Joseph N. Stember and Gregory S. Ezra- Isomerization kinetics of a strained Morse oscillator ring

8/3/2019 Joseph N. Stember and Gregory S. Ezra- Isomerization kinetics of a strained Morse oscillator ring

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Isomerization kinetics of a strained Morse oscillator ring

Joseph N. Stembera,1, Gregory S. Ezraa,2,

aDepartment of Chemistry and Chemical Biology

Baker Laboratory

Cornell University

Ithaca

NY 14853

Abstract

Isomerization kinetics are studied for a 3-atom linear Morse chain under

constant strain. Cyclic boundary conditions render the problem isomorphic toan isomerizing system of particles on a ring. Both RRKM (fully anharmonic,Monte Carlo) and RRK (harmonic appproximation) theories are applied to pre-dict isomerization rates as a function of energy for a particular strain value.Comparison with isomerization rates obtained from trajectory calculations offlux correlation functions shows that the harmonic approximation significantlyoverestimates the rate constant, whereas the anharmonic calculation comes con-siderably closer to the simulation result. The energy range over which a rateconstant has dynamical meaning is delineated.

Keywords:

Mechanochemistry, Isomerization kinetics, Statistical theoriesPACS: 82.20.Db, 34.10.+x, 83.20.Lp

1. Introduction

Isomerization reactions are of great importance in chemistry, and are centralto many condensed phase and biological processes. Study of the dynamics ofisomerization provides a venue for exploring issues of statisticality and the ef-fectiveness of statistical theories. (For a brief survey of some relevant literature,see [1].) For example, the classic work of De Leon and Berne [2] examined thedynamics of a symmetrical two-well potential, adjusting the energy and degreeof coupling in the system. For energies just above the activation barrier, RRKMbehavior [37] was found for sufficiently large coupling. However, for energiessignificantly above the barrier, even with large coupling, oscillatory recrossing

Corresponding authorEmail address: [email protected] (Gregory S. Ezra)1Present address: NYU Langone Medical Center, 550 First Ave, New York, NY 100162Tel: 607-255-3949; Fax: 607-255-4137

Preprint submitted to Elsevier September 13, 2010


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motions led to non-exponential decay of reactive flux, so that an isomerizationrate constant was no longer defined.

A fundamental understanding of the intramolecular dynamics and kinetics offragmentation (bond dissociation) of atomic chains subject to a tensile force isneeded to provide a solid foundation for theories of material failure under stress[8, 9], polymer rupture [1014], adhesion [15], friction [16], mechanochemistry[1719] and biological applications of dynamical force microscopy [2024]. (Formore detailed discussion of this topic, see the previous paper [25].)

In the present paper we study numerically the kinetics of bond breaking insingle atomic chains under stress. The dissociation of a 1-D chain subject toconstant tensile force is a problem in unimolecular kinetics, and a fundamentalissue concerns the applicability of statistical approaches such as RRKM [37]or transition state theory [26]. In previous work, we have studied the applica-bility of statistical theories to describe the dissociation rate of tethered linearchains under tensile stress [25]. In the present work we impose cyclic boundaryconditions 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 higherstrains a bifurcation occurs to yield several distinct minima. The reaction ofinterest now corresponds to isomerization between the various minima. Stan-dard methods involving the reactive flux can be employed to study the kineticsof isomerization [27]

Section 2 describes the system to be studied and the form of the potentialsurface as a function of the strain. In section 3 we discuss the methods usedto compute the isomerization rate constant based on the reactive flux approach[27]. Section 4 discusses the results of our trajectory simulations, while sta-tistical (RRK and RRKM) computations 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 linearchain treated previously [25]. Our system consists of P identical atoms on a lineconfined to a box of length L. We assume that adjacent atoms interact viaMorse potentials, and impose cyclic boundary conditions, so that our setup canbe mapped onto a system consisting of P atoms on a ring with circumferenceL. That is, we map a linear chain to a strained ring. Taking the length L to bea constant, we have P 1 = N degrees of freedom for a P-atom system.

