3038
J. Phys. Chem. 1983, 87, 3038-3042
Method To Determine the Number of Constants of the Motion for Multidimensional Systems J. R. Stlne” Los Akmos National Laboratory, Los Alamos, New Mexico 87545
and D. W. Nold” Chemistry Division, Oak Ridge National Laboratory, Oak Ridge. Tennessee 37830 and Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37916 (Received: December 3, 1982; In Final Form: January 25, 1983)
A simple numerical method is presented to determine the number of constants of the motion (integrals) for multidimensional nonseparable Hamiltonians with n degrees of freedom. This method is based on the fact that a trajectory with k constants of the motion will be confined to a 2n - k dimensional surface of phase space. The method is applied to model systems of two and three degrees of freedom.
I. Introduction The problem of energy flow in polyatomic systems is exceedingly complex. As is well-known, complex motions can occur even in model systems of two degrees of freedom. This complexity has caused most of the effort to date to be devoted to understanding model systems of two degrees of freedom such as the Henon-Heiles,l Barbanis,2 and Thiele-Wilson3 Hamiltonians. These Hamiltonian systems can display two types of classical motions-stochastic and quasiperiodic. Various methods have been developed to distinguish between these two types of motions; the construction of the Poincare surface-of-sectionhas proved particularly useful. Here the surface-of-sectionappears as a random scattering of points for the stochastic trajectory and a well-ordered locus of points for the quasiperiodic trajectory. Another useful method for distinguishing between the two types of motions is the calculation of the power spectrum. Here a stochastic trajectory results in a spectrum with many broad lines whereas the quasiperiodic trajectory gives a spectrum with a few sharp lines. Another method takes advantage of the fact that two initially very close trajectories separate a t an exponential rate if stochastic and a t a linear rate if quasiperiodic. Based on this phenomenon a “Kolmogorov entropy” can be defined where the entropy is nonzero for the stochastic case and zero for the quasiperiodic case. Descriptions of these methods and the types of systems to which they have been applied are given in some recent review^.^-^ These methods have been applied primarily to systems with two degrees of freedom, for which the trajectory is either totally quasiperiodic or totally stochastic. We believe that a polyatomic system will have n constants of the motion (where n is the number of degrees of freedom) at low energies and only one constant of the motion (namely, conservation of energy) a t high energies. The number of constants of the motion will be between one and n at intermediate energies. M. Henon and C. Heiles, Astron. J., 69, 73 (1964). B. Barbanis, Astron. J., 71, 415 (1965). E. Thiele and D. J. Wilson, J. Chem. Phys., 35, 1256 (1961). D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Annu. Reu. Phys. Chem., 32, 267 (1981). ( 5 ) M. Tabor, Ado. Chem. Phys., 46, 73 (1981). (6) S. A. Rice, Adu. Chem. Phys., 47, 117 (1981). (7) P. Brumer, Adu. Chem. Phys., 47, 201 (1981). (1) (2) (3) (4)
0022-3654/83/2087-3038$0 1.50/0
TABLE I: Expected Results for Various Methods for Determining t h e Number of Constants of t h e Motion for Systems with Three Degrees of Freedom spectrum Poincare surface-ofsection separation of adjacent trajectories Kolmogorov entropy dimensionality plot
k= 1
k = 2
Iz = 3
many broad lines scatter of points
m a n y broad lines scatter of points
few sharp lines orderly locus of points
exponential
exponential
linear
K
K#O
K=O
slope = 4
slope = 3
#
0
slope = 5
Although Poincare surfaces-of-section can be constructed for some systems with three degrees of freedom, the resulting section for a quasiperiodic trajectory, for example, will have a slight scatter of points, i.e., a “width”.8 In special cases, Poincare surfaces-of-section for multidimensional systems can be constructed with zero width, but the method becomes very difficult for higher dimension^.^ The concept of a Kolmogorov entropy has also been used to qualitatively determine the number of integrals of the motion for systems with three degrees of freedom.l0 In general, it appears that the above-mentioned methods for distinguishingbetween stochastic and quasiperiodicmotion cannot be used to quantitatively determine the number of constants of the motion at intermediate energies where some of the degrees of freedom will be stochastic while the rest are quasiperiodic. This is demonstrated schematically in Table I, where it is expected that, at best, only semiquantitative information can be obtained. Because the existing methods could not be applied to polyatomic systems, we developed an approach based on the trajectory being restricted to a lower dimensional surface in phase space depending on the number of constants of the motion. For example, for a system with three degrees of freedom, a trajectory with three constants of the motion is confined to a surface of lower dimensionality (8) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J . Chem. Phys., 73, 391 (1980). (9) J. R. Stine and D. W. Noid, J . Phys. Chem., 87, 3733 (1982). (10) G. Contopoulos, L. Galgani, and A. Giorgilli, Phys. Reu. A , 18, 1183 (1978).
