A new Gauss quadrature for multicentre integrals over STOs in the Gaussian integral transform approach
Bibliographic record
Abstract
When computing multicentre integrals over Slater-type orbitals (STOs) by means of the Shavitt and Karplus Gaussian integral transforms (Shavitt and Karplus 1962 J. Chem. Phys. 36 550), one usually ends up with a multiple integral of the form (Shavitt and Karplus 1965 J. Chem. Phys. 43 398) in which all the integrals are inter-related. The most widely used approach for computing such an integral is to apply a product of Gauss–Legendre quadratures for the integrals over [0, 1] while the semi-infinite term is evaluated by a special procedure. Although numerous approaches have been developed to accurately perform the integration over [0, ∞) efficiently, it is the aim of this work to add a new tool that could be of some benefit in carrying out the hard task of multicentre integrals over STOs. The new approach relies on a special Gauss quadrature referred to as Gauss–Bessel to accurately evaluate the semi-infinite integral of interest. In this work, emphasis is put on accuracy rather than efficiency since its aim is essentially to bring a proof of concept showing that Gauss–Bessel quadrature can successfully be applied in the context of multicentre integrals over STOs. The obtained accuracy is comparable to that obtained with other methods available in the literature.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".