Laurence E. Fried and Gregory S. Ezra- Generalized Algebraic Quantization: Corrections to Arbitrary Order in Planck's Constant

8/3/2019 Laurence E. Fried and Gregory S. Ezra- Generalized Algebraic Quantization: Corrections to Arbitrary Order in Planck's

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3144 J . Phys. Chem. 1988, 92 , 3144-3154We finally conclu de with some speculation concerning whenone might expect the extra quantum localization observed in thispaper to vanish and for classical and quantum mechanics tocorrespond in classically chao tic regions. Th e simple answer is.when flux across most curves approaches h (see, however, thequantum effects observed in ref 5 7 ) . But how does the quan tummecha nics approach this limit? The work of ref 57 suggeststunneling which increases with energy might be the route for sucha limit. Thi s would be manifested in qua ntu m eigenstates whichbecome more delocalized in phase space as energy is increased,though their peaks m ight rema in in the sam e location even after

a great de al of spreading. Such eigenstates would be consistentwith the work of Berry,94which suggests tha t Wigner transform sof chaotic eigenstate s would have essentially fla t distributions.However, we do not necessarily expect this flattening t o occur in(94) Berry, M. V. J . Phys. A 1977, I O , 2083.

a monotonic fashion, because of increased tunneling, thoughflattening may be monotonic over a large range of energies. Thelack of monotonic spreading might explain t he interference effectsobserved in ref 57. Perhaps we have observed extra spreadingin the 92nd and 93rd stat es studied in this paper and previously.62However, we expect that th e effects observed in the larger doubletspacings may be caused by a very weak avoided crossing whichis experienced more strongly in the 102nd and 104th states.Acknowledgment. It is a pleasure to thank Stephen Gray, RexSkodje, Gregory Ezra , Craig Ma rtens, and Nelson De Leon for

helpful discussions. I also acknowledge the many colla boratorswhose work I summarized in the paper. They are S. Gray, S. Rice,L. Gibson, G. Schatz, M. Ratner, R . Skodje, and R. Steckler. Thiswork was supported by the Office of Basic Energy Sciences,Division of Chemical Sciences,US. epartment of Energy, underContract N o. W-3 1-109-ENG-38.

Generalized Algebraic Quantization: Corrections to Arbitrary Order in Planck's ConstantLaurence E. Fried and Gregory S. Ezra*+Department of Chem istry, Baker Laboratory, Cornell University, Ithaca, New York 14853(Received: August 17, 1987)

The algebraic approach to semiclassical quantization uses a series of canonical transform ations to bring the classical Hamiltonianof interest into a standard, simplified form. A quantizatio n rule is then employed to convert the simplified classical Hamiltonianto a block diagonal quantum operator: Diagonalizing the blocks yields semiclassical eigenvalues. M any quan tization prescriptionsare ava ilable, but the resul ting semiclassical eigenvalues depend upon the rule used. We present a method for deriving correctionsin powers of h (Planck's co nstant ) that is applicable to any invertible quantization rule. The inclusion of these correctionterms decreases the dependence of energy eigenvalues on the quantizatio n rule used, and incorporates quantum effects analyticin h arising from the transforma tions n a controlled manner. For a Hamiltonian which is a polynomial in Cartesia n coordinatesand momenta, the series of h-dependent corrections truncates, so that results from algebraic quantization converge to thoseof its quantum analogue, Van Vleck perturbation theory. Using the Weyl quantization rule, we ca lculate vibrational eigenvaluesfor several multidimensional systems with PERTURB, a special-purpose algebraic manipulation package. It is shown that theinclusion of low-order corrections in h can lead to significant improvements in the accuracy of energy eigenvalues.

I. IntroductionThe development bf semiclassical methods for calculating highlyexcited energy levels of polyatom ic molecules has received muchattention in recent year~.l-~ he semiclassical approach is ofinterest both as a practical alternative to quantum variationalcalculation^^^^ and as a useful interpretive framework for un-ders tanding observed quan tum ph en ~ m e na .~ .~Most semiclassical techniques are based on the EBK quanti-zation approach'J*12 in which invariant tori with actions satisfyingcertain quantization conditions are sought. The primitive sem-iclassical approximation to the energy of a stafe is simply theenergy of a trajectory on the quantizing torus. EBK quantizationworks well in the regular regime, where trajectories are quasi-periodic and good action variables exist.I3 It is not directlyapplicable to strongly chaotic motion, since invariant tori are notpresent throughout most of phase space.C la ss ic al p e rt ur ba t io n t h e ~ r y ' ~ . ' ~rovides one route to se mi-classical in the chaotic regime.16.'7i19,30 orsystems that are effectively nonresonant, (Le., for which the smalldenominator problem is not apparent at the order of the per-turbative calc ulation performed), a series of canonical transfor-mations near the identity formally reduces the Hamiltonian toa function of action only, to the desired order in pe rt ~ rb a ti on . ' ~The actual Hamiltonian is thus replaced by an integrable ap-p r ~ x i m a n t ' ~ ~ ~ ~ . ~ 'hat can be quantized by imposing the EBKconditions on the good actions.+Alfred P Sloan Fellow

We know of no rigorous results justifying this procedure. Th erelation 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, 6 5 , 161.(5) Ezra, G.S.;Martens, C. C.; Fried, L. E. J . Phys. Chem. 1987,91, 3721( 6 ) Carney, G. D.;prandel, 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) Se e articles in NA TO Advanced Research Workshop on QuantumChaos: Chaotic Behauior in Quantum System s, Theory and Experiment;Casati, G . , Ed.; Plenum: New York, 1985.(10) Einstein, V. Vertsch. Drsch. Phys. Ges. 1917, 19, 82. An Englishtranslation by C. Jaff6 is available as JILA Report no. 116, University ofColorado, Boulder, C O .(11) Brillouin, M. L. J . Phy s . 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, 6 4 ,(16) Swimm, R. T.; Delos, J . B. J . Chem. Phys. 1979, 7 1 , 1706.(17) Shirts, R. B.; Reinhardt, W . P. J . Chem. Phys. 1982, 7 7 , 5204.(18) Schatz, G. .; Mulloney, T. J . Phys. Chem. 1979, 83 , 989.(19) Ja m , C.; Reinhardt , W . P. J . Chem. Phys. 1982, 7 7 , 5191.(20) Ramaswamy, R.; Siders, P.; Marcus, R. A. J . Chem. Phys. 1980, 7 3 ,

I. G., elleman, R. H. G., Eds.; North H olland: 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


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Generalized Algebraic Quantization Th e Jou rna l of Physical Chemistry, Vol. 92 , No. 11, 1988 3145well as the meaning of classical perturbation theory in th e chaoticre gi me , is re ce ivi ng in cr ea se d a t t e n t i ~ n . ~ ~ - ~ ~urchetti3 findstha t the ra dius of convergence of fixed-frequency perturbationtheory corresponds to the value of the p erturbation strength a twhich an invariant torus is destroyed. Graffi an d usingthe Bargmann representation of quantum mechanic^,^^ haveproved39 hat EBK quantization of the Birkhoff-Gustavson normalform is accurate to first order in h, and to all orders in pertur-bation, if the quantizing action values correspond to a KAMt o r ~ s . ~ 3 ~ ~ as recently introduced what he terms a quantumnormal form, defined by analogy with the classical Birkhoff-Gustavson normal form. To lowest order in h , the quantumnormal form is the same as the usual classical normal form.

