Exponential variational expansion in relative coordinates and scalar coupling coefficients for three-body systems
Bibliographic record
Abstract
The exponential variational expansion is applied to highly accurate computations of the ground states in the Ps− and ∞H− ions. The determined variational energies for these systems are −0.262 005 070 232 980 107 7335 and −0.527 751 016 544 377 196 5668 au respectively. These energies and corresponding wavefunctions are significantly more accurate than values known from earlier studies. A number of bound state properties are determined for the Ps− ion. The method of scalar coupling for three-body systems is developed. A general analytical expression is derived for the overlap integral between two scalar functions ϕ(r32, r31, r21) and ψ(r32, r31), written in relative coordinates and one-particle coordinates respectively. The case of bound S(L = 0) states in Coulomb three-body systems is discussed in detail. In this case, explicit analytical formulae for the three-body scalar coupling coefficients have been produced and tested in actual highly accurate calculations for the Ps− and ∞H− ions. The approach developed in this work can be applied to a large number of real three-body problems.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".