Carlos A. Arango, William W. Kennerly and Gregory S. Ezra- Classical and quantum mechanics of diatomic molecules in tilted fields

8/3/2019 Carlos A. Arango, William W. Kennerly and Gregory S. Ezra- Classical and quantum mechanics of diatomic molecule

1/15

Classical and quantum mechanics of diatomic molecules in tilted fields

Carlos A. Arango, William W. Kennerly, and Gregory S. Ezra a

Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca,New York 14853

Received 31 January 2005; accepted 17 February 2005; published online 6 May 2005

We investigate the classical and quantum mechanics of diatomic molecules in noncollinear tilted

static electric and nonresonant linearly polarized laser fields. The classical diatomic in tilted fieldsis a nonintegrable system, and we study the phase space structure for physically relevant parameterregimes for the molecule KCl. While exhibiting low-energy pendular and high-energy free-rotorintegrable limits, the rotor in tilted fields shows chaotic dynamics at intermediate energies, and thedegree of classical chaos can be tuned by changing the tilt angle. We examine the quantummechanics of rotors in tilted fields. Energy-level correlation diagrams are computed, and thepresence of avoided crossings quantified by the study of nearest-neighbor spacing distributions as afunction of energy and tilting angle. Finally, we examine the influence of classical periodic orbits onrotor wave functions. Many wave functions in the tilted field case are found to be highlynonseparable in spherical polar coordinates. Localization of wave functions in the vicinity ofclassical periodic orbits, both stable and unstable, is observed for many states. 2005 American Institute of Physics. DOI: 10.1063/1.1888574

I. INTRODUCTION

There has been considerable recent interest in the controlof atomic and molecular quantum dynamics see, for ex-ample Refs. 18 and references therein. Research on thecontrol of molecular rotational degrees of freedom9,10 in-cludes production of specific states11,12 or bands of highlyexcited rotational states,13 rotationally induced bonddissociation,1317 and coherent control of periodically kickedmolecules.1820 Particular attention has been given to the pro-duction of angularly localized aligned or oriented

states.2,4,9,21,22

For a recent survey, see Ref. 23.In the present paper we consider the rotational dynamicsof a diatomic molecule in a combination of strong staticelectric and pulsed nonresonant infrared laser fields. Thiscombination of fields, primarily in the collinear configura-tion, has been investigated as a means of producing orientedstates, both theoretically2427 and experimentally.2831 Herewe study both the classical and the quantum mechanics forthe case of tilted fields, where the classical dynamics is typi-cally nonintegrable.32 The properties of strongly mixed rota-tional quantum states produced by molecule-field interac-tions are a novel topic in the study of the classical-quantumcorrespondence for both integrable33 and nonintegrable34

systems. For example, both symmetric tops in static electricfields35 and diatomics in collinear static and laser fields36

exhibit quantum monodromy, a phenomenon recently studiedin both atomic and molecular systems.3740 The classicalphase space in the integrable collinear field case is organizedby relative equilibria,36 which are associated with stationarypoints of the effective angular potential.41 Here we investi-gate the classical-quantum correspondence for the noninte-

grable tilted field case, with special emphasis on the role ofperiodic orbits that evolve from the relative equilibria of thecollinear field limit.

In Sec. II we provide some relevant background on theproblem of rotational dynamics of molecules in strong exter-nal fields. The classical and quantum Hamiltonians for oursystem are introduced in Sec. III. Section IV presents ourresults on the classical phase space structure of the diatomicin collinear and tilted combined fields. In Sec. V we discussour analysis of the nearest-neighbor level spacing distribu-tion for the quantum problem, and correlate our findings withthe corresponding classical dynamics. In Sec. VI we examinerotor wave functions, paying particular attention to localiza-tion of eigenfunction probability densities in the vicinity ofclassical periodic orbits. Section VII concludes.

II. ROTATIONAL DYNAMICS OF MOLECULESIN STRONG EXTERNAL FIELDS:GENERAL BACKGROUND

Beams of oriented molecules have long been used asprobes of the orientation dependence of molecular collisiondynamics.2,21,42 Oriented states can be produced by brute

force techniques, in which strong static uniform externalE-fields induce mixing between J levels and lead to angularlocalization nonzero P1cos .

21,43

Field strengths used in brute force orientation require anonperturbative approach to state mixing. For polar diatom-ics, the resulting oriented pendular states have receivedextensive study.4346 Some studies of state mixing in sym-metric tops and asymmetric tops subject to intense externalfields have also been carried out.47,48 One focus of this workhas been on the evolution of eigenstates from the zero fieldto the strong-field limit; sharply avoided crossings of energyaElectronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 122, 184303 2005

0021-9606/2005/12218 /184303/15/$22.50 2005 American Institute of Physics122, 184303-1

Downloaded 10 May 2005 to 128.253.229.100. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
http://dx.doi.org/10.1063/1.1888574


2/15

levels as a function of static field strength can lead to incor-rect state labeling according to the zero-field limit parent inan adiabatic correlation scheme,47 although in such cases itmight be possible to provide physically significant labels interms of diabatic states.

Onset of field-induced ionization limits experimentalstatic electric-field strengths to values 5104 V cm1.This limitation on the static field strength has motivated the

development of alternative approaches to producing angu-larly localized states for systems with small dipole momentsand/or moments of inertia. Nonresonant linearly polarizedlaser fields oscillating at infrared frequencies lead, viasecond-order molecule-field interaction mediated by mo-lecular polarizability rather than dipole moment, to alignedstates.4,4951 These pendular states have been investigatedtheoretically.24,25,45,46,49

An important aspect of the use of pulsed fields is thetime dependence of alignment, both during26,5254 and fol-lowing the pulse, so-called field-free alignment.23 Studiesof the quantal time evolution of rigid rotors subject to singlenonresonant laser pulses have identified three regimes:53 anadiabatic regime, where the pulse length pulserot, whererot is the time scale for rotation; a short pulse regime, wherepulserot; and an intermediate regime, where pulse rot. Inthe adiabatic regime, the time dependence of the alignmentessentially follows that of the nonresonant laser pulse.

For small enough values of angular momentum compo-nent m see below, the effective potential in the coordinate the angle between the diatomic axis and the laser polariza-tion vector associated with the nonresonant oscillatory fieldhas the form of a double well, and closely spaced level dou-blets of aligned states are found at large field strengths.25,49,51

Friedrich and Herschbach noted that application of a collin-ear static field leads to a splitting of the doublets, and so tothe production of strongly oriented states.24,25 This methodfor the production of oriented molecules has been verifiedexperimentally for OCS molecules28,29 see also Ref. 30.