Resonant systems cannot be quantized by the perturbationprocedure described above, since the Ham iltonian ca nnot be madea functio n of actions without introducing infinite corrections.I3It is also possible tha t th e corrections, while finite, are stronglydivergent. W e will call such systems nearly resonant. To avoidsmall-denominator problems associated with exact and nearresonances, terms depending on resonant combinations of anglesmust be kept in the transformed Hamiltonian.13~19~2z~23~30~4zneffect, apparen t convergence (Le., convergence to a given orderin perturbation) is obtained at the price of a more complicatedform for the transformed H amiltonian. If a single resonantcombination of angles appears, the final Ham iltonian has onenonignorable coordinate. Th e system is therefore equivalent toa one-dimensional problem; one-dimensional semiclassical pro-cedures can be used to quantize the Hamiltonian in the newcoordinate system.16,19,21,27,43wim m an d Delos,16 using prim itivequantization, and JaffE and Reinhardt, lg using a uniform pro-cedure, showed this technique to be effective even when the actu alclassical motion is chaotic.For systems requiring the retention of one or more resonantterms in the transformed Hamiltonian, algebraic quantization(A Q ) has been ~ ~ e d , ~ ~ - ~ ~ , ~ ~n this procedure, a quantizationr ~ l e ~ ~ , ~ ~urns 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, . 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. Let t . 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 . Mat h . Phy s. 1985, 26, 10 .(33) Eckhardt, B. J . Phys. A 1986, 19, 2961.(34) W ood, W. R.; Ali, M. K. J . Phy s . A 1987, 20, 351.(35) Graffi, S.; aul, 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,94and that theperturbation is a polynomial potential. Other, more technical, conditions applyas 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. Let t . 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.

quan tum me chanical operator. If the number of linearly inde-pendent resonant com binations of frequencies encountered is lessthan the num ber of degrees of freedom, the classical Hamiltonianhas constants of the motion that are linear in zeroth-order actions.The corresponding quantum mechanical Hamiltonian is blockdiagonal.30 Th e blocks ar e often finite, so that the calculationof eigenvalues requires only the diago nalization of small ma trices.30In a previous paper,30we showed that A Q yields accura te andreliable eigenvalues for a variety of pe rturbed harmonic oscillatorswith up to five degrees of freedom. One unsatisfacto ry aspectof AQ, however, is that there are many quantization rules sat-isfying the basic requirement of linearity. Several rules werestudied, and it was found that the accuracy of resulting semi-classical eigenvalues depends on the particular quantizatio n ruleused.Wood and Ali34have also noted this dependence on quantizationrule. They have argued tha t the applicability of the Birkhoff-Gustavson app roach is consequently severely limited, since thenormal form cannot reproduce the series generated by Ray-leigh-Schrodinger perturbation theory. Th e Birkhoff-Gustavsoliprocedure can therefore never yield exact eigenvalues, even if itis resummed. While this is true in principle, numerical experienceshows that many anha rmonic systems can be quantized to ac-ceptable accuracy with AQ . Nevertheless, for some systems, suchas a three -mode model for 03,ignificant discrepancies betweenlow-lying quantum and A Q energies are found to persist to highorders in perturbation. Wood and Ali also point out that littleis known abou t the summability of classical normal forms. The ysuggest that normal forms ma y be summab le only for integrablesystems. The work of Graffi an d Paul mentioned above, howeuer,gives hope that classical normal forms are summable when thequantizing values of the action correspond to a K A M torus. Thesummability of classical normal farm s when the underlying motionis chaotic is poorly understood. Nonethe less, accurate semic lassicalenergies have been obtained without resumm ation for energiesin the chaotic regime.In the present work, we generalize AQ to incorporate correctionsto the transform ations in powers of h . For the systems studiedhere, the inclusion of these corrections in h decreases the de-pendence of energy eigenvalues on the quantization rule. Forpolynomial Hamiltonians, it is possible to remove al l dependenceon the qu antization rule by including sufficiently high orders ofh . In doing so we arrive at results equivalent to usual quan tummechanical Van Vleck perturbation theory.Ou r strategy is to replace the Poisson brackets used to carryou t canonical transformations in AQ with a representation of thequant um mechanical commutator in a mock phase space.46 W ethen derive corrections to AQ by expanding the m ock phase spacecomm utator in powers of h . The form of the phase space com-mut ator depends on the quantization rule used. If the Weylquantization ruleM s chosen, the m ock phase space representationof the comm utator is the Moyal bracket.47 In this paper weimplement the AQ algorithm, replacing Poisson brackets withMoyal bracke ts. W e find tha t the inclusion of corrections in hoften leads to markedly better results than a purely semiclassicalapproach (Le., no h-dependent corrections). For -polynomialHam iltonians, the comm utator is exactly represented by finitelymany terms. When enough orders in h ar e included, generalizedA Q exactly reproduces results derived from qu antu m perturbationtheory applied to polynomial Ha miltonian s.The structu re of the pape r is as follows: In sectio n 11, we reviewthe A Q approach, and point out the difficulties involved in a choiceof quantization rule, as well as problems associated w ith smalldenom inators. In section 111, a general proced ure for implementingAQ to higher order in h is given. We derive explicit formulasfor generalized A Q, using the Weyl quantization rule. SectionIV presents results obtained with PERTURB, a special purposealgebraic manipulation package, for a variety of resonant andnonresonant three-dim ensional 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. 949, 45 , 99 .


