jonathan tennyson and stavros c. farantos- routes to vibrational chaos in triatomic molecules

8/3/2019 Jonathan Tennyson and Stavros C. Farantos- Routes to Vibrational Chaos in Triatomic Molecules

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Anal>xis of both dassic+ and quantum caIcuhions on LiNC and- iCN shows dkonset of v ibrational dx& isdAyasociatd winzh he degree of bend&g acitation Corwasdy quasipaiodic~stietd&g states pa& ibove *e banie. komaizacion &d StlIdies on 03 gi& similar l-rsdu. In the light-of t&e92 results we r&map% the high-enaviiratiorwl data on 0, ind HCN aad,suggest thattlte obsemcd re@ar stretcbiq states @obabb are embedded in the cbaOFegioaWCdkUSSthCimportancc of mode coupiing by the potent ial -; :.

1. rI&vasim _-:me d i a t e /h i gh -e ne r gy re g i on . In parti*ar,- Lemann et al_ 123 ecorded the specuumof~~inRecently Hosz ark Taylor [l] have analysed the the range 15000-185~ cm-r in an attempt phenomenon of mode Iocalization in highly ex- observe .chaos.-AU the leveis they saw couldcited vibrational states_They argue that.overtone assigned and well fit ted by standard techniques stateshave a stability not. displaced by combina- localized states They were thus unabre to identtion states, and thus can exist as mode locahzed any chaotic states in this region, despite &&al(quasiperiodic) statesembedded in the mode mixed calcukions which predicted chaos above. 12 9(chaotic) qwsifxxnimllnn They use this argument cm-_ We note, however, that they failed to fito explain a variety of experimental data on poly- any state with two or more quanta &I the atomics in high-lying v ibrational states_ bending mode aud that in the energy range studiTriatomics are the simplest molecules that can one would expect the bending coordinate to bdisplay_ vibrational chaos_ Conventionally their come defocahzed [3]_ :!ocalized vibrational states, are .Iabekd as similarly, Imre et al 141n&&cd the vibr;u,, 4, u,), where ur and u3 represent stmtching tional spect rum of. 0, to within 500 cm:modes and o, is a bending mode. (degenerate for dkociation_ Their results have been Successfulinear m01ecuIes) _ In triatomics with more than fitted .using both Darling-Dennison. [S] and aIgone minim- it is the u2 bending coordinate braic approach [6] hamilto&ns, and all observwhich is hugely responsible for linking the.- states ckssified as regular- However, Imre et al [several triatomics-have-be+ studiedin the inter- observed no states involving bend@tg excitatio._-: : Classical t rajectory cakulations on Oaby Far$+toI pc t~$&&;of~*.&.&q&f and I+rrell : 7J. f&md i chaotic trajectorie$50-crrte,~crrrc.GrrsDc--- : - :G... : cmYL- below dissokitio~ I? lthqugh the_+pa&&~is~oofnD~n~rncbrs~:---l ..:: _ of Ikhmann et.ah [S] showsthat Such comp&s&. .~&1-0104/85/$03.30 @ Else&r Science publishers ti _V_ -~ :(North-HollannP~~~pUblishingDivisiOn) .~ .- ~: I : -: ~. . ; : .~ :


