R.M. Benito et al- Comparison of Classical and Quantum Phase Space Structure of Nonrigid Molecules, LiCN

8/3/2019 R.M. Benito et al- Comparison of Classical and Quantum Phase Space Structure of Nonrigid Molecules, LiCN

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Volume 161, number 1 CHEMICAL PHYSICS LETTERS I September 1989

COMPARISON OF CLASSICAL AND QUANTUM PH ASE SPACE STRUCTUREOF NONRIGID MOLECULES, LiCNR-M. BENITO , F. BORONDO *, J.-H. KIM, B.G. SUMPTER 3 and G.S. EZRA 4Department of Chemistry, Baker Laboratory, Cornell University, thaca, NY 14853, USAReceived 25 April 1989; in final form 7 July 1989

We examine the classical and quantum phase space structure of the nonrigid molecule LiCN. The quantum phase space densityfor many vibrational eigenstates exhibits marked localization in classically chaotic regions of phase space.

1. Intr oductionThere has been much recent progress in under-

standing the classical phase space structure of non-integrable two-mode coupled oscillator systems andassociated area-preserving mappings [ 11. The im-portant role of phase space features such as cantori[ 2,3] and partial separatrices [ 4,5 ] as bottlenecks inintramolecular energy transfer [61, isomerization[ 71, unimolecular decay [ 8- lo], and bimolecularreactions [ 11,121 has been demonstrated. Two ma-jor problems are currently outstanding. The firstconcerns the generalization of theories of transportbased on phase space bottlenecks [2,6,13] to mul-timode classical systems [ 14-171. The second con-cerns the relation between the classical and quantumphase space structure of nonintegrable systems[ 18,191. The present paper addresses the secondproblem.

Recent work has revealed a striking localization ofquantum eigenstates in both configuration space [20 ]and phase space [ 19,21-271 for many systems, evenin regions of phase space that are classically quite

Permanent address: Departamento de Fisica, ETSI Teleco-municacion, Politecnica University, 28040 Madrid, Spain.Permanent address: Departamento Quimica C-14, Universi-dad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain.Permanent address: Chemistry Division, Oak Ridge NationalLaboratory, Oak Ridge, TN 37831, USA.Alfred P. Sloan Fellow: Camille and Henry Dreyfus Teacher-Scholar.

chaotic. Unstable periodic orbits have been found toinfluence the form of individual eigenstates in an asyet incompletely understood fashion [ 19-301, whilequantum phase space densities trapped by classicalcantori [ 2 1,23 and localized along separatrix man-ifolds [ 19,22,24,27 ] have been observed.

In this Letter we compared the classical and quan-tum phase space structure of a two-mode model ofthe nonrigid molecule LiNC/LiCN, in which the CNbond is frozen. The LiCN molecule has served as atest case for quantum mechanical methods for cal-culation of vibration-rotation levels in floppy sys-tems [ 3 1,321, An ab initio potential surface (see fig.1) is available [33].

An earlier comparative study of the classical andquantum mechanics of two-mode LiCN has beenmade by Farantos and Tennyson (FT) [ 3 11. By ex-amining configuration space plots of the first 80 (ro-tationless) eigenstates of LiCN, FT classified the vi-brational states into 5 different types: regular stateslocalized around the LiNC minimum; irregular stateslocalized around the LiNC minimum; regular stateslocalized around LiCN; irregular, delocalized states;and free rotor or polytopic states. The regularityor otherwise of an eigenstate was judged by the com-plexity of its nodal pattern [ 341. Other previouslyproposed diagnostics of quantum chaos such aslevel spacing statistics [ 35 1, sensitivity of eigenval-ues to changes in the Hamiltonian [361, and avoidedcrossings of energy levels [37] were also studied.Corresponding studies of surfaces of section [ 1 ] and

60 0 009-2614/89/$ 03.50 0 Elsevier Science Publishers B.V.( North-Holland Physics Publishing Division )


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K-entropies [ 38 ] for classical trajectories were a lsomade, and revealed the early onset of widespreadnonquasiperiodic motion. FT concluded that thereis an early onset of quantum chaos in LiCN , andthat there is good qualitative agreement betweenquantum and classical mechanics for LiCN, albeitwith some quantum sluggishn ess.

In the present paper we examine both the classicaland quantum mechan ics of LiCN on the same foot-ing, i.e. in phase space. There is found to be markedlocalization of quantum phase space density in ap-parently highly chaotic regions of classical phasespace. Such a finding is in accord with recent workon the quantum /classical correspondence for othersystems [ 19-271; the present study extends thesefindings to a realistic model of the highly nonrigidmolecule LiCN.

2. Classical mechanics o f LiNC /LiCN &.(8)=4.1159+cK?551 COS8+0.4983COS28

Elucidation of the classical and quan tum phasespace structure of the nonrigid molecule LiCN in-volves several technical difficulties, as discussed indetail elsewhere [ 391. For present purposes, webriefly outline the method used to study the classicaldynamics of LiCN in this section, and the corre-sponding quantum calculation in section 3.

