Approximation Methods — formula sheet

The variation theorem: for any normalisable, well-behaved function φ that satisfies the boundary conditions of the problem, the expectation value of the Hamiltonian is never less than the true ground-state energy: W[φ] = ⟨φ|Ĥ|φ⟩ / ⟨φ|φ⟩ ≥ E_0. Equality holds if and only if φ is itself the ground-state eigenfunction.

The symmetry-restricted variation theorem: if φ belongs to a definite irreducible representation Γ (or a definite eigenvalue of some operator commuting with Ĥ), then W[φ] ≥ E_0^Γ, the lowest exact eigenvalue of that same symmetry. This is what makes variational calculations of excited states possible at all, and it is why quantum-chemistry programs ask you for the state symmetry before they start.

The Hylleraas–Undheim–MacDonald theorem: if the linear variation problem is solved in a basis of n functions, the n roots W_1 ≤ W_2 ≤ … ≤ W_n satisfy W_k ≥ E_k−1 for every k. That is, the k-th root is an upper bound to the k-th exact eigenvalue, not just the lowest. Adding a further basis function pushes every root down or leaves it where it was, and the new roots interlace the old ones.

Trial function (or trial wavefunction): a normalisable function φ that obeys the boundary conditions of the problem and contains one or more adjustable variational parameters. The parameters are chosen so as to make W as small as possible; the resulting minimum is the best estimate the chosen functional form can give.

Linear variation method: take the trial function to be a linear combination of a fixed set of basis functions f_1, f_2, …, f_n, with the expansion coefficients as the variational parameters: φ = ∑_i c_if_i. The basis functions are fixed and known; only the coefficients are varied.

Secular equations: the set of n simultaneous linear homogeneous equations ∑_i(H_ki − W S_ki)c_i = 0 obtained by minimising the linear variational energy. Secular determinant: the determinant of the coefficient matrix, which must vanish for a non-trivial solution to exist.

Unperturbed Hamiltonian Ĥ^(0): an operator whose eigenfunctions ψ_n^(0) and eigenvalues E_n^(0) are known exactly. Perturbation Ĥ′: the remainder, assumed small in the sense that its matrix elements are small compared with the spacings of the unperturbed levels. λ: a dimensionless bookkeeping parameter, run from 0 to 1, whose only job is to let terms of different orders be separated.

First-order energy correction: the expectation value of the perturbation over the unperturbed wavefunction. It requires no new wavefunction, no sum over states and no knowledge of any other level — just one integral.

The second-order ground-state theorem: the second-order correction to the energy of the ground state is always negative or zero, whatever the perturbation is. It is zero only if the perturbation couples the ground state to nothing at all. There is no corresponding rule for excited states.

Degenerate perturbation theory: when the level of interest is g-fold degenerate, form the g×g matrix of the perturbation within that degenerate set, H′_ij = ⟨ψ_i^(0)|Ĥ′|ψ_j^(0)⟩, and diagonalise it. Its eigenvalues are the first-order energy corrections; its eigenvectors are the correct zeroth-order wavefunctions.

Fermi's golden rule: the first-order transition rate from an initial state into a continuum of final states is (2π/ℏ) times the squared coupling matrix element times the density of final states at the energy of the initial state. Energy conservation is not assumed; it emerges from the long-time limit.

Transition dipole moment: the matrix element of the dipole operator between the initial and final states. If it vanishes the transition is forbidden in the electric-dipole approximation; if it is non-zero the transition is allowed, and its intensity is proportional to the square of the moment. Every selection rule in spectroscopy is a statement about when this integral vanishes.

Effective nuclear charge Z_eff and screening constant σ: the charge an electron behaves as though it sees, Z_eff = Z − σ. The variational calculation for helium gives σ = 5/16 = 0.3125 exactly, meaning that one 1s electron screens the nucleus from the other by just under a third of a proton.

Electron correlation: the tendency of electrons to avoid one another beyond what the average (mean-field) repulsion accounts for. Correlation energy: the difference between the exact non-relativistic energy and the Hartree–Fock limit — that is, the best energy obtainable from a single determinant of orbitals. It is always negative.

Validity condition for WKB: the local de Broglie wavelength must change little over one wavelength: |dλ̄/dx| ≪ 1 with λ̄ = ℏ/p, equivalently mℏ|dV/dx| ≪ p³. The condition fails wherever p → 0 — at every classical turning point — and wherever V changes abruptly.

Born–Oppenheimer approximation: because nuclei are far heavier than electrons, the electronic wavefunction can be solved with the nuclei held fixed, with the nuclear positions entering only as parameters. The electronic energies as a function of nuclear geometry constitute the potential-energy surface, on which the nuclei then move as though in an ordinary potential.