Jonathan Tennyson- Quantum vibrational chaos in the ArHCI van der Waals molecule

8/3/2019 Jonathan Tennyson- Quantum vibrational chaos in the ArHCI van der Waals molecule

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MOLECULAR PHYSICS, 1985, VOL. 55, NO. 2, 463- 473

Q u a n t u m v i b r a t i o n a l c h a o s i n t h e A rH C I v a n d e r W a a l sm o l e c u l eby JO N A T H A N T E N N Y S O N

Science and Engineering Research Council, Dar esbur y Laborator y,Daresbury, Warrington WA4 4AD, England

(Rece ived 3 f fanuary 1985 ; accepted 7 Feb ruary 1985)

Quantum mechanical ro-vibrational calculations are presented for theArHC1 van der Waals complex using Hutson and Howard's empirical M5potential. Analysis of the nodal structures, second differences and overlap-ping avoided crossings suggests that the higher bound states of ArHC1 arechaotic. This chaos is made evident by perturbing the angular part of thekinetic energy term, e.g. by isotopic substitution or vibrational excitation ofHC1. Calculations with J > 0 show increased level crossings but no signifi-cant increase in chaos because of an extra nearly good quantum number. Thelow-lying states of ArHC1 appear to be poorly modelled by either the harmo-nic oscillator or free rotor approximations. Comparison with other moleculesfor which vibrational chaos has been predicted are made.

1. INTRODUCTIONIn a recent series of papers [-1-3] Te nn ys on and Farantos studied the onset

and nature of vibrational chaos in the floppy K C N and L i C N molecules. Theydid this by performing quantum and classical calculations on realistic (ab in i t io )potential energy surfaces. Quantum chaos was identified by using a variety ofpreviously pro pos ed criteria [-4] including nodal stru cture [5], spectral distribu-tions [.6], seco nd differences [-7] and the o ccu rre nce of overl appi ng avoided cross-ings [8]. While no criterion was found to give a complete solution to the problemof characterising quantum chaos, the combination of indicators allowed chaos tobe identified. Qualitative agreement was also obtained between classical andquantum calculations, although some quantum sluggishness was observed. Theunusually early onset of chaos in both K C N and LiCN was found to be closelyassociated with the pres ence of low-lying barriers in the bend ing coordinate [-3].

Many van der Waals complexes have low-lying barriers in their intermolecu-lar bending coordinate(s). Furthermore there is evidence from both classical cal-culations [-9] and experi ment [-10] that A r clusters behave chaotically. As the lowdissociation energy of van der Waals systems means that most or all of theirvibrational levels are thermally occupied, they would appear promising candi-dates for the experimenta l study of chaos.

The simplest van der Waals complexes that could display chaos are atom-diatom systems. Complexes formed by light diatomics, especially H2, have onlysmall barriers in the bending coordinate. Such systems have been studied byLeRoy and co-workers [11], who were able to obtain entire anisotropic potentialsfor such systems by inversion of (welt resolved) infra red spectra. The isotropic


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464 J. Ten nys onnature of these systems means that they are very well represented by the freerotor approximation. Indeed a model based on this approximation has been suc-cessfully used for the analysis of H2 X ( X = rare gas) hyperfine spectra [12].Vibrational calculations on H 2 X van der Waals complexes u sing free rotor basisfunctions show that individual states are generally dominated by one basis func-tion [13] these states can thus be assigned regular (or 'm od e localized') by thedominant coefficient criteria of Hose and Taylor [14].

Conversely, van der Waals complexes involving heavy diatomics show con-siderable anisotropy. Their infra red spectra are complicated envelopes (forexample references [15, 16]) which require sophisticated averaging procedures tobe reproduced theoretically [17]. Without resolving the mass of competing tran-sitions only limited information about the potential can be obtained from thesespectra. However, empirical potentials can be obtained for such systems by thestudy of scattering data, for example [18, 19].

