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J. Phys. Chem. 1982, 86, 2118-2124
Criterla for Quantum Chaos Eric J. Heller Los Aiamos National Laboratory and Department of Ctmmistty, Universiiy of California at Los Angeks, Los Angeks, California 90024
and Mlchael J. Davls Department of Chemistry, The Universiiy of Texas, Austin, Texas 78712 (Received:July 31, 1081)
We review some of the opinions on quantum chaos put forth at the 1981 American Conference on Theoretical Chemistry and present evidence for our own point of view.
Introduction The 1981 American Conference on Theoretical Chemistry saw a great deal of discussion about classical chaos and the appropriate means of defining chaos quantum mechanically. Naturally, the degree of correspondence between classical and quantum onset and extent of chaos differs markedly according to the definition adopted for quantum chaos. At one extreme, a quantum generalization of the classical Kolmolgorov entropy as discussed by Rice’ gives zero entropy for quantum systems with a discrete spectrum, regardless of the classical properties. At the other, the quantum phase space definition adopted by our g r o ~ p (and ~ - ~a related one adopted by Kaf) shows generally excellent correspondence to the accepted classical phase space measures. The exceptions, as discussed in ref 2-5 and herein, are the fully expected effects of (1) quantum interference, (2) quantum tunneling, and (3) quantum smoothing. (See below.) To continue with the differences of opinion, there is the question of whether the spectrum of energy levels (or its variation with some parameter of the Hamiltonian) is enough to characterize the quantum chaos (or lack of it), or whether more information is needed (Le., eigenfunctions). For example, Marcus and co-workers’ have put forth a criterion based on level crossings as a function of a parameter. PercivaP and Pomphrepb have discussed a related property, namely sensitivity of eigenvalues to changes in an anharmonicity parameter. Be# has shown that chaotic dynamics leads to the Wigner surmise seT8‘/* dependence for the probability of finding an energy level a distance s away from a given energy level, but he also shows that one cannot go backwards from the distribution of energy levels to definite conclusions about the dynamics. HoltloJ1 has given arguments to show that any quantum spectrum of eigenvalues can be matched with a Hamilto(1)S.A. Rice, this issue; R. Kosloff and S. A. Rice, J. Chem. Phys., 74. 1340 (19811. .~ ,--.. (2) E. J. Heller, J . Chem. Phys.,72, 1337 (1980). (3)M. J. Davis, E. B. Stechel, and E. J. Heller, Chem. Phys. Lett., 76, 21 (1980). .~...,. (4)M. J. Davis and E. J. Heller, to be published. (5)E. J. Heller, Chem. Phys. Lett., 60, 338 (1979). (6)K. G.Kay, J. Chem. Phys., 72,5955 (1980). (7)R. A. Marcus, this issue; R. Ramaswamy and R. A. Marcus, J. Chem. Phys., 74,1379(1981);D. W. Noid, M. L. Koszykowski, M. Tabor, and R. A. Marcus, J. Chem. Phys., 72,6169(1980). (8)(a) I. C. Percival, J.Phys. A, 7,794(1974);(b) N. Pomphrey, J. Phys. B., 7,1909 (1974); (c) I. C. Percival and N. Pomphrey, ibid., 31, 97 (1976); (d) I. C.Percival, Adu. Chem. Phys., 36,1 (1977). (9)M. V. Berry, this issue; see also M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros, Ann. Phys. (N.Y.),122,26(1979);M. V. Berry and London, Ser. A, 349,101 (1976). M. Tabor, R o c . R. SOC. (10)C. Holt, Ph.D. Thesis, University of Colorado, 1981. (11)W. P. Reinhardt, this issue; C. Jaffe and W. P. Reinhardt, to be published; R. Shirts and W. P. Reinhardt, to be published. ~
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0022-365418212086-2118$01.25/0
nian that is totally integrable. We conclude that one does not want to rely upon eigenvalues alone to characterize the degree of chaos in the quantum dynamics. Still, the disparity of opinion is perhaps not as wide as might be implied by the above summary of the situation. For example, there is no disagreement, as far as we are aware, on the facts seen in Figure l.3 If we place a wavepacket (coherent state), which is well localized in both position and momentum, in a quasiperiodic regime in phase space, we get a sparse spectrum of Franck-Condon factors (wavepacket 1shown on the Henon-Heiles surface of section at an energy of 10.0). Wavepacket 2, placed in the (coexisting) chaotic regime, gives a much denser spectrum. (This observation generates our own definition of quantum chaos, discussed in ref 2 and below.) Huchinson and Wyatt12have seen excellent agreement between classical and quantum (Wigner) surfaces of section. Pechukas13has observed that other, nonzero quantum generalizations of the Kolmolgorov entropy are possible. How can agreement on the facts mentioned coexist with