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Intramolecular energy transfer in the isomerization of cyclobutanone
20 March 1998
Chemical Physics Letters 285 Ž1998. 143–149
Intramolecular energy transfer in the isomerization of cyclobutanone Hui Tang, Soonmin Jang, Meishao Zhao, Stuart A. Rice The James Franck Institute, The UniÕersity of Chicago, Chicago, IL 60637, USA Received 29 September 1997; in final form 8 December 1997
Abstract We report the use of the results of calculations of the rate constant for the unimolecular isomerization of cyclobutanone to test a modified version of the Zhao–Rice classical MRRKM theory. The modification consists of using, in place of the Zhao–Rice approximation for the rate of intramolecular energy transfer, the local Lyapunov function which describes the divergence of initially nearby trajectories started in the region of the cantorus. The rate of isomerization of cyclobutanone obtained from trajectory calculations is used as the standard for comparison of the two calculations of the rate of intramolecular energy transfer which are shown to yield very similar, although not identical, rate constants. q 1998 Elsevier Science B.V.
1. Introduction Gray and Rice w1x have developed a theory of the rate of unimolecular isomerization, based on classical mechanics, which differs in several important respects from the conventional RRKM theory of the rate of unimolecular isomerization. In addition to applying ideas from the theory of nonlinear mappings, which were introduced into unimolecular reaction rate theory by Davis and Gray w2x to identify and locate the important phase space bottlenecks in the isomerization reaction, Gray and Rice note that the topology of the molecular phase space differs from that assumed in the RRKM approach. Specifically, they show that it is necessary to use a three-state representation of the phase space rather than the conventional two-state representation. The necessity for this classification of the regions of phase space follows from the observation that as long as the molecule remains intact there must be, in addition to
the states we identify with isomers A and B, a third state of the system, with energy in excess of the barrier to isomerization, which is neither A nor B. The existence of these three-system states is clearly seen in trajectory studies of an isomerizing molecule. The analysis of the reaction dynamics using the three-state representation represents a qualitative change from the view posed by RRKM theory; it also leads to an improvement in the quality of the predicted rate constants even in the limit when the rate of intramolecular energy exchange is taken to be very large compared to the rate of reaction. In its original form, except for subtraction of the region of phase space in which there is quasiperiodic motion from the total available phase space, Gray and Rice did not include the competition between the rates of intramolecular energy exchange and reaction in their analysis. Zhao and Rice w3x improved the Gray–Rice theory by developing a useful approximation to the rate of intramolecular energy exchange.
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H. Tang et al.r Chemical Physics Letters 285 (1998) 143–149
Their approximation takes advantage of the identification of the barrier to intramolecular energy flow with the remnants of the loci of periodic motion associated with the most irrational frequency ratio in the system Žthe cantorus., which they replace with a curve defined by analogy with the analytic approximation to the separatrix. As expected, the inclusion of the intramolecular energy transfer dynamics in the analysis of the isomerization rate improves the quality of the prediction of the rate constant. The Zhao–Rice approximation to the rate of intramolecular energy exchange is plausible, but it is not obtained directly from dynamical considerations. In this Letter we examine a different approximation to the rate of intramolecular energy transfer. Specifically, we examine the local Lyapunov function which describes the divergence of initially nearby trajectories started in the region of the cantorus. We compare that rate of divergence with the rate generated by the Zhao–Rice approximation. The new analysis is applied to a calculation of the rate of isomerization of cyclobutanone. The results of our calculations of the rate of isomerization as a function of energy are compared with the results of trajectory calculations; the agreement is found to be good.
2. The model cyclobutanone system We consider first the model system used to test the two calculations of the rate of intramolecular energy transfer mentioned in Section 1. Since this model is similar to that in our previously reported study of the rate of isomerization of 3-phospholene w4x, our description of the methodology will be brief. We adopt the potential energy surface supplied by Laane and coworkers w5x for cyclobutanone. Let x 1 be the coordinate representing the carbonyl out-ofplane wagging and x 2 the coordinate representing the ring-puckering motion. The two-dimensional potential energy surface proposed by Laane et al. is V Ž x 1 , x 2 . s V1 Ž x 1 . q V2 Ž x 2 . q W Ž x 1 , x 2 .
Ž 2.1 .
with V1Ž x 1 . s a1 x 14 q b 1 x 12 ; V2 Ž x 2 . s a2 x 24 q b 2 x 22 ; W Ž x 1 , x 2 . s c12 x 1 x 23 . We display V1Ž x 1 . in Fig. 1; the barrier height is 0.0094 au. Unless stated otherwise specifically, we use atomic units Žau. for all quantities.
Fig. 1. The one-dimensional quartic two-well potential V1Ž x 1 . for the model cyclobutanone system.
