Critical nuclear charge and shape resonances for the two-electron Hamiltonian
Bibliographic record
Abstract
The hydrogen negative ion H$^-$ is the simplest two-electron system that exists in nature. This system is not only important in astrophysics but it also serves as an ideal ground to study electron-electron correlations. The peculiar balance of the correlations between the two electrons with the interaction of electron-nucleus in H$^-$ makes this system to have only two bound states, one being the ground state $1s^2\,^{1}\!S^e$ and the other the doubly-excited metastable state $2p^2\,^{3}\!P^e$ embedded below the hydrogen $n=2$ threshold. Here we report a calculation for the $2p^2\,^{3}\!P^e$ state of H$^-$ that yields the energy eigenvalue $E=-0.125\,355\,451\,242\,864\,058\,376\,012\,313\,25(2)$, in atomic units. Our result substantially improves the best available result by 16 orders of magnitude. We further study the critical nuclear charge $Z_{\rm cr}$, the minimum value of nuclear charge $Z$ that is required to bind a nucleus and two electrons. Our determination of $Z_{\rm cr}$ for the $2p^2\,^{3}\!P^e$ state of two-electron systems is $Z_{\rm cr}=0.994\,781\,292\,240\,366\,246\,3(1)$, corresponding to $1/Z_{\rm cr}= 1.005\,246\,085\,546\,985\,509\,4(1)$, which improves the best published value of $Z_{\rm cr}$ by about 10 orders of magnitude. We further investigate in a definitive way the unexplored regime of $Z < Z_{\rm cr}$ using the method of complex scaling and establish precise shape resonance poles for the state of $2p^2\,^{3}\!P^e$ in the complex energy plane.
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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".