such widely differing measures of quantum chaos? The reason, it seems, is that there is not yet a consensus on physical relevance of the measure of quantum chaos. Some workers adopt a more mathematical point of view, and this is understandable because the field is fraught with mathematical fine points and careful theorems. Others have a more “chemical” viewpoint and are willing to pay the price of reduced mathematical rigor in the hope of finding criteria with measurable consequences. This is not to say that rigor and relevance are incompatible, but that the two so far have not been successfully married in this field. The motivation for our point of view stems from the classical idea of flow in phase space and happens to result in observable consequences for molecular electronic spectra, both in absorption (emmission) and in Raman scattering. We will review the quantitative aspects in the next section, but first let us see how phase space flow can result in quantum spectral consequences. In classical mechanics, all regions in phase space consistent with the known constants of the motion are “fair game” for a trajectory (or a continuous classical phase space distribution) to sample in the course of dynamical evolution. One often defines a localized initial distribution (e.g., energy in one bond of a molecule) and time evolves the distribution to see where it goes. If it eventually evenly covers all regions that are allowed by the known constants (12)J. S. Hutchinson and R. E. Wyatt, Chem. Phys. Lett., 72,378 (1980);Phys. Reu. A , 23, 1567 (1981). (13)P. Pechukas, this issue.
0 1982 American Chemical Society
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Criteria for Quantum Chaos
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The Journal of Physical Chemistry, Vol. 86, No. 72, 1982 2119
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Flgure 2.
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54th eigenfunction of the potential V = '/g2 0 . 1 3 with ~ ~ two ~ Gaussian packets.
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(l.818181)2y2
which has all the same expectation values and dispersion as g,. (pa is the Wigner transform14 of g,.) From eq 1we
see that both the position and momentum of the Gaussian are under our control, and this gives us the flexibility to examine the phase space disposition of eigenstates. In the example of Figure 2, we used zero momentum packets and looked for probability near the classical turning points. This assumes that the average energy of the packet is close to the energy of the eigenfunction. (It would not be fair to expect a state of E = 10 to overlap a packet with ( E ) = 50.) More generally, we look in phase space, varying the position and momentum. The lesson we learn from Figure 2 is that very regular, nonchaotic eigenfunctions are localized to subregions of phase space, and that this shows up as a large variation in the overlap with the test Gaussian packet, depending on the packet's phase space location. Exactly the same holds true in classical mechanics, when wave functions would be replaced by trajectories lying on localized tori and the test state would be the pa density of eq 2. We can actually produce such test Gaussian packets in electronic absorption and emi~si0n.l~The wave packet is that of the ground vibrational wave function which finds itself placed upon a new potential surface after the absorption (or emission) of a photon. The overlap we have been discussing are then Franck-Condon factors (FCFs). If we fix g, and look at the variation of the FCF's as the eigenstate changes, we expect wild fluctuations in the FCF's, depending on whether the eigenstate happens to avoid the domain of the wavepacket test state. At least this will be the case in the regular regime, where the eigenstates are confined, as in Figure 2. In fact, as the number of coordinates increases, the quasiperiodic tori on which the wave functions are localized occupy a tiny fraction of the available phase space. The immediate conclusion is that, for three or more degrees of freedom, the vast majority of FCFs with a localized g, are essentially zero in the quasiperiodic domain. In the fully chaotic regime, it may be that the wave functions cover all phase space domains consistent with their total energy. This would imply that the FCF would vary much less as the wavepacket is moved around in phase space at constant average energy. Or, a given Gaussian would experience smoothly varying FCF's if all the eigenfunctions in the same energy domain were fully chaotic; none could "hide" from the Gaussian and fail to overlap it. Thus we conclude the following:
(14)See, e.g., E.J. Heller, J. Chem. Phys., 67 3339 (19771,and references therein.
