Relativistic corrections to vibrational states of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Mu</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> beyond the Born-Oppenheimer approximation
Bibliographic record
Abstract
We present the ab initio calculation of leading-order relativistic corrections to the pure vibrational energies of the muonium-muonium molecule (${\mathrm{Mu}}_{2}$), an exotic four-lepton system composed of two positive muons ${\ensuremath{\mu}}^{+}$ and two electrons. These corrections are evaluated perturbatively using variational wave functions constructed from an explicitly correlated Gaussian basis, within a fully non-Born-Oppenheimer framework. We accurately determine the rotationless vibrational spectrum, dissociation energy, and key spectroscopic constants, including the harmonic constant ${\ensuremath{\omega}}_{e}=13\phantom{\rule{0.16em}{0ex}}322.11\phantom{\rule{0.28em}{0ex}}{\text{cm}}^{\ensuremath{-}1}$ and the anharmonicity constant ${\ensuremath{\omega}}_{e}{x}_{e}=1097.37\phantom{\rule{0.28em}{0ex}}{\text{cm}}^{\ensuremath{-}1}$. Furthermore, we present a range of expectation values for various structural quantities that provide insight into the intrinsic properties of this system. Our results represent a meaningful step toward a more accurate theoretical description of ${\mathrm{Mu}}_{2}$.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".