Approximate Methods: Variation and Perturbation

Most real quantum systems, from the helium atom to molecules, cannot be solved exactly by the Schrödinger equation. Two approximate methods help. The variation method guesses a trial wavefunction with an adjustable parameter and tunes it for the lowest energy. Perturbation theory starts from an exactly solvable system and adds the real complexity as a small disturbance.

These methods matter because almost all of computational chemistry, from estimating molecular energies to predicting spectral shifts, is built on them. They explain where approximate atomic energies come from and why variational results are always upper bounds.

Key Concepts

The variational principle states that the energy computed from any trial wavefunction is always greater than or equal to the true ground-state energy E0: Etrial ≥ E0. A trial wavefunction φ(α) has an adjustable parameter α, varied until the energy is minimised. In perturbation theory, the Hamiltonian is split as H = H0 + λH′, where H0 is exactly solvable and λH′ is the small perturbation. The first-order energy correction is the average of the perturbation over the unperturbed state: En(1) = ⟨ψn0|H′|ψn0⟩. Both methods assume the trial guess is reasonable (variation) or the disturbance is genuinely small (perturbation).

Formulas

Variational energy: E(α) = ⟨φ(α)|H|φ(α)⟩ / ⟨φ(α)|φ(α)⟩ ≥ E0, minimised with respect to α. Perturbation expansion: En ≈ En0 + λ⟨n|H′|n⟩ to first order, where ⟨n|H′|n⟩ is the expectation value of the perturbation in the unperturbed state. Higher-order terms in λ2, λ3 are used only when first order is insufficient.

Worked Example

Use the variation method on a hydrogen-like ion with nuclear charge Z = 2 (He+), guessing the trial function φ = e-αr. Evaluating the energy functional and minimising with respect to α gives the optimum α = Z/a0 and a minimum energy E = -13.6 × Z2 eV. Substituting Z = 2: E = -13.6 × 4 = -54.4 eV. The exact ground-state energy of He+ from the Bohr formula is also -13.6 × Z2 = -54.4 eV. Here the variational method recovers the exact answer because the guessed exponential form happens to match the true shape of the 1s orbital. In most problems the guess is imperfect, so the variational energy sits slightly above the true value, as the principle guarantees.

Common Misconceptions

• The variational method gives the exact energy. It gives an upper bound, not the exact value. Only a perfect guess, as in the He+ example, hits the true energy exactly.
• Perturbation theory works for disturbances of any size. It works only when the perturbation is small compared to the energy gaps of the unperturbed system. A large disturbance needs a different approach.
• First-order corrections are always enough. They often suffice, but when the first-order shift is zero or the perturbation is borderline, second- and higher-order terms become essential.

Real-World Applications

The variational method underlies Hartree-Fock and density functional calculations used in drug design and materials science. Perturbation theory explains the Stark effect (spectral line splitting in electric fields) and the Zeeman effect (splitting in magnetic fields), and it gives the first estimate of the helium atom's ground-state energy. Time-dependent perturbation theory describes how atoms absorb and emit light.

When exact solutions fail, guess and optimise (variation) or start simple and correct (perturbation). Variational energies are always upper bounds; perturbations must stay small.