Variable-Metric\nLocalization of Occupied and Virtual\nOrbitals
Bibliographic record
Abstract
The key idea of the variable-metric\napproach to orbital localization\nis to allow nonorthogonality between orbitals while, at the same time,\npreventing them from becoming linearly dependent. The variable-metric\nlocalization has been shown to improve the locality of occupied nonorthogonal\norbitals relative to their orthogonal counterparts. In this work,\nnumerous localization algorithms are designed and tested to exploit\nthe conceptual simplicity of the variable-metric approach with the\ngoal of creating a straightforward and reliable localization procedure\nfor virtual orbitals. The implemented algorithms include the steepest\ndescent, conjugate gradient (CG), limited-memory Broyden–Fletcher–Goldfarb–Shanno\n(L-BFGS), and hybrid procedures as well as trust-region (TR) methods\nbased on the CG and Cauchy-point subproblem solvers. Comparative analysis\nshows that the CG-based TR algorithm is the best overall method to\nobtain nonorthogonal localized molecular orbitals (NLMOs), occupied\nor virtual. The L-BFGS and CG algorithms can also be used to obtain\nNLMOs reliably but often at higher computational cost. Extensive tests\ndemonstrate that the implemented methods allow us to obtain well-localized\nBoys–Foster (i.e., maximally localized Wannier functions) and\nPipek–Mezey, orthogonal and nonorthogonal, and occupied and\nvirtual orbitals for a variety of gas-phase molecules and periodic\nmaterials. The tests also show that virtual NLMOs, which have not\nbeen described before, are, on average, 13% (Boys–Foster) and\n18% (Pipek–Mezey) more localized than their orthogonal counterparts.
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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 | 0.001 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.581 | 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".