In the present study we focus our attention on the instan-taneous rigid-rotor eigenstates in the combined static andlaser fields, so that our considerations are relevant for theexperimentally accessible28,29 adiabatic regime.

A new level of complexity is attained by subjecting polarmolecules to tiltedstatic and oscillatory E-fields.24 This caseis now discussed in detail.

III. POLAR DIATOMIC MOLECULE IN TILTED FIELDS

In this section we consider the classical and quantumHamiltonians for a diatomic molecule in combined static andnonresonant linearly polarized laser fields. In the generalcase, where the static field and laser field polarization vectorsare not collinear, the classical problem is nonintegrable, andcan exhibit complicated dynamics.

A. Classical mechanics

1. Hamiltonian

The classical Hamiltonian for a rigid polar diatomic in astatic electric field S along the space-fixed z axis is

H= H0 SD cos

=

j2

2I

SD

cos =

1

2Ip

2

+

p2

sin2

SD

cos , 1

where j is the rotor angular momentum, I the moment ofinertia, D the magnitude of the dipole moment, the sphericalpolar coordinates , describe the orientation of the rotoraxis, and p,p are conjugate momenta. The angle betweenthe diatom axis and the external field is . In terms of free-rotor action-angle variables j , m , qj , qm,

55H=Hj , m , qj is

a function only of j, m, and qj,

H=j2

2I SD1 m2

j2cos qj . 2

The projection m p is therefore a constant of the motion.The dependence ofH on qj results in j mixing j is no longera constant of the motion, while the dependence on m meansthat the angular momentum vector j precesses around the zaxis with constant projection m.

Hamiltonians 1 and 2 describe the sphericalpendulum.33 The classical spherical pendulum, although anintegrable system,56 exhibits the phenomenon of mono-dromy, which is a topological obstruction to the existence ofa global set of action-angle variables.33 The quantum spheri-cal pendulum exhibits quantum monodromy.37 Eigenstates ofthe quantum spherical pendulum exhibit orientation of therotor by the field.43,45

For a polarizable diatomic in a rapidly oscillating non-resonant infrared frequency linearly polarized laser fieldLt along the space-fixed z axis, the relevant Hamiltonian isthe time-averaged interaction

H=j2

2I

Lt2

2 + cos

2 , 3

where , are components of the polarizability paralleland perpendicular to the diatomic axis, respectively, andLt

2 denotes the time-averaged laser field. If Lt=ftL cos2vt, where v is the laser frequency, L the

field amplitude, and ft a slowly varying envelope function,then Lt2 =f2L

2 /2. Note that the molecule-field interac-tion depends on cos2 . The projection m is a constant of themotion, and the effective potential in the orientation angle has a double minimum for small values of m2. Quantummechanically, for a strong field, extensive J mixing occurs toproduce aligned rather than oriented states;4,13,25,50 eigen-states of the double-well potential form closely spaced par-ity doublets.25 Conservation of m means that the classicalsystem is integrable. While there is no monodromy for therotor in a linearly polarized laser field with no additionalfields, the circularly polarized case does exhibit classical33

and quantum36 monodromy.

184303-2 Arango, Kennerly, and Ezra J. Chem. Phys. 122, 184303 2005



3/15

A more complicated but still integrable problem resultswhen collinear static and linearly polarized laser fields areapplied.25 The effective potential in has for small enoughm the form of an asymmetric double well.25 In terms ofdimensionless variables SD/B and Lt2/2B B

2/2I, and omitting a constant energy shift,

the scaled Hamiltonian HH/B is

H

= j2

cos cos2

= p2 + p2

sin2 cos cos2 , 4

where cos is the static field term, cos2 is the termdue to the laser field, and j is now a dimensionless angularmomentum measured in units of. The larger the parameter, the greater the number of aligned states trapped in thedouble well; the larger the value of, the more asymmetricthe well. The classical dynamics depends only on the dimen-sionless ratio A/, whereas the quantum problem de-pends also on the magnitude of through the rotational en-ergy B2. The quadratic spherical pendulum Hamiltonian 4shows classical and quantum monodromy.36,40,57

A weak static field desymmetrizes the laser field poten-tial and leads to a splitting of the parity doublets found in theabsence of the static field. The eigenstates can then exhibitorientation enhanced over the value obtainable with thestatic field only.25,51

The dynamics becomes more complex when the staticand laser field directions are tilted with respect to oneanother.24,25,36 Setting the static field S along the directionsin ,0,cos in the xz plane, the scaled classical Hamil-tonian is

H= j2 sin sin cos + cos cos cos2 ,

5

and is a function of all four free-rotor action-angle variables

H= j2 sin mjcos qm cos qj sin qm sin qj

+ cos 1 m2j2

cos qj 1 m2j2

cos2 qj . 6The downright botanical complexity of the quantum eigen-states previously noted in the tilted field case25 is a conse-quence of classical nonintegrability.32 The classical Hamil-

tonian 6 for the tilted field case no longer conserves m andthere is both j and m mixing, so that the classical motion canexhibit chaos. The complicated classical dynamics arisesfrom a competition between the tendency of the rotor toalign along the laser field axis and the tendency to orientalong the static field. Quantum mechanically, the presence ofthe cos term in Hamiltonian 5 leads to M mixing as wellas J mixing. We shall investigate numerically both the clas-sical and quantum mechanics for a rigid diatomic in tiltedfields, Eq. 5.

The model parameters we use are appropriate forthe diatomic KCl cf. Table III of Ref. 25: B =0.1286 cm1,=10.48 D, 3.1 3, so that =41.1 for S

=30 kV cm1 and =240 for L =1012 W cm2. For KClwe therefore have 240 and 50 for experimentallyattainable field strengths, so that the number of trapped pen-dular states created by both the static field and the laser pulsefor reasonable intensities is large enough for semiclassicalconsiderations to be appropriate that is, ,1. More-over, in this semiclassical regime the dimensionless ratio A/of the static field interaction energy to the laser field

interaction energy can be in the range 1/61/4, where sig-nificant classical chaos is present for large tilt angles seebelow.

