The Product Wavefunction: Hartree's Self-Consistent Field, Priced
Year: 2026
Type: preprint
Abstract: In 1928 Douglas Hartree gave physics its first working route from the Schrödinger equation to real many-electron atoms: take the many-electron wavefunction as a product of one-electron functions, each electron moving in the averaged field of the others, and iterate the field to self-consistency. I credit that paper at full strength: the self-consistent field founded computational atomic structure, stands underneath Hartree–Fock, density-functional practice and every mean-field method since, and its accuracy is no small thing — on my own bench the product at its one-parameter best stands within 1.9309% of the exact helium ground-state energy. Nothing in this paper falsifies the method or a number it returns. What I evaluate is the product form's derivation status. No step in 1928 derives the factorisation from the many-electron Schrödinger equation it is used on — the pair term 1/r₁₂ couples the coordinates inseparably, so the product is the separation, asserted — and self-consistency, a property of the iteration, is not a derivation of the ansatz. My bench evaluates what the assumption deletes, on declared inputs only, separated by successor addresses. The bare product returns −2.750000; screening inside the form (ζ* = 1.687500) recovers 0.097656 to −2.847656; and the form then stops 0.056068 E_h (1.525682 eV, 1.9309%) above the exact record −2.903724 — a gap separating into orbital shape 0.014024 E_h (repairable by the method's own iteration), exchange 0.000000 at this closed-shell singlet address, and correlation 0.042044 E_h (1.144076 eV, 74.9878% of the gap) that no iteration within the form can touch. Counted plainly: 1 product ansatz against 0 derivations in 1928. Grade for this census entry, printed at full size: **REASON FOUND / STILL TURNING** — PREMISE-ONLY, explicitly not a kill and not a falsification. The program headline stands unmoved: **vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.**
Author: Aldred, William Alexander
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225433285