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Fragmentation kinetics of a Morse oscillator chain under tension

Joseph N. Stember, Gregory S. Ezra *

Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, NY 14853, United States

Received 20 March 2007; accepted 6 June 2007Available online 21 June 2007

Abstract

The bond dissociation kinetics of tethered atomic (Morse potential) chains under tensile stress is studied. Both RRKM (fully anhar-monic, Monte Carlo) and RRK (harmonic appproximation) theory are applied to predict bond dissociation rate constants as a functionof energy and tensile force. For chains with N P 3 atoms a hybrid statistical theory is used involving a harmonic approximation formotion in the transition state for bond dissociation. For chains with N = 2–5 atoms, while the RRK approximation significantly over-estimates the dissociation rate constant, the fully anharmonic RRKM rate is quite close to simulation results. For the N = 2 chain, anovel approach to the extraction of decay rate constants based on the classical spectral theorem is implemented. Good agreementbetween the RRKM and dynamical rate constants is obtained for N = 2 despite the fact that the reactant phase space contains a signif-icant fraction of relatively short-lived trajectories.Ó 2007 Elsevier B.V. All rights reserved.

PACS: 82.20.Db; 34.10.+x; 83.20.Lp

Keywords: Mechanochemistry; Fragmentation kinetics; Statistical theories

1. Introduction

A fundamental understanding of the intramoleculardynamics and kinetics of fragmentation (bond dissocia-tion) of atomic chains subject to a tensile force is neededto provide a solid foundation for theories of material fail-ure under stress [1,2], polymer rupture [3–7], adhesion [8],friction [9], mechanochemistry [10–12] and biological appli-cations of dynamical force microscopy [13–17]. In chainscission models for polymer fiber failure, the primary fail-

ure events are breaking of covalent bonds within individualchains [1,2,18,19]. Although trajectory studies on modelpolyethylene chains suggest that complete intramolecularvibrational energy redistribution (IVR) can occur on apicosecond timescale [20,21], extensive theoretical workhas shown that the rate of bond breakage in single polymerchains under constant stress/strain can be up to severalorders of magnitude slower than calculated on the basis

of statistical approaches such as transition state theory[22–30]. Of course, studies of energy transfer and equipar-tition in single chains of coupled anharmonic oscillatorshave a long history, beginning with the seminal work of Fermi et al. [31,32]. (For some recent work on dynamicsof atomic chains see [33–38].)

In the present paper we study numerically the kinetics of bond breaking in single atomic chains under stress. Thedissociation of a 1D chain subject to constant tensile forceis a problem in unimolecular kinetics. A fundamental issue

in unimolecular kinetics concerns the applicability of statis-tical approaches such as RRKM [39–43] or transition statetheory [44]. Previous theoretical work has suggested thatdissociation of atomic chains under stress is not amenableto simple statistical approaches [22–30]. Early trajectorysimulations on the dynamics of Morse chains [22–25]showed that correlated motions of $5–10 chain atomsare necessary for bond breaking to occur. Moreover, sim-ple bond stretching or force criteria for bond rupture werefound to fail, in that apparently broken bonds wereobserved to reform (bond healing). A significant fraction

0301-0104/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2007.06.019

* Corresponding author. Tel.: +1 607 255 3949; fax: +1 607 255 4137.E-mail address: [email protected] (G.S. Ezra).

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Chemical Physics 337 (2007) 11–32

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of dissociative events occurred via a two-stage process: aninitial, relatively persistent local bond stretching, followedby rapid breaking of the stretched bond [24]. Simulationsof the fragmentation of 1D Lennard–Jones (LJ) chains atconstant strain with inclusion of a frictional damping termand a stochastic force modelling interaction with a heat

bath show that, while fragmentation rate constants fit anArrhenius form with activation energy consistent with astatic picture at constant strain [45], the magnitude of thepreexponential factor was at least three orders of magni-tude smaller than expected on the basis of naive dynamicalconsiderations [27] (see, however, Ref. [46]). Healing of incipient breaks is highly efficient. Comparison of polymerrupture with single-bond dissociation dynamics for thesame damping/fluctuating force via Kramers theory [47]shows that stochastic forces are not responsible for thebond healing. Bolton, Nordholm and Schranz (BNS) havestudied the dissociation of 1D Morse chains (N = 2–20)under stress [28], following earlier work on the classical

dynamics of unstressed chains [48–50]. Nonexponentialdecay, failure of RRKM theory, and extensive transitionstate recrossing effects were found. Interestingly, the pres-ence of lateral interchain coupling increased recrossingeffects. Standard harmonic classical TST has been appliedto the dissociation of a 1D Morse chain [29,30], with thetransition state for dissociation of a given bond locatedat the maximum of the effective potential (see below).The harmonic canonical TST rate constant did not agreewith molecular dynamics calculations, but effects of anhar-monicity [43,28] on the predictions of TST were not sys-tematically investigated. Molecular dynamics studies on

thermal degradation in polyethylene [51] indicate that thedependence of bond scission rate on average thermal exci-tation energy can be fit with RRK theory [52], suggestingthat bending and torsional modes might play an importantrole in establishing statistical behavior. (Refs. [53–56] arerepresentative studies of the role of torsional and bendingdegrees of freedom in intramolecular dynamics.)

Dissociative trajectories in 1D LJ chains do not exhibitthe sensitive dependence on initial conditions expected forchaotic systems [26]; this suggests a significant degree of regularity at the transition state (saddle point) for singlebond dissociation. Such regularity is analogous to thatfound for isomerization reactions in clusters (see, for exam-ple, [57,58] and references cited therein).

In the present work classical trajectory simulations areused to investigate the fragmentation kinetics and phasespace structure of short tethered atomic chains under con-stant tensile stress. Morse parameters in the simulationscan be chosen, for example, to model the effective CH2-CH2 bond potential in polyethylene [51]. We focus on non-statistical aspects of the dissociation dynamics and the roleof anharmonicity.

In Section 2 we introduce the model Hamiltonian to bestudied, which consists of a tethered chain of identicalatoms interacting via pairwise Morse potentials subject to

constant tensile stress. Section 3 discusses application of

statistical theories such as RRKM and RRK (harmonicapproximation to RRKM) to our model. We describe thenumerical procedures used to implement a fully anhar-monic version of RRKM theory using Monte Carlo inte-gration to determine reactant and transition state phasespace volumes as a function of energy. (See also recent

work by Zhao and Du [59].) Section 4 outlines the proce-dures used to extract dissociation rate constants from clas-sical trajectory simulations of chain fragmentation. InSection 5 we discuss in some detail the phase space struc-ture for the N = 2 case, as visualized through use of Poin-care surfaces of section [60]. For the N = 2 chain, there aretwo distinct transition states corresponding to dissociationof the two different bonds, and we examine intersections of the stable and unstable manifolds of the two transitionstates in order to give a phase space picture of short-time(direct) dynamics in terms of reactive cylinders [61–65].Section 6 compares the values of trajectory dissociationrate constants with those obtained using statistical

approaches for the N = 2 chain. In Section 7 we explorethe possibility of making a more stringent comparisonbetween dynamical and statistical rate constants by intro-ducing corrections to the RRKM expression based on theclassical spectral theorem [66–73]. Section 8 presents acomparison of statistical and dynamics rate constantsobtained for chains with N = 3–5 atoms, while Section 9concludes.

2. Model potential and Hamiltonian

2.1. Potential energy surface

We consider first a one-dimensional Morse oscillatorplus potential linear in bond coordinate, describing a par-ticle tethered to a wall (infinite mass) and acted on by aconstant tensile force f . The potential for this system takesthe form

V ðr ; f Þ ¼ V M ðr Þ À f ðr À r 0eqÞ ð1aÞ¼ D0½1 À expfÀbðr À r 0eqÞg2 À f ðr À r 0eqÞ: ð1bÞ

Unless otherwise specified, we shall measure length in unitsof r 0eq, the zero-force Morse oscillator equilibrium bond dis-tance, and energies in units of D0, the unperturbed Morsedissociation energy.

For the Br2 molecule, for example, r 0eq ¼ 2:28 A,D0 = 45.89 kcal/mol and b ¼ 4:432=r 0eq [28]. For the typicaltensile stress value of 25 kcal molÀ1 AÀ1 studied by Boltonet al., f ¼ 1:24 D0=r

0eq [28]. For the CC bond in model poly-

ethylene (pe), r 0eq ¼ 1:529 A, D0 = 83 kcal/mol andb ¼ 2:94=r 0eq [74,75,51,76]. At the representative tensilestress value of 1 GPa [2], f ¼ 0:053 D0=r

0eq. We have studied

Morse chains with b values in the range 1–3. All the calcu-lations reported here are carried out for the tensile stressvalue f = 0.02.

The potential (1a) is plotted versus r in Fig. 1 for various

f values and b = 1. For f 6 f crit ¼ b2, potential (1a) has two

12 J.N. Stember, G.S. Ezra / Chemical Physics 337 (2007) 11–32



stationary points, one at the potential minimum r = req andone at the saddle point, r = rà: In general, the critical valuesof the potential are given by

r eq ¼ 1

bln

bÀ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibðÀ2 f þ bÞp f

" #þ r 0eq ð2aÞ

r z ¼ 1

bln

bþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibðÀ2 f þ bÞp f

" #þ r 0eq: ð2bÞ

As f increases, rà decreases while req increases slightly. Thedepth of the potential well decreases from the zero-forcevalue D0 until f reaches the critical value f crit, at whichpoint the well disappears completely. For b = 1 and f = 0.02, we have req = 1.010 and rà = 5.595.