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3146discuss the usefulness of generalized AQ , both as a practical toolfor finding semiclassical eigenvalues and as a w ay of studying thesummability of classical perturbation theory when applied toquan tum problems.11. A Review of Algebraic Q uantizationA. Lie Transforms. In AQ, a classical Hamiltonian is subjectedto a sequence of canonical transformations before it is quantized.The f ir st 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 , N o . 11, 1988 Fried an d Ezra

and the new coordinates are denoted by bars. The functionalinversion required to obtain an explicit formula for one set ofvariables in terms of another is a principal difficulty in imple-menting the Birkhoff-Gustavson method.Lie transforms,% however, are more convenient than gene ratingfunctions for carrying out c anonical transforma tions, since theyeliminate the need for functional inversion. There a re severalapproaches to Lie transforms, and m any perturbative algorithmsbase d o n t he m h ave been p r o p o ~ e d . ~ ~ - ~ ~e briefly discuss themethod du e to Dragt a nd Finn,55,56 hich is among th e simplestof such techniques. Th e Dragt-Finn algo rithm is of special sig-nificance for th e present work, since it is the classical analogueof Van Vleck perturbation theory.58 Despite the variety ofmethods available, it should be stressed tha t all forms of classicalperturbation theory based on a direct expansion in a small pa-rameter give identical results when taken to the same order. Thus,the traditional Birkhoff-Gustavson technique, as well as Lietransforms, all give the sam e simplified H amiltonian.Lie transforms express a canonical transformation entirely interms of Lie operators. A Lie operator, tf,s defined by

L / = K*l (2 )where { , } enotes the Poisson bracket. Th e exponential of a Lieoperator generates a canonical transformation;

z + Z = exP(L/)z (3 )is canonical. This result is the basis of Dragt-Finn pertu rbati ontheory. The transformation 3 also includes a correspondingtransformation on functions. Let

dz) = g t W ) (4)The new function g can be expressed directly in terms of Lieoperators:

= e x P ( q g ( 4 ( 5 )Dragt and Finn introduce a sequence of L ie operators to simplifythe Hamiltonian

K ( z ) H ( Z ( z ) )= exp(ckFk) exp(ek-Fk-l)...exp( tF, )H(z) ( 6 )where Fk s the Lie operator LA. The sequence of transformationsis chosen to make t he new Ham iltonian a s close to integrable aspossible. If an int ege r linear combinatio n of zeroth-order fre-

(48) Birkhoff, G . D. Dynamical Systems; A. M. S . Colloquim Publications:(49) Gustavson, F. G. Astron. J . 1966, 7 1 , 670.(50) Cary, J . R . Phy s . R e p . 1981, 7 9 , 129.(51) Hori, G . Publ . Astron. SOC. pn. 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, . M . J . Mat h . Phy s . 1976, 17 , 2215.(56) Dragt, A. J. ; Finn, J. M . J . Mat h . Phy s . 1979, 20 , 2649.(57) Dragt, A. J .; Forrest, E. J . Mat h . Phy s . 1983, 2 4 , 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 anishes identically, a frequency commensurabilityis said to exist. Syst ems with frequency comme nsurabilities a recalled resonant. Th e sequence of transformations 6 cannot makea resonant Hamiltonian into a function of good actions only.Systems without exact frequency commensurabilities areclassified as either nonresonant or nearly resonant. Th e small-denominator problem ultimately destroys the convergence ofclassical canonica l perturbation theory,59so a strict distinctionbetween nearly resonant and nonresonant Hamiltonians is notpossible. Operation ally, we use the term s nearly resonant andnonresonant to indicate how quickly the perturbation seriesdiverges. If divergence is appa rent before the desired order inperturbation is reached, the system is termed nearly resonant;otherwise , it is called nonresonan t. Stric tly speaking, every systemis nearly resonant if one goes to sufficiently high order in per-turbation, or if the perturbation is sufficiently large.For nonresonant systems, all the angle dependence of th etransformed Hamiltonian K can by definition be eliminated toa given finite order without encounterin g divergence. Energy levelscan be found by substituting quantizing values of th e good actionsinto K . For nearly resonant systems, the angle dependence of Hcan be formally eliminated, at the cost of introducing rapidlydivergent corrections into the perturbation expansion. The sim-plified Hamiltonian which results from eliminating all angle-dependent terms except those which are exactly resonant is calledthe Birkhoff-Gustavson normal form.48.49Th e presence of an-gle-dependent terms or divergent corrections prevents perturbativeEBK quantization from being applied straightforwa rdly to reso-nant or nearly resonant systems.B . Quantizationof Resonant and Nearly Resonant System s.The re are several ways to quantize reson ant or nearly resonantsimplified Ham iltonians. Resum mation can beapplied to the divergent normal form s of nearly resonant systems.Ali, Wood, a nd D e ~ i t t , 6 ~sing a PadE app roxim ation ~ c h e m e , ~ ~and Arteca,66using a functional meth ~d,~ -~Oave been able toresum th e normal form of a one-dimensional qua rtic oscillator.Even though one-dimensional problems cannot have small-de-nomina tor problem s, the classical perturbation series of this systemhas a small , bu t f in ite, r adiu s of c o n ~ e r g e n c e , ~ ~ , ~ ~n contrast tothe zero radius of convergence generically found in multidimen-s iona l sys tems. Farr e lly and U ~ e r * ~ave applied Pad; approx-imation to several two-dimensional systems, obtaining good resultsfor some states, but poor agreement for others. More work clearlyneeds to be done on the resummation of classical perturbationtheory applied to multidimensionai Hamiltonians.Other procedures are appropriate if the number of linearlyindependent resonant or nearly resonant combinations of fre-quencies encountered in carrying out the calculation is less thanthe dimensionality of the original system. Suppose that t here areQ !inearly independent resonant com bina tions of frequencies.Then , if Q C N , where N is the number of degrees of fr e e d ~ m , ~the final Hamiltonian has Q nonignorable coordinates. If Q =

(59) Benettin, G.; Galgani, L.; Giorgilli, A. In Advances in NonlinearDynamics andStocha stic Processes; Livi, R ., Politi, A,, Eds.: World Scientific:Singapore, 1985.(60) Simon,B. Int . J. Quantum Chem. 1982, 2 1 , 3.(61) Bender, C. M . Int. J . Quantum Chem . 1982, 2 1 , 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.; G raves-Morris, P. Encyclopedia of Mathematics and(65) Baker Jr. , G. A. Essentials of Pad6 Approximants; Academic: Ne w(66) Arteca, G . A. Phys. Rev . A 1987, 35, 479.(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.; C astro, E. A. Phys. Let f .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 th e

i ts A pplications; Addison-Wesley: Reading, MA , 198 1; Volume 13-14.York, 1975.25, 2377.25 , 3492.1 1 1 , 269.Phys. Lett . A 1985, 103, 19 .system of interest.


