J. Phys. Chem. 1989,93, 6947-6957
6947
those sites on surface where there is an open bond emanating from a 4f (3f) monomer. At the threshold P,, the total number of monomers N in the whole clusters is
cupied/white sites), and then it is the same as that for the RSP. The above arguments demonstrating the equality of P,, D, and v for the two models and the numerical results shown in sections IIa and IIb indicate that the chemical Eden clusters (for large N) belong to the same universality class as the RSP clusters.
Since the fractal surface (perimeter in d = 2) is proportional to RD,where R is the average radius of the cluster, and since N w 3 G a N6 RDb,with 6 < 1, the overall behavior of N is dominated by the first two terms and one gets N RD,asymptotically. Thus the behavior of the chemical Eden clusters at P = P, reduces to RSP; Le., the fractal dimensions of RSP and the chemical Eden clusters are equal. Compared to the epidemics model, the difference in the chemical Eden model is that the former reduces to Eden model A and the latter to Eden model B in the limit P = 1. This difference is not expected to have any effect on the static universality class either in the Eden limit or at the phase transition point, because the only difference lies in the slight, local, random changes in the statistical weights of the growth sites making the surface smoother for Eden model B/chemical Eden model. Finally, it also appears that the correlation length exponents u of the RSP and the chemical Eden model are equal. For this, we need consider only P > P,, (as distinct from P = P,) where the clusters are Euclidean. The correlation length l characterizes the average distance over which two particles belong to the same cluster: Le., 5 is proportional to the radius of a typical cluster at a given concentration P . For the chemical Eden clusters to be Euclidean, most of the 1f monomers (forbidden/black sites) have to be on the cluster surface, and therefore they do not contribute as much to the average radius. If such is the case, the exponent v is largely determined by the 4f (3f> monomers (oc-
111. Summary In this paper, we have described a class of generalized Eden models which we call the chemical Eden model and in which the functionality of various monomers plays an important role. We have illustrated the models by examining the bulk properties of three different specific binary mixtures: 4f-1 f on a square lattice and 3f-1 f, 3f-2f on a honeycomb lattice. For the mixtures involving the 1f poisons, the model reduces to R S P in equilibrium near a threshold concentration P, of the 4f (3j) monomers: P,, D, and u are the same for both the models. In absence of the 1f monomers, the chemical Eden model reduces to the Eden model B. For the binary mixture 3f-2f on a honeycomb lattice, we also find percolative behavior with its P, at 0.225 f 0.01.
-
Acknowledgment. It is a pleasure to thank (i) Fereydoon Family and Dietrich Stauffer for initiating us into the world of Eden growth, and for many discussions, and (ii) Tony Guttmann and Naeem Jan for sharing the joys of honeycomb lattice. R.C.D. also thanks (i) Katja Lindenberg and John Wheeler for their hospitality a t the Institute for Nonlinear Science and the Department of Chemistry, University of California at San Diego, and (ii) Tony Guttmann and Colin Thompson for their hospitality at the Department of Mathematics, University of Melbourne; a significant portion of this work was done at these two institutions during R.C.D.'s research leave.
Hamiltonian Mapping Models of Molecular Fragmentation Pierre Gaspard+ and Stuart A. Rice* Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (Received: February 24, 1989; In Final Form: June 15, 1989)
We report the results of a study of the qualitative features of several Hamiltonian mapping models allowing unbounded motion; these mappings are designed to be surrogates for the equations of motion of fragmenting molecules. We compare similarities and differences in the dynamics of two- and four-dimensional mappings. These mappings have as common features the destabilizing influence of low-order resonances, the fractal nature of the repellor (displayed in a complex escape time function structure), and multiple-lifetime decay of an ensemble of particles distributed with respect to initial conditions. The intermolecular bottleneck to flow in phase space generates the fast decay in the early stage of fragmentation, and intramolecular bottlenecks control the slow decay. In the four-dimensional mapping, a quasiinvariant set survives for a very long time, illustrating the extremely slow rate of Arnold diffusion. Finally, we show how to generalize to multidimensional systems the concept of homoclinic orbit.
I. Introduction Recent advances in the classical mechanical analysis of the dynamics of conservative systems of nonlinear oscillators have led to insights concerning energy flow in a molecule and its influence on the rate of a unimolecular reaction. In general, a classical Hamiltonian system supports periodic, quasiperiodic, and stochastic motions; the different kinds of motion can be intermingled in the many-dimensional phase space of the system. We find it convenient to take advantage of the observation that the representation of the evolution of a classical mechanical system as a flow of points in phase space can be thought of as a Hamiltonian mapping of the phase space into itself. The features of this
0022-3654/89/2093-6947$01.50/0
mapping have been explored in great detail for systems with two degrees of freedom. Indeed, many of the important advances in our understanding of the relationship between regular and stochastic flow, in particular the discovery of the role of the cantorus as a bottleneck to the flow of phase points, have emerged from the study of Hamiltonian mappings in two-dimensional bounded systems.'#* In contrast, our knowledge of the properties of Hamiltonian mappings in unbounded systems is much less well developed than for the case of bounded systems. Of course, any discussion of unimolecular fragmentation must allow for separated fragment trajectories that go to infinity, Le., be based on the study of the Hamiltonian for an unbounded system. This paper is
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
