The Localization–Delocalization Matrix and the Electron-Density-Weighted Connectivity Matrix of a Finite Graphene Nanoribbon Reconstructed from Kernel Fragments
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Bibliographic record
Abstract
Bader's quantum theory of atoms in molecules (QTAIM) and chemical graph theory, merged in the localization-delocalization matrices (LDMs) and the electron-density-weighted connectivity matrices (EDWCM), are shown to benefit in computational speed from the kernel energy method (KEM). The LDM and EDWCM quantum chemical graph matrices of a 66-atom C46H20 hydrogen-terminated armchair graphene nanoribbon, in 14 (2×7) rings of C2v symmetry, are accurately reconstructed from kernel fragments. (This includes the full sets of electron densities at 84 bond critical points and 19 ring critical points, and the full sets of 66 localization and 4290 delocalization indices (LIs and DIs).) The average absolute deviations between KEM and directly calculated atomic electron populations, obtained from the sum of the LIs and half of the DIs of an atom, are 0.0012 ± 0.0018 e(-) (∼0.02 ± 0.03%) for carbon atoms and 0.0007 ± 0.0003 e(-) (∼0.01 ± 0.01%) for hydrogen atoms. The integration errors in the total electron population (296 electrons) are +0.0003 e(-) for the direct calculation (+0.0001%) and +0.0022 e(-) for KEM (+0.0007%). The accuracy of the KEM matrix elements is, thus, probably of the order of magnitude of the combined precision of the electronic structure calculation and the atomic integrations. KEM appears capable of delivering not only the total energies with chemical accuracy (which is well documented) but also local and nonlocal properties accurately, including the DIs between the fragments (crossing fragmentation lines). Matrices of the intact ribbon, the kernels, the KEM-reconstructed ribbon, and errors are available as Supporting Information .
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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 | 0.001 | 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it