The ArHC1 van der Waals system is an interesting example of a complex witha strong anisotropic component. It has been much studied theoretically [17-22]due, at least in part, to the accurate empirical potentials obtained by Hutson andHoward [18, 19]. In particular, their M5 potential [19] has been used suc-cessfully for theoretical calculations [17, 22]. This potential predicts minima forboth linear s tructures of the ArHC1 complex, separated by a barrier of 76 cm 1.As this is qualitatively the shape of potential which has been fo und to supportchaos in LiCN [2, 3], the purpose of this paper is to ask whether some of thevibrational states of ArHC1 are chaotic and if so how they might effect thepredicted behaviour of the molecules.

2. CALCULATIONSON ArH35C1The hamiltonian for an atom-diatom complex can conveniently be written in

body-fixed coordinates as [23]

H - 2#1R-------- OR R2 ~ 2#2 r 2 ~r r2 ~r q- 2#2 r 2 + -- 2 # R q- V ( R , r , 0 ) ,(1 )

where #a and #2 are the reduced masses, and J and j are -th e total angularmo me nt um operator s of the complex and diatom respe ctively; r is the diatomicbond length , R the separation of the atom from the diatom centre of mass and 0the angle betw een t and R. 0 is 0 ~ for linear ArHC1 and 180 ~ for ArC1H. The M5potential, V, of Hut so n and Howa rd [19] is only a two dimensional f unction of Rand 0. Following other workers [17, 20-22], the Hamiltonian (1) was initiallysolved with r fr ozen at the equili brium value of 2"435 a 0 .

Variational solutions of (1) were obtained using program A TO MD I AT [24]with the GENPOT addendum [25]. This method, due to Tennyson and Sutcliffe[26], uses a basis expansion of Morse-like oscillators for R and associated Leg-endre functions (free rotor functions) for 0. Trial calculations showed that 21Morse funct ion s with R1 = 9"8a0, D1 = 0"0004Eh and o51 = 5 x 10 4Eh and 12Legendre functions (for J = 0) converged 15 bound states for ArH35C1 to within0"05cm -1 for the highest states and considera bly better for the lower ones.Whilst the existence of further bound states cannot be ruled out on the basis of a


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Quantum chaos in ArHCl 4651 2

G ro u n d s t a t e

6 -

o 1 2r r

6 - 8 m E x c i t e ~

1 t E x c i te d s ta t e

0 9o 180 0 9o 1808 / ~

Figure 1. Amplitudes of 4 vibrational states of ArH35CI with J = 0. The contours linkpoints where the wavefunction has 4%, 8%, 16%, 32% and 64% of its maximumamplitude. Solid contours enclose regions of positive amplitude and negative con-tours regions of negative amplitude.

variational calculation, these are only likely with very low binding energy andlarge ( R ) . Addi tional states of this type will have little effect on the conclus ionsdrawn below.

Figure 1 shows plots of 4 typical ArH35C1 wavefunctions for J = 0 vibrationalstates. While the ground state shows some delocalization, it is substantially local-ized about the ArHC1 minimum. Conversely, the first excited state, a bendingexcitation, is clearly delocalized. Higher excited states are characterized by com-plicated nodal patterns at short R and simple delocalized behaviour at large R.These states cannot meaningfully be assigned quantum numbers by inspectionwhich is one indication of chaos [5].

The stability of levels with respect to a small perturbation is another meansthat has successfully been used"to detect qua ntu m chaos. Pomph rey 's use ofsecond differences [7] and Marcus et ars overlapping avoided crossings [8] arebased on this. In a real problem such as this, the appropriate perturbation is notreadily apparent. We thus chose to study 5 perturbations which in turn affect allparts of the Hamilt onian (1). Th e par ameters varied were mAr , the mass of Ar,and r, ~the HC1 separation, which perturb the kinetic energy operator, and e, Aeland Ae2 which be tween the m per turb all parts of the potential [19] and are


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466 J . Ten nys onTab le 1. Second differences for five paramete rs (see text) for the bo und states ofArH3SC1 with J = 0.