A unique equilibrium structure exists for L P req, req being the equilibriumbond distance in the unstrained Morse potential. All bond distances are equalin such a structure. As L increases in the strained chain to the point where the

value ofL/P significantly exceeds the equilibrium value req, a bifurcation occurswherein the number of equilibrium structures goes from 1 to P, at which pointthe initial (symmetric) equilibrium structure becomes a global maximum of thepotential [28]. A schematic of the two isomers forming in a strained N = 1 ringis shown in Figure 1.

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In terms of single-particle coordinates x = {xk}, the potential energy isgiven by

V({xi}Pi=1; L) =P1i=1

VM(xi+1 xi) + VM(L (xP x1)), (1)

where the Morse potential is

VM(r) = D0[1 exp((r req))]2. (2)For all calculations, we measure energy in units of the pairwise dissociationenergy D0, and length in units of the equilibrium bond distance req.

In bond coordinates the potential is

V({ri,i+1}P1i=1 ; L) =P1

i=1VM(ri,i+1) + VM(L

P1

i=1ri,i+1), (3)

where ri,j xj xi.2.1. Diatomic chain: N = 1

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

V(r12) = VM(r12) + VM(L r12). (4)As L increases from 1 to 10, the potential curve bifurcates, going from havingone well to two wells, both corresponding to a stable isomer (Figure 2). Thelarge L regime is of interest from the point of view of isomerization dynamics.Denoting the left well A and that on the right well B, the energetic barrier to

isomerization is given by V(r12) V(rA/B12 ) = EL, where rA/B12 could be theposition of either well A or well B (which have the same potential energy.)

To simplify our calculations, we transform coordinates x x such that thekinetic energy takes the simple diagonal form

T =1

2

Ni=1

p2i . (5)

The generating function [29] for the desired transformation to Jacobi coordinates(X, ) is

F(x1, x2, P, ) = a(x1 + x2)P + b(x2 x1), (6)which gives the relations

X =F

P= a(x1 + x2) (7a)

= F

= b(x2 x1) (7b)

P =1

2a(p1 + p2) (7c)

=1

2b(p2 p1). (7d)

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The kinetic energy is1

2 (p

2

1 + p

2

2) = a

2

P

2

+ b

2

2

, (8)

so that, with a = b = 1/

2, we have

T =1

2(P2 + 2), (9)

and the Hamiltonian takes the form

H = VM(

2) + VM(L

2) +1

2P2 +

1

22. (10)

Of particular note is that P = HX = 0, so that the momentum P conjugateto X is a constant of the motion. In other words, integration of the equationsof motion should ideally preserve the center of mass momentum P(t), so that if

we choose only trajectories such that P(t = 0) = 0, then every such trajectoryshould have P(t) = 0 t. For N = 2 we find numerically that our symplecticintegrator generates |P(t)| values on the order of 1018 or smaller.

Choosing P(0) = 0 = 12

(p1 +p2) implies that p1(0) =

T, p2(0) =

T.

Motion of the system consists of trivial oscillation between the two wells whenE > EL, and within one well when E EL.

2.2. Triatomic: N = 2

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

V(r12, r23) = VM(r12) + VM(r23) + VM(L (r12 + r23)). (11)As L increases from 2 to 10, we again see a bifurcation, this time into a three-well system (Figures 3 and 4). Each pair of wells is separated by an activationbarrier, as in the N = 1 case. All three activation energies are equivalent. Thereis also a global maximum in the potential, situated symmetrically with respectto the three wells, which represents the energetically unfavorable configurationwherein all three bonds are extended equally to a value considerably greaterthan req.

