Three‐center nuclear attraction, three‐center two‐electron Coulomb and hybrid integrals over B functions evaluated using the nonlinear <i>S</i>\documentclass{article}\pagestyle{empty}\begin{document}$\overline{D}$\end{document} transformation
Bibliographic record
Abstract
Abstract This work describes and applies the nonlinear \documentclass{article}\pagestyle{empty}\begin{document}$S\overline{D}$\end{document} transformation, recently developed by the first author, to accelerate the convergence of semi‐infinite integrals occurring in molecular structure electronic energy calculations. These energy terms must be evaluated precisely and quickly. The approach chosen involves expanding Slater type atomic orbital basis functions over B functions to exploit the compact Fourier transforms of B functions. The resulting integrals have been evaluated in previous work using the nonlinear D (due to Levin and Sidi) and \documentclass{article}\pagestyle{empty}\begin{document}$\overline{D}$\end{document} (due to Sidi) transformations and some improved nonlinear transformations HD and \documentclass{article}\pagestyle{empty}\begin{document}$H\overline{D}$\end{document} have been developed. These methods are precise and faster than alternatives for convergence acceleration of oscillatory integrals involved. In the present work a more recent, highly compact, and convenient nonlinear transformation \documentclass{article}\pagestyle{empty}\begin{document}$S\overline{D}$\end{document} is described and applied to the three‐center nuclear attraction integral common to ab initio and density functional theory work. Three‐center Coulomb and hybrid integrals which occur in ab initio self‐consistent field work are also evaluated. This approach is shown to represent a further substantial advance in efficiency and should lead to a definitive suite of ab initio Slater software. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".