The variational principle: rigorous guessing

Solving the Schrödinger equation exactly is hopeless for all but the simplest systems — so we guess. The variational principle makes guessing rigorous: for any trial wavefunction ψ, the energy you compute, E[ψ] = ⟨ψ|H|ψ⟩ / ⟨ψ|ψ⟩, is never below the true ground-state energy E₀.

That one-directional guarantee is the engine of quantum chemistry. Choose a trial form with a tunable parameter — for hydrogen, ψ(r) = e^{−ζ|r|} — and slide ζ to push the energy as low as it will go. The minimum you reach is your best estimate, and it is still an upper bound on E₀.

Everything ahead — basis sets, the Fock matrix, the SCF loop — is machinery for running this same minimization systematically, with many parameters at once.