Neglecting molecular rotation, the model cyclobutanone Hamiltonian has the form H Ž x 1 , x 2 . s 12 g 1 Ž x 1 . p x21 q 12 g 2 Ž x 2 . p x22 q V Ž x1 , x 2 . Ž 2.2 . Ž0. Ž2. 2 Ž4. 4 Ž . where g 1 x 1 s g 1 q g 1 x 1 q g 1 x 1 q g 1Ž6. x 16 ; g 2 Ž x 2 . s g 2Ž0. q g 2Ž2. x 22 q g 2Ž4. x 24 q g 2Ž6. x 26 with g 1Ž x 1 . and g 2 Ž x 2 . the coordinate expansions of the kinetic energies for the wagging and puckering motions, respectively. The coefficients appearing in Eq. Ž2.2. can be found in Ref. w5x. The Gray–Rice analysis of the rate of isomerization divides the energy surface into three regions in phase space by use of the dynamical separatrix rather than the two regions which are generated by the RRKM critical coordinate surface. The two closed regions inside the separatrix, which represent the two isomers, are denoted A and B; the exterior region between the separatrix and energy boundary, which represents an intermediate state of the molecular which cannot be identified with either isomer, is denoted C. The elementary rate constants describing phase point transfer between the states A, B and C are given by FAC FCA k AC s k CA s NA NC k BC s
FBC NB
k CB s
FCB NC
Ž 2.3 .
H. Tang et al.r Chemical Physics Letters 285 (1998) 143–149
where FAC is the forward flux from region A to region C Ž FCA , FBC and FCB are similarly defined. and NA is the phase space volume of region A. Microscopic reversibility requires that FAC s FCA . 3. Approximating the rate of intramolecular energy transfer A useful analytic approximation to the dynamical separatrix associated with the reaction coordinate x 1 by dropping the terms in the Hamiltonian which prevent the system from being integrable w1,3x. For our model of cyclobutanone this approximation to the dynamical separatrix is S Ž p x 1 , x 1 ,x 2 . s 12 g 1 Ž x 1 . p x21 q V1 Ž x 1 . q W Ž x 1 ,x 2 . y ´ E, Ž 3.1 . where x 2 is a fixed value of x 2 Žchosen to be at the saddle point of V Ž x 1 , x 2 .. and the energy E is scaled by a factor ´ to simulate the energy-dependence of the dynamical separatrix. Except for excluding the region of phase space in which there is quasiperiodic motion, the original Gray–Rice theory of the rate of isomerization neglects the influence of intramolecular energy on the reaction rate. To improve the Gray–Rice theory, Zhao and Rice introduced a semiclassical approximation for the representation of the intramolecular bottlenecks w3x. They suggested using Eq. Ž3.1. for the functional form of the intramolecular bottleneck dividing surface, which implies that the nth intramolecular bottleneck has the form w3,6x Sintra Ž p x 1 , x 1 ,x 2 . s 12 g 1 Ž x 1 . p x21 q V1 Ž x 1 . q W Ž x 1 ,x 2 . y E x 1Ž n .
Ž n s 1,2, . . . . .
Ž 3.2 .
When the motion in the degree of freedom x 1 is confined to the vicinity of the isomer equilibrium value a harmonic approximation should be valid. This assumption yields E x 1Ž n . s Ž n q 12 . v x 1
Ž 3.3 .
with
v x 1 s g 1Ž x 1 .
E 2 V Ž x1 , x 2 . E x 12
1r2
, x 1 sx 1 e s1.68; x 2 sx 2 e s0
Ž 3.4 .
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where the derivative is evaluated at the potential well minimum. We now examine a different view of the dynamics of intramolecular energy exchange. As originally suggested by Brumer and Shapiro w7x, the local expansion and contraction properties of an initial distribution of trajectories, as measured by the Lyapunov exponent, should provide an estimate of the relaxation time for systems whose dynamics is chaotic. It is uncertain, however, which aspect of the relaxation dynamics is properly represented by the Lyapunov exponent. For a Hamiltonian system the Lyapunov exponents are the measures of the average exponential rates of divergence of initially nearby trajectories in phase space; a Hamiltonian system of F degrees of freedom has 2 F Lyapunov exponents w8x. Liouville’s theorem Žthe conservation of phase space volume. implies that Lyapunov exponents occur in pairs l j s yl jqF , j s 1, . . . , F. There are 2 N vanishing Lyapunov exponents if there are N independent constants of motion w9x. In fact, exponential separation of trajectories can not prevail indefinitely; limits to the trajectory separation are set by Poincare recurrences or, in some cases, the accessible volume of phase space. Usually, however, such limits become effective on time scales much longer than that on which regular or chaotic behavior is manifest w10x. Brumer and coworkers w11–13x attempted to associate the Lyapunov exponent with the rate of intramolecular energy exchange in two model systems: a Henon–Heiles system at an energy in the chaotic regime and a coupled Morse oscillator system at an energy for which there is a mixture of Žpredominantly. chaotic motion and regular motion. They found that the time scale of intramolecular energy exchange, when the initial state is global, is consistent with the time scale defined by the maximum Lyapunov exponent. However, this correlation does not hold for a model of the OCS molecule, in which case quasiperiodic motion and resonance island chains occupy a considerable fraction of the energy surface. In fact, the rate of intramolecular energy exchange in OCS is found to be slower than expected from the rate of exponential divergence of nearby trajectories. This dilemma was resolved by Davis w14x, who used the theory of nonlinear mappings w15–18x to identify the cantorus as a bottleneck
