The Journal of
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0 Copyright, 1981, by the American Chemical Society
VOLUME 85, NUMBER 4
FEBRUARY 19,1981
LETTERS Quantum Dynamical Tunneling in Large Molecules. A Plausible Conjecture Eric J. Heiler’ and Michael J. Davls Department of Chemistry, University of California, Los Angeles, Callfornk, 90024 (Received: August 26, 1980; In Final Form: December IO, 1980)
Dynamical tunneling between distinct types of molecular motion, forbidden classidy but not involving a potential barrier, may occur in large isolated polyatomic molecules and cause profound changes in their motion long after they are initially excited. Classical mechanics offer significant advantages over quantum mechanics in terms of computationsand physical insight into the dynamics of molecules. Because molecules obey quantum mechanics, qualitative differences between quantum and classical dynamics, which may loosely be termed “quantum effects”, necessarily become important objects of study. Two important quantum effects are interference and tunneling. In the literature on scattering between atoms and molecules, these two effects play a prominent ro1e.I Indeed, although classical trajectory methods are often very successful for molecular collisions,2 interference and tunneling account for most of the discussion centering around deviations of classical and quantal cross sections.13 In the literature on bound states, the situation is quite different as regards the role played, at least so far, by interference and tunneling. Discussion concerning quan(1) W. H. Miller, Adv. Chern. Phys., 25, 69 (1974), has given an excellent review of semiclassical collision theory, emphasizing interference and tunneling effects. (2) See, e.g., R. N. Porter and L. M. Raff, and also W. L. Hase in “Dynamics of Molecular Collisions”, Part B. W. H. Miller, Ed., Plenum, New York, 1976. (3) See, e.g., M. S. Child and also P. Pechukas in “Dynamics of Molecular Collisions”, Part B, w. H. Miller, Ed., Plenum, New York, 1976. 0022-3654/81/2085-0307$01.00/0
tum effects has largely centered around the question of the transition to mostly stochastic motion, known to occur rather abruptly in classical, nonseparable system^.^ This question is not yet settled, and such differences in the quantum and classical manifestations of stochasticity as do exist have not been blamed particularly on interference or tunneling. As pointed out in an earlier publication,5the question of quantum-classical correspondence below the stochastic threshold has been somewhat neglected. In the regular region, the dynamics is perhaps very anharmonic but still nonstochistic and, for most initial conditions, N constants of the motion exist for N degrees of freedom. In ref 5, a close correspondence was noted between the quantum and classical manifestations of Fermi resonance in the regular region. It was found that not only did resonance set in at the same energy and to the same extent classically and quantally, but also that spectral combination-overtone spacings and post-resonant band(4) For review see s. A. Rice, “Advances in Laser Chemistry”, A. Zewail, Ed., Springer-Verlag, New York, 1978; M. Tabor, Adv. Chem. Phys., to be published; P. Brumer, ibid., to be published; M. V. Berry, in “Topics in Nonlinear Dynamics”, S. Jorna, Ed., American Institute of Ph sics Conference Proceedings. h - 4 6 , New York, 1978. &) E.J. H e E , E . B.Steche1, and M. J. Davis, J. Chern. Phys., 73, 4720 (1980).
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widths could be obtained from a few classical trajectories. Also, a large literature exists on finding eigenvalues of bound systems in the regular and semiclassical quantization often works quite well. Interference effects are at the heart of semiclassical quantization of energy levels, because constructive interference of the classical action is required for bound states to exist.g Further, barrier tunneling is well understood at least in one-dimensional systems, and tunneling factors, level splittings, etc. can be calculated semic1assically.l’ Thus, it may seem as though quantum effects are now rather well understood in the regular classical regime. However, the semiclassical treatment of the symmetric double wellll raises some interesting questions. If each well is treated separately, using the quantization conditions appropriate to a single well, then degenerate pairs of quantum levels are predicted. On the other hand, a full WKB treatment, with matching conditions connecting the wells and barriers, shows that the levels are split byll AI3 = a ex,( - h
Letters
I I
X
1’ dx) -a
A
(p(
.--_--
