Generalized Algebraic Quantization: Corrections to ... · Generalized Algebraic Quantization The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3145 well as the meaning of classical

3144 J . Phys. Chem. 1988, 92, 3144-3154

We finally conclude with some speculation concerning when one might expect the extra quantum localization observed in this paper to vanish and for classical and quantum mechanics to correspond in classically chaotic regions. The simple answer is .when flux across most curves approaches h (see, however, the quantum effects observed in ref 5 7 ) . But how does the quantum mechanics approach this limit? The work of ref 57 suggests tunneling which increases with energy might be the route for such a limit. This would be manifested in quantum eigenstates which become more delocalized in phase space as energy is increased, though their peaks might remain in the same location even after a great deal of spreading. Such eigenstates would be consistent with the work of Berry,94 which suggests that Wigner transforms of chaotic eigenstates would have essentially flat distributions. However, we do not necessarily expect this flattening to occur in

(94) Berry, M. V. J . Phys. A 1977, I O , 2083.

a monotonic fashion, because of increased tunneling, though flattening may be monotonic over a large range of energies. The lack of monotonic spreading might explain the interference effects observed in ref 57. Perhaps we have observed extra spreading in the 92nd and 93rd states studied in this paper and previously.62 However, we expect that the effects observed in the larger doublet spacings may be caused by a very weak avoided crossing which is experienced more strongly in the 102nd and 104th states.

Acknowledgment. It is a pleasure to thank Stephen Gray, Rex Skodje, Gregory Ezra, Craig Martens, and Nelson De Leon for helpful discussions. I also acknowledge the many collaborators whose work I summarized in the paper. They are S. Gray, S. Rice, L. Gibson, G. Schatz, M. Ratner, R. Skodje, and R. Steckler. This work was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, US. Department of Energy, under Contract No. W-3 1-109-ENG-38.

Generalized Algebraic Quantization: Corrections to Arbitrary Order in Planck's Constant

Laurence E. Fried and Gregory S. Ezra*+

Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, New York 14853 (Received: August 17, 1987)

The algebraic approach to semiclassical quantization uses a series of canonical transformations to bring the classical Hamiltonian of interest into a standard, simplified form. A quantization rule is then employed to convert the simplified classical Hamiltonian to a block diagonal quantum operator: Diagonalizing the blocks yields semiclassical eigenvalues. Many quantization prescriptions are available, but the resulting semiclassical eigenvalues depend upon the rule used. We present a method for deriving corrections in powers of h (Planck's constant) that is applicable to any invertible quantization rule. The inclusion of these correction terms decreases the dependence of energy eigenvalues on the quantization rule used, and incorporates quantum effects analytic in h arising from the transformations in a controlled manner. For a Hamiltonian which is a polynomial in Cartesian coordinates and momenta, the series of h-dependent corrections truncates, so that results from algebraic quantization converge to those ofits quantum analogue, Van Vleck perturbation theory. Using the Weyl quantization rule, we calculate vibrational eigenvalues for several multidimensional systems with PERTURB, a special-purpose algebraic manipulation package. It is shown that the inclusion of low-order corrections in h can lead to significant improvements in the accuracy of energy eigenvalues.

I. Introduction The development bf semiclassical methods for calculating highly

excited energy levels of polyatomic molecules has received much attention in recent y e a r ~ . l - ~ The semiclassical approach is of interest both as a practical alternative to quantum variational calculation^^^^ and as a useful interpretive framework for un- derstanding observed quantum p h e n ~ m e n a . ~ . ~

Most semiclassical techniques are based on the EBK quantization approach'J*12 in which invariant tori with actions satisfying certain quantization conditions are sought. The primitive semiclassical approximation to the energy of a stafe is simply the energy of a trajectory on the quantizing torus. EBK quantization works well in the regular regime, where trajectories are quasi- periodic and good action variables exist.I3 It is not directly applicable to strongly chaotic motion, since invariant tori are not present throughout most of phase space.

Classical perturbation t h e ~ r y ' ~ . ' ~ provides one route to semiclassical in the chaotic regime.16.'7i19,30 For systems that are effectively nonresonant, (Le., for which the small denominator problem is not apparent at the order of the perturbative calculation performed), a series of canonical transformations near the identity formally reduces the Hamiltonian to a function of action only, to the desired order in per t~rba t ion . '~ The actual Hamiltonian is thus replaced by an integrable ap- p r ~ x i m a n t ' ~ ~ ~ ~ . ~ ' that can be quantized by imposing the EBK conditions on the good actions.

+Alfred P Sloan Fellow

We know of no rigorous results justifying this procedure. The relation between quantum and classical perturbation theory, as

~ ~~ ~~~~~ ~~ ~~~

(1) Percival, I. C. Adu. Chem. Phys. 1977, 36, 1. (2) Berry, M. V. In Chaotic Behauior of Deterministic Systems; Gerard,

(3) Littlejohn, R. G. Phys. Rep. 1986, 138, 193. (4) Delos, J . B. Adu. Chem. Phys. 1986, 65, 161. (5) Ezra, G. S.; Martens, C. C.; Fried, L. E. J . Phys. Chem. 1987,91, 3721

( 6 ) Carney, G. D.; Sprandel, L. L.; Kern, C. W. Adu. Chem. Phys. 1978,

(7) Tennyson, J. Comput. Phys. Rep. 1986, 4 , 1. (8) Stechel, E. B.; Heller, E. J. Annu. Rev. Phys. Chem. 1984, 35, 563. (9) See articles in NATO Advanced Research Workshop on Quantum

Chaos: Chaotic Behauior in Quantum Systems, Theory and Experiment; Casati, G . , Ed.; Plenum: New York, 1985.

(10) Einstein, V. Vertsch. Drsch. Phys. Ges. 1917, 19, 82. An English translation by C. Jaff6 is available as JILA Report no. 116, University of Colorado, Boulder, CO.

(11) Brillouin, M. L. J . Phys. 1926, 7 , 353. (12) Keller, J. B. Ann. Phys. 1958, 4 , 180. (13) Lichtenberg, A. J.; Lieberman, M. A. Regular and Stochastic

(14) Born, M. Mechanics of the Atom; Ungar: New York, 1960. (1 5 ) Chapman, S.; Garrett, B. C.; Miller, W. H. J. Chem. Phys. 1976, 64,

(16) Swimm, R. T.; Delos, J. B. J . Chem. Phys. 1979, 71, 1706. (17) Shirts, R. B.; Reinhardt, W. P. J . Chem. Phys. 1982, 7 7 , 5204. (18) Schatz, G. C.; Mulloney, T. J . Phys. Chem. 1979, 83, 989. (19) J a m , C.; Reinhardt, W. P. J . Chem. Phys. 1982, 77, 5191. (20) Ramaswamy, R.; Siders, P.; Marcus, R. A. J . Chem. Phys. 1980, 73,

I. G., Helleman, R. H. G., Eds.; North Holland: New York, 1983.

and references within.

37, 305.

Motion; Springer-Verlag: New York, 1983.

502.

5400.

0022-3654/88/2092-3144$01.50/0 0 1988 American Chemical Society

Generalized Algebraic Quantization The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3145

well as the meaning of classical perturbation theory in the chaotic regime, is receiving increased a t t e n t i ~ n . ~ ~ - ~ ~ Turchetti3’ finds that the radius of convergence of fixed-frequency perturbation theory corresponds to the value of the perturbation strength at which an invariant torus is destroyed. Graffi and using the Bargmann representation of quantum mechanic^,^^ have proved39 that EBK quantization of the Birkhoff-Gustavson normal form is accurate to first order in h, and to all orders in perturbation, if the quantizing action values correspond to a KAM t o r ~ s . ~ 3 ~ ~ has recently introduced what he terms a quantum normal form, defined by analogy with the classical Birkhoff- Gustavson normal form. To lowest order in h, the quantum normal form is the same as the usual classical normal form.

Resonant systems cannot be quantized by the perturbation procedure described above, since the Hamiltonian cannot be made a function of actions without introducing infinite corrections.I3 It is also possible that the corrections, while finite, are strongly divergent. We will call such systems nearly resonant. To avoid small-denominator problems associated with exact and near resonances, terms depending on resonant combinations of angles must be kept in the transformed Hamiltonian.13~19~2z~23~30~4z In effect, apparent convergence (Le., convergence to a given order in perturbation) is obtained at the price of a more complicated form for the transformed Hamiltonian. If a single resonant combination of angles appears, the final Hamiltonian has one nonignorable coordinate. The system is therefore equivalent to a one-dimensional problem; one-dimensional semiclassical procedures can be used to quantize the Hamiltonian in the new coordinate system.16,19,21,27,43 Swimm and Delos,16 using primitive quantization, and JaffE and Reinhardt,lg using a uniform procedure, showed this technique to be effective even when the actual classical motion is chaotic.

