Gonzalo Garcia De Polavieja et al- Coarse grained spectra, dynamics and quantum phase space structures of H3^+

8/3/2019 Gonzalo Garcia De Polavieja et al- Coarse grained spectra, dynamics and quantum phase space structures of H3^+

1/19

MOLECULAR PHYSICS, 1994, VOL. 83, No. 2, 361-379

C o a r s e g r a i n e d s p e c t r a , d y n a m i c s a n d q u a n t u m p h a s e s p a c es t r u c t u r e s o f H 3By GONZALO GARCIA DE POLAVIEJA

Departamento de Quimica, C-IX. Universidad Autdnoma de Madrid,Cantoblanco 28049, Madrid, SpainNICHOLAS G. FULT ON and JONATHAN TENNYSON

Department of Physics and Astronomy, University College London, Gower St.,London WC1E6BT, UK(Received 26 May 1994; accepted 13 June 1994)

A fully quantum-based assignment system is presented and applied to 3DH~ vibrational calculations. By analysing eigenstates in phase space using theHusimi function, dynamical information is used to find hidden localizations ingroups of eigenfunctions. Localization onto several types of periodic orbit isdescribed, including the excited horseshoe motion which has been suggested asan explanation of the features in the H~ photodissociation spectrum ofCarrington, A., and Kennedy, R. A., 1984,J. chem. Phys., 81, 91. However, otherlocalizations are found at the dissociation energy which should not be ignoredin the explanation of the Carrington-Kennedy spectrum.

I . I n t r o d u c t i o nIn this journal in 1982 Carrington, Buttenshaw and Kennedy announced

preliminary results for the infrared photodissociation spectrum of H~ [1]. Thisspectrum contained some 27 000 well resolved transitions in a 220 cm-1 region atabout 1000 cm-1. Subsequent work [2, 3] has confirmed this spectrum and shownit to vary strongly with isotopic substitution and the kinetic energy of the photo-dissociated proton. The result which has stimulated most interest is the observationthat the coarse-grained photodissociat ion spectrum shows four peaks each separatedby approximately 50 cm- 1.

The last decade has seen many theoretical attempts to interpret this spectrum[4-8] , based largely on classical or semi-classical dynamics. These studies have shownthat, in the H~ dissociation region, classical phase space is dominated by chaotictrajectories. However, one series of periodic trajectories, named the horseshoe, hasbeen discovered [4] and used to explain the coarse-grained regularity in theexperimental spectrum [5].

Semiclassical calculations confirmed the existence of the horseshoe states [6], andfull three-dimensional (3D) quantal techniques have now been developed which yieldenergy levels and wavefunctions for the H~ system near dissociation [9, 10]. Thehorseshoe states have been shown [11] to give high transition probabilities undercertain conditions.

Much effort has been denoted to the study of thow eigenfunctions reflect theinvariant structures of classical results, and this is the thrust of the persent work.For integrable systems, eigenfunctions are associated with classical invariant tori.

0026-8976/94 Sl0.00 9 1 9 9 4 Tay lo r & Fr an c i s L td


2/19

3 62 G . G . d e P o l a v ie j a e t a l .For non- in t egrab le sys t ems , no s ing le a ssoc ia t ion wi th the c l a ss i ca l i nva r i an ts t ruc tu res has been dev i sed .

Shn i re lman [123 p rov ed tha t t he e igenfunc t ions o f t he Lap lac ian on a com pac thype rbo l i c su r face became un i fo rmly d i s t r ibu ted in the semic la ss ica l l imi t excep t fo ra se t o f dens i ty ze ro . He l ton a nd T ab or dem ons t ra t ed [13] tha t fo r gene ra l sys t emse igenfunc t ions loca li ze on to the reg ions o f phase space a ssoc ia t ed w i th c l ass i ca linvar iant s t ruc tures .

A m o r e i n t u i t i v e a p p r o a c h i s d u e t o Vo r o s a n d B e r r y [ 1 4 ] , wh o p r e se n t e d t h emic roca non ica l hypo thes i s t ha t desc r ibes e igenfunc t ions a s e rgod ic - like d i s t r ibu t ions .T h i s h y p o t h e s is h a s b e e n t e s te d b y M a c D o n a l d a n d Ka u f m a n i n t h e s t a d iu m b i ll ia r d[15] a nd has been sh ow n to fail because o f t he ex i s tence o f enhanc ed am pl i tude a longclassica l unstable per iodic orbi t s [16] .

Us ing Gutzw i l le r ' s semicla ssi ca l Green ' s fo rmu la expressed a s a sum ove r pe r iod icorb i t s o f t he sys t em, Bog om olny [173 and Ber ry [18] h ave show n the e f fec t o floca l i za tion on uns t ab le p e r iod ic o rb i t s i n e rgod ic sys t ems in conf igura t ion and phasespace , respect ive ly .

M any app l i ca t ions to rea l sys t ems have been pe r fo rmed where e igens t a te s loca l izeon to c l a ss i ca l i nva r i an t s t ruc tu res ( to r i , can to r i , uns t ab le and s t ab le pe r iod ic o rb i t s ,e tc .) [19 , 203 . Recen t ly , i t has been show n tha t t he in fo rma t ion conce rn ing c l a ss i ca luns t ab le o rb i t s i n c l a ss i ca l ly chao t i c sys t ems i s con ta ined , i n gene ra l , i n g roups o fe igenfunc t ions , and a me thod to i so l a t e tha t i n fo rma t ion has been g iven [21] .E igenfunc t ions p rev ious ly named as e rgod ic - l ike func t ions can then be shown tocon ta in in fo rma t ion abou t l oca l i za t ion .Tw o-d im ens iona l t ime-dep enden t ca l cu la t ions o f i n it ia l ly loca li zed wa vepa cke t shave he lped our unde rs t and ing o f l oca l i za tion phe nom ena [22] . Seve ra l mo lecu leshave bee n s tud ied in th is w ay , i nc lud ing H 2 0 [ -23] and CO2 [24-] , bu t w i th one o f t heth ree v ib ra t iona l coord ina te s f ixed .