Other molecules for which the dynamical regime inves-tigated here could be realized experimentally are bimetalliccompounds LiM.58

2. Classical trajectories and surfaces of section

To avoid singularities in the classical equations of mo-tion for the rotor expressed either in terms of spherical polarcoordinates ,,p,p 5 or free-rotor action-angle vari-ables j , m , qj , qm 6, we integrate trajectories using Carte-

sian coordinates x ,y ,z and associated conjugate momentapx ,py ,pz. The holonomic constraint of constant bondlength, x2+y2+z2=const, is imposed via a Lagrangianmultiplier.59 This constraint can easily be relaxed to allowbond stretching and dissociation. Integration of trajectories isperformed using a sixth-order hybrid Gear algorithm.60

At any time step along the trajectory, we can extractvalues of either spherical polar coordinates and their conju-gate momenta or of free-rotor action-angle variables, to-gether with the time derivatives of all these quantities. De-tails of the transformations involved are given elsewhere.61

The phase space for the diatomic rotor in tilted fields isfour dimensional. In the collinear field case =0 the exis-tence of two constants of the motion E and m means that thephase space explored by a typical trajectory is two dimen-sional. In the general tilted field case, however, when 0,the only constant of the motion is the total energy and mo-tion can occur on a three-dimensional manifold as well as ontwo-tori and periodic orbits.

We use Poincar surfaces of section32 SOS to explorethe classical phase-space structure as a function of energyand other parameters. Surfaces z are defined by the con-dition qi =const, where qi is a canonical variable. Values ofthe other conjugate pair of variables qj ,pj are recordedevery time a trajectory crosses with qi0.

3. Involutions, lines of symmetry, and periodicsolutions

Hamiltons equations of motion 5 are invariant underthe change of variables S,

S:t,,,p,p t,,2 , p,p . 7

The transformation S is an involution self-inverse operation

SS = I, 8

where I is the identity transformation. IfP1 is the inverse ofthe Poincar map P, then the product P1S is also an invo-

184303-3 Diatomic molecules in tilted fields J. Chem. Phys. 122, 184303 2005



4/15

lution, and P can be written as the product of twoinvolutions,32,62

P = SP1S . 9

A symmetry line L on a given SOS is the intersection of theSOS with the set of phase points invariant under the involu-tion . For =S, the symmetry line on the ,p SOS de-fined by =0 or , 0, is the line p=0. Periodic solu-

tions lying on symmetry lines can be found efficiently bylocating the intersections of the symmetry line with its iter-ates under the map P.62 Any phase point lying on such anintersection after a single iteration is either a period-1 or aperiod-2 periodic orbit.

Analysis of the iterated symmetry line LS is performed inCartesian coordinates. A one-parameter family of rotor phasepoints on LS at fixed energy is defined by the conditions y=0, z=0, x =0, y0. Both positive and negative values of xare allowed, so that the =0, 0 and =, 0 sur-faces of sections are treated together. Each point in this fam-ily of initial conditions is propagated forwards in time untilits trajectory intersects the xz plane y =0 again in the posi-

tive sense y0. Since xz for phase points in the xz planeexcept at the poles!, those phase points having z =0 againlie on the symmetry line LS. These points can therefore befound by one-dimensional interpolation.

B. Quantum mechanics

1. Eigenvalues and eigenvectors

Eigenvalues and eigenvectors for the quantum version ofHamiltonian 4 are computed by matrix diagonalization in abasis of spherical harmonics YM

J ,. Our computer pro-gram allows for nonconservation of the quantum number M,as required for the tilted fields case. For the tilting angle 0, eigenfunctions of the quantum Hamiltonian are su-perpositions

= J,M

c,JMYMJ ,. 10

Standard checks are carried out for convergence of the com-puted levels with respect to basis size.63 In the following,probability densities , =sin ,

2 associated witheigenstates 10 are plotted, and patterns of wave-functionlocalization in the vicinity of classical periodic orbitsscarring64 are investigated.

For 0/2, the only symmetry of the Hamiltonianis the reflection operation xz, with associated angle transfor-mation xz : ,,. A suitably symmetrized basis isthen

J, M, =1

2YM

J MYMJ , 1 MJ, 11

with J, 0 , + YM=0J . For =/2, the Hamiltonian is in ad-

dition invariant under reflection yz and rotation C2x.The quantum calculations of energy levels used for the

analysis of nearest-neighbor level spacing statistics given be-low are carried out in the symmetrized basis and thepositive-parity state level statistics analyzed.

IV. CLASSICAL DYNAMICS AND PHASE-SPACESTRUCTURE

In this section we investigate the classical phase space ofthe rotor in collinear and tilted fields. We identify key struc-tures in phase space low-order stable and unstable periodicorbits, resonance islands, and regions of chaotic motion andstudy their dependence on Hamiltonian parameters. We donot provide a comprehensive survey of the classical dynam-

ics, but rather briefly describe those classical phase spacefeatures that are important for understanding and interpretingthe properties of the corresponding quantum system. Moredetails on the classical dynamics can be found in Ref. 61.

The classical mechanical problem is integrable for col-linear fields =0. Our classical trajectory survey shows theexistence of a significant fraction of chaotic classical phasespace for ratios A0.25 and tilt angles / 4 see below.

A. Diatomic rotor in collinear fields: Integrabledynamics

For the diatomic molecule in collinear static and laserfields =0, with scaled Hamiltonian 4, is an ignorable

coordinate and p= m is a constant of the motion, as is H

itself. The static field case =0 is the well-known spheri-cal pendulum problem.33,56,65 A more extensive discussion ofthe classical mechanics of the rotor in the pulsed laser field=0 is given in Ref. 61. Here we mainly focus on thephysically and dynamically interesting ratio A=/=1/4.

The effective potential Veff; m for motion at fixed mis

Veff;m =m2

sin2

cos cos2 . 12

Effective potentials for several different values ofm2/ areshown in Figs. 1a A=0 and 1b A=1/ 4, respectively.Classical relative equilibria41 correspond to critical points ofVeff,

Veff

= 0. 13

Scaled energies EE/ vs values for relative equilibriaare shown in Fig. 1. Consider first the case with A=0, 0 laser field only, Fig. 1a. At energies just above E=0there are three curves, associated with the two symmetrically

located potential minima and a central potential maximum.The lowest-energy point on each of the three curves is arelative equilibrium with m =0. The two stable relative equi-libria with m =0 correspond to the rotor stationary either par-allel =0 or antiparallel = to the field; the unstablerelative equilibrium at = /2 corresponds to a one-parameter family of stationary rotor configurations in the fullphase space, parametrized by azimuthal angle =0. Form 0, the relative equilibria are associated with pairs ofperiodic orbits with m conical pendulum orbits66. Therelative equilibria merge at E= in a symmetric pitchforkbifurcation, a consequence of the invariance ofVeff under thetransformation ofVeff.




