A simultaneous probability density for the intracule and extracule coordinates
Bibliographic record
Abstract
We introduce the intex density X(R,u), which combines both the intracular and extracular coordinates to yield a simultaneous probability density for the position of the center-of-mass radius (R) and relative separation (u) of electron pairs. One of the principle applications of the intex density is to investigate the origin of the recently observed secondary Coulomb hole. The Hartree-Fock (HF) intex densities for the helium atom and heliumlike ions are symmetric functions that may be used to prove the isomorphism 2I(2R)=E(R), where I(u) is the intracule density and E(R) is the extracule density. This is not true of the densities that we have constructed from explicitly correlated wave functions. The difference between these asymmetric functions and their symmetric HF counterparts produces a topologically rich intex correlation hole. From the intex hole distributions (X(exact)(R,u)-X(HF)(R,u)), we conclude that the probability of observing an electron pair with a very large interelectronic separation increases with the inclusion of correlation only when their center-of-mass radius is close to half of their separation.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".