A contracted basis-Lanczos calculation of vibrational levels of methane: Solving the Schrödinger equation in nine dimensions
Bibliographic record
Abstract
We present a contracted basis-iterative method for calculating numerically exact vibrational energy levels of methane (a 9D calculation). The basis functions we use are products of eigenfunctions of bend and stretch Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wavefunctions well, yet are simple enough that matrix-vector products may be evaluated efficiently. We use Radau polyspherical coordinates. The bend functions are computed in a nondirect product finite basis representation [J. Chem. Phys. 118, 6956 (2003)] and the stretch functions are computed in a product potential optimized discrete variable (PODVR) basis. The memory required to store the bend basis is reduced by a factor of ten by storing it on a compacted grid. The stretch basis is optimized by discarding PODVR functions with high potential energies. The size of the primitive basis is 33 billion. The size of the product contracted basis is six orders of magnitude smaller. Parity symmetry and exchange symmetry between two of the H atoms are employed in the final product contracted basis. A large number of vibrational levels are well converged. These include almost all states up to 8000 cm−1 and some higher local mode stretch bands.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".