Critical Nuclear Charge for Two-Electron Atoms
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Bibliographic record
Abstract
The critical nuclear charge Z(c) required to bind a nucleus plus two electrons in a heliumlike atom has recently been an area of active study, resulting in a disagreement with earlier calculations and with the value obtained from the radius of convergence 1/Z* of a 1/Z expansion of the energy. In order to resolve the disagreement, have performed high-precision variational calculations in Hylleraas coordinates. With the double basis set method, we have been able to obtain good convergence for Z very close to Z(c), which together with the Hellmann-Feynman theorem yields the value Z(c) = 0.911,028,224,077,255,73(4), corresponding to 1/Z(c) = 1.097,660,833,738,559,80(5). This value is in agreement with the value obtained by Baker et al. [Phys. Rev. A 41, 1247 (1990)]. A significant feature of the results is that the outer electron remains localized near the nucleus, even at Z = Z(c), and the bound state evidently changes smoothly into a shape resonance for Z<Z(c). A qualitative polarization potential is proposed to account for the resonance, and the radial distribution function for the electron density is calculated.
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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 it