Us i n g i n f o rm a t i o n c o n t a i n e d i n th e sh o r t te r m q u a n t u m d y n a m i c s o f H ~ we h a v econs t ruc ted loca li zed wavefun c t ions on c l a ss ica l inva r i an t s t ruc tu res . T hese loca l i zedwav efunc t ions a re a ssoc ia t ed wi th the ban ds o f coa rse -g ra ined spec t ra . T ime-dependen t ca l cu la t ions a re pe r fo rmed us ing the fu l l 3D bound e igens t a t e ca l cu la t ionschoo s ing a s the in it ia l cen t re o f t he w avep acke t m axim a o f the e igenfunc t ions in theq u a n t u m p h a se sp a c e Hu s i m i r e p re se n ta t io n . An a u t o m a t i c a s s i g n m e n t o f e ig e n -func t ions is p rov ided fo r H ~ s t a r ting f rom the fu ll 3D d i sc re t e va r iab le represen ta t ion(DVR) ca l cu la t ion . No c l a ss i ca l gu idance i s necessa ry , a l though two-d imens iona l(2D) t r a j ec to r ie s a re sho wn to p rov ide co m plem enta ry ve r i fi ca tion o f t he re su lt s .

2. Theory2.1. Q u a n t u m p h a s e s p a c e d is t r ib u t i o n f u n c t i o n s

The so lu t ion o f a c l a ssi cal conse rva t ive N degrees o f f r eedom sys t em i s desc r ibedb y t h e p h a se t r a j e c t o r y ( q l ( t ) . . . . . q u ( t ) , p ~ ( t ) . . . . . p N ( t) ). Fr e q u e n t l y , 2 D d y n a m i cp r o b l e m s a r e s t u d i ed b y m e a n s o f t h e Po i n c a r 6 su r f a ce o f s e c ti o n ( PSOS ) , t h esuccessive in terac t ions of the ph ase s pace t ra jec tor ie s wi th a Sui table surface forseve ra l i n it ia l cond i t ions . G enera l sys t em s show a mixed P SO S of r egu la r Ko l -m o g o r o v , Ar n o l d a n d Mo se r ( KAM) t o r i a n d c h a o t i c s e a s .

The d i ff icu l ty o f r epresen t ing qu an tu m wav efunc t ions in phase space is due tothe unce r t a in ty p r inc ipl e . Ho w ever , func t ions re sembl ing phase space d i s t r ibu t ionshave been p roved to be use fu l i n quan tum mechan ica l s tud ie s .


3/19

Phase space analysis of H ~ 36 3Wig n e r [ 2 5 ] i n t r o d u c e d t h e f i r s t q u a n tu m p h a s e s p a c e d i s t r i b u t i o n , k n o w n b yh is n a m e , w h ic h f o r o n e d e g r e e o f f r e e d o m a n d f o r a p u r e s t a te h a s t h e f o r m

Pw (q, P) -- ~ dygJ*(q + y)Tt(q - y) e xp (2ipy/h). (1 )- o oO 'Co n n e l l a n d W ig n e r [ 2 6 ] s h o w e d th e p r o p e r t ie s o f th e d i s t ri b u t io n . A m o n g o th e r s,i t has the cor rec t marg ina l d i s t r ibu t ions .

f Pw dp = I ~ ( q ) l2,f Pw dq = (2)7'(p)[ z ,

w i thf Pw dp dq = 1.

However , the r e su l t can have nega t ive va lues . Wigner [27] has shown tha t any rea ld is t r ibu t ion w i th the co r rec t ma rg ina l d i s t r ibu t ions wi ll a lso ha ve n ega t ive va lues fo rsom e q and p d ue to the unc e r ta in ty p rinc ip le . A pos i tive d i s t r ibu t ion due to H us imi[ 2 8 ] c a n b e o b t a in e d b y G a u s s i a n s mo o th in g o f t h e Wig n e r d i s t ri b u t io n .1 f q)2 (p,i __ p )2 ])

Pn(q, P) = dp' dq' Pw (q',P') exp ( - r - (qi-~ + (3)\ L 2 (Aq) z 2 ( A p~ ] , ]1

- 2nh I (~ ' q P ( x ) ~ ( x ) ) 1 2 ' (4 )wi th ~9qp the m in im um wa vepa cke t tak ing the fo rm

1 i~tqp = (27c(Aq)2)TM e x p ( ~ p xw h e r e

( X - - q ) 2 ~4-(Aq ~ / t ' ( 5 )

S S ~ I .