The potential surface used for LiCN is that of ref.[ 331, expressed in scattering coordinates (R, 0):

9V(R, e) = 1 PA(COS e) K(R) . (1 )A= 0

R is the distance from Li to the center of mass of CN,8 is the angle between the CN axis and the Li to CNvector, and PA is the Legendre polynomial of order1. The potential is a sum of short-range and long-range parts, with coef5cients V, as a function of Rgiven in ref. [ 3 31.

A contour plot of the potential surface is show n infig. 1. The existence of minima at the linear config-urations LiNC and LiCN should be noted; the en-ergy difference E( LiCN) -E( LiNC) ~2 28 1 cm-The minimum energy path R, (13) connecting the twoisomers is show n as a dotted curve. W e have fittedthe m inimum energy path R,( 0) to a series in cos 8:

LiCN/LiNC

0 r/2 n0

Fig. 1 . Conto ur p lot of the potential energy surface for LiCN /LiNC of ref. [331.R is given in atomic units. !3=0 correspondsto the LiCN minimum , 0~ II o LiNC. The minimum energy pathconnecting he two isomers s shown as a dotted line (cf. eq. (2 ) 1 .

to.05343 COS 8-0.068124 COS 8+0.020578 COS58 , (2 )

in atomic units. The height of the LiNC /LiCN bar-rier is 3454 cm- LiCN is a very nonrigid m olecule,capable of complicated internal motions. At rela-tively m odest e nergies, the system can p ass betweenthe two isomeric forms, with the Li orbiting aroundthe CN [31,39].

The classical Ham iltonian for rotationless L iCN is

(3)where P R and pe are the momenta conjugate to R an d8, respectively, p, is the Li-CN reduced mas s, p2 isthe reduced mass of CN, and r is the CN bond length,frozen at its equilibrium value 2.186 bohr [ 3 11,

The surface of section (SOS) [ 1 ] is used to studythe classical phase space structure of two-mode LiCN .A very useful choice of SOS is defined by taking thesectioning coordinate to lie along the minimum en-ergy path connecting LiNC and LiCN. That is, thevalues of the coordinate (pe, 0) are noted every timethat a trajectory has R= R,(B). To ensure that theSOS is an area-preserving map [ 1J, it is necessary tomake a canonical transformation (R, 0, pR, pe) -+ (p,

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Volume 161, number 1 CHEMICAL PHYSICS LETTERS 1September 1989k/, p,,, p,), with generating function [ 391E(R, 0, P/HP,) =P,JR-R,(Q) 1+P,o. (4)The generating function (4) is of the F2 type [ 11, i.e.a function of the old coordinates and the new mo-menta. The new coordinates (p, w) are obtained interms of (R, 8, p,,, pv) by differentiation of F withrespect to the momenta (a,,, a,), while the old mo-menta (pR,ps) are obtained by differentiation withrespect to (R, 0). The coordinate p describes dis-placements transverse to the minimum energy path.The SOS used here is then a plot of the conjugatepair of variables ( y/, p,) at every intersection of thetrajectory with the reaction coordinate, where p=O(i.e. R=R,( v) ), and pP is chosen to be on a partic-ular branch of the momentum. The quantum wave-functions are defined on the coordinate range (seesection 3) 0


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Fig. 3. As for fig. 2, with E~2875.56 cm-, state 32.

y/(R, 8) are normalized with a weight functionR * sin 19, ppropriate for three-dimensional sphericalpolar coordinates. The wavefunctions y are mostusefully thought of as functions of three coordinates,(R, 8, $), where p is a second angular coordinate de-scribing rotation of Li around the CN axis, whichhappen to be independent of the coordinate 4. Thequantum solutions are inherently three-dimensional,and the restriction of the angle 0 to the interval0 < 8s n in the classical SOS defined in section 2 re-flects this fact.The appropriate arena for comparison of classicaland quantum mechanics is phase space. To define aphase space representation for the eigenstates ofLiCN, we use the Husimi [411 or coherent state [421representation. Here, the square magnitude of theoverlap of the eigenstate yl(R, 19)with a coherent state1a) [42] parametrized by a set of phase space co-ordinates & defines a nonnegative phase space den-sity, sections through which can be regarded as thequantum analogue of the classical SOS [43 1.

the bend and stretch frequencies are quite different,there is no obvious or unique way to define a set ofcoherent states with which to calculate the Husimifunction. In the present work, we use isotropic har-monic oscillator coherent states, with width param-eter corresponding to the geometric mean of that forthe LiNC bend and the LiNC/CN stretch modes[ 391. An obvious problem in defining the Husimifunction arises from the fact that the wavefunctionsyl(R, 0) are defined on a half-plane, whereas 2D OS-cillator coherent states are defined on the full (x, y)plane. It is therefore essential to embed the 2D quan-tum wavefunction in a three-dimensional Cartesianspace, and calcdate overlaps between wavefunctionsty(R, 19,$) (independent of angle 6) and 3D har-monic oscillator coherent states centered on the 2D(R, S) plane (9 constant ) with zero momentum outof the plane. The three-dimensional integrals re-quired are performed by numerical quadrature [391.A well defined nonnegative quantum phase spacedensity is thereby obtained.