(15) E. J. Heller, J. Chem. Phys., 68,2066,3891(1978);K.C.Kulander and E. J. Heller, ibid., 69, 2439 (1979).
ENERGY
Flgure 1. Wavepacket 1 , in quasiperiodic region, leads to sparse spectrum (middle) with many "missing" lines. Wavepacket 2, in chaotic region, gives denser spectrum but still is not fully chaotic due to the coexistence at this energy of the quasiperiodic domain. See ref 3 for details.
of the motion, then no other constants exist and we say the motion is chaotic or stochastic. If it avoids certain regions, then the motion is less than fully stochastic, and new constants of the motion restrict the flow. The motion may then be quasiperiodic or perhaps stochastic only in a subdomain of phase space, as in the "scattered" region in the surface of section of Figure 1. Consider Figure 2. I t shows a wave function for a Hamiltonian with a 1:2 resonance. The wave function is severely distorted from the form it would have if the coupling were reduced but it is still highly organized and covers only a fraction of the available coordinate space (shown by the interior to the contour line). At the top of Figure 2, a black area shows a Gaussian wavepacket (coherent state) placed at rest at the location shown. Clearly, the overlap of the eigenfunction and the Gaussian is essentially zero. Below, with the Gaussian placed on the wave function, the overlap is very large. Clearly, the Gaussian can act as a probe for the location of the wave function. In one dimension such a Gaussian has the form
This is a minimum uncertainty state with ( x ) = xa, ( p ) = p a , Ax = h'J2/2a,Ap = h'I2a/2, Ap = AxAp = h/2. The phase density corresponding to g, is
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(1)In the regular region (nonchaotic), the FCF’s with a particular g, fluctuate wildly from one eigenstate to the next, with many FCF’s essentially zero. (2) In the irregular regime (chaotic), the FCF’s vary smoothly. This is the basic idea that the FCF’s tell us about the dynamics. T h e y are measurables! Unfortunately, we cannot vary the state g, at will experimentally, but we can still look at the FCF’s for a particular g,. There is a caveat which should be mentioned: wave functions, no matter how stochastic, have nodes. If our test state g, had a large a,it would sense fluctuations in the FCF’s due to nodal properties of the wave function. Thus, an ideal theory would allow for some fluctuation, even in the fully chaotic case, due to quantum nodal structure. This can be dealt with in another way by coarse graining over several nearby states to wash out the nodes. This is something like what Kay has done.6 Unfortunately the coarse graining is not generally available experimentally, and we want to keep our theory as measurable as possible. It does seem that the Gaussian packets, with reasonable choices for a,are not sensitive to local nodal structure of the wave function, so with such test states we may not need to allow for nodal fluctuations. For example, the overlap of a displaced ground-state Gaussian with the eigenstates of the harmonic oscillator yield a smooth Poisson distribution, with no fluctuations in the FCF’s other than the smooth dependence necessary to reproduce the spectral envelope. However, if we shrink the packets down to position space 6 functions, the FCF’s would fluctuate due to the quantum nodes in the eigenstates.
