Reading and Using Character Tables

Character tables remain the daily working tool of vibrational and electronic spectroscopists. Fourier-transform infrared and Raman instruments generate spectra whose peak counts and mutual-exclusion patterns are predicted, before any experiment is run, purely from the point group and its character table — the classic diagnostic distinguishing cis from trans metal carbonyl isomers, or confirming octahedral versus square-planar geometry in coordination chemistry, rests on counting IR- and Raman-active modes from basis-function columns exactly as demonstrated for formaldehyde above. In transition-metal and lanthanide chemistry, the quadratic and cubic basis-function columns of $O_h$, $T_d$, and lower-symmetry tables underlie the entire framework of ligand-field and crystal-field theory, explaining d-orbital and f-orbital splitting patterns that determine magnetism, color, and reactivity in catalysts, pigments, and battery cathode materials. Photocatalysis and photovoltaic materials design leans on symmetry-derived selection rules to predict which electronic transitions will be optically bright versus symmetry-forbidden (and therefore long-lived, useful for triplet-harvesting phosphors and OLED emitters). Computational quantum chemistry packages (Gaussian, ORCA, Q-Chem) automatically detect molecular point groups and use character tables internally to block-diagonalize the Hamiltonian by symmetry species, dramatically reducing the cost of self-consistent-field and post-Hartree–Fock calculations — a direct industrial payoff of the abstract classification scheme described here. Structural biology and materials crystallography use analogous space-group tables (character tables generalized to include translational symmetry) to interpret X-ray diffraction patterns, again relying on the same conjugacy-class logic that governs point groups.

Ask the tutor: How do I use a character table with the reduction formula to find which irreducible representations are contained in the reducible representation built from a molecule's atomic motions or orbitals?

A character table is best understood as the finished product of asking, for a given point group, "how many fundamentally different ways can a physical quantity respond to the group's symmetry operations?" Each fundamentally different way is an irreducible representation, and the table records how that response looks for every operation in the group. Consider $C_{2v}$, the point group of water. It has four symmetry operations: the identity $E$, a $C_2$ rotation through the molecular plane bisector, and two reflections $\sigma_v(xz)$ and $\sigma_v'(yz)$. Because $C_{2v}$ is abelian (every operation commutes with every other), each operation sits in its own class — there are four classes, and consequently four irreducible representations, each one-dimensional. The rows of the $C_{2v}$ table are labeled $A_1$, $A_2$, $B_1$, $B_2$. These letters are not arbitrary — they are Mulliken symbols, a coding scheme invented by Robert Mulliken that lets a chemist infer the character values from the label alone, without ever looking at the numbers. The letter $A$ means "symmetric under the principal rotation axis" (character $+1$ under $C_2$); the letter $B$ means "antisymmetric under the principal axis" (character $-1$ under $C_2$). Subscripts $1$ and $2$ then distinguish behavior under the secondary reflection planes, and further subscripts $g$ (gerade, even) and $u$ (ungerade, odd) appear in groups with an inversion center. Two-dimensional irreducible representations are labeled $E$ (not to be confused with the identity operation!) and three-dimensional ones $T$ (or $F$ in older texts). This labeling convention is what makes character tables scannable: an experienced spectroscopist recognizes "$B_2$" and immediately knows its parity properties without re-deriving them. The columns tell a complementary story. Operations are grouped into classes — sets of operations conjugate to one another, meaning related by $R' = X R X^{-1}$ for some group element $X$. All operations in the same class necessarily have the same character in every irreducible representation, a direct consequence of the trace being invariant under similarity transformations (since a similarity transformation is exactly a change of basis, and traces are basis-independent). This is why a table for $C_{3v}$ lists only three columns — $E$, $2C_3$, $3\sigma_v$ — rather than six columns for six individual operations: the two rotations $C_3$ and $C_3^2$ form one class, and the three vertical mirror planes form another. The integer in front of each class ($2C_3$, $3\sigma_v$) is the number of operations in that class, and it is essential for the orthogonality sums to close correctly.

