J. Phys. Chem. 1994,98, 12481-12485
12481
ARTICLES (He, Hz’) Dynamics: A Phase Portrait Analysis? Asif Rahaman and N. Sathyamurthy* Department of Chemistry, Indian Institute of Technology, Kanpur 208 016, India Received: January 14, 1994; In Final Form: March 3, 1994@
By a detailed analysis of the phase portrait of collinear nonreactive (He, H2+) collisions plotted in terms of Poincare surface of section, we show that there exists an intermolecular bottleneck which can be identified with the zero-order separatrix. It is shown that the regular trajectories do not cross the intermolecular bottleneck, but the irregular trajectories do. While a large number of irregular trajectories enter through the flux-in region and exit through the flux-out region of the bottleneck, in conformity with the earlier findings of Davis and Gray for (He, 12) collisions, a significant number of trajectories enter directly, presumably due to the fact that these trajectories traverse through the product channel of the configuration space although the energy of the system is well below the reaction threshold. This is reinforced by the observation that, with a decrease in energy, a larger number of irregular trajectories enter through the flux-in region. The fractal characteristics of the action-angle plots revealed in our earlier study (J. Chem. Phys. 1991, 95, 4160) are related to the existence of concentric layers of “baseball bat” like structures in phase space. We also identify “islands” in the phase space corresponding to quasiperiodic motion of bound trajectories in the neighborhood of chaotic scattering trajectories.
Introduction The quasiclassical trajectory method’ has been used to study collinear collisions between an atom (A) and a diatom (BC) for the past three decades. In order to fully characterize the initial conditions for such a collisional system, one needs to specify the initial separation RsheU between A and the center of mass (c.m.) of BC, relative translational energy (Ems), the vibrational action (ni) or the vibrational energy (&it,) of BC, At a fixed EWms and Rsheu and its initial vibrational phase the dynamical outcome can be characterized in terms of final vibrational action (nf) of AB or BC (depending on the collision being reactive or nonreactive), and a plot of nf as a function of 4 can be revealing indeed. For a certain range of &, nf varies smoothly and the trajectories involve single collisions and are termed direct or regular. For a certain other range, the value of nf is highly sensitive to the choice of 4i and the collisions involve multiple encounters. These trajectories are referred to as indirect, snarled, chattering, or irregular? In the case of reactive systems for which the reaction probability is less than unity, nA&) invariably consists of reactive (R) and nonreactive (NR)bands and irregular regions in between. The switchover region, on increased resolution in &, reveals an array of alternating R and NR bands.3 During a detailed examination of the action-angle plot for a model nonreactive system, Gottdienefl discovered that the chattering region contained a pattern of “parabolas” which repeated itself endlessly with increased resolution dong @i. Similar parabolas were found to exist for a number of reactive as well as nonreactive system^.^-^ While investigating (He, 12) collisions, Noid et al.’ discovered similar such parabolas but called them “icicles” and pointed out that the scale invariance of such icicles is characteristic of fractal^.^ ~~~~~
t Dedicated to Professor C. N. R. Rao on his 60th birthday.
