Prove2Me

Overlap of Slater determinants is the determinant of orbital overlaps (Eqs. (7), (46))

Proved
KochHF.slater_overlap_det

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

hartree-fockmathematical-physicssecond-quantization

Let α1,…,αN\alpha_1,\dots,\alpha_N and β1,…,βN\beta_1,\dots,\beta_N be arbitrary orbitals in CK\mathbb C^K and ∣Φα⟩=cαN†⋯cα1†∣0⟩|\Phi_\alpha\rangle=c^\dagger_{\alpha_N}\cdots c^\dagger_{\alpha_1}|0\rangle, ∣Φβ⟩|\Phi_\beta\rangle likewise. Then

⟨Φα∣Φβ⟩=det⁡(⟨α1∣β1⟩⋯⟨α1∣βN⟩⋮⋱⋮⟨αN∣β1⟩⋯⟨αN∣βN⟩).\langle\Phi_\alpha|\Phi_\beta\rangle=\det\begin{pmatrix}\langle\alpha_1|\beta_1\rangle&\cdots&\langle\alpha_1|\beta_N\rangle\\ \vdots&\ddots&\vdots\\ \langle\alpha_N|\beta_1\rangle&\cdots&\langle\alpha_N|\beta_N\rangle\end{pmatrix}.

In particular, Slater determinants built from orthonormal orbitals are normalized, and determinants of orthonormal basis orbitals form an orthonormal basis.

Preamble
import Mathlib
import Definitions.Def_KochHF_FockSpace

open Matrix
Formal statement
namespace KochHF

theorem slater_overlap_det {K N : ℕ} (α β : Fin N → Fin K → ℂ) :
    fockInner (slater α) (slater β) = (Matrix.of fun i j => orbInner (α i) (β j)).det := 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. 1, p. 2.4, Eq. (7); Sec. 2.4, p. 2.13, Eq. (46).
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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 K,NK,N and all families α0,…,αN−1\alpha_0,\dots,\alpha_{N-1} and β0,…,βN−1\beta_0,\dots,\beta_{N-1} in CK\mathbb C^K (no orthonormality, independence or nonvanishing assumed):

⟨Φα∣Φβ⟩=det⁡(⟨αi∣βj⟩)i,j=0N−1.\langle\Phi_\alpha|\Phi_\beta\rangle=\det\bigl(\langle\alpha_i|\beta_j\rangle\bigr)_{i,j=0}^{N-1}.

For N=0N=0 both sides equal 11 (empty determinant, ⟨0∣0⟩\langle0|0\rangle). If N>KN>K both vectors are zero and the Gram-type determinant also vanishes.

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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