This d i s t r ibu t ion can be in te rpre ted a s the p roba b i l i ty o f f ind ing the sys tem in a boxcen tred on (q, p) o f size h/2. T a k a h a s h i [ 2 9 ] a n d H a r r i m a n a n d C a s i d a [ 3 0 ] h a v eshow n tha t W igner d i s t r ibu t ions r evea l a qu i te v io len t na tu re , bu t , on th e o the r hand ,the Hus im i has a g oo d cor re s pon den ce wi th coa r se -gra ine d c la ssica l ca lcu la tions .2.2. W avep acket dynam ics, pow er spectra and localization of wavefuntions

Q u a n t u m w a v e p a c k e t d y n a m i c s f o r a b o u n d e d s y st e m c a n b e w r i t te n i n t er m s o fth e eigenfunctions as~ ( t ) = ~ In>(nl~k(O)>exp(--i~" t ) . = o (6 )

w h e r e t h e in i ti a l w a v e p a c k e t ~ ( 0 ) c a n b e t a k e n t o b e t h e h a r m o n ic o s c i ll a to r c o h e r e n ts ta te , and In> i s the n th e igens ta te . Apply ing the Four ie r t r ans form to bo th s ides o f


4/19

3 64 G . G . d e P o l a v i e ja et al.(6 ) , w e c a n ob ta in the ru th e ige ns ta t e ( fo r h = 1 ) :

l i m d t 1 0 ( t ) ) e x p (iE~t) = I rn )


5/19

P h a s e s p a c e a n a ly s i s o f H ~ 3652.3. J a c o b i H a m i h o n i a n a n d H u s i m i f u n c t i o n s f o r t ri a t or n ic s

I n J a c o b i c o o r d i n a t e s c o n f i g u r a ti o n s o f a t r ia t o m i c c a n b e r e p r e s e n t e d b y t w ol e n g th s a n d a n a n g l e . T h e f i rs t l e n g t h r l d e s c ri b e s t h e d i s t a n c e b e t w e e n t w o o f th ea t o m s o f m a s s m 2 a n d m 3 . T h e s e c o n d l e n g t h r2 d e s c r i b e s t h e d i s t a n c e o f t h e t h i r da t o m , m a s s m l , f r o m t h e c e n t r e o f m a s s o f t h e o t h e r t w o . F i n a l ly , 0 d e sc r i b es t h ea n g l e b e t w e e n r I a n d r 2 . I n t h e s e c o o r d i n a t e s t h e c l a s s ic a l r o t a t i o n l e s s H a m i l t o n i a nc a n b e w r i t t e n a s

prH = - - + P ~ + + + V ( r ~ , r 2 , 0) ,2 / /1 2 / / 2 2 -w h e r e t h e r e d u c e d m a s s e s a r e

(12)

/ /1 1 = (m~- i + m~- 1) , / ./21 = ( m l 1 + (m2 + m 3 ) - 1 ) . (13 )T h e J a c o b i H u s i m i d i s t ri b u t io n c a n b e w r i t te n a s th e m o d u l u s s q u a r e d o v e r l a p o ft h e m i n i m u m w a v e p a c k e t ~r'~,r~,o' ,p;~,v;~,pb w i t h t h e e i g e n f u n c t i o n q ) i :

p n ( r ' l , r ' z , 0 ' , p ' r~ , P ' r~ , P 'O ) = ] ~ , ; , ~ , O ' , p ; , , p ; , , ~ l r (14)T h e e i g e n f u n c t io n s u s e d i n th is p a p e r a r e o b t a i n e d f r o m D V R c a l c u la t i o n s [ 1 0 ]. T h i sm e t h o d r e p r e s e n t s t h e w a v e f u n c t i o n s as a m p l i t u d e s a t p o i n ts , t h e p o i n t s b e i n g c h o s e na s t h e G a u s s i a n q u a d r a t u r e p o i n t s o f t h e b a s is fu n c ti o n s. T h e m i n i m u m w a v e p a c k e tis c o m p u t e d u s i n g t h e G a u s s i a n q u a d r a t u r e p o i n t s o f t h e D V R w a v e f u n c t i o n , w h i c hm e a n s t h a t t h e i n t e g r a t i o n is a s im p l e s u m o v e r a g r i d o f D V R p o i n t s . T h e s i ze o fthe g r id i s n~, nr~ no , w h e r e n ~, is t h e n u m b e r o f p o i n t s i n r l , nr2 t h e n u m b e r o fp o i n t s i n r 2 a n d n o t h e n u m b e r i n t h e a n gl e . T h i s i n t e g r a t i o n i s a s s u m e d t o b e a c c u r a t ef o r e n e r g i es o f t h e w a v e p a e k e t w h i c h a r e c l o s e t o t h e e i g e n e n e r g i e s o f t h e w a v e -f u n c t i o n s .

T h e H u s i m i v a l u e f o r a p o i n t i s t h e r e f o r e c a l c u l a t e d a s

pn (r ' l , ' , , =2 , 0 ' , p~a , p~ , , P 'o ) OPi ( r rJ2 , ok )i = 1 j = l k 1i j 2 .x O~ ; , r~,o , ,p ; , ,p ;~ ,pb(r~, r2 , O )

S i n c e t h e 0 c o o r d i n a t ec o o r d i n a t e s , q l , q 2 a n d

(15)

m o m e n t u m i s c o u p l e d t o t h e ra a n d r2 c o o r d i n a t e s , t h r e eq 3 a n d t h e ir c o r r e s p o n d i n g m o m e n t a a r e d e f in e d a s fo ll ow s :

q l = F 1 ,q 2 = r z c o s ( 0 ) ,q3 = r2 sin (0) ,

P q l = P r l ,P ~2 = (P r 2 + p g ) : / 2 c o s 0 - t a n - 1 \ P ~ / A

p q 3 - - ( p ~ 2 + p g ) a / E s i n I O - t a n - l ( P ~ ) 1 .