Parameter mar r go Agl Ag2P0 40 amu 2-35 a 0 ) )Ap 2 amu 0.05 a o 0"025 0.02 0"04State AJcm - 1

1 0-018 0'001 0"005 0'035 0"0042 0.017 0"039 0'008 0-015 0'0443 0.043 0"013 0"013 0'007 0-0144 0.037 0'022 0-026 0"004 0"0175 0'050 0"007 0'045 0'037 0"0046 0-026 0"021 0.012 0-012 0'0047 0.053 0.071 0"058 0 " 0 1 9 0-0248 0.055 0.087 0-002 0 " 0 1 3 0'0179 0.035 1"177 0'003 0'092 0"02010 0.049 0.268 0"024 0-039 0'00111 0.048 0.208 0.075 0'023 0.00412 0.030 0.313 0'009 0 " 0 0 5 0"0 0313 0.053 0-333 0.156 0'014 014 0.024 0-417 0.047 0.006 0'001t Th e potential parameters, which are functions of O , areThey were scaled in the perturbed calculations by (1 + Ap). as defined by reference [19].

increa singl y anisotropic. Ta ble 1 shows second differences obtai ned for the i thstate as a func tio n of parame ter p using the formu la [7]

A i = ] (El(P0 + Ap) -- E i ( P o ) ) - ( E i ( . P o ) - - E ~ ( P o - - Ap)) [. (2)Values of Ap were chosen so as to give second differences for the lower states ofapproximately the same magnitude. For one perturbation, r , the second differ-ences sh ow n in table 1 are significantly larger for the up per 7 states than thelower states. Th is is indicat ive of chaos [7].

Figure 2 shows the energy levels of the ArHC1 problem plotted as a functionof r. Insp ect ion of the f igure reveals that s tates bo un d by less than 40 cm -1undergo many avoided crossings. This again indicates that these s tates are chaotic[81. Conve rsely , plots of eigenenergy as a functi on of the other paramete rs c on-sidered in table 1 show no such erratic behaviour and imply that al l s tates areregular with respect to these perturbations.

In general the density of s tates of a system increases with total angularmo me nt um , J . T his has led to suggestions that chaos will manifest i tself morestrong ly for rot ationally excited s tates because of the increased probabi l i ty ofne ar -r es on an t inte racti ons. F igu re 3 depicts the J = 1 e and J = 2 r where e de-notes the posit ive pari ty co mbin ati on [271, levels as a functio n of r. T hese f iguresagain show m an y (overlapping) avoided crossings in the high ener gy region.

An addit ional indicator of chaos which has proved useful in previous calcu-lat ions is the use of spectral dis tr i bution s [4, 61. Th is measur e is based on theshape of his tograms of the spacings between levels of the same symmetry. Forgood statistics, however, many levels are required. With less than 15 levels in thechaotic regi on and fewer below, even for J = 2 e, ArHC1 supp orts insu ffici entlevels to mak e a dis tr ibuti onal analysis useful .


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Quantum chaos in ArHCI 4 6 7

F i g u r e 2 .

F i g u r e 3 .

7Eo

0

- 4 0 -

-80 -

A r H C [ J = 0

- 1 2 0 t2 .0 re 2 .8rH C [ / a o

Vibra t iona l energ ies as a func t ion of HCI separa t ion fo r ArH35C1 wi th J = 0.

0

- 4 0 -

- 8 0 -

- "f2 0

A r H C ! . J = l e 0

- 4 0 -

'T

m -

-8 0 -

- 1 2 0 -

A r H C L J = 2 e

f f2 .0 r e 2 .8 2 .0 r e 2 .8rH C [ / a o rH C t / a o

(a) (b)V i b r a t i o n a l e n e r g i e s a s a f u n c t i o n o f H C 1 s e p a r a t i o n f o r A r H 3 5 C [ w i t h ( a )

J = 1 e a n d ( b ) J = 2 e.


