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Isomerization kinetics of a strained Morse oscillator ring Joseph N. Stember 1 , Gregory S. Ezra Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, NY 14853, USA article info Article history: Received 14 September 2010 In final form 19 January 2011 Available online 1 February 2011 Keywords: Mechanochemistry Isomerization kinetics Statistical theories abstract Isomerization kinetics are studied for a 3-atom linear Morse chain under constant strain. Cyclic boundary conditions render the problem isomorphic to an isomerizing system of particles on a ring. Both RRKM (fully anharmonic, Monte Carlo) and RRK (harmonic approximation) theories are applied to predict isom- erization rates as a function of energy for a particular strain value. Comparison with isomerization rates obtained from trajectory calculations of flux correlation functions shows that the harmonic approxima- tion significantly overestimates the rate constant, whereas the anharmonic calculation comes consider- ably closer to the simulation result. The energy range over which a rate constant has dynamical meaning is delineated. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Isomerization reactions are of great importance in chemistry, and are central to many condensed phase and biological processes. Study of the dynamics of isomerization provides a venue for explo- ration of fundamental issues of statisticality and the effectiveness of statistical theories [1–5]. (A brief survey of some relevant literature is given in Ref. [6]. See also Refs. [7–11].) For example, De Leon and Berne [12] examined the dynamics of a symmetrical two-well potential, adjusting the energy and degree of coupling in the system. For energies just above the activation barrier, RRKM behavior [1–5] was found for sufficiently large coupling. However, for energies significantly above the barrier, even with large cou- pling, oscillatory recrossing motions led to non-exponential decay [13,14] of reactive flux [15], so that an isomerization rate constant was no longer defined. A fundamental understanding of the intramolecular dynamics and kinetics of fragmentation (bond dissociation) of atomic chains subject to a tensile force is needed to provide a solid foundation for theories of material failure under stress [16–18], polymer rupture [16,17,19–26], adhesion [27], friction [28], mechanochemistry [29–34] and biological applications of dynamical force microscopy [35–41]. The dissociation of a 1-D chain subject to constant tensile force is a problem in unimolecular kinetics, and a fundamental is- sue in unimolecular kinetics concerns the applicability of statistical approaches such as RRKM [1–5] or transition state theory [42]. Previous theoretical work has suggested that dissociation of atomic chains under stress is not amenable to simple statistical approaches [43–52]. Studies of energy transfer and equipartition in single chains of coupled anharmonic oscillators have a long his- tory, beginning with the seminal work of Fermi et al. [53] and Ford [54]. For other work on dynamics of atomic chains, see Refs. [49,55–66]. In the present paper we study numerically the kinetics of bond breaking in single atomic chains under stress. In previous work, we have studied the applicability of statistical theories to describe the dissociation rate of tethered linear chains under tensile stress [67]. In the present work we impose cyclic boundary conditions on the chain, so that it is mapped onto a ring polymer under strain. For small strains, the potential surface has a single minimum, whereas at higher strains a bifurcation occurs to yield several distinct min- ima [68]. The reaction of interest now corresponds to isomeriza- tion between the various minima. Standard methods involving the reactive flux can then be employed to study the kinetics of isomerization [15]. Section 2 describes the system to be studied and the form of the potential surface as a function of the strain. In Section 3 we outline the methods used to compute the isomerization rate constant based on the reactive flux approach [15]. Section 4 discusses the results of our trajectory simulations, while statistical (RRK and RRKM) com- putations of the rate are reported in Section 5. Section 6 concludes. 2. Isomerizing system The isomerizing system we consider is a cyclic version of the tethered linear chain treated previously [67]. Our system consists of P identical atoms on a line confined to a ‘‘box’’ of length L. We assume that adjacent atoms interact via Morse potentials, and im- pose cyclic boundary conditions, so that our setup can be mapped onto a system consisting of P atoms on a ring with circumference L. That is, we map a linear chain to a strained ring. Taking the length L 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.01.013 Corresponding author. Tel.: +1 607 255 3949; fax: +1 607 255 4137. E-mail address: [email protected] (G.S. Ezra). 1 Present address: NYU Langone Medical Center, 550 First Ave, New York, NY 10016, USA. Chemical Physics 381 (2011) 80–87 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Transcript
Page 1: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

Chemical Physics 381 (2011) 80–87

Contents lists available at ScienceDirect

Chemical Physics

journal homepage: www.elsevier .com/locate /chemphys

Isomerization kinetics of a strained Morse oscillator ring

Joseph N. Stember 1, Gregory S. Ezra ⇑Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, NY 14853, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 September 2010In final form 19 January 2011Available online 1 February 2011

Keywords:MechanochemistryIsomerization kineticsStatistical theories

0301-0104/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.chemphys.2011.01.013

⇑ Corresponding author. Tel.: +1 607 255 3949; faxE-mail address: [email protected] (G.S. Ezra).

