Calculation of the Gaunt coefficients over real spherical harmonics
Bibliographic record
Abstract
The Gaunt coefficients are numerical coefficients that describe the interaction between two different angular momentum quantum states. In the quantum physics, this interaction affects the energy levels and structures of atoms and molecules. The calculation of Gaunt coefficients plays a significant role in molecular integral evaluations, which emerge from the solving Schrödinger equation. This work presents the analytical evaluation methods for the Gaunt coefficients using real spherical harmonics and associated Legendre functions. In this paper, the general integral form of the Gaunt coefficients is written in two main parts as polar and azimuthal integrals. The new analytical relations and symmetry properties are generated for these integrals. The analytical expressions are obtained in terms of the beta functions, and these coefficients are calculated for different atomic orbitals of angular momentum using the Mathematica programming language. The obtained numerical results are compared with the values in the literature and we found completely agreement to 15 digits.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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".