Dr. Karl Sohlberg
Chemists frequently study the vibrational motions of a polyatomic molecule by couching the problem in terms of a set of classical coupled oscillators. See, for example, the classic text by Wilson, Decius, and Cross[1]. The representation of a polyatomic molecule as a set of classical coupled oscillators has found use in the assignment of vibrational modes to spectral features[1,2], semiclassical quantization of vibrational energy levels[3], and in the prediction of unimolecular decay rates[4]. Herein we will demonstrate a novel method of visualizing the topology of the motion of one of these coupled oscillator systems.
First consider a simple one dimensional classical oscillator. A model familiar to chemists is the Morse oscillator[5]. The classical Hamiltonian we will consider is:
Here x is the displacement coordinate and px is the
associated momentum coordinate. The kinetic (T ) andpotential (V ) energy terms are,
and
Figure 1.
respectively. This system is bound for all values of the total energy E such that -1.0 < E < 0.0. Suppose we make a phase-space plot of the motion of this system when the total energy is E = -0.1. Such a plot is shown in Fig. 1. Here the horizontal axis represents the value of x and the vertical axis represents the value of px. Note that the trajectory forms a simple closed curve in this, the (x, px) plane. At first glance this may be surprising. The motion of the system is confined to a one-dimensional structure in a two-dimensional space. Why is this so? The reason for this is that there is a conserved quantity in the system, the total energy. Since the total energy is a constant, only certain combinations of x and px are allowed. In Fig. 2 we show what happens to the phase space trajectory in the event that the total energy is not conserved. (In this case the system is gradually losing energy.) Note that in this case the trajectory is no longer confined to a simple closed curve
In a system of N coupled oscillators, phase space
has 2N dimensions. For each constant of the motion
the dimension of the set of phase space points which
the trajectory may sample is reduced by one[6].
If there are N constants of motion for a system of N
coupled oscillators, the system can be separated into
a superposition of N one-dimensional oscillators.
The motion of such a system is said to be "regular"
and such systems are particularly easy to analyze.
One identifying feature of such systems is that a
trajectory will be confined to an N-dimensional
surface embedded in a 2N-dimensional phase space.
The topology of this N dimensional object is that of a
torus and it is referred to as an "invariant torus."
Figure 2.
For some systems it is easy to see that the motion will be regular. Consider for example the system of two uncoupled linear oscillators governed by the Hamiltonian,
Since there are no terms coupling the x motion to the
y motion, it is clear that this can be rewritten as,
where,
and
Figure 3.
Clearly there are two constants of the motion, Hx and Hy, which are the total energies in the x and y oscillators respectively. A trajectory of this system will be confined to a two-dimensional surface in the four-dimensional phase space. The motion will be regular.
For other systems, there is no such obvious separation. One system, which has been rigorously shown to be nonseparable, is the Henon-Heiles Hamiltonian[7],
where b is a constant frequently taken to be about 0.11. Still, numerical experiments have determined that for certain energies, regular trajectories can be found[8]. If the trajectories are regular, they should be confined to a two-dimensional invariant torus in the four-dimensional phase space. Heretofore the most typical method of depicting this has been by the use of a surface of section plot. A trajectory is run and the values of y and py are tabulated every time the trajectory passes through the x=0 plane. This set of points (y, py) is then plotted. Such a plot is shown in Fig. 3. Note that there are two simple closed curves. One represents the set of points where the trajectory passed through the x=0 plane with positive momentum and the other represents the set of points where the trajectory passed through the x=0 plane with negative momentum. These curves are cross sections through the invariant torus.
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We have devised a more elaborate graphical representation of the invariant torus which comes closer to depicting its full dimensionality. Points (x, px, y, py) are generated on the invariant torus and saved as vectors in a 4-space whose axes we label x, y, z, and w. These points are then projected into a three dimensional space with axes x', y', and z'. Next the resulting three dimensional figure is displayed in two dimensions using perspective. Such a procedure gives a single view of the invariant torus. See, Fig. 4. (See top. Also discussed in reference[9].) To give a more complete view of the invariant torus, we have created a "movie." This movie is composed of 72 frames. The torus is rotated in the z-w plane, about the x-y plane in 5 degree increments. The 72 frames then represent a complete 360 degree rotation of the object. (Note that in a plane, rotations take place about a point, in 3 dimensions rotations take place about a line, and in 4 dimensions rotations are about a plane. This is why we say that the rotation is in the z-w plane about the x-y plane.) Play movie 1 --> and observe the invariant torus as it is rotated in 4-space. The checkerboard background is used to improve perspective and is not part of the object.
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Note that the torus intersects itself. On physical grounds, no intersections are allowed. The observed intersection is an artifact of the observer's viewpoint. To illustrate this point we have produced a small demo. A simple closed curve in three space may appear to have intersections when projected into a plane. Consider the curve defined by, x = cos(theta), y = sin(theta), and z = cos(2*theta) when theta is varied from 0 to 360 degrees. This is a circle in the x-y plane but oscillates in z back and forth across the x-y plane. For our demo, we have made another movie constructed from 72 individual frames. Each frame shows the curve projected into a plane but the viewpoint is rotated in 5 degree increments. Over the course of the 72 frames, the viewpoint moves all the way around the object. Play movie 2 --> and note that from some viewpoints, the curve appears to intersects itself but from others is appears to be a simple closed curve.
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We think that this new animated representation, used here to depict an invariant torus, is useful for visualizing objects of dimensionality greater than three.
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References cited:
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[1] E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, "Molecular Vibrations", (Dover, New York, 1980, republished from: McGraw-Hill, 1955).
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[2] Karl Sohlberg, Brian Baker, Scott Leary, Noel L. Owen, Julio Facelli, and Boris Trofimov, J. Molec. Struct., Vol. 354, 55 (1995).
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[3] T. J. Pickett and R. B. Shirts, J. Chem. Phys., Vol. 94, 6036 (1991).
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[4] W. Forst, Theory of Unimolecular Reactions, (Academic Press, New York, 1973).
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[5] P. M. Morse, Phys. Rev., Vol. 34, 57 (1929).
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[6] R. T. Swimm and J. B. Delos, J. Chem. Phys., Vol. 71, 1706 (1979).
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[7] M. Henon and C. Heiles, Astron. J., Vol. 69, 73 (1964).
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[8] F. G. Gustavson, Astron. J., Vol. 71, 670 (1966).
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[9] Karl Sohlberg and Randall B. Shirts, J. Chem. Phys., Vol. 101, 7763 (1994).