concerned with the insights into the dynamics of molecular fragmentation that can be gleaned from a study of model Hamiltonian mappings in unbounded systems. Consider the case of a system with two degrees of freedom, for which the PoincarB surface of section provides a visualization of the flow of phase points associated with the Hamiltonian mapping; indeed, the Poincar6 surface of section is an area preserving mapping. In the region of phase space where the motion is bounded, KAM tori (which support quasiperiodic motion) serve as natural boundaries between regions where the motion is chaotic. Only when such tori disappear can phase points leak from one region to another. MacKay, Meiss, and Percival’ and Bensimon and KadanoffZ have shown that the broken remnants of a KAM torus, specifically the cantorus whose image in the Poincard surface of section consists of periodic and nonperiodic unstable trajectories, serve as a leaky barrier to the flow of phase points. On the other hand, the region of phase space where the asymptotic motion is bounded is separated from the region in which the asymptotic motion is unbounded by the system ~eparatrix,~ which also serves as a partial barrier to the flow of phase points. Adopting the notion that the crossings of the cantorus and the separatrix are independent processes, Davis and Gray4 and Gray, Rice, and Daviss show that an improved theory of the rate of a unimolecular reaction can be constructed. In this new theory, the crossing of the cantorus is identified with the onset of rapid intramolecular randomization of the energy and the transition state is identified with the separatrix, so that crossing the latter leads to reaction (i.e., to fragmentation or isomerization). The fluxes of phase points across the cantorus and the separatrix are calculated from one step in their iterative Hamiltonian mappings. We note that although the cantorus (or, possibly, several cantori) and the separatrix of a system are defined by the properties of its Hamiltonian, the assumption that they serve as barriers that define the major dynamical features of the flow of phase points and that these barriers are independent monitors of the flow of phase points, and the method used to calculate the rates of crossing the cantorus and the separatrix, together generate a statistical approximation to the true dynamics of the system. A few details concerning aspects of the representation of dynamical systems as discrete mappings are pertinent to our discussion. We will be particularly concerned with chaotic motion associated with homoclinic orbits. These orbits destroy the integrability of the system dynamics6 and, thereby, permit the existence of a nontrivial escape dynamics, which we associate with occurrence of a reaction. In classical systems which have defocusing dynamics (e.g., scattering of a point particle from the hard convex surfaces of other particles), all of the trajectories are unstable and it is possible to construct a symbolic dynamics in one-to-one correspondence with the system trajectories. Such systems usually have a set of unstable trajectories trapped in the vicinity of the scattering centers for all time; this set is called the repellor. Trajectories with initial conditions infinitesimally displaced from those in the repellor can be detained in the vicinity of the scattering centers for a very long time, but they eventually escape to infinity. The flow on the set of unstable trapped trajectories can then be modeled by a probabilistic Markov chain characterized by a positive Kolmogorov-Sinai entropy per unit time. As one example we cite the recent analysis, by Gaspard and Rice, of the elastic scattering of a point particle from three disks in a plane.’ For this system, and related systems, the classical decay, Le., the rate of depopulation of an ensemble of (3) The system separatrix is a complete barrier only for the asymptotic motion, when the distance between the fragments is large. However, crossing this asymptotic separatrix is possible at short distance because of the nonintegrability of the fragmentation dynamics. Such a separatrix corresponds to the partial separatrixdefined by the following authors: MacKay, R. S.;Meiss, J. D.; Percival, I. C. Physica D 1987, 27, 1. Dana, I.; Murray, N. V.; Percival, I. C. Phys. Rev.Lett. 1988, 62, 233. (4) Davis, M. J.; Gray, S.K. J . Chem. Phys. 1986, 84, 5389. (5) Gray, S.K.; Rice, S.A,; Davis, M. J. J . Phys. Chem. 1986,90, 3470. ( 6 ) Moser, J. Stable and Random Motions in Dynamical Systems; Princeton University Press: Princeton, 1973. (7) Gaspard, P.; Rice, S.A. J . Chem. Phys. 1989, 90, 2225.
Gaspard and Rice trajectories with initial conditions infinitesimally displaced from the repellor, is a simple (one lifetime) exponentially decaying function of the time. However, in typical Hamiltonian systems the dynamics is sometimes focusing and sometimes defocusing. Consequently, such a Hamiltonian supports both stable periodic motion and unstable chaotic motion, and there is a complex hierarchical organization of the phase space into quasiperiodic islands of every size, separated by regions in which there is chaotic motion. This paper reports a study of the decay dynamics of several model unbounded Hamiltonian systems, with emphasis on mappings that describe escape of a trajectory to infinity. Although there is a large body of information concerning bounded systems whose dynamics can be described by the so-called standard map, very little is known of the properties of mappings in unbounded Hamiltonian systems. In this paper we propose several simple model mappings designed for the study of unbounded systems; we use these mappings as surrogates in the study of the Hamiltonian dynamics of molecular fragmentation. There are two principal motivations for this study. First, the properties of a mapping can often be determined from analytical considerations, thereby providing invaluable guidance to understanding the mechanics of the system. Second, the numerical calculation of the properties of a mapping is often several orders of magnitude more efficient than is integration of the equations of motion for which the mapping is surrogate. This feature allows us to explore complexities in the mechanics of the system which are more difficult to discover and analyze when direct integration of the equations of motion is employed. Mappings can be constructed in several different ways. In the case of autonomous systems with two degrees of freedom, with the representative Hamiltonian a mapping is generated by the flow in a Poincar6 surface of section
S ( X 1 , P l A 2 , P 2=) constant
(1.2)
in the constant-energy hypersurface H = E . The mapping is represented by
which is the action associated with the unique trajectory from q to q’, where q is the intrinsic position of the Poincard surface of section, and L is the Lagrangian corresponding to the Hamiltonian (1.1). The mapping in the @,q) plane is obtained from p = -aA/aq
(1.4)
p’ = dA/aq’
(1.5)
where p and q are the canonical momentum and coordinate used to define the Poincard surface of section. In principle, a twodimensional mapping can be generated in this fashion. However, in practice, development of the generating function (1.3) usually requires the introduction of analytical approximations. Thus, a mapping corresponding to (1.3) is usually generated numerically. The situation is simpler for the case of periodically perturbed (“kicked”) systems, which can be represented by the generic Hamiltonian
+-
H = Ho(X,P) + T G ( X ) n--” 6 ( t - nT)
(1.6)
In (1.6), C(X) is the amplitude of the perturbation (kick) at position X and T is the time between kicks. This Hamiltonian models, for example, the dynamics of a material system with Hamiltonian Ho interacting with a pulsed laser; the laser pulses are presumed to have a duration small with respect to the time between pulses. Of course, the material system evolves under the Hamiltonian Ho during the time between pulses. If this *zeroorder” Hamiltonian system is integrable, the equations describing the mapping can be written out explicitly. In general, the presence of the periodic external perturbation converts the regular motion of the “zero-order” system into irregular motion. As will be shown