(16)


6/19

3 6 6 G . G . d e P o l a v ie j a e t a l .S o t h e m i n i m u m w a v e p a c k e t is n o w d e fi n ed a s

~ ; , ~ , o ' , ~ , , p ; 1 . p ~ - - e x p ( i [ q l p q l + q z P q 2 + q 3 P o 3 ] )x exp ( [ (q l - q , ) 2 + ( q z - q~)2 + (q3 _ q ; )2 ] / 2 ) ' ( 17 )

t t tw h e r e q l , q z a n d q3 a re t h e t r a n s f o r m e d w a v e f u n c t i o n s D V R p o i n t s , a n d q a , q 2, q 3 ,P'q,, P'q2 a n d p'q3 a r e t h e t r a n s f o r m e d c o o r d i n a t e s a n d m o m e n t a d e f in i n g th e p o s i t i o na n d s h a p e o f t h e m i n i m u m w a v e p a c k et .

3 . R e s u l t s

I n t h i s p a p e r w e p r e s e n t r e s u l t s o b t a i n e d m a i n l y w i t h t h e H f p o t e n t i a l e n e r g ys u r fa c e o f M e y e r , B o t s c h w i n a a n d B u r t o n ( M B B ) [ 3 3 ] . W e h a v e a n a l y s e d fu ll 3 De i g e n v e c t o r s c a l c u l a t e d u s i n g th e D V R 3 D p r o g r a m [ 3 2 ] i n J a c o b i c o o r d i n a t e s . T h ed e t a i ls o f t h e c a l c u l a t i o n s c a n b e f o u n d i n [ 1 0 ] , w h i c h p r e s e n t s t h e e n e r g i e s a n ds y m m e t r y a s si g n m e n t s o f a ll t h e b o u n d s t at e s o f H r . W e u s e d e i g e n v e c t o r s f r o me x t e n d e d c a l c u l a t io n s w h i c h h a s f i na l H a m i l t o n i a n s o f si ze 8 5 0 0 a n d 1 0 0 0 0 f o r th ee v e n a n d o d d s y m m e t r ie s , r es p ec ti ve ly . A l t h o u g h H f h a s D 3 h s y m m e t r y t h e D V R 3 Dp r o g r a m u s e s C z v s y m m e t r y . T h i s s p l i t s t h e c a l c u l a t i o n s i n t o t w o s y m m e t r y b l o c k sb y c o n s i d e r i n g o d d a n d e v e n p a ri ti e s i n t h e a n g u l a r c o o r d i n a t e . M o s t o f t h e r e su l tsp r e s e n t e d h e r e a r e w i t h t h e a n g l e f i x e d o n t h e C zv s y m m e t r y p l a n e , 0 = 9 0 ~ w i t h n om o m e n t u m in t h e a n g le . T h i s m e a n s t h a t o n l y e v en s y m m e t r y e i g e n v ec t o rs n e ed t ob e s tu d ie d ; t he o d d s y m m e t r ie s r e q u ir e n o n - z e r o 0 m o m e n t u m f o r n o n - z e r o o v e r la po f t h e w a v e p a c k e t . F o r t h e o d d s ta te s , e q u i v a l e n t c u ts c a n b e p e r f o r m e d w i t h n o 0m o m e n t u m b y p e r m u t i n g t h e a t o m s s o t h e c u t n o l o n g e r l i e s a l o n g t h e s y m m e t r yp l a n e ; m o r e d i s c u s s i o n o f t h is c a n b e f o u n d i n s e c t i o n 3 .4 .

3.1. S e l e c t i o n o f in i t i a l p o i n t sA s d i s c u s s e d i n s e c t i o n 2 .2 , h i g h v a l u e s o f t h e H u s i m i f u n c t i o n , e q u a t i o n s ( 1 4 )

a n d ( 15 ), c a n b e c h o s e n a s c en t re s o f w a v e p a c k e t p r o p a g a t i o n . T h e H u s i m i h a s2 N - 1 d i m e n s i o n s , w h e r e N i s t h e n u m b e r o f w a v e f u n c t i o n d im e n s i o n s . O n ed i m e n s i o n i s l o s t s in c e e n e r g y m u s t b e c o n s e r v e d . W e h a v e s t u d i e d w a v e p a c k e tp r o p a g a t i o n w i t h i n it ia l p o i n t s c e n t r e d a t 0 = 9 0 ~ Po = O, r z = 10 .5 a o , Pr2 be ingo b t a i n e d f r o m e n e r g y c o n s e r v a ti o n .

I n f i g u r e 1 ( a ) w e p r e s e n t t h e r e s u l t f o r a f r e q u e n t p a t t e r n s e e n w i t h a c u t t h r o u g ht h e H u s i m i f u n c t i o n , u s i n g t h e p h a s e s p a c e s l i c e d e s c r i b e d a b o v e . I t i s c l e a r t h a tt h e r e i s a p e a k , r e p r e s e n t e d b y t h e d a r k e r r e g i o n , w h i c h i s c e n t r e d a t P r, = - 0 . 3 4 a u ,r l = 5 " 0 7 a o. F i g u r e 1 ( b ) s h o w s t h e e i g e n f u n c t i o n p l o t f o r a c u t a t 0 = 9 0 ~ T h i se i g e n s ta t e , t h e 5 6 7 t h e v e n s t a te w i t h a n e n e r g y 3 3 2 9 6.3 c m - 1 a b o v e t h e g r o u n d s ta t e,c o r r e s p o n d s t o t h e w e l l s t u d i e d h o r s e s h o e m o t i o n [ 4 , 6, 1 1 ]. F i g u r e 2 ( a ) p r e s e n t s t h es a m e H u s i m i c u t f o r a d if f e re n t r e c u r r i n g p a t t e r n , a g a i n w i t h t h e e i g e n f u n c t i o n p l o t t e di n fi g u r e 2 ( b) . T h i s i s th e 6 6 1 s t e v e n p a r i t y s t a t e w i t h a n e n e r g y 3 4 7 4 3 .2 c m - 1 a b o v et h e g r o u n d s t a t e . T h e r e a r e t w o c l e a r p e a k s , t h e h i g h e r o n e c e n t r e d a t Pr , = 0 . 3 4 a u ,r l = 3 .7 5 a o , t h e o t h e r a t p r, = 0 " 0 , r l = 2 .6 5 a o . A l t h o u g h t h e e i g e n f u n c t i o n d o e s n o ts h o w s u c h c l e a r l o c a l i z a t i o n , t h e r e is s o m e s t r u c t u r e a r o u n d r 2 = 2 .8 a o .