The two rightmost blocks of the table are frequently the most useful part for a working chemist and the part most often skipped by beginners. The linear function column shows how $x$, $y$, $z$ (and, separately, the rotations $R_x, R_y, R_z$) transform. Because the electric dipole moment operator has components proportional to $x, y, z$, this column directly tells you the symmetry of allowed dipole transitions — the essence of infrared and UV-vis selection rules. The quadratic function column shows how products like $x^2-y^2$, $z^2$, $xy$, $xz$, $yz$ transform, which is exactly the angular dependence of the $d$-orbitals, so this column tells you the symmetry labels of $d_{z^2}$, $d_{x^2-y^2}$, $d_{xy}$, $d_{xz}$, $d_{yz}$ in any point group — indispensable in crystal-field and ligand-field theory. Cubic functions ($x^3$, $xyz$, etc.) are sometimes listed too, relevant for $f$-orbitals and hyperpolarizability selection rules. A second figure makes concrete how a table like $C_{3v}$'s condenses six operations into three classes, and how the degenerate $E$ representation carries a character of $2$ under the identity because it is two-dimensional.

The structure of every character table follows from a small set of theorems, so before using one it is worth stating explicitly what guarantees its shape and consistency.

where: $\ell_i$ is the dimension of the $i$-th irreducible representation (1 for $A,B$; 2 for $E$; 3 for $T$), and $h$ is the order of the group (total number of symmetry operations). Reading: the "sizes" of all the fundamentally different symmetry behaviors, squared and summed, always exactly account for every operation in the group — nothing is left over and nothing overlaps.

where: a class is a set of operations mutually related by conjugation, $R' = XRX^{-1}$. Reading: a character table is always square in the sense that rows (species) and columns (classes) match in count, even though the operations themselves may greatly outnumber the classes (as in $O_h$, with 48 operations in 10 classes).

where: the sum runs over all $h$ individual operations $R$ (not classes), and $\chi_i(R)$ is the character of operation $R$ in irreducible representation $i$. Reading: this is the normalization form of the The Great Orthogonality Theorem applied to a single representation against itself — every row of the table, expanded operation-by-operation, has "length" exactly $\sqrt{h}$.

where: $\chi_i(R)$ and $\chi_j(R)$ are characters of two different irreducible representations for the same operation $R$. Reading: different rows of the table are mutually orthogonal vectors when expanded over the full set of operations — this is what allows the The Reduction Formula to cleanly extract how many times each irreducible representation appears inside a large reducible representation built from atomic motions or orbitals.

where: $\chi(E)$ is the character of the identity operation, always equal to the dimension $\ell$ of the representation. Reading: the identity column of a character table is simply the list of dimensions — $1,1,1,1$ for $C_{2v}$; $1,1,2$ for $C_{3v}$; $1,1,1,1,2,2,3,3$ for $O_h$'s point group — and this column alone lets you verify $\sum \ell_i^2 = h$ before doing anything else.

Example 1 — Foundational: classifying the 2p orbitals of nitrogen in ammonia (point group $C_{3v}$). Ammonia has $C_{3v}$ symmetry. Consider the nitrogen $2p_x$, $2p_y$, $2p_z$ orbitals and ask which irreducible representations they belong to, reading directly off the basis-function columns derived above. The $C_{3v}$ table lists $z$ under $A_1$, and $(x,y)$ together under $E$ (they transform as a degenerate pair because a $C_3$ rotation mixes $p_x$ and $p_y$ into linear combinations of each other rather than leaving each invariant). Therefore:

This immediately explains the bonding pattern in NH$_3$: the nitrogen lone pair, which points along the $C_3$ axis, is built predominantly from the $A_1$-symmetric $2p_z$ (mixed with $2s$, also $A_1$), while the two N–H bonding combinations that are degenerate by symmetry draw on the $E$-symmetric $(p_x,p_y)$ pair. Because $A_1 \ne E$, the lone pair orbital cannot mix with the degenerate bonding pair by symmetry alone — a result obtained without ever writing an explicit wavefunction or evaluating an overlap integral, purely from the table's basis-function column.

Limiting-case check: if the molecule flattens toward $D_{3h}$ (trigonal planar, as in a hypothetical planar NH$_3$ transition state), $p_z$ becomes $A_2''$ (antisymmetric under $\sigma_h$) while $(p_x,p_y)$ remain a degenerate $E'$ pair — the qualitative story survives, but with an additional gerade/ungerade-like label from the new horizontal mirror plane, exactly as expected when a $\sigma_h$ is added to the group.