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:238 X Tmyswr. SC F-mr , Vii~-onal chaas in rrimm-c mokde~must be made with caution, the two results wouldappear anomalous_Recently we have analysed the vibrationaI mo-tions of KCN and IiCN using both quantum andclassical mechanics [9.10]_ In that work. .vibra-t.ionaI states were characterised as regular or irrcg-ular a c c or d i n g t o t h e n o d a l p a :t e i r n s o f t h e i r w a v e -f u n c t i o n s , t h e a v o id e d c r o s s in g s a n d s e c on d d i f-f e re nc e s i n the energy levels, the distribution ofIeveI spacings and the occurrence of dominantcoeffki ents_ For the classicaI trajectories, we ccm-puted the power spectra of the momenta. PoincarEsurfaces of section, plots of the trajectories on thecoordinate phtne, and the rate of exponential di-vcrgcnce o f two initiahy adjacent trajectories_ Wepredicted that for both molecules an early onset ofvibrational chaos is to be expect ed_ The strongcoupling between the bending and stret chingmodes in KCN meant that we found no quasiperi-odic trajectories (ahhcugh KAM theorem predictsthat some must exist) or reguIar @caked) statesembedded in the chaotic region_ Conversely, inLiCN which displays weak couphng, trajectoriesor quantum states with regular motion persistedover the entire energy range studiedIn thy paper we turmer analyse the vrbrationalmotions of LiCN_ ParticuIar attention is paid tothe stretching versus bending energy distributionof the quasiperiodic trajectories and reguIar statesembedded in the chaotic region_ We show that inLiCN the onset of chaos is correlated with thedegree o f bending exci tation and hence that regu-Iar states with a high degree of stretching excita-tion but Iit tIe -or no bending exci tation can beexpected to persist wcII into the chaotic region andpossiily to di s soc ia t ion (depending on the degreeof coupling)_ This shows us to reinterpret theapparent confhct between the classicaI trajectorycalculations and the obsenred highly exci ted vibra-tionaI states of HCN and OS_

ZReStlItST h e numericaI methods used here to am&e thecIassicaI and q u a n t u m -m e c h a n i c a l m o t i o n o f L iC N

h a v e b e em descri&erI in previous publications [9,10].The potential energy surface used is the ab initio

surf2ce of Essers et &._ [111_ This poterit ial isfunction of two variables, the distance of the Lfrom the CN- cent&of -mass. R, and the anglebetween R and the CN_ bond, r_ It reproduces observed geometry-and vibrational fundamentaof LiNC(B= lSOo) (121 as welI as predicting local LiCN(8 = 00) minimum_ -~- -~ .~We analysed trajecto+with different enerpartitioning between t h e stretch (R) and bend (coordinates_ To do this trajectories were started the equilibrium geometry of LiNC (or LiCN)_ Tenergy in each mode was defined as the kineenergy of that mode at 2 = 0. Whilst altering initiaf phase of 2 trajectory can change in behaviour, we do not believe the conclusions drawbeIow are sensitive to the phase of indiv idut r a j e c t o r i es _ E a c h t r a j e c t o r y w a s i n t e g r a t e d= 10 ps_

Fig_ 1 shows two t ypicaI trajectories with toenergies of 3712 and 3394 cm-t which are abothe barrier t o isomer-k&on of 3377 cm-_ Tquasiperiodic trajectory started with most of energy in the stretching mode whereas the chaotrajectory had most in the bending coordinate- Wfound that this behaviour was typicaI of trajectties above the criticaI energy for the transition chaos (at = 1600 cm- [lo&The unshaded areas of fig_ 2a show the range energies in the bending and stktching modes whethe trajcctorics were assigned as quasiperiodic. Wfound quasipet iodic trajectories well above tbarrier to isomerisation provided only 2 Smauamount of energy was pIaced in the bending modOn the other hand, all trajectories with enermore than 1509 cm- in the bending mode azero-point energy in the stretching mode wefound to be chaotic. The singly crossed region fig_ 2a separates the quasiperiodic from the chao(shaded) domains. This domain encompasstrajectories which were dif ficuh to assign withfinite-time caIcuIation_

AsimiIaranaIysishasbeencarriedoutforthequantum mechan.icaI status of LiiC (fig_ 2b). Ts t a t e s h a v e b e e n a s s i g n e d - a s r e g u la r -or chaoaccording to our previous criteria [lo], IargeIy anaIysmg their nodaI structure_ For quantum stait is not po&Me to make a rigqro& decompositio f t h e e n - h o s t r e t c h i n g a n d b e n d i n g c o n t


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FREO_ 01Fig i- The pourr s&m and the projecti on of two typical trajectories started at LiNC n&mm and wilh enagksr E- = 3395 an-. E- = 317 cm-=_ (b) E, ,= 37 7 an -. Eap a = 3017 an-. Q1 and Q2 denote the bending coo&hate dqus and stm&ing coonihate in o,,. r a p u x i s d y _

