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

Derive the Fock Equations — Koopmans from the canonical equations

Exercises

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

Practice Problem 7 of 9

Koopmans from the canonical equations

Problem

In the canonical basis,

ε_{k} = ⟨ ϕ_{k} ∣ \hat{F} ∣ ϕ_{k} ⟩ = h_{k} + \sum_{j} (2 J_{kj} - K_{kj})

. Identify this as “the energy of an electron in orbital

k

interacting with everything,” and state why removing that electron (orbitals frozen) costs exactly

- ε_{k}

Solution

Predict before reading on. The worked example produced $ε$ as a Lagrange multiplier; here you read off its physical meaning.

Removing electron $k$ deletes its kinetic+nuclear energy $h_{k}$ and its interaction with every other electron $\sum_{j} (2 J_{kj} - K_{kj})$ — exactly the terms in $ε_{k}$ , and nothing else. So $E^{+} - E = - ε_{k}$ : Koopmans’ theorem.