Green's function method in quantum chemistry: New numerical algorithm for the Dirac equation with complex energy and Fermi‐model nuclear potential
Bibliographic record
Abstract
Abstract We present a new effective approach to construction of the electron Green function for the Dirac equation with a nonsingular central nuclear Fermi‐model potential and complex energy. We represent the radial Green function as a combination of two fundamental solutions of the Dirac equation. The approach proposed includes a procedure of generating the relativistic electron functions Ψ with performance of the gauge invariance principle. To reach the gauge invariance principle performance, we use earlier developed QED perturbation theory approach. In the fourth order of the QED perturbation theory (PT) there are diagrams, whose contribution into imaginary part of radiation width ImdE for the multielectron system accounts for many‐body correlation effects. A minimization of the functional ImdE leads to integral‐differential Kohn‐Sham‐like density functional equations. Further check for the gauge principle performance is realized by means of the Ward identities. In the numerical procedure we use the effective algorithm, within which a definition of the Dirac equation fundamental solutions is reduced to solving the single system of the differential equations. This system includes also the differential equations for the Fermi‐model nuclear potential and equations for calculating the integrals of the ∫ ∫ dr 1 dr 2 type in the Mohr formula for definition of the self‐energy shift to atomic levels energies. Such an approach allows to compensate a main source of the errors, connected with numerical integration ∫ d ξ and summation on χ in the Mohr expressions during calculating the self‐energy radiative correction to the atomic levels energies. Some numerical illustrations of applying the approach within QED PT to calculate the intermediate and high‐Z Li‐like ions transitions energies are presented. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
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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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".