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Reactive bands and rebounding trajectories in collinear F + H2
Vahxe 57, number2
REACm
15 July 1978
BANDS AND RJZBQUNDINC TRAJECTORIES IN COLLINEAR
Freya SCEBUBEL
F + Hz
and Sally CHAPMAN
Depatnzent of CRen&zs, lhrmz.~ Cdege, Cduntbia i&iyersity, New York, New York 10027. US.4 Received30 March 1978
Bandsof rezxctiwand unreactivetrajectorieshave been mapped for classicalcollinearF + Hz (u = 0). The edges of tie ‘band are citaracterizedby trajectodeswhi& undergoseveralreflectionsin the comer of the potentialener_q surface. MtdtipierefIectionsazeseen to lead to less &a.@~ defaed band edp.
mofecufar
Coffinear dynamics is an intriguingsubject. For a three atom system the motion takes place in two mathematicaldimensions,so it is easy to visualize the potential energy surface and the resulting particle trajectory_The quantum dynamical studies of KU~~SRISIII and co-workers and others have S~OWEIa number of interestingeffects including resonancesin the reactivity and vortices in the quantum flux [l--4]. M&y classicalcolhnear dynamical studies have been =zndertakenin order to compare with semiclassicalor quautum results [S-7], while others have focused on the classicalbehavior alone IS]. In particular, a number of studies have focused on the bands of reactive trajectoriesin the H i Hz system [9, TO]_While F + Hz is a system which has received considerable attention [4--6], we are not aware of a previous study of the band structurein ibis asymmetricexchange reaction. The phase space of a collinear triatomic system is four dimensional. Thus four variables must be applied to initiate a classicaltrajectory: typically l5e asymptotic separation of the reagent molecules, the diatomic vibrational energy, the relative translationalenergy, and the vibrational phase. if the first two are fixed, as in a usual quasiclassicaltrajectory study, the last two form a space in which reactiveand unreactivezones can be seen. In the H f Hz system, the boundaries of these bands am reIativelysharp, while in a system like H f Clj [ 1I 1, there are broad areas where the outcome of a trajectory is 3x5extremely sensitivefunction of the initial phase, giving a seemingly random pattern of tra-
jectory results. We have carried out a collinear quasiclassicaltrajectory study on F + Hz (u = 0) using an extended LEPS potential surface [la], with the parameters of Mucke,man’s surface five [4] _ This system has been studied before; we are simply looking at a different aspect of the dynamics. The reaction has a sharp threshold with the reaction probability rising to about 0.7 at 1 kcal/molgmlative translationalenergy- The probability f& roughiy linearly reaching 0.5 by S kcallmole of translation. Fig. 1 shows the f5na.laction variable for the product diatomic molecule [l 111as a function of the initial action angle determining the vibrational
phase for five energies. In view of the fact that the absolute phase is arbitrary, depending on the choice of the initial separation, there is a remarkable similarity in these curves. Most reactive trajectories fail on a smooth curve with the hugest action at the ends of the curve. At one end there is a fairly sharp transitionto an unreactivecurve, with the gap between, in which the outcome ffuctuates, 1% of the domain or less. The other boundary between the bands is much wider on the order of 13% of the domain - and includes rapid changeover from reactiveto unreactivebehavior. Within this “random phase” region appears to a short branch or a smooth reactivecurve, but without many more trajectoriesit is difficulr to make a deCn.itive statementabout the conthmity in this region. We were interestedin investigatingthe relationship between surface and dynamics which lrjves rise to this
Vclume 57. number2
0
_ -_-
0
CHEMICAL
.?,
r
NTIAL
AhQ-E
,
PKfSIcs
15 July 1978
I
360
8 (DEGREES)
FXg. I. Finalaction as a function of initialphaseangle for five choicesof relativetransIati~nal energyfor F + Hz (II= 0). Filledcircles represent reactivetrajectories,open circlesumeactive-Arrowspoint to trajectoriesshown in later figures.
behavior. We first observed that trajectories at the band edges and in the random phase region were particularly long, lasting on the average20 to 50% longer than the other trajectories. While the time is still very short compared to trde complex formation, it does suggestmultiple reflection in the reaction zone. In order to isolate the dynamical effects responaibie:,we looked at adjacent groups of trajectories (each offset 2.5” in the irStialphase angle)_The first set, taken at the aharperband edge at I kcal/moIe translation,is shown in fig_ 2. The potential has been shown in mass weighted coordinates, so the motion is as that of a particle of fmed mass sliding on tb%surface. The aimilarity of these trajectories up until their first encounter with the F-H repulsivewall is evident; On returning to the attractive wall, the difference ininitial phase resultsin a clear difference. One trajectory reacts directly, heating into the products channel, never to return The second startsback towards reactants_How-
190
LJFITERS
X Fig. 2. Particlemotion on the potentialeneqg surf&x for F + Hz (u = 0) at 1 kcal/mole reIative translational energy. The coordinates X and Y are defined so that the kineticenergy is (p’x + P$))IuI whereJVis the totalmass,allin atomic units.Thesaddle point is marked“‘P.