1 Present address: NYU Langone Medical Center, 510016, USA.

Isomerization kinetics are studied for a 3-atom linear Morse chain under constant strain. Cyclic boundaryconditions render the problem isomorphic to an isomerizing system of particles on a ring. Both RRKM(fully anharmonic, Monte Carlo) and RRK (harmonic approximation) theories are applied to predict isom-erization rates as a function of energy for a particular strain value. Comparison with isomerization ratesobtained from trajectory calculations of flux correlation functions shows that the harmonic approxima-tion significantly overestimates the rate constant, whereas the anharmonic calculation comes consider-ably closer to the simulation result. The energy range over which a rate constant has dynamical meaningis delineated.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Isomerization reactions are of great importance in chemistry,and are central to many condensed phase and biological processes.Study of the dynamics of isomerization provides a venue for explo-ration of fundamental issues of statisticality and the effectivenessof statistical theories [1–5]. (A brief survey of some relevantliterature is given in Ref. [6]. See also Refs. [7–11].) For example,De Leon and Berne [12] examined the dynamics of a symmetricaltwo-well potential, adjusting the energy and degree of couplingin the system. For energies just above the activation barrier, RRKMbehavior [1–5] was found for sufficiently large coupling. However,for energies significantly above the barrier, even with large cou-pling, oscillatory recrossing motions led to non-exponential decay[13,14] of reactive flux [15], so that an isomerization rate constantwas no longer defined.

A fundamental understanding of the intramolecular dynamicsand kinetics of fragmentation (bond dissociation) of atomic chainssubject to a tensile force is needed to provide a solid foundation fortheories of material failure under stress [16–18], polymer rupture[16,17,19–26], adhesion [27], friction [28], mechanochemistry[29–34] and biological applications of dynamical force microscopy[35–41]. The dissociation of a 1-D chain subject to constant tensileforce is a problem in unimolecular kinetics, and a fundamental is-sue in unimolecular kinetics concerns the applicability of statisticalapproaches such as RRKM [1–5] or transition state theory [42].Previous theoretical work has suggested that dissociation ofatomic chains under stress is not amenable to simple statistical

ll rights reserved.

: +1 607 255 4137.

50 First Ave, New York, NY

approaches [43–52]. Studies of energy transfer and equipartitionin single chains of coupled anharmonic oscillators have a long his-tory, beginning with the seminal work of Fermi et al. [53] and Ford[54]. For other work on dynamics of atomic chains, see Refs.[49,55–66].

In the present paper we study numerically the kinetics of bondbreaking in single atomic chains under stress. In previous work, wehave studied the applicability of statistical theories to describe thedissociation rate of tethered linear chains under tensile stress [67].In the present work we impose cyclic boundary conditions on thechain, so that it is mapped onto a ring polymer under strain. Forsmall strains, the potential surface has a single minimum, whereasat higher strains a bifurcation occurs to yield several distinct min-ima [68]. The reaction of interest now corresponds to isomeriza-tion between the various minima. Standard methods involvingthe reactive flux can then be employed to study the kinetics ofisomerization [15].

Section 2 describes the system to be studied and the form of thepotential surface as a function of the strain. In Section 3 we outlinethe methods used to compute the isomerization rate constant basedon the reactive flux approach [15]. Section 4 discusses the results ofour trajectory simulations, while statistical (RRK and RRKM) com-putations of the rate are reported in Section 5. Section 6 concludes.

2. Isomerizing system

The isomerizing system we consider is a cyclic version of thetethered linear chain treated previously [67]. Our system consistsof P identical atoms on a line confined to a ‘‘box’’ of length L. Weassume that adjacent atoms interact via Morse potentials, and im-pose cyclic boundary conditions, so that our setup can be mappedonto a system consisting of P atoms on a ring with circumference L.That is, we map a linear chain to a strained ring. Taking the length L

Page 2: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

Fig. 1. Schematic illustration of the formation of 2 ‘isomers’ via bifurcation for thestrained diatomic chain, P = 2, L > 2req.

