A new algorithm for accurate and fast numerical evaluation of hybrid and three-centre two-electron Coulomb integrals over Slater-type functions
Bibliographic record
Abstract
Recently we developed a new algorithm for a fast and accurate numerical evaluation of three-centre nuclear attraction integrals over Slater-type functions, the results obtained were very satisfactory. Now, it is shown that this new algorithm can also be applied to hybrid and three-centre two-electron Coulomb integrals over Slater-type functions. These integrals, which are numerous, are very difficult to evaluate to a high accuracy, because of the presence of spherical Bessel functions and hypergeometric series in the integrands. We have proved that the integrands that occur in the analytic expressions of the integrals under consideration satisfy all the conditions to apply the approach. The hypergeometric functions which occur in the semi-infinite integrals can be expressed as a finite expansion and the semi-infinite integrals involving the spherical Bessel functions can be transformed into semi-infinite integrals involving the simple sine function. The numerical results obtained with linear and non-linear systems illustrate clearly a further improvement of accuracy and a substantial reduction in calculation times. Comparisons with existing codes, STOP developed by Bouferguene et al (1996 Int. J. Quantum Chem. 57 801) and ADGGSTNGINT developed by Rico et al (1997 Comp. Phys. Commun. 105 216), are listed.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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".