For all our calculations, we use = 1 and L = 10. This yields the activationenergy EL=10 = 0.882, which is close to that of the tethered linear chain studiedpreviously with = 1 and f = 0.02 [25]. Potential energy surfaces ( =1, L = 10) in bond coordinates and Jacobi coordinates, together with saddlepoints, are displayed in Figures 4(a) and 4(b), respectively. We use the followingnomenclature for our wells: let A be the lower left well in Figure 4(a), B the

lower right well in Figure 4(a) and C the upper left well in Figure 4(a). Then,the transition state separating wells A and B for instance will be labeled AB.The full Hamiltonian in terms of bond coordinates is

H = VM(r12) + VM(r23) + VM(L (r12 + r23)) + p2r12 + p2r23 pr12pr23 , (12)

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

H = VM(x2 x1) + VM(x3x2) + VM(L (x3x1)) + 12

p21 + p

22 + p

23

. (13)

The generating function for the tranformation to Jacobi coordinates (X, 1, 2)is

F(x1, x2, x3, P, 1, 2) = a(x1 +x2 +x3)P+b(x2x1)1 +c(x3 12

(x1 +x2))2,

(14)so that we have

X =F

P= a(x1 + x2 + x3) (15a)

1 =F

1 = b(x2 x1) (15b)

2 =F

2= c(x3 1

2(x1 + x2)) (15c)

P =1

3a(p1 + p2 + p3) (15d)

1 =1

2b(p2 p1) (15e)

2 =1

3c(2p3 p2 p1). (15f)

The kinetic energy is

T =1

2 (p21 + p

22 + p

23) =

3a2

2 P2

+ b2

21 +

3c2

4 22, (16)

so that, setting a = 13

, b = 12

and c =

23 , we have T =

12 (P

2 + 21 + 22)

and resulting Hamiltonian

H =1

2(P2 + 21 +

22) + VM(

21) + VM

1

2(1 +

32)

+ VM

L 1

2(1 +

32)

. (17)

As for the case N = 1, the center of mass momentum is a constant of the motion,i.e., P = HX = 0.

3. Isomerization kinetics and rate constant for N = 2

We now analyze the kinetics of isomerization in the triatomic chain subjectto cyclic boundary conditions. We follow the derivation for the two-well systemgiven, for example, in Chandler [27].

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3.1. Phenomenological kinetics

Isomerization reactions occurring in our system are

AkAB B (18a)

AkAC C (18b)

BkBA A (18c)

BkBC C (18d)

CkCA A (18e)

CkCB B. (18f)

Since the shape and depth of each well is identical, all the rate constants (as-

suming we are in a parameter regime in which a rate constant has meaning) areequal, i.e. kAB = kBA = kAC = kCA = kBC = kCB = k.The concentrations ofA,B and C are therefore determined by the linear rate

equations

cA(t) = 2kcA(t) + kcB(t) + kcC(t) (19a)cB(t) = kcA(t) 2kcB(t) + kcC(t) (19b)cC(t) = kcA(t) + kcB(t) 2kcC(t). (19c)

In matrix notation we havec(t) = K c(t), (20)

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

K =

2k k kk 2k k

k k 2k

, (21)

is the rate constant matrix, with eigenvalues {0,3k,3k}. Differential equa-tion (20) has the solution

c(t) = exp(Kt)c0, (22)

where c0 = c(t = 0).Assuming an initial condition with all species in well A, and setting the

initial concentration equal to unity, i.e. c0 = [1, 0, 0], we obtain

c(t) = exp(

Kt)c

0 =

13 +

23 exp(3kt)

1

3 1

3 exp(3kt)13 1

3 exp(3kt)

. (23)

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3.2. Isomerization rates and concentration fluctuations

Standard arguments based upon the regression hypothesis [27] relate therelaxation behavior of the concentration cA(t) of species in well A, for example,to the decay of fluctuations of the corresponding equilibrium concentration:

cA(t)

cA(0)=

C(t)

C(0)=nA(0)nA(t)nA(0)2 , (24)

where cA(t) = cA(t)cA(t ) = cA(t)cA and C(t) nA(0)nA(t) isthe equlibrium concentration fluctuation correlation function. The microscopicquantity nA(t) is the average number of systems in well A as determined byaveraging over the (trajectory) ensemble, and nA(t) = nA(t) nA, wherenA is the long time (equilibrium) average value of nA(t).

