3.2 Point Groups and Character Tables

Symmetry Elements and Operations

Fundamental Concepts of Symmetry

A symmetry element is a geometric feature of a molecule (an axis, a plane, a point) about which a symmetry operation can be performed. A symmetry operation is the actual transformation you carry out: rotating, reflecting, or inverting the molecule so that the result is indistinguishable from the starting configuration.

Every molecule possesses at least one symmetry element (the identity). The complete collection of symmetry operations a molecule possesses defines its point group, and group theory provides the mathematical framework for working with these sets of operations to extract real chemical information.

Types of Symmetry Elements and Operations

There are five types you need to know:

• Identity ($E$): Does nothing. Every molecule has it. It's required for the set of operations to satisfy the mathematical definition of a group (closure, associativity, identity, and inverse).
• Proper rotation axis ($C_n$): Rotation by $360°/n$ about an axis leaves the molecule unchanged. The axis with the highest $n$ is the principal axis. Water has a $C_2$ axis, while benzene has a $C_6$ axis (and also $C_3$, $C_2$ as subgroups of that).
• Mirror plane ($\sigma$): Reflection through a plane that maps the molecule onto itself. These are classified as $\sigma_v$ (contains the principal axis), $\sigma_h$ (perpendicular to the principal axis), or $\sigma_d$ (contains the principal axis and bisects two $C_2$ axes).
• Center of inversion ($i$): Every atom at position $(x, y, z)$ is mapped to $(-x, -y, -z)$. Octahedral $\text{SF}_6$ has an inversion center; tetrahedral $\text{CH}_4$ does not.
• Improper rotation axis ($S_n$): A rotation by $360°/n$ followed by reflection through a plane perpendicular to that axis. Note that $S_1 = \sigma$ and $S_2 = i$, so these are special cases.

Importance of Symmetry in Chemistry

• Symmetry determines whether a molecule has a dipole moment. Only molecules in $C_1$, $C_s$, $C_n$, or $C_{nv}$ point groups can be polar.
• A molecule is chiral only if it lacks any $S_n$ axis (including $\sigma$ and $i$, since those are $S_1$ and $S_2$).
• Symmetry simplifies quantum mechanical calculations by block-diagonalizing Hamiltonians, reducing large problems into smaller, independent ones.
• It directly predicts spectroscopic selection rules and governs which orbital interactions are symmetry-allowed.

Point Group Classification

Assigning a molecule to its point group is a systematic process. Follow this flowchart-style procedure:

1. Check for special groups first. Is the molecule linear? If yes, it's either $C_{\infty v}$ (no inversion center, e.g., HCl) or $D_{\infty h}$ (has inversion center, e.g., $\text{CO}_2$). Does it belong to a high-symmetry cubic group ($T_d$, $O_h$, $I_h$)? Recognizing these early saves you from a long search through the flowchart.
2. Find the principal axis. Identify the highest-order $C_n$ axis.
3. Look for $n$ $C_2$ axes perpendicular to the principal axis. If they exist, you're in a $D$ group. If not, you're in a $C$ or $S$ group.
4. Check for a $\sigma_h$ (horizontal mirror plane). If present: $D_{nh}$ or $C_{nh}$.
5. Check for $\sigma_v$ or $\sigma_d$ planes. If present: $D_{nd}$ or $C_{nv}$.
6. If no mirror planes exist: $D_n$, $C_n$, or check for an $S_{2n}$ axis giving an $S_{2n}$ group.

Some common examples to anchor your understanding:

• $\text{H}_2\text{O}$: $C_{2v}$ (one $C_2$, two $\sigma_v$ planes, no $\sigma_h$, no perpendicular $C_2$ axes)
• $\text{NH}_3$: $C_{3v}$ (one $C_3$, three $\sigma_v$ planes)
• $\text{BF}_3$: $D_{3h}$ (one $C_3$, three perpendicular $C_2$ axes, one $\sigma_h$)
• $[\text{PtCl}_4]^{2-}$ (square planar): $D_{4h}$
• $\text{CH}_4$: $T_d$

Character Tables and Representations

Structure and Components of Character Tables

A character table is a compact summary of all the symmetry information for a given point group. Here's what each part means, using $C_{2v}$ as an example:

• Top row: The symmetry operations of the group, grouped into classes.
• Left column: The Mulliken symbols labeling each irreducible representation.
• Body of the table: The characters (traces of the transformation matrices) for each irreducible representation under each operation.
• Right columns: Show how the translational coordinates ($x, y, z$), rotations ($R_x, R_y, R_z$), and quadratic functions ($x^2, xy$, etc.) transform.

The order of the group ($h$) is the total number of symmetry operations. For $C_{2v}$, $h = 4$ (one $E$, one $C_2$, one $\sigma_v$, one $\sigma_v'$). You'll need $h$ whenever you use the reduction formula.

Mulliken Symbol Conventions

The labels follow specific rules:

• A: Symmetric with respect to rotation about the principal axis (character of +1 under $C_n$).
• B: Antisymmetric with respect to rotation about the principal axis (character of -1 under $C_n$).
• E: Doubly degenerate representation (dimension 2). Don't confuse this with the identity operation $E$.
• T: Triply degenerate representation (dimension 3).
• Subscripts 1/2 distinguish behavior under a perpendicular $C_2$ or $\sigma_v$. Subscript 1 means symmetric, 2 means antisymmetric.
• Subscripts g/u (gerade/ungerade) indicate symmetric/antisymmetric behavior under inversion. These only appear in groups that contain $i$.
• Primes (' and '') indicate symmetric/antisymmetric under $\sigma_h$.

