Molecular Symmetry and Point Groups: The Topic That Pays for Itself

The five symmetry elements

Mirror planes are further labelled by their orientation to the principal axis: σ_v contains it, σ_h is perpendicular to it, and σ_d contains it while bisecting two C₂ axes. Getting these labels right is necessary before a point group can be assigned.

Assigning a point group — the decision sequence

1. Is it linear? If yes, it is D_∞h when it has an inversion centre and C_∞v when it does not.
2. Does it have multiple high-order axes? If so it is one of the high-symmetry groups — tetrahedral, octahedral or icosahedral.
3. Find the principal axis — the C_n of highest n.
4. Are there n C₂ axes perpendicular to it? If yes the group is a D type; if no it is a C type.
5. Look for a σ_h. Present gives D_nh or C_nh.
6. Otherwise look for σ_v or σ_d, giving C_nv or D_nd. With no planes at all, D_n or C_n.

Common molecules and their groups

What symmetry immediately tells you

Polarity

A molecule can have a permanent dipole only if the dipole vector is unchanged by every symmetry operation. In practice that rules out any group with an inversion centre, and any group with more than one non-coincident axis. So C_nv and C_n molecules can be polar; D-type and O_h or T_d molecules cannot.

Chirality

A molecule is chiral if it has no improper axis of any kind. Since a mirror plane is S₁ and an inversion centre is S₂, the familiar rule — no mirror plane and no inversion centre — is a special case of the general statement. Framing it as "no S_n" is the version that handles awkward cases correctly.

Spectroscopic activity

A vibration is infrared active if it transforms like x, y or z in the character table, and Raman active if it transforms like a quadratic function such as x² or xy. In a centrosymmetric molecule the two sets never coincide, which gives the rule of mutual exclusion: no vibration is both IR and Raman active. Observing a band in both spectra therefore proves the molecule has no inversion centre, and that inference is a standard exam question.

Reading a character table

The rows are irreducible representations, labelled A, B, E and T. A and B are one-dimensional, E is doubly degenerate and T is triply degenerate. Subscripts g and u indicate symmetric or antisymmetric behaviour under inversion, and appear only in centrosymmetric groups.

The two right-hand columns are where most of the practical value sits: they list which functions transform as each representation, and those are what you match against to decide IR and Raman activity or to build symmetry-adapted orbitals.

Reducible representations, briefly

To find how a set of orbitals or vibrations behaves, build a reducible representation by counting how many basis vectors are unmoved by each operation, then reduce it using the standard formula. For vibrations, generate the representation for all 3N degrees of freedom and subtract translations and rotations — which are always listed in the table — leaving 3N − 6 vibrational modes, or 3N − 5 for a linear molecule.

Frequently asked questions

Why is the identity operation needed at all?

Because the set of symmetry operations must form a mathematical group, and a group requires an identity element. It is a formal requirement rather than a physical one, but omitting it makes the arithmetic fail.

How is S_n different from doing C_n and then σ_h separately?

An improper rotation is a single operation. A molecule can possess S_n without possessing either C_n or σ_h individually, and such cases appear in questions specifically to test whether the distinction is understood.

How much group theory does IIT-JAM need?

Generally element identification and point group assignment, with straightforward polarity and chirality inferences. CSIR-NET goes further into reducible representations, IR and Raman activity, and symmetry-adapted linear combinations.

Is memorising character tables necessary?

No — they are usually supplied where needed. What must be automatic is reading one: knowing what the labels mean and which column answers which question.