Our analysis of the phenomenological kinetics of isomerization shows that

cA(t)

cA(0) = exp(3kt), (25)so that the value of the isomerization rate coefficient k can in principle beextracted from a trajectory simulation evaluation of C(t).

The population nA(t) is defined by the integral over the trajectory ensemble

of the characteristic function HA = (r12r12)(r23r23), where r12, r23 mark

the positions of the saddle points dividing A from B and A from C, respectively.By definition, the quantity HA [r12(t), r23(t)] is equal to 1 when the trajectoryis in well A, and 0 when outside ofA. We observe that HA = H2A = xA = 13 ,the equilibrium fraction of A. We have

C(0) = nA(0)2 = (HA(0) xA)2 = xA(1 xA) = xA(xB + xC) = 29

. (26)

The numerator of the right hand side of Equation (24) is

nA(0)nA(t) = nA(0)nA(t) n2A = nA(0)nA(t) x2A. (27)Taking the time derivative of both sides of Equation (24) leads to the moreuseful form (cf. [27])

23

k exp(3kt) = HA(0)HA(t) = HA(0)HA(t). (28)

We now observe that

d

dtHA [r12(t), r23(t)] = r12

r12HA [r12(t), r23(t)] + r23

r23HA [r12(t), r23(t)] .

(29)

Then, recalling our definition HA [r12(t), r23(t)] (r12r12(t))(r23r23(t)),we have

HA(0) =

r12(0)(r23 r23(0))(r12(0) r12)

+ r23(0)(r12 r12(0))(r23(0) r23)

(30)

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so that

k exp(3kt) = 32

r12(0)(r23 r23(0))(r12(0) r12)+ r23(0)(r

12 r12(0))(r23(0) r23)

HA(t)

,

(31)

where the average over initial conditions is taken over the full available phasespace, spanning the 3 wells. Note that the minus sign in Equation (31) iscorrect; at very short times only those trajectories crossing into well A, forwhich r1k < 0, k = 2, 3, will contribute to the integral.

The microcanonical density of states for well A is

(E) =1

3

well A well B well C

dr12dpr12dr13dpr13(E H(r12, pr12 , r13, pr13)),(32)

where the microcanonical partition function is obtained by averaging over allthe available phase space (wells A, B and C), and is just three times the parti-tion function for an individual well, since all 3 wells are identical (as is evidentfrom Figure 4(b)). From Equation (31), we therefore obtain via standard ma-nipulations [27] the result

k exp(3kt) = 12(E)

AB ,H(z)=E

dr23dpr23sign[r12(0)] HA(t)

+

AC ,H(z)=E

dr12dpr12sign[r23(0)] HA(t)

.

(33)

4. Trajectory simulations and extraction of rate coefficient

To go from the integral form of k exp(3kt) given in Equation (33) to theexpression that we actually use to compute the fluxes, we sum over discretecontributions r12(0)HA(t) and r23(0)HA(t):

k(E)exp(3k(E)t) = 12(E)

TAB

NAB(E EL)

ntraj

sign[r12(0)] HA(r12(t), r23(t))

+TAC

NAC(E EL)

ntraj

sign[r23(0)] HA(r12(t), r23(t))

,

(34)

where Tij is the set of initial conditions on transition state ij. Nij(E EL) isthe phase space volume-like quantity that gives the area in the ( rij , prij ) 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.

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Since NAB(E EL) = NAC(E EL) by symmetry, and because we areinterested chiefly in the rate of exponential flux decay, we may disregard constant

factors and rewrite our rate constant equality more simply as a proportionalityrelation

k(E)exp(3k(E)t)

trajs,AB

sign[r12(0)] HA(r12(t), r23(t))

+

trajs,AC

sign[r23(0)] HA(r12(t), r23(t))

.

(35)

In a regime where the flux computed using Equation (35) exhibits exponentialdecay, the decay exponent is just 3k(E).