For systems requiring the retention of one or more resonant terms in the transformed Hamiltonian, algebraic quantization (AQ) has been ~ ~ e d , ~ ~ - ~ ~ , ~ ~ In this procedure, a quantization r ~ l e ~ ~ , ~ ~ turns the transformed classical Hamiltonian into a

(21) Uzer;T.; Noid, D. W.; Marcus, R. A. J . Chem. Phys. 1983, 79, 4412. (22) Sibert 111, E. L.; Hynes, J. T.; Reinhardt, W. P. J. Chem. Phys. 1982,

(23) Sibert 111, E. L. J . Chem. Phys., to appear. (24) Farrelly, D.; Uzer, T. J. Chem. Phys. 1986, 85, 308. (25) Sanders, J. A. J . Chem. Phys. 1981, 74, 5733. (26) Robnik, M. J . Phys. A 1984, 17, 109. (27) Uzer, T.; Marcus, R. A. J . Chem. Phys. 1584, 81, 5013. (28) Saini, S . Chem. Phys. Lett. 1986, 125, 194. (29) Farrelly, D. J . Chem. Phys. 1986, 85, 2119. (30) Fried, L. E.; Ezra, G. S . J . Chem. Phys. 1987, 86, 6270. (31) Reinhardt, W. P. J . Phys. Chem. 1982,86, 2158. (32) Ali, M. K. J . Math. Phys. 1985, 26, 10. (33) Eckhardt, B. J . Phys. A 1986, 19, 2961. (34) Wood, W. R.; Ali, M. K. J . Phys. A 1987, 20, 351. (35) Graffi, S.; Paul, T. Commun. Math. Phys. 1987, 108, 25. (36) Robnik, M. J . Phys. 1986, 19A, L841. (37) Turchetti, G. In Advances in Nonlinear Dynamics and Stochastic

Processes; Livi, R., Politi, A, , Eds.; World Scientific: Singapore, 1985. (38) Schulman, L. S. Techniques and Applications of Path Integration;

wiley: New York, 1981. (39) The proof assumes that the zeroth-order system is a harmonic os-

cillator with frequencies satisfying a Diophantine relation,94 and that the perturbation is a polynomial potential. Other, more technical, conditions apply as well.

(40) Arnold, V. I. Mathematical Methods of Classical Mechanics; Springer-Verlag: New York, 1978.

(41) M o w , J. Stable and Random Motions in Dynamical Systems; Princeton University Press: Princeton, NJ, 1973.

(42) Sibert, E. L. Chem. Phys. Lett. 1986, 128, 404. (43) Farrelly, D. J . Chem. Phys. 1986, 85, 2119. (44) deGroot, S . R.; Suttorp, L. G. Foundations of Electrodynamics;

(45) Abraham, R.; Marsden, J. E. Foundations of Mechanics; Benja-

77, 3595. Sibert 111, E. L. J . Chem. Phys. 1985, 83, 5092.

North-Holland: Amsterdam, 1972.

min/Cummings: Reading, MA, 1978.

quantum mechanical operator. If the number of linearly independent resonant combinations of frequencies encountered is less than the number of degrees of freedom, the classical Hamiltonian has constants of the motion that are linear in zeroth-order actions. The corresponding quantum mechanical Hamiltonian is block diagonal.30 The blocks are often finite, so that the calculation of eigenvalues requires only the diagonalization of small matrices.30

In a previous paper,30 we showed that AQ yields accurate and reliable eigenvalues for a variety of perturbed harmonic oscillators with up to five degrees of freedom. One unsatisfactory aspect of AQ, however, is that there are many quantization rules satisfying the basic requirement of linearity. Several rules were studied, and it was found that the accuracy of resulting semiclassical eigenvalues depends on the particular quantization rule used.

Wood and Ali34 have also noted this dependence on quantization rule. They have argued that the applicability of the Birkhoff- Gustavson approach is consequently severely limited, since the normal form cannot reproduce the series generated by Ray- leigh-Schrodinger perturbation theory. The Birkhoff-Gustavsoli procedure can therefore never yield exact eigenvalues, even if it is resummed. While this is true in principle, numerical experience shows that many anharmonic systems can be quantized to ac- ceptable accuracy with AQ. Nevertheless, for some systems, such as a three-mode model for 03, significant discrepancies between low-lying quantum and AQ energies are found to persist to high orders in perturbation. Wood and Ali also point out that little is known about the summability of classical normal forms. They suggest that normal forms may be summable only for integrable systems. The work of Graffi and Paul mentioned above, howeuer, gives hope that classical normal forms are summable when the quantizing values of the action correspond to a KAM torus. The summability of classical normal farms when the underlying motion is chaotic is poorly understood. Nonetheless, accurate semiclassical energies have been obtained without resummation for energies in the chaotic regime.

In the present work, we generalize AQ to incorporate corrections to the transformations in powers of h . For the systems studied here, the inclusion of these corrections in h decreases the dependence of energy eigenvalues on the quantization rule. For polynomial Hamiltonians, it is possible to remove all dependence on the quantization rule by including sufficiently high orders of h. In doing so we arrive at results equivalent to usual quantum mechanical Van Vleck perturbation theory.

Our strategy is to replace the Poisson brackets used to carry out canonical transformations in AQ with a representation of the quantum mechanical commutator in a mock phase space.46 We then derive corrections to AQ by expanding the mock phase space commutator in powers of h . The form of the phase space commutator depends on the quantization rule used. If the Weyl quantization ruleM is chosen, the mock phase space representation of the commutator is the Moyal bracket.47 In this paper we implement the AQ algorithm, replacing Poisson brackets with Moyal brackets. We find that the inclusion of corrections in h often leads to markedly better results than a purely semiclassical approach (Le., no h-dependent corrections). For -polynomial Hamiltonians, the commutator is exactly represented by finitely many terms. When enough orders in h are included, generalized AQ exactly reproduces results derived from quantum perturbation theory applied ’to polynomial Hamiltonians.

The structure of the paper is as follows: In section 11, we review the AQ approach, and point out the difficulties involved in a choice of quantization rule, as well as problems associated with small denominators. In section 111, a general procedure for implementing AQ to higher order in h is given. We derive explicit formulas for generalized AQ, using the Weyl quantization rule. Section IV presents results obtained with PERTURB, a special purpose algebraic manipulation package, for a variety of resonant and nonresonant three-dimensional systems. Finally, in section V we

(46) Balazs, N. L.; Jennings, B. K. Phys. Rep. 1984, 104, 347. (47) Moyal, J . E. Proc. Cambridge Philos. SOC. 1949, 45, 99.

3146

discuss the usefulness of generalized AQ, both as a practical tool for finding semiclassical eigenvalues and as a way of studying the summability of classical perturbation theory when applied to quantum problems.

11. A Review of Algebraic Quantization A. Lie Transforms. In AQ, a classical Hamiltonian is subjected

to a sequence of canonical transformations before it is quantized. The first work on AQZs2’ used the Birkhoff+ustavson m e t h ~ d ~ , ~ ~ to carry out the transformations. This algorithm uses as a sequence of polynomial generating functions F(p,q), where

The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 Fried and Ezra

and the new coordinates are denoted by bars. The functional inversion required to obtain an explicit formula for one set of variables in terms of another is a principal difficulty in implementing the Birkhoff-Gustavson method.

Lie transforms,% however, are more convenient than generating functions for carrying out canonical transformations, since they eliminate the need for functional inversion. There are several approaches to Lie transforms, and many perturbative algorithms based on them have been p r o p o ~ e d . ~ ~ - ~ ~ We briefly discuss the method due to Dragt and Finn,55,56 which is among the simplest of such techniques. The Dragt-Finn algorithm is of special sig- nificance for the present work, since it is the classical analogue of Van Vleck perturbation theory.58 Despite the variety of methods available, it should be stressed that all forms of classical perturbation theory based on a direct expansion in a small parameter give identical results when taken to the same order. Thus, the traditional Birkhoff-Gustavson technique, as well as Lie transforms, all give the same simplified Hamiltonian.

Lie transforms express a canonical transformation entirely in terms of Lie operators. A Lie operator, tf, is defined by

L/= K*l (2)

where { , } denotes the Poisson bracket. The exponential of a Lie operator generates a canonical transformation;

z + Z = exP(L/)z (3)

is canonical. This result is the basis of Dragt-Finn perturbation theory. The transformation 3 also includes a corresponding transformation on functions. Let

dz) = g t W ) (4)

The new function g can be expressed directly in terms of Lie operators:

= e x P ( q g ( 4 (5)

Dragt and Finn introduce a sequence of Lie operators to simplify the Hamiltonian

K ( z ) H(Z(z ) ) = exp(ckFk) exp(ek-’Fk-l)... exp(tF,)H(z) ( 6 )

where Fk is the Lie operator LA. The sequence of transformations is chosen to make the new Hamiltonian as close to integrable as possible. If an integer linear combination of zeroth-order fre-

(48) Birkhoff, G. D. Dynamical Systems; A. M. S . Colloquim Publications:

(49) Gustavson, F. G. Astron. J . 1966, 71, 670. (50) Cary, J. R. Phys. Rep. 1981, 79, 129. (51) Hori, G. Publ. Astron. SOC. Jpn. 1966, 18, 287. (52) Deprit, A. Celest. Mech. p 6 9 , 1 , 12. (53) Howland, Jr., R. A. Celest. Mech. 1977, 15, 327. (54) Henrard, J. Celesr. Mech. 1970, 3, 107. (55) Dragt, A. J.; Finn, J. M. J . Math. Phys. 1976, 17, 2215. (56) Dragt, A. J.; Finn, J. M. J . Math. Phys. 1979, 20, 2649. (57) Dragt, A. J.; Forrest, E. J . Math. Phys. 1983, 24, 2734. (58) PapouSek, D.: Aliev, M. R. Molecular Vibrational-Rotational

New York, 1927; Volume 9.

Spectra; Elsevier/North Holland: New York, 1982.

quencies m u vanishes identically, a frequency commensurability is said to exist. Systems with frequency commensurabilities are called resonant. The sequence of transformations 6 cannot make a resonant Hamiltonian into a function of good actions only.