Now consider a linear chain of N atoms. For this systemwith N P 2 degrees of freedom, the conversion from inter-nal (bond) coordinates r to external (lab-fixed) coordinatesx is given by (see Fig. 2)

xk ¼Xk j¼1

r j; k ¼ 1; . . . ; N : ð3Þ

We assume that the potential between atoms is pairwiseMorse (nearest neighbors only) and that the tensile forceacts only on the outermost atom, giving a potential func-tion for the chain

V ðx; f Þ ¼ V M ð x1Þ þ V M ð x2 À x1Þ þ Á Á Á þV M ð x N À x N À1Þ À f ð x N À Nr 0eqÞ ð4aÞ

¼X N i¼1

½V M ðr iÞ À f ðr i À r 0eqÞ: ð4bÞ

The force is distributed through the chain so that the po-

tential in each bond coordinate rk has the same functionaldependence on bond length as the one-particle potential(1a). Hence the equilibrium bond length between any adja-cent pair of atoms at the global minimum in the N P 2chain is just the equilibrium bond length for the one-dimensional case. A similar equivalence holds for the crit-ical bond length. For example, the equilibrium and saddleconfigurations corresponding to breaking bond 1 are

xk ;eq ¼ kr eq; k ¼ 1; . . . ; N ; ð5aÞ xk ;saddle ¼ r z þ ðk À 1Þr eq; k ¼ 1; . . . ; N : ð5bÞFor a single tethered atom, Fig. 1 shows the existence of a

well-defined configuration space transition state for bondbreaking located at the top of the effective potential energybarrier. For a chain of atoms under stress, there is an anal-ogous critical configuration associated with the dissocia-tion of each bond. Detailed static analyses of polymerchains subject to tensile force (both constant strain andconstant stress) have been given [45,77]. Although the staticanalysis leads to a clear view of the nature of certain impor-tant cuts through the multibody potential (for example,one bond stretched with all other bonds having equal dis-placements from equilibrium), it provides limited insightinto fracture dynamics.

Contours of the N = 2 potential energy surface (PES),Eq. (4a), as a function of the single-particle coordinatesx1 and x2 are plotted in Fig. 3a. There is a global minimumcorresponding to the bound chain and two saddle pointsassociated with the two possible bond dissociation path-ways. Along the minimum energy pathway for breakingof bond 1, both x1 and x2 coordinates increase, while thepath for breaking of bond 2 is approximately a verticalline, x1 $ const. Bound motions at the respective transitionstates are approximately perpendicular to the minimumenergy paths (see below).

The potential energy barrier to dissociation, i.e., theactivation energy E a

¼E Ã f , is defined to be

E Ã f ¼ V ðr z; f Þ À V ðr eq; f Þ: ð6ÞFor any particular value of the force f , all energies are mea-sured from the f -dependent minimum energy V (xeq( f )). Forb = 1 and f = 0.02 we have E a = 0.8881.

2.2. Hamiltonian

In single particle coordinates, the N -atom chain Hamil-tonian is simply

H

ðx; f

Þ ¼X N

i¼1

1

2mi

p 2i

þV

ðx; f

Þ;

ð7

Þ

2 4 6 8 10x

0.5

1

1.5

2

V

Fig. 1. Potential V (r, f ) for a tethered Morse particle under tensile stress,Eq. (1a). The unit of length is r 0eq, the zero-force Morse oscillatorequilibrium bond distance, and the unit of energy D0, the unperturbedMorse dissociation energy. Force parameter f : f = 0 (black), f = 0.02 (red), f = 0.04 (green), f = 0.06 (blue), f = 0.08 (purple), f = 0.1 (light blue). (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

1 2 3 4 f

x1

x2

Fig. 2. Definition of single-particle coordinates {xi } for a tethered chain

of N = 4 atoms under tensile stress.

J.N. Stember, G.S. Ezra / Chemical Physics 337 (2007) 11–32 13



where pi is the momentum of particle i . We shall take allparticles to have the same mass mi = m. Fixing the unitof mass specifies a unit of time; with D0 ¼ r 0eq ¼ m ¼ 1,the period of the harmonic oscillation at the bottom of the Morse potential well with b = 1 is sHO ¼ 2p=

ffiffiffi2

p .

Although the overall magnitudes of dissociation rate con-stants depend on the value of the particle mass m, the ratiosof rate constants computed from two different methods(e.g., trajectory and RRK) do not (see Section 6.1).

In terms of bond coordinates, the N -atom chain Hamil-tonian is

H ðr; f Þ ¼ 1

2 p 2r 1 þ

X N k ¼2

p 2r k ÀX N À1

k ¼1

p r k p r k þ1

þX N i¼1

ðV M ðr iÞ À f ðr i À r 0eqÞÞ: ð8Þ

In terms of single-particle coordinates the kinetic energy istherefore separable while the potential energy is nonsepara-ble, whereas in bond coordinates the potential energy sep-arates while the kinetic energy has coupling terms À p r k p r k þ1

.

3. RRKM and RRK dissociation rate constants

3.1. RRKM theory

The RRKM expression for the E and f -dependentmolecular fragmentation rate constant associated withbreaking of bond j is [39–43]

k RRKM j ð E ; f Þ ¼N

z jð E À E Ã f Þqð E Þ ; ð9Þ

where Nz jð E À E Ã f Þ is the classical sum of states (phase

space volume) at the transition state for breaking of bond

j , E Ã f is the force-dependent dissociation energy, and q(E ) is

the classical density of states for the reactant region of phase space,

qð E Þ ¼ dNð E Þd E

; ð10Þ

withNð E Þ the classical sum of states (phase space volume)at total energy E .

For an N -mode system, q has dimensions (qp)N /J = (Js)N /J = (Js)N À1

· s. The classical transition state

sum of states Nz is the phase space volume for an(N À 1)-degree-of-freedom system at the critical configura-tion, and has dimensions (qp)N À1 = (Js)N À1. The ratio of the two, which is the classical unimolecular rate dissocia-tion rate constant as defined in (9), then has units of inversetime, (Js)N À1/((J s)N À1

· s) = sÀ1, as it should. The unit of time used in our calculations, s0, is discussed in Section 4.1.

The quantum density of states qQ is obtained in the semi-classical limit by dividing q by the volume of a phase space‘‘cell’’. For an N mode system, qQ = q/hN . Similarly, thequantum transition state sum of states is Nz

Q ¼N

z=h N À1. We therefore have

k RRKM ¼Nzq

¼ hnÀ1NzQhnqQ

ð11aÞ

¼ 1

h

NzQ

qQ

; ð11bÞ

so that the usual expression for k RRKM, Eq. (11b), explic-itly contains Planck’s constant h [39–43]. The absence of Planck’s constant in the formula (9) expressed in terms of purely classical quantities is nevertheless appropriate andentirely correct.

Broadly speaking, in order for the RRKM rate constantexpression to be valid it is necessary that k rxn

(k IVR, i.e.,

the rate of intramolecular energy transfer should be greater

2 4 6 8 10 12

x1

2

4

6

8

10

12

x 2

2 4 6 8 10 12

X

2

4

6

8

10

12

Fig. 3. Contour plot of the N = 2-particle potential energy surface, Eq. (4a), with f = 0.02. Saddle points 1 (red) and 2 (green) are also shown. (a) Single-particle coordinates: {x1, x2}; (b) Jacobi coordinates: {n, X }. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)


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than the reaction rate. Many studies have been carried outon the validity of RRKM theory (9) and the connectionbetween deviations from statisticality and intramoleculardynamics (for example, the existence of bottlenecks tointramolecular energy flow) [66,78–83]. Our focus here ison the validity of expression (9) for the particular case of

atomic chains under tensile stress [28–30].We shall consider energies low enough so that the disso-ciation of two or more bonds is not energetically possible.If the overall statistical rate constant for chain fragmenta-tion via single bond dissociation is

k RRKM ¼X N j¼1

k RRKM j ð12Þ

where the total number of decay channels is equal to thenumber of bonds in the chain, then N (t), the number of bound molecules at time t, exhibits exponential decay,

N

ðt

Þ ¼N

ð0Þ

eÀk RRKM t :ð13

ÞIn general we define the overall dissociation rate constant,when it exists, by

k ¼ À 1

N ðt Þ_ N ðt Þ ð14aÞ

¼X N j¼1

k j ð14bÞ

¼ þ 1

N ðt ÞX N j¼1

_n jðt Þ; ð14cÞ

where n j (t) is the number of molecules that have dissociated

via channel j at time t. In a regime where the system exhib-its exponential decay, the rate constant for dissociation viachannel j is given in terms of the branching ratio b j as

k j ¼ n jðt ÞP N

m¼1nmðt Þk b jk : ð15Þ

3.2. RRK theory: a harmonic approximation to RRKM

theory

RRK theory is the harmonic limit of RRKM theory[52]. We approximate the potential function V (x) by a sec-

ond order expansion about the potential minimum andsaddle points. At these critical points, linear terms vanish,and we have

V crit ’ V ðxcritÞ þ 1

2

X N i¼1

X N j¼1

D xiD x jo

2V

o xio x j

x¼xcrit

þ Á Á Á

ð16ÞThere are N real normal mode (angular) frequencies for thereactant, fxig N i¼1, and N À 1 real frequencies, fxz

i g N À1i¼1 , for

the transition state, together with a single imaginary fre-quency associated with unstable motion off the saddle.For an r-dimensional harmonic oscillator, frequencies

fxigr

i¼1, the phase space volume Nð E Þ at energy E is

Nð E Þ ¼ ð2pÞr r !