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H. Tang et al.r Chemical Physics Letters 285 (1998) 143–149
to intramolecular energy transfer. The intramolecular energy exchange rate constant obtained by generating a dividing surface which is an approximation to the last broken KAM curve, then calculating the flux across this dividing surface by the difference between it and its iteration on the Poincare surface of section Žwhich generates a so-called turnstile.; this method leads to an accurate predication of the rate of intramolecular relaxation in OCS. Although there is not a simple relationship between the rate of intramolecular energy exchange and Lyapunov exponents in a system in which quasiperiodic and chaotic motion coexist, we suggest that the local Lyapunov function w8,19–22x, i.e. the finite time Lyapunov exponent, can provide insight into the correlation between the two time scales. In a system with mixed dynamics, chaotic trajectories can wander close to the KAM tori which bound the quasiperiodic motion. When they do so, such trajectories can become trapped in the vicinity of the KAM tori for a long time because the cantorus lying between a KAM torus and the stochastic region is ‘sticky’ by virtue of its fractal structure ŽCantor set. w8,23x. Since the Lyapunov exponent on the KAM tori is l1 s 0, initial conditions that lead to trajectories that spend a very long time near the KAM torus Žinside the cantorus. must have abnormally small values of l1. We suggest using this finite time l1 to characterize the intramolecular energy exchange time scale in a system with coexistent quasiperiodic and chaotic motion.
E s 0.005, 0.0075 and 0.01. The kinetics of the three-state mechanism of isomerization, with inclusion of the rate of intramolecular energy transfer, can be represented by a set of coupled master equations. The methodology used for extracting the isomerization rate constant from this set of equations can be found elsewhere w3x. We have also computed the local Lyapunov function for out model system by starting a trajectory inside the cantorus region. Note that the cantorus is essentially the quasi-periodic orbit with highest irrational winding number Žthe last broken KAM torus. and the long-time spent by a trajectory in the vicinity of the cantorus is due to quasi-periodic hopping on the broken torus. Therefore, for the purpose of calculating the Lyapunov function, where a trajectory is started on the cantorus has no effect on its value. The set of Lyapunov exponents for our two degree of freedom model of cyclobutanone is fully characterized by the largest, denoted l1 , because the conservation of total energy ensures that two of the Lyapunov exponents vanish. If the largest Lyapunov exponent vanishes we call the system regular; otherwise the system is chaotic. To compute the Lyapunov exponents we employed the algorithm of Wolf et al. w24x. In Fig. 2 we show the time dependence of l1 for a typical chaotic trajectory when E s 0.01; l1 converges to 0.00016, but the convergence is rather
4. Numerical calculations The multiple dimensional integrals which define the fluxes and the phase space volumes Žsee w7x., etc., needed to calculate the Zhao–Rice approximation to the isomerization rate constant were evaluated by Monte Carlo integration using 10 7 sets of random numbers. To improve the accuracy of the approximate representation of the dynamical separatrix the correction factor ´ in Eq. Ž3.1. was chosen to be 0.20 for E s 0.0025 and 0.005; 0.30 for E s 0.0075; and 0.50 for E s 0.01. To match the area of the region of quasiperiodic motion on the energy surface as closely as possible, the value of n in Eq. Ž3.2. was chosen to be n s 2 for E s 0.0025; n s 3 for
Fig. 2. Maximum Lyapunov exponent l1 versus time for Es 0.01. The starting point of the chaotic trajectory is the saddle point of V Ž x 1 , x 2 ..
H. Tang et al.r Chemical Physics Letters 285 (1998) 143–149
Fig. 3. The spectrum of the local Lyapunov function for the trajectory stuck in the cantorus region corresponding to very small l1Ž t ..