Systems without exact frequency commensurabilities are classified as either nonresonant or nearly resonant. The small- denominator problem ultimately destroys the convergence of classical canonical perturbation theory,59 so a strict distinction between nearly resonant and nonresonant Hamiltonians is not possible. Operationally, we use the terms ”nearly resonant” and “nonresonant” to indicate how quickly the perturbation series diverges. If divergence is apparent before the desired order in perturbation is reached, the system is termed nearly resonant; otherwise, it is called nonresonant. Strictly speaking, every system is nearly resonant if one goes to sufficiently high order in perturbation, or if the perturbation is sufficiently large.

For nonresonant systems, all the angle dependence of the transformed Hamiltonian K can by definition be eliminated to a given finite order without encountering divergence. Energy levels can be found by substituting quantizing values of the good actions into K. For nearly resonant systems, the angle dependence of H can be formally eliminated, a t the cost of introducing rapidly divergent corrections into the perturbation expansion. The simplified Hamiltonian which results from eliminating all angle- dependent terms except those which are exactly resonant is called the Birkhoff-Gustavson normal form.48.49 The presence of angle-dependent terms or divergent corrections prevents perturbative EBK quantization from being applied straightforwardly to resonant or nearly resonant systems.

B. Quantization of Resonant and Nearly Resonant Systems. There are several ways to quantize resonant or nearly resonant simplified Hamiltonians. Resummation can be applied to the divergent normal forms of nearly resonant systems. Ali, Wood, and D e ~ i t t , 6 ~ using a PadE approximation ~ c h e m e , ~ ~ , ~ ~ and Arteca,66 using a functional meth~d,~’-~O have been able to resum the normal form of a one-dimensional quartic oscillator. Even though one-dimensional problems cannot have small-denominator problems, the classical perturbation series of this system has a small, but finite, radius of c o n ~ e r g e n c e , ~ ~ , ~ ~ in contrast to the zero radius of convergence generically found in multidimensional systems. Farrelly and U ~ e r * ~ have applied Pad; approximation to several two-dimensional systems, obtaining good results for some states, but poor agreement for others. More work clearly needs to be done on the resummation of classical perturbation theory applied to multidimensionai Hamiltonians.”

Other procedures are appropriate if the number of linearly independent resonant or nearly resonant combinations of frequencies encountered in carrying out the calculation is less than the dimensionality of the original system. Suppose that there are Q !inearly independent resonant combinations of frequencies. Then, if Q C N , where N is the number of degrees of f r e e d ~ m , ’ ~ the final Hamiltonian has Q nonignorable coordinates. If Q =

(59) Benettin, G.; Galgani, L.; Giorgilli, A. In Advances in Nonlinear Dynamics andStochastic Processes; Livi, R., Politi, A,, Eds.: World Scientific: Singapore, 1985.

(60) Simon, B. Int. J . Quantum Chem. 1982, 21 , 3. (61) Bender, C. M. Int. J . Quantum Chem. 1982, 21, 93. (62) Wu, T. T. Int. J . Quantum Chem. 1982, 21, 105. (63) Mi, M. K.; Wood, W. R.; Devitt, J. S. J . Math. Phys. 1986, 27, 1806. (64) Baker Jr., G. A.; Graves-Morris, P. Encyclopedia of Mathematics and

(65) Baker Jr., G. A. Essentials of Pad6 Approximants; Academic: New

(66) Arteca, G. A. Phys. Rev. A 1987, 35, 4479. (67) Arteca, G. A.; Fernindez, F. M.; Castro, E. A. J . Math. Phys. 1984,

(68) Arteca, G. A,; Fernlndez, F. M.; Castro, E. A. J . Math. Phys. 1984,

(69) Arteca, G. A.; Fernlndez, F. M.; Castro, E. A. Phys. Letf. A 1985,

(70) Arteca, G. A,; Fernlndez, F. M.; Maluendes, S. A.; Castro, E. A.

(71) Fried, L. E.; Ezra, G. S., work in progress. (72) Henceforth, N will refer to the number of degrees of freedom of the

its Applications; Addison-Wesley: Reading, MA, 198 1; Volume 13-14.

York, 1975.

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25, 3492.

1 1 1 , 269.

Phys. Lett. A 1985, 103, 19.

system of interest.


1, one-dimensional uniform quantization can be applied to the system. For Q > 1, however, other methods must be used. Algebraic quantization, which was first applied to systems where Q = 1 and N = 2,25-27 has recently been extended to systems with more degrees of freedom and more resonance^.^^

AQ exploits the existence of N - Q ignorable angle variables. Each ignorable coordinate implies the existence of a classical invariant which is a linear function of action. A corresponding quantum problem can be derived by employing a quantization rule (such as the Weyl rule) to associate a quantum operator Ke with the transformed classical Hamiltonian K . Under mild assumptions about the quantization rule, it can be shown that KQ also has N - Q invariants which are linear functions of number operators 1373.~~ The existence of these quantum invariants implies that the operator KQ is block diagonal.30 The blocks are often small and can easily be diagonalized to yield approximate quantum eigenvalues of the original system.

There are several essential difficulties with AQ-the most significant of these is the appearance of small denominator^.^^^^^ Although AQ can forestall problems with small denominators, a t sufficiently high orders in perturbation N resonant or nearly resonant combinations of frequencies will have been encountered.

Van Vleck Genera l ized AQ AQ

Quantum Operator

Symbol Classical Funcl ion

Input Hamiltonian c ia~cai Classical Hamiltonian Symbol i ,m,t

Hamiltonian

Phase Space

Small Matrix Diagonaii2alion

Diagonal Hamiltonian

Resummation can sometimes be used in conjunction with AQ to overcome small denominators. The perturbation series for strongly anharmonic systems, or systems with many near resonances, are nevertheless difficult to resum. Therefore, AQ is a useful technique for treating a restricted, but important, class of Hamiltonians. More work on resummation is required before it can be suc- cessfully applied to highly anharmonic systems with many degrees of freedom.73

A second shortcoming of AQ is that different quantization rules can give different r e s ~ l t s . ~ ~ , ~ ~ At present, there exists no obvious way to determine an optimal quantization rule, although rules which reduce to EBK quantization when applied to diagonal Hamiltonians have given good results when applied to several molecular problems.30

We present a way of generalizing AQ to include corrections which are of higher order in h. This procedure allows results from AQ to be successively improved until convergence to quantum mechanical Van Vleck perturbation theory is achieved. There are several good reasons for implementing a scheme that allows a smooth transition to be made between the semiclassical and quantum limits. First, it is of some interest to incorporate certain quantum effects (those analytic in h arising from near-identity transformations) into a problem in a controlled manner. This is, as we show below, useful in solving vibrational problems to high accuracy. Note, however, that the entire problem is not subject to an expansion in h . Nonanalytic behavior in h can be reproduced by the small matrix diagonalization, a characteristic which allows our method to avoid the convergence problems usually associated with expansions in h. Second, we expect that insight into the relation between quantum and classical perturbation theory, and thereby into the fundamental problem of quantum i n t e g r a b i l i t ~ , ~ ~ . ' ~ ~ ~ can be gained by systematic studies of quantum corrections to classical perturbation theory.

111. Generalized Algebraic Quantization In this section corrections to AQ are derived. We first note

that the quantum analogue of Dragt-Finn perturbation theory is just the well-known Van Vleck algorithm. A mock phase space

(73) Quantum perturbation theory itself is often divergent. In the next section, It is shown that Van Vleck perturbation theory applied to harmonic oscillators has the same small denominator problem as classical perturbation theory. See, however, ref 36.

(74) Hietarinta, J . Phys. Lett. A 1982, 93, 55. (75) Korsch, H. J. Phys. Leu. A 1982, 90, 113. (76) Pechukas, P. J . Phys. Chem. 1984, 88, 4823.

formulation of Van Vleck perturbation theory is then presented. This formulation is particularly suitable for expansion in h . Ordinary AQ can be. recovered as the lowest order approximation to exact quantum mechanical perturbation theory in mock phase space.

The relation between Van Vleck perturbation theory, its mock phase space version, and AQ is summarized in Figure 1. Van Vleck perturbation theory is formulated in terms of quantum operators (left column of Figure 1). In the Van Vleck approach, an initial Hamiltonian is simplified by a sequence of unitary transformations. Each of these transformations can be expressed in terms of the commutator with a particular generating operator. The sequence of transformations is chosen to render the Ham- iltonian as nearly diagonal as possible. If generalized Fermi resonances are encountered, the normal operator will not be diagonal. Nonetheless, if sufficiently few resonances are found, the normal operator will have a block diagonal matrix. Therefore, only small matrix diagonalizations are necessary to produce a diagonal Hamiltonian.

A mock phase space version of Van Vleck perturbation theory is illustrated in the center column of Figure 1. In the phase space approach, the initial Hamiltonian operator is transcribed into a function of 2N variables, called a The rule for tran- scription is invertible, so that every quantum operator is associated with a unique symbol. The commutator used to generate unitary transformations in Van Vleck perturbation theory can be mapped into an operator on symbols. The Hamiltonian symbol can then be simplified by a sequence of transformations generated by phase space commutators, replacing the Van Vleck generating operators with generating symbols. By this sequence of transformations, a simplied Hamiltonian symbol is arrived at, which we term a normal symbol. The normal symbol is then mapped into a normal operator by the quantization rule. This phase space version of Van Vleck perturbation theory is entirely equivalent to the usual Hilbert space version.