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468 J. Ten nys on3. DIscussioN

The calculations presented in the previous section strongly imply that thehigher bound vibrational states in the M5 ArHC1 potential are chaotic. Not onlyare these states too irregular to make the assignment of qua ntu m n umbe rs on thebasis of their nodal structure meaningful, but also they behave erratically whenperturbed by changing r, the HC1 bondlength. The implication of this is thatvibrational chaos in ArHC1 is not induced by the potential as has been predictedfor KCN [1] and LCN [2, 3] but by the kinetic energy operator. As a check onthe effect of perturbing the kinetic energy operator in these systems, calculationswere performed for KCN as a function of CN bondlength, r. The results aregiven in figure 4, which, although it shows an increasing number of avoidedcrossings, does not show the large num be r of resonances from 40 0c m -1 abovethe g oun d state upwards, displayed when a pertu rbatio n of the potential was used[ i ] .

Chaos is usually understood both in classical and quantum mechanics throughthe partitioning of a given problem into a separable (non-chaotic) zeroth orderproblem, H0, which is then coupled by some chaos inducing perturbation.Indeed this concept is fundamental to both KAM theory and Hose and Taylor's' Quantum KAM theory ' [28].

In conventional (near rigid) molecules the harmonic oscillator model providesa good H 0 which usually allows man y vi brational states to be assigned. Fo rexample, this model provides a good description of the lowest (regular) states inK C N [29] and L iC N [30], although in the floppy molecules it soon breaks down[1, 2, 31].

Figure 4.

1 0 0 0

7~.Eo5 0 03 -

K C N J = O

2 .0 re 2 8rCN/ao

Vibrational energies as a function of CN separation for KCN with J = 0.


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Q u a n t u m c ha os i n A r H C I 4 6 9I n a t o m - l i g h t d i a t o m v a n d e r W a a l s d im e r s , s u c h a s H 2 X , ( X = H [ 1 3 ] ,

N o b l e g a s [ 1 1 , 2 6 ]) o r H e H F [ 3 2 ] , t h e fr e e r o t o r m o d e l p r o v i d e s a g o o d H 0 .W o r k o n a h e a v i e r c o m p l e x , t h e n i t r o g e n d i m e r , h a s s h o w n t h a t b o t h t h e h a r m o -n i c o s c i l l a to r [ 3 3 ] a n d f r e e r o t o r [ 3 4 ] m o d e l s p r o v i d e a p o o r d e s c r i p t i o n o f t h em o l e c u l a r v i b r a t i o n s , a n d t h a t e v e n t h e l ow e s t s ta t e s m u s t b e r e g a r d e d a s s o l u -t i o n s i n t e r m e d i a t e b e t w e e n t h e s e m o d e l s [ 3 5 ] .

I n s p e c t i o n o f t h e l o w e s t v i b r a t i o n a l w a v e f u n c t i o n s o f A r H C 1 s u g g e s t s ap i c t u r e s i m i l a r to t h a t f o u n d f o r ( N 2 ) 2 . T h e g r o u n d s t a te i s s u b s t a n t i a l l y l o c a l i z e da b o u t 0 = 0 ~ a l t h o u g h s o m e d e l o c a l i z a t i o n ( a n d f o r th i s s t a te t u n n e l l i n g ) c a n b eo b s e r v e d , s e e f i g u r e 1 . T h e f i rs t e x c i t e d s t a t e is a s i n g l e b e n d i n g e x c i t a t i o n a n d i sd e l o c a l i z e d . T h e s e c o n d e x c i t e d s t a t e i s, l a r g e l y , a s i n g l e s t r e t c h i n g e x c i t a t i o nl o c a l i z e d a b o u t A r H C 1 ( 0 = 0 ~ T h e t h i r d e x c i t e d s t a t e i s a c o m b i n a t i o n b e t w e e na d e l o c a l i z e d d o u b l e b e n d i n g e x c i t a t i o n a n d a s i n g l e s t r e t c h i n g e x c i t a t i o n lo c a l -i z e d a b o u t A r C 1 H ( 0 = 18 0~ A b o v e t h e s e i t b e c o m e s i n c r e a s i n g l y d i f f i c u l t t oa s s ig n s ta t e s b y i n s p e c t i o n . T h i s t r e n d a p p e a r s t o b e u n i f o r m a n d s u g g e s t th a tA r H C 1 i s n e a r e r t o K C N , w h e r e n o r e g u l a r s ta t e s c o u l d b e a s s i g n e d o r c l as s ic a lt r a j e c t o r ie s o b s e r v e d a b o v e t r a n s i t io n t o c h a o s [1 ] , t h a n L i C N w h e r e b o t h r e g u l a rs t a t e s a n d q u a s i p e r i o d i c t r a j e c t o r i e s w e r e f o u n d o v e r t h e e n t i r e r a n g e s t u d i e d [ 2 ,3 ] . A r H C 1 t h u s a p p e a r s t o b e s t r o n g l y m o d e c o u p l e d .

