A cusp criterion for a local phase-space bottleneck in a minimal adiabatic Marcus Hamiltonian
Abstract
Marcus--Hush theory describes electron transfer through reorganization energies, driving forces, and electronic coupling, but it does not by itself determine when an adiabatic model possesses a local Hamiltonian bottleneck. We study a minimal asymmetric two-degree-of-freedom lower-sheet Hamiltonian with an exactly separable harmonic transverse mode. For this conservative model, we establish a necessary-and-sufficient root count for the stationary points and recover the known cusp boundary for disappearance of the lower-sheet double well. We then interpret that boundary in Hamiltonian phase space: inside the cusp, the lower sheet has an index-one saddle and the associated local transition-state structures --- an unstable periodic-orbit NHIM, its stable and unstable manifolds, and an attached dividing surface --- exist; outside the cusp, this local saddle-mediated structure is absent. The result applies to this conservative adiabatic potential-energy Hamiltonian. It does not describe solvent free-energy dynamics, bath-induced mode coupling, or nonadiabatic transfer, and it does not predict Marcus rates for realistic condensed-phase electron-transfer systems.
Versions
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BibTeX
@misc{wiggins2026cusp,
title = {{A cusp criterion for a local phase-space bottleneck in a minimal adiabatic Marcus Hamiltonian}},
author = {Stephen Wiggins},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.17385},
eprint = {2605.17385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17385},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A cusp criterion for a local phase-space bottleneck in a minimal adiabatic Marcus Hamiltonian
AU - Stephen Wiggins
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.17385
UR - https://arxiv.org/abs/2605.17385
AB - Marcus--Hush theory describes electron transfer through reorganization energies, driving forces, and electronic coupling, but it does not by itself determine when an adiabatic model possesses a local Hamiltonian bottleneck. We study a minimal asymmetric two-degree-of-freedom lower-sheet Hamiltonian with an exactly separable harmonic transverse mode. For this conservative model, we establish a necessary-and-sufficient root count for the stationary points and recover the known cusp boundary for disappearance of the lower-sheet double well. We then interpret that boundary in Hamiltonian phase space: inside the cusp, the lower sheet has an index-one saddle and the associated local transition-state structures --- an unstable periodic-orbit NHIM, its stable and unstable manifolds, and an attached dividing surface --- exist; outside the cusp, this local saddle-mediated structure is absent. The result applies to this conservative adiabatic potential-energy Hamiltonian. It does not describe solvent free-energy dynamics, bath-induced mode coupling, or nonadiabatic transfer, and it does not predict Marcus rates for realistic condensed-phase electron-transfer systems.
ER -