ever, because the barrier is in rhe entrance channel the trajectory is reflected by the barrier, and requires an additional circuit before acquiring sufficient energy to surmount the barrier. With the initial phase offset by another 2-S”, the returning trajectory csosses the barrier cleanly. Thus we see that the narrow band edge is characterized by a fairly simple bifurcation between reactive and unreactive trajectories, determined by the exact position of first encounter with the steep repulsive wall, with the possibility of a second reflection behind the barrier. The second set of trajectories are also both at 1 kcal/mole translation,but are selected from the random phase region. Adjacent trajectories, one reactive, the other unreactive, are shown in fig. 3. Both of these co& lisions involve multiple reffections. It seems that the combination of the curvature of the reaction path primarily a mass effect - with the shape of the attractive wall on the inside of the curve gives rise to quite
CHEWCAL
PHYSECS LEITERS
15 July 1978
broader random phase regions are associated with periodic trajectories of greater stability. Previous work in the H f Hz system has shown that band edges are characterized by particularly long-lived trajectories which undergo multipIe reflections over the potential barrier_ We have found that in the asymmetric F + H2 system, which has an early barrier, similar multiple reflections occur, but without recrossing the barrier. These multiple reflections make clear the origin of apparently random regions at the band edges. We sumst that these trapped trajectories arc closeIy related to the periodic trajectories which Pechukas has shown to exist.
Y
The support of this research by the Research Corporation through a Cotrell College Science grant is gratefully acknowledged. 08
LO
09 X
Fig. 3. RuticIe motion on the potential energy surface for
F + H2 (u = 0) at 1 &xl/mole relative trans!.ational eners.
The trajectories are selected from the random phase repion. Contours are labeIIed in kcal/moIe relative to separated reactants.
complicated motion in the reactive zone. It is interesting to note that, in contrast to H f Hz, these multiple reflections do not imply multiple crossings over the potential barrier. Trajectories like those in fig. 3 make clear the origin of the random phase region: when trajectories make many collisions in the ieaCtiVeregion, a minute difference in initial phase propagates quickly into a large difference in outcome. Recently Pechukas and Pollak have given a proof that there exist periodic trajectories at the boundaries between reactive and unreactive bands in all systems 1131. Trajectories which enter the reactive zone from the asymptotic region may come arbitrarily close to these periodic orbits and thus have a very long lifetime. The trajectories seen above szem to approach this behavior- It is tempting to speculate that the
References [I]
A. Kuppermm, J.T. Adams and D.G. Truhlar, VlXI International Conference on the Physics of Electronic and Atomic Collisions, Belgrade, Yugoslavia (1971), Abstracts, p. 149. 121 E-A. hfcCulIough Jr. and R.E. Wyatt, J. Chem. Phys. 54 (1971) 3578. 131 J-T. Bowman and k Kuppermarm, J. Chem. Phys. 59 (1973) 6524.
141 G.C. Schatz, J-T. Bowman and A. Kuppermaun, J. Chem. Phys. 63 (1975) 674.
C51 G-C. Schatz, J-T. Bowman and A_ Kuppermann, J. Chem.
Phys. 63 (1975) 685. P-A. WhitIock and J.T. Muckerman. J. Chem. Phys. 61 (1974) 4618. 171 J.W. Duff and D-G. Truhiar, Chem. Phys. 4 (1974) 1. ra SF. Wu and RA_ Marcus, J- Chem. Phys- 53 (1970) 4026. PI J.S. Wright, K-G. Tan and KJ. Laidler, J. Chem. Phys. 64 (1976) 970. WI R.E. Howard, AC Yates andW.k Lester Jr., J. Chem. Phys. 66 (1977) 1960. WI CC Rankin and W-H. Miller, J. Chem. Phys. 55 (1971) 3150. WI PJ. Kuntz, EM. Nemeth, J-C. Polanyi, S.D. Romer and C-E. Young, J. aem. Phys. 44 (1966) 1168. 1131 P. Pechukas and E_ Pollak, J. Chem. Phys. 67 (1977) 5976.
