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Derive the Fock Equations — The FPS − SPF criterion

Exercises

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

Check Problem 8 of 9

The FPS − SPF criterion

Problem

Derive the matrix convergence criterion used by production codes: at SCF convergence the commutator

FPS - SPF = 0

(

F

Fock,

P

density,

S

overlap, all in the AO basis). Start from

FC = SCε

and

P = 2 C_{occ} C_{occ}^{T}

Solution

FPS = 2 F C_{occ} C_{occ}^{T} S = 2 S C_{occ} ε_{occ} C_{occ}^{T} S

using $F C_{occ} = S C_{occ} ε_{occ}$ . The right-hand side is symmetric, so it equals its own transpose $SPF$ , and the commutator vanishes. Off convergence, $F$ is built from a density inconsistent with its own eigenvectors and the symmetry argument breaks; the commutator’s magnitude is the DIIS error vector, the quantity real codes drive to zero.