E r Qr

i¼1xi

; ð17Þ

so that the harmonic (RRK) approximation to the RRKMrate constant (9) for dissociation via transition state a is

k RRK

f ð E ; a

Þ ¼1

2pQ

N

i¼1x f ;iQ N À1i¼1 xz

f ;iðaÞ E À E Ã f ;a

E

N À1

;ð18

Þwhere E Ã f ;a is the critical energy for dissociation of bond a

under external force f . It should be noted that in our pres-ent model with all atoms identical, E Ã f ;a E Ã f is the same forall transition states because b and r 0eq are the same for eachterm in potential (4a).

3.3. Computation of RRKM rate constant

3.3.1. Reactant phase space volume

To compute k RRKM f ð E Þ we must determine the reactant

phase space volumeNð E

Þ, from which the reactant density

of states can be obtained via Eq. (10).Nð E Þ is the volumeof the region of phase space consisting of all points withenergy less than or equal to E :

Nð E Þ ¼Z

d x1 d p 1 d x2 d p 2 Á Á Á d x N d p N Hð E À H ðx; pÞÞ ð19Þ

where H is the Heaviside step function. This integral has noanalytical solution for our system (cf. [84]), necessitatinguse of either a harmonic approximation to the potentialor numerical (Monte Carlo) integration. As the validityof the harmonic approximation versus the role of anhar-monicity in the reactant density of states is an importantquestion when assessing the validity of RRKM theory fortreating stressed polymer chains [28,29], for comparisonwith the harmonic approximation we shall evaluate thefully anharmonic reactant density of states and transitionstate sum of states essentially exactly using a numericalMonte Carlo approach.

To evaluate the phase space volume numerically, weconsider a region (‘‘hypercube’’) in phase space M 0 withvolume V 0 large enough to contain all reactant phasepoints with energy E 6 E MC

max. To computeNð E Þ, the phasespace volume as a function of energy, we determine the vol-umes V

aof subsets M

a& M 0 containing phase points with

E 6 E a 6 E MCmax, a = 1,2, . . . ,amax. If nr phase points

f xi; p ig N i¼1 2 M 0 are chosen randomly, and na of these pointslie within M

a, then, for large enough nr, the volume V

ais

V a ’ na

nr V 0: ð20Þ

From the set of pairs f E a; V agamax

a¼1 , we can fit the phasespace volume Nð E Þ to a polynomial in E ,

Nð E Þ ’ E N ðc0 þ c1 E þÁ Á Á Þ; ð21Þthat is, harmonic oscillator phase space volume plus anhar-monic corrections. The associated reactant density of statesis then obtained by differentiating the fittedNð E Þ with re-spect to E . Details of the sampling procedure are further

discussed in Appendix A.


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For N = 2, the configurational projection of M 0 is theunion of the rectangular region and adjacent triangularregion shown in Fig. 4. It should be noted that the sam-pling procedure used for N = 3–5, which employs a simplerphase space hypercube, gives approximately the same den-sity of states for N = 2 as that obtained using the moreelaborate shape shown in Fig. 4.

Harmonic and anharmonic reactant densities of statesare shown for N = 2–5 in Fig. 5.

As mentioned above, we consider energies low enoughsuch that multiple bond breakages (3-body dissociation

when N = 2) do not occur. For f = 0.02, b = 1, this meansthat the maximum energy we consider is E MC

max ¼ 1:5281(with respect to the f -dependent minimum energy).

3.3.2. Transition state sum of states

For determination of the phase space sum of states for

the transition state associated with channel 1, it is conve-nient to transform to Jacobi coordinates

X ¼ 1

2ð x1 þ x2Þ ¼ r 1 þ 1

2r 2 ð22aÞ

n ¼ x2 À x1 ð22bÞ P ¼ p 1 þ p 2 ð22cÞP ¼ 1

2ð p 2 À p 1Þ ð22dÞ

using the generating function [85,60]

F 2ð x

1; x

2;P; P

Þ ¼P

ð x

2 À x

1Þ þ1

2 P

ð x

1 þ x

2Þ:

ð23

ÞThe Hamiltonian in the transformed coordinates is then

H ð X ; P ; n;PÞ ¼ P 2

4þP2 þ V ðn; X ; f Þ: ð24Þ

The PES in Jacobi coordinates (Fig. 3b) indicates that theconfiguration space transition state 1 can approximately bedefined by X z ¼ 1

2ð x1 þ x2Þ ¼ 1

2ðr z þ ðr z þ r eqÞÞ ¼ r z þ 1

2r eq

(=6.10 for f = 0.02 and b = 1), which is just the harmonicapproximation nonreactive normal mode. In single particle

2 4 6 8 10 12x1

2

4

6

8

x2

Fig. 4. Configuration space projection (light blue) of the phase spaceregion used for sampling of coordinates in the Monte Carlo determinationof reactant phase space volume Nð E Þ. Contours of the potential energyand the location of saddles (saddle 1 (red) and saddle 2 (blue)) are alsoshown. (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

0.25 0.5 0.75 1 1.25 1.5E

20

40

60

80

100

120

140

N 2

0.25 0.5 0.75 1 1.25 1.5E

100

200

300

400

500

N 3

0.25 0.5 0.75 1 1.25 1.5E

200

400

600

800

1000

N 4

0.25 0.5 0.75 1 1.25 1.5E

250

500

750

1000

1250

1500

N 5

Fig. 5. Harmonic approximation (red) and fully anharmonic Monte Carlo (blue) reactant density of states versus energy for (a) N = 2, (b) N = 3, (c)

N = 4 and (d) N = 5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)




coordinates, the configuration space projection of transi-tion state 1 lies along the line x2 = Àx1 + 2rà + req.

The 2D ‘‘hypercube’’ M z0 used to sample transition state

1 is defined by conditions

X ¼ r z þ 1

2r eq ð25aÞ

n 2 ½0; r z ð25bÞ P ¼ 0 ð25cÞP 2 À

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E MC

max À E Ã f

q ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E MC

max À E Ã f

q h ið25dÞ

Thus the N = 2 transition state 1 hypercube has volume

V z0;1 ¼ 2r z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið E MC

max À E Ã f Þq

. The transition state 2 sampling

hypercube is defined as

x1 2 ½0; 3:303 ð26aÞ x2

¼r eq

þr z

ð26b

Þ p 1 2 À

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 E MC

max À E Ã f r

;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 E MC

max À E Ã f r !

ð26cÞ

p 2 ¼ 0; ð26dÞand has volume V

z0;2 ¼ 6:606

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð E max À E Ã f Þ

q . Again, the x1

range endpoint of 3.303 marks the position of the barrier to3-body dissociation products from transition state 2 for f = 0.02 and b = 1.

For both the reactant and transition state Monte Carlocalculations, we used amax = 100, with nrand = 107. Transi-tion state sums of states are fitted to functions of the form

Nð E À E z f Þ ¼ ð E À E z f Þ N

À1

½c0 þ c1ð E À E z f Þþ Á Á Á : ð27Þ

4. Classical trajectory simulations

4.1. Integration of trajectories

Classical equations of motion are

_z ¼ L H z fz; H ðzÞg: ð28Þwith z = (x1, . . . , xN , p1 . . . pN ) and Hamiltonian H (z) givenby Eq. (7). Here, {Æ,Æ} denotes the usual Poisson bracket[85,60]. We employ a simple reversible 2nd-order symplec-tic integrator [86,87] to integrate Eq. (28). A typical valueof the timestep used in our integrations is h = 0.1s0, wheres0 is the unit of time associated with our choice of units forenergy, length and mass. In units of s0, the period of mo-tion of a unit mass in the harmonic potential obtained byexpanding the Morse potential about its minimum issHO ¼ 2p=ðb ffiffiffi2p Þ. For example, using physical parametersfor model polyethylene [74,75,51,76] we have sHO ’1.5s0 ’ 46.4 fs. With h = 0.01 s0 and nstep = 8 · 103 time-steps, the length of our typical trajectory runs is thenapproximately 55 harmonic oscillator periods ’2.5 ps.For the tensile stress values used here, this timescale is long

enough to characterize the decay dynamics.

4.2. Sampling phase points on the energy shell

To compare the predictions of RRK and RRKM theoryat constant energy E with trajectory results, it is necessaryto sample trajectory initial conditions z = (x, p) distributedwith suitable measure on the energy shell H (z) = E . The

sampling procedure used is discussed in more detail inAppendix B.

4.3. Extraction of rate coefficient k(E)

Given a microcanonical ensemble of N (0) initial condi-tions fz jð0Þg N ð0Þ

j¼1 , and defining N (t) 6 N (0) as the numberof trajectories that have not reacted by time t, we can com-pute an effective rate constant k (E , f ) if we assume that thesystem exhibits exponential decay with time-independentrate constant. The associated lifetime distribution is

N ðt Þ ¼N ð0Þ exp½Àk ð E ; f Þt : ð29ÞFor each energy we run N (0) = 50,000 trajectories and re-cord the times at which dissociation of each trajectory oc-curs (if it ever does for that trajectory). For f = 0.02 andb = 1 the condition for dissociation of bond j is taken to be

r j > r z þ 2r 0eq; j 2 ½1; . . . ; N : ð30ÞThe resulting data are histogrammed. Transient behavior(the first 350 time steps for b = 1 and f = 0.02) is ignoredand an exponential fit to the histogram at longer timesgives approximate k (E , f ) values.

It should be noted that the bond dissociation criterion(30) is intended to ensure that recrossing effects are

excluded from the numerical determination of the long -time(asymptotic) decay rate, which is the quantity to be com-pared with statistical theories.

