eigenstate assignments and the quantum-classical correspondence for highly-excited ... ·...

Eigenstate assignments and the quantum-classical correspondencefor highly-excited vibrational states of the Baggot H 2O Hamiltonian

Srihari Keshavamurthya) and Gregory S. EzraBaker Laboratory, Department of Chemistry, Cornell University, Ithaca, New York 14853

~Received 24 September 1996; accepted 28 March 1997!

In this paper we study the classical and quantum mechanics of the 3-mode Baggot vibrationalHamiltonian for H2O. Our aim is to classify and assign highly-excited quantum states based upona knowledge of the classical phase space structure. In particular, we employ a classical templateformed by the primary resonance channels in action space, as determined by Chirikov resonanceanalysis. More detailed analysis determining the exact periodic orbits and their bifurcations andfamilies of resonant 2-tori for the Baggot Hamiltonian confirms the essential correctness of theChirikov picture. It is emphasized that the primary periodic orbits alone do not define a suitablephase space skeleton; it is important to consider higher dimensional invariant structures, such as2-tori and 3-tori. Examining the manifold of quantum states for a given superpolyad numberP5n11n21nb/2 reveals sequences of eigenstates that progress along the classical resonance zones.These sequences provide insight into the nature of strongly mixed states found in the vicinity of theresonance junction. To further explore the classical-quantum correspondence, we have alsocomputed eigenstate Husimi phase space distribution functions and inverse participation ratios. It isthereby possible to provide dynamically based assignments for many states in the manifold of stateswith superpolyad numberP516. © 1997 American Institute of Physics.@S0021-9606~97!01425-6#

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I. INTRODUCTION

Understanding the nature of highly-excited vibrationand rotation-vibration states of polyatomic molecules isproblem of central importance in chemical physics.1 Tradi-tional spectroscopic methods for assignment of levels baon harmonic oscillator~normal mode!-rigid rotor quantumnumbers and wavefunctions2 can break down for highly-excited rovibrational states as a result of strong mocoupling.3,4 Such mode coupling, due either to anharmocoupling terms in the potential or to rotation-vibration inteaction, leads to the phenomenon of intramolecular vibtional energy redistribution~IVR!,4 and results in compli-cated energy level and intensity patterns.

One source of apparent complexity in vibrational specis strong mixing of manifolds of near-degenerate states bsingle resonant coupling term; the corresponding classproblem is nevertheless integrable in this case. Another psibility is strong mixing of states due to the presence of tor more resonant coupling terms; classically, the problemnonintegrable and we have the possibility of chaos.5 Statemixing can also occur as a result ofdynamical tunneling,6 anonclassical mixing of states localized in different regionsphase space that is associated either with symmetry reor accidental degeneracies of levels.

Many approaches have been proposed for the studcomplicated spectra beyond traditional methods: we menstatistical Fourier transform analysis,7 hierarchical trees;8

periodic orbits;9,10 semiclassical propagator baseapproaches;11 and bifurcation theory.12,13Recently, Wolynes

a!Present address: Department of Chemistry, IIT Kanpur, U.P. 208India.

156 J. Chem. Phys. 107 (1), 1 July 1997 0021-9606/97/

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and coworkers have made efforts14–17to understand IVR andeigenstate localization in large systems by establishing cnections to the problem of Anderson localization in ttheory of disordered metals.18

Considerable progress has been made in the analystwo modes coupled by a single resonant term.12 In this casethere is a constant of the motion in addition to the toenergy, and the associated reduced phase space issphere, the so-called polyad phase sphere.19 Classically, onecan study the fixed points of a reduced 1 degree-of-freed~dof! spectroscopic Hamiltonian on the sphere, which corspond to periodic orbits~pos! in the full phase space. Thesfundamental periodic orbits serve as organizing centersthe classical and quantum phase space. As parametersas energy or polyad number are changed, these periodicbits will bifurcate or merge, leading to qualitative changesthe phase space structure. Each distinct arrangement ofodic orbits and associated stable and unstable manifolds~forunstable pos! defines azonein parameter space, and Kellmaand coworkers20 have systematically studied the bifurcatioof pos and the passage from one zone to another for bothand 2:1 resonant systems~see also the recent work oJoyeux21!. Periodic orbit bifurcations in 1:1 and 2:1 resonavibrational Hamiltonians have been studied using semicsical energy-time analysis22 of level spectra.23

It is of considerable importance to extend such aproaches to deal with multi-mode (N>3) systems. Moregenerally, we seek to develop qualitative methods for anasis of energy level patterns and wavefunctions in multimosystems based on the underlying classical nonlinear dynics.

One possibility, following the success of the polya6,

107(1)/156/24/$10.00 © 1997 American Institute of Physics

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157S. Keshavamurthy and G. S. Ezra: The Baggot H2O Hamiltonian

phase sphere approach for 2-mode systems, is to deterfundamental pos for multimode Hamiltonians~cf. Appen-dix!. Lu and Kellman24 have recently examined fundamentpos and their bifurcations in a classical version of t3-mode vibrational spectroscopic Hamiltonian for H2O dueto Baggot.25 The Baggot Hamiltonian25 has an extra constanof the motion in addition to the energy, the so-called suppolyad numberN 52(I 11I 2)1I b ~see next section!. In thepresence of this additional constant of the motion, the Bgot Hamiltonian has essentially 2 degrees of freedom. Flowing the procedure used for 2-mode resonant problemsand Kellman have determined critical points of the 2-moreduced Hamiltonian for H2O; these critical points correspond to the fundamental pos in the full phase space.24,26

One can then attempt to use the fundamental pos as a frwork for organizing the phase space structure and for unstanding the localization of quantum eigenstates.

Another possibility, given the existence of a superpolynumber, is to use the methods ofstratified Morse theory, asdeveloped in a series of papers by Zhilinskii acoworkers.13 In this approach a qualitative understandingthe families of pos and their possible bifurcations is obtainon rather general grounds using the powerful mathemaapparatus of Morse theory.

In the present paper, we aim at the construction o‘‘classical template,’’ based on an approximate partitionthe classical phase space, that can be used to understanorganize highly-excited quantum eigenstates. The partiting of classical phase space is achieved using Chrikov rnance analysis.27

An important aspect of the study of the phase spstructure of multimode systems is the search for wayspartition ~either approximately or exactly! phase space~fullor reduced, at constantE! into regions corresponding tqualitatively different dynamics. In the case that the Hamtonian of interest has two degrees of freedom, the stableunstable manifolds of unstable pos have the appropriatemensionality~codimension one! to divide the energy shelinto disjoint regions, and each zone in parameter spacthen associated with a different partitioning of the polyphase sphere.20 Such partitions have been used in theoriesphase space transport in systems with mixed phase space28,29

The relevant pos can be determined either as critical poon the polyad phase sphere for the case of a single resocoupling, or as fixed points on a Poincare´ section. ForN>3 mode systems, however, the stable and unstable mfolds of unstable pos do not have sufficiently high dimensto define a partition of phase space. The natural generation of an unstable periodic orbit forN>3 mode systems isa normally hyperbolic invariant manifold~NHIM ! ~Ref. 29!,which has codimension two. Thus the stable and unstamanifolds associated with a NHIM have the right dimensioality to partition phase space in a manner analogous totwo mode case. Mapping out the NHIMs in a general mumode system is a difficult task and there are essential difences as compared to the unstable po in two dimensiDetailed discussions of the dimensionality issues of dividsurfaces in multimode systems and appropriate genera

J. Chem. Phys., Vol. 10

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tions of unstable pos can be found in Refs. 29–31.In multimode nonintegrable systems, an approxim

partitioning of phase space can be obtained via Chrikov renance analysis.27 A resonance channelor resonance zone isregion of phase space of full dimensionality consistingtrajectories that are strongly affected by a coupling termthe Hamiltonian corresponding to a single resonance cotion on N zeroth-order frequencies.32 Resonance channeldefine regions of phase space characterized by particularnamical behavior; it is therefore natural to use them asbasis for a partitioning of phase space in multimode systeThe totality of resonance channels forms the Arnold web33

Phase points can drift along resonance channels,‘‘change direction’’ at intersections between channels.34 Inthree or more degrees of freedom systems this diffusionthe action variables in phase space leads to long tinstability.35,36

The location and width of resonance channels in a givsystem can be determined approximately using the stananalysis due to Chirikov.27 In their pioneering work, Oxtobyand Rice mapped out resonance zones for 2- and multi-mmodel molecular Hamiltonians in order to correlate the onof resonance overlap with statistical unimolecular decdynamics.37 Resonance channels have since been mapout for a number of molecular Hamiltonians,38–45 and con-nections with the corresponding quantum systeexamined.39,42,43,45Of particular interest in multimode classical systems is the possibility of chaotic motion at the intsection of resonance channels.40,41,43,46Atkins and Logan43

studied a three mode coupled Morse system with smallharmonicities and demonstrated the possibility of chaosthe intersection of lowest order resonances. For the speset of parameters in their paper, various measures of loization ~basis dependent! provided adequate means to distiguish between different types of quantum states. No signcant classical phase space transport of the Arnold diffustype was however observed in their system. Moreover,kins and Logan did not attempt to analyse in detail individuquantum states and correlate eigenstate features with inant structures in phase space.

The above studies have established close connectbetween classical overlapping resonance zones and thenomenon of IVR in the corresponding quantum systemsthe present paper, we further explore the correspondencthe level of analyzing individual quantum states based oclassical resonance template. Specifically, we examineproblem of assigning highly-excited eigenstates of3-mode Baggot vibrational Hamiltonian for H2O in the mani-fold of states withP5n11n21nb/2516. Results of ouranalysis forP58 have been presented elsewhere.47

Our analysis of the 45 eigenstates withP58 showedthat each state in the manifold could be given a dynamassignment based upon study of the distribution of amplitin action space and the quantum mechanical phase sdistribution function.47 The only ambiguity in assignmenwas associated with a narrowly avoided crossing of two lels of the same symmetry. The relatively straightforwardsignment process forP58 can be readily understood i

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158 S. Keshavamurthy and G. S. Ezra: The Baggot H2O Hamiltonian

terms of the underlying classical mechanics, in that theand 2:1 primary resonance channels hardly overlap.47

In the present paper we extend the Baggot Hamiltonbeyond the limit of strict applicability to H2O ~in particular,well above the energy at which the molecule can becolinear! in order to study the dynamically interesting case wP516. Not only does this manifold contain many stat~153!, but the primary resonance zones overlap, so thatare able to investigate assignment of states in the vicinitya resonance junction.43

Our approach is based upon a close study ofclassical-quantum correspondence, taking as our stapoint a Chirikov analysis of the resonance channels.27 Chir-ikov’s method is an approximate, perturbative approachthe phase space structure of multimode systems, in thadifferent resonances are analyzed independently. The Cikov resonance analysis can however be refined by detering the location of families of resonant (N21)-tori ~2-tori inthe case of the 3-mode Baggot hamiltonian! determined inthe single resonance approximation. We find that the resof the elementary Chirikov analysis and the more refinanalysis are in close agreement, confirming the essentialrectness of the Chirikov picture for the Baggot HamiltoniaUsing the classical resonance structure as a guide, weable to organize eigenstates into families that form rearecognizable sequencesalong the resonance channels. Thesequences of states are found to exhibit a distinct periodwith respect to the distribution of amplitude in action spathis periodicity has a simple origin in the slope of the clasical resonance line. The sequences provide one way otermining the ‘‘parentage’’ of mixed states near the renance junction~see below!.

