Solution of the Schrödinger equation for two electrons in axially symmetric cavities
Bibliographic record
Abstract
We present a method of solving the Schr\"odinger equation for two electrons confined in an almost arbitrary infinite-potential well of ${D}_{\ensuremath{\infty}h}$ symmetry. The boundary conditions for the wave function are imposed by devising a one-electron basis set that incorporates detailed information about the shape of the cavity. Two-electron integrals are evaluated using a fully separable expansion of the Coulomb operator similar to that developed earlier by the authors for a cylindrical cavity. To illustrate the capabilities of our approach, we present full configuration-interaction solutions for the lowest ${}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and ${}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ states of two electrons in four axially symmetric cavities: a sphere, a cylinder, a bulged cylinder, and a capsule. By comparing the ground-state electronic energies for different cavities of equal volume $V$, we find that spherical confinement is optimal for $V\ensuremath{\lesssim}1500{a}_{0}^{3}$; for larger volumes, the capsule-shaped system has a significantly lower energy. This switchover is driven by the same physical mechanism as Wigner crystallization and may serve as an indication of that process. The magnitude of the singlet-triplet gap strongly depends on the shape of the cavity and, for a fixed volume, decreases in the following order: bulged cylinder, sphere, cylinder, capsule.
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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".