Values of the decay rate constants obtained via trajec-tory simulation are compared with the predictions of statis-tical theory below. We also compare these dynamical decayrates with those obtained via an analysis based on the clas-sical spectral theorem [66–73].

5. Phase space structure and dynamics for N = 2

Before turning to a detailed comparison of RRKM and

trajectory dissociation rate constants, we examine thephase space structure for the N = 2 chain. For the tetheredchain with N = 2 atoms, we can use the standard Poincaresurface of section (SOS) construction [60] to examine thesystem phase space at constant energy E .

5.1. Surfaces of section

5.1.1. SOS along a dissociative coordinate: PODS and

turnstiles

At constant energy E , we can define a SOS by the con-dition (given in terms of Jacobi coordinates)

P ¼ 0; _P > 0: ð31Þ


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We will refer to this SOS as SOS1. Fig. 6 shows SOS1 for f = 0.02, b = 1 at energies E ¼ E Ã f À 0:4, E Ã f , E Ã f þ 0:01and E Ã f þ 0:1.

The overall shape of the separatrix [60] defining the reac-tant phase space region [79] is clearly visible in Fig. 6b, E ¼ E Ã f . The critical configuration for dissociation via tran-sition state 1 is located on SOS1 at P = 0, X = 6.10. Atenergies above the activation barrier, the periodic orbitdividing surface (PODS [88–90]; see below) appears as anunstable fixed point on SOS1 (Fig. 6c and d); the PODSand its associated stable/unstable manifolds are computedbelow. The points spilling out towards X

! 1are reactive

trajectories that dissociate via breaking of bond 1.

For E > E Ã f we also see ‘‘lobe’’ or ‘‘turnstile’’ structuresnear the boundaries of the reactive complex region. Theturnstiles mediate transport between phase space regions,and detailed analysis of turnstile dynamics can be used tounderstand rates of intramolecular energy flow and unimo-lecular dissociation [91,79,92]. Regions of lower trajectorydensity in SOS1 are associated with trajectories that disso-ciate through channel 2, and so do not return to the SOS(see below).

Also evident in the interior of the complex region is anarea of bounded, regular behavior, which appears as atransverse slice through a set of invariant tori [60]. Classical

trajectories with initial conditions on invariant tori do not

Fig. 6. Surfaces of section for the N = 2 chain ( f = 0.02). Conjugate variables (X , P ) are plotted when SOS1 conditions P = 0, _P > 0 are satisfied. (a) E ¼ E Ã f À 0:4; (b) E ¼ E Ã f ; (c) E ¼ E Ã f þ 0:01; (d) E ¼ E Ã f þ 0:1.




dissociate. The fact that a significant proportion of thetethered chain phase space volume is occupied by boundtrajectories means that, to obtain a more accurate predic-tion of the dissociation rate constant using statisticalassumptions, it is necessary to correct RRKM theory byremoving bound phase space regions from consideration

[93–95,79,96].SOS1 is symmetric about the line P = 0; this time-rever-sal symmetry can be exploited to determine relevant peri-odic orbits [97,98]. (An analogous SOS suitable forchannel 2, SOS2, is defined by the conditions p1 = 0,_ p 1 > 0. Whereas SOS1 is symmetric about the line P = 0,

SOS2 is symmetric about the line p2 = 0.)In the (X , P ) plane of SOS1, the unstable fixed point on

the line P = 0 is the intersection of the PODS [88–90] asso-ciated with the transition state for bond 1 cleavage with theSOS. This hyperbolic fixed point and its associated stable/unstable manifolds (plus the analogous structures for reac-tion 2) are phase space structures that determine the disso-

ciation dynamics of the N = 2 tethered chain.Consider determination of the PODS for channel 1. To

locate the PODS, we sample a line of initial conditions onSOS1 along the symmetry line P = 0. The SOS conditionson P and P imply that points on the symmetry line have p1 = p2 = 0, so that these trajectories start on the classicalzero-velocity manifold V (x) = E . Any trajectory thatreturns to the symmetry line in a single iteration of thePoincare map is a periodic orbit [97,98]. The location of the PODS on the symmetry line can therefore be deter-mined using a Newton–Raphson procedure [99].

Fig. 7 shows configuration space projections of PODS

associated with both transition states at several energies.The shapes of the PODS conform approximately to the har-monic prediction of a slanted line segment for transitionstate 1 and an approximately horizontal line segment fortransition state 2. Close inspection of the PODS reveals thatthey are in fact curved, an anharmonic dynamical effect.

Numerical approximations to the stable/unstable mani-foldsWu= s of the PODS are obtained by propagating phase

points close to the periodic orbit on the respective linear-ized manifolds (computed as outlined in Appendix C) for-wards/backwards in time [60]. To visualize Wu= s, we select1000 initial conditions along short segments of the linear-ized manifolds E u/s close to the PODS and propagate allthese phase points either forward or backwards in time.

The resulting manifolds for E Ã f þ 0:1 are shown in Fig. 8.Taken together, these two manifolds define a bound com-plex region. One possible definition of a direct trajectoryis one that does not enter or exit the bound separatrixregion via a turnstile [79].

5.1.2. SOS in reactant region (r1 = r 2): reactive cylinders

An informative view of the dynamics is provided by theSOS defined by crossings of the hypersurfaceR+ {z|r1 = r2, _r 2 > _r 1g. To formulate the crossing condi-tion in terms of a single variable, we consider a canonicaltransformation to coordinates

X ¼ 12 ð x1 þ x2Þ ð32aÞDr ¼ r 2 À r 1 ¼ x2 À 2 x1 ð32bÞwith generating function [60,85]

F ¼ 1

2ð x1 þ x2Þ P þ ð x2 À 2 x1Þ p Dr ; ð33Þ

from which we deduce the relation between conjugatemomenta

P ¼ 2

3ð p 1 þ 2 p 2Þ ð34aÞ

p Dr ¼1

3 ð p 2 À p 1Þ: ð34bÞ

1 2 3 4 5 6 7 8x1

6.4

6.6

6.8

7.2

7.4

7.6

7.8

x2

Fig. 7. PODS shown together with potential energy contours for f = 0.02.Saddle points 1 and 2 are marked as black dots, PODS1 and PODS2 areshown at energies E ¼ E Ã f þ 0:1 (red), E Ã f þ 0:2 (green), E Ã f þ 0:3 (blue), E Ã f þ 0:4 (yellow), E Ã f þ 0:5 (purple) and E max (black). (For interpretationof the references to colour in this figure legend, the reader is referred to the

web version of this article.)

Fig. 8. SOS (green) at E ¼ E Ã f þ 0:1 defined by P = 0, _P > 0; The stablemanifoldW s (blue) and unstable manifoldWu (red) of the PODS are alsoshown. (For interpretation of the references to colour in this figure legend,

the reader is referred to the web version of this article.)


http://-/?-

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(Note that the momentum P conjugate to X in the {X ,Dr}coordinate system differs from the momentum P in the{X ,n} coordinate system.) From the nondiagonal kineticenergy

T ¼ P 2

4À Pp

Dr

2þ 5 p 2

Dr

2; ð35Þ

we have

p Dr ¼

1

10 P Æ 1

10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi40T À 9 P 2

p : ð36Þ

We select the positive branch of solutions for pDr, which isequivalent to imposing the surface crossing condition p Dr >

110 P or

p 2 > 2 p 1; ð37Þwhich is just the condition p2 À p1 > p1 i.e. _r 2 > _r 1. Thus,an equivalent definition of our surface is R+ {z|Dr = 0, p Dr ¼ 1

10 P þ 1

10 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi40T À 9 P 2p

g. We examine the fate of trajec-tories initiated on a grid of initial conditions with

r 1ð0Þ ¼ r 2ð0Þ ¼ x1ð0Þ ¼ 1

2 x2ð0Þ 2 ½0; 3:5 ð38aÞ

P ð0Þ 2 ½À2:6; 2:6; ð38bÞwhere r(0) and P (0) values are evenly-spaced. Initial mo-menta { p1(0), p2(0)} are obtained from

p 1 ¼ P

2À 2 p Dr ð39aÞ

p 2 ¼ P

2þ p

Dr ; ð39bÞwhere the pDr(0) values are chosen to be on the positive

branch of (36) so as to place the initial conditions on R+.The resulting initial grid is shown in Fig. 9a, and the

SOS generated from this initial condition set is in Fig. 9b.A plot of the grid of initial conditions on R+ under one

iteration of the return map is shown in Fig. 9c. One note-worthy feature of Fig. 9b is the presence of ‘‘holes’’, regionswithin the SOS structure that are either devoid of or spar-sely filled by iterates.

To elucidate the dynamical significance of these holes,we initiate trajectories on transition states 1 and 2 (thatis, distributed along the corresponding PODS), propagatethem forward in time, and examine their intersection withR+. The successive intersections of these trajectories withthe SOS are shown in Fig. 10. Let Ru; j

m & Rþ be the regioncorresponding to the mth intersection of R+ by the set of trajectories initiated on transition state j . (The superscriptu indicates that the region Ru; j

m is bounded by the intersec-tion of the unstable mainfold of the PODS with R+.)Fig. 10a shows that Ru;1

1 and Ru;21 are the two holes toward

the bottom of the SOS structure that contain no phasepoints; that is, regions Ru;1

1 and Ru;21 are composed of phase

points with no pre-image on R+. The regions Ru;12 and R

u;22 ,

first visible in Fig. 10b, are cross sections of cylindricalstructures in phase space, so-called reactive cylinders [61– 65]. Upon further iterations, shown in Fig. 10c–e, the cyl-

inder cross sections stretch out, twist and generally spread

themselves through the reactive region. We infer that theset of reactive phase points on R+ is the closure of S2

j¼1

S1i¼1R

u; ji .