We note that Rose and Kellman have very recently psented a detailed study of eigenstate assignment inP58 manifold for the Baggot Hamiltonian.48 There are sev-eral points of similarity between the Rose-Kellman approaand the analysis given in the present paper and in Ref.For example, identification of resonance zones in the zerorder action space is central to both approaches~compareFigure 4 of Ref. 48 and Figure 1 of Ref. 47!; Rose andKellman locate resonance zones and assign states usingenvalue correlation diagrams and examination of polphase spheres in the single-resonance approximation; weploy Chrikov analysis and examination of action space aHusimi phase space distribution functions47 ~see below!.

Both Rose and Kellman48 and we47 agree on the straightforward assignability of states for theP58 manifold of theBaggot Hamiltonian. As noted above, the primary resonachannels do not intersect in this case, greatly simplifyingdynamics. In the present paper, we tackle the rather mchallenging case of theP516 manifold, in which the pri-mary resonance channels overlap. Eigenvalue correladiagrams for this case~cf. Figure 20! show many broadavoided level crossings, which render the diabat followemployed by Rose and Kellman problematic. Furthermarks on the relation between our approach and that of Rand Kellman are given below.

For the Baggot Hamiltonian, inverse participation ra


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~IPR! Ref. 49 studies do not show such a clear separatioeigenstates into various types~nonresonant, resonant, mixed!as observed in Ref. 43~see section IV!. Interestingly, IPRvalues seem to be sensitive to sharply avoided crossialthough small values of the IPR, which are associated wthe mixing of many zeroth-order eigenstates, do not necsarily imply that the relevant eigenstate is delocalizedphase space. The classical-quantum correspondenceeigenstates of the Baggot Hamiltonian is studied in detailthe Husimi phase space distribution function.50 We havefound several examples of eigenstates which are delocalin action space~and hence have small IPRs! but are quite‘‘clean’’ and localized in phase space. The Husimi functiotogether with the classical resonance analysis enable uprovide dynamical assignments for most states in theP516 manifold.

It is important to observe that, although the BaggHamiltonian studied here is an effectively 2 degree of fredom system due to the conservation ofN , the structure ofthe corresponding reduced phase space nevertheless exsome complexities not present in 2-mode systems typicstudied. In particular, the form of the contours ofH0 in theconstantN plane can lead to some unusual state mixeffects~discussed more fully below!.

This paper is organized as follows: In section II we dscribe the classical and quantum versions of the BagHamiltonian for H2O.

25 Section III provides a survey of theclassical phase space structure of the classical Baggot Hatonian. The locations of resonance zones, families of 2-tand periodic orbits are discussed. We also describe a usPoincare´ section that provides a global view of the classicphase space structure. Subtleties associated with2-sheeted nature of the surface of section are discussesection IV we outline the methods used for analysis of qutum eigenstates: action space projection; inverse partiction ratios; and Husimi phase space distributions. In secV we examine the quantum-classical correspondenceeigenstates of Hamiltonians that are special cases of thegot Hamiltonian, obtained by turning off one or more resnant coupling terms. Section VI is devoted to analysiseigenstates of the Baggot Hamiltonian itself, while sectVII concludes. Technical details concerning computationperiodic orbits and 2-tori are given in an appendix.

II. BAGGOT HAMILTONIAN FOR H 2O

A. Classical Hamiltonian

The classical version of the spectroscopic vibratioHamiltonian for H2O derived by Baggot25 is a three degreeof freedom local-mode/bend Hamiltonian which includtwo 2:1 stretch-bend resonant termsHs

2:1, s51 or 2, a 1:1stretch-stretch resonant termH1:1 and a 2:2 stretch-stretctermH2:2:

H5H01H1:11H2:21 (s51,2

Hs2:1, ~2.1!

whereH0 is the zeroth order Hamiltonian

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and the resonant interaction terms are

H1:15b128 ~ I 1I 2!1/2 cos~u12u2!, ~2.3a!

H2:25b22I 1I 2 cos@2~u12u2!#, ~2.3b!

Hs2:15bsb~ I sI b

2!1/2 cos~us22ub!. ~2.3c!

Here, ($I s ,us%,I b ,ub)[(I ,u) are canonical action-anglvariables for the two local mode stretches and the bmode, respectively, andb128 [b121l8(I 11I 2)1l9I b .

25

It is important to note that the resonance vectors~1,21,0!, (1,0,22) and (0,1,22) are linearly dependent,51 sothat there is a constant of the motionN in addition to theenergy:

N 52 (s51,2

I s1I b . ~2.4!

It turns out that the 2:2 stretch-stretch resonan~Darling-Dennison coupling52! does not significantly affecthe classical phase space structure in the regime of intefor this paper, and it is omitted from the Hamiltonian in thresonance analysis described below. It is however straiforward to include this coupling term if required. All of thexact numerical results presented in this paper, classicalquantum, do include the 2:2 stretch-stretch resonance.

B. Quantum Hamiltonian

The quantum version of the Baggot Hamiltonian is otained via the standard correspondence between action-avariables and creation-annihilation operators:

a†↔I 1/2eiu, a↔I 1/2e2 iu. ~2.5!

Using the above correspondence and expressing the resuHamiltonian in a basis of number statesun1n2nb&, we obtainBaggot’s Hamiltonian25 with the following mapping of pa-rameters:

~b12,b22,l8,l9!↔2~l,g,l8,l9!, bsb↔1

&

ksbb,

~2.6!

wherel andksbb are defined in Ref. 25. Values of the varous parameters are taken from Baggot’s fit to the vibratiolevels of H2O ~Table II of Ref. 25!.

The superpolyad number, P5(s51,2ns1nb/2, is a con-stant of the motion for the quantum Baggot Hamiltonian.given value ofP is associated with a corresponding classiinvariantN 52P15/2. Our eigenenergies are shifted wirespect to those reported in Ref. 25 by the ground stateergy

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Table I shows the correspondence between parametershere and Baggot’s parameters.

III. CLASSICAL MECHANICS OF 3-MODE H 2O

In this section we examine the classical mechanics of3-mode Baggot vibrational H2O Hamiltonian. The results oour analysis are summarized in Figure 1, which showsclassical phase space structure in the (I 1 ,I 2) plane forN 534.5 (P516). Figure 1 shows contours ofH0 , the lo-cation of periodic orbits, resonance channels, and familieresonant 2-tori. Note that the physical region of the plotthat for which the inequalityI 11I 2<34.5 is satisfied.

A. Periodic orbits and their bifurcations

In studying the phase space structure of H2O, a first stepis the determination of the primaryperiodic orbits~pos!. Fora 3-mode system with 2 linearly independent resonance cpling vectors, these pos can be located by finding the stat

FIG. 1. Classical phase space structure of the Baggot HamiltonianH2O. Primary periodic orbits~d! and resonant 2-tori~3! are shown. Themultimode resonances are indicated by thin lines.

TABLE I. Comparison of the parameters used in the paper and paramfrom Baggot’s fit to the spectroscopic Hamiltonian for H2O.

Baggota Paperb cm21

vs Vs 3885.57vb Vb 1651.72xs as 281.99xb* ab 218.91xss ess 212.17xsb* esb 219.12l b12 2112.96l8 l8 6.04l9 l9 20.16g b22 21.82

uksbbu bsb 18.79

aFrom Ref. 25.bMost of the values are same except forb12 , bsb , l8, l9, andg whichreflect the correct classical limit of the quantum Hamiltonian in Ref. 25

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ary points of the effectively two-degree-of-freedom Hamtonian expressed in terms of 2 resonant angles~cf.Appendix!.

The primary pos appear in 1-parameter families(P,I 1 ,I 2) space; the relevant bifurcation diagram has becomputed by Lu and Kellman.26 Following Lu and Kellman,we have determined the primary pos for the classical BagHamiltonian~see Appendix!. Figure 2 shows the families operiodic orbits and integrable limit 2-tori~see Section III C!in the (N ,I 1 ,I 2) space. For small values ofN (P) there arethree pos, which are the nonlinear normal modes whoseistence is predicted by the Weinstein-Moser theorem.53 Thefirst bifurcation~pitchfork! leads to the creation of two locamode stretches. The local mode stretches undergo furbifurcations for largerN due to the 2:1 resonance interations. The normal modes, meanwhile, also bifurcate duethe 2:1 resonance interaction with the bending mode. Mimportant for the classical-quantum correspondence ofBaggot Hamiltonian in the manifold of quantum states stied here is the periodic orbit structure for a given valueN .

The families of pos intersect constantN planes atiso-lated points, which are marked as circles in Figure 1. Theare 3 pairs of symmetry related (1↔2) pairs of pos on theI 1 and I 2 axes, respectively~filled circles!. Two are pairs of2:1 bond stretch-bend pos, one stable and one~singly! un-stable. The remaining pair consists of a pair of local mostretch orbits, where all energy is in one or the other bostretching mode. In addition, pos appear along the diagoI 15I 2 . For example, for all values ofN there aretwo poswith I 15I 25N /4 ~filled circle!, corresponding to stable anunstable~asymmetric and symmetric stretch, respective!normal mode pos. AtN 534.5, there are two other pos othe diagonal. One~filled circle! lies in the physical region oaction space, and is a 2:1 bend/normal mode resonant

FIG. 2. Periodic orbit bifurcation diagram for the Baggot Hamiltonian(P,I 1 ,I 2) space. The one parameter families of primary periodic orbitsshown as dark lines. The two parameter families of integrable limitresonant 2-tori are shown as thin lines.


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another po lies outside the physical action region, and is aa bend/normal mode po.

We see that, for fixedN , the primary pos appear aisolatedpoints in the action plane. The probability of findinone of the primary pos at an arbitrary value of the classenergy is zero. The primary pos alone do not therefore foa ‘‘skeleton’’ for the classical phase space structure. To mout the classical phase space structure in a way that is usfor studying the classical-quantum correspondence it is nessary to consider higher dimensional classical objects; sobjects are resonance channels, the object of standard Cikov analysis, and families of resonant 2-tori.

B. Chirikov resonance analysis for H0

Resonance conditions for the Baggot Hamiltoniandetermined in the usual way27 by analyzingH0 . Zeroth or-der frequencies are

vs[]H0

]I s5Vs12asI s1ess~12dss8!I s81esbI b ,

s51,2, ~3.1a!

vb[]H0

]I b5Vb12abI b1esb (

s51,2I s . ~3.1b!

