The energy of one determinant

A determinant is a sum of N! products, so you might expect its energy expectation to be a combinatorial nightmare. It collapses. For the closed-shell case — two electrons sharing one spatial orbital φ with opposite spins — expanding ⟨Ψ|H|Ψ⟩ (or invoking the Slater–Condon rules) leaves exactly two terms:

Two copies of the one-electron energy h_φ (kinetic + nuclear attraction, once per electron), plus a single Coulomb integral: the orbital's charge cloud repelling itself. Note what's missing — there is no exchange term. Exchange only couples electrons of the same spin, and our pair is α with β, so ⟨α|β⟩ = 0 kills it.

The Slater–Condon rules are the general version of this collapse: for any determinant, the energy reduces to sums of one-electron integrals plus Coulomb-minus-exchange pairs over the occupied orbitals. They are why Hartree-Fock is computable at all — the N! monster never has to be touched directly.