Of equal importance to the sets Ru; jm , j = 1, 2, which react

through transition states 1 and 2, respectively, in negativetime, is the set of trajectories R s; jm , which react in forward

time. We define R s; j

m to be the set of trajectories that reactthrough transition state j after undergoing m À 1 furtherintersections with R+. (The superscript s indicates thatthese trajectories are associated with the stable manifoldof the corresponding PODS; they are calculated by propa-gating backwards in time from the relevant transitionstate.) In particular, we consider the regions R

s; j1 , j = 1, 2,

which consist of phase points that react immediately in for-ward time, where immediate means without an interveningintersection of the SOS.

Fig. 11a displays R s;11

SR s;21 along with the grid of initial

conditions on R+ that intersect R+ for some positive timevalue. Note that R

s;11 SR

s;21 fills the ‘‘holes’’ in the SOS,

which consist of phase points that never return to the sur-face in positive time, and so react immediately.

It is informative to observe intersections between theforward and backward direct reactive tubes, R

u;12

TR s;11

and Ru;22

TR s;21 . (One could just as well inspect the intersec-

tions Ru;11

TR s;12 and R

u;21

TR s;22 , which appear in ure

Fig. 11b and are simply the time-evolved iterations of R

u;12

TR s;11 and R

u;22

TR s;21 , respectively for one mapping of

R+.) To illustrate the information afforded by these over-laps, the areaAðRu;1

2

TR s;11 Þ reflects the phase space volume

of trajectories that enter the reactant region through tran-sition state 1 and exit through transition state 1 after

2 + 1 À 1 = 2 intersections with the SOS R+. Hence,AðRu;1

2

TR s;11 Þ þAðRu;2

2

TR s;11 Þ should correlate with the

fraction P z1;d ð E À E Ã f Þ of transition state 1 belonging to

direct trajectories, where direct trajectories could bedefined as those having only two intersections with SOSR+, noting that since R

i;u1

TR s; j1 for i , j = 1, 2 is zero, no

reactive trajectories cross R+ only once.We observe from Fig. 12 that the overlap areas

Ru;12

TR s;11 and R

u;22

TR s;21 both increase with increasing

energy, while Ru;22

TR s;11 and R

u;12

TR s;21 are zero for most

energies except those near E max, where a thin strip of non-zero overlap R

u;22 T

R s;11 begins to emerge, as is evident in

Fig. 12d–f. Thus, there are virtually no trajectories thatenter through transition state m and exit through transitionstate n, m5n, without crossing SOS R+ at least three times.

6. Statistical theories compared with dynamical rate

constants

6.1. Ratios of rate constants are independent of choice of

mass unit

We shall compare absolute values of dissociation rateconstants obtained from classical trajectory simulationswith those calculated using RRKM-type theories. Recall

that our unit of time is defined in terms of the choice of




mass unit, s0 $ ffiffiffiffimp

. Although absolute values of dissocia-

tion rate constants depend upon the value of m used, ratiosof rates do not. Thus, from Eq. (18) for the RRK rate con-stant, and recalling the mass dependence of harmonic oscil-lator frequencies, we see that

k RRK f ð E Þ $

1 ffiffiffim

p N

1 ffiffiffim

p N À1

$ 1 ffiffiffiffim

p : ð40Þ

The RRKM rate constant is given by the ratio (9). Both thereactant phase space volume and the density of states q(E )scale in general as ð

ffiffiffiffim

p Þ N , where N is the number of degreesof freedom, while Nz

ð E

À E Ã f

Þscales as

ð ffiffiffiffimp

Þ N À1. The

anharmonic RRKM rate constant therefore scales in the

same way with mass as the RRK rate constant, as 1= ffiffiffiffimp

.

Finally, to determine the mass dependence of simulationresults k sim, we note that increasing m means that particlesare effectively moving more slowly by a factor of 1=

ffiffiffiffim

p ; we

therefore have k sim $ 1=ffiffiffiffim

p .

Rate constants k RRK, k RRKM, and k sim therefore allhave the same mass dependence and the ratio of any twoof these k values is mass-independent.

6.2. Comparison of RRK with RRKM and simulation rate

constants

RRK, RRKM and simulation rate constants for the

total dissociation rate for N = 2, f = 0.02, b = 1 are shown

Fig. 9. (a) Grid of initial conditions for E ¼ E Ã f þ 0:1 on SOS defined by r1 = r2, _r 2 > _r 1. (b) SOS for E ¼ E Ã f þ 0:1 defined by r1 = r2, _r 2 > _r 1. (c) Firstiterate under Poincare map of initial conditions in panel (a).




Fig. 10. The nth crossing of SOS r1 = r2, _r 2 > _r 1 for trajectories initiated on transition states 1 (red) and 2 (blue), E ¼ E Ã f þ 0:1. (a) n = 1; (b) n = 2; (c)

n = 3; (d) n = 4; (e) n = 10. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)




as functions of energy in Fig. 13a. While the harmonicRRK approximation significantly overestimates theRRKM rate constant (cf. Refs. [27–29]), the RRKM valueis only slightly larger than the simulation results, with thedisparity growing larger at higher energies. This level of agreement is perhaps surprising, given the non-negligiblefraction of regular phase space occupied by invariant tori,

as noted above. We comment further on this point below.

6.3. Relative rate calculations

As the phenomenon of reactive selectivity is of centralinterest in chemistry, it is important to examine the relativerates of competing reactions. For N = 2, the two compet-ing reactions are the cleavage of bonds 1 and 2, respec-tively. We therefore study the ratio of rate constants k 2/k 1. We consider the statistical predictions for the branchingratio k 2/k 1 and compare with simulation results.

6.3.1. RRK and RRKM calculations, N = 2From expression (9) for k RRKM(E , f ), the ratio of RRKM rate constants for reactions 1 and 2 is simply

k 2

k 1

RRKM

¼Nz2ð E À E Ã f Þ

Nz1ð E À E Ã f Þ

; ð41Þ

as the reactant density of states at energy E cancels out.The ratio in the harmonic approximation is then

k 2

k 1

RRK

¼ xz1

xz2

; ð42Þ

where xz j is the harmonic frequency of the nonreactive nor-

mal mode taken about the j th saddle point.

6.3.2. Trajectory results

If k sim is the total rate constant for dissociation as deter-mined from trajectory simulation, the rate constants forchannels 1 and 2, k 1,sim and k 2,sim respectively, are givenin terms of the respective branching ratios multiplied bythe overall rate constant:

k j;sim ¼n j

n1 þ n2 k sim; j ¼ 1; 2 ð43Þwhere n1, n2 are the number of trajectories that have re-acted through channels 1, 2 respectively at some relativelylong time (we use 2000 time steps) with all nonstatistical,transient trajectories omitted. The rate ratio is then

k 2

k 1

sim

¼ n2

n1

: ð44Þ

6.3.3. Periodic orbit dividing surface actions

If the transition state of (locally) minimal flux at energyE is the phase space region A, then for the N = 2 case itfollows from the variational principle of classical mechan-ics [100] that the boundary oA of the regionA is a periodicclassical trajectory defining the periodic orbit dividing sur-face (PODS) [88,89]. The associated flux is then the actionof the PODS

NzPODSð E À E Ã f Þ ¼

Z A

d p ^ dq ¼Z oA

p dq ð45Þ

defined as a line integral over one cycle of the PODS. Be-cause the PODS is a dynamically determined object andN

zPODSð E À E Ã f Þ reflects the full anharmonic dynamics in

the vicinity of the saddle point, this method for calculating

the sum of statesNzð E À E Ã f Þ provides a check on the accu-

Fig. 11. (a) Grid of initial conditions of trajectories started on the SOS r1 = r2, _r 2 > _r 1, E ¼ E Ã f þ 0:1, and returning to the SOS at least once, is shown ingreen. Trajectories in red and blue regions dissociate immediately through transition states 1 (red) and 2 (blue), respectively, in forward time. (b) Iterates of initial conditions in panel (a) are shown in green. Additional red and blue regions are preimages of regions in panel (a) under Poincare map; that is, theycontain initial conditions for trajectories that dissociate after crossing the SOS once in forward time. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)




Fig. 12. Red points are initial conditions for trajectories having at least one intersection with the SOS r1 = r2, _r 2 > _r 1. Green points are the iterates underthe Poincare map. E

¼E Ã f

þD E . (a) DE = 0.1; (b) DE = 0.2; (c) DE = 0.3; (d) DE = 0.4; (e) DE = 0.5; (f) DE = 0.6. (For interpretation of the references to

colour in this figure legend, the reader is referred to the web version of this article.)




racy of the corresponding harmonic and Monte Carlo cal-culations. We evaluate the PODS sum of states at energy E

asZ PODS

ð p 1 _q1 dt þ p 2 _q2 dt Þ ¼Z

PODS

ð p 21 þ p 22Þj H ¼ E dt ; ð46Þ

using PODS initial conditions obtained as described above.Fig. 14 showsNz

jð E À E Ã f Þ, j = 1, 2, versus energy E forthe harmonic approximation, Monte Carlo integration andthe PODS action method just described. We see that theanharmonic Monte Carlo areas agree essentially exactlywith the PODS actions (the true dynamical local fluxes).It is also evident that the harmonic approximation to thetransition sum of states is reasonably accurate, especiallyat lower energies.

