J . Phys. Chem. 1990, 94, 1847-1850 liquid. By contrast, the measurements, in conjunction with other data, yielded high-precision values for all three principal polarizabilities of fluorobenzene and pentafluorobenzene, and for both molecules the two in-plane components were found to be indistinguishable in magnitude. Trends in the mean polarizability, the in-plane and out-of-plane components, and the polarizability anisotropy in the molecules benzene, fluorobenzene, 1,3,5-tri-
1847
fluorobenzene, pentafluorobenzene, and hexafluorobenzene were also considered.
Acknowledgment. A Commonwealth Postgraduate Research Award (to 1.R.G.) and financial support from the Australian Research ~ o ~ n c(to i l G.L.D.R.1 are gratefully acknowledged. Registry No. C&F,
462-06-6; C6HF5, 363-72-4.
Correlation Function Diagnosis of Chaotic Vibrations in HCN Young June Cho, Paul R. Winter, Harold H. Harris,* Department of Chemistry, University of Missouri-St.
Louis, S t . Louis, Missouri 631 21
Eugene D. Fleischmann,t and John E. A d a m Department of Chemistry, University of Missouri-Columbia, (Received: July 3, 1989; In Final Form: September 2, 1989)
Columbia, Missouri 6521 1
Classical trajectory calculations on the potential energy surface of Murrell, Carter, and Halonen have been performed from semiclassical starting conditions between the zero-point energy and dissociation. Diagnosis of the onset of classical chaos has been accomplished in several ways, but especially by using as a criterion the amplitude of the four-mode instantaneous correlation function. The onset of chaos occurs first in the bending mode, at approximately 11 000 cm-I. At the dissociation limit, virtually all trajectories starting from overtones of the bend are chaotic within 41 ps of the start of the trajectory. On the other hand, the corresponding threshold when overtones of the stretching vibrations are initially excited is near 26 000 cm-’, and only a fraction of those trajectories just below dissociation become chaotic.
Introduction There is a continuing interest in our laboratories in the nature of intramolecular energy transfer in small molecules, especially through the application of classical trajectory methods. Since one of us (J.E.A.) earlier successfully applied the spectral intensity method of Noid et al.’ to the computation of rotational-vibrational spectra of models of both an isolated HCl molecule and one adsorbed on argon,2 one of our interests was the application of the method to a polyatomic. The simplest polyatomic is, of course, a linear triatomic. We have chosen to study H C N both because there is available a potential energy surface highly accurate up to the isomerization barrier3 and because there is a wealth of experimental information concerning vibrational f r e q ~ e n c i e s , ~ dipole moments,5 and IR integrated intensities? The apparently irregular/chaotic dynamics of HCN (earlier reported by Lehmann et al.7-9and by Founargiotakis et al.lO)led us further to consider criteria more convenient and less subjective than the spectral intensity method and more easily applicable to large-dimensional problems than the examination of Poincare surfaces of section. We have modified the correlation function approach first applied to a two-mode model of H 2 0 by Muckerman et al.” and applied it to HCN and also examined other correlation-function approaches to the transition from quasiperiodic to stochastic vibration. Calculational Details All of our computations employed the semiempirical potential energy surface (PES) devised by Murrell et aL3 which yields almost exactly the known vibrational frequencies of HCN, HNC, and their isotopic variants. Bacic and Light have computed accurate vibrational quantum levels in this potential, up to the isomerization barrier.I2 (Fleming and Hutchinson13have recently demonstrated the utility of a two-mode model for the high-energy vibrations of the molecule, if the two modes are in “optimal” Present address: Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306-3006. * Author to whom correspondence should be addressed.
