The quantum mechanical potential of mean force: density matrix formulation, estimators, and constrained path integral molecular dynamics integrators
Bibliographic record
Abstract
We obtain a formal expression for the quantum mechanical potential of mean force (PMF) in terms of the reduced density matrix associated with a reaction coordinate. We derive two path integral estimators for the derivative of the quantum mechanical PMF, which may be integrated to yield the PMF. For the first estimator, we perform the differentiation on the exact path integral, and for the second, the differentiation is performed after discretisation. These estimators are successfully validated for harmonic oscillator and Lennard-Jones dimer systems using constrained path integral Monte Carlo (PIMC) simulations at a series of temperatures. We propose two path integral molecular dynamics (PIMD) integrators, c-OBABO and c-BAOAB, to further test the estimators.Those are based on the path integral Langevin equation (PILE) with the addition of holonomic constraints. When the reaction coordinate is the distance between two centres of mass, we find that several exact expressions are accessible: the Fixman correction, the position constraint Lagrange multiplier, and various reaction coordinate derivatives. It is observed that c-BAOAB has a smaller time step error than c-OBABO. We show that both the PMF of a water dimer and its derivative obtained using c-BAOAB are in agreement with earlier path integral umbrella sampling results.
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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) | 0.000 | 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".