5/15

Relative equilibria for A=1/4 are shown in Fig. 1band, together with other periodic orbits, in Fig. 2a. Thesymmetry is now broken, and one of the stablerelative equilibrium periodic orbit pairs merges with the un-stable relative equilibrium orbit pair in a pair of saddle-nodebifurcations.67 In addition to the relative equilibria, there alsoexist one-parameter families of planar m =0 periodic orbitspassing through the poles. These orbits are librational pen-dular for energies below the barrier height, and rotational athigher energies. The transition from librational to rotational

motion occurs at precisely the energy at which the unstablerelative equilibrium appears. These additional orbits are rep-

resented by vertical traces at =0 and in Fig. 2a cf. Fig.3b. Also shown in Fig. 2a are 1:1 resonant parabolicneutrally stable periodic orbits that emerge from the planarpolar orbits and merge with the relative equilibria conicalpendulum orbits at higher energy. These resonant periodicorbits occur in one-parameter families, with each family fill-ing a rational torus. Particular orbits in the family are foundby iteration of the symmetry line p=0 on the =0, 0surface of section cf. Sec. III A 3.62 The extra lines in Fig.2a show the values for 1:1 resonant periodic orbits on the

symmetry line at energy E. Each resonant torus intersects thesymmetry line at two points, one on each side of a stable

FIG. 1. Effective potential Veff for dif-ferent values ofm2/2I, 0 seeEq. 12. a /=0. b /=1/4.

FIG. 2. Bifurcation diagrams showing

E vs for relative equilibria and reso-nant periodic orbits in combined staticand linearly polarized laser fields,/=1/4. a Collinear fields, =0.b Tilted fields with small tiltingangle, =0.01.




6/15

relative equilibrium. Unlike the periodic orbits associatedwith the relative equilibria, the angle is not constant alongthe 1:1 resonant periodic orbits see Fig. 3a.

The corresponding E vs m plot for the relative equilibriais the classical energy-momentum diagram.33 Classical E m diagrams for collinear fields have been given in Ref. 36.

Contours ofm2 in the p, plane for A=1 /4 at fixed E

are shown in Fig. 4a E=1 and 4c E=0.2. At the higher

energy there is only a single stable equilibrium point abovethe energy of the saddle-center bifurcation while at thelower energy there are two stable and one unstable equilib-rium points below the saddle-center bifurcation. We definea ,p surface of section by conditions =0 or and

0. The invariant curves in the ,p SOS for E=1.0Fig. 4b and E=0.2 Fig. 4d coincide with contours ofm2 for the same parameter values, thus confirming the cor-

FIG. 3. Periodic orbits for E=0.25, /=1/4. aCollinear fields, =0. Stable and unstable conical pen-

dulum orbits relative equilibria are shown, togetherwith 1:1 resonant periodic orbits. b Collinear fields,=0. Two marginally stable planar periodic orbits pass-ing through the poles are shown. c Tilted fields withsmall tilting angle, =0.01. Deformed conical pendu-lum orbits and resonant periodic orbits are shown. dTilted fields with small tilting angle, =0.01. Stableperiodic orbit rotating around x axis and unstable peri-odic orbit in xz plane are shown.

FIG. 4. a ,p reduced phase space, /=1/4,E=1. There is only one stable fixed point. b ,psurface of section, =0, 0. /=1/4, E=1. cp reduced phase space, /=0.25 and E=0.2.

There are two stable fixed points and one unstable fixedpoint. The separatrix curve delimits the different typesof motion pendular and free rotor. d ,p surfaceof section, =0, 0. /=1/4, E=0.2.




7/15

rectness of our trajectory numerics and the associated trans-formations from Cartesian to polar coordinates.

It is also interesting to examine the classical dynamics infree-rotor action-angle variables j , m , qj , qm,

55 in terms ofwhich the Hamiltonian is setting =0 in Eq. 5

H

qj,j,m = j2 1 m

2

j2 cosqj

1 m2j2

cos2 qj . 14Rearranging to get m2 as a function of qj, j2, qj, and E

E/ gives

m2 = j21 14 cos2 qj

A A2 4E+ 4j2

2 .15

There are two possible solutions corresponding to signs.

Since we must have sin 0, the physically relevant range

of qj for the first branch positive sign is 3/2qj/2mod 2, while that for the second negative sign is /2qj3/2.

Contours of m2 are shown in Fig. 5a for the case E

=1, A=1/4, as in Fig. 4a. The stable fixed point appears at

qj =0 mod 2. Figure 5c shows contours of m2 for E

=0.25. Corresponding SOSs in action-angle variables j , qjdefined by qm =0, qm0 are shown in Figs. 5b and 5d,respectively. In the SOS of Fig. 5b there is only one fixedpoint, a stable po at qj =0, corresponding to the single stablerelative equilibrium at this energy. Some invariant curves are

clearly associated with librations around qj =0, while the restcorrespond to rotations in qj. In Fig. 5d there are two stablefixed points relative equilibria, at qj =0 and qj =. Aroundeach of these stable pos there is an associated quasiperiodicregion, the region around qj corresponding to larger po-tential energy. The unstable relative equilibrium appears at

qj = /2,3/2. Again, comparison ofm2 contours with SOSshows that the transformation of coordinates along trajecto-ries has been correctly implemented.

B. Diatomic rotor in tilted fields: Nonintegrabledynamics

We now turn to the tilted field case, 0. The angularmomentum component m is no longer conserved, and thedynamics is no longer integrable. We shall examine severalSOSs for fixed A=/=1/4 as a function of energy Eandthe tilt angle , with a focus on the most important low-order

pos that evolve from relative equlibria in the collinear fieldcase, and on the onset of localized and global chaos.

1. Small tilting angle

When the tilting angle 0 there are new features in thedynamics. The only global constant of the motion is theHamiltonian

H,,p,p =1

2Ip2 + p

2

sin2 sin sin cos

+ cos cos cos2 . 16

FIG. 5. a j , qj reduced phase space, /=1/4, E=1. b j , qj surface of section, qm=0, qm0. /=1/4, E=1. c j,qj reduced phase space, /=0.25 and E=0.2. There are two stable fixed points andone unstable fixed point. The separatrix curve delimitsthe different types of motion pendular and free rotor.d j , qj surface of section, qm=0, qm0. /=1/4, E=0.2.




8/15

For =/100 and E=1.0 the ,p SOS Fig. 6a isvery similar in appearance to Fig. 4b, except near =0 and=. There are two new stable fixed points with p=0 closeto =0 and =, and Figs. 6b and 6c show these regionsin detail. The stable fixed points of Fig. 6b and 6c corre-spond to clockwise and counterclockwise rotations, respec-

tively, about the lab-fixed x axis in the yz plane. The ,pSOS for E=0.2 is shown in Fig. 6d. The conical pendulumperiodic orbits stable relative equilibria for =0 Fig. 3aare now deformed Fig. 3c, although they remain stable.Although invisible on the scale of the figure there are twostable fixed points near =0 and = as for E=1 cf. Figs.6e and 6f. These stable fixed points again correspond torotations about the x axis, clockwise and counterclockwise,respectively see Fig. 3d. Comparing Fig. 6d with Fig.4d, it is clear that even a small tilting angle =/100results in the onset of chaotic motion in the vicinity of theseparatrix dividing pendular from free-rotor states.

