Phase Space Bottlenecks in an Adiabatic Marcus Hamiltonian: Cusp Geometry, NHIMs, and Mixed Valence Electron Transfer

TL;DR

This paper derives a cusp condition for an adiabatic Marcus Hamiltonian that determines when a phase space transition state exists, providing a Hamiltonian perspective on electron transfer bottlenecks.

Contribution

It introduces a Marcus-specific cusp criterion that identifies the presence of a Hamiltonian bottleneck supporting a phase space transition state in a minimal model.

Findings

Derives an explicit cusp condition in parameter space for phase space transition states.

Shows the existence of a saddle-centre equilibrium and associated invariant manifolds inside the cusp.

Clarifies the role of phase space bottlenecks in adiabatic electron transfer models.

Abstract

Marcus--Hush theory explains electron transfer in terms of reorganization energies, driving forces, electronic couplings, and reduced free-energy or energy-gap descriptions. These descriptions do not by themselves determine when the underlying adiabatic dynamics possesses a genuine phase space transition state. We address this question for a minimal asymmetric two-degree-of-freedom adiabatic Marcus Hamiltonian obtained from two coupled diabatic harmonic surfaces. Passing to the lower adiabatic sheet gives a classical Hamiltonian with one electron-transfer coordinate and one transverse mode. We derive an explicit cusp condition in the plane of dimensionless asymmetry and coupling parameters that is necessary and sufficient for the lower sheet to possess an index-one saddle. This cusp criterion is the Marcus-specific result of the paper: it identifies when the lower adiabatic surface supports a local Hamiltonian bottleneck rather than only an energetic barrier in a reduced-coordinate picture. Inside the cusp, the corresponding Hamiltonian equilibrium is of saddle-centre type, and the standard local phase-space transition-state structures follow: in two degrees of freedom the normally hyperbolic invariant manifold is an unstable periodic orbit, with stable and unstable manifolds and an attached no-recrossing dividing surface. Outside the cusp, this lower-sheet local transition-state structure is absent. The construction provides a Hamiltonian complement to standard adiabatic Marcus theory, clarifies the role of the lower-sheet bottleneck in a minimal mixed valence setting, and separates the conservative adiabatic problem from dissipative solvent theories and nonadiabatic mixed quantum-classical formulations.