Compact wave functions for four-electron atomic systems
Bibliographic record
Abstract
A procedure is developed to construct improved wave functions for the bound states in various four-electron atoms and ions. The bound state wave functions are improved in the sense that our trial wave functions are compact and accurate. We employ two independent four-electron spin functions ${\ensuremath{\chi}}_{1}=\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}+\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}$ and ${\ensuremath{\chi}}_{2}=2\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}+2\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}$ in each trial wave function. Optimization of the nonlinear parameters is performed for each radial function. Our procedure is applied to obtain the total energies of the ground $^{1}S^{e}$ states in the following four-electron berylliumlike systems: ${\mathrm{Li}}^{\ensuremath{-}}$, Be, ${\mathrm{B}}^{+}$, ${\mathrm{C}}^{2+}$, ${\mathrm{N}}^{3+}$, ${\mathrm{O}}^{4+}$, ${\mathrm{F}}^{5+}$, ${\mathrm{Ne}}^{6+}$, ${\mathrm{Na}}^{7+}$, and ${\mathrm{Mg}}^{8+}$. All our trial wave functions used in this study are very compact. In fact, each of these functions contains only 300 basis (radial) functions which are constructed from ten-dimensional gaussoids of the five-body relative coordinates ${r}_{12}$, ${r}_{13}$, ${r}_{14}$, ${r}_{15}$, ${r}_{23}$, ${r}_{24}$, ${r}_{25}$, ${r}_{34}$, ${r}_{35}$, and ${r}_{45}$.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".