Criteria for Chaos Let us look at the consequences of classical chaos in terms of Gaussian classical test densities pa, pb, etc. (see eq 2). Even though h appears in eq 2, p is purely classical with h merely serving as a parameter to determine the spread in position and momentum. This spread causes in turn a dispersion in classical energy. We can easily imaging a whole set of “equivalent” p’s with different ( x ) and ( p ) , but with the same (or nearly the same-relaxing a little rigor!) average energy and dispersion in energy. If we suppose that energy is the only known constant of the motion, total classical chaos would imply that all such equivalent p’s would be equally sampled on the average in the time evolution of a single one of them. More precisely, we would have P(ala) = P(a(b) = P(a1b’) = P(b(b’) = ... (3) where
(4) In eq 4 Tr implies a classical phase space trace, p b ( t ) is a dynamically evolved classical distribution, and pa, Pb are two “equivalent” distributions. Translating this into more familiar language, a classical molecule whose phase space “cells” pa, pb, etc. obeyed eq 3 would also obey RRKM statistical behavior. Suppose we adopt eq 3 as our criterion for quantum chaos, where eq 4 is the same except that pa, &(t) are quantum density matrices (i.e., pa = k,) (gal)and Tr is now a standard quantum trace. Let us see where this criterion leads us. Define the FCF pna:
P n a = I(gaIn)12
Then P(alb) can be calculated as2J6 P(alb) = CPnaPnb n
(5) (6)
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Figure 3. The “stochastic ideal,” with two coherent states “a” (left) and “b” (right) having the same spectrum, which smoothly fills the envelope.
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Flgure 4. Spectra for two coherent states “a” and “b” (possesing the same energy envelope) in the nonchaotic regime. Note the fluctuating spectral intensities and “missing” lines, and note that the fluctuations in the case of these two states are out of “phase”
This is a fascinating formula, for it allows us to calculate the propensity of region “a” of phase space to evolve into region ”b”, simply from the spectra of a and b separately. If we consider the packets g,, gb, g< ...to all have the same low-resolution energy envelope, there are many possible sets of pn’s which could smooth to give this envelope. That is, the envelopes say nothing about the finer details as to how the spectrum fills in at high resolution. The ultrahigh resolution spectrum for g, is simply t a ( u ) = C6(u - En/h)pna (7) n
But combining the fully chaotic condition, eq 3, with eq 6, we must have, in the chaotic case pna = pnb = pnb’ = ... (8) Condition (8)still allows the pn)~ to fluctuate, but a simple argument shows2that they cannot and must instead be as smooth as possible consistent with the density of states and the energy envelope. This “stochastic ideal” is illustrated in Figure 3, which shows the (identical) FranckCondon spectra of two different packets. Using these spectra we would necessarily calculate P(ala) = P(bJb)= P(alb). In the quasiperiodic or nontotally chaotic domain, the pna’sfluctuate for every Gaussian g,. The reader can easily be convinced with a few moments’ thought that, in the fluctuating case, P(ala) will be larger than PTo(ala),where ”STO”refers to the stochastic case. This is an algebraic fact from eq 6 and the rule Cdna= 1. It also makes good physical sense since, if the dynamics is quasiperiodic, the evolving state g,(t) will spend more time in the region of g, since it is not sampling the whole phase space. A nonchaotic case is illustrated in Figure 4. Note that the two states chosen fluctuate out of phase, so that P(ala) P(blb) >> P(a1b). For this example, the state b is in a region only poorly sampled by dynamics emanating out of region a. Various measures of the departure from stochasticity could be taken. One is the “information entropy” (16) K. S. J. Nordholm and S. A. Rice, J.Chem. Phys., 61,203, (1976); 61, 768 (1974).