For collinear fields, =0, the ,p SOS was not

shown on account of its trivial structure. Since p is a con-stant of the motion the section =0 ,0, consists of in-variant curves with constant p=m. The invariant line m =0consists of a continuous set of period-1 fixed points. More-over, the conical stable periodic orbits typically do not ap-pear on the SOS as they have constant 0. The ,pSOS is of greater interest for A=1/4, 0. Sections with=/10, 0 are shown in Figs. 7a E=1.0 and 7bE=0.2. The line of fixed points with m =0 present for =0 disappears and is replaced in accord with the PoincarBirkhoff theorem by two stable and two unstable fixed pointsand associated homoclinic tangle.32 The stable fixed points at

= /2 and =3 /2 correspond to the same periodic tra-jectories associated with the stable fixed points in the psurface of section near =0 and =, respectively cf. Fig.6. The unstable fixed points in at =0 and = correspondto periodic orbits rotating about the y axis clockwise andcounterclockwise, respectively. Details of the sections nearp

0 are shown in Figs. 7c and 7d.

2. Larger tilting angles

More complete E vs plots for periodic orbits on thesymmetry line LS are given in Fig. 8 for =0 Fig. 8a,=0.01 Fig. 8b, =0.25 Fig. 8c, and =0.5Fig. 8d. These plots are obtained by propagation of pointson the symmetry line LS in Cartesian coordinates, as de-scribed in Sec. III A 3. The range of the coordinate is for-mally extended from 0 to 02, where 0 for =0 and 2 for =. An intricate bifurca-tion structure evolves from the integrable limit. Note that, for=/2, the po plots are invariant under the transformations 0, and 3 2 inducedby the symmetry xy :zz.

Several ,p SOSs for A=1/4 and =/4 are shownfor various energies in Fig. 9. At energies well below thebarrier height, E=0.5, Fig. 9f, the phase space primarilyexhibits regular pendular motion, with trajectories confinedto one well or the other. At energies well above the barrierheight E=1.5, Fig. 9a, the phase space mainly consists ofregular, free-rotor-type motion, although some irregular mo-tion is present near /2,p0. For energies betweenthese limiting values, the phase space shows large scale cha-otic dynamics Figs. 9b9e.

FIG. 6. ,p surfaces of section, defined by =0,

0. /=1/4, =/100. a E=1.0. b E=1.0. De-tail of the SOS near =0. c E=1.0. Detail near =.d E=0.2.e E=0.2. Detail of the SOS near =0. f

E=0.2. Detail near =.




9/15

In Sec. V, we examine quantum-mechanical energy-levelcorrelation diagrams and nearest-neighbor level spacing dis-tributions for the parameter values that correspond to theexistence of chaotic classical motion.

V. QUANTUM MECHANICS: ENERGY-LEVEL

CORRELATION DIAGRAMS AND LEVEL SPACINGDISTRIBUTION

A. Energy-level correlation diagrams

Energy-level correlation diagrams En versus Hamil-tonian parameter such as the tilting angle or field strengthare important in the study of molecule-field problems.Friedrich and Herschbach have calculated correlation dia-grams for low-energy eigenstates of the rotor in parallelfields =0.25 In Fig. 10a we show a correlation diagramfor 60 states spanning an energy range from the ground stateto well above the barrier for =60, =0, and =150300. At fixed , the energies of all eigenstates En decrease

as increases, in accord with the HellmannFeynmantheorem,68

En

= n

H

n = ncos2 n. 17

On the other hand, at fixed , energies either increase ordecrease as increases, depending on the expectation valuenH/n=ncos n.

Figure 10b is similar to Fig. 10a, but for tilted fields,= /4. Qualitatively, although still monotonically decreas-ing with , the levels appear to fan out more uniformlythan in the tilted field case, reflecting increased level repul-

sion in the nonintegrable case. This qualitative observation isquantified by the computation of nearest-neighbor spacingstatistics discussed below.

The existence of extensive avoided crossings in the non-integrable case is demonstrated more clearly in Fig. 11,which shows a correlation diagram at fixed =60,=240for the range of tilting angle =0/2. The limiting case= /2, although nonintegrable, possesses an additional re-flection symmetry xy, which results in closely spaced pairsof levels.

B. Analysis of nearest-neighbor spacing distributions

Strong state mixing found in systems with no rigorouslyconserved constants of the motion except the Hamiltoniannaturally leads to an association between the existence ofextensive avoided crossings and the presence of classicalchaos, where trajectories explore all of the available phasespaces.6971 The influence of avoided crossings on the quan-

tum level spectrum is quantified by the analysis of thenearest-neighbor spacing NNS distribution Ps, wherePsds is the probability of the energy difference betweentwo levels lying in the range ss +ds. When levels repeleach other, Ps will tend to vanish as s0. Otherwise, Pswill have its maximum value at s 0. Reviews of the basictheory can be found in Refs. 70 and 71.

The analytical form of Ps is known for ideal limitingcases. When the quantum levels are completely uncorrelatedPs= es, the exponential distribution, corresponding tocompletely regular classical phase space.72 Strong level re-pulsion, as found for Hamiltonians with random matrix ele-ments, leads to the Wigner distribution, Ps= 2 ses

2/4.70

FIG. 7. ,p surface of section defined by conditions

=/10, 0. /=1/4, =/100. a E=1.0. b

E=0.2. The central resonant chain and two chaoticstrips appear. c Details of the central resonant chain,

p0, E=1.0. d Details of the central resonant chain,

p0, E=0.2.




10/15

Note that Pss at small s. Much numerical evidence sup-ports the notion that level spacings for quantum counterpartsof classically strongly chaotic systems are described by theuniversal Wigner distribution.71

Classical systems with mixed regular/chaotic phasespace fall between these two extremes for example, Refs. 48and 73. Calculated level spacing distributions Ps for suchsystems can be fitted using one-parameter models that inter-polate between the exponential and Wigner distributions,such as the distributions of BerryRobnik74 and Brody et

al.

75

The BerryRobnik function is a semiclassical result ob-tained by analyzing the level spacings for a single connectedchaotic region in an otherwise regular phase space.74 A single

parameter q, varying between 0 and 1, is the fraction ofphase-space volume occupied by the chaotic region. The dis-tribution of Brody et al. simply interpolates between expo-nential and Wigner distributions,

PBrodys = 1 + qxq exp x1+q , 18

where

= q + 2q + 1

q+1 . 19The parameter q has no immediate physical interpretation,though it is presumably related to the chaotic phase space

FIG. 8. Energy E vs for periodic or-bits on symmetry line LS. Periodic or-bits on the symmetry line satisfy eitherof the conditions =0, 0 or =, 0. A=/=1/4. a =0;b =0.01; c =0.25; and d=0.5.