Originally
submitted for the C. N. R. Rao Festschrift (J. Phys. Chem. 1994,98 (37)). Abstract published in Advance ACS Abstracts, August 1, 1994. @
0022-365419412098-12481$04.50/0
Poincark surface of section (SOS) serves as a valuable diagnostic tool in understanding the detailed dynamics in phase space. The phase space for (He, 12) collisions in a T-shaped geometry, for example, is four-dimensional: (x,y,px,py)where n represents the center-of-mass separation between He and 12, y is the 1-1 bond distance, and p x and p y are the corresponding conjugate momenta. The SOS in this case is plotted in (x,px) space corresponding to y = ye and p y > 0. The fixed point for the system occurs at x = 00, p x = 0. For the uninitiated, we state that it is a point whose iterates are the same point. In this case, the fixed point for the system is a hyperbolic fixed point. That is, the fixed point has an unstable branch (moving away from it) along negative px and a stable branch (moving toward it) along positive p p The point at which these two branches meet is called a homoclinic point, beyond which each is unstable with respect to the other giving rise to what are called homoclinic oscillations. Davis and Graylo showed that there existed an intermolecular bottleneck which was identified with the zero-order separatrix (see below) for (He, 12) collisions. By evolving the zero-order separatrix for one time period across the SOS, they obtained the fist-order separatrix which revealed homoclinic oscillations resembling “turnstiles” in (x,p,) space. They also showed that the regular trajectories never entered the intermolecular bottleneck whereas the irregular trajectories did. The transport of the latter was shown to occur through the broken separatrix” (the turnstile portion): entry through the inner lobe (flux-in region) and exit through the outer lobe (flux-out region). They also showed that when evolved for additional times across the Poincart section, the separatrix yielded additional structures, and on superposition the two branches led to “noodle” like structures within the bottleneck and “finger” like structures without. The latter explained the origin of alternating regular 0 1994 American Chemical Society
12482 J. Phys. Chem., Vol. 98, No. 48, 1994 2.0
1.5
I
Rahaman and Sathyamurthy
,I
I
I
I
I
I
Zero order separatrix
First order separatrix 1.5
0.5 1 .o i
t
0
W
0.5
r; -05
I
I
1:
c
0.0
Reg u lo r
-0.5
-1 5
n
0
400
2
3
4i
4
5
44
6
7
Figure 2. Zero-order separatrix for collinear (He, Hzi) collisions, along with the fust-order separatrix at Etot= 0.6434 eV. Lobes 1 and 3 are flux-out regions and 2 and 4 are flux-in regions.
Results and Discussion For the purpose of the present study we have considered collinear He Hz+ (ni = 0) collisions at E@,,, = 0.5 eV with &hell = 10 au on the ab initio potential energy surface (PES) described elsewhere.16 The value of @, was sampled uniformly, and the Hamilton's equations were solved numerically using the fourth-order Runge-Kutta method with a step size of 0.215 48 fs. For collinear (He, H2+) collisions, the phase space is fourdimensional: (x,y,p,,p,), where x represents the distance of He from the center of mass (c.m.) of H2+, y represents the massweighted internal coordinate of H2+, and p x and p y are the conjugate momenta. Poincark surface of section was generated by marking the trajectories in (x,px)space whenever y = ye and p y > 0, and the results for a sample of 400 trajectories are shown in Figure l a along with the action-angle plot in Figure lb. There are essentially two types of dynamics in the phase space: (1) motion which is bound for all time and (2) motion which is unbound. The separatrix separates the two and represents the last bound curve corresponding to
+
-2 1
2
3
4
5
6
7
8
9 1 0 1 1 1 2
xiau) Figure 1. (a) Poincark section for 400 trajectories for collinear (He, Hz+) collisions on an ab initio potential energy surface with ni = 0 of Hzf and Etrans = 0.5 eV. (b) Action-angle plot for the system under the same conditions. and irregular trajectories and the existence of a self-similar pattern of icicles. Someda et a1.12have been able to explain the icicle structures through a decoupling surface analysis which separates the collision coordinate from the intramolecular coordinate in the spirit of vibrational adiabatic theories.13 We had demonstrateds earlier that the inelastic scattering of He by H2+ revealed regular and irregular regions in the actionangle plots. On further examination, the irregular region revealed the existence of fractal singulkties. Similar behavior was found in the case of reactive trajectories as well. In the present study we have undertaken an analysis of the phase portrait of the nonreactive trajectories by plotting PoincarC sections from the point of view of broken separatrices and transport across s turnstile^"^^^^^^^^ and relate them to the regular and the irregular trajectories. We must mention here an earlier study of HeH2+ trajectories using PoincarC sections by Mayne and Wolf.lS Their study was somewhat limited and on a different potential energy surface. Importantly, they showed that the quasiperiodic and the non-quasiperiodic trajectories occupied disjoint regions of phase space.