The Journal of Physical Chemisfry, Vol. 86, No. 12, 1982 2121
Criteria for Quantum Chaos
S a = C p n a In Pna n
to be compared with the stochastic entropy SaSTO = CpnSTO,a In p STO,a n
(9)
(10) I
The pnSToga is shown to be2 pnSTO,a = Sa@)/D(E)
(11)
where Sa(E)is the envelope function for the state a and
D(E)is the (smoothed) density of states. The reader can easily see that S attains the stochastic limit in Figure 3, and > Sa$for Figure 4. Another measure of the departure from fully chaotic dynamics is the root mean square deviation
62 = Tr
[((pa), - p,ST0)’I
(12)
where (pa) is the time-averaged density of the evolution of g,, i.e.
and pn is the density matrix for the nth eigenstates. The is defined to be stochastic density PaSTO
= CpnSTO,apn
(14)
n
The interesting aspect of this measure is that2 u,2 = P(ala) - PTo(ala)
(15)
so that these two measurables give the root mean square phase space deviation of the time-evolved density from the stochastic one. Exactly the same equations (12) and (15) hold classically as well. The reader may wish to examine Figure 1 again in the light of the picture which is emerging. Note that the packet placed in the chaotic region does not give a fully stochastic set of FCFs. This is because there are excluded, nonchaotic regions coexisting at the same energy. Detailed examination of the FCF’s in the two cases shows an anticorrelation: large FCFs in one case are small in the other. In ref 2 we were much more careful about defining envelopes, chaotic P(alb)’s, chaotic pna’s, etc., then we have been here. The interested reader is referred there for more details. One conclusion which is reached in ref 2 is that, if the dynamics is stochastic, then the quantum P(alb) should be close to
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Flgure 5. V = 1/2(1.44)x2 1/2(0.81)y2- 0 . 1 1 ~ ~Packet ~ . a: x o = 2.1, y o = 0. px, = 2.1, p,,, = 4.152. Packet b: x o = -2.1, y o = 0. px, = -2.1, pu, = 4.152. Packet b’: x o = -2.31, y o = 0. px, = 2.05 = 4.01416. Solld line: Quantum PT,dot-dash: classical PT. Piqila) is ~0.006 for this case.
It is often remarked that very long time, isolated molecule dynamics may be irrelevant to chemistry, so it is of interest to know how quickly PT approaches its asymptotic value, and whether the classical and quantal PT)s follow each other. In order to visually present our results, we show a potential contour at the energy of the wave packets, together with the packets themselves, as inset diagrams in each of the PT(alb) plots. Arrows indicate the momentum of the packets. The Hamiltonian used, together with the parameters of the packets, are described in each figure caption. In all cases, the relative position-momentum uncertainty (governed by a in eq 1) was chosen so that the packet was the ground state of the harmonic part of the potential in each dimension. The kinetic energy reads, in every case, T = px2/2 py2/2. A rather stunning, yet not atypical, example is shown in Figure 5. The quantum PT is shown as a solid line; the classical, a dot-dash line. The quantum result was determined by finding numerically converged eigenfunctions in our favorite basis set” &e., Gaussian wavepackets with various average positions and momenta) and employing the obvious formulae to determine Pp The classical results were obtained by sampling the (Gaussian) classical Wigner phase space distribution pa in a Monte Carlo way, propagating the resulting “swarm” of trajectories, overlapping this swarm p,(t) with the Gaussian analytic Pb, and using eq 17. This procedure is the exact analogue of the quantum determination of P(a1b). The initial and final state distributions pa, pb, etc. are exactly the same, via the “Wigner equivalent” formalism.’* The only difference is that, in the classical calculation, classical dynamics is used
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Note that this formula is consistent with the conclusion that P(a(b) = P(a(b’) = ... etc. for states b, b’, etc. with the same spectral envelope. From this qualitative introduction to our point of view, it is evident that it will be very interesting to examine P(alb)’s, both classical and quantal, in the chaotic and nonchaotic domains. Also it will be important to examine the spectral features to verify the qualitative correspondence between chaos and spectra. We proceed to examine these properties in specific cases. Numerical Results In addition to the classical and quantal P(alb)’s, which are infinite time-average quantities, one would also like to know the time evolution
(17) M. J. Davis and E. J. Heller, J. Chem. Phys., 71, 3383 (1979).