FIG. 9. ,p surfaces of section de-fined by condition =0, 0. A=/=1/4, =/4. a E=1.5; b

E=1.0; c E=0.25; d E=0.1; e E=0.1; and f E=0.5.




11/15

fraction. Both distributions reduce to the exponential distri-bution for q =0, and to the Wigner distribution for q =1.

The standard procedure for calculating Ps is as fol-lows. First, the calculated or experimental energy-level spec-trum is mapped unfolded to a new spectrum with unitmean level density to remove the effect of varying meanlevel density. The level spacings from the new spectrum arethen sorted into histogram bins. This histogram representa-

tion of Ps is then fitted to one of the model distributions,thus determining q. In the calculations reported here we usea fifth-order polynomial to fit the smoothed staircase func-

tion NE Refs. 70 and 71 and histogram bins with width=0.1.

Several features of the present problem render quantita-tive analysis of level spacing distributions difficult. Schlierhas convincingly demonstrated that q parameters are unreli-able when less than several hundred energy levels are used,and recommends using at least a thousand levels.76 At physi-cally relevant field strengths, our system has less than onehundred bound states in the energy range of interest. Thus we

combine smoothed spacings from many independent spectraacross a correlation diagram into one histogram.The tilted fields Hamiltonian 5 has reflection plane

symmetry. The existence of a good parity quantum numberresults in the existence of two uncorrelated level sequences.Only the positive-parity eigenstates are used in our levelspacing calculations.

We are particularly interested in learning how changes inclassical phase-space structure with energy influence thequantum level spectra. This influence can be monitored byselecting the same NL levels, from a specified base state tothe NL 1th level above that, from each spectrum in thecorrelation diagram as discussed above, and then calculating

q. The NL level window is then moved up to start at the nexthighest level so that the resulting q corresponds to a slightlyhigher-energy range. All NNS calculations presented here areof this form.

C. Results for tilted fields

When only one field is on, or the static and laser fieldsare collinear, the classical problem is integrable with twoconserved quantities H and m. In the absence of anomaliessuch as those associated with, e.g., harmonic-oscillator spec-tra, Ps is expected to be exponential, reflecting two inter-spersed independent spectra creating random spacings.72,74

Tilting the fields leads to a classically nonintegrable problemand typically mixed phase space. The quantum Ps exhibitsa smaller fraction of spacings in the smallest bins due tomore avoided crossings in the correlation diagram, and de-velops a maximum at s0.

Parameter regimes for NNS calculations must be chosencarefully to obtain meaningful results. We wish to show a

correlation between the nature of the classical phase-spacestructure in a given energy range and the quantum NNS dis-tribution. Lowest-energy states are in the two-dimensional2D harmonic-oscillator limit,43 so including them in theNNS analysis yields anomalous behavior, where Ps has amaximum near s =1 related to harmonic-oscillatorlike statesat the bottom of the potential well unrelated to any quantummanifestations of classical chaos.72 Eigenstates at energieswell above the top of the potential-energy barrier are ap-proximately free-rotor eigenstates, and the spectrum at theseenergies exhibits clusters of almost-degenerate levels. Ouranalysis is therefore restricted to intermediate energies be-tween these limits.

We have calculated both the parameters ofBerryRobnik.74 and Brody et al.75 We find that BerryRobnik q parameters obtained from our analysis are consis-tently 0.2-0.3 larger than the corresponding q values ofBrody et al., but exhibit the same trends. As we are onlyinterested in broad trends rather than quantitative interpreta-tion of the q parameters, we show only the parameters ofBrody et al. here.

Figure 12 presents plots of the parameter of Brody et al.versus base level. The lowest 300 energy levels are analyzedfor =60, with in the range of=150300. Figures12a12c show results for tilt angle =0, with NL =25Fig. 12a, 50 Fig. 12b, and 100 Fig. 12c levels per

FIG. 10. Energy-level correlation diagram showing lowest 60 energy levels,=60, =150300. a =0. b =/4.

FIG. 11. Energy-level correlation diagram for =60, =240, for range oftilt angles =0/2.




12/15

spectrum, respectively. Figures 12d12f show corre-sponding results for = /4. Each data point is calculatedfrom energy levels in the relevant correlation diagram, Fig.10a or 10b.

Our calculations show that: i while there is more

noise in the NL =25 and NL =50 results, the behavior of qversus state index is consistent for the three NL values foreach value of; ii a clear qualitative difference exists be-tween the energy dependence of q in the integrable =0and nonintegrable =/ 4 cases.

Our results are consistent with the underlying classicalphase-space structure. The first three panels Figs.12a12c show that q is small across the whole energyrange, reflecting the regular phase-space structure for collin-ear fields. The q values obtained for the nonintegrable case= /4 first increase then decrease at higher energies, re-flecting the change in character of the classical phase spacewith energy seen in the surfaces of section of Fig. 9. In

particular, the transition to regular free-rotor behavior athigh energies is apparent cf. Ref. 48. For a nonscaling sys-tem with a relatively small density of states, this is about asmuch information as can meaningfully be extracted concern-ing NNS distributions.76

A similar analysis of NNS distributions for the correla-tion diagram of Fig. 11 shows that the parameter q of Brodyet al. is uniformly larger for the spacings at large tilt angles=0.30.4 compared to those for small tilt angles=00.1.

Note that the shape of the BerryRobnik distributionchanges most rapidly from exponential to Wigner for q be-tween 0.5 and 1.0, so that the BerryRobnik distribution withq=0.5 is actually very close to an exponential distribution.The distribution of Brody et al. 18 changes most rapidly for0q0.5, so the distribution of Brody et al. with q=0.5 isessentially Wigner-type. Since the collinear field problem isintegrable, we should have q=0. We attribute discrepancies

FIG. 12. The parameters of Brody et al. extracted fromPs histograms vs base level. The lowest 300 energylevels are analyzed for =60, =150300. a Tiltangle =0, NL=25 levels per spectrum. b =0, NL=50. c =0, NL =100. d Tilt angle =/4, NL =25

levels per spectrum. e =/4, NL =50. f =/4,NL =100.




13/15

between our calculated q values and ideal exponential behav-ior to the difficulties noted above on the unreliability of qcalculations for nonscaling systems with small numbers oflevels. For =/4, q is well into the Wigner-type region forthis nonintegrable problem. As the base level increases, en-ergy increases and q decreases, consistent with increasingregularity of the classical phase space.

