The tight-binding formulation of the Kronig-Penney model
Bibliographic record
Abstract
Electronic band structure calculations are frequently parametrized in tight-binding form; the latter representation is then often used to study electron correlations. In this paper we provide a derivation of the tight-binding model that emerges from the exact solution of a particle bound in a periodic one-dimensional array of square well potentials. We derive the dispersion for such a model, and show that an effective next-nearest-neighbour hopping parameter is required for an accurate description. An electron-hole asymmetry is prevalent except in the extreme tight-binding limit, and emerges through a "next-nearest-neighbour" hopping term in the dispersion. We argue that this does not necessarily imply next-nearest-neighbour tunneling; this assertion is demonstrated by deriving the transition amplitudes for a two-state effective model that describes a double-well potential, which is a simplified precursor to the problem of a periodic array of potential wells. A next-nearest-neighbour tunneling parameter is required for an accurate description even though there are no such neighbours.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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".