Hamiltonian Mapping Models of Molecular Fragmentation in the following text, the dynamics of these symplectic mappings capture the qualitative features of Hamiltonian flows such as (1.1).8
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6949 G ( X ) = D(l
-
(2.3)
After the scaling
In the more general case of a system with N degrees of freedom, the 2N-dimensional phase space has intermingled regions where the trajectories are regular and chaotic. However, when N > 2 the resonance layers near the separatrices are not isolated from each other, since the N-dimensional surfaces that support quasiperiodic motion do not separate the (2N - l)-dimensional surface of constant energy into distinct regions, as they do when N = 2. For N > 2 all the stochastic regions of the phase space are connected in a single complicated web, the Arnold web.g For an initial condition within the web the trajectory will eventually intersect every finite region of the energy surface, a process called Arnold diffusion. In principle, Arnold diffusion implies that randomization of an initial energy distribution will always occur in a system with N > 2 degrees of freedom. In practice, the few numerical studies reported indicate that in the model systems studied Arnold diffusion is sufficiently slow that many of the qualitative features of regular and stochastic motion characteristic of an isolated two degree of freedom system survive for a very long time in a two degree of freedom subsystem of an N degree of freedom system.I0J' We shall show that a similar inference can be drawn from the properties of a model four-dimensional Hamiltonian mapping. These observations provide motivation for further examination of the dynamics of systems with only two degrees of freedom. All of the preceding refers to the classical mechanical description of a Hamiltonian system, whereas quantum mechanics is required for the proper description of the dynamics of molecular motion. At present, we have only a primitive understanding of the relationships between the quantum mechanical properties of a system which displays classical mechanical chaos and the characteristic features of the chaotic motion; we hope that the results described in this paper will provide a stepping stone, via quantum mechanical analysis of the properties of the mappings, to improved understanding of these relationships. As one example of the type of analysis envisioned, we cite the work of Gaspard and These investigators have shown that comparison of the scattering of a point particle from three disks in a plane using classical mechanics, semiclassical quantum mechanics, and quantum mechanics permits one to establish a relationship between the widths of quantum mechanical scattering resonances, and their distribution in energy, with characteristic features of the chaotic motion in the classical mechanical dynamics. 11. Free Point Particle in a Morse-like Periodic Perturbation
Consider a system with the Hamiltonian
+-
+ TG(x) c 2m
H = -p2
s(t
n=-m
- nn
(2.1)
which describes a free particle subject to periodic kicks. Let (Xn,Pn)be the position and the momentum of the particle before the nth kick. The coordinates of the particle before the next kick are then given by the area-preserving map
the map becomes pn+' = pn
qn+l = q n + Pn+l
where J is the Jacobian matrix of (2.5). A . The Fixed Point at q = p = 0. The eigenvalues of the fixed point with q = p = 0 are A* = '/2[2
T iPn+l
(2.2)
Since we are ultimately interested in molecular fragmentation dynamics, we assume that dG/dX vanishes at large distance; a function satisfying this condition is (8) A model of a Morse oscillator driven by impulsive interactions has also been considered by Poppe and Korsch (Poppe, D.; Korsch, J. Physica D 1987, 24, 367. In this model, the particle evolves in the Morse potential between the impulses but the latter occur when the particle makes a passage through the origin. Consequently, the external forcing is not periodic as in (1.6). (9) Arnold, V. I.; Avez, A. Ergodic Problems of Classical Mechanics; Benjamin: New York, 1968. (10) Chirikov, B. V. Phys. Rep. 1979, 52, 263. (1 1) Contopoulos, G.; Magnenat, P.; Martinet, L. Physica D 1982,6, 126. (12) Gaspard, P.; Rice, S . A. J . Chem. Phys. 1989, 90, 2242, 2255.
-d
f ( d ( d - 4)'f2]
(2.7)
which satisfy A + h = 1, showing that the mapping is area-preserving. When 0 < d < 4, the fixed point is a center with a rotation number p which means that its eigenvalues areI3 A+ = @*P (2.8) The value of d for which the center has this rotation number is
d = 2-2
COS
2rp
(2.9)
Clearly, the center has resonances when p is a rational number. These resonances are arranged in the parameter space in a monotone sequence between p = l / m and p = 112. The resonances of low order are the most important, namely, the 4-resonance at d = 2 and the 3-resonance at d = 3 . At d = 4 a period doubling occurs, and the fixed point (p = q = 0) becomes a saddle for d > 4 while a period-2 orbit is generated at the bifurcation. A calculation shows that the coordinates of this orbit are
= ft - (od - 4)
IJ2+
t o
= Xn +
(2.5)
Note that the map (2.5) is characterized by a single parameter, d . For positive q the perturbation vanishes, while for negative q it becomes very large. The kicks are then repulsive (Le., transfer momentum opposite to that of the free particle) if we assume the parameter d positive. If d were not chosen to be positive, the particle would be indefinitely accelerated for negative q. There is another reason for choosing d > 0. If we think of G ( X ) as a potential well in the limit where T goes to zero, then the mapping is equivalent to a flow and D > 0 defines the depth of the well relative to the energy zero at X = 0, P = 0. Defining w a(2D/m)'f2 to be the frequency associated with G ( X ) near X = 0, we see that d = w 2 p . The mapping (2.5) has two fixed points, q = p = 0 and p = 0, q = +a, whose stabilities are determined by the eigenvalues A of the linearized mapping obtained from (2.5). These eigenvalues satisfy det (J - AI) = 0 (2.6)
p = f -(d
Xn+l
+ d(e-2qn- e-*#)
O(d - 4),
- 4)
!I2
+ O(d - 4),
for d 2 4 (2.10)
This period-2 orbit is of center type after the bifurcation, but itself undergoes a period doubling at higher values of the parameter d . A period-doubling cascade follows, which transforms the invariant set into a Cantor-like repellor mostly composed of unstable trajectories. B. The Inuariant Set. For most of the values of the parameter d between 0 and 4, a set of bounded trajectories surrounds the center fixed point, which trajectories form the main quasiperiodic island of the mapping. This set is bounded by the largest invariant circle which bounds the inner trajectories. Outside the main island there exist smaller quasiperiodic islands around the high period orbits of center type (when such exist). All these quasiperiodic (13) Lichtenberg, A. J.; Lieberman, M. A. Regular and Stochastic Motion; Springer: New York, 1983.
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The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Gaspard and Rice
L
t
1
I
l " " l " " l " 1
-0.5
1
I ,
-0.5
0.0
I
I
,
,
,
I
I
I
0.5
9 Figure 1. Phase portrait of the mapping (2.5) for d = 1.8, in the (49)
plane. Fifty-four trajectories of 400 iterations are plotted.