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Generalized Algebraic Quantization Th e Journal of Physical Chemistry, Vol. 92 , No. 11, 1988 31471, one-dimensional uniform quantization can be applied to thesystem. For Q > 1, however, other methods must be used.Algebraic quantization , which was first applied to systems whereQ = 1 and N = 2,25-27as recently been extended to systems withmore degrees of freedom and m ore resonance^.^^A Q exploits the existence of N - Q ignorable ang le variables.Each ignorable coordinate implies the existence of a classicalinvariant which is a linear function of action. A correspondingquantum problem can be derived by employing a quantizationrule (such as the Weyl rule) to associate a quan tum ope rator Kewith the transformed classical Hamiltonian K . Under mild as-sumptions about the quantization rule, it can be shown that KQalso has N - Q invariants which ar e linear functions of numberoperators 1373.~~he existence of these qua ntum inv ariants impliesthat the operator KQ is block diagonal.30 The blocks are oftensmall and can easily be diagonalized to yield approxim ate quantumeigenvalues of the original system.Th ere ar e several essential difficulties with AQ-the mostsignificant of these is the appearance of small denominator^.^^^^^Although AQ can forestall problems with small denominators,at sufficiently high orde rs in perturbation N resonant or nearlyresonant com binations of frequencies will have been encountered.

Van V leck G en e r a l i z e d AQAQ

Q u a n t u mO perat or Symbol C l a s s i c a lFunc l i on

Input Hami l ton ian cia ~c ai Classica lHami l ton ianSymbo l i , m , tamiltonian

Phase Space

Small MatrixDiagonaii2alion

DiagonalHami l ton ian

Resum mation can sometimes be used in conjunction with A Q toovercome small denominators. The perturbation series for stronglyanharmo nic systems, or systems with m any near resonances, arenevertheless difficult to resum. Therefore, A Q is a useful techniquefor 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 degreesof freedom.73A second shortcoming of AQ is that different quantiza tion rulescan give different r e s ~ l t s . ~ ~ , ~ ~t present, t here exists no obviousway to determine an optimal quan tization rule, although ruleswhich reduce to EBK quantization when applied to diagonalHamiltonians have given good results when applied to severalmolecular problems.30We present a way of generalizing AQ to include correctionswhich are of higher order in h. This procedure allows results fromAQ to be successively improved until convergence to quantummecha nical Van Vleck perturb ation theory is achieved. Th ereare several good reasons for implemen ting a scheme tha t allowsa smooth transition to be made between the semiclassical andquan tum limits. First, it is of some interest to incorporate certainquantum effects (those analytic in h arising from near-identitytransformations) into a problem in a controlled man ner. Thi s is,as we show below, useful in solving vibrational problems t o highaccuracy. Note , however, tha t the entire problem is not subjectto an expansion in h . Nonanalytic behavior in h can be repro-duced by the small matrix diagonalization, a characteristic whichallows our method to avoid the convergence problems usuallyassociated with expansions in h. Second, we expect that insightinto the relation between quantum and classical perturbation

theory, and thereby into the fundam ental problem of quantumi n t e g r a b i l i t ~ , ~ ~ . ' ~ ~ ~an be gained by systematic studies of quantumcorrections to classical perturbation theory.111. Generalized Algebraic Quantization

In this section corrections to A Q are derived. W e first notethat th e quantum analogue of Dragt-Finn perturbation theoryis just the well-known Van Vleck algorithm. A mock phase space(7 3 ) Quantum perturbation theory itself is often divergent. In the nextsection, It is shown t hat Van Vleck perturb ation theory applied to harmonicoscillators h as the s ame small denominator problem as classical perturbationtheory. See, however, ref 36 .(74) Hietarinta, J . Phys. Let t . A 1982, 93, 55 .(75) Korsch, H. J. P h y s . L e u . A 1982, 90, 113.(76) Pechukas, P. J . Phy s . C he m. 1984, 8 8, 4823.

formulation of Van Vleck perturbation theory is then presented.This formulation is particularly suitable for expansion in h .Ordinary A Q can be. recovered as th e lowest order approxim ationto exact quantu m mechanical perturbation theory in mock phasespace.The relation between Van Vleck perturbation theory, its mockphase space version, and AQ is summarized in Figure 1. Va nVleck perturbation theory is formulated in terms of quantumoperators (left column of Figure 1) . In the Van Vleck approach,an initial Hamiltonian is simplified by a sequence of unitarytransformations. Each of these transformations can be expressedin 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 Fermiresonances ar e encountered, the norm al operator will not be di-agonal. Nonetheless, if sufficiently few resonances are found, thenormal operator will have a block diagonal matrix. Therefore,only small matrix diagonalizations are necessary to produce adiagonal Hamiltonian.A mock phase space version of Van Vleck perturbation theoryis illustrated in the center column of Figure 1. In the phase spaceapproach, the initial Hamiltonian operator is transcribed into afunction of 2N variables, called a Th e rule for tran-scription is invertible, so that every quan tum operator is associatedwith a unique symbol. The com mutator used to generate unitarytransformations in Van Vleck perturbation theory can be mappedinto an operator on symbols. Th e Hamiltonian symbol can thenbe simplified by a sequence of transformations generated by phasespace commutators, replacing the Van Vleck generating operatorswith generating symbols. By this sequence of transformations,a simplied Hamiltonian symbol is arrived at, which we term anormal sym bol. The normal symbol is then mapped into a normaloperator by the quantization rul e. This phase space version ofVan Vleck perturbation theory is entirely equivalent to the usualHilbert space version.The m otivation for introducing the phase space theory is thatits classical limit is precisely AQ. Thi s point is illustrated in therightmost column of Figure 1. AQ uses a sequence of canonicaltransformatio ns to simplify a classical Hamiltonian. Th e limitas h - of the Hamiltonian symbol is jus t the classical Ha m-iltonian. Thus, we arrive at the starting point of A Q by takingthe classical limit of fully quantum phase space perturbation

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


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Fried and Ezra148 The Journa l of Physical Chemistry, Vol. 92, No. 11, 1988theory. It is not always necessary to tak e this limit; if no mo-mentum-coordinate cross terms are present in the quantumHamilto nian, the initial Hamiltonia n symbol will be the same asthe classical Hamilton ian. Also, the classical limit of the phasespace commutator is the Poisson bracket. Unitary transbrmationsof the qua ntum theory therefore map into canonical transfor-mations. The simplified classical Hamiltonian is termed a nor ma lform. It can be viewed as an approximation to t he normal symbol.Hence, a quantization rule can be used to derive an approximatenormal operator corresponding to the normal form. In the fol-lowing subsections, we elaborate on the com ments made here andderive a means of approximating phase space Van Vleck per-turbation theory to arbitrary order in h .A. The Quantum Analogue of Dragt-Finn PerturbationThe ory. Th e Dragt-Finn algo rithm is an exp!icitly canonicalperturbation theory, Le., it is expressed entirely in terms of Poissonbrackets. Th e Poisson bracket is the classical limit of l / ( i h ) timesthe quantum mechanical con~mutator . ~This suggests that aqua ntu m analogue of Dragt-Finn perturbation theory can bewritten as