p32M
+ V(x,y=y,) = 0
(1)
where M is the reduced mass of He with respect to H2+ and V(X,y=ye) is the potential energy of the system with H2+ at its equilibrium position. This is referred to as the zero-order separatrix. Thus, for a range of n values, we solve eq 1 for px, and the resulting PoincarC section is shown in Figure 2. It is worth emphasizing that the zero-order separatrix does not depend on the total energy of the system. It depends on the PES only. From eq 1 it is clear that for a given value of x
Thus, for a particular value of x we have two values of p x which are same in magnitude but opposite in sign. In order to understand the dynamics at a particular energy, it becomes necessary to compute the exact separatrix. To a first approximation, this can be done by time evolving the zeroorder separatrix. For each set of (x,px) values on the zeroorder separatrix shown in Figure 2, we set y = y e and pr = where p is the reduced mass of H2+ and Evib is the
1IzFLE,b,
+
(He, H2+) Dynamics
J. Phys. Chem., Vol. 98, No. 48, 1994 12483 1.5
I
I
I
,
3
4
5
6
1 .o
1 -
0.5 n
A
3 0 -
3 0.0
0
U
W
W
dl
r;
-0.5
-
-1
-1
.o
-1.5
2
44
7
Figure 4. First-order separatrices resulting from forward (f) and backward (b) time propagation of the zero-order separatrix at Etot=
0.6434 eV.
1 .o
-
0.5
-
0.0
-
y-0.5
-
A
3
U
-2' 1
' 2
I
3
I
4
I
5
6
I
I
7
8
9
1
1
0
a
I
1
1
1
2
x(a4 Figure 3. Poincar6 section (a) of all regular trajectories (& = 0.531~ to 1.66~)in Figure lb plotted in conjunction with the zero- and the first-order separatrix at E,, = 0.6434 eV and (b) of all irregular trajectories (& = 0 to 0 . 5 2 5 ~and 1.66% to 2n).
vibrational energy, corresponding to a total energy (Et,,) of 0.6434 eV, and propagated the trajectory forward in time until it crossed the surface of section again. This exercise was repeated for the entire range of (x,px) on the zero-order separatrix. The resulting first-order separatrix is included in Figure 2, and as expected, it shows deviations from the zeroorder separatrix. It consists of two "flux-in" and two "fluxout" regions. Since we are dealing with an area preserving map (for a conservative system), the total area of the flux-in regions would equal that of the flux-out regions.14 As pointed out by Davis and Gray,l0 trajectories starting in the flux-in region get trapped inside the intermolecular bottleneck. In contrast, trajectories initiated in the flux-out region never enter the bottleneck region. Trajectories in the regular region never enter the intermolecular bottleneck as illustrated in Figure 3a. For instance, trajectories starting in region 1 pierce through regions 2, 3, and 4 successively and exit through regions 5, 6, 7, etc. The strikingly regular "baseball bat" like structures can be understood from the fact that these trajectories result in a range of final relative translational energies. That is, the separating species having a relatively low Ems (marked by low p x ) would have their trajectories piercing through the surface of section rather closely in the x direction whereas those having a large Ems (marked by large p z ) would have their marks far between.
-1.0
-
-1.5
-
1
2
3
4
5
6
7
8
9 1 0 1 1 1 2
x(a4 Figure 5. Poincar6 section of irregular trajectories (at E,, = 0.6434 eV) obtained by forward as well as backward time evolution along
with the zero- and first-order separatrices. Trajectories in the irregular region, on the other hand, are chaotic, and they invariably get trapped inside the bottleneck region as shown in Figure 3b. While some of them do enter through the flux-in region, undergo multiple encounters inside the bottleneck region, and exit through the flux-out region as suggested by Davis and Gray,l0 many of them enter into the bottleneck region without entering through the flux-in region fist. Characteristically, the baseball bat-like structures seen in Figure 3a are missing in Figure 3b, and there is a complementary dusty pattern. A proper understanding of the behavior of these trajectories becomes possible when we consider the fist-order separatrix obtained from backward time evolution of the zeroorder separatrix, in conjunction with the one obtained by forward time evolution. Superposition of the two fist-order separatrices reveals in Figure 4 the homoclinic oscillations of one branch with respect to the other, beyond the homoclinic point. Trajectories for (He, H2+) collisions starting with positive px and incremented in negative time yield a Poincark section similar to that for the trajectories with negative p x and incremented in
Rahaman and Sathyamurthy
12484 J. Phys. Chem., Vol. 98, No. 48, 1994
,
2.2
I
I
I
1
I
1
1
I
1
1
1 .o
.