slow. A spectrum of the local Lyapunov function, l1Ž t ., is shown in Fig. 3. Note that in the initial phase of the evolution of the trajectory it is stuck in the cantorus region Žfor about 2 = 10 6 au.. Consequently, the initial dynamical development of the energy exchange is characterized by a very small local Lyapunov function. At the end of this period the trajectory breaks through the cantorus and then wanders over the entire available region of chaotic motion. Associated with the breakout of the trajectory is an increase in the Lyapunov function, which eventually converges to its limiting value Žthe Lyapunov exponent.. If we assume that the minimum of the local Lyapunov function determines the effective rate of intramolecular energy exchange we find the values listed in the Table 1. These relaxation rate constants are nearly energy-independent, which we
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attribute to their dependence on the dynamic nature of the associated resonances. In Table 1 we also display the intramolecular energy exchange rate constants obtained from Zhao–Rice analysis. The agreement between the intramolecular energy exchange rate constants obtained from Zhao–Rice theory and from the local Lyapunov function approximation is remarkably good, especially in view of the semiclassical nature of the Zhao–Rice approximation. Trajectory based calculations of the population decay associated with the isomerization of our model cyclobutanone are shown in Fig. 4; these decay curves are, in general, nonexponential. An empirical estimate of the isomerization rate constant can be obtained from the time taken for the initial concentration of the reactant ensemble to decay 95% Žthe rate constant obtained in this fashion for E s 0.0025 agrees quite closely with that obtained from the best exponential fitting. w6x. The isomerization rate constants from RRKM theory, Gray–Rice theory, Zhao–Rice theory and our trajectory calculations are displayed in Table 2. Note that for this model system the RRKM theory generates about an order of magnitude overestimate of the reaction rate constant, that both the Gray–Rice and the Zhao–Rice theories predict rate constants with about the correct magni-
Table 1 Intramolecular energy exchange rate constants from Zhao–Rice theory and the local Lyapunov function at various energies Žthe unit used for all entries is =10y4 au. Energy
Zhao–Rice
Lyapunov function
25 50 75 100
0.111 0.125 0.184 0.264
0.13 0.23 0.18 0.16
Fig. 4. The reactant concentration as a function of time from RRKM-type trajectory calculations for several energies. The solid line is for Es 0.0025 while the other lines are, from top to bottom, for Es 0.005, 0.0075 and 0.01.
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Table 2 Classical isomerization rate constants from RRKM theory, Gray– Rice theory, Zhao–Rice theory and trajectory calculations Žthe unit used for all entries is =10y4 au. Energy
RRKM
Gray–Rice
Zhao–Rice
Trajectory
25 50 75 100
0.71 0.95 1.14 1.31
0.076 0.084 0.091 0.097
0.036 0.043 0.054 0.073
0.078 0.081 0.111 0.156
tude and the correct energy dependence. Overall, the Zhao–Rice theory rate constants agree within a factor of two with those derived from the trajectory calculations. This discrepancy is subject to the uncertainty associated with difference between the rate constants extracted from the three-state Zhao–Rice model and the two-state model used to analyze the trajectory studies. The entries in Tables 1 and 2 show that the isomerization rate constant is comparable in magnitude with the intramolecular energy exchange rate constant. An examination of Eq. Ž2.1. shows that our model system can be viewed as two coupled one-dimensional quartic two-well oscillators. The reduced masses of the oscillators are very different Ž m 1 s 7.408; m 2 s 190.84. w5x, which suggests that energy transfer between them can be very slow. It is then to be expected that the RRKM theory assumption that intramolecular energy exchange is much more rapid than reaction is not valid for the model cyclobutanone system we have studied, so it is not surprising that RRKM prediction of the isomerization rate constant is an order of magnitude too large.
tween the rate of intramolecular energy exchange estimated in this fashion and from the approximation treatment of Zhao and Rice, much more remains to be done to fully understand the complexities of this relaxation process. Indeed, it is necessary to justify analytically the correlation between the local Lyapunov function and intramolecular relaxation w28x. For the present we see our suggestion as a computationally feasible alternative to Davis’ elegant turnstile approach to the calculation of the rate of intramolecular reaction in a system with both quasiperiodic and chaotic motion.
Acknowledgements This research has been supported by a grant from National Science Foundation.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x
5. Closing remark In the early 1970s Oxtoby and Rice w25x demonstrated, from a study of coupled Morse oscillators, the importance of nonlinear resonances in effecting intramolecular energy transfer w26,27x. They pointed out that ‘‘isolated nonlinear resonances lead to trapping of the vibrational energy of the system, hence to slow vibrational relaxation’’. The use of the local Lyapunov function to define the time scale for intramolecular relaxation, proposed in this note, represents a quantification of the Oxtoby–Rice observation. Although there is reasonable agreement be-
w11x w12x w13x w14x w15x w16x w17x w18x w19x
w20x w21x
S.K. Gray, S.A. Rice, J. Chem. Phys. 86 Ž1987. 2020. M.J. Davis, S.K. Gray, J. Chem. Phys. 84 Ž1986. 5389. M. Zhao, S.A. Rice, J. Chem. Phys. 97 Ž1992. 943. M. Zhao, S.A. Rice, J. Chem. Phys. 98 Ž1993. 2837. J. Zhang, W. Chiang, J. Laane, J. Chem. Phys. 100 Ž1994. 3455. H. Tang, S. Jang, M. Zhao, S.A. Rice, J. Chem. Phys. 101 Ž1994. 8737. P. Brumer, M. Shapiro, in: I. Prigogine, S.A. Rice ŽEds.., Adv. Chem. Phys. LXX Ž1988. 365. E. Ott, Chaos in Dynamical System ŽCambridge, New York, 1993.. H. Meyer, J. Chem. Phys. 84 Ž1986. 3147. F. Haake, Quantum Signatures of Chaos ŽSpringer, BerlinrHeidelberg, 1991.. D. Carter, P. Brumer, J. Chem. Phys. 77 Ž1982. 4208. I. Hamilton, P. Brumer, J. Chem. Phys. 78 Ž1983. 2682. I. Hamilton, P. Brumer, J. Chem. Phys. 82 Ž1985. 1937. M.J. Davis, J. Chem. Phys. 83 Ž1985. 1016. Ch.F.F. Karney, Physica D 8 Ž1983. 360. R.S. Mackay, J.D. Meiss, I. Pricival, Physica D 13 Ž1984. 55. D. Bensimon, L.P. Kadanoff, Physica D 13 Ž1984. 82. J.D. Meiss, Rev. Mod. Phys. 64 Ž1992. 795. L.E. Reichl, The Transition to Chaos, in Conservative Classical System: Quantum Manifestations ŽSpringer, New York, 1992.. A.A. Zembekov, J. Chem. Phys. 101 Ž1994. 8842. M.A. Sepulveda, R. Badii, E. Pollak, Phys. Rev. Lett. 63 ´ Ž1989. 1226.