The motivation for introducing the phase space theory is that its classical limit is precisely AQ. This point is illustrated in the rightmost column of Figure 1. AQ uses a sequence of canonical transformations to simplify a classical Hamiltonian. The limit as h - 0 of the Hamiltonian symbol is just the classical Ham- iltonian. Thus, we arrive at the starting point of AQ by taking the classical limit of fully quantum phase space perturbation

(77) Voros, A. Ann. Inst. Henri Poincart 1976, 24, 3

Fried and Ezra 3148 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988

theory. It is not always necessary to take this limit; if no mo- mentum-coordinate cross terms are present in the quantum Hamiltonian, the initial Hamiltonian symbol will be the same as the classical Hamiltonian. Also, the classical limit of the phase space commutator is the Poisson bracket. Unitary transbrmations of the quantum theory therefore map into canonical transformations. The simplified classical Hamiltonian is termed a normal form. It can be viewed as an approximation to the normal symbol. Hence, a quantization rule can be used to derive an approximate normal operator corresponding to the normal form. In the following subsections, we elaborate on the comments made here and derive a means of approximating phase space Van Vleck perturbation theory to arbitrary order in h.

A. The Quantum Analogue of Dragt-Finn Perturbation Theory. The Dragt-Finn algorithm is an exp!icitly canonical perturbation theory, Le., it is expressed entirely in terms of Poisson brackets. The Poisson bracket is the classical limit of l / ( ih ) times the quantum mechanical con~mutator.’~ This suggests that a quantum analogue of Dragt-Finn perturbation theory can be written as

where H, and k are $11 quantum operators. I f f is assumed Hermitean, exp((l/ih)lf,*]) is a unitary transformation. I? (7), a sequence of uniiary operations transforms Hamiltonian H into the simpler form K, whereas in the classical version ( 6 ) a sequence of canonical transformations is used. The perturbation theory defined in (7) is, however, just the Van Vleck method of contact transformations. This is seen by rearranging (7) into a more familiar form, employing

expCi)H exp(-j) = exp( E * ) ] H (8)

to give

which is the familiar form of Van Vleck perturbation t h e ~ r y . * ~ - ~ ~ It is possible to develop semiclassical approximations to (7)

directly, by calculating the commutators, ordering terms with respect to noncommuting operators (for example, normal ordering), and truncating the result a t a given order in h. When taken to given order in perturbation parameter, the expansion in h truncates, provided that all quantities are polynomials in creation-annihilation operators (&,St). This is because the com- mutation relation

[&$+I = h (10)

implies that the commutator of two polynomials in (a,&+) will have finitely many powers in h . In such a scheme, the choice of quantization rule is only implicit in the ordering convention used. We prefer a mock phase space representation of quantum mechanics, such as the Wigner-Weyl formalism, and so obtain a direct quantum analogue of AQ.

The mock phase space techniques discussed here are most emphatically not mappings of quantum mechanics onto classical mechanics; rather, they are entirely equivalent to the usual Hilbert space quantum mechanic^!^.'^*^^ Moreover, a mock phase space description allows approximation of both mixed and pure states, which suggests that use of such spaces could expand the range

(78) Dirac, P. A. M. The Principles of Quantum Mechanics; Oxford University: Clarendon, U.K., 1935.

(79) Hillery, M.; OConnell, R. F.; Scully, M. 0.; Wigner, E. P. Phys. Rep. 1984, 106, 121.

(80) There are problems, however, in finding a simple definition of the class of physically admissable phase space distributions. In contrast, the class of physically admissable wavefunctions is easily defined. See Balasz and Jen- n i n g ~ ~ ~ for a discussion.

of problems to which semiclassical techniques can be fruitfully applied.

B. Mock Phase Space Representations of Quantum Mechanics. There are several equivalent ways to introduce a mock phase space representation of quantum mechanics. For instance, an expansion of the vector space of operators in a basis parametrized by c- numbers (a,b) induces a mock phase space representation of quantum mechanics; the mock phase space being given by the pair (a,b).46 An expansion of a given operator in terms of projectors onto coherent states is an example of this.81 We do not follow that route here. Rather, we use invertible quantization rules to motivate the use of a mock phase space. This approach is most natural in the context of AQ, where such rules are invoked to turn the transformed classical Hamiltonian into a quantum operator.

The problem of quantizing a classical Hamiltonian containing products of noncommuting operators is still largely open. A quantization rule gives a correspondence between a classical function and a quantum operator. For our purposes, we do not require this rule to be “correct”, in the sense of giving the quantum Hamiltonian corresponding to the classical system. Rather, we only desire uniqueness-every classical function should have a unique quantum counterpart. We will call rule inuertible if there exists an inverse operator which takes A(fi,fi) into a function As(p,q). Note that this inverse rule cannot be the same as merely taking the classical limit, for taking this limit would map many quantum operators into the same classical function.

An invertible rule allows the introduction of a mock phase space. Assume that we are given a quantum operator. Then the inverse rule provides us with a unique image of the operator in phase space. This phase space, however, is not the same as the phase space of classical mechanics, because the operations on it (multiplication, etc.) are different from the corresponding operations in classical mechanics. To make this distinction explicit, the space is referred to as a mock phase space, and the image of an operator is called a symbol. To lowest order in h, many quantities in mock phase space are the same as those in the phase space of classical mechanics, which is the justification for associating the words “phase space” with a quantum mechanical object.82

The only quantization rule explicitly used in this work is the Weyl quantization r ~ l e . ~ ~ , ~ ~ This rule associates an operator A(@,@ with a symbol As(p,q) according to the formula46

where h(p,q) is an operator defined by

h(p,q) ( h / 2 ~ ) ~ S e x p { i [ u . ( @ - p) + ~ ( f i - q)]) du dv (12)

Conversely, given an operator, the Weyl symbol can be found with the inverse relation46

&(PA) = Tr (4B4) & P d ) (13) More explicit formulas can be given for monomials in the

variables (p,q). For notational convenience, we restrict ourselves to a monomial in one dimension. The Weyl quantization procedure then gives

min(m,n) (ifj,)‘ W(p”q“) = i=n E -+y)(y)p-‘q-/ (14)

The inverse relation can also be derived-the Weyl symbol W, of a monomial is

The operator function jLj provides a specific illustration of the invertibility of the Weyl quantization rule. Its symbol is qp -

(81) Mizrahi, S. S. Physicu A (Amsterdam) 1984, 127, 241. (82) Some quantities in mock phase space, such as the Wigner function,

have essential singularities at h = 0. Here distribution-valued expansions in t2 are sometimes a p p l i ~ a b l e . ~ ~ . ~ ~

(83) Weyl, H. The Theory of Groups and Quantum Mechanics; Dover: New York, 1950.

Generalized Algebraic Quantization

( i h / 2 ) . Applying the Weyl quantization rule to this symbol recovers the original operator.

C. Symbol Calculus. As discussed in the previous section, invertible quantization rules allow operators to be represented by functions in a mock phase space. This is only one step toward a formulation of quantum mechanics based entirely on entities in a mock phase space. To complete this formulation, we must examine how operations in quantum mechanics, such as multiplication and addition, map into operations in mock phase space. These operations constitute a symbol calculus.77

Consider the addition of two operators

t = A + B (16)

As long as the quantization rule is linear, the symbol for e, Cs, is simply As + Bs. We will not consider nonlinear quantization rules, since they are rarely used.

As a consequence of the noncommutative nature of operator multiplication, the multiplication of two operators does not simply map into the multiplication of symbols. Nevertheless, a generalized multiplication can be defined which is the image of operator multiplication in mock phase space. This image is sometimes called a twisted m ~ l t i p l i c a t i o n , ~ ~ in reference to its noncommutative nature. For the operator product

= AB (17)

cs = As*& (18)

the corresponding symbol is denoted by

The twisted multiplication operator * is particularly simple for the Weyl quantization rule:46 we have

CS = exp ( - ( v ~ ' v p ~ i: - vc'Vp,))AS(q,4~P,4) BS(qB,PB) (19)

Equation 19 has a formal expansion in powers of h; in some cases, this expansion converges. The lowest order term in h is just ordinary multiplication. This result holds for any quantization rule which acts by reordering the terms of a noncommuting polynomial.

Once a generalized multiplication is found, a representation of the commutator follows trivially. If we define

(20) (As,BsJs = As*& - &*As

then the symbol of [A,&] is given by (As,Bs}s. For the Weyl quantization rule the phase space commutator is called a Moyal bracket.'" Using the explicit formula for the twisted multiplication of the Weyl quantization rule,19 we find that the Moyal bracket of two monomials is (4m~Pn~,4m2Pm)M =

min(ml+m2,n,+n2) ( ih)k c - C(k,ml ,m2,nl,n1)qml+m2-kpnl+n2-k (2 1) k = l ; k o d d 2k-1

where

(22)

The lowest order term in h of the Moyal bracket is given by ih times the Poisson bracket. The higher order terms in h can be viewed as quantum corrections to the Poisson bracket. Although the precise nature of these corrections will in general depend on the particular quantization rule used, many rules give mock phase space commutators that are equal to the Poisson bracket to lowest order in h.

