{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"fc597d706507494550e9c5ffae44af35435f1f07d921626bf3750677946ea9e8","created":"2026-10-03T07:17:58Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"69d5e22946860339503dd62f1fcfb4911a675ac66c02518da4c1bbdfda0ff88d","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"chem.reaction-dynamics.quantum-tst-deep-tunneling","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"Transition-state theory estimates a reaction rate from the flow across the top of the energy barrier without following full trajectories. Its quantum version, built on ring polymers (chains of beads that represent a quantum particle), is unique and exact in some limits but can fail when tunneling dominates.","posed_since":"","precise":"Ring-polymer molecular dynamics transition-state theory (RPMD-TST) is the unique $t\\to 0+$ limit of a quantum flux-side time-correlation function and is exact when no recrossing of the dividing surface or its ring-polymer generalizations occurs (Hele and Althorpe, J. Chem. Phys. 2013, https://doi.org/10.1063/1.4792697). Below the crossover temperature $T_{x}=\\hbar\\omega_{b}/(2\\pi)$ ($\\omega_{b}=\\text{barrier frequency},\\ k_{B}=1$) it deviates from exact quantum rates for asymmetric barriers. Derive a rate theory without real-time dynamics whose error vanishes both without recrossing and in the semiclassical instanton limit, or rigorous bounds on the RPMD-TST error versus $T/T_{x}$ and barrier asymmetry, tested on Eckart barriers and H + H2 type reactions.","problem_ref":null,"references":"","settled_by":"A derivation with error bounds verified against exact quantum rates for one-dimensional and multidimensional benchmark barriers.","status_note":"Uniqueness and exactness without recrossing were derived in 2013; no transition-state theory is known to be exact in deep tunneling through asymmetric barriers.","title":"A quantum transition-state theory valid in the deep-tunneling 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