62 1983 American Chemical Society
Constants of Motion for Multidimensional Systems
3039
The Journal of Physical Chemistty, Vol. 87, No. 16, 1983
than one with two constants of the motion; both would be confined to a surface of lower dimensionality than a trajectory for which conservation of energy is the only constant of the motion. The details of how the dimensionality of this surface is determined for various model systems of two and three degrees of freedom are given in the next section. The method is summarized and some additional applications and cautions are discussed in section 111. 11. Method and Applications If we consider a Hamiltonian system with two degrees of freedom and hence a four-dimensional phase space, we would expect a stochastic trajectory (a trajectory with only conservation of energy as a constant of the motion) to be confined to a three-dimensional surface. For model systems with two degrees of freedom, a quasiperiodic trajectory is restricted to travel on the surface of a torus, a surface of two dimensions in phase space. In this paper we exploit this difference in dimensionality of the surface in the phase space on which trajectories with different numbers of constants of the motion are confined. Now the actual trajectory is represented by a point moving in phase space and hence has unit dimensionality (a line). However, as this trajectory propagates, the surface to which it is restricted begins to take shape and eventually becomes well-defined. In general, we expect a trajectory with n degrees of freedom moving in a 2n-dimensional phase space to be confined to a surface of dimension 2n - k, where k is the number of constants of the motion. Determination of the dimensionality of this surface is based on dividing phase space into 2n-dimensional “hypercubes” or bins and noting those bins accessed by the trajectory. If the bin size is changed for this same trajectory, then the number of bins accessed should also change by an amount that depends on the dimensionality of the surface. If N represents the number of bins in each direction and Nmiq the number of unique bins accessed by the trajectory for a particular bin size, then
0
-I
-2
-3
I
3
2
The quantity Nunisrepresents the number of bins in phase space that have been accessed at least once and thus, for a finite bin size, this quantity will eventually approach a constant value. Thus, if a particular trajectory is binned according to various bin sizes, we expect In Nuniq = (2n - k) In N constant (2)
+
such that a “dimensionality” plot of In NUiqas a function of In N should yield a straight line with slope 2n - k. Alternatively, if Nunisis divided by the total number of bins in phase space, W”,then a plot could be made where the slope of the line would be k. It is expected that the trajectory will travel on a “well-behaved”multidimensional surface in phase space and hence k represents the Euclidean dimensionality and is an integer. We will call this integer value of k the “theoretical” dimensionality. Section 111contains a further discussion of this point. This procedure will be demonstrated first for a model Hamiltonian with two degrees of freedom. The HenonHeiles system was selected because its dynamics have been well characterized. The Hamiltonian for this system is HHH
=
f/203i2
+ pz2 + 4i2 + ~ z ’ ) + Xqi(q22 - Y&i2)
(3)
with X = 0.1118. Three typical trajectories have been selected for analysis: the first two can be shown to be quasiperiodic and have been described as “librating” and
1
1
i
1
,
~
1
2
,
1
1
0
,
1
1
(1
2
,
i
,
i
(
(
1
i
b
7
H
1
Figure 1. Plots of three typical trajectories for the Henon-Heiles Hamiltonian: (a) quasiperiodic “librating”, (b) quasiperiodic “precessing”,and (c)stochastic type trajectories.