Fig. 15 shows the RRK and PODS branching ratios k 2/k 1 versus energy compared with simulation results. Thevalue of the RRK branching ratio ð ffiffiffi2p Þ is essentially deter-mined by the ratio of the reduced masses for breakingbonds 1 and 2, and is independent of energy (cf. Ref.[28]). To the extent that motion along the PODS (thedynamically defined configuration space transition states)corresponds to separable motion along the bond coordi-nates, the RRKM/PODS branching ratios will be identicalwith the harmonic value. This is not quite the case, how-ever, as can be seen from the form of the PODS plottedin Fig. 7, so that the branching ratios calculated from the

PODS are close to but not identical with the RRK value.

Only the ratio obtained from simulations exhibits anysignificant energy dependence; further analysis of its

dynamical origins will require a more detailed investigationof ‘‘lobe dynamics’’ [92] via the surface of section shown inFig. 9.

7. Correcting RRKM theory

7.1. General considerations

While Fig. 6a–d show that a nonnegligible portion of thereactant phase space is occupied by regular, bound trajec-tories, RRKM rate constants for bond dissociation calcu-lated using an anharmonic density of states are in fairlyclose agreement with trajectory values, at least for N = 2(see Fig. 13a–d). Given that the reactant phase space is cer-tainly not globally chaotic, and given the probable exis-tence of barriers to IVR in the vicinity of the regularregions of phase space (cantori [91,79]), it is possible thatthe good agreement between RRKM and trajectory rateconstants is to some degree fortuitous.

In order to explore this point further, and to make aproper comparison between the ‘‘statistical’’ rate constantand the dynamical result, we need to ensure that the influ-ence of bound and direct trajectories on the RRKM-typecalculation is taken into account. A noteworthy discussionof this problem has been given by Berblinger and Schlier in

the context of the dissociation of the Hþ3 molecule [96].

0.1 0.2 0.3 0.4 0.5 0.6E Ef

0.025

0.05

0.075

0.1

0.125

0.15

k N 2

0.1 0.2 0.3 0.4 0.5 0.6E Ef

0.02

0.04

0.06

0.08

k N 3

0.1 0.2 0.3 0.4 0.5 0.6E Ef

0.005

0.01

0.015

0.02

0.025

k N 4

0.1 0.2 0.3 0.4 0.5 0.6E Ef

0.005

0.01

0.015

0.02

k N 5

a b

c d

Fig. 13. k RRKð E À E Ã f Þ (red), k RRKMð E À E Ã f Þ (blue) and k simð E À E Ã f Þ (green) for (a) N = 2, (b) N = 3, (c) N = 4 and (d) N = 5. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)




Barblinger and Schlier identify various properties of thereactant phase space that should be taken into accountwhen computing a statistical, RRKM-type dissociationrate constant. These factors are: (a) presence of direct reac-tive trajectories; (b) recrossing of the transition state(s); (c)existence of bound (nonreactive) regions of reactive phasespace. After corrections for all these factors have been

made, any remaining discrepancies between RRKM theory

and simulation results can be presumably be attributed todynamical effects such as slow IVR (existence of significantphase space bottlenecks) in the reactive portion of phasespace.

The analysis of Berblinger and Schlier can be summa-rized in the following formula for the corrected RRKM

dissociation rate constant [96,80]

k RRKMcorrected ¼ k RRKMj

ð1 À /zdirectÞ

ð1 À /bound À /directÞ; ð47Þ

where /bound is the fraction of the reactant phase space re-gion occupied by bound (unreactive) trajectories, /direct isthe fraction of phase space occupied by direct (‘‘nonstatis-tical’’) trajectories, /z

direct is the fraction of the phase spaceat the transition state occupied by direct trajectories, and j

is a factor that corrects for recrossing of reactive trajecto-ries at the transition state (the transmission coefficient).

Rather than attempt to implement a corrected statisticalrate theory in the form given by Berblinger and Schlier, wenow discuss a related approach for the N = 2 chain basedon the classical spectral theorem [66–73].

7.2. Computation of statistical and dynamical rate constants

via the classical spectral theorem

In classical unimolecular rate theory it has been knownat least since the work of Thiele [66] that the appropriateinvariant measure on the energy shell for reactive trajecto-ries is given by the volume element (see also refs [67–73])

dr ¼ dqz ^ d p z ^ dt ; ð48Þwhere (qà, pà) are canonical coordinates on the transitionstate (we consider the N = 2 case for simplicity), and t isthe time along a trajectory initiated on the transition statepassing into the interior of the reactant region. The totaldensity of non-bound trajectories on the energy shellH = E in the reactant region is therefore obtained by inte-grating the volume element dr over the transition state(s)and over the gap time for each trajectory; this is the contentof the classical spectral theorem [66–73]. Thus, trajectoriesinitated on transition states j = 1, 2 are run backwards intime until they either pass out of the reactant region or asuitably large cutoff time has elapsed. (Note that only a

set of measure zero of these trajectories initiated at thetransition state will remain trapped in the reactant regionforever.) The time interval between entry into and exit fromthe reactant region is the gap time [101,66]. Details of thesampling procedure and computation of gap times aregiven in Appendix D.

For N = 2, the total density of non-bound trajectorieson the energy shell for H = E is then

qnonboundð E Þ ¼Nz1ð E À E Ã f Þht i1 þNz

2ð E À E Ã f Þht i2; ð49ÞwhereNz

i ð E À E Ã f Þ is the sum of states for transition state i

and

ht

ii is the average gap time for trajectories that leave

the reactant region via transition state i (in forward time)

0.1 0.2 0.3 0.4 0.5 0.6E Ef

0.5

1

1.5

2

2.5

1‡

0.1 0.2 0.3 0.4 0.5 0.6E Ef

1

2

3

4

2‡

Fig. 14. (a) Sum of states for transition state 1. Nz1;HOð E À E Ã f Þ (red),

Nz1;anhð E À E Ã f Þ (blue) and Nz

1;PODSð E À E Ã f Þ (black circles) versus E ; (b)Sum of states for transition state 2. Nz

2;HOð E À E Ã f Þ (red), Nz2;anhð E À E Ã f Þ

(blue) andNz2;PODSð E À E Ã f Þ (black circles) versus E . (For interpretation of

the references to colour in this figure legend, the reader is referred to theweb version of this article.)

0.1 0.2 0.3 0.4 0.5 0.6E–Ef

*

0.8

1.2

1.4

1.6

1.8

k2

k1

Fig. 15. Branching ratio k 2/k

1as a function of energy above threshold,

E À E Ã f , f = 0.02. HO approximation (red); anharmonic Monte Carlo(blue); PODS actions (black circles); trajectory simulation (green circles).(For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)


http://-/?-

http://-/?-



having entered through either channel. Eq. (49) may beinterpreted as an expression for the combined volume of two (bifurcated) reactive cylinders [61–63] with cross-sec-tional areasNz

i ð E À E Ã f Þ and lengths htii , i = 1, 2.The density of states defined in Eq. (49) excludes bound

regions of phase space, as bound trajectories by definition

do not pass through the transition state; direct trajectories,however defined, are still included.By initiating trajectories on transition state j , we can

determine N ( j )(t), the number (or fraction) of unbound spe-cies left at time t that will ultimately decay through channel j . It is straightforward to show that, whereas products inchannel j appear at rate characterized by the constant k j ,the subpopulation of reactive species that dissociatethrough channel j undergoes the same exponential deple-tion exp(Àk t) as the overall reactant concentration.

For those reactant species destined to dissociate throughchannel j , a suitably normalized distribution of gap times is(assuming an exponential distribution [66])

f jðt ; E Þ ¼ expðÀk ð E Þt ÞR 1t c

exp½Àk ð E Þt 0dt 0 ; ð50Þ

where tc > 0 is a cutoff time beyond which all transient,nonexponential behavior has died out. The average gaptime for reaction through channel j is then given by

ht i jð E ; t cÞ ¼Z 1

t c

t 0 f jðt 0; E Þdt 0 ¼ 1

k ð E Þ þ t c; ð51Þ

and depends on the cutoff time tc. The values of k (E ) ob-tained with this method are denoted by k PODS j

1 ð E Þ, wherethe notation indicates that the intercept 1/k is in practice

obtained by fitting the observed linear behavior of hti j atlarge cutoff times tc ! 1 and extrapolating back to findthe intercept. A representative plot of hti j (E ) versus tc(Fig. 16) shows transient, nonexponential behavior forshorter tc values and linear behavior – corresponding toexponential decay – for larger tc values.

Also of interest is the quantity k 0(E ), the statistical dis-sociation rate constant corrected for the presence of bound

regions of reactant phase space (and obtained via the spec-tral theorem with tc = 0). By definition of the average gaptime, q( j )(E ), the portion of the reactant density of statesdestined for dissociation through channel j , is given by

qð jÞð E Þ ¼Nz jð E À E Ã f Þht i jð E Þ; ð52Þ

where hti j (E ) is obtained with tc = 0, so that all ‘‘direct’’trajectories are included in the average. The rate constantk 0(E ) is obtained from

k 0ð E Þ ¼

P N j¼1

Nz jð E À E Ã f Þ

P N j¼1

qð jÞð E Þ: ð53Þ

7.3. Comparison of statistical and dynamical dissociation

rates for N = 2

Total dissociation rate constants for N = 2, f = 0.02 andb = 1 obtained using various methods are shown in Fig. 17.We can see that there is a significant (roughly threefold)disparity between the RRK and the RRKM predictions,representing a qualitative failure of the harmonic approxi-mation. This is not unexpected, as the harmonic approxi-mation is especially inadequate for the reactant region(see Fig. 5a–d). At fixed energy per mode, e = E /N , the dis-parity between harmonic and anharmonic statistical ratetheory increases with the number of atoms N (see below).

