UHF: breaking symmetry on purpose
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Unrestricted Hartree-Fock makes one structural change: α and β electrons get separate orbital sets, hence separate Fock matrices (Pople–Nesbet). Each spin's exchange couples only to its own density; Coulomb still sees everyone:
Pople-Nesbet UHF builds two Fock matrices. In Fα, the exchange term subtracts only Pα — never Pβ. Why?
The freedom matters only when the molecule wants it. Stretch H₂ and watch the bifurcation in real numbers: at R = 2.0 the broken solution doesn't exist (UHF = RHF, ⟨S²⟩ = 0); at R = 2.2 it appears (⟨S²⟩ = 0.036, energy 0.07 mHa below); by R = 6.0 UHF sits at −0.9332 — two free atoms, the curve repaired — with ⟨S²⟩ = 0.999. The onset is the Coulson-Fischer point, ≈ 2.1 bohr here.
The Coulson-Fischer point of H₂ (≈2.1 bohr in this basis) marks the geometry where…
The repair has a price and a procedure. Price: the 'singlet' is now half triplet — fine for the dissociation energy, poisonous for spin-sensitive properties and for perturbation theory built on top. Procedure: the broken solution is invisible to a symmetric SCF — you must seed the asymmetry (α localized left, β right). Project 10 walks you through both.
UHF dissociates H₂ to the right energy (two free atoms) with the wrong wavefunction (⟨S²⟩ ≈ 1). For which prediction would this contamination actually bite?