The resonance conditions for the 1:1 and 2:1 cases aretained by settingv1(I )5v2(I ) andvs(I )52vb(I ), respec-tively, yielding immediately the resonance conditions in ation space

2~esb2as!I s1~2esb2ess!I s8~12dss8!

5Vs22Vb1~esb24ab!I b , ~3.2a!

I 15I 2 , ~3.2b!

for the 2:1 and 1:1 cases, respectively, withs,s851,2.Each of the resonance conditions defines a plane in

3D action space. These planes intersect the constantN

plane in a line, which defines the center of the resonachannel. All three resonance lines intersect at a pointI r ,which lies either inside or outside the physically allowed pof the action plane,I 11I 2<N /2, depending on the value oN .

To estimate the resonance channel widths in action spconsider the generating function

F 5 (s51,2

usI sr 1ubI b

r

1 (s51,2

~us22ub!Js1SAub1B (s51,2

usD J, ~3.3!

which generates the canonical transformation (I ,u)→(J,c)[(Js ,cs ,J,c) given by

cs5us22ub ; I s5I sr 1Js1BJ ~3.4a!

c5Aub1B (s51,2

us ; I b5I br 1AJ22(

sJs . ~3.4b!

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Bilinear terms of the formJsJ in the new Hamiltonian canbe eliminated by suitable choice of the parametersA andB. In terms of the new variables defined by the generatfunction F , we transform our original Baggot Hamiltoniato a new Hamiltonian centered atI r :

H→H0~ Ir !1 c (

s51,2Js21 c12J1J21g~J!

1b128 @ I 1~J!I 2~J!#1/2 cos~c12c2!

1 (s51,2

bsb@ I s~J!I b2~J!#1/2 coscs , ~3.5!

where

c[4ab22esb1as , ~3.6a!

c12[8ab24esb1ess. ~3.6b!

The resulting Hamiltonian is cyclic in the anglec, so that theconjugate action variableJ is conserved. Expressing the coserved actionJ in terms of the original set of action variableI leads to a constant of motion in addition to the total ene

N 52 (s51,2

I s1I b , ~3.7!

which is the classical analog of~twice! the superpolyadquantum numberP. Conservation ofN means that we canfollow the classical evolution of action variables for a giv(N ,E) in the 2-dimensional action plane (I 1 ,I 2). Moreover,we can eliminateI b to obtain equations for the resonanlines in terms ofI 1 and I 2

I 25z

c1222c

c12I 1 , ~3.8a!

I 25z

2c2c122c

I 1 , ~3.8b!

I 25I 1 , ~3.8c!

wherez[2Vb2Vs1(4ab2esb)N , and the location of theresonance intersection is given by

I 1r 5I 2

r 5z

2c1 c12. ~3.9!

In accordance with standard approximations,27 we ignorethe J1J2 coupling and assume that the coefficients ofresonance terms are slowly varying functions ofJ. The latterassumption is reasonable in the vicinity of the resonaintersection pointI r and leads to the following resonanHamiltonian:

H r' c (s51,2

Js21b128 ~ I 1

r I 2r !1/2 cos~c12c2!

1 (s51,2

bsb~ I sr !1/2I b

r coscs , ~3.10!

where I br 5N 22(s51,2I s

r . At this stage, averagingH r

over eitherc1 or c2 leads to a Hamiltonian of the familia


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pendularform governing the motion ofJ2 or J1 respectively.Thus the resonance widths for the two 2:1 resonances aI r

are given by

DI s2:1'F8c bsb~ I s

r !1/2I br G1/2. ~3.11!

The resonance width for the 1:1 channel can be estimatea similar fashion by considering a further canonical transfmation (J,c)→(K ,f) generated by

G ~K ,c!5 (s51,2

csJsr 1~c12c2!K11~c11c2!K2 ,

~3.12!

and averaging the resultant Hamiltonian over thef2 vari-able. The resonance width is obtained as

DI 1:1'F 8

2c2 c12b128 ~ I 1

r I 2r !1/2G1/2. ~3.13!

Resonance widths at points along the resonance lines afrom the intersection pointI r are determined, approximatelyby substitutingI r→I in the above expressions for the width~cf. Ref. 43!. Thus we have approximately determined tlocation and extent of the relevant classical resonance cnels. Note that, sinceb12,0 andl8.0, the 1:1 resonancewidth actuallydecreasesas the actionsI 1 ,I 2 get larger, i.e.,for smaller values of the bend actionI b .

Resonance lines and corresponding channels are ploin Figure 1. The diagonalI 15I 2 is the 1:1 resonance line; inthe absence of any other resonances, the region of pspace inside the associated resonance channel corresponormal mode trajectories, the region outside to local motrajectories. The lineI 11I 25const is then a projection of thpolyad phase sphere for the 1:1 resonance~constantI b! intothe (I 1 ,I 2) plane. The two symmetry related 2:1 resonanzones slope inwards towards the 1:1 resonance zone. Inabsence of other resonances, each 2:1 resonance channregion of phase space in which the bend mode and one olocal stretch modes are strongly coupled. The widths of2:1 resonance channels decrease to zero near the centhe action plane.

Of particular interest for the vibrational dynamics of thH2O molecule are the regions of phase space where themary resonance channels intersect. Such intersectionscharacteristic of the dynamics of multimode systems, givrise to chaotic classical dynamics.32,34 For N 518.5 (P58), Chirikov resonance analysis shows that the primresonance channels hardly intersect; consistent with thithe relative ease with which eigenstates for theP58 mani-fold can be assigned.47

Assignment of eigenstates forP58 has recently beenconsidered by Rose and Kellman.48 Eigenvalue correlationdiagrams are used in the single resonance approximatiolabel each zeroth-order state as resonant or nonresonantrespect to stretch-stretch and stretch-bend couplinrespectively.48 This ‘‘schematic’’ Chirikov-type analysis~cf.Figure 4 of Ref. 48! should be contrasted with the classicresonance analysis presented here, where the classical a

7, No. 1, 1 July 1997

p

oor

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space is partitioned into resonance zones by explicit comtations using the classical Baggot Hamiltonian.

In order to arrive at an unambiguous assignmentstates, Rose and Kellman had to assume that normal mclassification~N-N-B in their notation! took precedence ovelocal mode~L-L-B ! assignment.48 Moreover, it was notedthat no states were assigned both normal mode~N-N-B! andFermi resonant~L-R-R! labels. The explicit Chirikov analy-sis presented here and in Ref. 47 provides a clear dynamjustification for these empirical observations in terms oflocation of the 1:1 and 2:1 classical resonance channelspecially the fact that forP58 the resonance zones do noverlap appreciably. On the other hand, forN 534.5 (P516), the case studied here, three resonance channelssect in the region aroundI 15I 258. It is in this region ofaction space that ‘‘mixed’’ or nominally chaotic states afound ~see below!.

In addition to the primary 1:1 and 2:1 resonance zonthe location of higher-order multimode resonances canbe calculated at the next order of perturbation theory. InBaggot Hamiltonian, Poincare´ surfaces of section~discussedin section III D! show a particular multimode resonance ocupying a significant region of phase space at energies bethat at which the primary 2:1 resonance islands appear. Smultimode resonances arise due to the interaction betwlocal modes and 1:1 resonant modes. They are indicateFigure 1 as a pair of thin resonance zones. By Fourier ansis of classical trajectories located in the relevant regionphase space,54,55 we have determined that these multimoresonances correspond to the~symmetry related! resonancevectors (2,21,22) and (21,2,22). We have also estimated the resonance width and location for the particumultimode resonance by removing the 1:1 resonance toorder56 ~see also Ref. 57!. This calculation is the first step ina possible renormalization analysis58 of higher order reso-nances, which could lead to an accurate determination ofthreshold for onset of large-scale stochasticity. The widththe corresponding multimode resonances are:

Dm'F 8bsbb12

a1~9a11a2!

I bI 1,21/2

~ I 12I 2!G1/2, ~3.14!

wherea1,2[2c7 c12. Note that this perturbative result doenot hold as one approaches the resonance intersectionI r inthe (I 1 ,I 2) plane.

C. Families of resonant 2-tori

Chirikov resonance analysis provides an approximpicture of the phase space structure of the H2O molecule. Itis a perturbative approach in that each resonance coupterm is analysed independently. Moreover, the phase sstructure in the vicinity of each resonance line is assumebe pendulum-like.27 For systems with weak anharmonicitiethere may be ranges ofE andN over which the local phasespace structure is more complicated.

A more detailed analysis of the phase space strucinvolves determination of the location of families ofreso-nant 2-tori. Consider the classical H2O Hamiltonian with a


u-

fde

ales-

ter-

s,oe

-wcheniny-f

rst

hef

e

ngceto

re

single resonance coupling term. There are in this case 2norable angles, and 2 constants of the motion in additionthe energy; for example, if only the 1:1 resonance couplterm is present,I b and I 11I 2 are constants of the motionThe dynamics is then described by a 1 degree of freedomreduced Hamiltonian, and stationary~equilibrium! points ofthis Hamiltonian correspond to 2-tori in the full phase spa~In the absence of any coupling, almost all trajectories lienonresonant 3-tori.! Details of the computations of resona2:1 tori for the 1:1 and 2:1 resonances in the Baggot Hamtonian are given in the Appendix.

At fixed superpolyad numberN , the resonant 2-tori appear in continuous one-parameter families, as shown in Fure 1 ~two parameter families in (N ,I 1 ,I 2) space as shownin Figure 2!. For the 1:1 resonant case, there are 2 familiesnormal mode 2-tori along the diagonalI 15I 2 , and a sym-metry related pair of local mode 2-tori along theI 1 and I 2axes. In addition, there are two families of resonant 2-tassociated with bend-local resonant motion; these famare marked with crosses in Figure 1, and are located closthe centers of the 2:1 resonance channels determinedChirikov analysis. We stress again that our computationsthe resonant 2-tori are not exact, but are carried out witthe single resonance approximation.

Two conclusions are evident from our computationthe resonant 2-tori. First, from Figure 2 it is clear that tprimary pos appear where families of 2-tori meet. Moreovthe location of the primary pos is given quite accuratelythe intersection of the loci of the families of 2-tori detemined in the single-resonance approximation. Second,apparent that the 2-tori form the ‘‘skeleton’’ of the classicphase space, and in fact follow very closely the Chirikresonance channels. All this suggests that the Chirianalysis is meaningful for the Baggot Hamiltonian.

D. Poincare surface of section

In addition to the calculations outlined above, we hacomputed classical surfaces of section for the Baggot Hatonian. These surfaces of section are useful in order to demine the significance of the classical phase space structdiscussed above, and also to compare with quantumchanical phase space~Husimi! distribution functions.

