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Derive the Fock Equations — Brillouin as a unit test

Exercises

Quantum Chemistry I › Unit 7 · The Fock problem › Derive the Fock Equations › all problems

Practice Problem 4 of 9

Brillouin as a unit test

Problem

Using the converged H₂ quantities, compute the occupied–virtual Fock element

F_{o v} = \sum_{ij} c_{i}^{(o)} F_{ij} c_{j}^{(v)}

. What value must it have, and what does a nonzero value diagnose?

Solution

Predict before reading on. The worked example’s stationarity, turned into a numerical test. Which derivation statement is being checked?

Exactly 0 (the reference run prints 0.0e+00). A nonzero $F_{o v}$ means the energy is not yet stationary — the SCF has not converged. The statement “the occupied–virtual Fock block vanishes” is Brillouin’s theorem, and production codes use precisely this block (or the commutator in Check 1) as the convergence test.