F i g u r e 3 s h o w s t h a t i n c r e a s i n g t h e r o t a t i o n a l e n e r g y o f t h e s y s t e m d o e s i n d e e di n c r e a se t h e n u m b e r o f l e v el s a n d h e n c e t h e n u m b e r o f a v o i d e d c r o s s i n g s .H o w e v e r , c o m p a r i s o n o f f i g u re 2 a n d f i g u r e 3 g i ve s l it t l e e v i d e n c e f o r t h e l o w e ro n s e t o f c h a o s i n t h e r o t a t i o n a l l y e x c i t e d s y s t e m s . T h i s i s b e c a u s e a l t h o u g h t h e r ea r e m o r e a v o i d e d c r o s s i n g s i n t he s e s y s t e m s , m a n y o f t h e c r o s s i n g s a r e o n l yw e a k l y a v o i d e d . T h i s i s d u e t o th e p r e s e n c e o f a n e a r - g o o d q u a n t u m n u m b e r , K ,t h e p r o j e c t i o n o f J a l o n g R , w h i c h m e a n s t h a t le v e ls w i t h d i f f e r e n t K o n l y w e a k l yi n t e r a c t . T h i s b e h a v i o u r i s w e l l k n o w n i n v a n d e r W a a l s s y s t e m s a n d h a s b e e nu s e d i n a p p r o x i m a t i o n s w h i c h n e g l e c t C o r i o l i s i n t e r a c t i o n s [ 2 0 , 2 1 , 2 6] .

O f c o u r s e t h e a b i l i t y t o p e r t u r b t h e A r H C 1 c o m p l e x b y c h o o s i n g a v a l u e o f t h eH C I b o n d l e n g t h , r , i s o n e w h i c h i s o n l y a v a i la b l e t o t he t h e o r i s t . T h e r e a r e ,h o w e v e r , w a y s i n w h i c h s e n s i t i v i t y t o r m i g h t m a n i f e s t i t s e l f e x p e r i m e n t a l l y .F i r s t l y , i n th e r e a l A r H C 1 m o l e c u l e , t h e H C 1 c o o r d i n a t e i s n o t f r o z e n ; d i f f e r e n tv i b r a t i o n a l s t a te s o f H C 1 h a v e d i f f e r e n t ( r ) . S e c o n d l y , i s o t o p i c s u b s t i t u t i o n i nH C I p e r t u r b s t h e s a m e t e r m s i n th e H a m i l t o n i a n a s c h a n g i n g r .

T a b l e 2 g iv e s e n e r g i e s o f t h e v i b r a t i o n a l l e v e ls o f t h e A r H C 1 m o l e c u l e r e l a t i v et o H C 1 i n i ts g r o u n d a n d f i rs t e x c i t e d s t a te . F o r c o m p a r i s o n t h e r e s u l t s o f ac a l c u l a t i o n w i t h H C 1 f i x e d a t it s e q u i l i b r i u m s e p a r a t i o n a r e a l so g i ve n . T h e rd e p e n d e n t c a l c u l a t i o n s w e r e p e r f o r m e d u s i n g a n e m p i r i c a l l y d e r i v e d H C I p o t e n -t ia l d u e t o O g i l v i e [ 36 ] . T h r e e M o r s e o s c i l l a t o r - l i k e f u n c t i o n s w e r e u s e d t o r e p -r e s e n t t h is c o o r d i n a t e w i t h r z = 2 " 4 6 5 a 0 , (-o 2 = 0 " 0 1 2 E h a n d D 2 = 0 " 1 0 4 5 E h [ 2 4 ] .T h i s b a s i s r e p r o d u c e d t h e o b s e r v e d H C 1 v ib r a t i o n a l f u n d a m e n t a l t o w i t h i n 0 -0 1p e r c e n t ( 0 "2 c m - 1). N o p o t e n t i a l c o u p l i n g b e t w e e n r a n d ( R , O ) w a s a l l o w e d f or .