As the Baggot Hamiltonian has one ignorable angle,can define a 2D surface of section~sos! by plotting N1

[(I 12I 2)/2 andf1[u12u2 at constant values of the sectioning anglef2[u11u224ub with f2.0. The value ofthe ignorable angle conjugate to the superpolyad action~N !is irrelevant. All trajectories in a given sos have the saenergyE; for givenN there is a range of possible sos eergies.

The general structure of the classical sos is as follownormal-mode resonant region is present in the middle ofsos (I 1.I 2), corresponding to ‘‘librational’’ motion.5 Localmode~above barrier! regions are found above and below thcentral normal mode region. Large 2:1 bend/local-streresonant islands appear in the local mode regions~e.g., Fig-ure 3b!. Higher-order resonances also appear. At low en

7, No. 1, 1 July 1997

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gies, the center of the normal mode islands corresponstable resonant 2-tori, while a ‘‘whiskered’’~unstable!2-torus appears at the center of the sos. Similarly, thebend/local-stretch resonance is defined by the intersectiostable and unstable resonant 2-tori with the sos.

The classical surface of section described above haadditional feature which manifests itself at values of theperpolyad number for which the resonance intersectionI r

lies in the physical region of the action plane. This featurean apparently unavoidable ‘‘double-sheetedness’’ of theface of section due to the closing of the contours ofH0

around I r ~cf. Figure 1!. Such a topological change in thcontours ofH0 , and thence of the phase space of the efftive 2D reduced system, leads to the possibility of tcreation/destruction of fixed points in phase space, as follin general from the analysis of stratified Morse theories.13 Aninteresting aspect of the two-sheeted structure of the surof section is that some quantum states can selectively loize on one sheet or the other whilst other states can deloize over both sheets. Examples of such states are showndiscussed below. Moreover, interaction between such stcan lead to very highly mixed states, thus complicatingspectral assignments.

In our calculations points on the classical sos are demined to lie on the upper or lower sos via a simple criterioThe value of the bend action (I b

r ) corresponding to the resonance intersection is obtained asI b

r 5N 22(I 1r 1I 2

r ). Theintersection point of the classical trajectory with the sos lon the lower or upper sos if the associated value ofI b is lessthan or greater thanI b

r , respectively. In what follows, bothsheets of the surface of section are shown~when necessary!with u and l denoting upper and lower sheets respectiveAll ( N1 ,f1) surface of sections shown in this paper corspond to the anglef1 measured in units of 8p.

In Figure 3 a series of 4 surfaces of section are sho(N1 versusf1/8p! as a function of increasing energy forP516. At low energies~Figure 3a! most of the surface osection is occupied by regular normal mode orbits, toget

FIG. 3. Lower Poincare´ surface of sections forP516 as a function ofincreasing energy.~a! 41892 cm21. ~b! 43488 cm21. ~c! 48970 cm21. ~d!49157 cm21.


to

:1of

an-

sr-

-es

cel-al-ndese

r-.

s

.-

n

er

with large islands corresponding to the (2,21,22) and(21,2,22) multimode resonances as discussed in the prous section~cf. section III B!. At moderate energies~Figure3b!, the main features are the 2:1 and 1:1 islands withdestruction of the multimode resonances. Although thereincreased stochasticity due to the higher order resonanthere are still invariant curves serving as barriers to transin phase space. In Figure 3c the surface of section cosponds to energies close to that of the bifurcation ofnormal mode periodic orbits due to the 2:1 resonancesthese high energies, most of the sos is occupied by stochorbits with the relatively small 2:1 islands embedded in tstochastic sea. The islands do not have enough phase sarea to support even a single quantum state. At even higenergies as shown in Figure 3d the entire sos is filled wstochastic orbits. The 1:1 regions are completely destrowith small islands of 2:1 resonances still persisting. At thighest energy almost all of the invariant curves in theare destroyed in marked contrast to the analogous caseP58.

The surfaces of section shown in Figures 3c and 3dat high enough energy so that the sos is double-sheeteFigure 4 we show the corresponding upper sos. At the hiest energies all of the invariant structures on the upperare also destroyed.

FIG. 4. Poincare´ surfaces of section forP516 at energies corresponding tFigure 3.~a! 48970 cm21. ~b! 49157 cm21.

7, No. 1, 1 July 1997

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IV. ANALYSIS OF EIGENSTATES: METHODS

Quantum eigenfunctions were generated by diagonaing the Baggot Hamiltonian for a given polyad numberP inthe number basisun1 ,n2 ,nb&, wheren1 ,n2 are the quantumnumbers for the anharmonic local O-H stretch modesnb denotes the bend quantum number. The Hamiltoniablock diagonal inP and the total number of states at fixedPis (P11)(P12)/2. To analyze the eigenstates of the BagHamiltonian, it is natural to project them onto the (n1 ,n2)quantum number~action! plane. The physical points in th(n1 ,n2) lattice are those points for whichP2n12n2 is non-negative. We represent the eigenstates by plotting at ephysical lattice point a circle with radius equal to the squof the coefficient of the corresponding zeroth-order bastate in the eigenstate of interest. In the present paperdiscuss the manifold of 153 states withP516. Analysis ofthe caseP58 has been presented in an earlier paper.47 In therest of this section we give a brief description of the methoused for studying the classical-quantum correspondence

A. Inverse participation ratios (IPR)

To analyze the quantum wavefunctions, and to aidcorrelating with the classical resonance analysis, we hcomputed inverse participation ratios~IPR! for eigenstates.49

If the stateuCa& is expanded in an orthonormal basis$f j% as

uCa&5(j

uf j&cja , ~4.1!

then the corresponding IPR is defined by

La5(j

ucjau4. ~4.2!

The IPR is a measure of the delocalization ofuCa& in thebasis$f j%: if uCa& is essentially composed of a single bastate, then the IPR will be about unity; on the other handuCa& is a superposition ofN basis functions with comparablcoefficients, then the value IPR will be about 1/N. Stateswith low IPR values are therefore delocalized over the ba$f j%.

The value of the IPR is of course by definition basisdependent. In addition to obtaining a rough measureeigenstate delocalization, we are interested in using IPRsculated in terms of one or more physically appropriate basets to distinguish between different classes of eigensFollowing the analysis by Atkins and Logan of a 3-mosystem with two resonant coupling terms,43 we compute theIPR in two different basis and average over them. In ocase, we compute the IPR in a basis of eigenfunctions ofzeroth-order Hamiltonian plus the 1:1 resonant couplterm, and in the basis of eigenstates ofH0 plus both 2:1resonant coupling terms. We then calculate an averagedefined by

L51

2~L1:11L2:1!, ~4.3!

where


z-

dis

t

ryeise

s

nve

if

is

tfal-iste.

reg

R

L1:15(j

ucja1:1u4,

~4.4!

L2:15(j

ucja2:1u4.

Here,cja1:1 and cja

2:1 are the coefficients in the expansionthe eigenfunctionsuCa& in the 1:1 and 2:1 resonance baserespectively. It follows from the definition of the IPR thastates with mostly 1:1 resonant character will have high vues ofL1:1 and low values ofL2:1 and vice versa. Stronglymixed states influenced by both 1:1 and 2:1 resonancesexpected to have low values of the IPR in both bases. Onother hand, states that are completely outside either ofresonance zones will have the highest values of the IPRboth bases. Thus we provisionally associate low valuesthe average IPR,L, with mixed states, moderate values wistates being in either the 1:1 or both of the 2:1 resonazones and high values with states outside the influenceany of the resonance zones.

Note that, with our present definitions, low values ofLshould only be found for states that are strongly mixed byallresonant coupling terms. Thus, states that are mixeddominantly by the two 2:1 resonances, and which mightin some sense ‘‘chaotic’’ in the semiclassical limit~as thecorresponding 2-resonance classical system is nonintegra!will nevertheless have high values ofL2:1.

Figure 5 summarizes our IPR computations for the fBaggot Hamiltonian~all coupling terms included! with N532. The results are displayed as a smoothed histogshowing the number of states within a given range ofL. It isapparent from the histogram that the IPR values do not ya clean separation of eigenstates into 3 classes: nonresosingly-resonant and mixed, as they do for the case studieAtkins and Logan.43 The IPR appears to be too crude aindicator of eigenstate delocalization to provide a basisclassification of eigenstates of the Baggot Hamiltonian.

Also shown in the Figure 5 are the effects of setticoupling parametersl8 andg to zero. The number of state

FIG. 5. Smoothed histogram of average IPRs,L, for the full Baggot Hamil-tonian ~solid!, l850 ~dotted!, andb2250 ~dashed!. P516.

7, No. 1, 1 July 1997

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with very low values of the IPR increases sharply whenl8 isset equal to zero, indicating a significant increase in mixof zeroth-order states; Baggot25 has stressed the importancof the parameterl8. If the 2:2 resonant coupling term iremoved by settingg50, however, there is only a minochange in the form of the histogram.

B. Husimi distribution functions

In order to explore the quantum-classical correspdence for the Baggot Hamiltonian in more detail, andprovide a dynamical basis for assignment of eigenstates,useful to have aphase spacerepresentation of the quantumwavefunctions; such a representation facilitates comparof the structure of the quantum state with the important csical phase space structures.9 Two well known representations are the Wigner59 and the Husimi50 functions. We usethe Husimi representation since the Wigner representacan take negative values, thereby invalidating its use aprobability distribution, and also exhibits oscillations onfiner and finer scale as\→0.59 The Husimi function is aGaussian smoothed version of the Wigner function. It is nnegative, and is well suited for study of the classicquantum correspondence.60 Other related approaches to thdefinition of quantum phase space involve the construcof generalized coherent states61 which take into account theunderlying dynamical symmetry group of the problem. Fpresent purposes, however, we use the following simmethod to obtain the necessary distribution function.

We start by laying down a grid in the (N1 ,f1) coordi-nates on the Poincare´ surface of section~i.e., for a fixedvalue of the sectioning anglef25f2!. For a given point onthis grid, denoted by (N1

(k) ,f1(k)), and the chosen value o

f2 , we calculate the associated values of the original cancal variables (I (k),u(k)). To compute (I (k),u(k)) we must pickan arbitrary value~equal to zero for results reported in thpaper! of the ignorable anglec conjugate to the superpolyaaction J, and ensure that the energy of the classical phspace point equals that of the eigenstate under consideraCartesian coordinates (pm

(k) ,qm(k)), m51,2,b, associated with

the grid point (N1(k) ,f1

(k)) are then calculated via the usuharmonic action-angle transformations. The value of the Hsimi function for eigenstateuC& at phase space poin(p(k),q(k)) is calculated by taking the square of the overlapuC& with a product of minimum uncertainity Gaussians

x~qm!5S cm2

p D 1/4 expF2cm2

2~qm2qm

~k!!21i

\pm

~k!qmG ,~4.5!

for each of the three modesm51,2,b. Diagonalization of theBaggot Hamiltonian in a number basis gives the eigenstuC& as linear combinations of a finite number of productsharmonic oscillator basis functionswnm

(qm):

C~qm!5 (n1n2nb

Cn1n2nbwn1

~q1!wn2~q2!wnb

~qb!. ~4.6!