“precessing” trajectories, respectively; the third trajectory can be shown to be stochastic.* These trajectories have initial parameters given in Table I, where q1 = qz = 0, p1 = ( ~ E H H ~ X )and ~ ’ ~p, 2 = ( 2 E H H ( 1 - fX)j1/’, and are shown in Figure 1. Hamilton’s equations
P1
=
Pz
41
= P1
42
= Pz
-Q1
=
-Q2
- u q 2 2 - 412) - X(2QlQ2)
(4)
are integrated and the coordinates and momenta are stored
3040
The Journal of Physical Chemistry, Vol. 87, No. 16, 1983
Stine and Noid
11.0 10.5
.-V
5
9.5
c
9.0
z
3
1200
10.0
8.5
-
8.0
-
7.5 lOG0 -______ 7
I
I
i 00
100
i‘J0
100
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
500
\I ~ T h o ~ ~ i r ~ d - ) Figure 2. Plot of the number of unique bins in the phase space accessed for the trajectory shown in Figure l a after M time increments. Here the number of bins per direction equals 20.
after a time increment At. The maximum and minimum are determined for each of the coordinates and momenta during integration of this trajectory. Now each of the four dimensions of phase space is divided into N bins where the extent of available phase space is determined by the maxima and minima of the coordinates and momenta found. Thus, the phase space under consideration contains N4bins. A unique integer, L, was assigned to each of these bins by
L = 14P+ I J P
+ 12N + I ,
-
9.5
.-0C. 9.0
2
W
c
3
8.5
8.0
(5)
I
I
where Iirefers to the number of the bin in the i-th direction where 0 I Ii I N - 1 and i = 1, 2, 3, 4. At each time step along the trajectory, the immediate bin number is calculated and the number of unique bins is determined by sorting all the bin numbers generated during the course of the trajectory and then eliminating any duplicates. For the present cases, At has arbitrarily been taken to be 0.1. Figure 2 shows how the number of unique bins converges as a function of the number of time steps for N = 20. In the early stages the trajectory will access new bins before returning to the original bin. However, in the later stages it becomes increasingly difficult for the trajectory to find bins that it has not accessed before; therefore, a relatively large number of time steps should be generated to assure convergence of the number of unique bins. We found 5 X lo5 time steps to be sufficient. Figure 3a shows a “dimensionality” plot of In N W i ,vs. In N for the trajectory shown in Figure la. A linear least-squares analysis of the data yields a slope of 2.10, which agrees favorably with the theoretical value of 2 for this trajectory that is known to have two constants of the motion. Figure 3b shows a similar dimensionality plot for the trajectory shown in Figure lb. This quasiperiodic trajectory is qualitatively different from that shown in Figure la, but it too possesses two constants of the motion. This is also indicated by the dimensionality plot shown in Figure 3b, where the least-squares line has a slope of 1.99, in close agreement with the theoretical value of 2. The third trajectory has only one constant of the motion (conservation of energy), and the dimensionality plot should produce a line with a theoretical slope of 3. This dimensionalityplot is shown in Figure 3c for this stochastic trajectory; the calculated slope of 2.87 agrees quite well with the theoretical value.
I
I
I
I
I
I
I
1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6
10.0
j
C
9.5 n
.-0C. 9.0
c
8.5
M
8.0 7.5
1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3
WN) Flgure 3. Dimensionality plots corresponding to the three trajectories shown in Figure la-c. The straight lines have the theoretical slopes of 2, 2, and 3 for plots a-c, respectively.