0022-3654/90/2094-1847$02.50/0
coordinates in the potential energy surface of B0ts~hwina.l~) Lehmann, Scherer, and Klemperer had used the Murrell surface for classical trajectory calculations but had provided excitation only to the C-H bond, while the C-N bond and the bend were given zero-point vibrational energy. In their earlier ~ o r k , ~ , ~ Lehmann et al. reported that the transition to chaotic motion is rather abrupt, occurring low in the vibrational well at 12 990 cm-’ ( I ) (1) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J . Chem. Phys. 1977, 67, 404. Koszykowski, M. J.; Noid, D. W.; Marcus, R. A. J . Phys. Chem. 1982,86,2113, and references cited therein. Some molecule systems are also described in: Koszykowski, M. L.; Pfeffer, G. A,; Noid, D. W. In Chaotic Phenomena in Astrophysics; Eichhorn, H.; Ed.; Ann. N.Y. Acad. Sci. 1987, 497, 127. (2) Adams, J. E. J . Chem. Phys. 1986, 84, 3589. (3) (a) Carter, S.; Mills, I. M.; Murrell, J. N. J . Mol. Spectrosc. 1980,81, 1 IO. (b) Murrell, J. N.; Carter, S.; Halonen, L. 0.J . Chem. Phys. 1982, 93, 307. (4) Douglas, A. E.; Sharma, D. J . Chem. Phys. 1953,21,448. Rank, D. H.; Skorinko, G.; Eastman, D. P.; Wiggins, T. A. J . Opt. SOC.Am. 1960, 50, 421. Rank, D. H.; Schearer, J. N.; Wiggins, T. A. Phys. Reu. 1954, 94, 575.
Rank, D. H.; Guenther, A. H.; Schearer, J. N.; Wiggins, T. A. J . Opt. SOC. A m . 1957,47, 148. Rank, D. H.; Skorinko, G.; Eastman, D. P.; Wiggins, T. A. J. Mol. Spectrosc. 1960, 4, 518. Dagg, I. R.; Thompson, H. W. Trans. Faraday SOC.1956, 52, 455. Checkland, P. B.; Thompson, H. W. Trans. Faraday SOC.1955, 51, 1. Allen, H. C.; Tidwell, E. D.; Plyler, E. K. J . Chem. Phys. 1956, 25, 302. Maki, A. G.; Blaine, L. R. J. Mol. Spectrosc. 1964, 12, 45. (5) Ebenstein, W. L.; Muenter, J. S. J . Chem. Phys. 1984,80, 89. DeLeon, R. L.; Muenter, J. S. J. Chem. Phys. 1984, 80, 3992. (6) Foley, H. M. Phys. Reo. 1946, 69, 628. Hyde, G. E.; Hornig, D. F. J . Chem. Phys. 1952, 20, 647. Finzi, J.; Wang, J. H. S.; Mastrup, F. N. J . A D D ~Phvs. . 1977.48,2681. Kim, K.: Kina. W. T. J. Chem. Phvs. 1979, 71. 1967. Smith, I. W. M. J. Chem. SOC.,Faraday Trans. 2 1981, 77, 2357. (7) Lehmann, K. K.; Scherer, G. J.; Klemperer, W. J . Chem. Phys. 1982, 76, 644 I . (8) Lehmann, K. K.; Scherer, G. J.; Klemperer, W. J . Chem. Phys. 1982,
77, 2853. (9) Lehmann, K. K.; Scherer, G. J.; Klemperer, W. J . Chem. Phys. 1983, 78, 608. (IO) Founargiotakis, M.; Farantos, S. C.; Tennyson, J. J. Chem. Phys. 1988, 88, 1598. (I 1) Muckerman, J. T.; Noid, D. W.; Child, M. S. J . Chem. Phys. 1983, 78, 3981. (12) Bacic, Z.; Light, J. C. J . Chem. Phys. 1987, 86, 3065. (13) Fleming, P. R.; Hutchinson, J. S . J . Chem. Phys. 1989, 90, 1735. (14) Botschwina, P. J. Chem. SOC.,Faraday Trans. 2 1988, 84, 1263.
