Fourth-order constant denominator perturbation theory
Bibliographic record
Abstract
A constant denominator perturbation theory based on a zeroth-order Hamiltonian characterized by degenerate subsets of orbitals is developed to fourth order. This formulation allows a decoupling of the numerators of the perturbation sequence, allowing for a much more rapid evaluation of the resulting sums. Although the theory is general, a constant denominator is chosen as a scaled difference between the average occupied and average virtual orbital energies. The correction for this choice is folded back into the perturbation. The scale can be chosen such that the first-order wave-function yields the lowest possible variational bound, or it can be chosen based upon the sequence of energy terms and the size of the wave function corrections. Results are presented within the localized bond model utilizing both the Pariser-Parr-Pople and CNDO model Hamiltonians. When the scale is chosen variational the second-order theory yields a useful bound, usually containing most of the basis set correlation. The third order appears as a scaled Langhoff-Davidson correction. The fourth-order theory displays remarkable stability, even in those cases in which Nesbet-Epstein partitioning seems poorly converged and Moller-Plesset theory uncertain. Other possible scalings are introduced and discussed, but the utility of these must await the more accurate predictions of fifth order for comparison.
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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 | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 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".