If we consider the reactant density of states for N = 2, itshould apparently always be the case that

qð1Þð E Þ þ qð2Þð E Þ 6 qð E Þ; ð54Þas the RHS of (54) includes bound as well as non-boundphase points, so that we expect the inequality

k 0ð E ÞP k RRKMð E Þ ð55Þto hold. As we can see in Fig. 17, inequality (55) holds onlyup to a certain energy, E À E Ã f ’ 0:4, above which the rate

20 40 60 80 100tc

50

100

150

200

250

300

⟨t⟩1

Fig. 16. Plot of average gap time hti1(tc) as a function of cutoff time tc fortrajectories initiated on PODS1, E

¼E Ã f

þ0:4. Note linear behavior

consistent with exponential decay as tc ! 1.

0.1 0.2 0.3 0.4 0.5 0.6E–Ef

*

0.025

0.05

0.075

0.1

0.125

0.15

k

Fig. 17. Rate constants k RRKð E À E Ã f Þ (red), k RRKMð E À E Ã f Þ (blue),k simð E À E Ã f Þ (green circles), k 0ð E À E Ã f Þ (black circles) and k PODS1

1 ð E À E Ã f Þ(purple) versus energy above threshold, E À E Ã F , for N = 2, f = 0.02 andb = 1. (For interpretation of the references to colour in this figure legend,

the reader is referred to the web version of this article.)




constant k 0(E ) actually becomes smaller than the RRKMrate constant. We resolve this apparent paradox by notingthat q(E ) is calculated using the fixed configuration spacesampling region shown in Fig. 4, while the densities q( j )

are obtained via the spectral theorem using the PODS.As the PODS move progressively outwards from their

respective saddle points with increasing energy, the increaseof the associated densities of states with energy is sufficientto outweigh the effects of the presence of bound regions,resulting in a violation of the inequality (55).

The anharmonic RRKM results are close to total disso-ciation rate constants determined via simulation. However,the RRKM results begin to disagree with the simulationrates with increasing energy, the disparity eventually reach-ing a factor of roughly three. This is perhaps not surpris-ing, given the lack of global chaos seen in the SOS forN = 2. At the very highest energies, it becomes difficult tofit the lifetime distributions to a pure exponential decay;nevertheless, the fits obtained for k sim and k 0 agree at these

energies.Dissociation rate constants obtained using the classical

spectral theorem essentially match those obtained by simu-lation. Of course the spectral theorem approach, which isbased upon the distribution of gap times, must reflect thesame dynamical information as a standard trajectory sim-ulation. It is nevertheless gratifying to see that the twoapproaches agree so well. It seems likely that the spectraltheorem approach will prove useful in the multimode case(N P 3 dof) in combination with the normal form basedanalysis of the transition state of Uzer et al. [102,103].

8. N > 3 atom chains

A significant jump in dynamical complexity occurs ingoing from N = 2 to N P 3-atom chains. For N > 2, thedynamics and phase space structure cannot easily be visu-alized via a SOS [60,104]. Further, the transition statesfor chain fragmentation are themselves (N À 1)P 2-degreeof freedom dynamical systems, making visualization andaccurate anharmonic sampling difficult using theapproaches described above for N = 2 atom chains.

Accurate determination of transition states in N -atomchains under tensile stress is a problem that seems well-sui-ted to the application of recent advances in normal formexpansion techniques recently developed by Uzeret al.[102,103,105]. In the present work, however, werestrict ourselves to the computation of a ‘‘hybrid’’ har-monic–anharmonic rate constant for our statistical predic-tions of chain fragmentation rates.

As Fig. 14 shows, for N = 2 at least the harmonic tran-sition state sum of states is a close approximation to theanharmonic result at energies close to the dissociationthreshold E Ã f . We therefore combine a harmonic oscillatorapproximation for the transition state sum of states witha full numerical determination of the reactant density of states to give our RRKM expression for N P 3-atom

chains:

k RRKM=HOð E Þ ¼NzHOð E À E Ã f Þqanhð E Þ

: ð56Þ

Total bond dissociation rate constants obtained usingRRK, RRKM and simulation with f = 0.02 and b = 1are plotted as a function of energy E for N = 2–5 inFig. 13. To further examine the validity of the harmonicand statistical approximations to the dynamical rate con-stant, we plot the ratios of harmonic (RRK) to anharmonic(hybrid RRKM) and anharmonic to simulation results as afunction of energy per degree of freedom, E /N , in Figs. 18and 19, respectively.

The ratios of RRK to hybrid RRKM rate constants aresimply the ratios of harmonic and anharmonic reactantdensities of states. Fig. 18 shows that, while the discrepancybetween the RRK and RRKM rate constants grows withthe number of atoms N at fixed energy per mode, e = E /N , at constant total energy E the ratio of RRK and RRKMrate constants approaches unity (harmonic limit) as N

increases. For example, for total energy E = 1 the ratiosare 3.24(N = 2), 2.46(N = 3), 2.12(N = 4) and 1.89(N = 5). RRKM total dissociation rate constants com-

0.3 0.4 0.5 0.6 0.7

E

N

1.5

2

2.5

3

3.5

kRRKM

ksim

Fig. 19. Ratio k RRKM(E )/k sim(E ) as a function of energy per mode, E /N ,for N = 2 (red), N = 3 (green), N = 4 (blue) and N = 5 (purple). (Forinterpretation of the references to colour in this figure legend, the reader is

referred to the web version of this article.)

0.3 0.4 0.5 0.6 0.7

E

N

2.5

3

3.5

4

4.5

kRRK

kRRKM

Fig. 18. Ratio k RRK(E )/k RRKM(E ) as a function of energy per mode, E /N ,for N = 2 (red), N = 3 (green), N = 4 (blue) and N = 5 (purple). (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)




puted with an anharmonic reactant density of states arewithin a factor of 2 of the simulation result. The ratio of RRKM to simulation rate constant shows an overallincreasing trend with increasing E /N .

Finally, Fig. 20 shows the ratios (k RRK/k RRKM) and(k RRKM/k sim) as a function of E /N , N = 2–5, for tetheredchains with potential parameters corresponding to modelpolyethylene [74,75,51,76] and tensile force f = 0.02.

9. Summary and conclusions

In this work we have studied the bond dissociationkinetics of tethered atomic (Morse potential) chains undertensile stress. Our focus has been the nonstatistical aspectsof the dissociation kinetics and the role of anharmonicity.Statistical theories of unimolecular dissociation rate con-stants, both RRKM and RRK (harmonic appproximation)have been applied to predict bond-dissociation rates as afunction of energy and tensile force. A fully anharmonicversion of RRKM theory is implemented using MonteCarlo integration to determine reactant and transition statephase space volumes as a function of energy. Comparisonof statistical predictions with dynamical dissociation rate

constants obtained using trajectory simulations for chains

with N = 2–5 atoms shows that, while the RRK approxi-mation significantly overestimates the dissociation rate,the fully anharmonic RRKM theory is quite accurate.The effects of anharmonicity are at least partly able torationalize discrepancies between trajectory results andclassical transition state theory noted previously [29].

For the two-atom chain, N = 2, we also carry out adetailed examination of phase space structure using Poin-care surfaces of section. We note that good agreementbetween the RRKM and dynamical rate constants isobtained despite the fact that the reactant phase space con-tains a significant fraction of relatively short-lived trajecto-ries (as judged by the number of intersections with theSOS). For the N = 2 chain, we also implement a novelapproach to the extraction of decay rate constants basedon the classical spectral theorem. This approach correctsfor the presence of invariant bound (nonreactive) regionsof reactant phase space, and yields total dissociation ratesin close agreement with dynamical simulation.

Our statistical predictions for chains with N P 3 atomsare obtained using a hybrid statistical theory involving aharmonic approximation for motion in the transition statefor bond dissociation. Further exploration of the dynamicsin the vicinity of the transition state in multimode chainsalong the lines of the recent work by Uzer and Jaffe[102,103,105] would clearly be worthwhile, as would aninvestigation of the applicability of the spectral theoremapproach.

Appendix A. Determining the reactant phase space volume

N ðE Þ

For N = 2, the configurational projection of M 0 is theunion of the rectangular region and and adjacent triangularregion shown in Fig. 4. The shape of this region ensuresthat phase points in the vicinity of transition states 1 and2 are sampled appropriately while avoiding ‘‘leakage’’ of sampled points into nearby exit channels.

For example, for f = 0.02 and b = 1 the line x1 = 3.303is taken to define a suitable boundary between samplingregions in the vicinity of transition states 1 and 2 (seeFig. 4). The line x2 = Àx1 + 2rà + req defining transitionstate 1 intersects this line at (3.303,8.898). The phase spacesampling for the rectangular region is then

x1 2 ½0; 3:303 ðA:1aÞ x2 2 ½0; r eq þ r z ðA:1bÞ p j 2 À

ffiffiffiffiffiffiffiffiffiffiffiffi2 E max

p ; ffiffiffiffiffiffiffiffiffiffiffiffi

2 E max

p h i; j ¼ 1; 2; ðA:1cÞ

while that for the triangular region is

x2 2 ½0; 8:898 ðA:2aÞ x1 2 ½3:303;À x2 þ 2r z þ r eq ðA:2bÞ p j 2 À


p ;


p h i; j ¼ 1; 2: ðA:2cÞ

The phase space region M 0 therefore has volume

0.3 0.4 0.5 0.6 0.7

E

N

2.5

3

3.5

4

4.5

5

kRRK

kRRKM

0.4 0.5 0.6 0.7

E

N

0.5

1.5

2

2.5

kRRKM

ksim

a

b

Fig. 20. (a) Ratio k RRK(E )/k RRKM(E ) as a function of energy per mode E /N for N = 2 (red), N = 3 (green), N = 4 (blue) and N = 5 (purple). (b)Ratio k RRKM(E )/k sim(E ) as a function of energy per mode E /N for N = 2(red), N = 3 (green), N = 4 (blue) and N = 5 (purple). Morse parameterscorrespond to model polyethylene. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of thisarticle.)