P r e v i o u s c a l c u l a t i o n s o n th e b i n d i n g e n e r g y o f v a n d e r W a a l s d i m e r s a s af u n c t io n o f m o n o m e r v i b r a t i o n a l s t a t e h a v e b e e n p e r f o r m e d o n t h e H e H F m o l e -c u l e [ 3 2 ] . T h i s c o m p l e x , w h i c h i s h i g h l y i s o t r o p i c , s h o w e d s y s t e m a t i c ( b u t s m a l l)i n c r e a s e s i n b i n d i n g e n e r g y a s f i rs t ly t h e H F c o o r d i n a t e w a s u n f r o z e n a n d s e c -o n d l y H F v i b r a t i o n a l e x c i t a t i o n i n c r e a s e d . S i m i l a r b e h a v i o u r c a n b e s e e n f o r t h el o w - l y i n g l e v e ls o f A r H C 1 s h o w n i n t a b l e 2. T h i s i s d u e t o i n c r e a s e s i n ( r ) w i t hH C 1 v i b r a t i o n a l r e l a x a t i o n a n d e x c i t a t i o n . H o w e v e r , t h e 6 t h s t a t e o f A r H C 1 i s


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4 7 0 J . Te n n y s o n

Ta b l e 2. V i b r a t i o n a l e n e r g y l ev e l s ( i n c m - 1 ) r e l a t i v e t o f r ee A r a n d H a s CI ( v ) fo r A r H 3 5 CIas a func t ion o f HCI v ib ra t io na l s ta te v . Resu l t s fo r a f ixed nuc le i ca l cu la t io n areg i v e n f o r c o m p a r i s o n .

Sta tes Fi xe d~ v = 0 v = 11 - t 1 7 .2 4 - 1 1 7 .3 0 - 1 1 7 - 4 92 - -92 '33 9 - 92 '38 -9 2-6 13 - 83. 76 - 85-8 0 - 83 '984 -- 65-04 -- 65 '07 -- 65"425 -- 58' 70 -- 58"76 -- 58"976 - -47"88 - -47 ' 71 - -48 .857 - - 4 0 '5 6 - - 4 0 .5 8 - - 4 0 '9 58 - - 3 6 - 6 0 - - 3 6 '6 5 - - 3 6 .8 99 - - 2 4 '9 8 - - 2 4 .9 3 - - 2 5 .6 4

10 - -22"01 - -22 . 10 -22" 1911 -- 16-81 -- 16-73 -- 17-4912 - - 11 .09 - -11"17 - -11 '3013 - -6 '68 - -6 .6 6 - -7"2214 - -5 -17 - -5"28 - -5 ' 2715 - - 0 '36 - -0"43 - -0 ' 69

~" r = 2. 4 35 a 0.

a c t u a l l y l e s s b o u n d i n t h e f u l l c a l c u l a t i o n s , t h i s m u s t b e a s s o c i a t e d w i t h t h ea v o i d e d c r o s s i n g u n d e r g o n e b y t h i s s t a t e a s r i s v a r i e d , s e e f i g u r e 2 . A b o v e t h i sl e v e l t h e r e i s i n c r e a s i n g l y e r r a t i c b e h av i o u r w i t h v i br a t i o n a l e x c i t a t i o n . S o m es t a t e s b e c o m e m a r k e d l y m o r e b o u n d ( e . g . n u m b e r s 9 a n d 1 1) w h i l s t s t a t e 1 4a c t u a l l y h a s a s l i g h t d e c r e a s e i n b i n d i n g .

