On 1/<i>Z</i> expansion, the critical charge for a two-electron system, and the Kato theorem
Bibliographic record
Abstract
The 1/Z expansion for the ground state energy of the Coulomb system of an infinitely massive center of charge Z and two electrons (two-electron ionic sequence) is studied. A critical analysis of the 1/Z coefficients presented in Baker et al. (Phys. Rev. A, 41, 1247 (1990)) is performed and its numerical deficiency is indicated, leading, in particular, to unreliable decimal digits beyond digits 11–12 of the first coefficients. We made a consistency check of the 1/Z-expansion with accurate energies for Z = 1–10: the weighted partial sums of the 1/Z expansion with Baker et al. coefficients reproduce systematically the ground state energies of two-electron ions with Z ≥ 2 up to 12 decimal digits and for Z = 1 up to 10 decimal digits calculated by Nakashima and Nakatsuji (J. Chem. Phys. 127, 224104 (2007)) with unprecedented accuracy. This rules out the presence of non-analytic terms at Z = ∞ contributing to the first 10–12 decimal digits in the ground state energy; it agrees with the Kato theorem about convergence of the 1/Z expansion within that accuracy. The ground state energy of two-electron ions Z = 11 (Na9+) and Z = 12 (Mg10+) is calculated with 12 decimal digits. This study can be considered as the independent confirmation of the correctness of 10 decimal digits in all 401 coefficients of 1/Z-expansion printed in Baker et al. (Phys. Rev. A, 41, 1247 (1990)).
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| 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".