Spin: what determinants get wrong

The exact wavefunction is an eigenfunction of total spin: S²Ψ = S(S+1)Ψ — singlets have S = 0, triplets S = 1. Determinants, the workhorses of everything we've built, only sometimes manage this. A closed shell is a clean singlet. But put one α and one β electron in different orbitals and the single determinant |φ₁α φ₂β| is no spin eigenfunction at all — it is a 50/50 mix of singlet and triplet.

The true eigenfunctions take two determinants each:

And their energies differ by a quantity you already own. For H₂'s σ→σ* excited states, the gap is twice the exchange integral between the orbitals — with the real numbers, 2K = 2 × 0.1813 = 0.3625 hartree, triplet below singlet. Same-spin electrons carry a built-in Fermi hole; exchange is its energy signature; Hund's first rule is this lesson run in reverse.

The practical diagnostic, which you will compute in Project 10: ⟨S²⟩ = Ms(Ms+1) + N_β − Σ|⟨φᵢ^α|φⱼ^β⟩|². Perfectly paired spin orbitals give the exact value; any mismatch leaves contamination behind.