0 1990 American Chemical Society
1848 The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
above the ground state. A later report9 of results obtained by using an improved PES indicated that the transition is much higher in the well, beginning at 40462 cm-I. We calculated classical trajectories for a number of different total energies of HCN up to the hydrogen dissociation limit and with different initial energy distributions. The PES employed in the current calculation^^^ is an improved version of one published earlier. refined so that the variationally calculated frequencies better match the observed frequencies, while maintaining good agreement between observed and variationally calculated spectroscopic constants. Initial position coordinates and momenta for these trajectories were obtained by using a method first employed by Chapman and Bunker15 and later by Sloane and Hase,I6 in which vibrational phases are chosen at random and the atoms are displaced along their normal-mode directions; the degenerate bending modes were assigned the same phase so as to eliminate vibrational angular momentum in the initial conditions. As will be seen later, the distribution of the initial energy among coordinates and momenta is at least as important to the trajectory's denouement as is the total energy. Integration of Hamilton's equations of motion was performed using a Runge-Kutta-Gill algorithm for the first five steps and then a (faster) AdamsMoulton fifth-order predictor, sixth-order corrector algorithm for the rest of the trajectory. The trajectories each ran for 40.957 ps unless prematurely interrupted, with an integrator step size of 1.O X 10-4 ps. They typically conserved energy to within 2 parts in IO5, which is the uncertainty in the trajectory of highest energy. The C-H and C-N bond lengths and the angle were computed every 25 integration steps. Results If one computes the time-dependent dipole moment of the molecule during a trajectory and then calculates the intensity of the Fourier transform of that moment, one obtains the classical analogue of the infrared spectrum of the motion.' Given the detailed information provided by a classical trajectory, one can also examine the frequency spectrum of an individual bond or of group vibrations. The criterion for chaos employed by Lehman et involved the examination of the power spectra of coordinate displacements. A quasiperiodic trajectory produces a frequency spectrum consisting of discrete lines which represent the fundamentals, and possibly overtones and combinations, whereas a chaotic trajectory produces a power spectrum having clumps of lines which become increasingly dense as the length of the trajectory increases. For power spectrum calculations, we used a standard CooleyTukey algorithm to calculate fast Fourier transforms, either on autocorrelated displacements or on the displacements themselves. s between calculations of the The time interval of 2.5 X molecular coordinates yields a maximum linear frequency in the power spectrum, u, of 6667 cm-I, while the total length of the trajectories affords a theoretical resolution, Au, of 0.81 cm-'. Figure 1 shows power spectra of the three mode types of HCN, at the energy of the zero-point vibrations. As one would expect, the spectra are very simple, consisting essentially of the frequencies of the normal modes of the m o l e c ~ l e . ' ~But, since the normalmode approximation is exact only with infinitesimal vibrational amplitudes, when the same spectra are plotted on a logarithmic scale, one can see evidence for substantial coupling between the modes, even at the energy of the zero-point vibrations. In HCN, the integrated intensities of the minor features are several orders of magnitude smaller than those of the fundamentals at the zero point, but it has been reported that calculations on LiCNi8 and KCNI9 indicate that those molecules may have very strong coupling even at the energy of the zero-point vibration.
Cho et al.
PT
I
II
7
3-5
1
1
7360
2000 -re:-ei-c,
'cr-
4500
I
Figure 1. Intensities of power spectra in the normal modes of HCN, at the zero-point energy. With linear ordinates (in arbitrary units, on the right side, and the lower curve in each of the graphs), only the fundamentals and a few overtone/combinationsare discernible, but logarithmic intensity axes (arbitrary units, left side) show that there is significant interaction between modes, even at the minimum semiclassical energy.
em-'
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b. E=30,93G c i n - '
i
~
1
L . .
1 ' ~ ' ' " ' ' ' I ~ ' ' ' l ' ' " ~ ' ' ' ' l " ' ' l ' " '
(15) Chapman, S.; Bunker, D. L. J . Chem. Phys. 1975, 62, 2890. (16) Sloane, C. S.; Hase, W. L. J . Chem. Phys. 1977, 66, 1523.
( I 7) The forms of the normal vibrations for HCN are given in: Herzberg, G . Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand Reinhold: New York, 1945; p 174. (18) Farantos, S. C.; Tennyson, J. J . Chem. Phys. 1985, 82, 800. (19) Tennyson, J.; Farantos, S. C. Chem. Phys. Lett. 1984, 109, 160.
53cc
500
1000
1500
2000
2500
3000
3500
1000
1 4500
Frequency ( CIII-') Figure 2. Power spectra in the CN vibration at two energies.
Using subjective evaluation of power spectra, we confirm that HCN does exhibit chaotic motion at higher vibrational energies.