J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87 81

to be a constant, we have P � 1 = N degrees of freedom for a P-atomsystem.

A unique equilibrium structure exists for L � P req, req being theequilibrium bond distance in the unstrained Morse potential. Allbond distances are equal in such a structure. As L increases inthe strained chain to the point where the value of L/P significantlyexceeds the equilibrium value req, a bifurcation occurs wherein the

a b

c d

e

Fig. 2. 2-Atom potential curves for b = 1 and (a) L = 2, (b) L = 3, (c) L = 4, (d) L = 5 and (e) Ldissociation energy D0, and length in units of the equilibrium bond distance req (cf. Eq.

number of equilibrium structures goes from 1 to P, at which pointthe initial (symmetric) equilibrium structure becomes a maximumof the potential [68]. A schematic of the two isomers forming in astrained N = 1 ring is shown in Fig. 1.

In terms of single-particle coordinates x = {xk}, the potentialenergy is

VðfxigPi¼1; LÞ ¼

XP�1

i¼1

VMðxiþ1 � xiÞ þ VMðL� ðxP � x1ÞÞ; ð1Þ

where the Morse potential is

VMðrÞ ¼ D0½1� expð�bðr � reqÞÞ�2: ð2Þ

For all calculations, we measure energy in units of the pairwise dis-sociation energy D0, and length in units of the equilibrium bond dis-tance req.

In bond coordinates the potential is

V fri;iþ1gP�1i¼1 ; L

� �¼XP�1

i¼1

VMðri;iþ1Þ þ VM L�XP�1

i¼1

ri;iþ1

!; ð3Þ

where ri,j � xj � xi.

2.1. Diatomic chain: N = 1

For the diatomic case (P = 2,N = 1) the potential is simply

Vðr12Þ ¼ VMðr12Þ þ VMðL� r12Þ: ð4Þ

As L increases from 1 to 10, the potential curve bifurcates, goingfrom having one well to two wells, both corresponding to a stableisomer (Fig. 2). The large L regime is of interest from the point of

= 10. For all calculations, we measure energy in units of the pairwise Morse potential(2)).

Page 3: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

82 J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87

view of isomerization dynamics. Denoting the left well A and thaton the right B, the energetic barrier to isomerization is given byVðrz12Þ � VðrA=B

12 Þ ¼ E�L , where rA=B12 denotes the position of either well

A or well B (which have the same potential energy).To simplify our calculations, we transform coordinates x! ~x

such that in the new coordinates the kinetic energy takes the sim-ple diagonal form

T ¼ 12

XN

i¼1

~p2i : ð5Þ

The generating function [69] for the desired transformation toJacobi-type coordinates (X,n) is

Fðx1; x2; P;PÞ ¼ aðx1 þ x2ÞP þ bðx2 � x1ÞP; ð6Þ

which gives the relations

X ¼ @F@P¼ aðx1 þ x2Þ; ð7aÞ

n ¼ @F@P¼ bðx2 � x1Þ; ð7bÞ

P ¼ 12aðp1 þ p2Þ; ð7cÞ

P ¼ 12bðp2 � p1Þ: ð7dÞ

The kinetic energy is

12ðp2

1 þ p22Þ ¼ a2P2 þ b2P2; ð8Þ

so that, with a ¼ b ¼ 1=ffiffiffi2p

, we have

T ¼ 12ðP2 þP2Þ; ð9Þ

a

c

Fig. 3. 3-Atom potential energy surfaces for b = 1 and (a) L = 3, (b) L = 4, (c) L = 5 and (d) Ldissociation energy D0, and length in units of the equilibrium bond distance req (cf. Eq.

and the Hamiltonian takes the form

H ¼ VM

ffiffiffi2p

n� �

þ VM L�ffiffiffi2p

n� �

þ 12

P2 þ 12

P2: ð10Þ

We have _P ¼ � @H@X ¼ 0, so that the momentum P conjugate to X is a

constant of the motion. In other words, integration of the equationsof motion should ideally preserve the center of mass momentumP(t), so that if we choose only trajectories such that P(t = 0) = 0, thenevery such trajectory should have P(t) = 0 "t. For N = 2 we findnumerically that our symplectic integrator generates jP(t)j valueson the order of 10�18 or smaller.

