Correlation: what the mean field leaves out
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Hartree-Fock electrons feel each other only on average, so they wander too close too often. The energy HF misses has a name and a sign:
It comes in two temperaments. Dynamic correlation is the short-range dodge — countless small corrections from electrons swerving around each other; one determinant is still a fine starting point, it just needs polish. Static correlation is structural: two or more determinants become nearly degenerate and the single-determinant ansatz is wrong from the first line. The diagnostic case is stretched H₂.
Stretching H₂ toward dissociation makes the correlation problem qualitatively harder than it is at equilibrium. Which kind of correlation dominates there?
Pull the atoms apart and restricted HF, forced to keep both electrons in one doubly-occupied σ orbital, dissociates H₂ into a 50/50 mix of two neutral atoms and an H⁺/H⁻ ion pair — the curve sails high. You met this failure as a fact in the Fock lesson; now you can name its cause: missing static correlation.
Restricted Hartree-Fock (RHF) famously fails as H₂ is pulled apart. The reason is that…
UHF patches the curve by letting the α electron settle on one atom and the β on the other. The energy improves; the spin symmetry breaks. The repaired 'singlet' is contaminated with triplet character — ⟨S²⟩ drifts from 0 toward 1.
UHF rescues the H₂ dissociation curve by letting the α and β electrons take different spatial orbitals. What does it pay for this?
Every post-HF method — CI, coupled cluster, perturbation theory — is a strategy for buying back E_corr at acceptable cost. The next two skills are the workhorse and the price-is-no-object benchmark.