Molecular Symmetry & Point Groups

H₂O possesses the identity E, a C₂ axis (bisecting the HOH angle), and two vertical mirror planes: σ_v (the molecular plane) and σ_v′ (perpendicular to the molecular plane, containing the C₂ axis). There is no horizontal mirror plane σ_h because that would require a plane perpendicular to C₂ through the O atom, which does not leave the molecule unchanged. This set of elements defines the C_2v point group.

A symmetry element is a geometric entity — a point, line, or plane — with respect to which a symmetry operation is carried out. For example, a mirror plane (σ) is the element; reflection through that plane is the operation. Similarly, C_n is a rotational axis (element), and rotation by 360°/n about that axis is the operation. Every symmetry operation leaves at least one point unchanged, which is why molecular symmetry groups are called point groups.

Methane (CH₄) belongs to the T_d point group, which contains three S₄ axes. Each S₄ axis bisects two HCH bond angles: rotating by 90° and then reflecting through the perpendicular plane maps the molecule onto itself. Note from the figure that S₁ ≡ σ and S₂ ≡ i, so these are not counted separately. Neither H₂O, NH₃, nor BF₃ possess an S₄ axis.

The C_3v point group has 6 symmetry operations: E (identity), C₃ (rotation by 120°), C₃² (rotation by 240°), and three vertical mirror planes σ_v, σ_v′, σ_v″. The order of the group (h = 6) equals the total number of symmetry operations, which is also the sum of the squares of the dimensions of its irreducible representations (1² + 1² + 2² = 6).

NH₃ is not linear and not a special cubic group. The highest-order rotation axis is C₃ (through the N atom and the centre of the H₃ triangle). There are no C₂ axes perpendicular to C₃, so it cannot be a D group. There is no σ_h, but there are three σ_v planes, each containing the N–H bond and the C₃ axis. This leads unambiguously to C_3v.

XeF₄ has a C₄ principal axis perpendicular to the molecular plane, four C₂ axes perpendicular to C₄ (two through opposite F atoms, two bisecting F–Xe–F angles), a σ_h (the molecular plane), two sets of σ_v planes, and a centre of inversion. The presence of C₄ with perpendicular C₂ axes and a σ_h defines the D_4h point group.

Like XeF₄, the square-planar [PtCl₄]²⁻ ion belongs to D_4h. The key symmetry elements are the C₄ axis perpendicular to the molecular plane, four perpendicular C₂ axes, a horizontal mirror plane σ_h, and a centre of inversion. Square-planar MX₄ species (where all X are identical) always belong to D_4h, regardless of whether the central atom is a transition metal or a main-group element.

CHCl₃ has a C₃ axis along the C–H bond direction. There are three vertical mirror planes, each containing one C–Cl bond and the C₃ axis. There is no σ_h and no perpendicular C₂ axes. Following the decision tree: not linear, not cubic, highest C_n = C₃, no perpendicular C₂, has σ_v → C_3v. This is the same point group as NH₃, which shares the same pyramidal topology.

For the two O–H stretching modes of H₂O in C_2v, we count bonds unmoved: under E both bonds are unmoved (χ = 2), under C₂ neither is unmoved (χ = 0), under σ_v both are unmoved (χ = 2), under σ_v′ neither is unmoved (χ = 0). The reducible representation Γ = (2, 0, 2, 0) decomposes into A₁ + B₁. The A₁ mode is the symmetric stretch; the B₁ mode is the antisymmetric stretch. Both are IR-active and Raman-active since C_2v has no centre of inversion.

For a vibration to be IR-active, it must cause a change in the dipole moment. In the symmetric stretch (ν₁) of CO₂, both C=O bonds extend equally — the dipole moment remains zero, so it is IR-inactive (but Raman-active). The antisymmetric stretch (ν₃) causes one bond to extend while the other compresses, producing an oscillating dipole — it is IR-active. Since CO₂ is centrosymmetric, the mutual exclusion rule applies: a mode that is IR-active is Raman-inactive, and vice versa.

In a molecule with a centre of inversion (i), each vibrational mode is either symmetric (g, gerade) or antisymmetric (u, ungerade) with respect to inversion. IR activity requires a change in dipole moment (u symmetry), while Raman activity requires a change in polarizability (g symmetry). Since no mode can be both g and u simultaneously, no mode can be active in both IR and Raman. This is a powerful diagnostic: if a molecule has coincident IR and Raman bands, it lacks a centre of inversion.

In O_h symmetry, the five d orbitals transform as two irreducible representations: the triply degenerate t_2g set (d_xy, d_xz, d_yz) and the doubly degenerate e_g set (d_z², d_x²−y²). The t_2g orbitals point between the ligand axes and are lower in energy in a crystal field, while e_g orbitals point directly at the ligands and are higher in energy. The energy gap between them is the crystal field splitting parameter Δ_o.

In linear CO₂, the two C=O bond dipoles are equal in magnitude and exactly opposite in direction — they cancel by the D_∞h symmetry of the molecule, giving a net zero dipole moment. In bent SO₂ (C_2v), the two S=O bond dipoles are directed at an angle to each other and do not cancel. Formally, only molecules belonging to the point groups C_n, C_nv, or C_s can have a permanent dipole moment — any higher symmetry forces cancellation.

In T_d symmetry, the five d orbitals split into e (d_z², d_x²−y²) and t₂ (d_xy, d_xz, d_yz). The four ligand σ group orbitals transform as a₁ + t₂. Since only orbitals of matching symmetry can form bonding combinations, the d orbitals that participate in σ bonding are the t₂ set. The e set is non-bonding with respect to σ interactions, which is opposite to the octahedral case where e_g is the σ-bonding set.

A molecule is optically active (chiral) if and only if it is not superimposable on its mirror image. The necessary and sufficient symmetry condition is the absence of any improper rotation axis S_n. This includes S₁ (which is a mirror plane σ) and S₂ (which is a centre of inversion i). Chiral molecules therefore belong only to point groups C₁, C_n, D_n, T, O, or I — groups that contain only proper rotations.