Choosing Pð0Þ ¼ 0 ¼ 1ffiffi2p ðp1 þ p2Þ implies that p1ð0Þ ¼ �

ffiffiffiTp

;

p2ð0Þ ¼ �ffiffiffiTp

. Motion of the system consists of trivial oscillation be-tween the two wells when E > E�L , and within one well whenE 6 E�L .

2.2. Triatomic: N = 2

In terms of bond coordinates, the potential for P = 3, N = 2 is

Vðr12; r23Þ ¼ VMðr12Þ þ VMðr23Þ þ VMðL� ðr12 þ r23ÞÞ: ð11Þ

As L increases from 2 to 10, we again see a bifurcation, this time intoa three-well system (Figs. 3 and 4). Each pair of wells is separatedby an activation barrier, as in the N = 1 case. All three activationenergies are equivalent. There is also a maximum (index 2 saddle)in the potential, situated symmetrically with respect to the threewells, which represents the energetically unfavorable configurationwherein all three bonds are extended equally to a value consider-ably greater than req.

For all our calculations, we use b = 1 and L = 10. This yields theactivation energy E�L¼10 ¼ 0:882 D0, which is close to that of thetethered linear chain studied previously with b = 1 and f = 0.02

b

d

= 6. For all calculations, we measure energy in units of the pairwise Morse potential(2)).

Page 4: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

a

b

Fig. 4. 3-Atom potential energy surface for b = 1 and L = 10 with saddles �AB (redcircle), �AC (green square) and �BC (blue triangle). The local maximum in thepotential is denoted by a purple diamond. (a) V as a function of single particlecoordinates (r12,r23). (b) V as a function of Jacobi coordinates (n1,n2). For allcalculations, we measure energy in units of the pairwise Morse potential dissoci-ation energy D0, and length in units of the equilibrium bond distance req (cf. Eq. (2)).(For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87 83

[67]. The potential barrier associated with the centrally located lo-cal maximum is at energy E = 1.4482 D0.

Potential energy surfaces (b = 1, L = 10) in bond coordinates andJacobi coordinates, together with saddle points, are displayed inFig. 4(a) and (b), respectively. We use the following nomenclaturefor our wells: A is the lower left well in Fig. 4(a), B is the lower rightwell in Fig. 4(a) and C the upper left well in Fig. 4(a). The transitionstate separating wells A and B (for instance) is then denoted by �AB.

The full Hamiltonian in terms of bond coordinates is

H ¼ VMðr12Þ þ VMðr23Þ þ VMðL� ðr12 þ r23ÞÞ þ p2r12

þ p2r23� pr12

pr23; ð12Þ

where we have set the center of mass momentum P equal to zero. Interms of single-particle coordinates, the Hamiltonian is given by

H ¼ VMðx2 � x1Þ þ VMðx3 � x2Þ þ VMðL� ðx3 � x1ÞÞ

þ 12

p21 þ p2

2 þ p23

� �: ð13Þ

The generating function for the tranformation to Jacobi coordinates(X,n1,n2) is

Fðx1; x2; x3; P;P1;P2Þ ¼ aðx1 þ x2 þ x3ÞP þ bðx2 � x1ÞP1

þ cðx3 �12ðx1 þ x2ÞÞP2; ð14Þ

so that we have

X ¼ @F@P¼ aðx1 þ x2 þ x3Þ; ð15aÞ

n1 ¼@F@P1

¼ bðx2 � x1Þ; ð15bÞ

n2 ¼@F@P2

¼ c x3 �12ðx1 þ x2Þ

� �; ð15cÞ

P ¼ 13aðp1 þ p2 þ p3Þ; ð15dÞ

P1 ¼1

2bðp2 � p1Þ; ð15eÞ

P2 ¼13cð2p3 � p2 � p1Þ: ð15fÞ

The kinetic energy is

T ¼ 12ðp2

1 þ p22 þ p2

3Þ ¼3a2

2P2 þ b2P2

1 þ3c2

4P2

2; ð16Þ

so that, setting a ¼ 1ffiffi3p ; b ¼ 1ffiffi

2p and c ¼

ffiffi23

q, we have T ¼ 1

2P2 þP2

1 þP22

� �and resulting Hamiltonian

H ¼ 12ðP2 þP2

1 þP22Þ þ VM

ffiffiffi2p

n1

� �þ VM

1ffiffiffi2p ð�n1 þ

ffiffiffi3p

n2Þ� �

þ VM L� 1ffiffiffi2p ðn1 þ

ffiffiffi3p

n2Þ� �

: ð17Þ

As for the case N = 1, the center of mass momentum is a constant ofthe motion, i.e., _P ¼ � @H

@X ¼ 0.

