Using rectangular collocation with finite difference derivatives to solve electronic Schrödinger equation
Bibliographic record
Abstract
We show that a rectangular collocation method, equivalent to evaluating all matrix elements with a quadrature-like scheme and using more points than basis functions, is an effective approach for solving the electronic Schrödinger equation (ESE). We test the ideas by computing several solutions of the ESE for the H atom and the H2+ cation and several solutions of the Kohn-Sham equation for CO and H2O. In all cases, we achieve millihartree accuracy. Two key advantages of the collocation method we use are (1) collocation points need not have a particular distribution or spacing and can be chosen to reduce the required number of points - they need not converge any quadrature; (2) the better the basis is, the less sensitive the results are to the choice of the point set. The ideas of this paper make it possible to use any basis functions and thus open the door to using basis functions that are not Gaussians or plane waves. We use basis functions that are similar to Slater-type orbitals. They are rarely used with the variational method, but present no problems when used with collocation.
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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 | 0.001 | 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".