El
19i
REACm
15 July 1978
BANDS AND RJZBQUNDINC TRAJECTORIES IN COLLINEAR
Freya SCEBUBEL
F + Hz
and Sally CHAPMAN
Depatnzent of CRen&zs, lhrmz.~ Cdege, Cduntbia i&iyersity, New York, New York 10027. US.4 Received30 March 1978
Bandsof rezxctiwand unreactivetrajectorieshave been mapped for classicalcollinearF + Hz (u = 0). The edges of tie ‘band are citaracterizedby trajectodeswhi& undergoseveralreflectionsin the comer of the potentialener_q surface. MtdtipierefIectionsazeseen to lead to less &a.@~ defaed band edp.
mofecufar
Coffinear dynamics is an intriguingsubject. For a three atom system the motion takes place in two mathematicaldimensions,so it is easy to visualize the potential energy surface and the resulting particle trajectory_The quantum dynamical studies of KU~~SRISIII and co-workers and others have S~OWEIa number of interestingeffects including resonancesin the reactivity and vortices in the quantum flux [l--4]. M&y classicalcolhnear dynamical studies have been =zndertakenin order to compare with semiclassicalor quautum results [S-7], while others have focused on the classicalbehavior alone IS]. In particular, a number of studies have focused on the bands of reactive trajectoriesin the H i Hz system [9, TO]_While F + Hz is a system which has received considerable attention [4--6], we are not aware of a previous study of the band structurein ibis asymmetricexchange reaction. The phase space of a collinear triatomic system is four dimensional. Thus four variables must be applied to initiate a classicaltrajectory: typically l5e asymptotic separation of the reagent molecules, the diatomic vibrational energy, the relative translationalenergy, and the vibrational phase. if the first two are fixed, as in a usual quasiclassicaltrajectory study, the last two form a space in which reactiveand unreactivezones can be seen. In the H f Hz system, the boundaries of these bands am reIativelysharp, while in a system like H f Clj [ 1I 1, there are broad areas where the outcome of a trajectory is 3x5extremely sensitivefunction of the initial phase, giving a seemingly random pattern of tra-
jectory results. We have carried out a collinear quasiclassicaltrajectory study on F + Hz (u = 0) using an extended LEPS potential surface [la], with the parameters of Mucke,man’s surface five [4] _ This system has been studied before; we are simply looking at a different aspect of the dynamics. The reaction has a sharp threshold with the reaction probability rising to about 0.7 at 1 kcal/molgmlative translationalenergy- The probability f& roughiy linearly reaching 0.5 by S kcallmole of translation. Fig. 1 shows the f5na.laction variable for the product diatomic molecule [l 111as a function of the initial action angle determining the vibrational
phase for five energies. In view of the fact that the absolute phase is arbitrary, depending on the choice of the initial separation, there is a remarkable similarity in these curves. Most reactive trajectories fail on a smooth curve with the hugest action at the ends of the curve. At one end there is a fairly sharp transitionto an unreactivecurve, with the gap between, in which the outcome ffuctuates, 1% of the domain or less. The other boundary between the bands is much wider on the order of 13% of the domain - and includes rapid changeover from reactiveto unreactivebehavior. Within this “random phase” region appears to a short branch or a smooth reactivecurve, but without many more trajectoriesit is difficulr to make a deCn.itive statementabout the conthmity in this region. We were interestedin investigatingthe relationship between surface and dynamics which lrjves rise to this
Vclume 57. number2
0
_ -_-
0
CHEMICAL
.?,
r
NTIAL
AhQ-E
,
PKfSIcs
15 July 1978
I
360
8 (DEGREES)
FXg. I. Finalaction as a function of initialphaseangle for five choicesof relativetransIati~nal energyfor F + Hz (II= 0). Filledcircles represent reactivetrajectories,open circlesumeactive-Arrowspoint to trajectoriesshown in later figures.
behavior. We first observed that trajectories at the band edges and in the random phase region were particularly long, lasting on the average20 to 50% longer than the other trajectories. While the time is still very short compared to trde complex formation, it does suggestmultiple reflection in the reaction zone. In order to isolate the dynamical effects responaibie:,we looked at adjacent groups of trajectories (each offset 2.5” in the irStialphase angle)_The first set, taken at the aharperband edge at I kcal/moIe translation,is shown in fig_ 2. The potential has been shown in mass weighted coordinates, so the motion is as that of a particle of fmed mass sliding on tb%surface. The aimilarity of these trajectories up until their first encounter with the F-H repulsivewall is evident; On returning to the attractive wall, the difference ininitial phase resultsin a clear difference. One trajectory reacts directly, heating into the products channel, never to return The second startsback towards reactants_How-
190
LJFITERS
X Fig. 2. Particlemotion on the potentialeneqg surf&x for F + Hz (u = 0) at 1 kcal/mole reIative translational energy. The coordinates X and Y are defined so that the kineticenergy is (p’x + P$))IuI whereJVis the totalmass,allin atomic units.Thesaddle point is marked“‘P.