Initial conditions are selected at random on the two configuration space

transition states, i.e., with equal probability for phase points lying on eithertransition state. Two line segments define the transition states AB and AC,and are obtained by solving the equations

V(r12, r23) Emax (36a)

andV(r12, r

23) Emax, (36b)

for r23 and r12, respectively, where Emax is our maximum energy of interest,which we set equal to 1.0 for = 1 and L = 10. The configuration spacetransition states are shown in Figure 5. Associated conjugate momenta pr23or pr12 are sampled uniformly in the appropriate range corresponding to total

energy E.In practice, for parameters = 1 and L = 10 we use a modified characteristicfunction HA(t) = (r

12r12(t))(r23r23(t))(8 (r12(t) + r23(t))) to ensure

that only phase points actually in well A are included by the characteristicfunction.

Having selected r12, r23, pr12 and pr23 we convert to single-particle coordi-nates via

x1 = 0 (37a)

x2 = x1 + r12 (37b)

x3 = x2 + r23 (37c)

p1 = pr12 (37d)

p2 = pr12 pr23 (37e)p3 = pr23 , (37f)

where we have arbitrarily set x1 = 0 since the potential depends only on relativeparticle displacements (bond distances). Trajectories are integrated using asecond-order symplectic integrator as in our previous work on linear tethered

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chains [25], and the integration algorithm is simpler in single-particle coordinatesdue to the diagonal kinetic energy. A sample trajectory initiated near transition

state AB is shown in Figure 6. We can see various bonds forming, breaking andreforming.

Initial positions on transition states AB, AC are chosen with probability 12 ,and corresponding momenta are taken to have random signs (). We calculatethe reactive flux at energies {E = EL + iE}Mi=1, where M = 10 and E =(Emax EL)/M. For each energy, we integrate ntraj = 3 103 trajectories fornstep = 6104 time steps. Our trajectory time step value is = 102 0, where0 is the unit of time associated with our choice of units for energy, length andmass.

As we can see from Figure 7, for lower energies the nontransient flux showsexponential decay, but the dynamics become more oscillatory with increasingenergy. For these higher energies, the rate constant ceases to have meaning,so we restrict ourselves to energies less than E

L+ E. We calculate fluxes for

{E = EL+iE}Mi=1, where M = 10 and E = E/10, the fine-grained energyincrement. The flux decay for one of these fine-grained energies, E = EL+4E

,is plotted in Figure 8.

5. RRK and RRKM calculations

For comparison with our simulation rate constant, we compute RRK (har-monic) and RRKM (anharmonic) rate constants following the procedures de-scribed previously [25]. The region of reactant configuration space sampled forthe reactant sum of states calculation is shown in Figure 9. Since kAC = k, theRRKM rate constant is given by

kRRKM = NAC(E EL)A(E)

, (38)

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

is given in Figure 10. We find good agreement between the statistical andsimulation rate constants for energies near the activation barrier, with somedivergence at larger energies.

6. Summary and conclusions

We have computed harmonic, anharmonic and trajectory rate constants forisomerization of a triatomic Morse chain under strain, subject to cyclic bound-

ary conditions. Computation of the rate coefficient via trajectory simulationrequires us to determine the rate of exponential decay of fluxes across the twotransition states of interest. Initial conditions are sampled uniformly on theenergy shell, and trajectories integrated using a second-order symplectic inte-grator.

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For our microcanonical calculations, we see good agreement between thestatistical and simulation rate constants for energies near the activation barrier,

with some divergence at larger energies. Compared to the tethered chain dis-cussed previously [25], recrossing is more significant for the ring system treatedhere. An isomerization rate constant exists only within a somewhat narrowenergy range above the isomerization barrier. At higher energies, the simula-tions reveal oscillating patterns of flux among the wells. In other words, as wasobserved for the tethered chain [25], the dynamics are statistical only withina narrow range of energy per mode. For both systems, reactive events havenon-exponential lifetime distributions at high energies.