3. Isomerization kinetics and rate constant for N = 2

We now analyze the kinetics of isomerization in the triatomicchain subject to cyclic boundary conditions. We follow the deriva-tion for the two-well system given, for example, in Chandler [15].

3.1. Phenomenological kinetics

Isomerization reactions occurring in our system are

A!kAB B; ð18aÞ

A!kAC C; ð18bÞ

B!kBA A; ð18cÞ

B!kBC C; ð18dÞ

C!kCA A; ð18eÞ

C!kCB B: ð18fÞ

Since the shape and depth of each well is identical, all the rate con-stants (assuming we are in a parameter regime in which a rate con-stant has meaning) are equal, i.e., kAB = kBA = kAC = kCA = kBC = kCB = k.

The concentrations of A, B and C are therefore determined bythe linear rate equations

_cðtÞ ¼ KcðtÞ; ð19Þ

where c(t) = [cA(t),cB(t),cC(t)] is the concentration vector and

K ¼�2k k k

k �2k k

k k �2k

264

375 ð20Þ

is the rate constant matrix, with eigenvalues {0,�3k,�3k}. Assum-ing an initial condition with all species in well A, and setting the ini-tial concentration equal to unity, i.e., c0 = [1,0,0], the appropriatesolution is

cðtÞ ¼ expðKtÞc0 ¼

13þ 2

3 expð�3ktÞ13� 1

3 expð�3ktÞ13� 1

3 expð�3ktÞ

264

375: ð21Þ

Page 5: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

84 J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87

3.2. Isomerization rates and concentration fluctuations

Standard arguments based upon the regression hypothesis [15]relate the relaxation behavior of the concentration cA(t) of speciesin well A, for example, to the decay of fluctuations of the corre-sponding equilibrium concentration:

DcAðtÞDcAð0Þ

¼ CðtÞCð0Þ ¼

hdnAð0ÞdnAðtÞihdnAð0Þ2i

; ð22Þ

where DcA(t) = cA(t) � cA(t ?1) = cA(t) � hcAi and C(t) � hdnA(0)dnA(t) i is the equilibrium concentration fluctuation correlationfunction. The microscopic quantity nA(t) is the average number ofsystems in well A as determined by averaging over the (trajectory)ensemble, and dnA(t) = nA(t) � hnAi, where hnAi is the long time(equilibrium) average value of nA(t).

Our analysis of the phenomenological kinetics of isomerizationshows that

DcAðtÞDcAð0Þ

¼ expð�3ktÞ; ð23Þ

so that the value of the isomerization rate coefficient k can in prin-ciple be extracted from a trajectory simulation evaluation of C(t).

The population nA(t) is defined by the integral over the trajec-tory ensemble of the characteristic function HA ¼ Hðrz12 � r12ÞHðrz23 � r23Þ, where rz12, rz23 mark the positions of the saddle pointsdividing A from B and A from C, respectively. By definition, thequantity HA[r12(t), r23(t)] is equal to 1 when the trajectory is in wellA, and 0 when outside of A. We observe that hHAi ¼ hH2

Ai ¼ xA ¼ 13,

the equilibrium fraction of A. We have

Cð0Þ ¼ hdnAð0Þ2i ¼ hðHAð0Þ � xAÞ2i ¼ xAð1� xAÞ

¼ xAðxB þ xCÞ ¼29: ð24Þ

The numerator of the right hand side of Eq. (22) is

hdnAð0ÞdnAðtÞi ¼ hnAð0ÞnAðtÞi � hn2Ai ¼ hnAð0ÞnAðtÞi � x2

A: ð25Þ

Taking the time derivative of both sides of Eq. (22) leads to the moreuseful form (cf. [15])