where the interval (-a,a) is the tunneling region, IpI is the absolute value of the momentum, and w is the classical frequency of the wells, at the energy of the level. The splitting (1) may be viewed as arising from a complex classical trajectory which connects two classically allowed regions.12 This example raises the intriquing question of the “other kind” of tunneling, namely, dynamical tunneling. Dynamical tunneling is simply a classically forbidden process which does not happen to involve an energetically insurmountable potential barrier. Many examples are known in collisions.12 A trivial example of a dynamically forbidden process is the exchange of energy between separable degrees of freedom in a multidimensional system. In this case, there is no quantum dynamical tunneling. Of course, good action-angle variables4 exist in a separable case, as well as for nonseparable systems in the regular regime. Yet, in the latter case, dynamical tunneling is permitted! Very recently, the present authors found unambiguous evidence for bound state dynamical tunneling between two classically distinct “local mode” vibration^.'^ A very close analogy was shown to exist with barrier tunneling in a symmetric double well (Figure la). When the symmetry of the potential was broken with the addition of a small term to the potential, the situation became analogous to an asymmetric double well (Figure lb), with greatly reduced tunneling, a consequence of the breaking of zeroorder degeneracy of the left- and right-hand well states.14 From the point of view of semiclassical mechanics, the dynamical tunneling and consequent energy and wave function changes would be found by complex classical (6)(a) R. A. Marcus, Discuss. Faraday SOC.,55, 34 (1973); (b) W. Eastes and R. A. Marcus, J.Chem. Phys., 61,4301(1974);(c) D. W.Noid and R. A. Marcus, ibid., 62, 2119 (1975);(d) 67,559 (1977). (7)K.S.Sorbie and N. C. Handy, Mol. Phys., 32,1327 (1976);K.S. Sorbie, ibid., 32,1572 (1976). (8)S.Chapman, B. Garrett, and W. H. Miller, J. Chem. Phys., 64,502 (1976);W.H. Miller, ibid., 63,996 (1975). London, Ser. A , 349,101 (9)M.V.Berry and M. Tabor, Proc. R. SOC. (1976). (10)M. C.Gutzwiller, J. Math. Phys., 12,343 (1971);8,1976(1967); 10,1064 (1969). (11)See, e.g., D. ter Haar, Ed., “Problems in Quantum Mechanics”, Pion, London, 1975,p 13. (12)W. H. Miller and T. F. George, J. Chem. Phys., 56,5668(1972); T.F. George and W. H. Miller, ibid., 57,2458 (1972). (13)M. J. Davis and E. J. Heller, J. Chem. Phys., submitted for publication. (14)R. A. Harris and L. Stobolsky, Phys. Lett., 78B,313 (1978).
X Figure 1. [a) Symmetric double well, wtth ground state wave function shown. ( I Asymmetric double well, showing reduced tunneling of the wave fur tion. (c) Asymmetric double well, with phase space of ”accepting” (rlght-hand) well much larger than that of the “donating” (left-hand) well. Tunneling occurs, and both wells contain significant probability.
trajectories connecting the classically forbidden regions. Noid and MarcusGdrecognized that such complex trajectories could improve their eigenvalues, but did not calculate the corrections. Lawton and Child15speculated on the symmetric double-well analogy in connection with the local modes in HzO, and also suggested complex trajectories would be required. In the symmetric dynamical tunneling between local modes discussed in ref 13, a wave function confined initially to one (classicallytrapped) region tunnels completely to the symmetrically equivalent trapped region, after hundreds or thousands of vibrational periods. The barrier analogy (Figure la) also exhibits complete tunneling. This type of degenerate dynamical tunneling is possible for example whenever there are two or more equivalent atoms in a molecule, and trapped “local mode” motion exists. However, a more profound type of tunneling would exist if say a trapped local C-H stretch could tunnel to become a C-C stretch. This would certainly be an asymmetric dynamical tunneling event. Several workers have seen trapped classical motion even at high energies in polyatomic systems, including above d i s s o c i a t i ~ n , ~so~ that J~ the significance of dynamic tunneling, if it exists, may be considerable. Our question is: would a quantum wave function, confined initially to a classically trapped region in the phase space of the molecule, eventually leak out completely into other regions corresponding to qualitatively different types of motion? (What we have in mind is not the type of exponentially small tunneling which, for example, occurs in Figure lb. While exponentially small dynamic tunneling will surely occur in nonseparable bound systems, we refer to significant tunneling, in which the bulk of the proba(15)R.T.Lawton and M. S. Child, Mol. Phys., 37, 1799 (1979). (16)P. J. Nagy and W. L. Hase, Chem. Phys. Lett., 54, 73 (1978); erratam, ibid., 58,482 (1978). (17)(a) E.Pollak and P. Pechukas, J. Chem. Phys., 69,1218(1978); (b) P. Pechukas and E. Pollak, ibid., 71,2062 (1979).
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bility distribution flows into qualitatively distinct motion in a classically forbidden way.) We do not know the answer to this question, but there is good reason, presented below, for suspecting the answer will be yes, but perhaps only for large polyatomic molecules. A more formal statement of the problem is as follows: suppose that one has good action variables (Il,...,I N ) , and another set (Il’,..., IN’),such that H(I1, ...,I N ) H(I