VI. QUANTUM MECHANICS: WAVE FUNCTIONS

Energy eigenstates , for the rotor in collinearfields are separable in coordinates and , due to conserva-tion of m. They can, however, exhibit localization in ori-entation; see Ref. 25.

When the fields are tilted, eigenstates are no longer sepa-rable. Examination of rotor eigenfunctions , for thetilted field case reveals localization of a significant fractionof rotor eigenstates in the vicinity of classical periodic orbitssee Figs. 13 and 14. These periodic orbits can be stable cf.Fig. 13, in which case the localized quantum states are as-sociated with a quantizing torus in a regular region of therotor phase space, or unstable cf. Fig. 14, in which case wehave the phenomenon of scarring.64 The stable periodic or-bits can be those orbits that evolve from the stable relativeequilibria in the collinear field case for example, Figs. 13aand 13c, leading to states localized in , or new stableorbits associated with regular states localized in Figs.

13b and 13d. Scarring by unstable pos leads to localiza-tion in angles Figs. 14c and 14d, Fig. 14e, orboth Figs. 14a, 14b, and 14d.

More definitive characterization of eigenstate localiza-tion will require the computation of associated phase-spacedensities. It is possible to define rotational coherent states,which are parametrized by four classical action-angle vari-ables and are localized in phase space. These states are ob-tained by suitably modifying the coherent states for diatomicrotors defined by Morales et al.,77 and can be used to definea rotational phase-space density Husimi function78 associ-ated with rotor eigenstates.63 The rotational Husimi functioncan then be used to investigate phase-space localizationproperties of rotor eigenstates.63

VII. SUMMARY AND CONCLUSION

In this paper we have investigated the classical andquantum mechanics of diatomic molecules in noncollineartilted static and nonresonant linearly polarized laser fields.The classical diatomic in tilted fields is a nonintegrable sys-tem, and we have examined in some detail the phase-spacestructure for physically relevant parameter regimes for themolecule KCl. While exhibiting low-energy pendular andhigh-energy free-rotor limits, the rotor in tilted fields showschaotic dynamics at intermediate energies in the vicinity ofthe potential barrier to rotation. The degree of classical chaos

FIG. 13. Probability densities ,2 sin for tilted rotor eigenstates

=60, =240, =/4. Scaled eigenvalues are E=E/. Stable classicalperiodic orbits at the same energy are also shown superimposed on quantumprobability densities.

FIG. 14. Probability densities ,2 sin for tilted rotor eigenstates.

=60, =240, =/4. Scaled eigenvalues are E=E/. Unstable classi-cal periodic orbits at the same energy are also shown superimposed onquantum probability densities.




14/15

can be tuned by changing the tilt angle, with tilting angles /4 leading to widespread chaos over a wide energyrange.

We have examined the quantum mechanics of rotors intilted fields. Energy-level correlation diagrams have beencomputed over physically significant ranges of fieldstrengths. Tilting the fields from a collinear configurationleads to the appearance of avoided crossings in the level

correlation diagram. The presence of avoided crossings wasquantified by the study of nearest-neighbor spacing distribu-tions as a function of energy and tilting angle.

Finally, we examined the influence of classical periodicorbits on rotor wave functions. Many wave functions in thetilted field case are found to be highly nonseparable inspherical polar coordinates ,. Localization of wavefunctions in the vicinity of classical periodic orbits, bothstable and unstable, was observed for many states. Such lo-calization facilitates assignment of the strongly mixed rotoreigenstates obtained with tilted fields.

Nonintegrable dynamics of rotors in various externalfield configurations provides a rich area for exploration of

the classical-quantum correspondence. In model Hamilto-nians for molecular vibrational problems, it is found thatessential information on the underlying classical mechanics,such as the existence of resonances, is encoded in the leveldynamics.7982 The same is likely to be true for the problemof rotors in external fields. The connection between phase-space localization and molecule-field level dynamics83 andthe importance of chaotic/dynamical tunneling84,85 in thetilted strong-field problem are topics deserving further study.

1R. J. Gordon and S. A. Rice, Annu. Rev. Phys. Chem. 48, 601 1997.2J. P. Simons, Faraday Discuss. 113, 1 1999.3R. G. Gordon, L. Zhu, and T. Seideman, Acc. Chem. Res. 32, 1007

1999.4P. B. Corkum, C. Ellert, M. Mehendale, P. Dietrich, S. Hankin, S. Aseyev,D. Rayner, and D. Villeneuve, Faraday Discuss. 113, 47 1999.

5H. Rabitz and W. S. Zhu, Acc. Chem. Res. 33, 572 2000.6M. Shapiro and P. Brumer, Adv. At., Mol., Opt. Phys. 42, 287 2000.7S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics Wiley,New York, 2000.

8M. Shapiro and P. Brumer, Principles of the Quantum Control of ChemicalReactions Wiley-Interscience, New York, 2003.

9R. S. Judson, K. K. Lehmann, H. Rabitz, and W. S. Warren, J. Mol. Struct.223, 425 1990.

10L. Shen and H. Rabitz, J. Phys. Chem. 95, 1047 1991.11R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 1992.12J. Li, J. T. Bahns, and W. C. Stwalley, J. Chem. Phys. 112, 6255 2000.13J. Karczmarek, J. Wright, P. Corkum, and M. Ivanov, Phys. Rev. Lett. 82,

3420 1999.14

D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner, M. Y. Ivanov,and P. B. Corkum, Phys. Rev. Lett. 85, 542 2000.15M. Spanner and M. Y. Ivanov, J. Chem. Phys. 114, 3456 2001.16M. Spanner, K. M. Davitt, and M. Y. Ivanov, J. Chem. Phys. 115, 8403

2001.17R. Hasbani, B. Ostojic, P. R. Bunker, and M. Y. Ivanov, J. Chem. Phys.116, 10636 2002.

18R. Blumel, S. Fishman, and U. Smilansky, J. Chem. Phys. 84, 26041986.

19J. Ortigoso, Phys. Rev. A 57, 4592 1998.20J. Gong and P. Brumer, J. Chem. Phys. 115, 3590 2001.21H. J. Loesch, Annu. Rev. Phys. Chem. 46, 555 1995.22A. Auger, A. B. Yedder, E. Cances, C. L. Bris, C. M. Dion, A. Keller, and

O. Atabek, Math. Models Meth. Appl. Sci. 12, 1281 2002.23H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, 543 2003.24B. Friedrich and D. Herschbach, J. Chem. Phys. 111, 6157 1999.

