A Phase-Space Electronic Hamiltonian for Molecules in a Static Magnetic Field. I: Conservation of Total Pseudomomentum and Angular Momentum
Bibliographic record
Abstract
We develop a phase-space electronic structure theory of molecules in magnetic fields. For a system of electrons in a magnetic field with vector potential A ( r ̂), the usual Born–Oppenheimer Hamiltonian is the sum of the nuclear kinetic energy and the electronic Hamiltonian, ( P – q A ( X )) 2 /2 M + Ĥ e ( X ) (where q is a nuclear charge). To include the effects of coupled nuclear-electron motion in the presence of a magnetic field, we propose that the proper phase-space electronic structure Hamiltonian will be of the form ( P – q eff A ( X ) – e Γ ̂) 2 /2 M + Ĥ e ( X ). Here, q eff represents the screened nuclear charges and the Γ ̂ term captures the local pseudomomentum of the electrons. This form reproduces exactly the energy levels for a hydrogen atom in a magnetic field; moreover, single-surface dynamics along the eigenstates are guaranteed to conserve both (i) the total pseudomomentum and (ii) the total angular momentum in the direction of the magnetic field. This Hamiltonian form can be immediately implemented within modern electronic structure packages (where the electronic orbitals will now depend both on nuclear position ( X ) and nuclear momentum ( P )). One can expect to find novel beyond Born–Oppenheimer magnetic field effects for strong enough fields and nonadiabatic systems.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".