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

Derive the Fock Equations — Varying the Coulomb functional

Exercises

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

Practice Problem 3 of 9

Varying the Coulomb functional

Problem

The Coulomb functional is

J_{kj} = \iint ϕ_{k} (1)^{2} r_{12}^{- 1} ϕ_{j} (2)^{2}

. Vary

ϕ_{k}

and show the result is

2 ⟨ δ ϕ ∣ \hat{J}_{j} ∣ ϕ_{k} ⟩

, with

\hat{J}_{j}

the potential of orbital

j

’s charge cloud.

Solution

Predict before reading on. Same expansion as the previous problem, but the operator that emerges is built from the other orbital. What plays the role of $\hat{h}$ here?

δ J = 2 \iint δ ϕ (1) ϕ_{k} (1) r_{12}^{- 1} ϕ_{j} (2)^{2} = 2 ⟨ δ ϕ ∣ \hat{J}_{j} ∣ ϕ_{k} ⟩, \hat{J}_{j} (1) = \int ϕ_{j} (2)^{2} r_{12}^{- 1} d 2

$ϕ_{k}^{2}$ varies to $2 ϕ_{k} δ ϕ$ at first order. The mean field is literally the electrostatic potential of the other electrons’ density.