V 0 ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffi

2 E max

p 2

3:303ðr eq þ r zÞþ1

28:898ð2r z þr eq À 3:303ÞÂ Ã& '

:

ðA:3ÞFor N = 2 the phase space volume Nð E Þ is fitted to theform

Nð E Þ ’ E 2

ðc0 þ c1 E þ . . .Þ; ðA:4ÞFor N P 3, dim(M 0) = 2N and the region M 0 is defined bythe following ranges of phase space coordinates:

x j 2 ½ x jÀ1; x jÀ1 þ r z; j ¼ 1; . . . ; N ðA:5aÞ p j 2 À


p ; ffiffiffiffiffiffiffiffiffiffiffiffi

2 E max

p h i; j ¼ 1; . . . ; N : ðA:5bÞ

with x0 = 0. Hence for the reactant region of phase space,V 0 ¼ ð2r z ffiffiffiffiffiffiffiffiffiffiffiffi

2 E max

p Þ N , and the function Nð E Þ is fitted to theform

N

ð E

Þ ’E N

ðc0

þc1 E

þ. . .

Þ;

ðA:6

Þthat is, harmonic oscillator phase space volume plus anhar-monic corrections.

Appendix B. Sampling phase points on the energy shell

For N degrees of freedom, the phase space volumeelement is

dz ¼ dxd p ¼ d x1 Á Á Á d x N d p 1 Á Á Á d p N : ðB:1ÞIntroducing polar coordinates in momentum space, thisbecomes

dz ¼ d x1 Á Á Á d x N p N

À1

d p dx ðB:2Þwhere dx is the associated angular volume element. ForN = 2, dx = d/, with 0 6 / 6 2p, while for N = 3,dx = sinhdh d/, with 0 6 h 6 p and 0 6 / 6 2p. To spec-ify phase points on the energy shell, we change variablefrom p to E , using

o H

o p ¼ oT

o p ¼ p ðB:3Þ

(setting all masses m = 1) to obtain

dz ¼ d x1 . . . d x N p N À2d E dx: ðB:4Þ

On the energy shell, the (2N À 1)-dimensional element of hypersurface area d r d z|H (z )=E is therefore

dr ¼ d x1 Á Á Á d x N p N À2 dx: ðB:5Þ

The task is then to sample points distributed according tothe distribution function implied by (B.5) (see also[106,28]).

B.1. N = 2 degrees of freedom

The situation for N = 2 dof is very straightforward. Thevolume element is

dr N ¼2 ¼ d x1d x2 d/: ðB:6Þ

To sample phase points, we simply choose points uni-formly in some region R of configuration space,x ¼ ð x1; x2Þ 2R. If V (x) 6 E , we determine the value of the momentum magnitude p required to satisfy H = E .Momentum components are then defined according to p = ( psin/, pcos/), with angle / chosen at random in the

interval 0 6 / 6 2p.

B.2. N P 3 degrees of freedom

The volume element in this case is

dr ¼ d x1 d x2 d x3 p dx: ðB:7ÞIt is therefore necessary to weight momentum valuesaccording to p rather than the natural 3D weighting p2.To accomplish this we use a rejection method [99].

Points are sampled uniformly at random inside a regionR of x = (x1,x2,x3) space. For any given energy E , there is a

maximum possible momentum magnitude P , which will bedetermined by the value of the kinetic energy at the poten-tial minimum. For any sampled point x, providedV (x) 6 E , we determine the associated scaled momentummagnitude,

g p

P ; 0 6 g 6 1: ðB:8Þ

Now choose a random number q between 0 and 1.

(1) If g < q, we reject the point.(2) If g > q, we accept the point. Gaussian variates in 3D

are then used to compute a random orientation for

the momentum vector in 3D. If n is a randomly cho-sen vector on the 2-sphere, the chosen phase point isz ¼ ðx; p ¼ p n ¼ g P nÞ.

Analogous rejection procedures are used for samplingthe energy shell for N > 3.

Let N trial be the number of sampled points satisfyingV (x) 6 E ; this is the total number of points sampled onthe energy shell, some of which will be rejected as above.Let the number of accepted points {zk } be N accept. Theaverage value of any property F (z) is then

h F

i ¼Pk F ðzk Þ N accept ð

B:9Þ

where the sum is over accepted points.

Appendix C. Stable and unstable manifolds of the PODS

The first step in computing the full nonlinear stable andunstable manifolds W s=u of the PODS is determination of the linearized manifolds, E s and E u, respectively [107].The monodromy matrix M corresponding to one triparound the unstable periodic orbit PODS1 provides a line-arized approximation to the iterated dynamics of a phasepoint on the SOS displaced from the PODS by

Dz = (DX ,DP ),




M Dz ¼ M XX M XP

M PX M PP

!D X

D P

!¼ D X 0

D P 0

!¼ Dz0: ðC:1Þ

To determine M we propagate the displaced phase pointsDzDX = (DX ,0) and DzDP = (0,DP ) for one iteration of theSOS mapping. If DzDX and DzDP evolve to Dz 0 and Dz00

respectively under the mapping, thenD

X M XX = X 0,D

X M PX = P 0 and DP M XP = X 00, DP M PP = P 00. The matrixM thus obtained can then be diagonalized to obtain eigen-values and eigenvectors.

For example, for E ¼ E Ã f þ 0:1, eigenvalues arek1 = 0.715, k2 = 1.399 with corresponding stable andunstable eigenvectors zs = (X s,P s) = (À0.9808,0.1948) andzu = (X u,P u) = (À0.9808, À 0.1948), respectively. Thedeterminant of M is 1.00006, verifying symplecticity (areaconservation) of the linearized dynamics [60] to within rea-sonable numerical accuracy.

Points on the (nonlinear) manifolds Wu= s are obtainedby propagating phase points close to the periodic orbit

on the respective linearized manifolds forwards/backwardsin time. To visualizeWu= s, we select 1000 initial conditionsalong short segments of E s/u close to the PODS and prop-agate all these phase points either forward or backwards intime. The resulting manifolds for E Ã f þ 0:1 are shown inFig. 8. The bound complex region can be defined to bethe phase space volume enclosed by these two manifoldsbetween the PODS and their first intersection [79].

Appendix D. Computing average gap times

To calculate hti1(E ), the average gap time for trajectories

exiting via transition state 1, initial coordinate values ontransiton state 1 are selected according to

nð0Þ 2 ½0; r z ðD:1aÞ X ð0Þ ¼ X PODS1

ð E Þ; ðD:1bÞwhere X PODS1

ð E Þ is the average value of coordinate X forPODS 1 at energy E . The initial momentum variableP(t = 0) is selected in the range

Pð0Þ 2 À ffiffiffiffiffiffiffiffiffiffiT ð0Þ

p ; ffiffiffiffiffiffiffiffiffiffiT ð0Þ

p h i; ðD:2Þ

where the kinetic energy is T (0) = E

ÀV (X (0),n(0)). The

momentum P (0) is then P ð0Þ ¼ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

4ðT ð0Þ ÀPð0Þ2Þq

, wherethe positive root is taken so that the trajectory will enterthe reactant region in negative time.

To calculate the average gap time hti2(E ), we samplex1(t = 0) and p1(t = 0) on transition state 2 by selecting ini-tial coordinate values in the ranges

x1ð0Þ 2 ½0; xmax1 ð E Þ; ðD:3aÞ

x2ð0Þ ¼ xPODS22 ð E Þ ðD:3bÞ

where xmax1 ð E Þ is a suitably chosen maximum value of coor-

dinate x1, and xPODS22

ð E

Þis the average x2 value for

PODS2(E ). We sample initial momenta according to

p 1ð0Þ 2 À ffiffiffiffiffiffiffiffiffiffiffiffi

2T ð0Þp

; ffiffiffiffiffiffiffiffiffiffiffiffi

2T ð0Þp h i

; ðD:4aÞ

p 2ð0Þ ¼ Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2T ð0Þ À p 1ð0Þ2q

; ðD:4bÞwhere T (0) = E À V (x1(0),x2(0)). We select only the posi-tive solution for p2(0) so that again our trajectories will

propagate into the reactant region in negative time.For both average gap time calculations, the dissociationcriterion is that either X ðÀt Þ > X PODS1ð E Þ (reaction 1) or x2ðÀt Þ > x

PODS22 ð E Þ, x1ðÀt Þ < xmax

1 ð E Þ (reaction 2). Forchannel i , we therefore compute the average gap time fortrajectories that, in forward time, enter the reactant regionfrom either transition state and exit only through transitionstate i . For both calculations we integrate ntraj = 104 trajec-tories for energies E ¼ E Ã f þ 0:1; E Ã f þ 0:2; E Ã f þ 0:3; E Ã f þ0:4; E Ã f þ 0:5 and E max. Each trajectory is integrated withtime step s = À10À1s0 for nstep = 5 · 105 time steps or untilreaction has occurred. A negligible number of trajectoriesfail to react within nstep timesteps.

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