D . Van Vleck Perturbation Theory in Mock Phase Space. Given the concepts presented above, a mock phase space formulation of Van Vleck perturbation theory can be derived. The starting point is the quantum mechanical Hamiltonian H defining the problem of interest. We take H to be a function of harmonic oscillator creation-annihilation operators. The Hamiltonian can

The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3149

then be transformed into a symbol Hs(a*,a), where a and a* are the classical analogues of creation-annihilation operators. These are defined as

and ai* is the complex conjugate of ai. Let us further assume that the quantization rule used is of the following form:

min(m,n)

/=0 a*man - c C ( m , n , l ) h ' a + " - ' a " - ' (24)

(Both the Weyl quantization rule and the symmetrization rule are of this form.) Then a-Hamiltonian symbol Hs will correspond to a diagonal operator H if and only if Hs is a function of the products ai*ai only.

We therefore seek a sequence of unitary transformations which render the Hamiltonian as nearly diagonal as possible. This is expressed in mock phase space as

The procedure for choosing the generating function fk is similar to that of the classical theory. On the kth iteration, we choose fk to make the transformed Hamiltonian a function of ai*ai. Resonances between the zeroth-order frequencies will lead to complications, just as they do classically.

Explicit formulas can be derived by expanding all quantities in powers of e. Let Hk be the Hamiltonian symbol after k transformations. Expanding Hk in powers of c yields

Hk = C$H,k (26)

Inserting this into eq 25 gives

The transformations are carried out by expanding the phase space commutator in powers of h. The structure of the resulting perturbation theory can be understood for the Weyl quantization rule by examining eq 21. A term in the expansion of ( l / i h ) ( }s which is of j t h order in h is 2j polynomial degrees less than the classical term. The integer j must be even, because the Moyal bracket has an expansion in odd powers of h . The first nonclassical term is therefore a polynomial of four degrees less than the classical series. In the large quantum number limit, the nonclassical terms become negligible compared with the classical terms, because they involve lower powers of quantum number.

Since orders in h are in 1 : 1 correspondence with polynomial degrees, we can estimate the maximum number of terms that could possibly be generated by including a higher order correction in h. A polynomial of degree m, in N degrees of freedom, can have

( N + m - l)! (28) ( N - l)!m!

terms. Therefore, the ratio of the number of terms of thejth-order correction in h to that of the classical series is, a t order ( m - 2) in perturbation (polynomial degree m)

( N + m - 2j - l ) ! ( m - 2j )! ( N + m - l)!m! (29)

Apparently, very high order corrections in ti will produce many fewer terms than the classical theory. Therefore, we expect that inclusion of low-order corrections in h will require a significant fraction of the amount of work required for a fully quantum calculation. Conversely, it is shown below that inclusion of just the first correction in h can lead to substantial improvement in the accuracy of energy eigenvalues. Mixed approximations, where high orders in t are calculated to low order in h , and vice versa,

3150 The Journal of Physical Chemistry, Vol. 92, No. 1 1 , 1988 Fried and Ezra

TABLE I: Perturbation Coefficients for the Anharmonic Oscillator of Eq 31 sco hlb ti" ti21 OM'

3.750 000 00E-1 7.500 000 00E-1 7.500 000 OOE- 1 7.500 000OOE-1 7.500 000 OOE- 1 5.3 1 2 500 00E-1 3.187 50000E+O 2.625 000 OOE+O 2.625 00000E+O 2.625 00000E+O 1.464 843 75E+O 5.219238 28E+O 2.137 426 75E+ 1 9.559323 12E+1 4.543 799 91E+2 2.258 5 3 4 6 6 E f 3 1.161 71906E+4 6.1 39 452 94E+4 3.316555 31E+5 1.82442985E+6 1.019 052 00E+7 5.766 587 07E+7 3.300043 77E+8 1.907 098 52E+9 1.1 11 646 18E+10 6.529 424 70E+10 3.861 371 59E+11 2.297 548 35E+12

3.5 15 625 OOE+ 1 6.263 085 94E+2 1 S 3 8 947 26E+4 4.817 898 85E+5 1.832060 13E+7 8.195 77058E+8 4.215646 15E+10 2.450 673 15E+ 12 1.588635 30E+14 1.136 076 26E+16 8.883921 17E+17 7.540818 10E+19 6.904612 24E+21 6.783 309 67E+23 7.117 17429E+25 7.942 725 23E+27 9.394 338 70E+29 1.173 839 10E+32

2.081 25000E+1 2.412 89063E+2 3.58098047E+3 6.398 281 35E+4 1.329 733 72E+6 3.144821 47E+7 8.335 41604E+8 2.447 89407E+10 7.893 333 16E+11 1.890 288 55E+13 3.779 395 70E+ 15 1.586 51031E+18 4.402 956 89E+20 1.067 904 22E+23 2.420 939 30E+25 5.294 153 48E+27 1.137741 90E+30 2.431 850 38E+32

2.081 25000E+1 2.412 89063E+2 3.58098047E+3 6.398 281 35E+4 1.329 733 72E+6 3.144821 47E+7 8.335 41604E+8 2.447 89407E+10 7.893 333 16E+11 2.773 877 69E+13 1.05564666E+15 4.326 81068E+16 1.900817 19E+18 8.912 101 78E+19 4.442 550 86E+21 2.346 464 34E+23 1.309 150 19E+25 7.694 000 92E+26

2.08 1 250 OOE+ 1 2.41289063E+2 3.580980 47E+3 6.398 281 35E+4 1.329 733 73E+6 3.444821 47E+7 8.335 41603E+8 2.447 89407E+10 7.893 333 16E+11 2.77387770E+13 1.055 646 66E+15 4.326 810 68E+16 1.900 817 20E+18 8.912 101 78E+19 4.44255089E+21 2.34646431E+23 1.309 150 26E+25 7.693 999 85E+26

"Semiclassical perturbation expansion. E-1 ( E + l l and so on stand for X10-' (10') etc. * Weyl quantization rule used in ti expansions. cQuantum mechanical perturbation theory of ref 89

could possibly come close to the accuracy of a fully quantum calculation, with only slightly more work than a classical calculation. Such an approach would be consistent, because high-order terms in E are expected to be quite small, and thus can be calculated more approximately than low-order terms. The utility of such approximation schemes is under current investigation.

The expression determining fk is of special interest, since it sheds light on the relation between the small denominator problem in classical and quantum mechanics:

Consider this equation when the Weyl quantization rule is used. The Moyal bracket will reduce to i h times the Poisson bracket if the zeroth-order system is a harmonic oscillator. Therefore, f k is determined by the same equation as in the classical theory. This implies that the two perturbation series have precisely the same small denominator problem. As a partial solution to this problem, we do not eliminate any terms that would produce denominators small enough to destroy the apparent convergence of the series.

After the desired number of transformations is carried out, the appropriate quantization rule is employed to convert the simplified Hamiltonian symbol into an operator. For nonresonant systems, a diagonal Hamiltonian operator is obtained; energy eigenvalues are determined simply by evaluating the diagonal matrix elements. For resonant systems, the quantized Hamiltonian is not diagonal. If the number of linearly independent resonances found is less than the number of degrees of freedom, however, it will be block diagonal. Only a small matrix diagonalization is then required.

By expanding the phase space commutator in powers of h , the transformations can be carried out to a given power of h. If only the lowest term in h is kept, the phase space commutator becomes a Poisson bracket, and ordinary AQ is obtained. Only part of the full h dependence of the quantum problem, however, is described by the expansion in powers of h . The small matrix diagonalizations required for resonant systems introduce possibly nonanalytic behavior in h. This explains the success of AQ in reproducing splittings which are entirely quantum mechanical-the h dependence associated with such splittings is well reproduced by the small matrix diagonalizations. In the next section we apply the generalized AQ procedure to a variety of problems. It will be shown that inclusion of corrections in h often leads to substantial improvements in the accuracy of energy eigenvalues.

IV. Results In this section, four systems are treated with the generalized

AQ procedure described above. The one-dimensional quartic

oscillator is studied as an example of a system with a divergent perturbation series. Three multidimensional systems are then examined. A three-mode model for rotationless SOZa4 is treated as an example of an effectively nonresonant system. The resonant three-dimensional system first treated by Noid, Koszykowski, and Marcusa5 is quantized next. Finally, we calculate vibrational energy levels for ozone (03), a nearly resonant system with strong anharmoni~i t ies .~~ We show that generalized AQ is able to reproduce the results of extremely large variational calculationsa6 very well. The addition of corrections in h is found to improve the accuracy of energy eigenvalues considerably.

A. The Quartic Oscillator. As an example of generalized AQ applied to a simple system, we treat the quartic oscillator. This system is given by the Hamiltonian

H = '/2(q2 + p 2 ) + q4

Despite its apparent simplicity, this system has been studied in- tensively, since it can be viewed as a quantum field theory in one space-time dimension with a s e l f - i n t e r a c t i ~ n , ~ ~ ~ ~ ~ The perturbation expansion of the quartic oscillator has therefore been examined to gain insight into the convergence properties of perturbative techniques in quantum field theory.60,61 For the purposes of this paper, however, we simply regard it as a well- established example of a system with a divergent Rayleigh- Schrodinger perturbation series.

Bender and Wusg390 found the perturbation expansion of this system's ground-state energy to be of the form

m

E = 1/2 + ~( -1 )"- ' t"A, (32)

They found the coefficients A,, for n I 75 by iterating a difference equation. Table I compares the first 20 coefficients found by Bender and Wu with those obtained by ordinary semiclassical quantization and generalized AQ. We obtained the first column by semiclassical quantization of the Birkhoff-Gustavson normal form. The semiclassical perturbation expansion is clearly much

n= 1

(84) Kuchitsu, K.; Morino, Y . J . Chem. SOC. Jpn. 1965, 38, 805. Barbe,

( 8 5 ) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J . Chem. Phys.

