115 research outputs found
Index k saddles and dividing surfaces in phase space, with applications to isomerization dynamics
In this paper we continue our studies of the phase space geometry and
dynamics associated with index k saddles (k > 1) of the potential energy
surface. Using normal form theory, we give an explicit formula for a "dividing
surface" in phase space, i.e. a co-dimension one surface (within the energy
shell) through which all trajectories that "cross" the region of the index k
saddle must pass. With a generic non-resonance assumption, the normal form
provides k (approximate) integrals that describe the saddle dynamics in a
neighborhood of the index k saddle. These integrals provide a symbolic
description of all trajectories that pass through a neighborhood of the saddle.
We give a parametrization of the dividing surface which is used as the basis
for a numerical method to sample the dividing surface. Our techniques are
applied to isomerization dynamics on a potential energy surface having 4
minima; two symmetry related pairs of minima are connected by low energy index
one saddles, with the pairs themselves connected via higher energy index one
saddles and an index two saddle at the origin. We compute and sample the
dividing surface and show that our approach enables us to distinguish between
concerted crossing ("hilltop crossing") isomerizing trajectories and those
trajectories that are not concerted crossing (potentially sequentially
isomerizing trajectories). We then consider the effect of additional "bath
modes" on the dynamics, which is a four degree-of-freedom system. For this
system we show that the normal form and dividing surface can be realized and
sampled and that, using the approximate integrals of motion and our symbolic
description of trajectories, we are able to choose initial conditions
corresponding to concerted crossing isomerizing trajectories and (potentially)
sequentially isomerizing trajectories.Comment: 49 pages, 12 figure
Roaming at Constant Kinetic Energy:Chesnavich's Model and the Hamiltonian Isokinetic Thermostat
We consider the roaming mechanism for chemical reactions under the
nonholonomic constraint of constant kinetic energy. Our study is carried out in
the context of the Hamiltonian isokinetic thermostat applied to Chesnavich's
model for an ion-molecule reaction. Through an analysis of phase space
structures we show that imposing the nonholonomic constraint does not prevent
the system from exhibiting roaming dynamics, and that the origin of the roaming
mechanism turns out to be analogous to that found in the previously studied
Hamiltonian case.Comment: arXiv admin note: text overlap with arXiv:1909.0555
Multiple Transition States and Roaming in Ion-Molecule Reactions: a Phase Space Perspective
We provide a dynamical interpretation of the recently identified `roaming'
mechanism for molecular dissociation reactions in terms of geometrical
structures in phase space. These are NHIMs (Normally Hyperbolic Invariant
Manifolds) and their stable/unstable manifolds that define transition states
for ion-molecule association or dissociation reactions. The associated dividing
surfaces rigorously define a roaming region of phase space, in which both
reactive and nonreactive trajectories can be trapped for arbitrarily long
times.Comment: 20 pages, 6 figure
Bulgac-Kusnezov-Nos\'e-Hoover thermostats
In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in
phase space by means of non-Hamiltonian brackets. Two generalized versions of
the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are
globally controlled by means of a single additional Nos\'e variable, and
another where each demon is coupled to an independent Nos\'e-Hoover thermostat.
Numerically stable and efficient measure-preserving time-reversible algorithms
are derived in a systematic way for each case. The chaotic properties of the
different phase space flows are numerically illustrated through the
paradigmatic example of the one-dimensional harmonic oscillator. It is found
that, while the simple Bulgac-Kusnezov thermostat is apparently not ergodic,
both of the Nos\'e-Hoover controlled dynamics sample the canonical distribution
correctly
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