7/19

P h a s e s p a c e a n a l y s i s o f H ~ 3 6 7

5 . 0 -

2 . 5 -

( a )

. . . . . . . % .....

( b )

0 . 0 , , ~ i I- 2 0 - 1 0 0 1 0 2 0 1 2 3 4 5 6

p ~ l / a . u r 2 / a oF i g u r e 1 . ( a ) r~ v e r s u s P r, H u s i m i p l o t o f t h e 5 6 7 t h e v e n p a r i t y H ~ w a v e f u n c t i o n w i t h

a n e n e r g y 3 3 2 9 6 '3 c m 1 f r o m t h e g r o u n d s t a t e. F i x e d c o o r d i n a t e s a r e 0 = 9 0 ~ Po = 0,r 2 = 1 0 5 a o a n d P ,2 o b t a i n e d f r o m t h e e n e r g y . D a r k e r r e g i o n s i n t h e p l o t i n d i c a t e h i g h e ra m p l i t u d e s . ( b ) r 1 v e r s u s r 2 f o r th e s a m e w a v e f u n c t i o n a n d 0 = 9 0 ~ P o s i t i v e a n d n e g a t i v ea m p l i t u d e s a r e i n d i c a t e d w i t h t h e s o l i d a n d b r o k e n c o n t o u r s , r e s p e c t i v e l y , a n d t h e f a i n tl i n e i n d i c a t e s t h e c l a s s ic a l t u r n i n g p o i n t f o r t h e e n e r g y .

.-......

2 . 5 -

( a ) I - ". . . . (b)::::::~

.. .. .. .. ....... iiiiiii

0 o 0 "

- 2 0 - 1 0 0 1 0 2 0 1 2 3 4 5 6P r , / a . u r 2 / a 0

F i g u r e 2 . ( a ) r l v e r s u s P rl H u s i m i p l o t o f t h e 6 6 1 s t e v e n p a r i t y H ~ w a v e f u n c t i o n w i t ha n e n e r g y 3 4 7 4 3 .2 c m - 1 f r o m t h e g r o u n d s t a te . F i x e d c o o r d i n a t e s a r e 0 = 9 0 ~ Po = O,r 2 = 1 0 - 5 a o a n d P r2 o b t a i n e d f r o m t h e e n e r g y . D a r k e r r e g i o n s in t h e p l o t i n d i c a t e h i g h e ra m p l i t u d e s . ( b) r l v e r s u s r 2 f o r th e w a v e f u n c t i o n a n d 0 = 9 0 ~ C o n t o u r s a s i n f i g u re 1.

3 .2 . C o r r e l a t i o n f u n c t i o n s a n d p o w e r s p e c t raR e s u l ts c o n c e r n i n g w a v e p a c k e t p r o p a g a t i o n c a n b e s t u d i e d u s i n g t h e c o r r e l a t i o n

f u n c t i o n , e q u a t i o n (8 ). F i g u r e 3 ( a , b ) p r e s e n t s t h e c o r r e l a t i o n f u n c t i o n s f o r w a v e -


8/19

3 6 80.3-

G . G . d e P o l a v i e j a e t a l .

0.2

0.1

0.075

0 . 0 5

(a )

( b )

0 . 0 2 5

0 . 00 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0

T i m e / f sFi g u r e 3 . C o r r e l a t i o n f u n c t io n s f o r ( a ) H + e v e n p a r i t y w a v e p a c k e t w i t h m o m e n t aPrl = -0 .3 4 au , P ,2 = 7 .62 au , Po = 0 cen t red a t r 1 = 5 '07 ao , r2 = 10 .5 ao , 0 = 90 ~ a nd(b) H + even pa r i ty wa vepac ke t wi th m om en ta Pr, = 0 .34a u , Pr2 = 15 .40 au , Po = 0cen t red a t r I = 3.75 ao , r2 = 10 -5 ao , 0 = 90 ~ N ot e tha t the ver t ica l sca le is d if ferentfo r ( a) and (b) ; bo t h h ave an o f f - sca le m ax i m um a t 1 .0 fo r the T = 0 peak . R ecu r rencesi n t h e w a v e p a c k e t p r o p a g a t i o n a p p e a r a s p e a k s .

