“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Hamiltonian — and Slater-Condon for free

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Lesson 3 of 5 standard ~6 min

In this language the Hamiltonian is universal — it does not care which determinant you apply it to:

\hat{H} = pq \sum h_{pq} a_{p}^{†} a_{q} + \frac{1}{2} pq rs \sum ⟨ pq ∣ rs ⟩ a_{p}^{†} a_{q}^{†} a_{s} a_{r}

The one-electron part moves an electron from orbital q to p with amplitude h_pq; the two-electron part annihilates a pair (r,s) and creates a pair (p,q). Every matrix element ⟨Φ|H|Φ'⟩ is now a vacuum expectation value of strings of operators, evaluated by the anticommutators alone.

And here is the payoff: the Slater-Condon rules you memorized in QC I are no longer rules to remember — they fall out. Push the annihilation operators right; a term survives only if every operator finds a partner, which happens precisely when the two determinants differ in zero, one, or two spin-orbitals. The three cases, the signs, the spectator sums: all are forced by the algebra. The same diagrammatic machinery scales to coupled cluster, where hand-expanding determinants would be hopeless.

This is the language the rest of QC II is written in. You will not need to compute with it by hand much — that is what the algebra spares you — but reading a CI or CC derivation requires fluency in these three lines.