0.5
1
t
1 0.001
0.0001 o~~
L
1.0~10-~
-0.5
0.0
0.5
q Figure 2. Same as Figure 1 for d = 2 with 48 trajectories.
islands form an invariant set of positive Lebesgue measure in the two-dimensional phase space. Besides this set, there exists a Cantor-like invariant set of unstable trajectories wandering between the quasiperiodic islands. This latter set is the repellor of the system, and it controls the escape dynamics. We have numerically constructed the global invariant set, plotting only the trajectories which remain at finite distance over a long time. This invariant set is shown for different values of the parameter d in Figures 1-3. There is a catastrophic collapse of the global invariant set when the center p = q = 0 undergoes a low-order resonance, in particular the 4-resonance at d = 2 (see Figure 2). The main island is large at d = 1.8 (Figure l), its size is reduced for d = 2.2 (Figure 3), but it increases thereafter. At the 4-resonance, when d = 2, the invariant set has a qualitatively different dustlike structure with an 8-fold figure near p = q = 0 (Figure 2). To understand this figure, we need to study the nonlinear stability of the fixed point. A Taylor expansion of (2.5) shows that
0
1
3
2
4
5
d Figure 4. Area of the invariant set in the (49)plane versus the parameter d . The area vanishes near d = 0 according to (2.15), as well as at the 4-resonance when d = 2 and at the 3-resonance when d = 3. After d = 4, a period-doubling cascade occurs, which transforms the invariant set into a dustlike repellor.
values of X such that = Xqn+4 if pn = hq,. Substituting this condition in (2.1 1) and (2.12), we obtain a polynomial equation of degree 4 which admits four real roots: (-1.75, 0.64, 1.36, 3.75)
(2.13)
The roots (2.13) define the eight arms seen near the center of Figure 2. Along these arms, the motion is alternately converging to p = q = 0, or diverging. We have studied the area of the global invariant set versus the value of the parameter d . The area was numerically calculated by a Monte Carlo integration. A large number of initial conditions (typically lo6 or 10') were taken with a uniform probability distribution in a rectangle containing the invariant set. The trajectories which remain trapped a t finite distance were then counted after a long time. The area so computed is represented in Figure 4. Note that, after an initial increase for small values of d, the area decreases as d increases, vanishing for d > 4.53. Now, in the limit d 0 the amplitude of the kicks is small and the mapping is nearly equivalent to a two-dimensional Hamiltonian flow in the (p,q) plane, Le.
-
dp/dt = d(e-*q - e-9) After four iterations, the motion is thus purely nonlinear, but it leaves invariant several radial motions. Thus, we look for the
dq/dt = p
(2.14)
Hamiltonian Mapping Models of Molecular Fragmentation
50
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6951
400
0.2 -
4 .3
a
0.0
3001
:
-
Q)
(d
m
W -0.2 -
loo/
I
I
-0.4 0.0
-0.1
-0.2
0.1
0.2
0.0
0.4
0.2
0.6
P Figure 5. Phase portrait of (2.5) ford = 4.1 just after the period doubling of p = q = 0 occurring at d = 4. Ten trajectories of 1000 iterations are plotted.
Figure 6. Escape time function of (2.5) for d = 1.8. The function is constructed with 6000 trajectories of 500 iterations with initial conditions: q = 0 and variable p . Escape is assumed to occur when q > 50. 500
This approximation is valid for d smaller than 0.1, and with it we obtain the asymptotic behavior of the area near d = 0. We find that
A
E
27rd1I2
The area of the global invariant set also vanishes, for different reasons, near the 3- and 4-resonances, at which there is collapse of the main island because of the breaking up of the invariant circles at the resonances. This phenomenon has been described by Moser, Henon, and Amold;14we rediscover it here in a mapping modeling molecular fragmentation, where it plays an important role. The period doubling which occurs at d = 4 is illustrated in Figure 5 , which represents the invariant set at d = 4.1 just after the bifurcation. Note the ordering around the period-2 orbit of center type. C. The Escape Time Function and the Decay of an Ensemble of Particles. The time evolution of the mapping can be studied with two methods: the construction of the escape time function and the study of the decay of an ensemble of particles. Consider, first, the escape time function. The time for a trajectory to reach a large distance ( q f = 50) is plotted versus the initial condition taken on a set of dimension 1. This latter was chosen as (40
= 0, 0
PO
< 0.6)
400
(2.15)
(2.16)
Examples of this function are shown in Figures 6 and 7. For d = 1.8 (Figure 6), all the trajectories with initial conditions inside the maximal invariant circle have an infinite escape time. Outside this set, the escape time sharply decreases, but with a complex structure with many peaks, which is reminiscent of the fractal structure of the repellor. At the 4-resonance for d = 2 (see Figure 7), the escape time function changes drastically. Because of the breakup of the invariant circles, particles escape even from close to the fixed point p = q = 0. Furthermore, the decrease of the escape time is slower in the average than it is off resonance. A structure which is common to both figures is the appearance of intervals where the escape time varies smoothly and presents a sharp peak at both ends. At most of these peaks the initial condition reaches the stable manifold of the unstable fixed point at infinity, and the trajectory undergoes a critical slowing down. Peaks are also expected near the stable manifold of a periodic or nonperiodic unstable orbit, but such peaks are rare. (14) Moser, .I. Astronom. J . 1958, 63, 439. Henon, M.Q. Appl. Math. 1%9,27, 291. Arnold, V. I. Mathematical Methods of Classical Mechanics; Springer: New York, 1978; Appendixes 7 and 8.
i
j"
300
Q)
a (d
0 VI
200
w 100
c 0.0
0.4
0.2
0.6
P Figure 7. Same as Figure 6 for d = 2.
