Computation, Spectroscopy & the Quantum Toolkit — formula sheet

Basis set: the fixed set of one-electron functions from which every molecular orbital in a calculation is built. It is the vocabulary of the calculation. The variational principle guarantees the best sentence that can be written in that vocabulary — and nothing about whether the vocabulary was adequate.

The Gaussian product theorem: the product of two Gaussian functions centred on different points is a single Gaussian centred on a point between them. Symbolically, e^−α|r−A|² × e^−β|r−B|² = K e^−(α+β)|r−P|² with P the weighted midpoint of A and B and K a constant.

Contracted Gaussian function (CGF): a fixed linear combination of primitive Gaussians, χ = Σ_k d_kg_k, in which the contraction coefficients d_k and the exponents α_k are determined in advance and held fixed during the molecular calculation. Only the coefficients c_i multiplying whole contracted functions are varied.

Basis-set superposition error (BSSE): in a calculation on a complex A⋅B, the basis functions belonging to B are physically present near A and improve A's description, and vice versa. Each monomer is therefore described better inside the complex than alone, which artificially stabilises the complex. The error is largest for small basis sets and for weak interactions — precisely where interaction energies are being computed.

Correlation energy: E_corr = E_exact − E_HF, where both are evaluated non-relativistically, within the Born–Oppenheimer approximation, and in the same basis set. It is negative by definition, since Hartree–Fock is variational and therefore an upper bound. It is typically about 1% of the total energy — and it is of the same order as an entire chemical bond.

Size-consistency: E(A⋅⋅⋅B at infinite separation) = E(A) + E(B). A property of the method applied to a supersystem. Size-extensivity: the correlation energy scales linearly with the number of electrons for a system of N non-interacting identical units. The two are closely related and often used interchangeably; strictly, extensivity is the more fundamental diagrammatic property (no unlinked diagrams) and consistency is its physical consequence. HF, MPn and CC are size-extensive at every order; truncated CI is not.

Hohenberg–Kohn theorem I (existence): the ground-state electron density ρ(r) of a system of interacting electrons determines the external potential v_ext(r) uniquely, up to an additive constant. Since v_ext and the electron count N (which is ∫ρ dr) fix the entire Hamiltonian, ρ(r) determines everything — the ground-state energy, the wavefunction, and every observable.

Hohenberg–Kohn theorem II (variational): there exists an energy functional E[ρ] such that, for any trial density ρ′(r) that is non-negative, integrates to N and is v-representable — that is, is the ground-state density of some external potential (see the constrained-search note in the advanced layer, which removes this restriction) — E[ρ′] ≥ E_0, with equality only for the true ground-state density. So the ground state can be found by minimising a functional of a three-variable function.

Frequency calculation: evaluation and diagonalisation of the mass-weighted matrix of second derivatives (the Hessian) at a stationary point. The eigenvalues give the squared harmonic vibrational frequencies and the eigenvectors give the normal modes. A negative eigenvalue corresponds to an imaginary frequency, conventionally printed as a negative wavenumber, and means the surface curves downwards along that mode.

Transition dipole moment: μ_fi = ∫ψ_f* μ̂ ψ_i dτ, where μ̂ = Σ_j q_jr_j is the electric dipole moment operator. It is a vector. The intensity of the transition is proportional to |μ_fi|². If μ_fi = 0 the transition is forbidden.

The vanishing-integral rule: an integral over all space is zero unless the integrand is unchanged by every symmetry operation of the system — that is, unless the direct product of the irreducible representations of the three factors contains the totally symmetric representation. In one dimension this reduces to: the integral of an odd function over a symmetric interval is zero.

Point group: the complete set of symmetry operations that leave a molecule looking unchanged and leave at least one point fixed. The operations are the identity E, proper rotations C_n, reflections σ (σ_v containing the principal axis, σ_h perpendicular to it, σ_d bisecting two C_2 axes), the inversion i, and improper rotations S_n = C_n followed by σ_h. Water is C_2v (E, C_2, two σ_v); ammonia is C_3v; benzene is D_6h; methane is T_d; SF_6 is O_h.

Irreducible representation (irrep): a representation that cannot be reduced to block-diagonal form by any change of basis. Every group has a small, fixed set of them, and every representation is a sum of them. The character table lists the characters of every irrep for every class of operation. The labels are Mulliken's: A and B for one-dimensional irreps (A symmetric, B antisymmetric under the principal rotation), E for two-dimensional and T for three-dimensional ones — so an E or T label is a statement of orbital degeneracy; subscripts g and u for symmetric and antisymmetric under inversion; subscripts 1 and 2 for symmetric and antisymmetric under a σ_v or a perpendicular C_2; primes for behaviour under σ_h.

The direct product. For one-dimensional irreps, multiply the characters operation by operation: Γ_a ⊗ Γ_b has χ(R) = χ_a(R)χ_b(R). Three consequences carry almost all the exam questions. (i) Γ ⊗ Γ is always totally symmetric for a one-dimensional irrep, so ∫ψ²dτ never vanishes — as it must not, since it is a normalisation integral. (ii) A direct product contains the totally symmetric irrep if and only if the two irreps are the same. (iii) In a group with an inversion centre, g ⊗ g = g, u ⊗ u = g, g ⊗ u = u, which is the whole of the Laporte rule.

Franck–Condon factor: the squared vibrational overlap |⟨v′|v″⟩|². It multiplies the intensity of the v″ → v′ band. The whole intensity distribution across a vibrational progression is determined by these factors, and they in turn are determined by how much the potential curve has shifted between the two electronic states.

Koopmans' theorem: within the Hartree–Fock approximation, and assuming that the remaining orbitals do not change on ionisation (the ‘frozen orbital’ assumption), the ionisation energy from orbital i is IE_i = −ε_i, minus the orbital energy.

Atomic units: the system in which m_e = e = ℏ = 4πε_0 = 1. Length is then measured in bohr (a_0 = 52.9177 pm), energy in hartree (E_h = 27.2114 eV), and the hydrogen-atom Hamiltonian collapses to −½∇² − Z/r. Essentially every computational output is in atomic units.