Molecular Quantum Mechanics — formula sheet

Born–Oppenheimer approximation: the total molecular wavefunction is written as a product Ψ(r, R) ≈ ψ_el(r; R) χ(R), in which the electronic wavefunction depends on the nuclear coordinates R only parametrically. Equivalently: the nuclear kinetic-energy operator is neglected when the electronic problem is solved, and its eigenvalue E_el(R) then acts as the potential energy for nuclear motion.

Potential-energy surface (PES): the total energy of a molecule — electronic energy plus nuclear repulsion — as a function of nuclear geometry, with the electrons in a specified state (usually the ground state). It has 3N − 6 dimensions for a non-linear molecule of N atoms, 3N − 5 if linear.

LCAO (Linear Combination of Atomic Orbitals): the approximation that a molecular orbital can be written as a weighted sum of atomic orbitals centred on the constituent atoms. It is a basis-set approximation, not a physical claim: the atomic orbitals are simply a convenient, chemically meaningful set of functions in which to expand the unknown MO.

Conical intersection: a point (strictly, a (3N − 8)-dimensional seam) at which two Born–Oppenheimer surfaces of the same spin multiplicity are exactly degenerate, and in whose vicinity the electronic and nuclear motions cannot be separated. Population transfers between the surfaces on a timescale of tens of femtoseconds — faster than a single vibration.

Jahn–Teller theorem: any non-linear molecule in a spatially degenerate electronic state is unstable with respect to some non-totally-symmetric distortion that removes the degeneracy. The theorem states that such a distortion exists; it does not predict which one, or how large.

Bond order: BO = ½(number of electrons in bonding MOs − number in antibonding MOs). It is a bookkeeping device, not an observable, but it correlates strongly and monotonically with bond length (shorter for higher BO) and with dissociation energy (larger for higher BO).

The heteronuclear polarity rule: the bonding MO has the larger coefficient on the more electronegative (lower-energy) atom, and the antibonding MO has the larger coefficient on the less electronegative atom. Electron density in the occupied bonding orbital therefore sits preferentially on the electronegative atom — which is the bond dipole, derived rather than assumed.

Hybrid orbital: a normalised linear combination of atomic orbitals on the same atom, chosen so that it points along a bond direction. Hybridisation is a unitary transformation within one atom's valence space. It changes no energy, creates no new orbital, and describes no physical process.

α (the Coulomb integral): H_rr = ⟨φ_r|Ĥ^eff|φ_r⟩, roughly the energy of an electron in an isolated 2p orbital on carbon. It is negative (a bound electron) and is the zero of the energy scale.

β (the resonance integral): H_rs = ⟨φ_r|Ĥ^eff|φ_s⟩ for bonded r and s. It is negative and is the interaction energy. Both are treated as empirical parameters and are never evaluated; results are quoted in units of β.

Alternant hydrocarbon: a conjugated system whose atoms can be divided into two sets (‘starred’ and ‘unstarred’) such that no two members of the same set are bonded to each other. Equivalently, the molecular graph is bipartite — it contains no odd-membered ring. Benzene, butadiene, naphthalene and allyl are alternant; azulene and fulvene are not.

Delocalisation energy (DE): the difference between the actual Hückel π energy and the π energy of a reference structure built from isolated, localised double bonds. DE = E_π(actual) − E_π(localised reference). Because β is negative, a negative DE coefficient means stabilisation.

Frost–Musulin circle: inscribe the regular n-gon of the ring in a circle of radius 2|β| centred at α, with one vertex pointing down. The vertical position of each vertex is the energy of one MO. It is not a mnemonic but an exact geometric statement of x_k = 2cos(2πk/n).