Chemical Bonding and Group Theory: VB and MO Theory, Hückel Theory, Point Groups and Selection Rules

Nuclei are thousands of times heavier than electrons and move far more slowly, so in the Born–Oppenheimer approximation the electronic Schrödinger equation is solved with the nuclei held fixed. Repeating this for many nuclear arrangements gives the potential-energy curve or surface on which the nuclei then vibrate and rotate; bond lengths, force constants and dissociation energies are all read from it. The approximation fails where two electronic surfaces come close, as at conical intersections in photochemistry.

The Heitler–London valence-bond treatment of H₂ pairs one electron on each atom: ψ_VB = 1s_A(1)1s_B(2) + 1s_A(2)1s_B(1), multiplied by the singlet spin function. It gives binding because of the exchange (resonance) integral, and it omits the ionic terms H⁺H⁻ entirely; adding them with a small coefficient improves the energy. Hybridisation mixes orbitals on one atom to point along bonds: sp (linear, 180°), sp² (trigonal, 120°), sp³ (tetrahedral, 109.47°), dsp² (square planar), sp³d (trigonal bipyramidal) and sp³d² (octahedral). For two equivalent s–p hybrids with fractional s-character α the interorbital angle obeys cos θ = −α/(1 − α), so more s-character means a wider angle — Bent’s rule: more electronegative substituents take orbitals richer in p-character.

In the LCAO-MO method a molecular orbital is a linear combination of atomic orbitals, and the variational principle fixes the coefficients through a secular determinant. For H₂⁺, ψ± = (1s_A ± 1s_B)/√(2(1 ± S)) with energies E± = (α ± β)/(1 ± S), where α is the Coulomb integral, β the (negative) resonance integral and S the overlap. Because of the 1 ± S denominators the antibonding orbital is raised more than the bonding orbital is lowered, so filling both bonding and antibonding orbitals (He₂) is net repulsive. H₂ puts two paired electrons in σ1s; the simple MO function contains ionic and covalent terms in equal weight, the opposite fault to the VB function, and configuration interaction repairs both.

For second-row homonuclear diatomics the valence MOs are σ2s, σ2s, σ2p, π2p (×2), π2p (×2) and σ*2p. s–p mixing pushes σ2p above π2p for Li₂ to N₂, while for O₂ and F₂ the order is σ2p below π2p. Bond order = ½(bonding − antibonding electrons). O₂ has two unpaired electrons in the degenerate π* pair and is paramagnetic, which the Lewis structure cannot explain; B₂ is paramagnetic for the same reason in its π2p pair; C₂ and N₂ are diamagnetic. In heteronuclear molecules the more electronegative atom’s orbitals lie lower and dominate the bonding MOs: in CO (isoelectronic with N₂) the HOMO is a σ orbital concentrated on carbon, which is why CO binds metals through C; NO has one electron in π* and bond order 2.5; in HF the bonding σ is mostly F 2p.

Hückel theory treats only the π electrons, one p orbital per carbon, and makes drastic simplifications: every Hᵢᵢ = α; Hᵢⱼ = β for bonded neighbours and 0 otherwise; Sᵢⱼ = 0 for i ≠ j. Writing x = (α − E)/β, the secular determinant has x on the diagonal and 1 for each bonded pair. Both α and β are negative, so E = α + kβ with k > 0 is bonding. For ethylene E = α ± β. For a linear chain of N atoms the roots are E_k = α + 2β cos(kπ/(N + 1)), and for a ring E_k = α + 2β cos(2πk/N).

The delocalisation energy is the π energy minus that of the same number of isolated ethylenic bonds (2β per bond). Benzene’s 2β is the Hückel measure of aromatic stabilisation. The ring formula explains Hückel’s 4n + 2 rule: a ring has one lowest orbital and then degenerate pairs, so 4n + 2 electrons fill a closed shell, while 4n electrons half-fill a degenerate pair — cyclobutadiene is predicted to be a diradical with no delocalisation energy, i.e. antiaromatic. The coefficients also give π-bond orders and charge densities: in butadiene the C1–C2 π bond order is 0.894 and C2–C3 is 0.447, matching the short–long–short bond pattern.