H. Tang et al.r Chemical Physics Letters 285 (1998) 143–149 w22x T. Horita, H. Hata, R. Ishizaki, H. Mori, Prog. Theor. Phys. 83 Ž1990. 1065. w23x A.D. Perry, S. Wiggins, Physica D 71 Ž1994. 102. w24x A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16 Ž1985. 285.
w25x w26x w27x w28x
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D.W. Oxtoby, S.A. Rice, J. Chem. Phys. 65 Ž1976. 1676. T. Uzer, J.T. Hynes, Chem. Phys. Lett. 113 Ž1985. 483. T. Uzer, W.H. Miller, Phys. Rep. 199 Ž1991. 74. R.S. Dumont, P. Brumer, J. Chem. Phys. 87 Ž1987. 6437.
Chemical Physics Letters 285 Ž1998. 143–149
Intramolecular energy transfer in the isomerization of cyclobutanone Hui Tang, Soonmin Jang, Meishao Zhao, Stuart A. Rice The James Franck Institute, The UniÕersity of Chicago, Chicago, IL 60637, USA Received 29 September 1997; in final form 8 December 1997
Abstract We report the use of the results of calculations of the rate constant for the unimolecular isomerization of cyclobutanone to test a modified version of the Zhao–Rice classical MRRKM theory. The modification consists of using, in place of the Zhao–Rice approximation for the rate of intramolecular energy transfer, the local Lyapunov function which describes the divergence of initially nearby trajectories started in the region of the cantorus. The rate of isomerization of cyclobutanone obtained from trajectory calculations is used as the standard for comparison of the two calculations of the rate of intramolecular energy transfer which are shown to yield very similar, although not identical, rate constants. q 1998 Elsevier Science B.V.
1. Introduction Gray and Rice w1x have developed a theory of the rate of unimolecular isomerization, based on classical mechanics, which differs in several important respects from the conventional RRKM theory of the rate of unimolecular isomerization. In addition to applying ideas from the theory of nonlinear mappings, which were introduced into unimolecular reaction rate theory by Davis and Gray w2x to identify and locate the important phase space bottlenecks in the isomerization reaction, Gray and Rice note that the topology of the molecular phase space differs from that assumed in the RRKM approach. Specifically, they show that it is necessary to use a three-state representation of the phase space rather than the conventional two-state representation. The necessity for this classification of the regions of phase space follows from the observation that as long as the molecule remains intact there must be, in addition to
the states we identify with isomers A and B, a third state of the system, with energy in excess of the barrier to isomerization, which is neither A nor B. The existence of these three-system states is clearly seen in trajectory studies of an isomerizing molecule. The analysis of the reaction dynamics using the three-state representation represents a qualitative change from the view posed by RRKM theory; it also leads to an improvement in the quality of the predicted rate constants even in the limit when the rate of intramolecular energy exchange is taken to be very large compared to the rate of reaction. In its original form, except for subtraction of the region of phase space in which there is quasiperiodic motion from the total available phase space, Gray and Rice did not include the competition between the rates of intramolecular energy exchange and reaction in their analysis. Zhao and Rice w3x improved the Gray–Rice theory by developing a useful approximation to the rate of intramolecular energy exchange.
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Their approximation takes advantage of the identification of the barrier to intramolecular energy flow with the remnants of the loci of periodic motion associated with the most irrational frequency ratio in the system Žthe cantorus., which they replace with a curve defined by analogy with the analytic approximation to the separatrix. As expected, the inclusion of the intramolecular energy transfer dynamics in the analysis of the isomerization rate improves the quality of the prediction of the rate constant. The Zhao–Rice approximation to the rate of intramolecular energy exchange is plausible, but it is not obtained directly from dynamical considerations. In this Letter we examine a different approximation to the rate of intramolecular energy transfer. Specifically, we examine the local Lyapunov function which describes the divergence of initially nearby trajectories started in the region of the cantorus. We compare that rate of divergence with the rate generated by the Zhao–Rice approximation. The new analysis is applied to a calculation of the rate of isomerization of cyclobutanone. The results of our calculations of the rate of isomerization as a function of energy are compared with the results of trajectory calculations; the agreement is found to be good.
