Universal mathematical identities in density functional theory: Results from three different spin-resolved representations
Bibliographic record
Abstract
This paper supersedes previous theoretical approaches to conceptual DFT because it provides a unified and systematic approach to all of the commonly considered formulations of conceptual DFT, and even provides the essential mathematical framework for new formulations. Global, local, and nonlocal chemical reactivity indicators associated with the "closed-system representation" ([N(alpha),N(beta),nu(alpha)(r),nu(beta)(r)]) of spin-polarized density functional theory (SP-DFT) are derived. The links between these indicators and the ones associated with the "open-system representation" ([mu(alpha),mu(beta),nu(alpha)(r),nu(beta)(r)]) are derived, including the spin-resolved Berkowitz-Parr identity. The Legendre transform to the "density representation" ([rho(alpha)(r),rho(beta)(r)]) is performed, and the spin-resolved Harbola-Chattaraj-Cedillo-Parr identities linking the density representation to the closed-system and open-system representations are derived. Taken together, these results provide the framework for understanding chemical reactions from both the electron-following perspective (using either the closed-system or the open-system representation) and electron-preceding perspective (density representation). A powerful matrix-vector notation is developed; with this notation, identities in conceptual DFT become universal. Specifically, this notation allows the fundamental identities in conventional (spin-free) conceptual DFT, the [N(alpha),N(beta)] representation, and the [N=N(alpha)+N(beta),N(S)=N(alpha)-N(beta)] representation to be written in exactly the same forms. In cases where spin transfer and electron transfer are coupled (e.g., radical+molecule reactions), we believe that the [N(alpha),N(beta)] representation may be more useful than the more common [N,N(S)] representation.
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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 | 0.008 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| 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".