( 8 6 ) Frederick, J., private communication, 1987; Frederick, J. H.; Heller,

(87) Simon, B. Phys. Reu. Lett. 1970, 25, 1583. (88) Bender, C. M.; Wu, T. T. Phys. Rev. Lett. 1968, 21 , 406. (89) Bender, C . M.; Wu, T. T. Phys. Reo. 1969, 184, 1231. (90) Bender, C. M.; Wu, T. T. Phys. Reo. D 1973, 7 , 1620.

A,; Secroun, C.; Jouve, P. J . Mol. Spectrosc. 1974, 49, 17 1.

1980, 73, 391.

E. J . J . Chem. Phys. 1987, 87, 6592.


TABLE II: Parameters for the SO2 and O3 Hamiltonians of Ref 84 (in cm-I)

parameter SO, 0, parameter SO, 0, "I 1171 "2 525 " 3 1378 k l , , 44

k122 -1 2 k112 -19

k133 159 k222 -7.0

11 34.9 k233 4.7 -59.3 716.0 kllll 1.8 2.2 1089.2 k1122 -3.1 -1.8 -48.1 k1133 15 28.3 -29.7 k2222 -1.4 0.6 -25.5 k2233 -6.5 -5.9 -225.8 k3333 3.0 6.7 -19.2

less divergent than the quantum expansion. This is expected-the semiclassical expansion is known to have a small but finite radius of convergence, whereas the quantum expansion has a zero radius a f c o n ~ e r g e n c e . ~ ~ , ~ ~

Table I gives results obtained with generalized AQ in the next three columns of Table I. The h1 column corresponds to expansion of the Moyal bracket to first order in h , and is therefore equivalent to ordinary AQ implemented with the Weyl quantization rule. This yields the exact quantum result for the first correction, but all subsequent corrections are very different from the quantum values. Note also that the series obtained by means of the Weyl quantization rule diverges faster than the quantum series. This suggests that perturbation series derived with the Weyl quantization rule might be harder to resum than those derived semi- classically, at least for one-dimensional systems. The summability of classical perturbation theory when applied to multidimensional systems is largely unexplored.

The column labeled h" gives results derived by expanding the Moyal bracket to 11th order in h. The first 11 expansion coefficients are nearly identical with those given by Bender and Wu. The 12th expansion coefficient, however, is markedly different. This is because the order a t which the h expansion of the Moyal bracket truncates is dependent upon the polynomial degree of the normal symbol. Thus, incorporating corrections to given order in h makes a finite number of perturbation coefficients match the quantum values. The nonmatching coefficients, however, are not necessarily close to the quantum values. Given that enough orders in h are included, generalized AQ will match quantum perturbation theory to any given order in e, but only because the series of corrections truncates in h. It is in this sense that the expansion in h converges to the quantum result.

Finally, we display coefficients that are accurate to 21st order in h in Table I. These match the calculations of Bender and Wu closely for almost all the coefficients given. Our results demonstrate that generalized AQ becomes equivalent to quantum perturbation theory, provided that enough orders in h are included. The agreement is slightly worse for the last few coefficients. We believe this is due to roundoff error. The formula for the Moyal bracket of two polynomials (21) contains factorial terms that make it susceptible to roundoff error. The Weyl quantization rule also contains such factorial terms. Thus, generalized AQ is best used to relatively low order in h , if numerical problems are to be avoided. If very high order corrections in h are required, it is probably best to do fully quantum perturbation theory directly. Nonetheless, for many systems the use of just the first few corrections in h yields very accurate results. We give examples of three such systems below.

B. A Nonresonant System. The application of generalized AQ to effectively nonresonant, weakly coupled systems is straight- forward. To demonstrate this, we have quantized the vibrations of rotationless SO2, a nonresonant system with three degrees of freedom. The Hamiltonian for this system iss4

The parameter values are given in Table 11. The anharmonic part of the potential is ordered according to polynomial degree. Thus, the cubic terms were taken to be of order e, while the quartic

TABLE III: Energy Eigenvalues for the SO, Hamiltonian of Ref 84

NI N2 N3 Est" Elb E,' ESd EQMe

0 0 0 1528.93 1526.17 1530.27 1530.27 1530.27 0 1 0 2044.82 2041.93 2046.15 2046.15 2046.15 0 2 0 2555.00 2551.96 2556.32 2556.32 2556.32 1 0 0 2684.47 2681.93 2685.96 2685.96 2685.96 0 0 1 2888.21 2885.67 2889.50 2889.50 2889.50 0 3 0 3059.22 3056.01 3060.52 3060.52 3060.52 1 1 0 3197.53 3194.86 3199.02 3199.03 3199.03 0 1 1 3398.82 3396.14 3400.09 3400.09 3400.09 0 4 0 3557.17 3553.77 3558.45 3558.45 3558.45 1 2 0 3704.77 3701.94 3706.25 3706.25 3706.25 2 0 0 3832.69 3830.39 3834.33 3834.33 3834.33 0 2 1 3903.64 3900.81 3904.89 3904.89 3904.89 1 0 1 4031.72 4029.41 4033.14 4033.14 4033.14 0 5 0 4048.50 4044.88 4049.75 4049.75 4049.74 1 3 0 4205.90 4202.89 4207.37 4207.37 4207.37 0 0 2 4238.76 4236.45 4240.02 4240.02 4240.02 2 1 0 4342.57 4340.14 4344.22 4344.22 4344.22 0 3 1 4402.42 4399.41 4403.64 4403.64 4403.64 0 6 0 4532.79 4528.91 4534.01 4534.01 4533.97 1 1 1 4538.93 4536.49 4540.33 4540.33 4540.33 1 4 0 4700.60 4697.39 4702.07 4702.07 4702.06

2 5 1 7621.20 7618.06 7622.59 7622.59 7622.49 3 0 2 7621.92 7620.27 7623.43 7623.42 7623.51 0 7 2 7638.33 7634.48 7639.24 7639.25 7638.79 0 13 0 7668.14 7660.64 7669.45 7669.46 7712.23 0 10 1 7688.39 7683.06 7689.21 7689.22 7687.43 1 2 3 7688.62 7686.50 7689.91 7689.91 7689.91 2 8 0 7723.07 7718.54 7724.67 7724.65 7724.29 5 1 0 7733.28 7731.65 7735.26 7735.26 7737.42 3 3 1 7778.19 7775.84 7779.75 7779.74 7779.78 1 5 2 7813.01 7809.90 7814.18 7814.18 7814.10 2 0 3 7825.18 7823.55 7826.57 7826.57 7826.59 0 2 4 7896.00 7893.91 7897.22 7897.22 7897.21 1 8 1 7901.73 7897.22 7902.82 7902.81 7902.45 3 6 0 7912.02 7908.60 7913.88 7913.86 7913.65 4 1 1 7919.55 7917.79 7921.19 7921.19 7921.67 1 11 0 7938.30 7931.88 7939.68 7939.67 7943.79

Semiclassical quantization using a Birkhoff-Gustavson normal form. bCalculation done to 12th order in perturbation and 1st order in h. 'Calculation done to 12th order in perturbation and 3rd order in h . dCalculation done to 12th order in perturbation and 5th order in h. eVariational calculation of ref 86.

. . . . . . . . . . . .

terms were assigned order e2 , This is consistent with the size of the anharmonicities in Table 11.

Table I11 compares results derived by EBK quantization of a classical normal form and generalized AQ with a large-scale variational calculation.86 All perturbative calculations were done to 12th order in e. The variational calculation86 used a 11 by 24 by 10 (2640 function) Cartesian harmonic oscillator basis set. Energies derived by semiclassical quantization of a classical normal form are presented in the first column Table 111. These energies are typically within a few wavenumbers of the variational results, even for the group of highly excited states given.

In the next three columns of Table 111, we present results found by AQ. The column labeled hl gives results for ordinary AQ implemented with the Weyl quantization rule. The semiclassical results are generally better than those derived by the Weyl rule. The situation changes dramatically, however, as soon as higher order corrections in h are included. We achieve agreement with the quantum results for the first group of states to within 0.01 cm-I. Good agreement is also seen for the group of excited states. Convergence to the variational results, however, is nonuniform. For instance, the state (1,2,3) shows agreement to within 0.01 cm-'; a much larger discrepancy is seen with the state (0,13,0). We believe that the quantum calculation may not be completely converged for states with high bending excitation. An estimate of the error of our result can be obtained by taking the difference between the present 12th-order calculation and a 10th-order calculation. For the state (0,13,0) we estimate our error to be about 3 cm-I. Table I11 also presents results for a calculation done to fifth order in h. These are nearly identical with the calculation

3152

TABLE IV: Energv Eigenvalues for the Three-Dimensional Hamiltonian of Ref 85

The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 Fried and Ezra

0 1 0 0 2 I 1 0 3 0 2 0 2 1 4 I 0 3 I 0

2 1 5 8 I 0 4 3 7 10 2 5 I 0 1 4

0 0 0 1 0 0 1 0 0 I 0 2 1 0 0 1 0 0 2 I

1 0 I 1 4 3 0 3 0 0 2 2 1 0 5 I

0 0 1 0 0 1 0 2 0 1 1 0 0 2 0 1 3 1 0 2

4 6 2 0 1 3 4 1 2 0 3 1

7 0 3

<

1.492 96 2.18435 2.48474 2.771 17 2.872 31 3.175 81 3.450 75 3.472 79 3.55673 3.754 19 3.863 42 4.036 25 4.12674 4.163 33 4.237 51 4.432 56 4.456 87 4.547 48 4.703 34 4.732 96