.- .~ ( i but i ons - How&r , : o r r e g u l a r S t & e s a n a p p r o x i - E-( u l, 4) = A!&,( u Ir 0),mi t e de c omp os i t ion wa s a c h i e ve & Thi s wa s --don e %+&+ 3) =%&I. 4%++(Y&:~ $1b y a s s ig n i n g t h e s t r e t c h i n g fu n d a m e n t a l a n d o v e r -t o n e s , u 1 a n d t h e n w r i t i n g - &he r s e .t i b l e d&i &i ons b f E & & , , an d k-1 _&e


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Fig_ Z Quxipaiodic -chmtic domains inLiNC~~ftl!Xtkl of bending-smztching sxrgy_ (a) c%sskal rcsuhs for fifty trajccto ric(b) Quamom nsulrs Energies arc rdatisxz O LiiC minimum.

no qualitatiwz difference to fig_ 2b_In fig . 2b the open circks mark the reguIarstates and crosses denote those which were dif-

ficult to designate as chaotic or regular_ Closedcircks represent chaotic states. for the majority ofchaotic states no method of partitioning the energycouId be found as not even approximate (0,. c~a)scouId be assigned_ The chaotic states not includedin fis 2b should I& to the upper right of thefigure_ Typical nodal pattern for re@Iar, uncertainand chaotic states are shown in fig- 3_

States IoeaIised in the LiCN minimum have alsobeen observed j10,12]. By assigning approximatebending and stretching quantum numbers to thesestates we found that regtdadty is again associatedwith a low degree of bending exci tation_ Fig 4shows a &ssieaI and quantum analysis .ofbend/stretch excit ation in LiCN. In some regionsuncertain/chaotic LiCN states were observed out-side the eIassieaIIy chaotic domain This is due totunnehng.

J- ttte tight of these results we have reexaminedthe cIassieaI trajectory results for 0, [q obtainedusing the p o t e n t i a l e n e rg y s u r f a c e of MurreII andFarantos [13j_ We have examined severaI -zajecto-

ries distributing t he energy between the bendingand two stretching modes. We found that exci tingthe symmetric stretch while keeping the bendingand asvmnxtric stretch at their zero-point enetgiesgave &guIar (Iocihzed inside the enek-geticaUavailable coordinate space) trajectories for totaenergies up to 6000 cm- (fig. Sa). This energy imeasured from the minimum of the potentiai whichis 8800 cm- beIow dissociation. The criti cal energy for the transition to chaotic has been locatedat = 3000 cm-t above the minimum [7]_ Regulartrajectories were also observed by exci ting thasymmetric stretch for total energies up to 500cm-.Combinations obtained by exci ting- bothstretching modes by one or two quanta ako resulted in locahzed trajectories. On the other handexcit ation of the bending-mode at &se energiegives chaotic behaviour (fig_ 5b). Our resuhs thuconfirm the conclusions drawn for Ji iNC/LiCNthat exeitz ftion of the stretching modes results inquasiperiodic trajecto ries whereas excit ation of thbending mode is associated with chaotic beIlaviour. . :_


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E~3_Thrcctypicalvi.i~wa~unctiionrThecontourslinkpoints~~~.~~~~~has4~.88.1656.32% and64% ofitsmaximum am$icud&SoEdams datotcp&irivcamplitude and d+d _lincs negative amplitude. (a) Regular(3.2). (b) unartam (2,12)?. and (c) dtaotic state