�23

k expð�3ktÞ ¼ hHAð0Þ _HAðtÞi ¼ �h _HAð0ÞHAðtÞi: ð26Þ

We now observe that

ddt

HA½r12ðtÞ; r23ðtÞ� ¼ _r12@

@r12HA½r12ðtÞ; r23ðtÞ�

þ _r23@

@r23HA½r12ðtÞ; r23ðtÞ�: ð27Þ

Then, recalling our definition HA½r12ðtÞ; r23ðtÞ� � Hðrz12 � r12ðtÞÞHðrz23 � r23ðtÞÞ, we have

_HAð0Þ ¼ � _r12ð0ÞHðrz23 � r23ð0ÞÞdðr12ð0Þ � rz12Þ

þ_r23ð0ÞHðrz12 � r12ð0ÞÞdðr23ð0Þ � rz23Þ

ð28Þ

so that

k expð�3ktÞ ¼ � 32

_r12ð0ÞHðrz23 � r23ð0ÞÞdðr12ð0Þ � rz12Þ�

þ_r23ð0ÞHðrz12 � r12ð0ÞÞdðr23ð0Þ � rz23ÞHAðtÞ

�; ð29Þ

where the average over initial conditions is taken over the full avail-able phase space, spanning the 3 wells. Note that the minus sign inEq. (29) is correct; at very short times only those trajectories cross-ing into well A, for which _r1k < 0, k = 2,3, will contribute to theintegral.

The microcanonical density of states for well A is

qðEÞ ¼ 13

Zwell A [ well B [ well C

dr12dpr12dr13dpr13

dðE� Hðr12; pr12; r13;pr13

ÞÞ; ð30Þ

where the microcanonical partition function is obtained by averag-ing over all the available phase space (wells A, B and C), and is justthree times the partition function for an individual well, since all 3wells are identical (as is evident from Fig. 4(b)). From Eq. (29), wetherefore obtain via standard manipulations [15] the result

k expð�3ktÞ ¼ � 12qðEÞ

ZzAB ;HðzÞ¼E

dr23dpr23sign½_r12ð0Þ�HAðtÞ

"

þZzAC ;HðzÞ¼E

dr12dpr12sign½_r23ð0Þ�HAðtÞ

#: ð31Þ

4. Trajectory simulations and extraction of rate coefficient

To go from the integral form of k exp(�3kt) given in Eq. (31) tothe expression that we actually use to compute the fluxes, we sumover discrete contributions _r12ð0ÞHAðtÞ and _r23ð0ÞHAðtÞ:

kðEÞ expð�3kðEÞtÞ ¼ � 12qðEÞ

XT AB

N yABðE� E�LÞntraj

!sign½_r12ð0Þ�

"

HAðr12ðtÞ; r23ðtÞÞ þXT AC

N yACðE� E�LÞntraj

!sign _r23ð0Þ½ �

HAðr12ðtÞ; r23ðtÞÞ�; ð32Þ

where T ij is the set of initial conditions on transition state �ij.Nyij E� E�L� �

is the phase space area in the ðrij;prijÞ plane of phase

points that satisfy the microcanonical condition H(z) = E, where ijis the degree of freedom that lies ‘‘along’’ the ij transition state.

Since N yAB E� E�L� �

¼ N yAC E� E�L� �

by symmetry, and because weare interested primarily in the rate of exponential flux decay, wemay disregard constant factors and rewrite our rate constantequality more simply as a proportionality relation

kðEÞ expð�3kðEÞtÞ /X

trajs;AB

sign _r12ð0Þ½ �HAðr12ðtÞ; r23ðtÞÞ"

þX

trajs;AC

sign½_r23ð0Þ�HAðr12ðtÞ; r23ðtÞÞ#: ð33Þ

In a regime where the flux computed using Eq. (33) exhibits expo-nential decay, the decay exponent is just 3k(E).

Initial conditions are selected at random on the two configura-tion space transition states, i.e., with equal probability for phasepoints lying on either transition state. Two line segments definethe transition states AB and AC, and are obtained by solving theequations

V rz12; r23� �

6 Emax ð34aÞ

and

V r12; rz23

� �6 Emax; ð34bÞ

for r23 and r12, respectively, where Emax is our maximum energy ofinterest, which we set equal to 1.0 for b = 1 and L = 10. The config-uration space transition states are shown in Fig. 5. Associatedconjugate momenta pr23

or pr12are sampled uniformly in the appro-

priate range corresponding to total energy E.In practice, for parameters b = 1 and L = 10 we use a modified

characteristic function HAðtÞ ¼ H rz12 � r12ðtÞ� �

H rz23 � r23ðtÞ� �

Hð8� ðr12ðtÞ þ r23ðtÞÞÞ to ensure that only phase points actually inwell A are included by the characteristic function.