where H, and k r e $11 quantum operators. If f is assumedHermitean, exp(( l/ih )lf ,* ]) is a unitary transformation. I? (7) ,a sequence of uniiary operations transforms Ha miltonian H intothe simpler form K , whereas in the classical version ( 6 ) a sequenceof canonical transformations is used. Th e perturbation theorydefined in (7) is, however, just the Van Vleck method of contacttransformatio ns. This is seen by rearranging (7) into a morefamiliar 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 approxima tions to (7 )directly, by calculating the commuta tors, ordering terms withrespect to noncommuting operators (for example, normal or-dering), and truncating the result at a given order in h. Whentaken to given order in perturbation p aramete r, the expansion inh truncates, provided t hat all quantities a re polynomials in cre-ation-annihilation operators (&,St). This is because the com-mutation relation[&$+I = h (10)

implies that the commu tator of two polynomials in (a,&+)will havefinitely many powers in h . In such a scheme, the choice ofquantization rule is only implicit in th e ordering convention used.W e prefer a mock phase space representation of quantum me-chanics, such as the Wigner-Weyl formalism , and so obtain adirect quantum analogue of AQ.The mock phase space techniques discussed here are mostemphatically not mappings of quantum mechanics onto classicalmechanics; rather, they ar e entirely equivalent to the usual Hilbertspace quantum mechanic^!^.'^*^^ Moreover, a mock phase spacedescription 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; OxfordUniversity: Clarendon, U.K., 1935.(79) Hillery, M.; Connell, R. F.; Scully, M . 0.;Wigner, E. P. Phys. R e p.1984, 106, 121.(80) There ar e problems, however, in finding a simple definition of the classof physically admissable phase space distributions. In contrast, the class ofphysically admissable wavefunctions is easily defined. See Balasz and Jen-n i n g ~ ~ ~or a discussion.

of problems to which semiclassical techniques can be fruitfullyapplied.B. Mock Phase Space Representations of Quantum Mechanics.The re are several equivalent ways to introduce a mock phase spacerepresentation of quan tum mechanics. For instance, an expansionof the vector space of operators in a basis parametrized by c-numbers (a,b) induces a mock phase space representation ofquan tum m echanics; the mock phase space being given by the pair(a,b).46 An expansion of a given operator in terms of projectorsonto coherent states is an example of this.81 We do not followtha t route here. Ra the r, we use invertible quantization rules tomotivate th e use of a mock phase space. This approac h is mostnatu ral in the context of AQ, w here such rules are invoked to turnthe transformed classical Ha miltonian into a quan tum operator.Th e problem of quantizing a classical Hamiltonian containingproducts of noncommuting operators is still largely open. Aquantization rule gives a correspondence between a classicalfunction and a quantum operator. For our purposes, we do notrequire this rule to be correct, in th e sense of giving the qu antu mHam iltonian corresponding to the classical system. Ra the r, weonly desi re uniqueness-every classical functio n should have aunique quan tum counterpart. We will call rule inuertible if thereexists an inverse operator which takes A(fi,fi) into a functionAs(p,q). No te that this inverse rule cannot be the same as merelytaking the classical limit, for taking this limit would ma p manyquantum operators into the same classical function.

An inve rtible rule allows the introdu ction of a mock phase space .Assume that we ar e given a qua ntum operator. Then the inverserule provides us with a unique image of the operator in phase space.This phase space, however, is not th e sa me as th e phase space ofclassical mechanics, because the operations on it (multiplication,etc.) a re different from the corresponding operations in classicalmechanics. To m ake this distinction explicit, the space is referredto as a mock phase space, and the image of an operator is calleda symbol. To lowest order in h , many quan tities in mock phasespace are the sam e as those in the phase space of classical me-chanics, which is the justification for associating the words phasespace with a quantum mechanical object.82Th e only quantization rule explicitly used in this work is theWeyl quantization r ~ l e . ~ ~ , ~ ~his rule associates an operatorA(@,@with a symbol As(p,q) according to the formula46

where h(p,q) is an operator defined byh ( 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 withthe inverse relation46

&(PA) = T r ( 4 B 4 ) & P d ) ( 1 3 )More explicit formulas can be given for monomials in thevariables (p,q). For notational convenience, we restrict ourselvesto a monomial in one dimension. The Weyl quantization procedurethen gives

min(m,n) (ifj,)W(pq) = i=n -+y)(y)p- q-/ (14)

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

The operator function jLj provides a specific illustration of theinvertibility of the Weyl quantization rule. Its symbol is q p -(81) M izrahi, S. . Physicu A (Am sterdam) 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 a re so metim es a p p l i ~ a b l e . ~ ~ . ~ ~(83) Weyl, H. The Theory of Groups and Quan tum Mechanics; Dover:New York, 1950.


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Generalized Algebraic Quantization( i h / 2 ) . Applying the Weyl quantization rule to this symbolrecovers the original operator.

C. Symbol Calculus . As discussed in the previous section,invertible quantization rules allow operators to be represented byfunction s in a mock phase space. This is only one step towarda formulation of q uan tum m echanics based entirely on entitiesin a mock phase space. To complete this formulation, we mustexamine how operations in qu antu m mechanics, such as multi-plication and addition, m ap into operations in mock phase space.These operations constitute a symbol calculus.77Consider the addition of two ope ratorst = A + B (16)

As long as th e quantization rule is linear, the symbol for e, Cs,is simply A s + Bs. W e will not consider nonlinear qua ntizationrules, since they are rarely used.As a consequence of the noncommutative nature of operatormultiplication, the multiplication of two operato rs does not simplyma p into the multiplication of symbols. Nevertheless, a gener-alized multiplication can be defined which is the imag e of operatormultip lication in mock phase space. This imag e is sometimescalled a tw is te d m ~ l t i p l i c a t i o n , ~ ~n reference to its noncommu-tative nature. For the operator product= AB (17)

cs = As*& (18)the corresponding symbol is denoted by

The twisted multiplication operator * is particularly simple forthe W eyl quantization rule:46 we haveCS = exp( ( v ~ ' v p ~: - 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. Th e lowest order term in h is justordinary multiplication. This result holds for any quantizationrule which acts by reordering the terms of a noncommutingpolynomial.Once a generalized multiplication is found, a representationof the commutator follows trivially. If we define

(20)As,BsJs= As*& - &*A sthen the symbol of [A,&]s given by (As , Bs }s . For the Weylquantization rule the phase space commu tator is called a Moyalbracket.'" Using the explicit form ula for the twisted multip licationof the Weyl q uantizatio n rule,19 we find tha t the M oyal bracke tof two mon omials is( 4 m ~ P n ~ , 4 m 2 P m ) M=

min(ml+m2,n,+n2) ( i h ) kc - ( k , m l m2,nl,n1)qml+m2-kpnl+n2-k2 1)k = l ; k o d d 2k-1where