The resulting Gaussian integrals are performed analyticto give the Husimi distribution function asuAu2, where


g

-

is

n-

na

--

n

rle

i-

seon.

-

f

esf

ly

A~p~k!,q~k!!5 (n1n2nb

Cn1n2nb)1

Anm

zmnm

3expF21

2uzmu21

i

2\pm

~k!qm~k!G . ~4.7!

In the above equation we have defined

zm[cm

&

qm~k!1

i

cm\&pm

~k! , ~4.8!

with cm2 the mode harmonic frequencies.

Note that the Husimi functions need only be computfor one quadrant of the (N1 ,f1) surface of section due to thsymmetry of the full Baggot Hamiltonian. To explore thclassical-quantum correspondence for eigenstates, conof the Husimi distribution functions are superimposed onclassical surface of section at an energy correspondingthe particular quantum state. In computing the Husimi funtions contributions from both upper and lower sheets ofsos~when present! are added incoherently.

V. BAGGOT HAMILTONIAN: SPECIAL CASES

Before tackling the full Baggot Hamiltonian, it is usefuto examine eigenstates of Hamiltonians obtained by keeponly selected resonant coupling terms~see also Ref. 48!. Inthe case that a single resonant coupling term is presentHamiltonian is classically completely integrable. When twresonant coupling terms are present~e.g., 1:1 plus 2:1, orboth 2:1 terms!, the Hamiltonian is nonintegrable, but posesses a constant of the motion in addition to the eneThis constant of the motion persists in the Baggot Hamtonian when all three coupling terms are present, as the tresonance vectors are linearly dependent.51

A. H0 plus a single 2:1 stretch-bend resonance

Figure 6 shows projections onto the action lattice of seral eigenstates of the quantum Hamiltonian obtainedadding a single 2:1 stretch-bend coupling term toH0 . Theeigenstates are superimposed upon a plot of the 2:1 rnance zone in the (I 1 ,I 2) action plane. In action space, theigenstates are delocalized along the linen25const, i.e.,N 1b[2n11nb is conserved in accord with the classical bhavior of the action variables for a single 2:1 resonance.

Figure 6 shows action projections for a sequencestates at constantn2 , corresponding to increasing excitatiotransverse to the resonance channel. A precise dynamicbased assignment of these eigenstates is obtained by eining associated Husimi functions: for resonant states,quantum phase space density is localized inside the releresonance channel in the surface of section, whereas forresonant states the quantum phase space density lies ou~‘‘separatrix’’ states62 occur at the boundary between theclasses of state, and have phase space density localizeunstable objects@fixed points# in the surface of section!.Each resonant state can be assigned three quantum num^N ,N sb ,n&, where the quantum numbern50,1,2..., mea-sures the degree of excitation within the resonance. For

7, No. 1, 1 July 1997

ance


FIG. 6. Action space projections for eigenstates of the single 2:1 resonchannel system with constantn250, increasingn. ~a! n50. ~b! n51. ~c!n52. ~d! n53. ~e! n54 ~separatrix state!.

re

arthota

out-roth-

e.

ose

ample, for the sequence of resonant states shown in Figuthere are five resonant states with^N 532, N 1b532, n50,1,2,3,4&. Note that the state withn54 is a separatrixstate. The associated Husimi distribution functionsshown in Figure 7. Note the movement of the maxima ofHusimi distributions outwards from the center of the resnance as one increases the excitation, with the final sclearly localized about the unstable 2-torus.


6,

ee-te

Nonresonant states, whose phase space density liesside the classical resonance channel, can be assigned zeorder quantum numbers (N ,n1 ,n2), corresponding to thedominant zeroth-order state contributing to the eigenstat

Our set of resonant quantum numbers^N ,N s,b ,n& forthe integrable 2:1 case is closely related to that used by Rand Kellman forL-R-R states:nl , $nR1 ,nR2%.

48 The super-polyad quantum number isN 52nl12nR112nR2 , the 2:1

7, No. 1, 1 July 1997

n in


FIG. 7. Husimi distribution functions for the resonant eigenstates showFigure 6.

ont

any

ho

resonance quantum number isns,b52nR11nR2 , while thequantum numbern can presumably be identified withnR1 .

There has been much discussion in the literature ccerning the correspondence between classical and qua


n-um

resonances~see, for example Ref. 63!. The 2:1 stretch-bendresonance channel studied here is large and contains mstates~as does the 1:1 resonance!. The situation is thereforeanalogous to that studied by Ramchandran and Kay, w

7, No. 1, 1 July 1997

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examined the connection between the existence of avoilevel crossings and classical resonant dynamics.64 Concep-tual problems concerning the classical-quantum correspdence for narrow, high-order resonances~width of order\!have been discussed by Roberts and Jaffe´.45 In particular,these authors define quantum resonant eigenstates asformed by strong interaction between zeroth-order states tmay lie outside the~narrow! classical resonance channel.45

From our perspective, such states have~distorted but! topo-logically nonresonant phase space densities~see, for ex-ample, Figure 2d of Ref. 47!.

States localized in the 2:1 resonance channel with diffent values ofn2 form sequences, where each sequencecharacterized by a particular value of the excitation quantunumbern. In Figure 8 we show a sequence of states whiprogresses along the 2:1 resonance zone. The 4 stateFigure 8, for example, are assigned quantum numbers^N532,N 1b532,n50&, ^32,26,0&, ^32,20,0& and^32,14,0&, re-spectively. The Husimi distributions for the above sequenof states are all localized in the vicinity of a stable resona2-torus~cf. Figure 7a!.

We shall investigate below the fate of these eigenstaas the remaining resonant coupling terms are turned on.

An important observation is thatn1 /nb resonant stateswith similar distributions of amplitudes over the resonaregion of action space, and the same value ofn, appear atintervals ofDI 253 in the actionI 2 ~cf. Figure 8!. This pe-riodicity is simply a consequence of the fact that the slopethe resonance line in the action plane is nearly23; hence,the form of the stretch-bend Hamiltonian in the zeroth-ordbasis near the center of the resonance zone is approximainvariant asn2→n213.65 The periodicity persists even uponinclusion of additional resonant couplings.

The size of the intervalDn253 is an accidental conse-

FIG. 8. Action space projections for a sequence of eigenstates in the si2:1 resonance system with constantn50 andn250,3,6,9.


ed

n-

oseat

r-ismhin

et

es

t

f

rely

quence of the particular parameter values in our Hamtonian. For example, if we setess5esb50 then the observedperiodicity isDn252. In each case, however, the appareperiodicity in the location of isomorphic resonance statesbe understood in terms of the slope of the classical resonaline. An analysis based solely on the properties of primperiodic orbits is, as far as we can see, unable to providcomparably simple explanation of these relatively subproperties of the eigenstate densities in action space.

To clarify the significance of the periodicity associatwith the slope of the resonance line in theI 1 ,I 2 action plane,consider a simple, integrable Fermi resonant 2-mode Hatonian,

H5VsI s1VbI b1asI s21esbI sI b1abI b

2

1b12I bI s1/2 cos~us22ub!, ~5.1!

with s51 or 2. This Hamiltonian describes an integrabsubsystem of the Baggot Hamiltonian. The resonance lin(I s ,I b) space is found by setting the frequency ratio equa2, vs52vb . This equation may be written:

I b5`~ I s2I s!, ~5.2!

where

`[2~esb2as!

~esb24ab!, ~5.3!

and

I s[~Vs22Vb!

2~esb2as!, ~5.4!

denote the periodicity and one of the intercepts of the renance line, respectively. The phase space structure ofHamiltonian has been studied in some detail including podic orbit bifurcations66 and the quantum (E,t) characteriza-tion of the energy spectrum.67 Performing a canonical transformation (I s ,I b ,us ,ub)→(I z ,c;I ,C) ~Refs. 66, 67! andusing the variables

Q52as28ab , ~5.5a!

g15as14ab12esb , ~5.5b!

g25as14ab22esb , ~5.5c!

P5Vs22Vb , ~5.5d!

it follows that the primitive periodic orbits~apart from theones at North and South poles on the polyad phase sph!can be obtained from the cubic equation:

Z36e1Z21e2Z7e350. ~5.6!

In the above equation we have definedZ[(I1I z)1/2 and

e156bsb

23/2g2, ~5.7a!

e25P1~Q22g2!I

2g2, ~5.7b!

gle

7, No. 1, 1 July 1997

m

ac

Tt

ac

eem

r

etnt

a

areates

in-

1:1r the1:1e caners

nant

s of

o-

1

1:1

thea

-onhee of

akions

e


e352bsbI

21/2g2. ~5.7c!

One can express the coefficients of the cubic equationterms of the variables (,I s ,P) using the following rela-tions:

Q5~22` !

`I sP, ~5.8!

g252~21` !

2`I sP. ~5.9!

We now study the evolution of the bifurcation diagrawith varying ` and keepingI s constant. Contact with theBaggot Hamiltonian studied in this paper is made by repling P by P8[P1(ess22esb)K2 , whereK2 is the conservedaction corresponding to the nonresonant stretch mode.superpolyad number for the Baggot Hamiltonian is relatedthe Fermi resonant polyad andK2 throughN 54I12K2 .Figure 9 shows the bifurcation plot~f[cosc(Iz2I)/2I ver-susN , with c50, p! as one varies the periodicity forfixed value ofI s andK2 . For the physical value of ~solidline!, we have three roots forN greater than 12, signalingthe presence of pendular resonance regions in phase sp67

The three roots correspond to a local mode~stable! 2-torusand a pair of stable/unstable 2:1 resonant 2-tori, in agreemwith our analysis of the 2-tori of the integrable subsystgiven earlier~cf. also Appendix!. On reducing` by half~dots! we are still in the Chirikov regime. For the particulachoice of Baggot parameters the periodicity`.0 in the(I s ,I b) plane. Eliminating the bend anharmonicity, i.e., sting ab50 in the Baggot Hamiltonian leads to a differebifurcation structure as shown in Figure 9~filled circles!.Now one clearly sees values ofN for which there are fewerthan three roots and hence a non-pendular phase sp

FIG. 9. Bifurcation diagram~f[cos@(Iz2I)/2I # versusN ! as a function ofthe periodicity` of the single 2:1 resonance line with fixedI s , K2 . Physi-cal ~solid line!, reduced ~solid dots! and zero anharmonicity of bend mod~open circles!.


in

-

heo

e.

nt

-

ce.

Quantum eigenfunctions for reduced bend anharmonicitystrongly mixed and bear little resemblance to the eigenstanalysed in this paper.