8.021 52 8.068 97 8.081 55 8.091 13 8.11524 8.16436 8.16808 8.191 86 8.228 52 8.238 66 8.262 74 8.293 98 8.312 85 8.349 71 8.367 44 8.388 30

1.490 42 2.181 66 2.482 19 2.768 35 2.869 45 3.173 09 3.447 74 3.470 19 3.553 69 3.751 33 3.860 5 1 4.033 09 4.12351 4.16054 4.234 28 4.429 50 4.454 21 4.544 37 4.699 94 4.73003

8.017 65 8.065 75 8.076 88 8.085 78 8.11057 8.16033 8.16445 8.18691 8.224 36 8.233 67 8.258 46 8.288 80 8.309 16 8.346 60 8.362 13 8.383 77

1.493 76 2.185 16 2.485 68 2.771 87 2.873 12 3.17672 3.45 1 46 3.473 86 3.557 55 3.755 07 3.864 32 4.036 82 4.12747 4.16437 4.238 35 4.433 40 4.458 09 4.548 38 4.703 96 4.73401

8.022 49 8.070 89 8.081 94 8.091 75 8.11593 8.16570 8.1 70 01 8.19256 8.23021 8.239 81 8.263 50 8.294 46 8.314 18 8.351 73 8.367 58 8.388 78

1.493 75 2.185 15 2.485 67 2.771 86 2.873 1 1 3.17672 3.451 45 3.473 85 3.557 54 3.755 06 3.86431 4.036 81 4.127 46 4.16436 4.238 34 4.433 39 4.458 08 4.548 37 4.703 94 4.73400

8.022 49 8.070 86 8.08 1 93 8.091 7 5 8.1 15 90 8.16567 8,16998 8.19254 8.230 18 8.239 81 8.263 50 8.294 46 8.314 18 8.351 69 8.367 54 8.388 78

1.49375 2.185 15 2.485 67 2.771 86 2.873 11 3.17672 3.451 45 3.473 85 3.557 54 3.75506 3.864 31 4.036 81 4.12746 4.16436 4.238 34 4.433 39 4.458 08 4.548 37 4.703 94 4.73400

8.022 49 8.070 86 8.081 93 8.091 75 8.11590 8.16567 8.16998 8.19254 8.230 18 8.239 80 8.263 50 8.294 46 8.314 18 8.351 69 8.367 54 8.388 78

1.493 75 2.185 15 2.485 67 2.771 86 2.873 11 3.17672 3.451 45 3.473 85 3.557 54 3.755 06 3.864 31 4.036 81 4.12746 4.16436 4.238 33 4.433 39 4.458 08 4.548 37 4.703 94 4.734 00

8.02246 8.07091 8.081 89 8.091 79 8.116 16 8.16548 8.170 24 8.19235 8.230 28 8.242 89 8.262 96 8.293 87 8.314 13 8.351 78 8.369 54 8.388 80

”Ordinary AQ of a 12th-order normal form, using Robnik’s approximation to the Weyl rule. *Calculation done to 12th order in perturbation and 1st order in f i . ‘Calculation done to 12th order in perturbation and 3rd order in f i . dCalculation done to 12th order in perturbation and 5th order in h . eCalculation done to 12th order in perturbation and 7th order in h . /Variational calculation of ref 86.

done to third order in ti. Thus, inclusion of the first correction to ordinary AQ is sufficient to reproduce the results of a variational calculation well; all other corrections to ordinary AQ are insig- nificant.

C. A Resonant System. Generalized AQ can be applied to resonant as well as nonresonant systems. For resonant systems, however, the simplified Hamiltonian symbol corresponds to a block diagonal Hamiltonian operator. The blocks are then diagonalized numerically. This is a very effective means of treating resonant systems. As an illustration, we have quantized the three-dimensional system first treated by Noid, Koszykowski, and Marcws5

3 2

r = l ,=1 H = XY2(pl2 + wiqI2) - O.lXql(q l+~z + 0.1q12) (34)

The frequencies are

wI = 0.7, w2 = 1.3, w j = 1.0 (35) so the system exhibits an exact 1:l:-2 resonance. Table IV presents energy eigenvalues for this system. The first column is obtained by AQ of a 12th-order normal form, where Robnik’s approximation to the Weyl quantization rule was used. In the first group of states, the average disagreement with the variational results is 0.00085. In the second group, the discrepancy has increased to approximately 0.0013. The second column of Table IV shows results for Weyl-based AQ of the same normal form. The Weyl results agree less well with the quantum calculation, showing an average error of 0.0038 for the first group of states, and 0.0056 for the second. This is in accord with previous ob- servations that Robnik‘s quantization rule generally yields more accurate results than the Weyl quantization rule when applied to weakly coupled o s ~ i l l a t o r s . ~ ~ ~ ~ ’

The use of generalized AQ leads to a remarkable improvement in the energies of the first group of states. For instance, the calculation done to third order in h shows an average error of 1 X in the first group of states. This error is decreased to 5 X lo-’ by including the fifth-order correction in h. In the second group of states, the average error for the calculation done to third order in h is about 0.00046, a substantial improvement over the semiclassical results. The higher order corrections in h are small compared with the typical energy for this group of states; including them does not lead to substantial improvements in energies. If required, further improvement in energies could probably be achieved by including higher orders or perturbation.

D. A Nearly Resonant System. As a final example of generalized AQ, we apply it to a realistic model of 03.84 The Hamiltonian for this system has the same form as that for SO2, with parameters given in Table 11. The O3 potential, however, is more anharmonic than that of SO2. O3 also has several near resonances-if applied straightforwardly, the perturbation theory will diverge. Rather than attempting to resum this divergent series, we included certain nearly resonant terms in the normal symbol.

The choice of which terms to keep in the normal symbol, and which to eliminate, is delicate. A general term in the Hamiltonian can be written as

C( a*)jak (36) where multiindex notation is used (Le., d a?’ ... adN). Eliminating this term gives a denominator (j - k ) w . If the term is treated as resonant, we call the vector j - k a resonance uector. In order

(91) There are counter examples to this. The quartic oscillator, for example, is quantized exact1 to first order in perturbation by the Weyl rule,

has a vanishing cubic term. but not by Robnik’s rule.’ r It may be significant that the quartic oscillator


TABLE V: Energy Eigenvalues for the Oq Hamiltonian of Ref 84 __ - N , N2 N3 ESSCFa E X b ElC E3d ESC E7f E Q M ~ 0 0 0 1459 1449.93 1446.10 1456.49 1456.47 1456.47 1456.47 0 0 1 0 0 1 0 0 1 2 0 1 0 1 0 2 0 0 1 1 2 3 0 1 0 2 0 0 1

1 0 0 2 1 1 0 3 0 0 2 2 1 1 4 1 3 0 0 3 0 0 2 2 5 2 4 1 1

0 1 0 0 1 0 2 0 1 0 1 0 2 1 0 0 1 3 2 0 1 0 2 1 0 0 1 3 2

2148 251 1 2590 2832 3166 3265

3510 367 1 371 1 3814 3934 4158

2 147.46 2492.84 2 5 5 2.7 9 2841.1 3 3163.76 3236.94 3519.94 3530.43 3563.66 3648.66 3827.28 3915.02 4160.61 4211.44 4214.73 4318.93 448 1.84 4529.28 4568.35 4585.91 4629.88 4741.89 4789.06 4845.44 4893.30 4979.95 5125.60 5138.32 5173.07

2142.91 2489.16 2549.8 1 2835.63 3 159.72 3232.45 3510.98 3523.61 3562.44 3650.25 3822.55 3908.50 4152.21 4209.28 4205.79 4318.37 4475.5 1 45 12.19 4559.25 4575.91 4642.10 4746.34 4780.64 4841.35 4880.56 4978.13 5113.16 5 124.36 5165.38

2155.04 2498.92 2560.1 1 2850.16 3172.30 3244.38 3519.82 3541.49 3572.65 3659.58 3839.10 3923.02 4164.03 4222.47 4228.48 4329.04 4497.71 4523.03 4567.02 4594.53 4653.14 475 1.66 4797.16 4858.80 4910.34 4990.84 5147.58 5133.20 5176.94

21 54.97 2498.75 2560.20 2850.02 3172.12 3244.41 3519.42 3541.25 3572.68 3 6 5 9.7 6 3838.87 3922.97 4163.63 4222.46 4228.13 4329.17 4497.47 4522.35 4 5 6 6.9 6 4594.38 4653.15 4752.00 4796.73 4858.74 4909.85 4990.88 5 147.15 5132.58 5176.84

2154.97 2498.75 2560.20 2850.02 3172.12 3244.42 3519.42 3541.25 3572.68 3 6 5 9.7 6 3838.87 3 9 2 2.9 7 4163.64 4222.47 4228.13 4329.17 4497.47 4522.35 4566.96 4594.38 4653.15 4752.00 4796.73 4858.74 4909.85 4990.88 5147.15 5132.59 5176.84

2154.97 2498.74 2560.20 2850.00 3172.07 3244.39 3519.37 3541.18 3572.58 3659.66 3838.46 3922.46 4163.54 4222.15 4227.80 4328.85 4495.28 4522.41 4565.57 4592.20 465 1.92 475 1.04 4796.02 4855.21 4908.58 4988.20 51 44.14 5126.11 5172.28

“Semiclassical self-consistent field (ref 92). bOrdinary AQ of a 10th-order normal form, using Robnik‘s approximation to the Weyl rule. cCalculation done to 10th order in perturbation and 1st order in h. dCalculation done to 10th order in perturbation and 3rd order in h . ‘Calculation done to 10th order in Derturbation and 5th order in h . fcalculation done to 10th order in perturbation and 5th order in h . ZVariational calculation of ref 86.

for the normal symbol to map into a block diagonal matrix, a maximum of N - 1 linearly independent resonance vectors can appear in it. More than N - 1 resonance vectors are easily found for 03. Care must be taken to include in the normal symbol only those terms which would otherwise destroy the convergence of the perturbation series.