2. The model cyclobutanone system We consider first the model system used to test the two calculations of the rate of intramolecular energy transfer mentioned in Section 1. Since this model is similar to that in our previously reported study of the rate of isomerization of 3-phospholene w4x, our description of the methodology will be brief. We adopt the potential energy surface supplied by Laane and coworkers w5x for cyclobutanone. Let x 1 be the coordinate representing the carbonyl out-ofplane wagging and x 2 the coordinate representing the ring-puckering motion. The two-dimensional potential energy surface proposed by Laane et al. is V Ž x 1 , x 2 . s V1 Ž x 1 . q V2 Ž x 2 . q W Ž x 1 , x 2 .
Ž 2.1 .
with V1Ž x 1 . s a1 x 14 q b 1 x 12 ; V2 Ž x 2 . s a2 x 24 q b 2 x 22 ; W Ž x 1 , x 2 . s c12 x 1 x 23 . We display V1Ž x 1 . in Fig. 1; the barrier height is 0.0094 au. Unless stated otherwise specifically, we use atomic units Žau. for all quantities.
Fig. 1. The one-dimensional quartic two-well potential V1Ž x 1 . for the model cyclobutanone system.
Neglecting molecular rotation, the model cyclobutanone Hamiltonian has the form H Ž x 1 , x 2 . s 12 g 1 Ž x 1 . p x21 q 12 g 2 Ž x 2 . p x22 q V Ž x1 , x 2 . Ž 2.2 . Ž0. Ž2. 2 Ž4. 4 Ž . where g 1 x 1 s g 1 q g 1 x 1 q g 1 x 1 q g 1Ž6. x 16 ; g 2 Ž x 2 . s g 2Ž0. q g 2Ž2. x 22 q g 2Ž4. x 24 q g 2Ž6. x 26 with g 1Ž x 1 . and g 2 Ž x 2 . the coordinate expansions of the kinetic energies for the wagging and puckering motions, respectively. The coefficients appearing in Eq. Ž2.2. can be found in Ref. w5x. The Gray–Rice analysis of the rate of isomerization divides the energy surface into three regions in phase space by use of the dynamical separatrix rather than the two regions which are generated by the RRKM critical coordinate surface. The two closed regions inside the separatrix, which represent the two isomers, are denoted A and B; the exterior region between the separatrix and energy boundary, which represents an intermediate state of the molecular which cannot be identified with either isomer, is denoted C. The elementary rate constants describing phase point transfer between the states A, B and C are given by FAC FCA k AC s k CA s NA NC k BC s
FBC NB
k CB s
FCB NC
Ž 2.3 .
H. Tang et al.r Chemical Physics Letters 285 (1998) 143–149
where FAC is the forward flux from region A to region C Ž FCA , FBC and FCB are similarly defined. and NA is the phase space volume of region A. Microscopic reversibility requires that FAC s FCA . 3. Approximating the rate of intramolecular energy transfer A useful analytic approximation to the dynamical separatrix associated with the reaction coordinate x 1 by dropping the terms in the Hamiltonian which prevent the system from being integrable w1,3x. For our model of cyclobutanone this approximation to the dynamical separatrix is S Ž p x 1 , x 1 ,x 2 . s 12 g 1 Ž x 1 . p x21 q V1 Ž x 1 . q W Ž x 1 ,x 2 . y ´ E, Ž 3.1 . where x 2 is a fixed value of x 2 Žchosen to be at the saddle point of V Ž x 1 , x 2 .. and the energy E is scaled by a factor ´ to simulate the energy-dependence of the dynamical separatrix. Except for excluding the region of phase space in which there is quasiperiodic motion, the original Gray–Rice theory of the rate of isomerization neglects the influence of intramolecular energy on the reaction rate. To improve the Gray–Rice theory, Zhao and Rice introduced a semiclassical approximation for the representation of the intramolecular bottlenecks w3x. They suggested using Eq. Ž3.1. for the functional form of the intramolecular bottleneck dividing surface, which implies that the nth intramolecular bottleneck has the form w3,6x Sintra Ž p x 1 , x 1 ,x 2 . s 12 g 1 Ž x 1 . p x21 q V1 Ž x 1 . q W Ž x 1 ,x 2 . y E x 1Ž n .
Ž n s 1,2, . . . . .
Ž 3.2 .
When the motion in the degree of freedom x 1 is confined to the vicinity of the isomer equilibrium value a harmonic approximation should be valid. This assumption yields E x 1Ž n . s Ž n q 12 . v x 1
Ž 3.3 .
with
v x 1 s g 1Ž x 1 .
E 2 V Ž x1 , x 2 . E x 12
1r2
, x 1 sx 1 e s1.68; x 2 sx 2 e s0
Ž 3.4 .