p a c k e t s c e n t r e d a t p o i n t s P r , = - 0 " 3 4 a u , r 1 = 5 ' 0 7 a o a n d pr~ = - 0 " 3 4 a u , r 1 = 3 .7 5 a of o u n d i n t h e p r e v i o u s s e ct io n . A l t h o u g h t h e y b o t h s h o w d e c a y , i n fi g u re 3 ( a) t h er e c u r r e n c e o f p e r i o d T = 18 f s f o r s h o r t t i m e s g i v e s ri se t o b r o a d p e a k s ; h o w e v e r ,a f t e r a l i fe t i m e ~ ~ 7 0 fs a m o r e c o m p l i c a t e d s t r u c t u r e a r is e s . I n f i g u r e 3 ( b) a l o n gw i t h t h e r e c u r r e n c e o f p e r i o d T = 1 8 fs t h e r e i s a s h o r t e r p e r i o d r e c u r r e n c e T = 1 6.5 fs,a n d a g a i n t h e d e c a y h a s a l if et im e z ~ 7 0 fs. T h e c o r r e s p o n d i n g p o w e r s p e c tr a ,e q u a t i o n ( 9) , a r e s h o w n i n f i gu r e 4 ( a , b ) . T h e l o w e r s p e c t r u m , f i g u r e 4 ( c) , is d u e t oL e S e u e r e t a l . [ 1 1 ] , a n d s h o w s t h e E i n s t e i n A c o e f f i c i e n t o f t r a n s i t i o n f r o m a ll t h eh i g h e r e v e n v i b r a t i o n a l s t a t e s t o t h e s e c o n d e x c i t e d e v e n - p a r i t y s t a t e , a s s i g n e d ( 0 , 1 1 ) ,3 1 7 8. 36 c m - 1 a b o v e t h e g r o u n d s ta te .

T h e s i m i l a r i t y o f t h e p o s i t i o n o f t h e p e a k s i n f i g u r e 4 ( a ) a n d ( c ) is c o n v i n c i n g ,a n d c o n f i r m s t h e i d e a t h a t t h e h ig h p r o b a b i l i t y t r a n s i t i o n s f r o m a l o w l y in g s t a te a r ed u e t o h o r s e s h o e s ta te s . W e h a v e a l so s u p e r i m p o s e d , o n t h e p o w e r s p e c t ra , th es m o o t h e d s p e c t r u m , e q u a t i o n ( 1 0 ), f o r t i m e T = 2 4 fs, c h o s e n a s a t i m e s l ig h t l y g r e a t e rt h a n t h e f i r st p e a k i n t h e c o r r e l a t i o n f u n c t i o n , f i g u r e 3 ( a) . F o r f i g u r e 4 ( b) a t l o we n e r g y t h e p e a k s c o i n c i d e w i t h t h o s e i n f ig u r e 4 ( a) , b u t a t h i g h e r e n e r g i e s t h e p e a k sa r e s e p a r a t e d b y a l a rg e e n e r g y g a p , w h i c h g i ve s t h e s h o r t e r p e r i o d r e c u r r e n c e s ee ni n f i gu r e 3 ( b ). A s m o o t h e d s p e c t r u m is a d d e d , e q u a t i o n ( 10 ), a g a i n f o r a t i m eT = 2 4 fs. A s d i s c u s s e d i n s e c t i o n 2 . 2, s h o r t r e c u r r e n c e s i n t h e c o r r e l a t i o n f u n c t i o n


9/19

Pha se space analysis o f H ~ 3 6 9

0 .75

~ ' 0 .5

0 . 2 5

0.020 22 24 26 28 30 32 34 36 38 40

E n e r g y f r o m g r o u n d s t a t e / 10 3 c m ~F i g u r e 4. P o w e r s p e c tr a f o r H ~ e v e n p a r i t y w a v e p a c k e t s w i t h ( a) m o m e n t a p ~, = - 0 . 3 4 a u ,

Pr2 = 7.63 au, Po = 0 ce n t r ed a t r 1 = 5 '0 7 a o , r 2 = 1 0 -5 ao , 0 = 9 0 ~ an d (b ) P r l = 0 . 3 4 au ,Pr2 = 15.40 au, Po = 0 cen t r ed a t r l = 3 '7 5 a o , r 2 = 1 0 5 ao ' 0 = 9 0 ~ A l s o p l o t t e d i s t h es m o o t h e d s p e c t r u m f o r a t i m e T = 24 fs . ( c ) S h o w s E i n s t e i n A c o e f f i c i e n ts fo r H ~t r a n s i t i o n s t o t h e ( 0, 1 1) s t a te f r o m h i g h e r e v e n p a r i t y v i b r a t i o n a l s t a t e s [ 1 1 ] . N o t e t h ec l o s e c o i n c i d e n c e o f p e a k s i n a ll t h r e e s p e c t r a f o r e n e r g ie s le ss t h a n 3 0 0 0 0 c m - 1 . A b o v e3 0 0 0 0 c m 1 t h e p e a k s i n p l o t (b ) d i f fe r f r o m t h e o t h e r t w o . T h e s m a l l a r r o w m a r k sd i s s o c i a t i o n ; i t i s c l e a r th a t t h e p e a k s c o n t i n u e a b o v e t h u s e n e r g y . T h e n u m b e r s o n t h ep e a k s i n ( a) a n d ( b) m a r k t h e p e a k s w h i c h w e r e c o m b i n e d t o f o r m l o c a l iz e d w a v e -f u n c t i o n s , f i g u r e 5 .

g i v e r i s e t o b a n d s t r u c t u r e i n t h e p o w e r s p e c t r a , w i t h p e a k s e p a r a t i o n 2 n k / T , w h e r ec o n v e r s i o n f r o m w a v e n u m b e r s t o f e m t o s e c o n d s g i v e s k = 5 3 0 8 .7 2 c m - 1 f s - 1 . T h ew i d t h o f a b a n d i s r e l a t e d t o t h e l i f e t im e s i n a s i m i l a r f a s h i o n .

3 .3 . Localized wavefunctionsA u t o m a t i c a s s i g n m e n t o f t h e b a n d s i n t h e c o a r s e - g r a i n e d p o w e r s p e c t r a c a n b e

p e r f o r m e d u s i n g th e b a n d w a v e f u n c t i o n s d e f i n e d i n e q u a t i o n (1 1) . U s i n g t h e


10/19

3 7 0 G . G . d e P o l a v i e ja e t a l .

t .

