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 H 2 O. 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 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 | 0.000 | 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".