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where the derivative is evaluated at the potential well minimum. We now examine a different view of the dynamics of intramolecular energy exchange. As originally suggested by Brumer and Shapiro w7x, the local expansion and contraction properties of an initial distribution of trajectories, as measured by the Lyapunov exponent, should provide an estimate of the relaxation time for systems whose dynamics is chaotic. It is uncertain, however, which aspect of the relaxation dynamics is properly represented by the Lyapunov exponent. For a Hamiltonian system the Lyapunov exponents are the measures of the average exponential rates of divergence of initially nearby trajectories in phase space; a Hamiltonian system of F degrees of freedom has 2 F Lyapunov exponents w8x. Liouville’s theorem Žthe conservation of phase space volume. implies that Lyapunov exponents occur in pairs l j s yl jqF , j s 1, . . . , F. There are 2 N vanishing Lyapunov exponents if there are N independent constants of motion w9x. In fact, exponential separation of trajectories can not prevail indefinitely; limits to the trajectory separation are set by Poincare recurrences or, in some cases, the accessible volume of phase space. Usually, however, such limits become effective on time scales much longer than that on which regular or chaotic behavior is manifest w10x. Brumer and coworkers w11–13x attempted to associate the Lyapunov exponent with the rate of intramolecular energy exchange in two model systems: a Henon–Heiles system at an energy in the chaotic regime and a coupled Morse oscillator system at an energy for which there is a mixture of Žpredominantly. chaotic motion and regular motion. They found that the time scale of intramolecular energy exchange, when the initial state is global, is consistent with the time scale defined by the maximum Lyapunov exponent. However, this correlation does not hold for a model of the OCS molecule, in which case quasiperiodic motion and resonance island chains occupy a considerable fraction of the energy surface. In fact, the rate of intramolecular energy exchange in OCS is found to be slower than expected from the rate of exponential divergence of nearby trajectories. This dilemma was resolved by Davis w14x, who used the theory of nonlinear mappings w15–18x to identify the cantorus as a bottleneck
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to intramolecular energy transfer. The intramolecular energy exchange rate constant obtained by generating a dividing surface which is an approximation to the last broken KAM curve, then calculating the flux across this dividing surface by the difference between it and its iteration on the Poincare surface of section Žwhich generates a so-called turnstile.; this method leads to an accurate predication of the rate of intramolecular relaxation in OCS. Although there is not a simple relationship between the rate of intramolecular energy exchange and Lyapunov exponents in a system in which quasiperiodic and chaotic motion coexist, we suggest that the local Lyapunov function w8,19–22x, i.e. the finite time Lyapunov exponent, can provide insight into the correlation between the two time scales. In a system with mixed dynamics, chaotic trajectories can wander close to the KAM tori which bound the quasiperiodic motion. When they do so, such trajectories can become trapped in the vicinity of the KAM tori for a long time because the cantorus lying between a KAM torus and the stochastic region is ‘sticky’ by virtue of its fractal structure ŽCantor set. w8,23x. Since the Lyapunov exponent on the KAM tori is l1 s 0, initial conditions that lead to trajectories that spend a very long time near the KAM torus Žinside the cantorus. must have abnormally small values of l1. We suggest using this finite time l1 to characterize the intramolecular energy exchange time scale in a system with coexistent quasiperiodic and chaotic motion.
E s 0.005, 0.0075 and 0.01. The kinetics of the three-state mechanism of isomerization, with inclusion of the rate of intramolecular energy transfer, can be represented by a set of coupled master equations. The methodology used for extracting the isomerization rate constant from this set of equations can be found elsewhere w3x. We have also computed the local Lyapunov function for out model system by starting a trajectory inside the cantorus region. Note that the cantorus is essentially the quasi-periodic orbit with highest irrational winding number Žthe last broken KAM torus. and the long-time spent by a trajectory in the vicinity of the cantorus is due to quasi-periodic hopping on the broken torus. Therefore, for the purpose of calculating the Lyapunov function, where a trajectory is started on the cantorus has no effect on its value. The set of Lyapunov exponents for our two degree of freedom model of cyclobutanone is fully characterized by the largest, denoted l1 , because the conservation of total energy ensures that two of the Lyapunov exponents vanish. If the largest Lyapunov exponent vanishes we call the system regular; otherwise the system is chaotic. To compute the Lyapunov exponents we employed the algorithm of Wolf et al. w24x. In Fig. 2 we show the time dependence of l1 for a typical chaotic trajectory when E s 0.01; l1 converges to 0.00016, but the convergence is rather
4. Numerical calculations The multiple dimensional integrals which define the fluxes and the phase space volumes Žsee w7x., etc., needed to calculate the Zhao–Rice approximation to the isomerization rate constant were evaluated by Monte Carlo integration using 10 7 sets of random numbers. To improve the accuracy of the approximate representation of the dynamical separatrix the correction factor ´ in Eq. Ž3.1. was chosen to be 0.20 for E s 0.0025 and 0.005; 0.30 for E s 0.0075; and 0.50 for E s 0.01. To match the area of the region of quasiperiodic motion on the energy surface as closely as possible, the value of n in Eq. Ž3.2. was chosen to be n s 2 for E s 0.0025; n s 3 for
Fig. 2. Maximum Lyapunov exponent l1 versus time for Es 0.01. The starting point of the chaotic trajectory is the saddle point of V Ž x 1 , x 2 ..