ever, because the barrier is in rhe entrance channel the trajectory is reflected by the barrier, and requires an additional circuit before acquiring sufficient energy to surmount the barrier. With the initial phase offset by another 2-S”, the returning trajectory csosses the barrier cleanly. Thus we see that the narrow band edge is characterized by a fairly simple bifurcation between reactive and unreactive trajectories, determined by the exact position of first encounter with the steep repulsive wall, with the possibility of a second reflection behind the barrier. The second set of trajectories are also both at 1 kcal/mole translation,but are selected from the random phase region. Adjacent trajectories, one reactive, the other unreactive, are shown in fig. 3. Both of these co& lisions involve multiple reffections. It seems that the combination of the curvature of the reaction path primarily a mass effect - with the shape of the attractive wall on the inside of the curve gives rise to quite
CHEWCAL
PHYSECS LEITERS
15 July 1978
broader random phase regions are associated with periodic trajectories of greater stability. Previous work in the H f Hz system has shown that band edges are characterized by particularly long-lived trajectories which undergo multipIe reflections over the potential barrier_ We have found that in the asymmetric F + H2 system, which has an early barrier, similar multiple reflections occur, but without recrossing the barrier. These multiple reflections make clear the origin of apparently random regions at the band edges. We sumst that these trapped trajectories arc closeIy related to the periodic trajectories which Pechukas has shown to exist.
Y
The support of this research by the Research Corporation through a Cotrell College Science grant is gratefully acknowledged. 08
LO
09 X
Fig. 3. RuticIe motion on the potential energy surface for
F + H2 (u = 0) at 1 &xl/mole relative trans!.ational eners.
The trajectories are selected from the random phase repion. Contours are labeIIed in kcal/moIe relative to separated reactants.
complicated motion in the reactive zone. It is interesting to note that, in contrast to H f Hz, these multiple reflections do not imply multiple crossings over the potential barrier. Trajectories like those in fig. 3 make clear the origin of the random phase region: when trajectories make many collisions in the ieaCtiVeregion, a minute difference in initial phase propagates quickly into a large difference in outcome. Recently Pechukas and Pollak have given a proof that there exist periodic trajectories at the boundaries between reactive and unreactive bands in all systems 1131. Trajectories which enter the reactive zone from the asymptotic region may come arbitrarily close to these periodic orbits and thus have a very long lifetime. The trajectories seen above szem to approach this behavior- It is tempting to speculate that the
References [I]
A. Kuppermm, J.T. Adams and D.G. Truhlar, VlXI International Conference on the Physics of Electronic and Atomic Collisions, Belgrade, Yugoslavia (1971), Abstracts, p. 149. 121 E-A. hfcCulIough Jr. and R.E. Wyatt, J. Chem. Phys. 54 (1971) 3578. 131 J-T. Bowman and k Kuppermarm, J. Chem. Phys. 59 (1973) 6524.
141 G.C. Schatz, J-T. Bowman and A. Kuppermaun, J. Chem. Phys. 63 (1975) 674.
C51 G-C. Schatz, J-T. Bowman and A_ Kuppermann, J. Chem.
Phys. 63 (1975) 685. P-A. WhitIock and J.T. Muckerman. J. Chem. Phys. 61 (1974) 4618. 171 J.W. Duff and D-G. Truhiar, Chem. Phys. 4 (1974) 1. ra SF. Wu and RA_ Marcus, J- Chem. Phys- 53 (1970) 4026. PI J.S. Wright, K-G. Tan and KJ. Laidler, J. Chem. Phys. 64 (1976) 970. WI R.E. Howard, AC Yates andW.k Lester Jr., J. Chem. Phys. 66 (1977) 1960. WI CC Rankin and W-H. Miller, J. Chem. Phys. 55 (1971) 3150. WI PJ. Kuntz, EM. Nemeth, J-C. Polanyi, S.D. Romer and C-E. Young, J. aem. Phys. 44 (1966) 1168. 1131 P. Pechukas and E_ Pollak, J. Chem. Phys. 67 (1977) 5976.
El
19i