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Figure 6. Potential is ‘/2(1.21)x2 4- ’/2y2 - 0.11yx2. Packet a: x o = 3.4, y o = 0,px, = 2.0, pv = 3.16424. Packet b: x o = 0,y o = 07 Px, = 0,P 0 = 5.2915. br(alb) is shown at the top. Solid line: quantum Pr(a6); dot-dash: classical Pr(alb). P(alb) quantum is shown as a horizontal dashed line. The spectrum resulting from packet “a” Is shown at the bottom. A surface of section with pa shown as a black dot is shown in the mlddle panel.
to propagate the phase space distributions. Thus the quantum and classical results differ only in their dynamics; all else is identical. The agreement between classical and quantal P(alb)’s is remarkable. Note that no coarse graining, as employed by Kay,6 is used here. The differences are on the order of the Monte Carlo sampling error. Note that P&a(b)for the middle panel is the same a t long times as P(a1a). Indeed, classical trajectories appropriate to pa quickly sample the phase space domain of Pb, and since Pb is symmetrically equivalent, the P(ala) = P(alb) in this case. (Symmetrical equivalence of a and b is not enough to guarantee P(ala) = P(a1b). See below.) The state b’ in the bottom panel is not symmetrically equivalent but it has the property Sa(E)= Sb(E),both classically and quantally. Thus,under stochastic dynamics, we would have P(ala) = P(a(b’) here too,but this is far from the case. Indeed, P(ala) >> P(alb’), and both classical and quantal mechanics agree on this. Inspection of the surface of section (not shown) reveals that d three packets, g,, gb,gb’are in the quasiperiodic regime. The region of phase space appropriate to b’ is quite inaccessible to trajectories starting near a. For comparison, PTo(ala) is r0.006. Next examine Figure 6. These packets start in the chaotic regime of phase space (see insert), but note that quasiperiodic regions coexist at this energy. The fully stochastic Pm(alb) is ~0.004.The horizontal dashed line in the bottom panel shows the eventual P(ala) in the quantal case which is not attained in the 300 time units shown. As expected, P(ala) more closely approaches the stochastic limit here than in the previous case, but still lies above because of the excluded quasiperiodic region of phase space. The spectrum is very dense with lines but
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Figure 7. Henon-Heiles Hamiltonian (see ref 3). Packet a: x,, = 5.096, y o = -0.57, px, = py, = 0. Packet b: x o = -2.054, y o = 4.698, px, = py, = 0. This example illustrates the quantum interference effect (due to spectral degeneracy) discussed in ref 5. The packets are in the partly chaotic regime.
not fully chaotic, again because of the excluded regions of phase space. The final two cases show decided classical-quantal differences, having to do with interference and tunnelling effects. In Figure 7, the Henon-Heiles potential is examined. It has C3”symmetry and the two packets shown in the bottom panel are equivalent via 120’ rotation, a group operation. Note that P(ala) = 2P(alb) in the quantum case, even through these packets start in a stochastic subdomain of phase space. The 1:2 ratio for the quantum case is predicted from the interference effect discussed in ref 5 and proven via rigorous group theory arguments. The classical P(a(a)lies significantly below the quantal; it does not suffer the interference effect. It is expected that coarse graining this case (by averaging over many nearby “b” states) would remove the classical-quantum differences, but we have not attempted this. The classical P(alb) does not quite equal the classical P(a1a); this is because quasiperiodic domains of the phase space again coexist at the same total energy. Finally, Figure 8 shows a dramatic tunneling effect. Full appreciation of this phenomena can be gained by examining ref 18, but we summarize the situation here. This potential has a 1:l resonance above a certain energy, wherein classical stable local modes exist. The packets are placed directly in these local mode regions. The local modes manifest themselves as two equivalent stable periodic trajectories which cannot evolve into each other via classical dynamics. The path of this trajectory, leading from the region of the “a” packet, is shown as a dashed line in the top insert. The classical PT(ala) is quite large, since the motion is confined to the region near the dashed line. (18)M. J. Davis and
E. J. Heller, J. Chem. Phys., 75,
246 (1981).
The Journal of Physical Chemistry, Vol. 86, No.