We now describe a variation of the above method that makes use of the conservation of energy restraint. As with any time-independent system, the Henon-Heiles system is a conservative Hamiltonian. Therefore, the energy is always a constant of the motion and not all phase-space variables are independent. For example, we can write where V(q,, q2)is the potential energy. This implies that only a three-dimensional space (e.g., q l , q2, and pl) need actually be binned. Dimensionalityplots can still be made
The Journal of Physical Chemistry, Vol. 87, No. 16, 1983
Constants of Motion for Multidimensional Systems
3041
8.0 7.6 h
n
.o
.-P
7.2
G
a
z
z
v
6.8
v
C 3
C
d
6.4 6.0 5.6 I
I
I
I
I
I
I
I
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
W) 8.81
1
b
: I //
8.4n
.-P G
8.0-
z
7.6 7.2-
C
7.2-
-
v
6.86.4I
I
I
2.5
3.0
3.5
I
2.0
1
4.0
I
I
I
I
I
I
I
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
MN)
-
9.5 n
.o 9.0-
-i C
-
8.5
3
2.0
2.2
2.4
2.6
2.8
3.0
0
Flgure 4. Reduced dimensionality plots corresponding to the three trajectories shown in Figure la-c. The straight lines have the theoretical slopes of 2, 2, and 3 for plots a-c, respectively.
Flgure 5. Reduced dimensionality plots corresponding to the three trajsctories described in the text. The straight lines have the theoretical slopes of 3, 3, and 4 for plots a-c, respectively.
where now Nmiq is the number of unique bins accessed by the trajectory in the reduced dimensional space. The theoretical slope of the line should still be 2n - k. Figure 4a-c shows these "reduced dimensionality plots" for the three trajectories shown in Figure la-c. Here again, the linear least-squares slopes of 2.11,2.05, and 2.74 agree well with the theoretical values of 2,2, and 3, respectively. This latter method leads to a slightly more efficient analysis as fewer points need be binned and sorted as the convergence is slightly accelerated. However, for systems with many degrees of freedom this savings becomes less significant.
We now test this method on a model system of three degrees of freedom (n = 3). So that the number of constants of the motion remains known, we consider where the total energy E = E H H + EHO. This system consists of the Henon-Heiles Hamiltonian and an uncoupled (separable) harmonic oscillator; it can now have two or three constants of the motion depending on whether the Henon-Heiles portion is stochastic or quasiperiodic, respectively.
J. Phys. Chem. 1983, 87, 3042-3048
3042
TABLE 11: Initial Parameters f o r t h e Three Shown in Figure 1 f, = 0 . 3 4 7 EHH= 4 . 3 3 1 trajectory a E,, = 7.000 f, = 0.650 trajectory b f x = 0.347 E,, = 12.50 trajectory c
Trajectories
E H O= 0.669 EHO= 3 . 0 0 0 E H O = 2.500
The initial conditions for the Henon-Heiles portion of eq 7 are the same as given in Table 11. The harmonic oscillator portion of eq 7 has energies, E H O , as given in Table I1 for trajectories a-c. For the reduced dimensionality plots only the harmonic oscillator coordinate (q3)need be calculated. It is given by
t (8) These reduced dimensionality plots for the three trajectories are shown in Figure 5. These lines should have theoretical slopes of 3 for the two totally quasiperiodic trajectories and 4 for the mixed stochastic-quasiperiodic trajectory. The least-squares slopes for these data are 3.05, 2.98, and 3.82, respectively. q 3 = (2EHo)’i’
COS
111. Discussion and Summary We have presented a numerical method to determine the number of constants of the motion for Hamiltonian systems. The method has been demonstrated for systems with two and three degrees of freedom but can, in principle, be easily extended to systems with larger numbers of degrees of freedom. The method determines the number of degrees of freedom from the dimensionality of the surface in phase space on which the trajectory moves. This number is determined from the slope of the line produced in a dimensionality plot. Because this is a numerical procedure, some consideration has to be given to the bin size and to the number of time steps needed to establish convergence of the number of unique bins through which the trajectory travels. If the number of bins is large, then the trajectory will not have the opportunity to access all of the bins that may be available. Because the trajectory has a dimen-