(22)The lowest order term in h of the Mo yal brack et is given by i htimes the Poisson bracket. Th e higher order terms in h can beviewed as quan tum corrections to the Poisson bracket. Althoughthe precise natu re of these corrections will in general depend onthe particular quantization rule used, many rules give mock phasespace commutators that are equal to the Poisson bracket to lowestorder in h .D . Van Vleck Perturbation Theory in Mock Phase Space.Given the concepts presented above, a mock phase spac e formu-lation of Van Vleck perturbation theory can be derived. Th estarting point is the quantum mechanical H amiltonian H definingthe problem of interest. W e take H to be a function of harmonicoscillator creation-annihilation operators. The Ham iltonian can

The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3149then be transformed into a symbol Hs(a*,a),where a and a* a rethe classical analogues of creation-annihilation operators. Theseare defined as

an d ai* is the complex conjugate of ai. Le t us further assume thatthe quantization rule used is of the following form:min(m,n)

/= 0a*man- c C (m , n , l ) h ' a + " - ' a " - ' (24)(Both the Weyl quantization rule and the symmetrization ruleare of this form.) The n a-Hamiltonian sym bol H s will correspondto a diagonal operator H if and only if H s is a function of theproducts ai*aionly.We therefore seek a sequence of unitary transformations whichrender the H amiltonian as nearly diagonal as possible. This isexpressed in mock phase space as

Th e procedure for choosing th e generating function fk is similarto that of the classical theory. On the k th iteration , we choosefk to make the transformed Hamiltonian a function of ai*ai .Resonances between the zeroth-order frequencies will lead tocomplications, just as they d o classically.Explicit formulas can be derived by expanding all quantitiesin powers of e. Let H k be the Hamiltonian symbol after ktransformations. Expanding H k in powers of c yields

H k = C$H,k (26)Inserting this into eq 25 gives

Th e transformations are carried out by expanding the phasespace commu tator in powers of h . The structu re of the resultingperturbation theory can be understood for the W eyl quantizationrule by examining eq 21 . A term in t he expansion of ( l / i h ) ( swhich is of jt h order in h is 2 j polynomial degrees less tha n theclassical term. Th e integer j must be even, because the Moyalbracket has an expansion in odd powers of h . The first non-classical ter m is therefore a polynomial of four degrees less thanthe classical series. In the large quantum number limit, thenonclassical terms become negligible compared with the classicalterms, because they involve lower powers of quantum number.Since orders in h are in 1 :1 correspondence with polynomialdegrees, we can estimate the maximum number of term s that couldpossibly be generated by including a hig her order correction inh. A polynomial of degree m , in N degrees of freedo m, can have

( N + m - l ) ! (28)( N - l ) !m!terms. Therefore, the ratio of the number of terms of thejth- ord ercorrection in h to th at of th e classical series is, a t order ( m - 2)in perturbation (polynomial degree m )

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

App arently, very high order corrections in ti will produce manyfewer terms than the classical theory. Therefore, we expect thatinclusion of low-order corrections in h will req uire a significantfraction of the amount of work required for a fully quantumcalculation . Conversely, it is shown below that inclusion of justthe first correction in h can lead to substantial improvement inthe accuracy of energy eigenvalues. M ixed approximations, wherehigh ord ers in t are calculated to low order in h , and vice versa,


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3150 The Journal of Physical Chemistry, Vol. 92, N o. 1 1 , 1988 Fried and EzraTABLE I: Perturbation Coefficients for the Anharmonic Oscillator of Eq 31sco h l b ti" ti21 OM'

3.750 000 00E-1 7.500 00000E-1 7.500 000 OOE- 1 7.5 00 000OOE-1 7.5 00 000OOE- 15 .3 12 500 00E-1 3.187 50000E+O 2.625 000OOE+O 2.625 0000 0E+O 2.625 0000 0E+O1.464 843 75E+O5.21923 8 28E+O2.137 426 75E + 19 .559323 12E+14.543 799 91E+ 22 .2 58 5 3 4 6 6 E f 31.161 71906E +46.1 39 452 94 E+43.316555 31E+51 . 8 2 4 4 2 9 8 5 E + 61.019 052 00E+ 75.766 587 07E+ 73.300043 77E+81.907 098 52 E+91.1 11 646 1 8E+1 06.529 424 70E+1 03.861 371 59E+112.297 548 35E+12

3.5 15 6 25 OOE+ 16.263 085 94E +21S 3 8 9 4 7 2 6 E +44.817 898 85E +51 . 8 3 2 0 6 0 1 3 E + 78.195 77058E+ 84.215646 15E+102.450 673 15E+ 121.588635 30E+141.136 076 26E+1 68.883921 17E+177.540818 10E+196.904612 24E+216.783 309 67E+ 237.117 17429E+ 257.942 725 23E+279.394 338 70E+2 91.173 839 10 E+32

2.081 25000E+ 12.412 89063 E+23 . 5 8 0 9 8 0 4 7 E + 36.398 281 35E +41.329 733 72E +63.144821 47E+78.335 41604E+82.447 89407E+1 07.893 333 16E+1 11.890 288 55E +133.779 395 70E + 151.586 51031 E+184.402 956 89E+2 01.067 904 22E+2 32.420 939 30E+2 55.294 153 48E+ 271.137741 90E+302.431 850 38E+32

2.081 2500 0E+12.412 89063E +23 . 5 8 0 9 8 0 4 7 E + 36.398 281 35E+ 41.329 733 72E +63.144821 47E+78.335 41604 E+82.447 89407E +107.893 333 16E+ 112.773 877 69E+131 . 0 5 5 6 4 6 6 6 E + 1 54.326 81068E+1 61.900817 19E+188.912 101 78E+194.442 550 86E+212.346 464 34E+231.309 150 19E+257.694 00092E+26

2.08 1 2 50 OOE+ 12 . 4 1 2 8 9 0 6 3 E + 23.580980 47E+36.398 281 35E+41.329 733 73E +63.444821 47E+78.335 41603E+82.447 89407E+1 07.893 333 16E+112 . 7 7 3 8 7 7 7 0 E + 1 31.055 646 66E+1 54.326 810 68E+161.900 817 20E +188.912 101 78E+194 . 4 4 2 5 5 0 8 9 E + 2 12 . 3 4 6 4 6 4 3 1 E + 2 31.309 150 26E+257.693 999 85E+26