Reducible and Irreducible Representations

Irreducible representations are the simplest, most fundamental ways a function or set of orbitals can transform under the symmetry operations of a group. They cannot be broken down further.

A reducible representation is a larger representation that can be decomposed into a sum of irreducible representations. This decomposition is central to applying group theory in chemistry. You'll generate reducible representations constantly when working with sets of bonds, orbitals, or vibrational modes.

To decompose a reducible representation, use the reduction formula:

$n_i = \frac{1}{h} \sum_R N(R) \cdot \chi_{\Gamma}(R) \cdot \chi_i(R)$

where:

• $n_i$ = number of times irreducible representation $i$ appears
• $h$ = order of the group (total number of symmetry operations)
• $N(R)$ = number of operations in class $R$
• $\chi_{\Gamma}(R)$ = character of the reducible representation for class $R$
• $\chi_i(R)$ = character of irreducible representation $i$ for class $R$

Apply this formula once for each irreducible representation in the character table. A quick check: the sum of all $n_i$ multiplied by the dimension of each irreducible representation should equal the dimension of your reducible representation (the character under $E$).

Interpreting the Right-Side Columns

The columns on the right side of a character table tell you how specific mathematical functions transform under the group's symmetry. This has direct physical meaning:

• Linear functions ($x, y, z$): These transform like $p$ orbitals and like translational motion. An irreducible representation that lists $z$ means a $p_z$ orbital (or a dipole moment component along $z$) transforms as that representation. These are also what you check for IR activity.
• Rotations ($R_x, R_y, R_z$): These transform like angular momentum components and are relevant for magnetic properties and certain spectroscopic transitions.
• Quadratic functions ($x^2, xy$, etc.): These transform like $d$ orbitals and polarizability tensor components. They're directly used to determine Raman activity.

Applications of Symmetry

Molecular Orbital Construction

Symmetry-adapted linear combinations (SALCs) are the standard method for building molecular orbitals in polyatomic molecules. The process works like this:

1. Assign the point group of the molecule.
2. Choose a basis set of atomic orbitals (e.g., the four H 1s orbitals in $\text{CH}_4$, or the ligand orbitals in a coordination compound).
3. Generate the reducible representation by tracking how many basis functions are unmoved by each symmetry operation. Each basis function that doesn't move contributes +1 to the character; each one that moves contributes 0.
4. Decompose the reducible representation into irreducible representations using the reduction formula.
5. Use projection operators to find the explicit form of each SALC.
6. Combine SALCs with central atom orbitals of matching symmetry to form bonding and antibonding molecular orbitals.

The key principle: only orbitals belonging to the same irreducible representation can have nonzero overlap. If two orbitals transform as different irreducible representations, their overlap integral is exactly zero by symmetry, and no bonding interaction occurs.

Spectroscopic Selection Rules

Symmetry provides direct yes-or-no answers about whether a spectroscopic transition is allowed.

For infrared (IR) activity: A vibrational mode is IR-active if it belongs to the same irreducible representation as $x$, $y$, or $z$ (the translational coordinates in the character table). This is because IR absorption requires a change in dipole moment, and $x$, $y$, $z$ represent the dipole moment components.

For Raman activity: A vibrational mode is Raman-active if it belongs to the same irreducible representation as a quadratic function ($x^2$, $xy$, $xz$, etc.). Raman scattering requires a change in polarizability, and the quadratic functions represent polarizability tensor components.

A useful check: in molecules with an inversion center, the mutual exclusion rule applies. No vibrational mode can be both IR-active and Raman-active. IR-active modes are ungerade ($u$), and Raman-active modes are gerade ($g$).

For electronic transitions: A transition between two states is allowed if the direct product of the initial state representation, the dipole operator representation, and the final state representation contains the totally symmetric representation ($A_1$, $A_{1g}$, etc.). In practice, you multiply the characters of the three representations operation by operation and then decompose the result.

Chemical Reactivity and Bonding

Frontier molecular orbital (FMO) theory uses symmetry to determine whether the HOMO of one reactant can interact with the LUMO of another. If their symmetry representations are incompatible under the relevant symmetry of the reaction coordinate, the interaction is symmetry-forbidden.

The Woodward-Hoffmann rules for pericyclic reactions (electrocyclic, cycloaddition, sigmatropic) are grounded in orbital symmetry conservation. Whether a reaction proceeds thermally or photochemically depends on the symmetry of the frontier orbitals involved.

In coordination chemistry, symmetry analysis of metal $d$ orbitals under the ligand field determines the splitting pattern. For example, in $O_h$ symmetry, the five $d$ orbitals split into $t_{2g}$ ($d_{xy}$, $d_{xz}$, $d_{yz}$) and $e_g$ ($d_{z^2}$, $d_{x^2-y^2}$) sets. You can confirm this by looking at where the quadratic functions appear in the $O_h$ character table.