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990 1849
Chaotic Vibrations in H C N But the transition from motion that is clearly quasiperiodic to that which is clearly chaotic takes place over a range of energies, and the energy of the transition depends strongly upon the mode excited. An example of this phenomenon is seen in Figure 2, which displays power spectra in the asymmetric stretch vibration at two energies. While there may be little objection to designating the trajectory at 18 162 cm-' as "regular" or "quasiperiodic", the higher energy spectrum, at 30936 cm-I, is not so complicated as to make its designation as "chaotic" unquestionable. Further, one cannot tell from the spectrum whether the vibration was only slightly perturbed throughout the trajectory or departed dramatically from regular motion, but only during a fraction of the trajectory. We searched for a less subjective criterion for the onset of chaos that would also provide temporal resolution and be applicable to larger systems. The customary cross-correlation approach often used for analysis of mechanical vibrations uses the formula (for discrete functions)20
-20
2 20 0
30
10
40
Time (ps)
Figure 3. Correlation function for a trajectory starting from a semiclassical initial state corresponding to (1 8,0,0).
N
where X i and Yiare amplitudes of displacements at time step i , ( ) denotes the mean, N is temporal length over which the averaged product is integrated, and C(n) is the cross-correlation function. The spectrum of such a function shows the frequencies that the vibrations X and Y have in common, as long as the matching vibrations occur within n steps of one another. Others have used various types of correlation functions to investigate intramolecular energy transfer. Koszykowski et aL2*used a variety of correlation functions to diagnose the onset of chaotic motion in the Henon-Heiles system, and Sumpter and ThompsonZ2used power spectra of normalized auto- and cross-correlation functions (coherency spectra), two modes at a time, to investigate intramolecular energy transfer in several four-atom systems. Muckerman et al." approached the question of a criterion for chaotic vibration by computing the number of peaks in the power spectrum of the instantaneous (n = 0) correlation function of classical trajectories of a two-mode model of water as a function of trajectory length. For approximately normal vibrations, the number of peaks is small and nearly independent of the length of trajectory transformed. For quasiperiodic vibrations (often called "local modes") not well described within "normal" coordinates, the number of peaks in the power spectrum typically is much larger and increases with the length of trajectory which is Fouriertransformed, because the resolution of the spectrum increases. However, the number of peaks in chaotic modes increases much faster with trajectory length than is the case in quasiperiodic modes, because the molecule explores dynamics having different characteristic frequencies in different temporal portions of the trajectory. Muckerman, Noid, and Child used as criterion for chaos that the number of peaks in a long trajectory exceed 170 and that the sum of the number of peaks in its spectrum exceed by more than a factor of 2 the number of peaks in the power spectra of individual "half-trajectories". We tried to apply such criteria to our fully three-dimensional H C N vibrational simulations, but we were not able to find a set of two-or-more-dimensional spectral correlations that reliably distinguished chaotic vibrations from quasiperiodic ones. We found, however, that the amplitude of the instantaneous multiple cross-correlation function, defined as (where X ( t ) is the amplitude of the doubly degenerate bending mode, and Y(t)and Z ( t ) represent the "normal" stretches) changes dramatically (typically by 2 or more orders of magnitude) when a trajectory goes from being quasiperiodic to being chaotic. ~~~
(20) Lichtenberg, A. J.; Lieberman, M. A. Regular and Stochastic Motion; Springer-Verlag: New York, 1983; p 419. (21) Koszykowski, M. L.; Noid, D. W.;Tabor, M.; Marcus, R. A. J . Chem. Phys. 1981, 74, 2530. (22) Sumpter, B. G.; Thompson, D. L. J . Chem. Phys. 1985, 82, 4557. Sumpter, B. G . ;Thompson, D. L. J . Chem. Phys. 1987, 86, 2805.
I
0
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-12u
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Figure 5. As Figure 3, except for (0,0,12).
