The use of so(2,1) algebra for the evaluation of atomic integrals: The study of two-electron atoms
Bibliographic record
Abstract
The use of the so(2,1) algebra for the study of the two-electron atoms is suggested. The radial part of the two-electron function is expanded into the products of the one-electron functions. These one-electron functions form complete, entirely discrete set and are identified as the eigenfunctions of one of the generators of the so(2,1) algebra. By applying this algebra we are able to express all the matrix elements in analytic and numericaly stable form. For matrix elements of the two-electron interaction this is done in three steps, all of them completely novel from the methodological point of view. First, repulsion integrals over four radial functions are written as a linear combination of the integrals over two radial functions and the coefficients of the linear combination are given in terms of hypergeometric functions. Second, combining algebraic technique with the integration by parts we derive recurrence relations for the repulsion integrals over two radial functions. Third, the derived recurrence relations are solved analytically in terms of the hypergeometric functions. Thus we succeed in expressing the repulsion integrals as rational functions of the hypergeometric functions. In this way we resolve the problem of the numerical stability of calculation of the repulsion integrals. Finally, as an illustration, the configuration interaction calculation of the lowest lying states of the He atom is discussed.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".