Tailored Gauss quadratures, a promising route for an efficient evaluation of multicenter integrals over B functions
Bibliographic record
Abstract
In the framework of the Fourier integral transform, complicated multicenter integrals, e.g., three-center nuclear attraction and exchange integrals, over B functions involve a multiple integral (double or triple), the innermost of which is a Hankel transform of an exponentially decreasing term. Because of the oscillatory nature of the Hankel transform and the order in which it occurs in the definition of multicenter integrals, i.e., innermost, an efficient evaluation of such a quantity requires highly performant algorithms. In this context, extrapolation techniques emerged, during the past decade, as a possible solution to the problem of evaluating the oscillating semi-infinite integral. With a view to improving the efficiency of future algorithms, this contribution introduces a new technique for the evaluation of the oscillating integral by means of a tailored Gaussian quadrature. Using the case of three-center nuclear attraction integrals as a working example, it is shown that the new approach allows the semi-infinite integral to be evaluated accurately if not exactly. Further, when the roots and weights of the quadrature are available, a complexity analysis of our algorithm shows encouraging results compared to nonlinear extrapolation techniques.
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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 | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".