Correlated exponential-basis integrals with logarithmic integrands
Bibliographic record
Abstract
Formulas are derived for three-body integrals containing all the interparticle coordinates as exponentials and with one coordinate also entering logarithmically. These integrals are relevant to the computation of relativistic and quantum electrodynamic effects in two-electron systems, and can contribute to an assessment of current approaches for the treatment of such effects. The present work is a generalization of earlier studies by others using Hylleraas wavefunctions (those in which only two of the three interparticle coordinates exhibit exponential dependence). Alternate formulations are developed to yield stability for all values of the parameters. The formulas have been checked by intercomparison of different methods, by agreement with three-dimensional numerical integration, and by their reduction to limiting cases corresponding to Hylleraas integrals. Representative integrals have been tabulated to illustrate the issues governing the choice of optimum formulas.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".