Example 2 — Applied: infrared activity of the bending mode of formaldehyde (point group $C_{2v}$). Formaldehyde H$_2$CO belongs to $C_{2v}$. Its three atoms other than carbon and the molecular geometry give it six vibrational degrees of freedom (six normal modes, since $3N-6 = 3(4)-6=6$). Suppose group-theoretic reduction of the full $3N$-dimensional Cartesian representation (a calculation that uses The Reduction Formula and is not repeated here) yields the vibrational representation

To decide which of these six modes are infrared active, compare each species against the linear ($x,y,z$) basis-function column of the $C_{2v}$ table constructed earlier: $A_1$ carries $z$, $B_1$ carries $x$, $B_2$ carries $y$. A vibrational mode is infrared active if and only if its symmetry species matches the symmetry of at least one dipole component, because the transition moment integral

is nonzero only when the integrand as a whole contains the totally symmetric representation $A_1$ — guaranteed automatically whenever the symmetry species of $\psi_f$, of $\hat\mu$'s relevant component, and of $\psi_i$ (usually the totally symmetric ground vibrational state) multiply out to include $A_1$. Since all three species present — $A_1$, $B_1$, $B_2$ — appear in the linear basis-function column ($z$, $x$, $y$ respectively), all six normal modes of formaldehyde are infrared active. This matches experimental IR spectra of formaldehyde, which show absorptions corresponding to C=O stretch ($A_1$), symmetric and antisymmetric CH stretches, scissor bend, and two out-of-plane/rocking modes, with none symmetry-forbidden.

Limiting-case check: for a hypothetical isotopic substitution that leaves the point group unchanged (e.g. D$_2$CO in place of H$_2$CO), the symmetry species assignments are identical because point group depends only on nuclear geometry, not on isotope mass — the selection-rule conclusion is unaffected, only the vibrational frequencies shift according to reduced-mass considerations from Characters and Reducible vs Irreducible Representations-based mode analysis, not the symmetry labels.

Example 3 — Exam/competition level: identifying the symmetry of the $t_{2g}$ and $e_g$ d-orbital sets in an octahedral complex ($O_h$). For an octahedral transition-metal complex such as $[\mathrm{Cr(NH_3)_6}]^{3+}$, the relevant point group is $O_h$, order $h = 48$, with ten classes: $E$, $8C_3$, $6C_2$, $6C_4$, $3C_2(=C_4^2)$, $i$, $6S_4$, $8S_6$, $3\sigma_h$, $6\sigma_d$. The quadratic-function column of the $O_h$ character table assigns:

Since $d_{x^2-y^2}$ and $d_{z^2}$ transform as the doubly degenerate $E_g$ representation, and $d_{xy}, d_{xz}, d_{yz}$ transform as the triply degenerate $T_{2g}$ representation, the familiar ligand-field splitting diagram — two orbitals raised in energy ($e_g$), three lowered ($t_{2g}$) — is a direct consequence of the $O_h$ character table's quadratic basis-function column, not an independent postulate. To verify dimension bookkeeping: $E_g$ contributes $\ell^2 = 2^2 = 4$ and $T_{2g}$ contributes $\ell^2 = 3^2 = 9$ to the sum rule; across all ten $O_h$ irreducible representations ($A_{1g}, A_{2g}, E_g, T_{1g}, T_{2g}, A_{1u}, A_{2u}, E_u, T_{1u}, T_{2u}$) the dimensions are $1,1,2,3,3,1,1,2,3,3$, and

This confirms that exactly ten irreducible representations correctly span the group of order 48, consistent with $O_h$ having ten conjugacy classes, and confirms that $E_g$ (dimension 2, matching the two $e_g$ orbitals) and $T_{2g}$ (dimension 3, matching the three $t_{2g}$ orbitals) are legitimate, correctly-sized irreducible representations rather than accidental degeneracies.

Limiting-case check: if the octahedron is tetragonally distorted (e.g. by a Jahn–Teller elongation along $z$, lowering the symmetry to $D_{4h}$), the $O_h$ label $E_g$ splits into $A_{1g}(d_{z^2})$ and $B_{1g}(d_{x^2-y^2})$, while $T_{2g}$ splits into $B_{2g}(d_{xy})$ and $E_g(d_{xz},d_{yz})$ — exactly the correlation table result expected when descending from a higher-symmetry to a lower-symmetry point group, and it correctly reproduces the observed splitting pattern of $e_g$ and $t_{2g}$ levels under Jahn–Teller distortion.