Page 6: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

0 2 4 6 8 100

2

4

6

8

10

r12

r23

Fig. 5. Configuration space projections of phase points sampling transition states�AB (vertical) and �AC (horizontal) for b = 1, L = 10 and E = Emax = D0.

1 2 3 4 5 6 x

20

40

60

80

100t

Fig. 6. Single particle coordinates x1 (red), x2 (green) and x3 (blue) versus time t fora single trajectory with N = 2, b = 1 and L = 10. The trajectory was initiated neartransition state �AB with energy E ¼ E�L þ 0:03 D0 ¼ 0:912 D0. Time is measuredin units of s0, the time associated with our choice of units for energy, length andmass. (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87 85

Having selected r12, r23, pr12and pr23

we convert to single-parti-cle coordinates via

x1 ¼ 0; ð35aÞx2 ¼ x1 þ r12; ð35bÞx3 ¼ x2 þ r23; ð35cÞp1 ¼ �pr12

; ð35dÞp2 ¼ pr12

� pr23; ð35eÞ

p3 ¼ pr23; ð35fÞ

where we have arbitrarily set x1 = 0 since the potential dependsonly on relative particle displacements (bond distances). Trajecto-ries are integrated using a second-order symplectic integrator asin our previous work on linear tethered chains [67], and the integra-tion algorithm is simpler in single-particle coordinates due to thediagonal kinetic energy. A sample trajectory initiated near transi-tion state AB with energy E ¼ E�L þ 0:03 D0 ¼ 0:912 D0 is shownin Fig. 6. This plot provides an informative representation of thecomplexity of the intramolecular/isomerization dynamics in thestrained chain; bond breaking and forming is clearly apparent asthe 3 particles execute their apparently irregular dance.

Initial positions on transition states AB, AC are chosen withprobability 1

2, and corresponding momenta are taken to have ran-dom signs (±). We calculate the reactive flux at energies

E ¼ E�L þ jDE, where j = 1,2,. . .M, M = 10, DE ¼ Emax � E�L� �

=M, andEmax = D0. For each energy, we integrate ntraj = 3 103 trajectoriesfor nstep = 6 104 time steps. Our trajectory time step value iss = 10�2 s0, where s0 is the unit of time associated with our choiceof units for energy, length and mass.

As we can see from Fig. 7, for lower energies the nontransientflux shows exponential decay, but the dynamics become moreoscillatory with increasing energy. For these higher energies, therate constant ceases to have meaning [12–14], so for computationsof the rate constant we restrict ourselves to energies in the rangeE�L 6 E 6 E�L þ DE. We therefore calculate fluxes for E ¼ E�L þ jDE0,where j = 1,2, . . .,M = 10 and DE0 = DE/10 is a ‘fine-grained’ energyincrement. The flux decay for the particular energy E ¼ E�L þ 4DE0 isplotted in Fig. 8.

5. RRK and RRKM calculations

For comparison with the values of the rate constant obtainedfrom trajectory simulations, we compute RRK (harmonic) andRRKM (anharmonic) rate constants following the procedures de-scribed previously [67]. The region of reactant configuration spacesampled for the reactant sum of states calculation is shown inFig. 9. Since kAC = k, the RRKM rate constant is given by

kRRKM ¼N zAC E� E�L

� �qAðEÞ

; ð36Þ

where qA(E) is the density of states for well A.A comparison between the flux rate constants and the RRK,

RRKM results is given in Fig. 10, while the ratios kRRK/kflux andkRRKM/kflux are tabulated in Table 1. As is well known (see, forexample, [5]), RRK theory overestimates the rate due primarily tothe harmonic density of states for the reactant being too small.We find good agreement between the RRKM estimate (which usesthe correct numerically determined anharmonic classical reactantdensity of states) and simulation rate constants for energies nearthe activation barrier, with some divergence at larger energies.