"Semiclassica l per turbat ion expansion. E-1 (E + ll an d so on stand for X10-' (10') etc. * Weyl quantization rule used in ti expansions. cQuantummechanical perturbation theory of ref 89could possibly come close to the accuracy of a fully quantumcalculation, with only slightly more work th an a classical calcu-lation. Suc h an approach would be consistent, because high-orderterms in E are expected to be quite small, and thus can be cal-culated more approximately than low-order terms. The utilityof such approximation schemes is under current investigation.The expression determining k is of sp ecial interest, since it she dslight on the relation between the small denominator problem inclassical an d quantum mechanics:

Consider this equation when the W eyl quantization rule is used.The Moyal bracket will reduce to i h times the Poisson bracketif th e zeroth-order syste m is a harmonic oscillator. Therefore,f k is determined by the sam e equation as in the classical theory.This implies that the two perturb ation series have precisely th esam e small denominator problem. As a partial solution to thisproblem, we do not eliminate any term s tha t would produce de-nominators small enough to destroy t he ap parent convergence ofthe series.After the desired number of transformations is carried out, theappropriate quantiz ation rule is em ployed to convert the simplifiedHam iltonian symbol into an operator. For nonresonant systems,a diagonal Hamiltonian operator is obtained; energy eigenvaluesare determined simply by evaluating the diagonal matrix elements.For resonant systems, the quantized Hamiltonian is not diagonal.If th e numbe r of linearly independent resonances found is less thanthe number of degrees of freedom, however, it will be blockdiagonal. Only a small matrix diagonaliza tion is then required.By expanding the phase space commutator in powers of h , hetransformations can be carried ou t to a given power of h . If onlythe lowest term in h is kept, the phase space commutator becomesa Poisson bracket, and ordinary AQ is obtained. Only part ofthe full h dependenc e of the qu ant um problem, however, is de-scribed by the expansion in powers of h. The small matrixdiagonalizations required for resonant systems introduce possiblynonanalytic behavior in h . This explains t he success of AQ inreproducing splittings which a re entirely quantum mechanical-theh dependence associated with such splittings is well reproducedby the small matrix diagonalizations. In the next section we applythe generalized AQ procedure to a variety of problems. It willbe shown tha t inclusion of correc tions in h often leads to sub-stantial improvements in the accuracy of energy eigenvalues.IV. Results

In this section, four systems ar e treate d with th e generalizedAQ procedure described above. Th e one-dimensional qua rti c

oscillator is studied as an e xample of a system with a divergentperturbation series. Thre e multidimensional systems are thenexamined. A three-mode model for rotationless SO Za 4s treatedas an example of an effectively nonresonant system. Th e resonantthree-dimensional system first trea ted by N oid, K oszykowski, andMa rcusa5 is quantized next. Finally, we calculate vibrationalenergy levels for ozone (03) ,nearly resonant system with stronga n h a r m o n i ~ i t i e s . ~ ~e show tha t generalized AQ is able to re-produce the results of extremely large variationa l calculationsa6very well. Th e addition of corrections in h is found to improvethe accuracy of energy eigenvalues considerably.

A. The Quartic Oscillator. As an exam ple of generalized AQapplied to a simple system, we treat the qua rtic oscillator. Thissystem is given by the HamiltonianH = '/2(q2+ p 2 ) + q4

Despite its appare nt simplicity, this system has been studied in-tensively, since it c an be viewed as a qu antum field theory in ones pa ce -tim e d im en si on w it h a s e l f - i n t e r a c t i ~ n , ~ ~ ~ ~ ~he per-turbation expansion of the quartic oscillator has therefore beenexamined to gain insight into the convergence properties ofperturbative techniques in qua ntum field theory.60,61 For th epurposes of this paper, however, we simply regard it as a well-established example of a system with a divergent Rayleigh-Schrodinger perturbation series.Bender an d Wusg390 ound th e perturb ation expansion of thissystem's ground-state energy to be of the formmE = 1/2 + ~ ( - 1 ) " - ' t " A , (32)

They found the coefficients A,, for n I 5 by iterating a differenceequation. Table I compares the first 20 coefficients found byBender and Wu with those obtained by ordinary semiclassicalquantization and generalized AQ. W e obtained the first columnby semiclassical quantization of the Birkhoff-Gustavson normalform. Th e semiclassical pertu rbati on expansion is clearly much

n= 1

(84) Kuchitsu, K. ; Morino, Y . . C he m. SOC . pn . 1965, 38 , 805. Barbe,( 8 5 ) Noid, D. W .; Koszykowski, M . L. ; Marcus, R. A . J . Chem. P h y s .( 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. . hys. Rev . Let t . 1968, 2 1 , 406.(89) Bender, C . M .; W u , T. . hys. Reo. 1969, 184, 1231.(90) Bender, C. M .; W u, T. T. hys. 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.


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Generalized Algebraic Quantization The Journal of Physical Chemistry, Vol. 92, N o. 11, 1988 3151TA BLE II: Parameters for the SO2 and O3 Hamiltonians of Ref 84(i n cm-I)

parameter SO, 0, parameter SO , 0,"I 1171"2 525" 3 1378k l , , 44k122 -1 2k112 -19k 1 3 3 15 9k222 -7.0

1134.9 k233 4.7 -59.3716.0 kl l l l 1.8 2.21089.2 k1122 -3.1 -1.8-48.1 k1133 15 28.3-29.7 k 2 2 2 2 -1.4 0.6-25.5 k2233 -6.5 -5.9-225.8 k3333 3.0 6.7-19.2less divergent than the qu antu m expansion. This is expected-thesemiclassical expansion is known to have a sm all but finite radiusof convergence, whereas the quantum expansion has a zero radiusa f c o n ~ e r g e n c e . ~ ~ , ~ ~Table I gives results obtained with generalized A Q in th e nextthree columns of Table I. The h1column corresponds to expansionof the Moyal bracket to first order in h, and is therefore equivalentto ordinary AQ implemented with th e Weyl quantization rule.This yields the exact qu antum result for the first correction, butall subsequent corrections are very d ifferent from the q uantumvalues. No te also tha t the series obtained by means of the Weylquantizatio n rule diverges faster than the quantu m series. Thissuggests tha t perturbation series derived with the W eyl quanti-zation rule might be harde r to resum th an those derived semi-classically, at least for one-dimensional systems. The sum mabilityof classical perturbation theory when applied to multidimensionalsystems is largely unexplored.The column labeled h" gives results derived by expandin g theMoyal bracket to 11th order in h . The first 11 expansioncoefficients ar e nearly identical with those given by Bender an dWu . Th e 12th expansion coefficient, however, is marke dly dif-ferent. This is because the order a t which the h expansion of theMoyal bracket truncates is dependent upon the polynomial degreeof the n ormal symbol. Thus, incorporating corrections to givenorder in h makes a finite number of perturbation coefficientsmatch the quantu m values. The nonm atching coefficients, how-ever, are not necessarily close to the qu antum values. Given thatenough orders in h are included, generalized AQ will matchquantum perturbation theory to any given order in e, but onlybecause the series of corrections trunc ates in h. It is in this sensethat t he expansion in h converges to the quant um result.Finally, we display coefficients t hat are a ccur ate to 21st orderin h in Table I. These match the calculations of Bender and Wuclosely for almost all the coefficients given. O ur results demon-strat e that generalized AQ becomes equivalent to quantum per-turbation theory, provided th at enough orders in h ar e included.The ag reeme nt is slightly worse for the last few coefficients. W ebelieve this is due to roundoff error. Th e formula for the Moyalbracket of two polynomials (21) contains factorial terms that makeit susceptible to roundoff error. Th e Weyl quantization rule alsocontains such factorial terms. Thus, generalized A Q is best usedto relatively low order in h , if numerical problems are to beavoided. If very high order corrections in h are required, it isprobably best to d o fully quantu m perturbation theory directly.Nonetheless, for many systems the use of just the first few cor-rections in h yields very accu rate resu lts. W e give examples ofthree such systems below.B . A Nonresonant Syste m. Th e application of generalized AQto effectively nonresonant, weakly coupled systems is straight-forwar d. To dem onstrate this, we have quantized th e vibrationsof rotationless SO2,a nonresonant system with three degrees offreedom. The H amiltonian for this system iss4