Figures 3-5 depict correlation functions for specimen trajectories (in each case, at energies just above the threshold in the corresponding manifold for chaotic vibration). It might at first seem unexpected that this so-called "correlation" increases so sharply, at the point in the trajectory when the vibrations explore more of the energetically allowed phase space. At energies well below the threshold for HCN-HNC isomerization, the amplitudes of the bends are much smaller than when "H-(CN) orbiting" motion is possible. At high energies, the amplitude of the bends can dominate the magnitude of the multiple instantaneous correlation function. Examination of individual trajectories led us to discover that it is possible to interpret the vibration at a particular time period as "nearly normal" if the amplitude of the correlation was very small. Local modes result in a correlation function that displays considerable regularity and does not change sign. This behavior is exemplified in Figure 4 between 12 and 23 ps. Chaotic motion results in a correlation function that changes sign often (every few vibrational periods). There were no essential differences between correlation functions when only one of the degenerate
1850 The Journal of Physical Chemistry, Vol. 94, No. 5 , 1990
3G
c
I
2E4
3E4
4E4
5E4
Energy ( c m - ’ )
Figure 6. Fraction of trajectories in each manifold that become chaotic anywhere within the 41 -ps trajectory length. Diamonds represent ensembles at each energy of trajectories within the bending manifold, circles the asymmetric stretch, and triangles the symmetric stretch.
bending modes was included in the correlation, but this conclusion could be different in cases where the phases of degenerate modes were not initially identical (and therefore vibrational angular momentum is nonzero). When we examined individual trajectories as they became chaotic, it became clear that chaotic motion was associated with motion in which the hydrogen atom “orbited” the CN radical, which is consistent with the implication of bending overtones in the onset of chaos. We observed no specimen trajectories in which apparently chaotic motion was not associated with some kind of orbiting motion. Except at energies almost sufficient to dissociate the molecule, and after excitation of bending overtones, our trajectories always displayed an interval of low Cm amplitude. We attribute this to our initial conditions being chosen (above) from nearly “normal” ensembles. This “induction” time was found not to be very energy dependent, suggesting that this transition might be modeled as a passage through a phase space “ b ~ t t l e n e c k ” . ~ ~ Figure 6 displays the results of our search for thresholds for chaotic motion in each of the vibrational manifolds. At least 50 trajectories were computed a t each of the energies which would correspond to harmonic levels of HCN within the three manifolds. The ordinate represents the fraction of those trajectories that became chaotic (within 41 ps). The lines through the data are
Cho et al. fifth-order regression of the data and do not represent a theoretical model. It is clear that the onset of chaos occurs much lower in energy when the molecule is excited in the bending mode (O,n,O) than when either the asymmetric stretch (O,O,n)or the symmetric stretch (n,O,O)is a recipient of equivalent energy. This behavior was presaged by an earlier, full-dimensional study by Liu et al.24 of vibrations of the water molecule, in which it was found that the energy threshold for chaotic motion was much higher when the bending degrees of freedom were eliminated than when they were fully active. The lowest energy at which chaotic motion is observed is 12 500 cm-’ in overtones of the bends, 23 300 cm-’ in the asymmetric stretch, and 28000 cm-’ in the symmetric stretch. The threshold for chaotic vibration in the bends is just above the 12 179-cm-I isomerization barrier for the potential energy surface used in the calculation. Further, all observed trajectories excited from the bend were chaotic just below the dissociation energy, whereas only approximately 85% of those excited from the symmetric stretch were. No more than approximately 20% of trajectories excited from the asymmetric stretch ever became chaotic, regardless of the energy. Summary
We have found that, for the molecule HCN, instantaneous correlation of the displacements in all the four molecule modes provides a convenient, relatively objective, and computationally efficient means of diagnosing for classically chaotic vibrations. The method seems likely to be generalizable to arbitrarily large molecules, unlike most of the methods previously reported in the literature. Excitation of the three different types of HCN vibrational modes shows that chaos is induced at lower energies when the bending modes are excited than when equivalent amounts of energy are supplied to either of the bends. Within the stretching manifolds, chaotic motion does not necessarily appear within 40 ps, even at energies almost sufficient to dissociate the molecule.
Acknowledgment. This work was supported in part by a grant from the Weldon Spring Research Fund of the University of Missouri and by the Research Experiences for Undergraduates Program of the National Science Foundation, which supported P.R.W. during the summer of 1988. (24) Liu. W.-K.: Noid. D. W.: Koszvkowski. M. L. In Intramolecular Ed;; D. Rdidel: Dordrecht, Holland, 1982; pp 191-203.
Dynamics; Jortner, J., Pullman, B.,
(23) Davis, M. J . J . Chem. Phys. 1985, 83, 1016.