Criteria for Quantum Chaos
SI>
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Figure 8. V = 1/2(0.95065)x* -I-l/g2 - 0.08yx2. Packet a: x o = 5.504, y o = 3.459, pxo= pyo= 0. Packet b: x o = -5.504, y o = 3.459, px, = pyo= 0. This example shows the effects of dynamical tunneling.
The quantum Pdala) shows a steady downward trend from the classical, eventually (at very long time) to come down to horizontal dashed line at almost exactly half of the classical value. Meanwhile, the quantum Pdalb), however, shows a steady increase until, at long time, it reaches the quantal P(ala) value at 0.064. As shown in ref 18 and discussed also by Lawton and Child,lg there is quantum dynamical tunnelling between the two, classically unconnected local mode regions. The sharp contrast between the classical and quantal P(ala) and P(alb) shows this tunneling dramatically.
Conclusions Except for cases of interference (removable by coarse graining) and tunneling, it would seem that a very excellent classical-quantum correspondence exists for our measure of chaos. In a mathematical sense, we can even get rid of the tunneling by taking the limits h 0, T -, in that order.2 In a physical sense, the tunneling remains and it is one of the potentially most interesting of quantum effects. We20 have speculated on the possible importance of dynamical tunneling in large molecules, where the tunneling might occur between qualitatively different kinds of vibrations of the polyatomic molecules. This could have important consequences for unimolecular dynamics. It seems too that, whenever the packet is placed in a chaotic regime, the quantum spectrum is dense, and when it is placed in a quasiperiodic regime, the spectrum is sparse. Subsequent to our initial study of this effect, Weissman and J o r t n e P have verified this to be the case in a large number of trials. Earlier than these general
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(19)R.T.Lawton and M. S. Child, Mol. Phys., 37,1799(1979);40,733 (1980). (20)E. J. Heller and M. J. Davis, J. Phys. Chem., 85, 307 (1981). (21)Y.Weissman and J. Jortner, Phys. Lett. A, 83,55 (1981).
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studies, Marcus and co-workers22had investigated the purely classical Fourier transform spectrum of a single trajectory. They found indeed a dense spectrum for the chaotic case, and a sparse one for the quasiperiodic case. Thus the extension of this notion to quantal FranckCondon spectra would seem natural. Our basic goal has been to come up with a useful measurable criteria for quantum chaos. In this regard, we agree with the philosophy of Waite and MilleP who placed emphasis on measurable rates of decay from various quasibound levels. (However, it is our opinion that at least some of their conclusions may be due to quantum smoothing, since their system had only 36 "bound" states, or 6 per dimension.) In this paper, we have used Gaussian packets as a test state for P(a(b)'s and spectra. There remains the question of what Brumer has called "basis set disease".24 Are our conclusions dependent upon our own predisposition toward Gaussians? First, we have established that there is excellent quantum-classical correspondence, even better than Kay saw: where exactly analogous classical Gaussian distributions are used. (Kay wed only approximately analogous classical distributions.) Second, the Gaussians are very close to the experimental situation of a Franck-Condon transition out of the ground vibrational level of a polyatomic. Third, as stated in ref 2, a more careful analysis shows that the definition of quantum chaos must necessarily change with the distributions used as test states. This is a subtle point, and the reader is referred to the complete paper.2 The Gaussians are the most convenient test states. The possibility exists, for example, that excellent correspondence would hold for Gaussians but not for some local mode overtone test states, even in the same molecule at the same total energy. Also, we have mentioned the problem of the nodal structure (really an interference effect) which shows up for some test states. It still seems to us that all deviations from classical and quantal chaos will arise ultimately from identifiable interference, tunneling, and smoothing effects, but we are careful not to dismiss these as unimportant. They may be quintessential to some measurements, and, after all, they make life more interesting. We have not fully discussed the phenomenon of quantum smoothing. What role does it play in the correspondence of quantum and classical chaos? The smoothing phenomenon is familiar from optics; we cannot, e.g., expect to see a virus in an ordinary microscope. Likewise, if the phase space cell of volume hN is larger than certain classical details, then these details will be smoothed out. The methods of Reinhardt and co-workers" depend on this fact for finding eigenvalues in the chaotic regime. In the classical mechanics of coupled anharmonic oscillators, the chaotic subdomains grow in, as energy is increased, from negligible volume in phase space to a large fraction of the volume. Several workers6J2have observed generally good quantum classical correspondence for chaos, but note that the quantum mechanics is sluggish in its response to classical chaos. In our opinion, this is a result of quantum smoothing; if the calculations had been made with h effectively smaller, the quantum response to classical chaos would have been sharper. Still, the sluggishness occurs on some scale for every h > 0, and this could be an important physical effect. (22)D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys., 67,404(1977);G.E.Powell and I. C. Percival, J.Phys. A, 12,2053 (1979),have used a classical spectral antropy measure. (23)B.A. Waite and W. H. Miller, J. Chen. Phys., 74,3910 (1981). (24)P. Brumer, this issue.