Th e parameter values ar e given in Table 11. The anharmonicpart of the potential is ordered according to polynomial degree.Thus, the cubic terms w ere taken to be of order e, while the quartic

TA BLE III: Energy Eigenvalues for the SO , Hamiltonian of Re f 84NI N2 N3 Est" E lb E,' ESd E Q M e0 0 0 1528.93 1526.17 1530.27 1530.27 1530.270 1 0 2044.82 2041.93 2046.15 2046.15 2046.150 2 0 2555.00 2551.96 2556.32 2556.32 2556.321 0 0 2684.47 2681.93 2685.96 2685.96 2685.960 0 1 2888.21 2885.67 2889.50 2889.50 2889.500 3 0 3059.22 3056.01 3060.52 3060.52 3060.521 1 0 3197.53 3194.86 3199.02 3199.03 3199.030 1 1 3398.82 3396.14 3400.09 3400.09 3400.090 4 0 3557.17 3553.77 3558.45 3558.45 3558.451 2 0 3704.77 3701.94 3706.25 3706.25 3706.252 0 0 3832.69 3830.39 3834.33 3834.33 3834.330 2 1 3903.64 3900.81 3904.89 3904.89 3904.891 0 1 4031.72 4029.41 4033.14 4033.14 4033.140 5 0 4048.50 4044.88 4049.75 4049.75 4049.741 3 0 4205.90 4202.89 4207.37 4207.37 4207.370 0 2 4238.76 4236.45 4240.02 4240.02 4240.022 1 0 4342.57 4340.14 4344.22 4344.22 4344.220 3 1 4402.42 4399.41 4403.64 4403.64 4403.640 6 0 4532.79 4528.91 4534.01 4534.01 4533.971 1 1 4538.93 4536.49 4540.33 4540.33 4540.331 4 0 4700.60 4697.39 4702.07 4702.07 4702.062 5 1 7621.20 7618.06 7622.59 7622.59 7622.493 0 2 7621.92 7620.27 7623.43 7623.42 7623.510 7 2 7638.33 7634.48 7639.24 7639.25 7638.790 13 0 7668.14 7660.64 7669.45 7669.46 7712.230 10 1 7688.39 7683.06 7689.21 7689.22 7687.431 2 3 7688.62 7686.50 7689.91 7689.91 7689.912 8 0 7723.07 7718.54 7724.67 7724.65 7724.295 1 0 7733.28 7731.65 7735.26 7735.26 7737.423 3 1 7778.19 7775.84 7779.75 7779.74 7779.781 5 2 7813.01 7809.90 7814.18 7814.18 7814.102 0 3 7825.18 7823.55 7826.57 7826.57 7826.590 2 4 7896.00 7893.91 7897.22 7897.22 7897.211 8 1 7901.73 7897.22 7902.82 7902.81 7902.453 6 0 7912.02 7908.60 7913.88 7913.86 7913.654 1 1 7919.55 7917.79 7921.19 7921.19 7921.671 11 0 7938.30 7931.88 7939.68 7939.67 7943.79Semiclassical quantiza tion using a Birkhoff-Gustavson normalform. bCalculationdone to 12th order in perturbation and 1st order inh . 'Calculation done to 12th order in perturbation and 3rd order in h .dCalculation done to 12th order in perturbation and 5th order in h .eVaria tional calculation of ref 86.

. . . .. . .. . .

terms were assigned order e 2 , This is consistent with the size ofthe anharm onicities in Table 11.Tab le I11 comp ares results derived by EBK quantizatio n of aclassical normal form and generalized AQ with a large-scalevariational calculation.86 All perturbative calculations were doneto 1 2th order in e. The v ariational calculation86 used a 11 by 24by 10 (2640 function) Cartesian harmonic oscillator basis set.Energie s derived by semicla ssical quantiz ation of a classical normalform ar e presented in the first column Tab le 111. These energiesar e typically within a few wavenumbers of the variational results,even for the group of highly excited states given.In th e next th ree columns of T able 111, we present results foundby AQ . Th e column labeled hl gives results for ordinary AQimplem ented with the Weyl quantization rule. Th e semiclassicalresults ar e generally better th an those derived by th e Weyl rule.The situation changes dramatically, however, as soon as higherorder corrections in h are included. W e achieve agreem ent withthe quantum results for the first group of states to within 0.01cm-I. Good agreeme nt is also seen for the group of excited states.Convergence to th e variational results, however, is nonuniform.For instance, the state (1,2,3) shows agreement to within 0.01cm-'; a m uch larger discrepancy is seen with the sta te (0,13,0).W e believe tha t the qua ntum calculation m ay not be completelyconverged for states with high bending excitation. An estimateof the error of our result can be obtained by taking th e differencebetween the present 12th-order calculation and a 10th-ordercalculation. Fo r the sta te (0,13,0) we estimate our error to beabout 3 cm-I. Table I11 also presents results for a calculation doneto fifth order in h . These ar e nearly identical with the calculation


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3152T A B L E IV: Energv Eigenvalues for the Three-Dim ensional Hamiltonian of Ref 85

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Laurence E. Fried and Gregory S. Ezra- Generalized Algebraic Quantization: Corrections to Arbitrary Order in Planck's Constant