Variational Hirshfeld Partitioning: General Framework and the Additive Variational Hirshfeld Partitioning Method
Bibliographic record
Abstract
We introduce the general mathematical framework of variational Hirshfeld partitioning, wherein the best possible approximation to a molecule’s electron density is obtained by minimizing the f -divergence between the molecular density and a non-negative linear combination of (normalized) basis functions. This framework subsumes several existing methods that variationally optimize their pro-atoms, like (Gaussian) iterative stockholder analysis (ISA and GISA) and minimal basis iterative stockholder partitioning (MBIS), and provides a solid foundation for developing mathematically rigorous partitioning schemes. In this paper, we delve into the mathematical underpinnings of Hirshfeld-inspired partitioning schemes and show that among all the valid f -divergence measures only the extended Kullback–Leibler is a suitable choice. This led us to develop a novel partitioning scheme, called additive variational Hirshfeld (AVH), which constructs the pro-molecular density as a convex linear combination of the densities from selected states of isolated atoms and atomic ions. The AVH method is size-consistent with a unique solution and provides a straightforward approach for adding constraints for fragment properties. It also results in an intuitively appealing valence-bond-like decomposition of the molecular density as a weighted average of the densities of the atomic states in the molecule; that is, the AVH atomic density is a minimal deformation of the corresponding isolated atomic reference state’s density. Compared to other variational Hirshfeld variants, our numerical results show that AVH yields chemically interpretable and sensible atomic charges that are not excessively large and demonstrate computational robustness.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 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".