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Fig. 3. The spectrum of the local Lyapunov function for the trajectory stuck in the cantorus region corresponding to very small l1Ž t ..
slow. A spectrum of the local Lyapunov function, l1Ž t ., is shown in Fig. 3. Note that in the initial phase of the evolution of the trajectory it is stuck in the cantorus region Žfor about 2 = 10 6 au.. Consequently, the initial dynamical development of the energy exchange is characterized by a very small local Lyapunov function. At the end of this period the trajectory breaks through the cantorus and then wanders over the entire available region of chaotic motion. Associated with the breakout of the trajectory is an increase in the Lyapunov function, which eventually converges to its limiting value Žthe Lyapunov exponent.. If we assume that the minimum of the local Lyapunov function determines the effective rate of intramolecular energy exchange we find the values listed in the Table 1. These relaxation rate constants are nearly energy-independent, which we
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attribute to their dependence on the dynamic nature of the associated resonances. In Table 1 we also display the intramolecular energy exchange rate constants obtained from Zhao–Rice analysis. The agreement between the intramolecular energy exchange rate constants obtained from Zhao–Rice theory and from the local Lyapunov function approximation is remarkably good, especially in view of the semiclassical nature of the Zhao–Rice approximation. Trajectory based calculations of the population decay associated with the isomerization of our model cyclobutanone are shown in Fig. 4; these decay curves are, in general, nonexponential. An empirical estimate of the isomerization rate constant can be obtained from the time taken for the initial concentration of the reactant ensemble to decay 95% Žthe rate constant obtained in this fashion for E s 0.0025 agrees quite closely with that obtained from the best exponential fitting. w6x. The isomerization rate constants from RRKM theory, Gray–Rice theory, Zhao–Rice theory and our trajectory calculations are displayed in Table 2. Note that for this model system the RRKM theory generates about an order of magnitude overestimate of the reaction rate constant, that both the Gray–Rice and the Zhao–Rice theories predict rate constants with about the correct magni-
Table 1 Intramolecular energy exchange rate constants from Zhao–Rice theory and the local Lyapunov function at various energies Žthe unit used for all entries is =10y4 au. Energy
Zhao–Rice
Lyapunov function
25 50 75 100
0.111 0.125 0.184 0.264
0.13 0.23 0.18 0.16
Fig. 4. The reactant concentration as a function of time from RRKM-type trajectory calculations for several energies. The solid line is for Es 0.0025 while the other lines are, from top to bottom, for Es 0.005, 0.0075 and 0.01.
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Table 2 Classical isomerization rate constants from RRKM theory, Gray– Rice theory, Zhao–Rice theory and trajectory calculations Žthe unit used for all entries is =10y4 au. Energy
RRKM
Gray–Rice
Zhao–Rice
Trajectory
25 50 75 100
0.71 0.95 1.14 1.31
0.076 0.084 0.091 0.097
0.036 0.043 0.054 0.073
0.078 0.081 0.111 0.156
tude and the correct energy dependence. Overall, the Zhao–Rice theory rate constants agree within a factor of two with those derived from the trajectory calculations. This discrepancy is subject to the uncertainty associated with difference between the rate constants extracted from the three-state Zhao–Rice model and the two-state model used to analyze the trajectory studies. The entries in Tables 1 and 2 show that the isomerization rate constant is comparable in magnitude with the intramolecular energy exchange rate constant. An examination of Eq. Ž2.1. shows that our model system can be viewed as two coupled one-dimensional quartic two-well oscillators. The reduced masses of the oscillators are very different Ž m 1 s 7.408; m 2 s 190.84. w5x, which suggests that energy transfer between them can be very slow. It is then to be expected that the RRKM theory assumption that intramolecular energy exchange is much more rapid than reaction is not valid for the model cyclobutanone system we have studied, so it is not surprising that RRKM prediction of the isomerization rate constant is an order of magnitude too large.
tween the rate of intramolecular energy exchange estimated in this fashion and from the approximation treatment of Zhao and Rice, much more remains to be done to fully understand the complexities of this relaxation process. Indeed, it is necessary to justify analytically the correlation between the local Lyapunov function and intramolecular relaxation w28x. For the present we see our suggestion as a computationally feasible alternative to Davis’ elegant turnstile approach to the calculation of the rate of intramolecular reaction in a system with both quasiperiodic and chaotic motion.
Acknowledgements This research has been supported by a grant from National Science Foundation.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x
5. Closing remark In the early 1970s Oxtoby and Rice w25x demonstrated, from a study of coupled Morse oscillators, the importance of nonlinear resonances in effecting intramolecular energy transfer w26,27x. They pointed out that ‘‘isolated nonlinear resonances lead to trapping of the vibrational energy of the system, hence to slow vibrational relaxation’’. The use of the local Lyapunov function to define the time scale for intramolecular relaxation, proposed in this note, represents a quantification of the Oxtoby–Rice observation. Although there is reasonable agreement be-
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