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

Derive the Fock Equations — Why “density stopped” equals “energy stationary”

Exercises

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

Practice Problem 6 of 9

Why “density stopped” equals “energy stationary”

Problem

The first-order energy change under an occupied→virtual rotation by angle

θ

δ E = 4 θ F_{o v} + O (θ^{2})

. Use it to explain why “the density stopped changing” and “the energy is stationary” are the same convergence statement.

Solution

Predict before reading on. Combines the previous problem’s $F_{o v}$ with the worked example’s stationarity logic.

The SCF diagonalization picks orbitals with $F_{o v} = 0$ for the current $F$ ; if the resulting density rebuilds the same $F$ , then $F_{o v}$ stays 0 and no rotation can lower $E$ to first order — stationary. A changing density means the new $F$ ’s occupied–virtual block is nonzero against the old orbitals: not yet stationary. Fixed point ⇔ stationarity.