5 . 0 -

2 . 5

0.0-

5 . 0 '

2 . 5 -

0.025.02

2.52

0 . 0 - -

1 2

4 5 6

5 . o - ~ : r ~ o2.52

0 . 0 - -5-02 ~ ~ IPr ';"

2.5 2 i

0.0 2.5 5.0 0.0 2.5 5.0 0.0 2.5 5.0

r~ /ao

3 ( a )

(b)

Figure 5 . Band wavefunctions cor respo nding to the numb ered peaks in the power spectraare sh ow n in plots (a) an d (b) for f igure 4(a ) and (b ) , respectively. In (a) thewavefunct ions are horseshoes and a classical ho rsesho e t rajectory is added with anenergy cor responding to the cent re of the band. Th e c lass ica l turning point a t the ban dcen tre is also given. Th is shows the m otio n is localized and does n ot fil l all the availablespace. In (b) n e w 'e lepha nt foot ' wavefunct ions are shown and again the correspo ndin gclassical t rajectory is added. The local izat ion is different from the horseshoe motion,being wider and flatter.

a p p r o x i m a t i o n o f c o m b i n i n g o n l y t h e s ta t e s u n d e r a s in g le b a n d w e h a v e u s e dl i f e t i mes z = 30 f s an d z = 22 f s f o r f i gu re 4 , pa r t s ( a ) an d (b) , re spec t i ve l y . F i gu re5 ( a ) p r e s e n t s t h e b a n d w a v e f u n c t i o n s c o r r e s p o n d i n g t o n i n e b a n d s , n u m b e r e d i ne n e r g y o r d e r , in t h e h o r s e s h o e s m o o t h e d p o w e r s p e c t r u m s h o w n i n fi g u re 4 ( a) . W eh a v e p e r f o r m e d c l a ss ic a l tr a j e c t o r y a n a ly s i s o f H + u s i n g t h e M B B p o t e n t i a l e n e r g ys u r f a c e , a n d t h e c la s s ic a l h o r s e s h o e p e r i o d i c o r b i t i s. a d d e d t o t h e b a n d w a v e f u n c t i o n s ,i ts e n e r g y b e i n g t a k e n a s t h e c e n t r e o f t h e b a n d . F u l l c l a s si c a l a n a l y s i s i n c l u d i n gb i f u r c a t i o n d i a g r a m s [ 3 4 ], P o i n c a r 6 s u r fa c e s o f s e c t io n a n d o t h e r p e r i o d i c o r b i t s w i llb e p r e s e n t e d e l s e w h e r e [ 3 5 ] .

T h e w a v e f u n c t i o n c a n b e s e e n t o l o c a l i z e s t r o n g l y a l o n g t h e p e r i o d i c o r b i t ;h o w e v e r , a t e n e rg i es a b o v e d i s s o c i a ti o n , 3 6 3 9 7 . 1 c m - 1 a b o v e t h e g r o u n d s ta t e,


11/19

P h a s e s p a c e a n a l y s is o f H ~ 371

2 . 5 -

0 . 0 -

(b )

0 . 0 '0.0 2 . 5 5 . 0 0 . 0 2 . 5 5 . 0 - 15 0 15 - 15 0 15

r 2 / a o p ~ / a . uFigure 6 . Th e four s ta tes g iv ing the h ighes t cont r ibut ion to pea k 3 in f igure 4 (b) . The y areeven par i ty s ta tes nu mb ers 647, 657, 661 and 663, wi th energies 34 552 '0 , 34673 '3 ,34 743 '2 a nd 34 770.4 c m - 1 abo ve the groun d s ta te . The wavefunc t ion p lo ts (a) do notseem to ha ve m an y similari ties. In plot (b) the r I versus Pr, H usim i plots are s how n forthe four states, with f ixed co ord inate s 0 = 90 ~ Po = 0, r 2 = 10 .5 a o a nd Pr2 ob taine dfrom the energy. The four Husimi p lo ts do have s imi lar s t ruc ture wi th two peaks a tPr, = 0, r x ~ 3"75 a 0 and r I ~ 2'5 a 0.m a r k e d b y a n a r r o w o n t h e p o w e r s p e c t r a , t h e q u a n t u m b e h a v i o u r c a n n o t b ee x p l a i n e d a d e q u a t e I y a s l o c a l i z a t i o n o n a s in g l e p e r i o d i c o r b i t [ 3 5 ] . A l s o , o u r q u a n t a lr e s u l t s a b o v e t h e d i s s o c i a t i o n l i m i t s h o u l d b e c o n s i d e r e d w i t h c a u t i o n , a s t h ew a v e f u n c t i o n s c a l c u l a t e d a t t h e s e e n e r g i e s a r e b o u n d , s i n c e o u r D V R c a l c u l a t i o n su s e d a f in i te g r i d a n d t h e r e f o r e d o n o t i n c lu d e t r u e c o n t i n u u m c o n t r i b u t i o n s .F u r t h e r m o r e , c o m p a r i s o n s w i t h se m i - c l a s s ic a l d e n s i t y o f s t a t e c a l c u l a t i o n s [ 3 6 ]s u g g e s t t h a t t h e D V R c a l c u l a t i o n s a r e s i g n i fi c a n tl y le ss w el l c o n v e r g e d i n t h is r e g i o n .

