Prove2Me

Koopmans' theorem: removing an electron costs −εa-\varepsilon_a (Eq. (66))

Proved
KochHF.koopmans_theorem

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

hartree-fockmathematical-physicssecond-quantization

With H^\hat H, TT, UU as in the definition file and Unn′,mm′=Un′n,m′mU_{nn',mm'}=U_{n'n,m'm}, let SS be the occupied reference orbitals and a∈Sa\in S. Removing the electron from orbital aa changes the energy expectation value by minus the Hartree-Fock orbital energy:

⟨S∖{a}∣H^∣S∖{a}⟩−⟨S∣H^∣S⟩=−εa,εa=Taa+∑m′∈SΔam′.\langle S\setminus\{a\}|\hat H|S\setminus\{a\}\rangle-\langle S|\hat H|S\rangle=-\varepsilon_a,\qquad \varepsilon_a=T_{aa}+\sum_{m'\in S}\Delta_{am'} .

Neglecting orbital relaxation, −εa-\varepsilon_a is thus the ionization energy from orbital aa.

Preamble
import Mathlib
import Definitions.Def_KochHF_FockSpace
import Definitions.Def_KochHF_Hamiltonian

open Matrix
Formal statement
namespace KochHF

theorem koopmans_theorem {K : ℕ} (T : Matrix (Fin K) (Fin K) ℂ)
    (U : Fin K → Fin K → Fin K → Fin K → ℂ) (hU : ∀ n n' m m', U n n' m m' = U n' n m' m)
    (S : Finset (Fin K)) (a : Fin K) (ha : a ∈ S) :
    detEnergy T U (S.erase a) - detEnergy T U S = -hfOrbitalEnergy T U S a := by sorry

end KochHF
Source
E. Koch, *Mean-Field Theory: Hartree-Fock and BCS*, in E. Pavarini, E. Koch, J. van den Brink, G. Sawatzky (eds.), Quantum Materials: Experiments and Theory, Modeling and Simulation Vol. 6, Forschungszentrum Jülich, 2016, ISBN 978-3-95806-159-0, http://www.cond-mat.de/events/correl16, Sec. 3.2, p. 2.18, Eqs. (65), (66).
Read-back

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Non-blind read-back — not independent testimony. This read-back was written by the same agent that drafted the Lean statement, with full knowledge of the source text and of the intended meaning. It is not a blind audit and must not be mistaken for independent testimony; reviewers should compare the Lean code against the source themselves.

Throughout, K∈NK\in\mathbb N is the number of orthonormal reference orbitals ϕ0,…,ϕK−1\phi_0,\dots,\phi_{K-1}; an orbital is a coefficient vector φ∈CK\varphi\in\mathbb C^K and ⟨φ∣χ⟩=∑μφμ‾χμ\langle\varphi|\chi\rangle=\sum_\mu\overline{\varphi_\mu}\chi_\mu. Fock space is C2K\mathbb C^{2^K} with orthonormal basis ∣S⟩|S\rangle, S⊆{0,…,K−1}S\subseteq\{0,\dots,K-1\}; ci†∣S⟩=(−1)#{j∈S: j>i}∣S∪{i}⟩c^\dagger_i|S\rangle=(-1)^{\#\{j\in S:\,j>i\}}|S\cup\{i\}\rangle for i∉Si\notin S and 00 for i∈Si\in S; ci=(ci†)†c_i=(c_i^\dagger)^\dagger (conjugate transpose); cφ†=∑μφμcμ†c^\dagger_\varphi=\sum_\mu\varphi_\mu c^\dagger_\mu, cφ=∑μφμ‾cμc_\varphi=\sum_\mu\overline{\varphi_\mu}c_\mu; ∣0⟩=∣∅⟩|0\rangle=|\emptyset\rangle; for orbitals α0,…,αN−1\alpha_0,\dots,\alpha_{N-1} the vector ∣Φα⟩=cαN−1†⋯cα0†∣0⟩|\Phi_\alpha\rangle=c^\dagger_{\alpha_{N-1}}\cdots c^\dagger_{\alpha_0}|0\rangle (for N=0N=0 this is ∣0⟩|0\rangle). For all KK, every complex K×KK\times K matrix TT, every complex 4-index array UU with Unn′mm′=Un′nm′mU_{nn'mm'}=U_{n'nm'm} for all indices, every S⊆{0,…,K−1}S\subseteq\{0,\dots,K-1\} and every a∈Sa\in S:

E(S∖{a})−E(S)=−(Taa+∑m′∈S(Uam′am′−Uam′m′a)),E(S\setminus\{a\})-E(S)=-\Bigl(T_{aa}+\sum_{m'\in S}\bigl(U_{am'am'}-U_{am'm'a}\bigr)\Bigr),

where E(R)=⟨R∣H^(T,U)∣R⟩E(R)=\langle R|\hat H(T,U)|R\rangle is the diagonal matrix element in the basis state ∣R⟩|R\rangle.

Human review

  • Endorsed by Shuze Chen · Oct 10, 2026

    Confirmed by the moderator at approval.

  • Endorsed by Lucas · Oct 10, 2026

    Confirmed by the mission captain (proposal self-audit).

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