. ..-~/&&+ov& d& i&&l-a& &e iq&&$~f~~.the b e n d i n g ~ m o d e i n r e a c h i n g t h e ch a o t i c & g G ono f . p h ase sp ace f or a ~ t r i a to m ic mo I ec$ e_ :Qr i !th eo t h e r . h a & d . h i g h l y e x c it e d s t r e t c h i n g s t a t e s t i t h l it t l e b e n d i n g e x c it a t i o n s h o w a r e g u I a r b e h a v i on rAl t h o u g h . t h e r o I e of - mo d e co u p l in g cannbt bi g n o r e d , w e b e h e v e t h i s i s d u e t o t h e m c r e a s e da n h a r m o n i c it y a n d r e d u c e d v i ir a t i o n & s p a c i n g i nt h e r e g io n o f a p o t e n t i a l b a r r i e r b e t w e e n h o m i n i m a I n t r i a t o m i c s t h i s e ff ec t d o m i n a t e s t e r mi n t h e p o t e n t ia t a s s oc ia t e d w i t h t h e b en d i n g r a t h et h a n s t r e t c h i n g c oo r d i n a t e s _ T h i s s u p p o r t s H o s ean d Tay lo r s th es i s [ I ] t h a t ex t r em e d i f fe r e n ces inm o d e m o t i o n s w i h r c s u h i n r & &r b e h a v i o t iH o w e v e r , O U T e s u l t s s u g g e st t h a t t h i s t h e s i s i s O n Iv a i id , i n t r i a t o m i c s G i t h I o w b a r r i e r s , fo r e x t r e m es t r e t c h i n g m o d e s -

I n a c c or d a n c e w i t h r e f_ [ I] , w e a r g u e t h a t I d l y -a t o m i c m o le c u k s w i l l s h o w n o n -e r g o d i c b e h a v i o u ri f t h e y a r e e x c it e d i n s t a t e s w i t h e x t r e m e m o d em o t i o n s . F o r e x a m p l e , a r g o n c lu s t e r s a t t a c h e d t oexcited ch em icah y in t e r e s t in g sp ec ie s [ &Z-1 5] sh o wh i g h l y n o n -s t a t i s t i c a l t m i m o I e c n I a r d i s s oc i a t i onp r o p e r t i e s . N o f r a g m e n t a t i o n o f a r g o n .c I u s t e r $ C a nb e o b s e r v e d d e s p i t e a n e n e r g y e xc e s s i n t h e ch e m ica l +xz ie s o f two o r d e r s o f ma g n i tu d e [ l s ] . Ofc o u r s e , s u c h s y st e m s a r e e xp e c t e d t o sh o w e x t r e m em o d e m o t i o n

H o w e v e r , f or sy s t e m s w i t h m o r e t h a n t ie d eg r ees o f fr ee d o m ( N > 2 ), i t i s n o t n ea x sa r y f orr e g u l a r i t y t o b e a s s o c i a t e d w i t h q u a s i p e r i o d i ct r a j e c t o r i e s . T h e r e m a y b e o n I y M cokants om o t i o n w h e r e 1 c M c AT_ Numer ical studies e ont h r e e +3 im e n s i on a I s y s t e m s h a v e s h o w n t h a t i r a j e cto r i e swi th M=2 o n b ef o u n d [ 1 6 ] . l I n th i scaset h e c h a o t i c r e g i on o f p h a s e s p a c e i s d i v id e d i n t od i f f e r en t e r god ic c o m p o n e n t s a c c o r d i n g : to thc o n s t a n t s o f mo t io n ,

H i g h l y e x c it e d r e g u I a r s t a t e s h a v e b e e n ~ ,o b -seqe&irt FCN [2] and ~_o, I+]. How ever i t~is . .n o t i ceab le lt a l l J = 0 s t a t e s o b s e r v e d h a v e t h eb en d in g mo d e in i t s g r &n d r ;~te An a ly s i s o f o u r


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~-.E,-358c m -.~~545c m -_~)~-M8q~ -~.E~=3938cm -~E,=54~cm - an d 6th & es; (a) E, - 3%_ Q1 and Q, denote rhe asymmetricaspStXiCsnclcba~y_resuits for LiCN and 0, suggests that the exci ted intensity, because the transition dipoles are smbending states in the region sampled are chaotic PI-These chaotic states would appear to. have km- Jiehmam et als [ZS] classical trajectory stud