Consider, now, the decay of an ensemble of particles initially close to the global invariant set. The number of particles N ( t ) remaining at finite distance at time t is counted. In a two-dimensional mapping, we know that trajectories may be trapped inside the invariant circle so that the decay is obtained by subtracting from N ( t ) the number of particles still at finite distance after a long time tf: M ( t ) = N ( t ) - N(tf). This function is plotted in Figure 8 for d = 1.8. In contrast with the behavior of the point particle-three hard disk system,' the decay of the ensemble is characterized by several time scales. This observation suggests that the asymptotic behavior is very difficult to study numerically. The results of numerical simulation seem consistent with a biexponential decay, as already observed by Davis and Gray4 for a model of the fragmentation of He-I2 in which both intermolecular and intramolecular bottlenecks were included. A similar situation exists in the present two-dimensional mapping. The short time scale is due to the (intermolecular) bottleneck defined by the homoclinic orbit of the fixed point at infinity. D. The Fixed Point at Inznity and Its Associated Homoclinic Orbit. A homoclinic orbit is a trajectory that goes to an unstable fixed point in the past as well as the future. Such a trajectory is thus at the intersection between the unstable and the stable manifolds of the fixed point. These manifolds generate a homoclinic ~ e b . ~ , ~
6952
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 1.0~10~
I "
"
I '
Gaspard and Rice 2
" ' 1
1
a
0
-1
-2
1 10000 I I I I I I I I I I I I I I I I / t
0
2000
1000
3000
Time Figure 8. Decay of an ensemble of lo7particles under the mapping (2.5)
for d = 1.8. The initial ensemble is uniformly distributed in the rectangle (91 = 0, 92 = 0.01) X (pI = 0.275,p z = 0.375). N ( t ) is the number of particles still with q < 10 after t iterations. N(5000) = 1 110789 and M(r) = N(r) - N(SOO0)is plotted.
-2
4
Figure 10. Stable (W,) and unstable (W,) manifolds of (2.5)for d = 1.8. Their intersections a and b define two distinct homoclinic orbits. Another homoclinic orbit is defined by the point t where W, is tangent to W,. 2
(
I
,
,
i
'
'
'
1-
1
a
0
0-
-1 -
-1
-2
-2 -2
2
9
2
a
0
0
2
4
9 Figure 9. Formation of a Smale horseshoe after six iterations starting from the domain labeled 0 under the mapping (2.5)with d = 1.8. The
curves are the stable and the unstable manifolds forming the homoclinic web. When the fixed point is a saddle, it can be proved that there exists an invariant set of trajectories close to the homoclinic orbit which form a symbolic dynamics on an infinite alphabet.6 Such an infinite symbolic dynamics is a generalization of the Smale horseshoe, which is in one-to-one correspondence with a twosymbol dynamics. Symbolic dynamics have, in general, a positive topological entropy per unit time, and as a corollary, chaotic behavior is possible on this invariant set.I5 The mere existence of the homoclinic orbit thus precludes the integrability of the system.6 Here we expect similar phenomena to occur, associated with the unstable fixed point at infinity (p = 0, q = +-). For instance, Figure 9 shows a Smale horseshoe giving a two-symbol subdynamics. However, we choose to emphasize some important differences between the present case and others. The stability eigenvalues of the fixed point at infinity are both equal to 1. This fixed point is thus not a saddle. Nevertheless, it is still unstable with distinct stable and unstable manifolds, but with a much slower dynamics than for a saddle. The homoclinic phenomena associated with such fixed points at infinity are new, (15)Smale, S.Bull. Am. Math. SOC.1967, 73, 141.
'
'
'
'
'
'
'
'
'
'
Figure 11. Construction of the partial separatrix associated with the fixed point at infinity. S is the domain bounded by the portion of W,, from (p = o , q = + m ) to the homoclinic intersection 6 and by the portion of W, from b to (p = 0,9 = +m). The turnstile q5 is adjacent to S and delimited by the portion of W, from a to 6. The coordinates of the homoclinic intersectionsare a = (p = 0, 9 = -0.40) and b = (p = 1.37,q = 0.77).
and they have not yet been studied very much. Let us first construct the stable and unstable manifolds. The fixed point at infinity is reached very slowly because the orbit of the stable manifold makes very small jumps at each iteration. As a consequence, the mapping may be replaced asymptotically by the differential equation system (2.14). We emphasize that this replacement is valid here for any value of the parameter d and not only near d = 0 as was the case in the discussion of the fixed point p = q = 0. The asymptotic motion along the stable manifold is thus q(t)
N 1-+-
In [ d t 2 / 2
+ d1/2(2e40- 1)1/2t + e401
(2.17)
The invariant manifolds are, asymptotically, W,:
p = +(de-q(2 - e-q))1/2
w,,:
p = -(dey(2
- e-q))'l2
(2.18)
These formulas define the seeds of the invariant manifolds. They are then propagated numerically from initial conditions at large values of q on the curves (2.18). We have carried out this numerical construction; the results are shown in Figure 10. At the intersections of W, and W, there are two distinct homoclinic orbits, given by the points a and b and their forward and
Hamiltonian Mapping Models of Molecular Fragmentation
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6953 I
-2
I
I
-2
I
I
’
f
i
I
I
I
’
2
0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
4
0
Figure 12. Stable (W.) and the unstable (W,,) manifolds of (2.5) now for d = 2. a’ and b’ are two new homoclinic intersections born at the homoclinic tangency t of Figure 10. Another homoclinic tangency occurs between d = 1.8 and d = 2, generating a further pair of homoclinic intersections a” and b”.
backward iterates. The flux across the intermolecular bottleneck is given by the area of the turnstile between W, and W,. See Figure 11. This area is equal to the difference of actions of the two homoclinic orbits
+-
2
1
3
q Figure 13. Phase portrait of the mapping (3.3) for d = 2, with 113 trajectories of 400 iterations plotted.
These tangent bifurcations occur here at parameter values accumulating on the homoclinic tangency d = d, like d , - d, E constant/n2 (2.25) This behavior is in contrast to the scaling behavior near a homoclinic tangency to a saddle of eigenvalue A > 1 where16 d , - d, constant/A” (2.26)
n=-m
where A(q,q’) is the generating function of the mapping 1 (2.20) A(q,q? = $4 - q12- g(q)
g(q) = :(I-
e-q)2
(2.21)
According to this simple model, with only an intermolecular bottleneck, the ensemble of particles decays like
where A is the area of the global invariant set and S is the area bounded by the truncated invariant manifolds W . and W,, as shown in Figure 1 1. For d = 1.8, the numerical values of these quantities are (P = 0.54, A = 0.29, s = 8.1 f 0.2, 7 t h = 0.072 f 0.002 (2.23) to be compared with the value from the early time decay of an ensemble of particles ynum 0.04 (2.24) Clearly, the correct order of magnitude of y is obtained from this simple model. One or several intramolecular bottlenecks should be incorporated to improve the predicted decay rate. E. Homoclinic Tangency. An interesting phenomenon, known as homoclinic tangency, exists in the present model. See Figure 10. Near d = 1.8 the unstable manifold becomes tangent to the stable manifold, and they form a tangent homoclinic orbit t. This phenomenon occurs at a critical parameter value d, = 1.8. After this bifurcation, two transverse homoclinic orbits a’ and b’ are generated. At d = 2 (see Figure 12), a second tangency of the manifolds occurs, which generates another pair af’ and b” of transverse homoclinic orbits. The homoclinic tangency is important because, in the vicinity of the tangency, there exists a sequence of tangent bifurcations each giving birth to a pair of periodic orbits, one of which is a saddle and the other a center. As the sequence is followed, the periods of these successively generated orbits increase indefinitely.