B. H0 plus the 1:1 and 2:2 stretch-stretchresonances

The Hamiltonian consisting ofH0 plus the 1:1 and 2:2stretch-stretch resonant coupling terms is also completelytegrable. Eigenstates are delocalized along the linen11n25const. Just as for the 2:1 resonance, we can classify thestates as resonant or nonresonant according to whethephase space densities are localized inside or outside theresonance channel. States outside the 1:1 resonance zonbe assigned zeroth-order local mode quantum numb(N ,n1 ,n2). The line n11n25const is the projection intothe action plane of the polyad phase sphere for 1:1 resosystems studied by Kellman and coworkers.20 We note thatthe width of the 1:1 resonance decreases at higher valueI 15I 2 .

Figure 10 shows a sequence of 1:1 resonant~normalmode, separatrix! states which progress along the 1:1 resnance channel and can be assigned resonant~normal mode!quantum numbers:@nb ,N 125n11n2 ,n#. The particular se-quence shown in Figure 10 can be assigned as@4k,2l ,2# withk50,1,...,7 andl582k. ~The correspondence with the 1:resonant N-N-B quantum numbers of Rose and Kellman48 isas follows:N 125ns1na , n5ns .!

Isomorphic normal mode states appear along theresonance zone at intervalsDN 1252, where all states in agiven sequence have either a node or an antinode alongdiagonal. This well known alternation can be viewed asconsequence of the slope11 of the 1:1 resonance line. Figure 11 shows 4 of the corresponding Husimi distributifunctions. The Husimi functions are all localized around tunstable normal mode 2-torus. The characteristic x-shapHusimis localized in the vicinity of an unstable fixed point60

is clearly evident in Figures 11c, 11d. The effect of the we2:2 resonance can be clearly seen in Figure 11d. Bifurcat

FIG. 10. Action space projections for a sequence of 1:1 resonant~normalmode! states for the 1:112:2 integrable subsystem andP516.

7, No. 1, 1 July 1997

coinfuinng

tes

cinick

cnwoiose

evin

sna:1

ssnt

erlle

ri-ateal-is-te-d,

it

2:1

ses.eso-


occur which affect the topology of phase space and theresponding Husimi distributions. Although we have notvestigated the possible bifurcations in this subsystem indetail, we note that the form of the Husimi distributionFigure 11d is consistent with a bifurcation in the underlyiclassical surface of section.

C. H0 plus two 2:1 stretch-bend resonant couplingterms

The Hamiltonian consisting ofH0 plus two 2:1 resonancoupling terms is classically nonintegrable. It neverthelconserves the quantityN 52(n11n2)1nb . In the rest ofthe figures involving action space projections for sequenof eigenstates we show the different states with varythickness. The terminal state is shown with the largest thness in the figures.

We now show action space projections for 3 sequenof eigenstates that progress along the 2:1 resonance chaFor example, in Figure 12 we show a sequence of statesn50 and symmetric with respect to reflection in the diagnal I 15I 2 . Proceeding inwards from the edges of the actplane, the first two states in the sequence bear a close reblance to the first two states in Figure 8 atI 250 and I 253, respectively. The next state in the sequence is howof symmetric ‘‘normal mode’’ character, and is localizedthe vicinity of the resonance intersectionI r . There is there-fore an effective 1:1 mixing along the directionN 12

5const arising from the two 2:1 stretch-bend coupling termIn fact, analysis of the interaction of the two 2:1 resonacoupling terms using classical perturbation theory givesestimate of the effective strength of the induced 1coupling.56

One can assign these states using labels which are etially the same as in the integrable 2:1 case, but augmewith a 6 subscript and ‡ superscript. The6 label describesthe state symmetry with respect to exchange 1↔2 while the‡ symbol denotes a state with ‘‘normal mode’’ charactThe three states in Figure 12 can therefore be labe^32,32,0&1 , ^32,26,0&1 and ^32,20,0&1

‡ , respectively.

FIG. 11. Husimi distribution functions for the normal mode sequence wn52 shown in Figure 10.


r--ll

s

esg-

esnel.ith-nm-

er

.tn

en-ed

.d

In Figure 13 we show the corresponding Husimi distbutions. Note that the Husimis, including the terminal stshown in Figure 13c, all show the common feature of locized density atf156p. The terminal state phase space dtribution also has some similarity to the corresponding ingrable limit normal mode state Husimi shown in Figure 11albeit somewhat distorted and phase shifted inf1 , and couldbe assigned the normal mode quantum numbers@4,14,2#1 .The terminal state is thus a mixed state for whichmore than

h

FIG. 12. Sequence of action space projections for eigenstates of tworesonance channel Hamiltonian withn50 andN 532. In order to distin-guish the states in a sequence we plot them with different line thicknesTerminal states have the thickest line, and the same holds for all multirnance action space plots shown in the rest of the paper.

FIG. 13. Husimi distributions corresponding to then50 2:1 resonancesequence shown in Figure 12.

7, No. 1, 1 July 1997

tee

rensici

acas

e’et-tein

ndclada

ater

ioplsi

o

f theels.la-

go

il-ur

o


one set of quantum number labels is possible.We observethat the resonance excitation quantum numbern is differentin each set of quantum numbers, i.e.,n50 ~2:1 label! versusn52 ~1:1 label!.

Figure 14 shows a sequence of 2 states^32,30,1&1 ,^32,24,1&1 that leads to a mixed normal mode-type sta^32,18,1&1

‡ in the vicinity of I r . The associated Husimis arshown in Figure 15. The parent state~at n251! Husimishown in Figure 15a corresponds ton51 as is clearly seenby comparing to the integrable case in Figure 7b. The pastate is relatively far away from the resonance junction ahence the close resemblance of the corresponding Hudistributions. The Husimi for the next state in the sequenFigure 15b, is localized in angle space but delocalizedaction. We provisionally associate this delocalization intion space with the destruction of barriers to classical phspace transport.

The terminal state, which resembles a ‘‘normal modstate in action space, localizes entirely on the upper shethe sos~Figure 15c!. The Husimi function, however, is significantly distorted as compared to the corresponding ingrable normal mode Husimi. Most noticeable is a shiftphase anglef1 from the integrable 1:1 case.

For the Hamiltonian with two 2:1 resonant stretch-becoupling terms, we are therefore able to identify sequenof resonant states that define the ‘‘parentage’’ of particumixed states. The terminal states are mixed, as suggestetheir Husimi distribution functions, and can be viewedpart of one or more sequences. This leads to anonuniquelabelling of the mixed states. For example, the terminal stof the sequence shown in Figure 12 can be assigned eith^32,20,0&1 or as@4,14,2#1 . The basis for our identificationof sequences is a study of amplitude distribution in actspace, guided by the classical resonance channel temobtained via Chirikov analysis. It is not clear how an analybased upon primary periodic orbit bifurcations alone20 wouldenable such patterns to be identified.

We emphasize that the nonunique labelling of states

FIG. 14. Sequence of action space projections for eigenstates of the twresonance channel Hamiltonian withn51.


ntdmie,n-e

’of

-

esrbys

eas

nates

b-

tained here and in the next section is a consequence ointersection of the primary 1:1 and 2:1 resonance channThis level of dynamical complexity is not present in the retively straightforwardP58 case treated previously by us47

and others.24,48 Our analysis and assignments thereforebeyond this earlier work.

VI. FULL BAGGOT HAMILTONIAN

The Chirikov resonance analysis of the Baggot Hamtonian given in section III B suggests the existence of fo

2:1

FIG. 15. Husimi distributions corresponding to then51 2:1 resonancesequence shown in Figure 14.

7, No. 1, 1 July 1997

,sihuas

n

e

eduny-ifi

a

inialinthdpadbee

desenathigsoo

oein

e-samne

d

e

bsec-

ress-tivethe

eirgainlessif-antis inized

own-ingla-

gme-partfull-

s ofanceil-nofgen-depre-se-t of

mil-


classes of eigenstate~cf. Ref. 37!: ~1! Nonresonant statesunaffected by any primary resonance. Phase space denfor such states are located outside any of the resonance cnels, and good quantum numbers are the zeroth-order qtum numbers n1 , n2 and nb . States are labelled a(n1 ,n2 ,nb). ~2! Normal mode~1:1! resonant states, withphase space densities located in the 1:1 resonance chaApproximate good quantum numbers areN , N 125n11n2and a third quantum numbern describing the level of exci-tation inside the 1:1 resonance. The relevant states arbelled as@N ,N 12,n# or @nb ,N 12,n#. ~3! Local-stretch/bend resonant states, with phase space densities locatthe 2:1 resonance channel. Approximate good quantum nbers areN ,N sb52ns1nb , and a resonant excitatioquantum numbern. States are therefore labelled b^N ,N sb ,n&. ~4! ‘‘Mixed’’ eigenstates, located in the vicinity of the resonance channel intersection. A similar classcation has been proposed by Lu and Kellman24 and by Roseand Kellman,48 in either case without reference to the actulocation of the resonance channels in action space.

Our classification scheme provides a framework withwhich to analyze the eigenstates of the Baggot HamiltonIn regions of phase space in which there is strong coupbetween modes, the zeroth-order invariant structuresprovide the basis of the classification scheme may bestroyed. Nevertheless, examination of quantum phase sdensities often enables us to provide an unambiguousnamical assignment. Such an approach has previouslysuccessful for theP58 manifold,47 and we examine here thset of states withP516.

A. Progressions of states along resonance channels

In the case of the integrable single resonance casescussed in the previous section, the zeroth-order and/or rnance quantum numbers provide a rigorous and completof state labels. For the full Baggot Hamiltonian, which cotains three linearly dependent resonance coupling termswhich is classically nonintegrable, states far away fromresonance intersection region of action space can be assusing labels appropriate to the first 3 classes discusabove. As one approaches the overlap region the apprmate labels may break down and we have a possibilitystates belonging to class 4.

Nevertheless, as noted in the previous section, it is psible to identify sequences of states that progress along rnance channels, so that specific states in the resonancesection region can be identified as the terminus of one~ormore! sequence~s! of regular~assignable! states. It is impor-tant to note that it is quite possible for two different squences to have a common terminal state; the two setassociated labels then approximately describe the seigenstate. Such ambiguity in the labelling of eigenfunctiodistinguishes theP516 case currently under study from thdynamically simpler P58 manifold analyzedpreviously.24,47,48

We now proceed to combine insights gained from stuof surfaces of section, Husimi phase space densities andclassical resonance template to analyze eigenstates of th


tiesan-n-

nel.

la-

inm-

-

l

n.gate-cey-en

is-o-set-ndenededxi-f

s-so-ter-

ofes

ythefull

Baggot Hamiltonian withP516. We focus our attention onthe fate of the three sequences presented in the last sution as we turn on the remaining primary 1:1 resonance.

