Two electrons in a cylindrical box: An exact configuration-interaction solution
Bibliographic record
Abstract
The problem of two Coulombically interacting electrons confined in a cylindrical potential box with impenetrable walls is solved by the method of full configuration interaction (CI) for states with zero axial component of the total orbital angular momentum, including the ground state ${}^{1}{\ensuremath{\Sigma}}_{g}^{+}$. The number of variables in the Schr\"odinger equation is reduced from six to five by separating the ${\mathrm{L\ifmmode \hat{}\else \^{}\fi{}}}_{z}$ operator from the complete Hamiltonian. Two-electron basis sets of symmetry-adapted configuration state functions are constructed from real eigenfunctions of the one-electron Hamiltonian for a cylindrical potential box. Kinetic energy integrals are evaluated analytically. Electron repulsion integrals are computed by developing a separable expression for the $1/{r}_{12}$ operator in the form of a double Fourier-Bessel-Fourier-cosine series and using enough terms to achieve a relative accuracy of ${10}^{\ensuremath{-}10}$ for the eigenvalues. Energy-level ordering and electron densities of selected electronic states are reported for cylindrical cavities of varying size with a fixed 1:1 length-to-diameter ratio. A remarkable feature of this system is that convergence of the full CI energy with respect to the one-electron basis set varies dramatically with the extent of confinement: For small cavities, the rate of approaching the basis-set limit is similar to that for two-electron atoms but is significantly faster for large cavities. Formation of singlet and triplet Wigner molecules is observed in cylindrical cavities with radii greater than about 5 bohrs.
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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 | 0.001 |
| 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".