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Figure 10. Semiclassically generated (top row) and quantum (bottom row) wave functions for an anharmonic potential. See ref 26.
Figure 9. Semiclassical (top) and quantum (bottom) spectra for “local mode” displaced wavepackets. See ref 25.
We should not be surprised to see such excellent quantum-classical correspondence on the question of chaos as has been demonstrated here. It is certainly true that excellent molecular spectra and wave functions can be obtained from trajectories only. Figure 9, for instance, shows a quantum and semiclassical spectrum obtained for the quasiperiodic region of an anharmonic potential.25 Figure 10 shows semiclassically obtained (top row) and quantum wave functions, also in the quasiperiodic region.26 Figure 11shows a highly excited state (approximatelythe 1800th level) obtained by our semiclassical method25 and the (25) E. J. Heller, J . Chem. Phys., 75, 2923 (1981). (26) M. J. Davis and E. J. Heller, J . Chem. Phys., 75, 3916 (1981).
Figure 11. Highly excited semiclassical state and analogous classical distribution. See ref 26.
analogous purely classical density. The correspondence principle, which is the foundation of our methods, is hard at work here. The correspondence principle should work for dynamical chaos too.
Relaxation Rates in Model Hamiltonian Systems, with Remarks on OCS I. Hamilton, D. Carter, and P. Brumer” Department of Chemistty, University of Toronto, Toronto, Ontario, Canada (Received: July 23, 1981)
Two studies in intramolecular dynamics are described. The first deals with the dynamics of OCS, a molecule which has been the subject of recent collisionless multiphoton dissociation experiments. The second describes results which show that the K-entropy provides an excellent basis-free route to evaluating relaxation times in N = 2 systems in the highly irregular energy regime.
I. Introduction Two major present-day problems in the area of intramolecular dynamics are readily identified. The first deals with the relationship between classical and quantal dynamics in the irregular regime (defined later below). Several ideas regarding this relationship are emerging1 although this problem, and related issues of classicalquantum correspondence in the irregular regime, will most certainly undergo continued study. The second problem, the primary subject of this paper, is the rate of relaxation in energy-rich molecules.
* Alfred P. Sloan Foundation Fellow. 0022-3654/82/2086-2124$01.25/0
Rapid, efficient, essentially statistical intramolecular energy “flow” in highly energized systems is a generally accepted,2 albeit not absolutely e~sential,~ aspect of statistical theories of unimolecular decay and chemical reactions. Under most circumstances the relevant issue is the relative rate of intramolecular dynamics and other competitive processes, e.g., unimolecular decay rates, time delay in chemical reactions, predissociation rates: etc. Of (1) D. W. Noid, M. L. Koszykowski,and R. A. Marcus, Annu. Rev. Phys. Chem., 32, 267 (1981). (2) See, e.g., W. Font, “Theory of Unimolecular Reactions”,Academic Press, New York, 1973. (3) K. Freed, Discuss.Faraday SOC.,67, 231 (1979).
0 1982 American Chemical Society
