Direct unconstrained variable-metric localization of one-electron orbitals
Bibliographic record
Abstract
Spatially localized one-electron orbitals, orthogonal and nonorthogonal, are widely used in electronic structure theory to describe chemical bonding and speed up calculations. In order to avoid linear dependencies of localized orbitals, the existing localization methods either constrain orbital transformations to be unitary, that is, metric preserving, or, in the case of variable-metric methods, fix the centers of nonorthogonal localized orbitals. Here, we developed a different approach to orbital localization, in which these constraints are replaced with a single restriction that specifies the maximum allowed deviation from the orthogonality for the final set of localized orbitals. This reformulation, which can be viewed as a generalization of existing localization methods, enables one to choose the desired balance between the orthogonality and locality of the orbitals. Furthermore, the approach is conceptually and practically simple as it obviates the necessity in unitary transformations and allows to determine optimal positions of the centers of nonorthogonal orbitals in a unconstrained and straightforward minimization procedure. It is demonstrated to produce well-localized orthogonal and nonorthogonal orbitals with the Berghold and Pipek-Mezey localization functions for a variety of molecules and periodic materials including large systems with non-trivial bonding.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".