Figures 16 and 17 show the sequence of states proging along the 2:1 resonance channels and their respecHusimi distributions. This sequence is to be compared tosimilar one shown in the previous section~cf. Figures 12,13!. The first and second states are quite similar to thpartners in the previous sequence. The terminal state is aa mixed state with nonunique labels; there are nevertheclear differences in the forms of the Husimi due to the dferent phases of the induced 1:1 and primary 1:1 resoncoupling terms. In the full Hamiltonian the terminal Husimis almost completely localized on the upper sos whereathe 232:1 resonance case the terminal state is delocalon both sheets of the sos.

In Figures 18 and 19 we show then51 sequence andthe corresponding Husimis analogous to the sequence shin Figure 14. For the full Hamiltonian, the form of the Husimi of the second state is indicative of an avoided crosswith another state. Computation of an energy level corretion diagram as a function ofbsb confirms this surmise. Theresonant1 parity state is involved in an avoided crossinwith a nearby normal mode state. The terminal state is sowhat more localized in phase space than the countershown in the previous section. The terminal state for theHamiltonian also looks ‘‘cleaner’’ in action space as compared to the 232:1 case.

The sequences considered here show how groupquantum states can organize around the classical resontemplate in action space. In the case of the Baggot Hamtonian, explicit computations of the Husimi distributiofunctions further illuminate the phase space significancesuch sequences. The local-bend resonant sequences areeralizations of the more familiar sequence of normal mostates. We have studied sequences in addition to thosesented in this paper. For example, it is possible to findquences which show an initially resonant state going ou

FIG. 16. Sequence of action space projections for the full Baggot Hatonian withn50.

7, No. 1, 1 July 1997

llyhton

aaete

acethinasednces

che lie

lvesso-pri-nc-nto-

fortert ishat

eamerpatheof

pi-

d intheothm-rgeee

Fi

mil-


resonance and becoming a local nonresonant state~cf. Ref.62!. Aided by the Husimi distributions, one can usefuidentify the parentage of very highly mixed states. Insigcan be thereby obtained into the nature of mixed states invicinity of the resonance intersection by their identificatias terminal states of a sequence~s! of regular states.

It is also apparent from the above examples that theproximate periodicity of the form of resonant eigenstatessociated with the value of the slope of the resonance linan important aid in identification of sequences of rela

FIG. 17. Husimi distributions corresponding to the sequence shown inure 16.


the

p-s-isd

states. The width of a particular resonance in action spdetermines the number of quantum states that can fit withe the resonance channel. In the Baggot case we have bour classical resonance analysis on the primary resona~2:1 and 1:1!, which have significant widths~compared to\!.High-order resonances will have narrow widths, in whicase quantum states strongly influenced by the resonancoutside the classical resonance zone.45 However, such inter-acting quantum states will nevertheless organize themsein accordance with the approximate periodicity of the renance line, just as for the sequences of states inside themary resonance channels. The corresponding Husimi futions will exhibit distorted but topologically non-resonafeatures. The periodicities in the form of the ‘‘quantum resnant’’ states analyzed by Roberts and Jaffe´ are clearly ob-servable in Figure 6 of Ref. 45.

B. Energy level correlation diagrams and avoidedcrossings

We have studied energy level correlation diagramsthe Baggot Hamiltonian under variation of the paramebsb , which controls the strength of the 2:1 resonances. Ifound that almost all of the narrowly avoided crossings toccur close to or at the physical value ofbsb , correspondingto the full Baggot Hamiltonian studied in the rest of thpaper, involve three states. Two of the states have the s@6# symmetry and are involved in either a broad or shacrossing, while the third state, of opposite symmetry, ispartner state to one of the two states. Interaction betweentwo states of the same symmetry results in delocalizationthe Husimi distribution for one of the partner states. Tycally, at high energies and values ofN such thatI r lies inthe physical region of action space, the two states involvethe crossing are localized on one of the two sheets ofsurface of section, with the other state delocalized over bsheets. An example would be a state with a very small nuber of bend quanta that interacts with a state with a lanumber of bend quanta. Another possibility is that all thr

g-

FIG. 18. Sequence of action space projections for the full Baggot Hatonian withn51.

7, No. 1, 1 July 1997

en

thf

uai

dery

rlts.

tionen-

theithf. 64reso-an-or-edxhibit20am-ne.areonces.’’s ofusly.singso-nyingsss-mo-

tespre-

Fi

got


states can localize/delocalize on one sheet of the surfacsection. An example for the latter case has been preseearlier ~cf. Figure 4 in Ref. 47!.

A considerable amount of work has been devoted inrecent literature to determining the quantal signatures oclassical nonlinear resonance.63 Most of the studies haveconcentrated on the evolution of the eigenenergies of a qtum system under the variation of a parameter of the Hamtonian. Attempts to correlate the type and number of avoicrossings to classical resonances have been unsatisfacto

FIG. 19. Husimi distributions corresponding to the sequence shown inure 18.


ofted

ea

n-l-d. In

fact, as pointed out by Berry,68 the fragility of one parametecrossings is partially responsible for the inconclusive resuThe results of Ramachandran and Kay64 and Ozorio deAlmeida69make it clear that there is no necessary connecbetween the existence of avoided crossings of quantumergy levels and classical resonant dynamics.

It is nevertheless interesting to ask which features ofenergy level correlation diagram might be associated wclassical resonances. The results in this paper and in Resuggest such a feature in the case when the classicalnances have significant widths in action space. Ramachdran and Kay noted that, under variation of a parameter cresponding to the frequency of one of 2 oscillators involvin a resonance, states deep inside the resonance zone enearly parallel curves in the correlation diagram. Figureshows the energy correlation obtained by varying the pareter,bsb , that controls the width of the 2:1 resonance zoVariations of the states in the classical resonance zoneshown as thick lines~diabats!. Note that these states undergthe largest amount of variation upon changing the resonawidth parameter; groups of such states form a set of ‘‘fanThe states within each group correspond to different levelexcitation inside the resonance zone, as discussed previoThe number of states in each group decreases with increaenergy, correlating with the decrease in the classical renance width. It is significant to observe that a state from agiven group of resonant states has many avoided crosswith local modes and normal modes. These avoided croings are not in general associated with classical resonanttion as also pointed out by several authors.70

It is also relevant to point out that the sequences of stathat progress along the 2:1 resonance channel consist

g-

FIG. 20. Evolution of the energy spectrum as a function of the Bagparameterbsb . ‘‘Diabats’’ are shown as thick lines.

7, No. 1, 1 July 1997

ne

adw

tofuc

tuorvf trtuotiofi-olaod

glliethfua

o

aicnesotabicaley

ofmoftru

esrli

nrgeaat

acedthstud-mil-

elysivetheenceofhib-

the.alforarepri-ul-ngical

eenelo-r too-da-lt

2.theSF

he


cisely of those states corresponding to nearly parallel lifrom different groups in Figure 20.

Finally, we note that the existence of many broavoided crossings in the eigenvalue correlation diagramshave computed forP516 means that it is often impossiblefollow diabats in order to correlate eigenstates of theHamiltonian with those of zeroth-order single resonanHamiltonians, as done by Rose and Kellman forP58.48

VII. CONCLUSIONS

In this paper we have studied the classical and quanmechanics of the Baggot vibrational Hamiltonian fH2O.

25 We have shown first of all that standard Chirikoresonance analysis provides a useful approximate map oclassical phase space structure. This conclusion is suppoby more rigorous analysis of classical phase space strucin terms of families of resonant 2-tori, and by examinationclassical surfaces of section. We point out that determinaof the location of primary periodic orbits alone is not sufcient to establish detailed correlations between the formthe quantum eigenstates of the Baggot Hamiltonian and csical phase space structure. Whereas the primary periorbits intersect constantN ~superpolyad number! surfaces inisolated points, resonant 2-tori determined within the sinresonance approximation appear in 1-parameter famiThe resonant 2-tori are thus suitable generalizations ofprimary periodic orbits to 3D systems, and form a useframework for understanding both the classical phase spand quantum-classical correspondence of the system.

An essential aspect of our study is the examinationquantum phase space~Husimi! distribution functions foreigenstates of the Baggot Hamiltonian. Comparison of qutum phase space distributions with corresponding classsurfaces of section enables us to define resonant andresonant states as localized inside or outside a given rnance channel, respectively. Quantum states are found tganize into sequences of states that progress alongresonance channels. An important observation is that thetion space amplitudes for states in such sequences exhiperiodicity, which is determined by the slope of the classiresonance line. The existence of these sequences enabto provide labels for terminal ‘‘mixed’’ states in the vicinitof resonance intersections.

For the manifold of states examined~all states withP516!, it is found that most states can be assigned to onedistinct classes of eigenstate; these assignments arewith the aid of the Husimi function and the identificationthe various sequences. The study of the energy specwith the variation of a resonant coupling parameter~whichdetermines the width of a resonance channel! provides acharacteristic quantum signature of states deep in the rnance regions. A similar signature had been studied eaby Ramachandran and Kay in a simpler system.64 Our stud-ies strengthen the notion that there is no necessary contion between the existence of avoided crossings in enelevel correlation diagrams and particular classical non-linresonances. However, the approximate periodicity associ


s

e

lle

m

hetedrefn

fs-ic

es.elce

f

n-alon-o-or-hec-t als us

4ade

m

o-er

ec-yred

with the slope of the classical resonance line in action spdoes influence the quantum states irrespective of the wiof the resonances, and is an important parameter in the sies of quantum-classical correspondence of resonant Hatonians.

The existence of the superpolyad quantum numberN

reduces the 3-mode Baggot Hamiltonian to an effectivtwo degree of freedom problem, and we have made extenuse throughout of the technology developed for studyingclassical mechanics and classical-quantum correspondof such systems. Despite being effectively a two degreefreedom problem, however, the phase space structure exits complexities not usually found in 2-mode systems.13 Fu-ture work will investigate the consequences of breakingconstancy ofN , thereby obtaining a true 3-mode problemVery little is currently known concerning detailed classicdynamics and/or quantum-classical correspondence3-mode molecular Hamiltonians. Most techniques thatuseful for 2 degree of freedom systems are either inapproate or need to be suitably generalized in order to study mtimode problems. A number of very rich problems pertainito the nature of classical phase space transport, dynamtunneling, chaos assisted tunneling71 and multimode effectsarise, however, and the undoubtedly subtle interplay betwclassical transport and quantum mechanisms leading to dcalization and/or localization has to be understood in ordegain insights into the spectrum and dynamics of realistic mlecular Hamiltonians. The present work provides the fountion for future work towards these important yet difficugoals.

ACKNOWLEDGMENTS

This work is supported by NSF Grant CHE-940357Computations reported here were performed in part onCornell Supercomputer Facility, which is supported by Nand IBM corporation.

APPENDIX: PRIMARY PERIODIC ORBITS ANDRESONANT 2-TORI

1. Bifurcation analysis of the Baggot Hamiltonian:Periodic orbits

In terms of variables (N,f), with N5K1Jr , the BaggotHamiltonian is

H52vN21a1N121a2N2

21b12~N2!