Tab l e 3. Vib r a t i ona l ene r gy leve ls ( in cm -1 re la t ive to f ree Ar + HC1) fo r the m ajo ri so to p ic spec ies o f ArHC 1 w i th J = 0.

S ta tes ArH3SC1 ArH37C1 ArD35C1 ArD37C11 -117" 24 - -117 .53 - -125"13 - -12 5 '462 - -92 .33 - -92 .58 - -98"64 - -98"873 - 83- 76 -- 84"4 0 -- 91 '01 -- 91 "654 -- 65.0 4 -- 65'5 5 -- 80"08 -- 80' 445 -- 58"70 -- 59-48 -- 69-42 -- 70.016 - -47 '88 - -48 '37 - - 61 -74 - - 62"597 - -40-56 - -41-31 - -55-84 - -56 '328 - 36.60 -- 37-45 --4 7 '1 5 --47 .739 -- 24' 98 -- 25' 74 - 44"56 -- 45' 39

10 - -22 .01 - -22"83 - -37 '31 - -3 8 '1911 -- 16"81 -- 17"57 -- 33 '3 4 -- 34-0 412 -- 11.09 -- 11 '75 -- 25.44 -- 26.331 3 --6" 68 -- 7-51 -- 24"06 -- 24"821 4 - - 5 - 1 7 - - 5 ' 7 1 - 1 8 "7 7 - - 1 9 .4 91 5 - - 0-36 -- 0-99 -- 15"68 -- 16-4616 - -12"22 - -12 ' 911 7 - - 8'65 -- 9.4318 --5"83 --6"3519 - -3"70 - -4 . 4220 -- t "77 -- 2'4 321 --0"13 --0-7 9


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Quantum chaos in ArHCl 471

Figure 5.

TEoL0

-4 0 -

-8 0 -

Ar HCI J =0

-1 2 0 -

1 .0 2. 0mH/a.m.u.

Vibrational energies as a function of hydrogen mass for ArH35C1 with J = 0.

This behaviour suggests that vibrational chaos in the higher states of ArHC1could be observed experimentally by studying the spectrum as a function of HC1excitation. However, the relatively small fluctuations in binding predicted willrequire experimental resolution well beyond that which has been achieved forthese systems.

Table 3 gives the vibrational levels of the major isotopic species of the ArHC1complex. To illustrate any erratic behaviour with respect to isotopic substitution,figure 5 shows the correlation of ArH35CI levels with those of ArD35C1 obtainedby systematically increasing the mass of hydrogen. The figure is qualitativelysimilar to figure 2 and suggests that one would indeed expect erratic isotope shiftsfor the weakly bound levels of the complex. This is in contrast to calculations ofKCN which indicate that the isotope shift is likely to be regular [2].

4 . CONCLUSIONSCalculations performed on the ArHC1 van der Waals molecule indicate that

the vibrational states which are bound by less than 40 cm -1 are chaotic. Thetransiti on to chaos thus occur s at abou t 2/3 of the way dissociation, this is similarto the figure suggested for conven tiona l molecules, such as SO 2 [37], on the basisof classical calculations, but considerably higher than the transition in the floppyKCN and LiCN molecules [1, 2].

Chaos in ArHC1 appears to be driven by the kinetic energy operator. This isin contrast to KCN and LiCN where chaos was found to be driven by the


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4 7 2 J . T e n n y s o np o t e n ti a l [-1 , 2 ]. A l t h o u g h m o d e l c a l c u la t io n s h a v e b e e n p e r f o r m e d o n p r o b l e m sw h e r e c h a o s i s c a u s e d b y k i n e t ic c o u p l i n g [ 8 ] , w e k n o w o f n o p r e v i o u s c a s e w h e r et h is h a s b e e n p r e d i c t e d f o r a p h y s i c a l s y s t e m .