5
1
1
1
i
i
i
1
1
1
1
1
L
1
1.0
-
-1.6 3 0
W
x1.4
-
1.2
-
0
-0.5
-1.0
-1.5
1 I
1
2
3
4
5
6
7
8
9
1 0 1 1
Figure 7. Poincark section for 4000 trajectories in the irregular region (+i = 0 . 1 to ~ 0 . 2 ~at) a 200-fold increase in resolution when compared
-1 5
1
1
10
9
I
I
1
2
3
4
5
,
s X! 3J)
1
\
I
7
to Figure la.
11.1
I
1
8
9
;
0
Figure 6. (a) Illustration of an irregular trajectory at E,,, = 0.6434 eV
entering the intermolecular bottleneck without entering through the fluxin region. (b) Plot of the same trajectory in configuration space indicating the tendency to traverse into the product space. positive time. A superposition of the PoincarC section for the regular trajectories in the reverse direction on the corresponding first-order separatrix confirms that they never enter the bottle neck region. A similar plot for the irregular trajectories shows that the trajectories in the forward direction which do not seem to enter through the flux-in region (see Figure 3b) are actually lying in the region of the phase space corresponding to the complex forming trajectories in the reverse direction as illustrated in Figure 5. The reason for the apparent breakdown of the generalization by Davis and Graylo can be traced to the difference in the nature of (He, 12) and He, H2+) systems. While the former is an exclusively nonreactive system, the latter is potentially reactive with the result that even at energies below the reaction threshold, many trajectories traverse into what would be close to the product space before returning into the reactant space as illustrated in Figure 6. In addition to the regular and the irregular behavior of trajectories, it was shown earlieI.8 that the irregular region, on increased resolution along @i, revealed additional structures lying within, characteristic of fractals. In order to understand the
origin of such fractal characteristics, we examined the PoincarC section for a large number of trajectories in the irregular region, at an increased resolution. The PoincarC section for 4000 trajectories with @i in the range 0 . 1 ~ - 0 . 2for ~ example, is shown in Figure 7. It corresponds to a 200-fold increase in resolution along @i, when compared to Figure 1. It can be seen from Figure 7 that these are the complex forming trajectories entering the bottleneck through the flux-in region, and yet the dusty pattern in Figures l a and 3b becomes a layered structure, revealing a large number of concentric baseball bat like structures, thus explaining the origin of the self-similar pattern of parabolas in the action-angle plot (see Figure 1 of ref 8). An examination of PoincarC sections for different (nl,&,,) values confirms our finding that the regular trajectories never enter the intermolecular bottleneck while the irregular trajectories do. In general, we find that there is an increase in the number of trajectories entering the bottleneck region and hence in the extent of complex formation, with decrease in E,,,. A careful examination of the PoincarC section (see Figure 7 for example) reveals that regardless of the number of trajectories run, some portions of the phase space never get filled. In other words, scattering trajectories never enter into those regions of phase space at that energy. This has been noticed in some of the earlier studies, and the portion of the configuration space that was never visited by the trajectories was referred to as a “dynamical white spot”.17 Many authors have noticed islands in the phase space and have related them to quasiperiodic trajectories (for example, see refs 15 and 18). We have examined the fate of a trajectory that gets started in one such island at Et,, = 0.6434 eV. Sure enough, that trajectory remains confined to a portion of the phase space within the bottleneck region, and to the extent that the trajectory was integrated, it remained bound to that region, as illustrated in Figure 8. Presumably, this trajectory would correspond to a classical resonance. A more detailed investigation of such trajectories is presently being pursued.