sionality of unity, a slope of unity for the line in a dimensionality plot would result if an insufficient number of time steps were generated. On the other hand, if the number of bins per direction were too small, then all the bins in phase space could be accessed and any trajectory would appear to be ergodic. These are extreme cases, and in fact a considerable range of bin sizes are available such that a valid slope in the dimensionality plot can be obtained. We have assumed here that the dimensionality plot should yield an integer value for k, Le., the Euclidean or topological dimensionality. Indeed, within the error limits of the calculations, the least-squares slopes agree with the theoretical values. It is known, however, that noninteger values for dimensionalities are possible for various physical systems. Examples are given in the theory of “fractals” as discussed in the very readable book by Mandelbrot.” In fact, the value of 2n - k calculated here represents the Hausdorff-Besicovitch dimensionality. If the trajectory travels on a multidimensional surface that is not “wellbehaved”, the dimensionality of this surface (2n - k) would not necessarily be an integer but would be greater than the topological dimension. Certain mappings exist for which fractional dimensionalities are indeed the case. However, for dynamical systems, because of their complexity, no such results can be proven. It is known that intricate phase-space structures occur in simple model Hamiltonians and indeed these structures may lead to noninteger dimensionalities. We are extending this method by applying it to realistic triatomic systems. Acknowledgment. This research was sponsored by the U.S. Department of Energy under contract W-7405-eng-36 with the University of California and under contract W-7405-eng-26 with the Union Carbide Corp. (11) B. B. Mandelbrot, ‘Fractals-Form, H. Freeman, San Francisco, CA, 1977.
Chance, and, Dimension”,W.
Stereochemistry of the Four-Centered Gas-Phase Dehydrohalogenation Reaction Steven D. Palsky+ and Bert E. Holmes’ Department of Chemistry, Ohio Northern Universi?v, Ada, Ohio 45810 (Received December 7, 1982)
The stereochemistry of the gas-phase four-centered dehydrohalogenation reaction has been determined. Deuterium analysis of the 2-butene isomers formed by the thermal decomposition of threo- and erythro-2bromo-3-deuteriobutae proves that elimination is principally via a syn transition state; i.e., the hydrogen and halogen are removed from the same side of the carbon-carbon bond. The results show that the contribution of anti stereochemical transition state complexes is small, perhaps zero. Primary and secondary deuterium kinetic isotope effects were 2.11 f 0.31 and 0.98 f 0.12 respectively for erythro-2-bromo-3-deuteriobutane, and 2.07 0.24 and 1.02 0.13 respectively for threo-2-bromo-3-deuteriobutane at 590 K.
*
*
Introduction One of the prototype gas-phase unimolecular processes is the a$-dehydrohalogenation Recent interest has centered on energy disposal,3-’ laser activation by absorption of multiple infrared photons,G6 surprisal analysis of the hydrogen halide vibrational state distriCurrent address: Chemistry Department, University of Wisconsin, Madison, WI. *Permanent address: W. C. Brown, Sr. Chair of Chemistry, Arkansas College, Batesville, AR 72501. 0022-3654~03l2087-3042$01.5010
bution: and the molecular reorganizationsat the transition ~tate.~!*’l In solution the nature of the transition state (1) D. W. Setaer in ‘International Review of Science. Physical Chemistry: Reaction Kinetics”, J. C. Polanyi, Ed.,Butterworths, London, 1972, Series 1, Vol. 9, Chapter 1. (2) A. Maccoll, Chem. Reu., 69,33 (1969). (3) B. E. Holmes and D. W. Setser in ‘Physical Chemistry of Fast Reactions”, Vol. 2, I. W. M. Smith, Ed., Plenum Press, New York/London, 1980, Chapter 2. (4) J. C. Stephenson, D. S. King, M. F. Goodman, and J. Stone, J. Chem. Phys., 70,4496 (1979).
0 1983 American Chemical Society