For the nonrigid molecule LiCN/LiNC, in which Quantum surfaces of section [ 431 (that is, con-

Fig. 4. Asfor fig. 2, withEz4019.99 cm-, state 71.

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Fig. 5. As for fig. 2, with E=4367.01 cm-, state 88.

tour plots of a particular section through the Husimifunction) defined in a fashion analogous to the clas-sical SOS of section 2 are superimposed on the cor-responding classical SOS in figs. 2a, 3a, 4a and 5a,respectively. Contours of the Husimi function sec-tions are drawn with a thick solid line, to distinguishthem from, e.g. islands in the classical SOS.The task of .comparing rotationless ( J = O ) clas-sical and quantum mechanics would be considerablysimplified if a planar quantum Hamiltonian were tobe used, with the 8 coordinate describing motion ofa particle on a ring. The DVR-DGB program of Bacicand Light used to calculate the quantum states treatedhere employs the full three-dimensional Hamilto-nian, however, necessitating the relatively compli-cated procedures discussed above.

4. Comparison of classical and quantum phasespace structure

In this section we compare the classical and quan-

tum SOS for several representative states of LiCN.More extensive discussion will be given elsewhere, aswill a more detailed examination of the classical me-chanics [ 391.

State 24. The classical SOS, quantum SOS, andconfiguration space wavefunction are shown in fig.2. The quantum phase space density is highly local-ized, and lies in a region of phase space that is clas-sically chaotic (fig. 2a). Note that, although smallresonant islands are apparent in the stochastic regionof the classical SOS, the quantum phase space den-sity is not localized in the vicinity of these islands,but rather between the islands, presumably in the vi-cinity of the associated unstable periodic orbit. Thenodal pattern for this state is not particularly ir-regular [31] (cf. fig. 2b). States 29, 44, 50, 62, 69and 77 are also of this type.

State 32. The classical SOS at the energy of eigen-state 32 shows widespread stochasticity (fig. 3a).Nevertheless, the wavefunction has a quite regularnodal pattern, and the quantum phase space densityoccupies a very compact region in the midst of thelarge classically chaotic region. The quantum phasespace density sits at the leftmost tip of a dark areaof the classical chaotic region; such a darkening isindicative of the presence of a phase space bottle-neck, which we suspect is associated with the stableand unstable manifolds of a periodic orbit lying alonga ridge in the potential analogous to that noted bySmith and Shirts in HCN [44]. This point is cur-rently under investigation. States 30, 48, 53 and 75are also of this type.

State 71. This state was classified as a regular freerotor state by FT [31]. The wavefunction for state71 is delocalized and indeed has a regular nodalpattern (fig. 4b). The corresponding classical SOSreveals global stochasticity (fig. 4a), while the quan-tum phase space density is localized at the peripheryof the classically irregular region. Construction of thestable and unstable manifolds of the unstable peri-odic orbit located at the top of the isomerization bar-rier shows that the quantum phase space density forstate 7 1 is localized outside the separatrix manifoldsforming the boundaries of both the LiNC and LiCNisomers [39]. States 67, 70, 74, 78 and 82 are alsoof this type. State 65 is a so-called separatrix state[ 191; that is, the phase space density is localizedalong the separatrix manifolds emanating from the

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periodic orbit at the top of the isomerization barrier. ReferencesState 88. This state is localized in the LiCN well.The wavefunction has a very regular nodal pattern

(fig. 5b). The quantum phase space density is never-theless localized at the edge of the chaotic region ofclassical phase space (fig. 5a). States 58 and 76 arealso of this type.

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5. Discussion and conclusions

We have compared the classical and quantumphase space structure of a two-mode model of thenonrigid molecule LiNC/LiCN. The quantum phasespace densities for many vibrational eigenstates ofthe LiCN molecule are localized in regions of phasespace that are classically completely chaotic. In thislimited sense, we do not find any direct qualitativecorrespondence between the classical and quantummechanics of LiCN [ 3 11. Moreover, eigenstates withapparently complicated or irregular configurationspace nodal patterns [ 3 1 ] often have very localizedquantum mechanical phase space densities [45 ] (forexample, state number 50, not shown here).

Nonetheless, classical objects such as separatrixmanifolds clearly have a profound influence on thequantum phase space density [ 191. Further under-standing of the relation between the quantum andclassical phase space structure of LiCN requires a de-tailed study of the classical phase space, and such astudy is now in progress [ 391,

AcknowledgementThis work was supported by NSF Grant CHE-

8704632. RMB and FB acknowledge the support ofthe Comunidad Autonoma de Madrid and CICYT,Spain (contracts PB86/540and PB87/112). We arevery grateful to Z. Bacic and J. Light for providinga version of their DVR-DGB code, and to J. Ten-nyson for providing a potential subroutine. Com-putations reported here were performed in part onthe Cornell National Supercomputer Facility, whichis supported by the NSF and IBM corporation.

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R.M. Benito et al- Comparison of Classical and Quantum Phase Space Structure of Nonrigid Molecules, LiCN