Summary and Conclusion In order to understand the origin of the regular and the irregular behavior of trajectories for the system, we have generated the PoincarC surface of section for collinear (He, Hzf (ni=O)) collisions on an ab initio surface at E,,, = 0.5 eV,
(He, Hz+) Dynamics
J. Phys. Chem., Vol. 98, No. 48, 1994 12485 region and thus the fractal characteristics noted in our earlier work. We have also identified islands in the phase space revealing quasiperiodic motion of bound trajectories in the neighborhood of chaotic scattering trajectories. It would be worth extending the present phase portrait analysis to include reactive collisions. Such a study is currently in progress.
Acknowledgment. This study was supported in part by a grant from the Indo-US subcommission. We are grateful to Dr. R. Ramaswamy for valuable discussions and for a critical reading of the manuscript. We thank Mr. Atul Bahel and Mr. Debanand Das for their contributions in the early part of the work. References and Notes (1) Thompson, D. L.;Raff, L.M. In Theories of Chemical Reaction 1.15
' """"' '" 2.7 2.9
" "
""
" " "
'
3.1
x
"
"
'"
"
' ""' 3.5
"" "
3.7
(a$
Figure 8. Plot of a typical quasiperiodic bound trajectory located in an island of the phase space at E,, = 0.6434 eV.
well below the reaction threshold. We have shown that the regular trajectories never crossed the intermolecular bottleneck, identified with the zero-order separatrix. In contrast all the irregular trajectories crossed-some through the flux-in region of the first order separatrix and some directly. While the former behavior is in conformity with the observation by Davis and Gray, for (He, 12) collisions, the latter is not, presumably due to the fact that these trajectories traversed into the product region of the configuration space. This is reinforced by the fact that, at lower energies, more trajectories enter the bottleneck region only through the flux-in portion and exit through the flux-out portion of the fist-order separatrix. Increased resolution along revealed concentric layers of "baseball bat" like structures in phase space explaining the existence of quasi-regular trajectories buried in the irregular
Dvnamics: Baer.. M.., Ed.:, CRC Press: Boca Raton.. FL.. 1985: Vol. 3. Chapter 1: (2) Rankin, C.C.:Miller. W. H. J . Chem. Phvs. 1971.55. 3150. (3) Stine, J. R.; Marcus, R. A. Chem. Phys. L;?n.1974,29, 575. (4) Gottdiener, L. Mol. Phys. 1975,29, 1585. (5) Polanyi, J. C.; Wolf, R. J. Ber. Bunsen-Ges. Phys. Chem. 1982,8,
356. (6) Agmon, N. J . Chem. Phys. 1982,76, 1309. (7) Noid, D.W.; Gray, S. K.; Rice, S. A. J. Chem. Phys. 1986,84, 2649. (8) Balasubramanian, V.; Mishra, B. K.; Bahel, A.; Kumar, S.; Sathyamurthy,N. J. Chem. Phys. 1991,95,4160. (9)Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: New York, 1982. (10) Davis, M. J.; Gray, S. K. J . Chem. Phys. 1986,84, 5389. (11) (a) Mackay, R. S.; Meiss, J. D.; Percival, I. C. Physica 1984,013, 55. (b) Bensimon, D.; Kadanoff, L. P. Physica 1984,013,82. (12) Someda, K.;Ramaswamy, R.; Nakamura, H. J. Chem. Phys. 1993, 98, 1156. (13) Pollak, E.In Theories of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985;Vol. 3,Chapter 2. (14) Wiggins, S. Chaotic Transpol? in Dynamical Systems; SpnngerVerlag: New York, 1991. (15) Mayne, H.R.; Wolf, R. J. Chem. Phys. Len. 1981,81,508. (16) (a) McLaughlin, D. R.; Thompson, D. L. J. Chem. Phys. 1979,70, 2748. (b) Joseph, T.;Sathyamurthy, N. J . Chem. Phys. 1987,86, 704. (17) Gertitschke, P. L.;Kiprof, P.; Manz, J. J . Chem. Phys. 1987,87, 941. (18) Kob, W.; Schilling, R. J. Phys. A: Math. Gen. 1989,22, L633.
