An extremely efficient approach for accurate and rapid evaluation of three-centre two-electron Coulomb and hybrid integrals over<i>B</i>functions
Bibliographic record
Abstract
Analytic expressions of three-centre two-electron Coulomb and hybrid integrals over B functions are obtained using the Fourier transform method thoroughly explored by Steinborn's group. These analytic expressions involve semi-infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical Bessel functions in the integrands. We have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the H approach which greatly simplifies the application of the nonlinear -transformation. This work presents a rapid and accurate evaluation of these integrals, obtained by using a new approach, which we called S . This new method is based on the H and methods and some practical properties of spherical Bessel, reduced Bessel and sine functions. The S method has greatly simplified the calculations, avoiding the long and difficult implementation of the successive zeros of the spherical Bessel function and a method for solving linear systems, which are required by H and .
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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 | 0.001 |
| 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 | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".