25B. Friedrich and D. Herschbach, J. Phys. Chem. A 103, 10280 1999.26L. Cai, J. Marango, and B. Friedrich, Phys. Rev. Lett. 86, 775 2001.27L. Cai and B. Friedrich, in Laser Control and Manipulations of Molecules,

edited by A. D. Bandrauk, Y. Fujimura, and R. J. Gordon Americanchemical society, Washington, DC, 2002, pp. 286303.

28H. Sakai, S. Minemoto, H. Nanjo, H. Tanji, and T. Suzuki, Phys. Rev. Lett.90, 083001 2003.

29S. Minemoto, H. Nanjo, H. Tanji, T. Suzuki, and H. Sakai, J. Chem. Phys.118, 4052 2003.

30N. H. Nahler, R. Baumfalk, U. Buck, Z. Bihary, R. B. Gerber, and B.

Friedrich, J. Chem. Phys.119

, 224 2003.31B. Friedrich, N. H. Nahler, and U. Buck, J. Mod. Opt. 50, 2677 2003.32A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics

2nd ed. Springer, New York, 1992.33R. H. Cushman and L. M. Bates, Global Aspects of Classically Integrable

Systems Birkhauser, Basel, 1997.34M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics Springer,

New York, 1990.35I. N. Kozin and R. M. Roberts, J. Chem. Phys. 118, 10523 2003.36C. A. Arango, W. W. Kennerly, and G. S. Ezra, Chem. Phys. Lett. 392,

486 2004.37R. Cushman and J. J. Duistermaat, Bull. Am. Math. Soc. 19, 475 1988.38D. Sadovskii and B. I. Zhilinskii, Phys. Lett. A 256, 235 1999.39M. P. Jacobson and M. S. Child, J. Chem. Phys. 114, 262 2001.40K. Efstathiou, M. Joyeux, and D. A. Sadovskii, Phys. Rev. A 69, 032504

2004.41

V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects ofClassical and Celestial Mechanics Springer, New York, 1988.42F. J. Comes, Angew. Chem., Int. Ed. Engl. 31, 516 1992.43J. M. Rost, J. C. Griffin, B. Friedrich, and D. R. Herschbach, Phys. Rev.

Lett. 68, 1299 1992.44B. Friedrich and D. R. Herschbach, Nature London 353, 412 1991.45B. Friedrich and D. Herschbach, Int. Rev. Phys. Chem. 15, 325 1996.46S. C. Ross and K. M. T. Yamada, Mol. Phys. 102, 1803 2004.47J. Bulthuis, J. Moller, and H. Loesch, J. Phys. Chem. A 101, 7684 1997.48T. P. Grozdanov and R. McCarroll, Z. Phys. D: At., Mol. Clusters 38, 45

1996.49B. Friedrich and D. Herschbach, J. Phys. Chem. 99, 15686 1995.50J. J. Larsen, H. Sakai, C. P. Safvan, I. Wendt-Larsen, and H. Stapelfeldt, J.

Chem. Phys. 111, 7774 1999.51B. Friedrich and D. Herschbach, Z. Phys. D: At., Mol. Clusters 36, 221

1996.52

A. R. Kolovsky, Opt. Commun. 82, 466 1991.53J. Ortigoso, M. Rodriguez, M. Gupta, and B. Friedrich, J. Chem. Phys.110, 3870 1999.

54R. Escribano, B. Mat, F. Ortigoso, and J. Ortigoso, Phys. Rev. A 62,023407 2000.

55M. S. Child, Semiclassical Mechanics with Molecular Applications Ox-ford University Press, New York, 1991.

56P. H. Richter, H. R. Dullin, H. Waalkens, and J. Wiersig, J. Phys. Chem.100, 19124 1996.

57M. Joyeux, D. A. Sadovskii, and J. Tennyson, Chem. Phys. Lett. 382, 4392003.

58E. Benichou, A. R. Allouche, R. Antoine et al., Eur. Phys. J. D 10, 2332000.

59T. C. Bradbury, Theoretical Mechanics Wiley, New York, 1968.60C. W. Gear, SIAM Soc. Ind. Appl. Math. J. Numer. Anal. 2, 69 1965.61C. A. Arango, Ph.D thesis, Cornell University, 2005.62

J. M. Greene, R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Physica D3, 468 1981.

63W. W. Kennerly, Ph.D. thesis, Cornell University, 2005.64E. J. Heller, Phys. Rev. Lett. 53, 1515 1984.65M. P. Jacobson and M. S. Child, J. Phys. Chem. A 105, 2834 2001.66J. L. Synge and B. A. Griffith, Principles of Mechanics unit, 3rd ed.

Mc-Graw Hill, New York, 1959.67M. Joyeux, S. C. Farantos, and R. Schinke, J. Phys. Chem. A 106, 5407

2002.68R. P. Feynman, Phys. Rev. 56, 340 1939.69D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Annu. Rev. Phys.

Chem. 32, 267 1981.70O. Bohigas and M.-J. Giannoni, in Mathematical and Computational

Methods in Nuclear Physics, Lecture Notes in Physics Vol. 209, edited byJ. S. Dehesa, J. M. G. Gomez, and A. Polls Springer, Berlin, 1984, pp.199.




15/15

71F. Haake, Quantum Signatures of Chaos Springer, Berlin, 1991.72M. V. Berry and M. Tabor, Proc. R. Soc. London, Ser. A 356, 375 1977.73J. Main, M. Schwacke, and G. Wunner, Phys. Rev. A 57, 1149 1998.74M. V. Berry and M. Robnik, J. Phys. A 17, 2413 1984.75T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M.

Wong, Rev. Mod. Phys. 53, 385 1981.76C. Schlier, J. Chem. Phys. 117, 3098 2002.77J. A. Morales, E. Deumens, and Y. Ohrn, J. Math. Phys. 40, 766 1999.78K. Takahashi, J. Phys. Soc. Jpn. 55, 762 1986.

79S. Keshavamurthy, J. Phys. Chem. A 105, 2668 2001.80N. R. Cerruti, S. Keshavamurthy, and S. Tomsovic, Phys. Rev. E 68,

056205 2003.81A. Semparithi, V. Charulatha, and S. Keshavamurthy, J. Chem. Phys. 118,

1146 2003.82A. Semparithi and S. Keshavamurthy, Chem. Phys. Lett. 395, 327 2004.83S. Tomsovic, Phys. Rev. Lett. 77, 4158 1996.84E. J. Heller, J. Phys. Chem. A 103, 10433 1999.85S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 1994.


Carlos A. Arango, William W. Kennerly and Gregory S. Ezra- Classical and quantum mechanics of diatomic molecules in tilted fields

Documents