3@~N21BJ!22N12#1/2 cosf11b22@~N21BJ!22N1

2#

3cos 2f11bb~AJ24N2!@~N21N11BJ!1/2

3cosf11~N22N11BJ!1/2 cosf2#, ~A1!

with b12(N2)[b128 1m1N21m2 , f6[(f16f2)/2, andb1b5b2b[bb . The various constants are functions of toriginal Baggot parameters:

v5Vs22Vb , ~A2a!

a1,252c7 c12, ~A2b!

7, No. 1, 1 July 1997

he

ith

s to

l

re,

to

s

b

h

in


m152~l822l9!, ~A2c!

m25~2l8B1l9A!J. ~A2d!

In terms of the original action variables, (AJ24N2)5I b and(N26N11BJ)5I 1,2.

Hamilton’s equations of motion are:

f152a1N12N1b12@~N21BJ!22N12#21/2

3cos~f1!22N1b22 cos~2f1!11

2bb~AJ24N2!

3@~N21N11BJ!21/2 cos~f1!

2~N22N11BJ!21/2 cos~f2!#, ~A3a!

f252v12a2N212b22~N21BJ!cos~2f1!1@m1@~N2

1BJ!22N12#1/21b12~N21BJ!@~N21BJ!22N1

2#21/2#

3cos~f1!24bb@~N21N11BJ!1/2 cos~f1!1~N2

2N11BJ!1/2 cos~f2!#11

2bb~AJ24N2!@~N21N1

1BJ!21/2 cos~f1!1~N22N11BJ!21/2 cos~f2!#,

~A3b!

N15b12@~N21BJ!22N12#1/2 sin~f1!

12b22@~N21BJ!22N12#sin~2f1!

11

2bb~AJ24N2!@~N21N11BJ!1/2 sin~f1!

1~N22N11BJ!1/2 sin~f2!#, ~A3c!

N251

2bb~AJ24N2!@~N21N11BJ!1/2 sin~f1!

2~N22N11BJ!1/2 sin~f2!#. ~A3d!

a. Normal mode periodic orbits

We are interested in finding the equilibrium points of tabove set of equations~Nk505fk , k51,2! since they cor-respond to periodic orbits of the full system, i.e., 1-tori wthe angle variable being the ignorable anglec[Aub1B(u11u2) conjugate to the polyad numberJ. We firstlook for solutions corresponding toN150, which are normalmode type solutions in the full system. If we setN150 in theabove equations then we get the system of equations:

f152bb~AJ24N2!~N21BJ!21/2 sin~f1/2!sin~f2/2!,~A4a!

f252v12a2N21@m1~N21BJ!1b12#cos~f1!

12b22~N21BJ!2 cos~2f1!2@8bb~N21BJ!1/2

1bb~AJ24N2!~N21BJ!21/2#cos~f1/2!cos~f2/2!,

~A4b!


N15b12~N21BJ!sin~f1!12b22~N21BJ!2 sin~2f1!

1bb~AJ24N2!~N21BJ!1/2 sin~f1/2!cos~f2/2!,

~A4c!

N25bb~AJ24N2!~N21BJ!1/2 cos~f1/2!sin~f2/2!, ~A4d!

where we have used the standard trigonometric identitierewrite terms involvingf6 . Note thatb12 is a function ofN2 . Let us consider the equation forN2 and note that itvanishes for the following cases:

~a! N25AJ/4,~b! N2ÞAJ/4, f156p,~c! N2ÞAJ/4, f250,62p.

We have not considered the possibility ofN21BJ50 sincethis corresponds toI b5J, I 15I 250, i.e., a pure bend criticapoint.

Case a: In this caseN25AJ/4 which implies that theaction in the bend mode is zero. Withf150,6p we see thatNk505f1 (k51,2) and

f252v11

2~a2A1b22!J1

1

4m1J1b12

24bbJ1/2 cos~f2/2!. ~A5!

Although the time derivativef2 is in general nonzero, thesolution withI b50 is nevertheless a periodic orbit, andf2 isjust the rotation rate around the periodic orbit. There ahowever, certain values off2[f2

(1) such thatf250, signal-ling the bifurcation of the normal mode periodic orbit due2:1 resonance with the bend mode. This value off2 is de-termined by:

cos~f2~1!!5

1

4bbJ1/2 ~C01C1J!, ~A6!

where

C052v1b128 1m2 , ~A7a!

C151

2~a2A1b22!1

1

4m1~11A!. ~A7b!

If both f1505f2 when I b50 thenf15u12u2 and f2

5u11u224ub . Thus, vanishing of both time derivativeimplies thatu15 u252ub .

Case b: It is easy to convince oneself that in case!bothN1 andf1 cannot vanish with a single choice off2 andhence we disregard this case.

Case c: In this case with a choice off150,62p wehaveNk505f1 , k51,2, and the equation we are left witis as follows:

f252v12a2N21m1~N21BJ!1b1212b22~N21BJ!

78bb~N21BJ!1/26bb~AJ24N2!~N21BJ!21/2.

~A8!

We cannot haveN25AJ/4 here since this was consideredthe previous case and moreover, this solution forN2 gives us

7, No. 1, 1 July 1997

iosetio

a-

o-

e2:

onwesfo

-

as

n of

ith

nd

,

esy

the

:1for-


precisely the periodic orbits corresponding to the bifurcatof the normal mode periodic orbit family resulting from caa! due to the 2:1 resonances. Simplifying the above equaand denotingx[(N21BJ) we get a cubic equation inx1/2:

x3/27px1qx1/26r50, ~A9!

where

p512bb

2~a21m11b22!, ~A10a!

q5C02~m112a2!BJ

2~a21m11b22!, ~A10b!

r5bbJ

2~a21m11b22!, ~A10c!

with the condition thatxÞ0.

b. Local mode periodic orbits

Referring back to the full system of Hamilton’s equtions of motion we observe that withN25AJ/4 andf150,6p we haveNk50, and the equation forf1 becomes:

f15N1F2~a12b22!7b12S J2162N12D 21/2G . ~A11!

For N156N1 with

N15F J2162 b122

4~a12b22!2G1/2, ~A12!

we havef150 and so obtain local mode periodic orbit slutions. Note, again that the finite value off2 here does notmatter asI b50. As before there are certain values off2

[f2(2) such thatf250 and this indicates that the local mod

periodic orbits can bifurcate due to the influence of theresonances. In the case whenf150 we have:

cos~f2~2!/2!5

f 1~J!

f 2~J!, ~A13!

where

f 1~J!52v11

2~a2A1a1!J1

m1b12

2~a12b22!, ~A14a!

f 2~J!54bbF S J41N1D 1/21S J42N1D 1/2G , ~A14b!

with a similar equation for the case whenf156p. In orderto calculate the periodic orbits resulting from the bifurcatiof the local mode solutions due to the 2:1 resonanceshave to consider the full system of equations and solve thwith the conditionN2ÞAJ/4. Let us consider the equationfor Nk (k51,2) and remember that the allowed rangesthe conjugate angles aref1P(22p,2p) and f2

P(28p,4p). Of all the possible combinations, it is sufficient to consider two cases with a! f1505f2 and b! f1

50, f252p. These choices for the angles imply thatNk

50; k51,2 and the relevant equations for the time derivtive of the angles reduces to a pair of nonlinear equation


n

n

1

em

r

-in

the variablest5x11x2 , s5x12 1x2

2 with x6[(N26N1

1BJ)1/2. The two cases are related by changes in the sigbb and the nonlinear equations to be solved are:

d1t31d2ts1d3t62bbs6bbJ50, ~A15a!

e1t472bbt

31e2st22e2s

21e3t26bbts1e4s6

1

2bbJt50,

~A15b!

where

d15a12b22, ~A16a!

d252Fd11 1

2m1G , ~A16b!

d35m1BJ2b128 2m2 , ~A16c!

e151

4m1 , ~A16d!

e251

2~a21b222m1!, ~A16e!

e35v2a2BJ, ~A16f!

e4521

2d32e3 . ~A16g!

The pair of nonlinear equations are solved numerically wthe conditionsx6Þ0.

2. Integrable limit 2-tori

Consider the Baggot Hamiltonian with only the 1:1 athe 2:2 resonance interactions included, i.e.,bb50. Usingthe generating function:

F r5~u12u2!L11~u11u2!L21ubLb , ~A17!

the Hamiltonian becomes:

H int52VsL21VbLb1~2as2ess!L121~2as1ess!L2

2

1abLb212esbLbL21b12~L2

22L12!1/2 cos~x1!

1b22~L222L1

2!cos~2x1!. ~A18!

It is clear that bothL2 andLb are constants of the motionand the superpolyad numberJ54L21Lb . Thus, finding theequilibrium points of the above Hamiltonian determinresonant 2-tori since there aretwo ignorable angles. It is easto solve the pair of equationsL1505x1 and the solutionsare withx150,6p:

a) L150, ~A19a!

b) L156FL222 b122

4~2as2ess2b22!2G1/2. ~A19b!

The first solution corresponds to normal mode 2-tori andsecond solution represents local mode 2-tori.

Now, consider the Baggot Hamiltonian with a single 2resonance interaction only and perform a canonical transmation

7, No. 1, 1 July 1997

i

ri.

oriot

in

ic-mi-baov

m

n

em

.

ed

m.

st,

-

nd

.


G r5~u122ub!K11u2K21~u112ub!Kb , ~A20!

which gives the transformed Hamiltonian:

H int5CK11BK121h~K2 ,Kb!12bb~Kb2K1!

3~Kb1K1!1/2 cos~z1!. ~A21!

Here, B[as22esb14ab , C[V1AKb1(ess22esb)K2 ,V[Vs22Vb , A[2(as24ab), and h(Kb ,K2) is a con-stant due to the fact that the new Hamiltonian is ignorablethe angles conjugate toKb ,K2 .

Again, solving the pair of equationsK1505 z1 is rela-tively straightforward and one obtains:

a) K15Kb , cos~z1!51

2bb~2Kb!1/2 ~C12BKb!,

~A22a!

b) 2By3/273bby1~C22BKb!y1/262bbKb50,

x150,6p. ~A22b!

Solution a! corresponds to 2-tori with no bend quanta (I b50), while solutions b! are local-mode/bend resonant 2-toIn this case one hasJ52K214Kb .

Thus, one gets 2-parameter families of resonant 2-tand we can plot the location of these solutions in actspace as a function of one of the constants of motion andpolyad number. The results are summarized in Figurewhere we show the families of periodic orbits and 2-tori(N ,I 1 ,I 2) space.

From Figure 2 it is clear that the local mode periodorbits ~I 150 or I 250! of the full system lie at the boundaries of the families of 2-tori of the integrable subsysteMoreover, the locus of periodic orbits resulting from the bfurcation of the normal mode solutions is given to reasonaapproximation by the intersection of the sheets of reson2-tori. This confirms the appropriateness of the Chirikanalysis for the Baggot Hamiltonian.

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