On the importance of the “density per particle” (shape function) in the density functional theory
Bibliographic record
Abstract
The central role of the shape function sigma(r) from the density functional theory (DFT), the ratio of the electron density rho(r) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential micro of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined.
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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 | 0.008 | 0.019 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.009 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.003 | 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".