F r o m t h e p o i n t o f v i e w o f e x p e r i m e n t , t h i s k in e t i ca l ly d r i v e n c h a o s w o u l da p p e a r t o h a v e t he a d v a n t a g e t h a t t h e r e a r e d i s ti n c t m e c h a n i s m s b y w h i c h c h a o sm a y b e i d e n ti f ie d , n a m e l y e r r a t i c b e h a v i o u r w i t h r e s p e c t to i s o t o p i c s u b s t i t u t i o na n d t h e d e g r e e o f H C 1 e x c i ta t io n . H o w e v e r , t h e c o m p l e x i t y o f t h e A r H C 1 s p e c-t r u m is s u c h [ 1 7 ] , t h a t t h e r e s o l u ti o n r e q u i r e d t o r e s o lv e s u c h e r ra t i c b e h a v i o u rm a y p r o v e d i f f ic u l t to a c h ie v e . T o t h is e n d f u r t h e r w o r k i s r e q u i r e d o n t h eo c c u r r e n c e o f v i b r a t i o n a l c h a o s in v a n d e r W a a l s s y s t e m s , f i rs t ly to d e t e r m i n ew h e t h e r i ts o c c u r r e n c e c a n b e p r e d i c t e d i n a n y s y s t e m a t i c w a y a n d s e c o n d l y tof in d e x p e r i m e n t a l l y f a v o u r a b l e c a s e s f o r in v e s t i g a ti o n .

I a m g ra t e f u l t o D r . S t a v r o s F a r a n t o s a n d J o a q u i n C a m a c h o f o r m a n y h e lp f u ld i s c u s s io n s d u r i n g t h e c o u r s e o f t h is w o r k .

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M ., a nd REUSS, J., 1981, Chem. Phys. , 63, 263.[13 ] TENNYSON, J., 1 981, Chem. Phys . Le t t . , 86, 181.[14] HOSE, G ., a nd TAYLOR, H . S. , 1982, J . chem. Phys. , 76, 5356.[15] RANK, D. H. , RAO, B. S . , an d WICGENS, T. A. , 196 2,J . c he m : Phy s . , 39, 2673.[16] MtZlOLEK, A. W ., a nd PIMENTEL, G . C . , 1976, J . chem. Phys . , 65, 4462.[17] BERNSTEIN,L. S., and WORMHOUDT,J . , 1984 , J . chem. Phys. , 80, 4630.[18] HUTSON,J. M ., a nd HOWARD, B. J . , 1981, da/lolec. P hy s. , 43, 493.[19] HUTSON, J . M ., an d HOWARD, B. J . , 1982, Molec . Phys . , 4 5 , 7 6 3 .[20] KIDD, I . F. , BALINT--KuRTI,G . G ., an d SHAPIRO, M ., 1981, J . chem. Soc . FaradayDiscuss., 7 1 , 2 8 7 .[21] ASHTON, C. J . , CHILD, M . S. , an d HUTSON, J . M ., 1983, J . chem. Phys . , 78, 4025.[22] HUTSON, J . M ., 1984, J . chem. Phys. , 81, 2357.[2 3] BROCKS, G ., VAN DER AVOIRD, A ., SUTCLIFFE,B. T . , and TENNYSON, J. , 1983, Molec.Phys . , 1025.[24 ] TENNYSON, J., 1983, Comput. Phys. Commun. , 29, 307.[25 ] TENNYSON, J., 1984, Comput. Phys. Commun. , 32, 109.[26] TENNYSON,J., an d SUTCLIEEE, B. T ., 1982, ft. chem. Phys. , 71, 4061.[27] BROWN, J. M ., HOIJCEN, T . J . , HURER, K .-P . , JOHNS, J . W . C. , K op p, I . , LEEEBVRE-BRm N, H ., MERER , A. J., RAMSAY, D . A ., ROSTOS, J., and ZARE, Z. N ., 1975, A~.molec. Spectrosc., 5 5 , 5 0 0 .


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Jonathan Tennyson- Quantum vibrational chaos in the ArHCI van der Waals molecule

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Transcript of Jonathan Tennyson- Quantum vibrational chaos in the ArHCI van der Waals molecule