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on HCN Ising diffe?xzas potaM !Ea%gy i?a!k-demonstrate the sensitivit y of the onset of chaos tothe potential_ They studied the exn-eme motion ofthe H-CN stretch and found that the onset ofchaos varied by a factor of three with the potentialfunction used. We would expect the bending modein HCN to show chaotic behaviour in the region ofthe HCN/HNC barrier_ Lehmann et aLs [S]calculations thus reflect the variation in barrierheight in the potent ials used and the strength of.the resulting mode coupling We note that none ofthe potekials used were fit ted to spectroscopicdata in the high-energy region_

In OJ we observe mode Iocalized (regular)stretches well into the chaotic region; these corre-spond with the stretching states which have beenobserved almost to dissociation_ Although quan-tum %luggishness~ [9,10] cannot be rukd out. wefeel that the earlier onset of chaos in the classicaltrajectory results is due not to faihn-e of classicalmechanics but to the inadequate .-epresentation ofmode coupling at high energies by the potentialfunction. Comparison of our calculation on floppymokcules [9,10] shows a qualitati ve agreement be-tween classical and quantum mechanics_ In partic-ular the strength of mode coupling by a givenpotential is reflected by both mechanics [lo].

4. Conchn5onIt has been shown for LiiC. LiCN and 0, that

exci tation of the bending motion is the primaryroute to chaos This conch&on is supported forLiNC/LiCN by both quantum and classical calcu-lations, It is our conjecture that for most tri-atomis espeklly those with more than oneminimum or barrier in the potential, bending exci -tation will lead to chaos at lower total energy thanby excitation of a stretching mode- Except ions tothis wiil be molecules, such as KCN [9,10], wherestrong mode coupling causes a near uniform onsetof chaos.

The impor?tance of the coupl& between bend-ing and stretching modes at intermediate an~bigbenergy means that one-dimensional hamiltonianswhich have been used to describe the vibrations efnon-rigid triatomics -(eg. refs. [3,18D should he

a- ** &$&_ - 5. a &_&_&&&ference between one-dimensional ap~rokiniationswhere the motion. is always regular and. two- -othree-dimetisionaI _caIcuIations for ~which chaos ipossible_ In -this context the use cf couplehstretching functions such as Morse osciRatOr[8,19], whilst mathematically interesting_does noprovide a good model of vibrational chaos~ intriatomics which we believe to be strongly associ-ated with the bending motion_

There is also a need for realistic poteritiais forthe interpretation of vibrational spectra. Manypotential energy surfaces are produced by fitt ingexperimental data which describe a iimited regionof nuclear configuration space_ The remainingspace is then described by extrapolating the func-tions into the unknown regions. Our analysis ofthe 0, and HCN problems suggests that accuratepotential functions should reproduce mode-cou-pling behaviour as a function of energy_ This i sundoubtedly a diff icult t ask_

We speculate that the diffi culty in observinghighly excited bending states for HCN and 0, isassociated with the chaotic character o f these statesWe feel that these bending states, and perhaps thebends in floppy molecules which have an earlyonset of chaos, provide good candidates for ex-perimental investigation of chaos [20]_

References[ll G- Hose and 3LT Taylor.C&n_ Phyx 84 1984) 375;

G- Hose, Hs Taylor and Y-Y_ Bai, J. Chun Phys SO(1984) 4363.121KK- Lehmann.GJ. Sdrerer nd W. Kkmpcrer, J_ GhanPhys 77 (l9S2) 2853.[3] M_ Pet%, M MadcnoviS S-D_ Pcyaimhoff and RJ_Bumker. than Phyr 82 (1963) 317_[4] D-G. Imm, J-L. Kinscy, RW_ Field and Del-L Katayama.I Phyr Chem 86 (1982) 2564_[S] RR Lehmans I Phys Ghan 88 (19%4) 1047.[6] L Eknjamin, RD. Levine and J-L Kinsq, J. Phys t han87 0983) 727.[?] SC F-tos and J_N- Mundl.

jonathan tennyson and stavros c. farantos- routes to vibrational chaos in triatomic molecules

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