This result represents yet another difference between standard homoclinicity to a saddle and the presently described homoclinicity to an unstable fixed point at infinity.16 Together with the period-doubling sequence, the homoclinic tangency is one of the basic mechanisms generating periodic orbits in a dynamical system. In this sense, it plays an important role in the transition from regular to stochastic regimes. Furthermore, locating homoclinic tangencies may be useful in the identification of high-order quasiperiodic islands and the classification of them according to their origins.
111. Mappings with Bottlenecks of Several Kinds Choosing a perturbation amplitude in eq 2.1 with several minima and maxima, we construct mappings with several bottlenecks, each associated with fixed points. The existence of several bottlenecks allows this model to mimic some properties of a molecule with intramolecular restrictions to energy flow. For instance, the following kick amplitude function dg/dq = -de-q(e-q - r ) ( e q - s ) ( e y - t ) (3.1) (with d
> 0) generates a mapping with four fixed points q =
+a,
1 In -
t’
1 In -
S’
1 In -
(3.2)
all at p = 0; these fixed points are centers if 0 < d2g/dq2 < 4 and are saddles otherwise. Figure 13 shows the global invariant set of the mapping pn+l= pn
+ deqn(e-qn - l)(e-qn
- Y2)(eqn- y4)
(3.3) qn+l = q n + Pn+l with two centers at p = 0, q = 0 and p = 0, q = In 4 and an intermediate saddle at p = 0, q = In 2, representing an intramolecular bottleneck. When t = 0, as in the mapping pn+l= pn + de-2qn(e-qn- l)(e-qn- y2)
qn+l = q n + Pn+l there are only two fixed points. For 0 < d
(3.4)
< 8, there is a center
(16) For the dissipative case see: Curry, J. H.; Johnson, J. R. Phys. Lett. A 1982, 92, 217. Gaspard, P.; Wang, X.-J. J . Stat. Phys. 1987, 48, 151.
6954
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 I " " 0.21'
'
This Hamiltonian represents a rotator in free motion between kicks of amplitude G(0,X);the kick amplitude is assumed to vanish at large distance X . Note that the rotator can escape from the kicking field. We consider a kicking field defined by
""'-3
'
Gaspard and Rice
G ( 0 A = D[(1
+ g cos
- 2e-uX]
(4.2)
In order that the motion be stable in the repulsive part of the field, we assume D > 0 and < 1 . After a scaling of the coordinates, the following mapping is obtained:
+ c sin 0, e-2qn
I,+1 = I,
= 0, + 1,+i
o,+i
+ g cos 0,)e-2qn - e-qn]
pn+l = pn + d [( 1
-0.2
4n+1 = 4, + P n + l 0.0
1 .o
0.5
q
(4.3)
This four-dimensional mapping is pseudosymplectic in the sense that its Jacobian matrix
Figure 14. Phase portrait of the mapping (3.4)for d = 1, with 29 trajectories of 400 iterations plotted.
J=
a(ln+i,Pn+ I r0n+i?qn+l)
(4.4)
~(~,9Pfl,~fl*qfl)
satisfies J'wJ = w
(4.5)
with the matrix w defined by
r o .; I; 0 0
0.10
24
-p
~
0
O 0
oD ]
0
0
1
(4.6)
where 2ca = dgp. Furthermore, (4.3)is volume preserving because ldet JI = 1 , from ( 4 . 5 ) . A . Fixed Points. The mapping under consideration has two fixed points at finite distance, namely, 1 = 0, 8 = 0, p = 0, q = In ( 1 g ) , whose four stability eigenvalues are given by
0.05
+
0.00 0.05
4
0.08
0.07
0.08
~
4 Figure 15. Phase portrait of the mapping (3.4)ford = 5 . Enlargement of the invariant set. Fifty-three trajectories of loo0 iterations are plotted.
IV. Free Rotator in a Morse-like Kicking Field The two-dimensional mappings discussed in the previous sections are obtained from periodically forced one degree of freedom Hamiltonians. This class of systems has special properties, e.g., complete barriers formed by invariant circles. In three degree of freedom Hamiltonian systems this particular situation no longer prevails because of the existence of Arnold diffusion. One consequence of the possibility of Arnold diffusion is that much less is known about the escape dynamics of the system. In order to study the class of systems with Arnold diffusion, we propose a four-dimensional mapping obtained from the following Hamiltonian: L2 P2 +H(L,B,P,X,t) = - + - TG(0,X) 6 ( t - n7') (4.1) 21 2m n=-'
(*[:a)'])
112
C
1 + 2(1 +g)Z A3.4
at p = q = 0 and a saddle at p = 0, q = In 2. A standard homoclinic structure is now associated with this saddle, and it forms an intramolecular bottleneck. (See Figure 14.) Qualitatively, the same sequence of bifurcations of the phase portrait around q = p = 0 is observed in the present model as for the model described in section 11; there are resonances of the center, incuding catastrophic collapses of the invariant set and a period-doubling bifurcation at d = 8 generating a period-2 orbit of center type. Figure 15 displays an example of formation of small quasiperiodic islands which is favored in this system.
