A fresh look at the computation of spherically averaged electron momentum densities for wave functions built from Gaussian‐type functions
Bibliographic record
Abstract
Abstract A new method for deriving the spherical average of the product of the Fourier transforms of two Gaussians is described. The method generates the result in terms of a few spherical Bessel functions leading to expressions that are much more compact than those in the literature. These integrals are required in the computation of spherically averaged electron momentum densities, and rotationally averaged X‐ray and high‐energy electron scattering crossections. All integrals needed for spherically averaged momentum densities are tabulated for s, p, d, and f‐type Gaussians. As an illustration of the method, we apply it to calculate spherically averaged electron momentum densities and their moments for H2O. The calculations are performed at the Hartree–Fock and 2nd‐ and 4th‐order Møller–Plesset perturbation theory levels with the aug‐cc‐pVDZ and aug‐cc‐pVTZ basis sets. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| 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 | 0.001 |
| 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".