3N, then subtract translations and rotations coefficient by coefficient. In C2v, translations are A1+B1+B2; rotations are A2+B1+B2. The remainder is Γvib. Add its multiplicities, weighted by row dimensions, to count modes. For SO2 in yz with C2 along z, Γvib=2A1+B2: symmetric stretch and bend are A1; antisymmetric stretch is B2. The two A1 modes share a label, not necessarily a frequency.

A fundamental is allowed in electric-dipole infrared absorption when its mode changes the dipole to first order; its symmetry matches x, y or z. First-order Raman activity needs a first-order polarizability change; the mode's symmetry matches quadratic functions such as x² or yz. A row containing both a coordinate and a quadratic function permits both activities. All SO2 modes meet both tests. Allowed activity does not guarantee a detectable peak.

An inversion centre gives mutual exclusion: the same mode cannot be both electric-dipole infrared active and first-order Raman active. The g labels are even under inversion; u labels are odd. Dipole components are odd, while polarizability components are even. Different modes may share a frequency. The rule neither forbids that coincidence nor requires every mode to be active in one method.