There are several possibilities for future work. One might involve computingisomerization rate constants for longer chains, P > 3, N > 2. One complicationthat arises for P > 3 is the presence of nonequivalent rate coefficients for inter-well transitions. Another possibility is to consider chains composed of differentatoms, for which the symmetry of the problem is reduced.

References

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Figure captions

Figure 1: Schematic illustration of the formation of 2 isomers via bifurcation for the straineddiatomic chain, P = 2, L > 2req.

Figure 2: 2-atom potential curves for = 1 and (a) L = 2, (b) L = 3, (c) L = 4, (d) L = 5and (e) L = 10

Figure 3: 3-atom potential energy surfaces for = 1 and (a) L = 3, (b) L = 4, (c) L = 5 and(d) L = 6.

Figure 4: 3-atom potential energy surface for = 1 and L = 10 with saddles AB (red), AC(green) and BC (blue). (a) Single particle coordinates. (b) Jacobi coordinates.

Figure 5: Sampling of transition states AB (vertical) and AC (horizontal) for = 1, L = 10and E = Emax.

Figure 6: Coordinates x1 (red), x2 (green) and x3 (blue) versus time t for N = 2, = 1 andL = 10. The trajectory was initiated near transition state AB.

Figure 7: Isomerizing flux versus time for N = 2, = 1, L = 10. (a) E = EL

+ E, (b)E = E

L+ 3E, (c) E = E

L+ 5E, (d) E = E

L+ 7E, (e) E = E

L+ 9E and (f)

E = EL

+ 10E. The energy parameters are as defined in the text.

Figure 8: Isomerizing flux versus time for N = 2, E = EL

+ 4Efg, = 1 and L = 10.

Figure 9: Configuration space projection of the Monte Carlo sampling for the reactant region(well A) sum of states (light blue) in Jacobi coordinates, N = 2, = 1 and L = 10. Alsoshown are AB (red), AC (green), BC (blue) and the global potential maximum (purple).

Figure 10: Rate coefficients kRRK (red), kRRKM (blue) and kflux (green) versus energy aboveactivation for N = 2, = 1 and L = 10.

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FIGURE 1 (J. N. Stember & G. S. Ezra)

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0 2 4 6 8 10r12

1

2

3

5

V

0 2 4 6 8 10r12

1

2

3

5

V

0 2 4 6 8 10r12

1

2

3

5

V

0 2 4 6 8 10r12

1

2

3

5

V

0 2 6 8 10r12

1

2

3

5

V


16


17/24

0 1 2 3 5 6

0

1

2

3

4

5

6

r12

r23

0 1 2 3 4 5 6

0

1

2

3

4

5

6

r12

r23

0 1 2 3 5 6

0

1

2

3

4

5

6

r12

r23

0 1 2 3 4 5 6

0

1

2

3

4

5

6

r12

r23


17


18/24

0 2 4 6 8 10

0

2

4

6

8

10

r12

r23

0 2 4 6 8

0

2

4

6

8

1

2


18


19/24

0 2 4 6 8 10

0

2

4

6

8

10

r12

r23


19


20/24

1 2 3 4 5 6x

20

40

60

80

100

t


20


21/24

100 200 300 400 500 600t

500

0

500

1000

1500

flux

E EL E

100 200 300 400 500 600t

500

0

500

1000

1500

flux

E EL 3E

100 200 300 400 500 600

t

500

0

500

1000

1500

flux

E EL

5E

100 200 300 400 500 600

t

500

0

500

1000

1500

flux

E EL

7E

100 200 300 400 500 600t

500

0

500

1000

1500

flux

E EL 9E

100 200 300 400 500 600t

500

0

500

1000

1500

flux

E EL 10E


21


22/24

100 200 300 400 500 600t

500

0

500

1000

1500

flux


22


23/24

0 2 4 6 8

0

2

4

6

8

1

2


23


24/24

0.002 0.004 0.006 0.008 0.010 0.012

EEL

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

k


24

Date post:	06-Apr-2018
Category:	Documents
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Joseph N. Stember and Gregory S. Ezra- Isomerization kinetics of a strained Morse oscillator ring

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