We have devised an empirical criterion for deciding whether a given term is to be treated as resonant or nonresonant. On the nth transformation, a term is considered resonant if

(37)

This attempts to force the calculation to converge geometrically with a rate l / y . A very small value of y will produce a rapidly convergent perturbation expansion. It is very likely, however, that enough linearly independent resonant terms will arise to leave the final Hamiltonian with no invariants a t all. On the other hand, if y is chosen to be too large, the perturbation expansion will converge too slowly (if at all) to yield useful information on excited states. As a general rule, we find that y should be roughly the size of the perturbation parameter e, given that dimensionless coordinates are used and that the zeroth-order Hamiltonian is of order unity. This represents the convergence rate one would naively expect to occur if small-denominator problems were not present. To ensure that the final Hamiltonian will be block diagonal, we eliminate a maximum of N - 1 linearly independent resonances.

With this approach, we were able to eliminate divergence from a 10th-order calculation. This does not imply, however, that the choice of terms retained in the final Hamiltonian will produce convergent results for arbitrary orders in perturbation. Ultimately, one must choose between convergence and block diagonal structure. The following vectors span the set of resonances eliminated:

(1 $0,-1) (38)

Table V gives energy eigenvalues for O3 derived by ordinary and generalized AQ of a 12th order normal symbol. Also shown are the semiclassical self-consistent field (SCF) results of Smith, Liu, and Noid?2 along with results from a large-scale quantum variational calculation done by Frederick.86 The ordinary AQ results are less accurate than those given in the previous examples. For instance, the ground-state energy of O,, as calculated by Robnik‘s approximation, disagrees with the variational calculation by about 6 cm-’. This disagreement cannot be improved by calculating more orders in E . Larger discrepancies can be found elsewhere. For instance, the state (0,4,1) has an error of about 18 cm-I. The results obtained from the Weyl quantization of a classical normal form show similar disagreements. The relatively poor performance of semiclassical approximations is probably due to this system’s substantial anharmonicity. The AQ results are, however, generally better than the self-consistent results of Smith et al. In contrast to the SCF approach, AQ does not make any assumptions concerning separability.

Including corrections in h dramatically improves the quality of the eigenvalues. For instance, the first correction to ordinary AQ results in a ground-state energy within 0.03 cm-’ of the quantum value. The addition of higher order corrections gives even better results. A similar improvement is observed for all other states. The last few states shown in Table V have larger errors than the low-energy states. This is because the perturbation series shows a small radius of apparent convergence. A combination of higher order calculation and resummation is probably necessary to produce better results for highly excited states. The convergence of the variational calculation also was slow; only 40 states could be reliably calculated with a 2640-state matrix diagonalization. Highly anharmonic systems provide a challenge to both perturbation theory and variational calculations. The application of recently developed distributed Gaussian-discrete variable representation algorithms to the variational problem could lead to a larger number of converged quantum levels for a given basis size.93

(39) (92) Smith, A. D.; Liu, W. K.; Noid, D. W. Chem. Phys. 1984, 89, 345.

3154 J. Phys. Chem. 1988, 92. 3154-3163

V. Conclusion We have demonstrated that generalized AQ is a useful tech-

nique for finding accurate vibrational eigenvalues. For all the systems considered (except the anharmonic oscillator), including just the first correction in h leads to a significant improvement in energy eigenvalues. Including corrections of high enough order to generate fully quantum results takes little more time than calculating the first correction. As discussed above, this is because high-order corrections in h are polynomials of relatively low degree. We are currently investigating ways of incorporating corrections in h with less effort. For systems with apparently convergent perturbation expansions, it would be consistent to calculate low-order corrections in 6 quantum mechanically, while finding high-order corrections in 6 classically. Such a scheme would probably take only slightly longer than an entirely classical calculation, while generating much better results. Furthermore, generalized AQ could profitably be used to calculate the per-

(93) BaEiE, Z.; Light, J. C. J . Chem. Phys. 1987, 86, 3065. (94) Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical

Systems, and Bifurcations of Vector Fields; Springer-Verlag: New York, 1983.

(95) Voros, A. Ann. Inst. Henri Poincare 1977, 26, 343. (96) Voros, A . In Lecture Notes in Physics; Casati, G., Ford, J. , Eds.;

Springer-Verlag: New York, 1977. No. 93.

turbation expansions of Hamiltonians with complicated functional forms. Since the Moyal bracket is expressed entirely in terms of derivatives, it is much easier to calculate the Moyal bracket of two complicated functions to a given order in h than it is to calculate the commutator exactly.

Our motivation for exploring the technique, however, transcends the purely computational. Generalized AQ offers a framework within which both classical and quantum perturbation theory can be studied on the same footing. This helps to illuminate the close relationship between the theories, and enables h-dependent effects to be introduced in a controlled manner. Mock phase space is conceivably the appropriate setting for deriving a “quantum KAM theorem”, which would be, to lowest order in h, the classical KAM theorem. Finally, we expect generalized AQ to be of use in examining the summability of classical vs quantum perturbation theory. An examination of these problems, as well as the con- struction of wavefunctions with generalized AQ, is under way.

Acknowledgment. We are very grateful to John Frederick for communicating the results of his variational calculations. This work was supported by NSF Grant CHE-8410685. Computations reported here were performed in part on the Cornel1 National Supercomputer Facility, which is supported in part by the NSF and IBM Corp. G.S.E. acknowledges partial support of the Alfred P. Sloan Foundation.

From Classical to Quantum Mechanics with Hard Chaos

Martin C. Gutzwiller IBM T. J . Watson Research Center, Yorktown Heights, New York 10598 (Received: June 15, 1987; In Final Form: October 14, 1987)

Most of the interesting dynamical systems in physics and chemistry are characterized by an intimate mixture of two extremes: integrability on one hand, and “hard chaos” on the other. The first of these is well understood, particularly as one tries to connect classical and quantum mechanics. The second, however, has been the exclusive preserve of the mathematicians and suffers, moreoever, from a dearth of good physical examples, although it is probably more frequent in nature than its integrable counterpart. The energy surface in the phase space of a harshly chaotic system foliates into two sets of smooth manifolds, the stable and the unstable ones, each of dimension equal to the number of degrees of freedom. A classical trajectory is the intersection of a stable and an unstable manifold. Its physical properties can be described by a simple coding scheme that gives a unique account of its whole history. As an example, the anisotropic Kepler problem (AKP), isotropic Coulomb potential with anisotropic mass tensor, is discussed in some detail, because it allows the two sets of manifolds to be constructed by varying only a single parameter and because its code is the simplest imaginable, consisting of all binary sequences. The transition to quantum mechanics, using Green’s function and its trace, leads to expressions that resemble very closely the grand-canonical partition function of a one-dimensional king chain. The energy levels are the zeros in the complex ( l / k T ) plane and should be located on the imaginary axis. This analogy is directly related to the underlying hard chaos and is followed up by studying the locations of these zeros for various coupling parameters in the Ising chain. It appears that some further restrictions have to be invoked, in order to guarantee a reasonable transition from classical to quantum mechanics in the case of hard chaos.

1. Introduction The term “hard chaos” is meant to describe one of the two

extreme conditions that are encountered in the study of phase- space dynamics. The first of these is the regular behavior which all of us have been taught in school as if there was nothing else to be expected in a mechanical system. It is characterized by the occurrence of invariant tori (cf. ref 1) to which the trajectories in phase space are bound, a situation typical of all integrable systems such as the electron attracted by two fixed nuclei. While this property of a dynamical system allows it to be effectively solved, both classically and quantum mechanically, it turns out to be highly exceptional.

( 1 ) Arnold, V. I . ; Avez, A. Ergodic Problems of Classical Mechanics; Benjamin: New York, 1968. Arnold, V. I. Mathematical Methods of Classical Mechanics; Springer: New York, 1978. Lichtenberg, A. J. , Lie- bermann, M. A. Regular and Stochastic Morion; Springer: New York, 1983.

A similarly unusual, but basically simple behavior forms the opposite extreme of utter randomness, no better than the throw of a dice although completely deterministic. This condition, to be called “hard chaos” with the adjective “harshly chaotic”, will be discussed in the present paper because it is not as well-known as the first. Most mechanical systems show a mixture between integrability and hard chaos which I like to call “soft chaos”. It is like a very intimate mix of the two extremes which seem to coexist in very close, complementary regions of phase space. The result is most often understood as a perturbation of the regular behavior, but it could equally well be viewed as a corruption of hard chaos.

Before going into any detailed discussion, however, some natural boundaries have to be mentioned within which these ideas can be applied. They may have a much wider domain, but there is no point in claiming more territory than can be staked out with some well-understood examples. There will be no dissipation; the

0022-3654/88/2092-3154$01.50/0 0 1988 American Chemical Society

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