+
=2
0.09
1 +4(i
'
=1
hlh2
=
1 and 1=0,
0=s,
p=O,
q=ln(l-g)
with = 1 -
C
C
112
-f (A[ -- 1 1 ) 2(1 - g ) 2
( 1 - g ) 2 4(1 - g ) 2
The stability diagram of these two fixed points when g = 0.1 is represented in Figure 16. B . Depletion Plots and Quasiinvariant Sets. We have investigated the fragmentation dynamics of the system defined by ( 4 . 1 ) numerically. Phase portraits of four-dimensional phase spaces are not as useful as for two-dimensional phase spaces. We turn thus to a different type of diagram which is convenient for the study of the escape dynamics, namely, a depletion plot. In a two-dimensional subspace of the four-dimensional phase space
-
Hamiltonian Mapping Models of Molecular Fragmentation
a
Er
4
HE-HH
HE-HE
0
1
1
1
1
I
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6955 0.5
1
EH-HH
HH-EH
HH-HH
EE-HH
HE-EH
HE-HH-
0.4
-
0.3 d
0.2 EE-HE
HE-HE
HE-EE
L-5
0.1
1
1
1
1
I
1
5
0
0.0
C
1
0
Figure 16. Stability diagram of the fixed points of the four-dimensional mapping (4.3) in the plane of parameters d and c for g = 0.1. In each domain, the left-hand (right-hand) stability symbol concerns the 8 = 0 (8 = 7)fixed point. The stability symbols are EE for a center-center, EH for a center-saddle, and HH for a saddle-saddle point.
2
3
2
3
8 Figure 18. Same as Figure 17 for d = 2.5.
0.4 -
0.3 d
X
a
0.2 a
x
-I#
x
0.1 X X
0
xx
0.0 0
I
1
8
*' 2
1
3
Figure 19. Same as Figure 17 for d = 2.7.
8
. 0.4 ....,:......... ...... ... .. . ! .
Figure 17. Depletion plot for the four-dimensional mapping (4.3) for parameter values d = 2, c = g = 0.1. The quasiinvariant set at lo3 iterations is shown with initial conditions q = p = 0 and variable 1 and 8.
1
,
,
. I
.. ,. .! . .
I
I
I
..I
.. .. . .1.
"
,
0.2
of initial conditions, we plot the initial conditions of the trajectories which remain trapped in the system after a long time. We call this set of initial conditions the quasiinvariant set. Some examples of depletion plots for 1000 iterations are presented in Figures 17-19 for c = g = 0.1 and varying d. For d = 2 (Figure 17) and d = 2.5 (Figure 18), the fixed point at 0 = 0, 1 = 0 is a saddle-center which explains why the depletion plot is empty in this region. On the other hand, the fixed point at 0 = a, 1 = 0 is a center-center so that this region is able to trap the trajectories for a very long time. At d = 2.7 (Figure 19), a transition occurs and the region around 8 = a, 1 = 0 becomes unstable. Indeed, a 3-resonance occurs just at d = 2.7, which leads to a catastrophic collapse of the quasiinvariant set. Some three frequency quasiperiodic motions are observed at d = 1, near the point 0 = a,as shown in Figure 20, which represents the projections of several trajectories on the ( I $ ) plane. C. Decay of an Ensemble of Particles. We have analyzed the escape dynamics for the system defined by (4.1) using the escape time function to investigate whether the quasiinvariant set will slowly disappear after a very long time. Figure 21 shows the escape time function at d = 2 for 8 = 1 and varying initial angular momentum 1 over lo3 iterations. As for two-dimensional mappings, the escape time function is complex due to the presence
d
0.0
-0.2
-0.4
2
4
3
8 Figure 20. Projection in the (1,8) plane of several trajectories of the four-dimensional mapping (4.3) for d = 1, c = g = 0.1. Note that there are two trajectories on 3-frequency tori.
of a fractal repellor. Figure 22 depicts the same function over the period corresponding to lo5iterations. Some parts of the lo3 quasiinvariant set are depleted after lo5iterations, but some other parts (around 1 = 0 and 1 = 0.35) remain, showing that the
6956 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Gaspard and Rice
1000
8
1
800
E"
.3
4
600
.3
4
a,
a 0
I
600 -
a,
Ld
-I
a
400
-
400
v)
GI
200
0 0.0
0.2
04
0.6
0.8
0.0
1.0
0.2
I
80000
.$ 4
0
0.8
1.0
1
Figure 23. Escape time function of the four-dimensional mapping (4.3) for d = 3, c = g = 0.1. The initial conditions are the same as in Figure - 1
LI.
50000
2
600001
a,
a d
0.6
1
Figure 21. Escape time function of the four-dimensional mapping (4.3) for d = 2, c = g = 0.1, constructed with 500 trajectories with initial conditions p = q = 0,O = 1, and variable I . The trajectories are followed over lo3 iterations. 100000
0.4
20000 40000
G 20000
z
-
10000
0 0 0.0
50000
100000
Time 0.2
0.4
0.6
0.8
1.o
1 Figure 22. Same as Figure 21, but the trajectories are now followed over lo5 iterations.
depletion dynamics is extremely slow. An important difference with respect to two-dimensional maps is that escape of a trajectory may be rapid between two regions in which it is extremely slow. This novel feature occurs because of the four-dimensional topology and is a general phenomenon in many degree of freedom systems. After the 3-resonance, at d = 3 the escape time is much shorter (Figure 23). The function is more regular, showing peaks at the ends of intervals of regularity. These peaks were explained previously in subsection 1I.C. We have also calculated the decay of an ensemble of particles for d = 2. As for two-dimensional mappings, the decay occurs over two different time scales, and it may be approximated here by a biexponential curve for intermediate times. However, an extremely slow decay still occurs after a very long time, as shown in Figure 24, which is due to the slow depletion of the quasiinvariant set. As a consequence, we may not define either a global invariant set or its volume for this system. As for the decay in two-dimensional mappings, it is very difficult to describe any characteristic features of the long-time behavior of the system. Over very long time, periods of fast decay alternate with periods of slow decay. The origin of this phenomenon can, presumably, be traced back to Arnold diffusion. In molecular fragmentation, only the short-time dynamics of these classical
Figure 24. Decay of an ensemble of 5 X lo4 particles under the fourdimensional mapping (4.3) with d = 2 and c = g = 0.1. The initial ensemble is uniformly distributed in the rectangle (0, = -1, O2 = 1 ) X ( I , = 0, l2 = 2r) of the plane q = p = 0.
