Using C3v symmetry with polyspherical coordinates for methane
Bibliographic record
Abstract
It is well known that the group of operators that commutes with the Hamiltonian operator can be used to facilitate the calculation of energy levels. Due to numerical errors in the computation of Hamiltonian matrix elements, it may happen that the matrix representation of a group operator does not commute with the Hamiltonian matrix although the group operator does commute with the Hamiltonian operator. We demonstrate that it is possible, even in this case, to use the single-symmetry and multisymmetry symmetry-adapted Lanczos (SAL) methods to efficiently compute energy levels. The two SAL methods are applied to the calculation of the bend levels of methane using the G6 symmetry group and polyspherical angles. We show that although potential matrix elements are corrupted by quadrature error, it is nonetheless possible to take advantage of the full symmetry of the polyspherical basis. For a CX3Y-type molecule the symmetry-adapted method of this paper would enable one to exploit all of the symmetry of the molecule.
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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 | 0.000 | 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".