Local Exchange Potentials for Electronic Structure Calculations
Bibliographic record
Abstract
The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock model as well as in some instances of the density functional theory. In a number of applications, it is convenient to approximate this integral operator by a multiplication operator, i.e. by a local potential. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator including the Slater potential, the optimized effective potential (OEP), the Krieger-Li-Iafrate (KLI) approximation and the common-energy denominator approximation (CEDA) to the OEP, and the effective local potential (ELP). In particular, we show that the Slater, KLI, CEDA and ELP potentials all can be defined as solutions of certain variational problems, and we provide a rigorous derivation of the OEP integral equation. We also establish an existence result for a coupled system of nonlinear partial differential equations introduced by Slater to approximate the Hartree-Fock equations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".