A Three-Degree-of-Freedom Chesnavich Model for Roaming: Derivation, Phase-Space Geometry, NHIM-Anchored Dividing Surfaces, and Roaming Transport

Abstract

Roaming reactions, in which a dissociating fragment moves through a flat region of the potential surface rather than down the minimum-energy path, lie outside the assumptions of conventional transition state theory. The phase-space theory of roaming -- unstable periodic orbits and their invariant manifolds organizing transport -- has been developed for the Chesnavich model of $\mathrm{C}{\mathrm{H}}_4^{+}\to \mathrm{C}{\mathrm{H}}_3^{+}+\mathrm{H}$, which is cylindrically symmetric and reduces to two degrees of freedom (2-DoF). We construct and analyze a three-degree-of-freedom (3-DoF) extension. From the rigid-body formulation of Ezra and Wiggins, we break the symmetry with an azimuthal coupling respecting the three-fold ($C_3$) symmetry of the methyl fragment, obtaining a family $H_b$ whose planar reduction at $b=0$ is the 2-DoF model exactly and which is genuinely 3-DoF for $b>0$. This activates the out-of-plane degree of freedom at once: with the physical planar-top inertia ratio $I_z=2I_x$, arbitrarily weak coupling makes the periodic orbit on the roaming shelf transversely unstable, opening an escape route out of the reaction plane. Apart from a narrow elliptic window $0.58\lesssim b\lesssim 0.63$, the instability persists across the range studied, changing type through a period-doubling at $b_c\approx 0.63$. Because a periodic orbit cannot anchor a dividing surface in three degrees of freedom, we construct the objects that do -- three three-dimensional normally hyperbolic invariant manifolds, one per transition state -- at $b=0$, and prove that every compact interior piece of each persists for sufficiently small $b>0$. At $E=0.5\mathrm{kcal}\mathrm{mo}{\mathrm{l}}^{-1}$ the coupling lowers the direct non-reactive fraction of a microcanonical ensemble of incoming trajectories by $0.032$ and raises the two roaming fractions by $0.040$; the effect decreases as the energy increases.

43 pages, 13 figures