Although it is well known that the value of the reactive fluxthrough the transition state at very short times is equal to the sta-tistical value, it is by no means the case that the rate constant ex-tracted from the decay of the reactive flux is necessarily equal tothe RRKM value [15]. In this case, we have good agreement closeto threshold.

6. Summary and conclusions

We have computed harmonic, anharmonic and trajectory rateconstants for isomerization of a triatomic Morse chain understrain, subject to cyclic boundary conditions. Computation of therate coefficient via trajectory simulation requires us to determinethe rate of exponential decay of fluxes across the two transitionstates of interest. Initial conditions are sampled uniformly on theenergy shell, and trajectories integrated using a second-order sym-plectic integrator.

For our microcanonical calculations, we see good agreement be-tween the statistical (anharmonic RRKM) and simulation rate con-stants for energies near the activation barrier, with somedivergence at larger energies. Compared to the tethered chain dis-cussed previously [67], recrossing is more significant for the ringsystem treated here. An isomerization rate constant exists onlywithin a somewhat narrow energy range above the isomerizationbarrier. At higher energies, the simulations reveal oscillating pat-terns of flux among the wells. In other words, as was observedfor the tethered chain [67], the dynamics are apparently statisticalonly within a narrow range of energy per mode. For both systems,

Page 7: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

100 200 300 400 500t

500

0

500

1000

1500

flux

E EL 10 E

a b

c d

e f

Fig. 7. Isomerizing flux versus time for N = 2, b = 1, L = 10. (a) E ¼ E�L þ DE, (b) E ¼ E�L þ 3DE, (c) E ¼ E�L þ 5DE, (d) E ¼ E�L þ 7DE, (e) E ¼ E�L þ 9DE and (f) E ¼ E�L þ 10DE. Theenergy parameters are as defined in the text. Time is measured in units of s0, the time associated with our choice of units for energy, length and mass.

100 200 300 400 500 600t

500

0

500

1000

1500

flux

Fig. 8. Isomerizing flux versus time for N = 2, E ¼ E�L þ 4DE0 , where b = 1, L = 10 andDE0 is the fine-grained energy increment defined in the text. Time is measured inunits of s0, the time associated with our choice of units for energy, length and mass.

Fig. 9. Configuration space projections of phase points obtained by Monte Carlosampling of the reactant region (well A) sum of states (light blue) in Jacobicoordinates, N = 2, b = 1 and L = 10. Also shown are �AB (red circle), �AC (greensquare), �BC (blue triangle) and the potential maximum (purple diamond). (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

86 J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87

reactive events have non-exponential lifetime distributions at highenergies.

There are several possibilities for future work. One might in-volve computing isomerization rate constants for longer chains,P > 3, N > 2. One complication that arises for P > 3 is the presenceof nonequivalent rate coefficients for interwell transitions. Anotherpossibility is to consider chains composed of different atoms, forwhich the symmetry of the problem is reduced. Molecular dynam-ics studies on thermal degradation in polyethylene [58] have sug-gested that bending and torsional modes might play an important

role in the kinetics of chain fragmentation, and such degrees offreedom should be included in future work (see, for example, Refs.[70–73]).

Page 8: Isomerization kinetics of a strained Morse oscillator … of isomerization dynamics. Denoting the left well A and that on the right B, the energetic barrier to isomerization is given

0.003 0.006 0.009E EL

0.001

0.002

0.003

k

Fig. 10. Rate coefficients kRRK (red squares), kRRKM (blue triangles) and kflux (greencircles) versus energy above activation for N = 2, b = 1 and L = 10. Units of rate arethe inverse of the unit of time associated with our choice of units for energy, lengthand mass. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Table 1Ratios of statistical isomerization rate coefficents totrajectory values as a function of energy above thresholdenergy E�L . Ratios kRRKM/kflux and kRRK/kflux versus E, withE ¼ E�L þ iDE0 , i = 2–10. Energy parameters E�L and DE0 aredefined in the text.

i Ratio kRRK/kflux Ratio kRRKM/kflux

2 3.75 1.343 2.38 0.934 2.87 1.175 2.93 1.226 3.53 1.497 3.69 1.578 3.70 1.589 3.42 1.46

10 3.27 1.40

J.N. Stember, G.S. Ezra / Chemical Physics 381 (2011) 80–87 87

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