models need be considered because the longtime dynamics will be governed by quantum mechanics, emission of radiation, or collisions. D. Invariant Set at Infinity and Its Associated Homoclinic Manifold. The shortest time scale in the dynamical evolution is due to the intermolecular bottleneck defined by the transverse homoclinic manifold associated with the invariant set at infinity. In systems with more than two degrees of freedom, the state at infinity is not a fixed point but, rather, a whole invariant set. For the four-dimensional mapping (4.3), it is given by p = 0 , q = + m , I = 0, 8 E [0, 2 r [ (4.9) The rotator at infinity may have any orientation. The state is an invariant set of dimension 1 with four stability eigenvalues equal to 1 . Consequently, the unstable and stable manifolds are of dimension 2. We construct them as follows. At large distance, the four-dimensional mapping may be replaced by a flow described by the differential equations
9 q
N
= c sin 8,e-'4
d [ ( l + g cos 8m)e-2q- e-41
(4.10)
where 8, is the orientation of the rotator at infinity. These equations can be integrated to obtain the asymptotic stable and unstable manifolds
J . Phys. Chem. 1989, 93, 6957-6961 (4.1 1) (4.12) 0
N
c sin 6, 0, -k -e-9 3d
(4.13)
dt2 In2
(4.14)
and q
N I-*-
with the plus sign for the stable manifold W, and the minus sign for W,, in eq 4.1 1 and 4.14. These equations form the seeds of the invariant manifolds which serve as the initial conditions for the numerical construction of W,,and W,at finite distance. The homoclinic manifold will then be the intersections of the stable and the unstable manifolds. The homoclinic manifold forms a set of dimension 1 in the four-dimensional phase space. In this way, the method of Davis, Gray, and Rice4” for the calculation of flux a c r m the separatrix can be extended to systems with more than two degrees of freedom.
V. Conclusions Our study of Hamiltonian mapping models of fragmentation displays the differences and similarities between several systems. The two-dimensional mappings of periodically perturbed systems have Poincart portraits which are comparable to the Poincart portraits of two degree of freedom Hamiltonian flows. Integrability is achieved when the time interval between the kicks vanishes or when the amplitude of the kicks goes to zero. A similar property has been recently observed by Tersigni, Gaspard, and Rice1’ in numerical studies of Hamiltonian flows as well as in a system consisting of a periodically kicked particle in a Morse potential. In this latter model, the limit were the kicking period goes to zero is equivalent to the limit where the ratio between the Morse
6957
oscillator intrinsic frequency divided by the kicking frequency goes to zero. The adiabatic hypothesis is valid in this limit, as shown in section I1 and in the discussion by Tersigni, Gaspard, and Rice” of the mapping behavior of a kicked Morse oscillator. In two-dimensional mappings, a global invariant set with a positive area is preserved by the escape dynamics. However, in four-dimensional mappings Arnold diffusion precludes the existence of complete barriers formed by invariant tori. Accordingly, no invariant set of positive Lebesgue measure is expected to exist. Nevertheless, numerical integration shows that a quasiinvariant set persists for a very long time. This quasiinvariant set shows a property similar to the invariant set of two-dimensional mappings: it collapses near low-order resonances. Furthermore, an ensemble of particles decays with two widely different time scales at intermediate times in four-dimensional as well as two-dimensional mappings. This important observation implies that a two degree of freedom subsystem of a larger system retains many of the properties of an isolated two degree of freedom system for a time long enough for physical processes of interest to occur. An obvious corollary of this observation is that two degree of freedom models of more complex systems can incorporate the major features of the larger system dynamics and are, therefore, worthy of detailed study. Finally, our work suggests the desirability of generalizing the concept of homoclinic orbit to four-dimensional mappings, with the goal of calculating the escape rates in models of molecular fragmentation with more than two degrees of freedom. Acknowledgment. Pierre Gaspard is “Chargt de Recherches” of the National Fund for Scientific Research (Belgium). This research was supported by a grant from the National Science Foundation (NSF C H E 86- 16608). (17) Tersigni, S. H.; Gaspard, P.; Rice, S. A. Influence of Vibrational Frequency Mismatch on Phase Space Bottlenecks to Intramolecular Energy Redistribution and Molecular Fragmentation; preprint, 1989.
Renormallzed Equations in Fokker-Pianck Models J. Javier Brey* and M. Morillo Fisica TeBrica, Universidad de Sevilla. Apdo. Correos, 1065 Sector Sur, 41080 Sevilla. Spain (Received: February 27, 1989)
A projection operator method is developed for the derivation of nonlinear equations for a Markov process described by a
nonlinear Fokker-Planck equation. The formalism leads to the fluctuation renormalization of the evolution equations for the averages and the correlations. The general equations are applied to a simple model. In a well-defined approximation and for small fluctuations the results agree with those derived by means of van Kampen’s Q-expansion and Kubo’sansatz.
I. Introduction The purpose of this paper is to study the effect of fluctuations on nonlinear equations for the averages and the correlations in a system described by a nonlinear Fokker-Planck equation. As is well-known,’ the nonlinearity gives rise to a dynamical coupling between ensemble-averaged properties and their ensemble fluctuations. This coupling leads to a renormalization of the nonlinear transport equations. Several approaches have been introduced to develop decoupling methods in order to obtain renormalized transport equations. Nordholm and Zwanzigl in a very nice paper proposed several approximation methods that seem useful when the system is in the vicinity of an equilibrium state or when a local equilibrium distribution is adequate. Nevertheless, it is not clear
that those approximation schemes correspond to systematic expansions in powers of a well-defined parameter of the system.l Near-equilibrium situations have also been considered by Mori and F ~ j i s a k a using , ~ projection operator techniques. The same problem has been studied by van Kampen? who proposed an asymptotic evaluation method based on his famous Q-expansion, and that essentially corresponds to the application of the central limit theorem. The theory is formulated in a quite general way and can be applied to equilibrium and far from equilibrium situations. Another way of approaching the effect of fluctuations on nonlinear equations is based on Kubo’s an sat^^,^ (2) Rodriguez, R. F.; van Kampen, N. G. Physica 1976, 85A, 374. Mori, H.; Fujisaka, H. Prog. Theor. Phys. 1972, 49, 764. (4) van Kampen, N. G. Adu. Chem. Phys. 1976, 34, 245. (5) Kubo, R. In Synergetics; Haken, H., Ed.; Teubner: Stuttgart, 1973.
(3)
(1) Nordholm, K.
S.J.; Zwanzig, R. J . Star. Phys. 1974, Zl,
143.
0022-3654/89/2093-6957$01 SO10
0 1989 American Chemical Society