A symmetry operation moves a molecule into an indistinguishable orientation; the symmetry element is the point, line or plane about which it acts. There are five kinds: the identity E; a proper rotation Cₙ by 360°/n; a reflection σ (σ_h perpendicular to the principal axis, σ_v containing it, σ_d containing it and bisecting two C₂ axes); inversion i through a centre; and an improper rotation Sₙ, rotation by 360°/n followed by reflection in the perpendicular plane (S₁ = σ, S₂ = i). The operations of a molecule form a group: the product of any two is a member (closure), multiplication is associative, E is present, and every operation has an inverse. The order h is the number of operations. A multiplication table lists every product, and operations that are conjugate (Y = X⁻¹AX for some X in the group) form a class — in C₃v the two C₃ rotations are one class and the three σ_v another.

Classification follows a fixed sequence: look first for a linear or high-symmetry (T_d, O_h, I_h) molecule; otherwise find the principal axis Cₙ; ask whether there are n C₂ axes perpendicular to it (the D groups) or not (C and S groups); then look for σ_h (Dₙh, Cₙh), for n σ_d or σ_v (Dₙd, Cₙv), or for an S₂ₙ alone. Two consequences are examined repeatedly: a molecule is chiral if and only if it has no improper axis Sₙ (no σ, no i, no S₄ …), and it can have a dipole moment only in the groups C₁, Cₛ, Cₙ and Cₙv.

A set of matrices that multiply like the group’s operations is a representation. Using any set of vectors as a basis (atomic displacements, orbitals, bonds) gives a reducible representation Γ, which can be broken into irreducible representations — the smallest blocks, listed with their characters (traces) in the character table. The number of irreducible representations equals the number of classes, and Σ dᵢ² = h. Mulliken labels: A and B are one-dimensional (symmetric or antisymmetric to the principal rotation), E two- and T three-dimensional; subscript g/u marks symmetric/antisymmetric under inversion, 1/2 under a C₂ or σ_v, and ′/″ under σ_h.

The reduction formula gives how many times each irreducible representation i occurs in Γ: nᵢ = (1/h) Σ_R g(R) χ_Γ(R) χᵢ(R), summed over classes with g(R) operations each. For the 3N displacements of H₂O the characters are found from the number of atoms left in place by each operation times the contribution per unshifted atom (E: 3, C₂: −1, σ: +1): Γ₃N = (9, −1, 3, 1). Reduction gives Γ₃N = 3A₁ + A₂ + 3B₁ + 2B₂. Removing translations (A₁ + B₁ + B₂, the functions x, y, z) and rotations (A₂ + B₁ + B₂) leaves Γ_vib = 2A₁ + B₁: two totally symmetric modes (symmetric stretch and bend) and one antisymmetric stretch. With the molecule placed in the yz plane instead, the last label is B₂; the physics is unchanged.

A transition between states ψᵢ and ψ_f is allowed only if the transition-moment integral ∫ψᵢ* μ̂ ψ_f dτ is non-zero, which requires the direct product Γᵢ ⊗ Γ_μ ⊗ Γ_f to contain the totally symmetric representation. For a fundamental vibration from the totally symmetric ground state this becomes simple: a mode is IR active if it transforms as x, y or z and Raman active if it transforms as a quadratic function (x², xy, …). In a molecule with a centre of inversion the x, y, z functions are u and the quadratic ones g, so no mode is both — the rule of mutual exclusion. CO₂ (D∞h) has 3N − 5 = 4 modes: the symmetric stretch Σg⁺ (Raman only), the antisymmetric stretch Σu⁺ and the doubly degenerate bend Πu (IR only). For electronic transitions in a centrosymmetric complex, g → g d–d bands are forbidden by the same argument — the Laporte rule.

Internal coordinates (bond stretches, angle bends) used as the basis give the symmetry of the stretching and bending modes directly: the two O–H stretches of water span A₁ + B₁, the bend A₁. A symmetry-adapted linear combination (SALC) is a combination of equivalent basis functions that transforms as one irreducible representation, obtained with the projection operator; only orbitals of the same symmetry can combine into MOs. The same logic constructs hybrids: the three σ bonds of BF₃ span A₁′ + E′ in D₃h, matched by s (A₁′) and (pₓ, p_y) (E′) — sp²; the four σ bonds of CH₄ span A₁ + T₂ in T_d, matched by s + (pₓ, p_y, p_z) — sp³ — or, in principle, s + (d_xy, d_xz, d_yz) — sd³.