F i g u r e 5 ( b ) p r eg e n ts b a n d w a v e f u n c t i o n s c o r r e s p o n d i n g t o s ix b a n d s o f t h es m o o t h e d p o w e r s p e c t r u m i n f i g u r e 4 ( b ) . A g a i n a c l a s s i c a l p e r i o d i c o r b i t i sa d d e d . T h i s n e w p e r i o d i c o r b i t , w h i c h w e h a v e c h r i s t e n e d t h e e l e p h a n t f o o t a si t is f a t t e r t h a n a h o r s e s h o e , a r i se s f r o m a b i f u r c a t i o n , a t ~ 2 8 3 0 0 c m - 1 a b o v et h e g r o u n d s t a t e [ 3 5 ] . T h e q u a n t u m b e h a v i o u r a g a i n l o c a l i z e s s t r o n g l y o n t h ep e r i o d i c o r b i t , a n d a p p e a r s t o s t a y l o c a l i z e d a b o v e d i s s o c i a t i o n .

I n f i g u r e 6 ( a ) t h e f o u r e i g e n s t a t e s c o n t r i b u t i n g m o s t t o b a n d 3 in f i g u r e 4 ( b) a r es h o w n . F i g u r e 5 ( b) s h o w s t h a t t h e b a n d g i v es ri se t o a s t r o n g l y l o c a l i z e d w a v e -f u n c t io n ; h o w e v e r : t h e e i g e n s t a t e s d o n o t h a v e v e r y c l e a r l o c a l i z a ti o n . I n t h eH u s i m i d i s t r i b u t i o n o f th e s e s t a t e s , f i g u r e 6 ( b ) , f o r a c u t a t 0 = 9 0 ~ Po = O,r2 = 1 0 - 5 a o it is c le a r t h a t t h e y s h a r e t h e s a m e m a j o r f e a t u r e s w i t h m a x i m a a tP r, = 0 " 0, r~ = 3 .4 % , a n d p , , = 0 .0 , r l = 2 .7 5 % . T h e e x i s t e n c e o f t w o m a x i m a i n a llo f t h e p l o t s i s d u e t o t h e f a c t t h a t t h e b i f u r c a t i o n g a v e r i se t o t w o , s l i g h t l y d i f f e re n tp e r i o d ic o r b it s , w h i c h b o t h g i v e m a x i m a o n t h e H u s i m i p l o ts . W e h a v e s t u d i e d p o w e rs p e c t r a f o r a w a v e p a c k e t c e n tr e d o n o n e o f t h e m a x i m a w h i c h g i v e s a c l e ar p o w e rs p e c t r u m . T h e l o c a li z ed w a v e f u n c t i o n s f o r th e o t h e r m a x i m u m c a n b e f o r m e d , b u t t h el o c a l i z a t i o n i s l es s s t r o n g . T h i s d e c o u p l i n g o f p e r i o d i c o r b i t s f r o m t h e r e s t o f p h a s es p a c e i n q u a n t u m m e c h a n i c s h a s b e e n s tu d i e d i n th e t im e i n d e p e n d e n t p i c t u re b ym e a n s o f F e s h b a c h t h e o r y [ 3 7 ] .


12/19

3 72 G . G . d e P o l av i ej a et al .N o t e t h a t i n b o t h H u s i m i f u n c t i o n p l o t s t h e m a x i m u m a m p l i t u d e d i d n o t o c c u r

e x a c t l y a t Prl = 0 "0 w h i c h w e w o u l d e x p e c t f r o m t h e s h a p e o f t h e l o c a l i z a t i o n . T h ea m o u n t o f c o m p u t e r t im e t h a t i t t a k es t o c o m p u t e t h e H u s i m i f u n c t i o n m a d es a m p l i n g v e r y l a rg e n u m b e r s o f p o i n t s p r o h i b i t i v e l y e x p e n si v e , a n d t h e r e w a s n o ta s a m p l i n g p o i n t a t P r, = 0 .0 . If t h e r e g i o n a r o u n d t h e p e a k is i n v e s t i g a t e d i n m o r ed e t a i l, t h e m a x i m u m d o e s o c c u r a t P rl = 0 " 0 ; h o w e v e r , t h e d i f fe r e n c e d o e s n o t a f f e c tt h e p o w e r s p e c t r u m o r t h e s h a p e o f t h e l o c a l iz e d w a v e f u c ti o n s .

W e h a v e f o u n d s im i l ar l o c a l iz a t i o n i n th e d i a t o m i c s i n m o l e c u l e ( D I M ) [ 3 8 ] H ~p o t e n t i a l e n e r g y s u r f a c e t o t h o s e p r e s e n t e d h e r e , w i t h q u a l i t a t i v e h o r s e s h o e a n de l e p h a n t f o o t f e a t u r e s b e i n g r e p r o d u c e d . C l a s s i c a l t r a j e c t o r y s t u d i e s a l s o l e a d t o t h es a m e c o n c l u s i o n t h a t d e t a il s o f t h e p o t e n t i a l d o n o t a f f ec t t h e b a n d l o c a l i z a ti o n a st h e d y n a m i c s a r e a f fe c te d b y m a i n l y t h e g e n e r a l t o p o l o g y o f t h e p o t e n t ia l . C l a s s ic a lc a l c u l a t i o n s u s i n g a v e r y s i m p l e p o t e n t i a l f o r m e d f r o m t h r e e M o r s e o s c i l l a t o r sb e t w e e n t h e a t o m s d i s p l a y t h e s a m e q u a l i t a t i v e f e a t u r e s ; s h o w i n g t h a t t h e h o r s e s h o ea n d e l e p h a n t f o o t p e r i o d i c o r b i t s a r e r o b u s t , a n d i n s e n s i t i v e t o c h a n g e s i n t h ep o t e n t i a l .

3.4. S y m m e t r y c o n s i d e r a t i o n sF o r a ll th e r e s u l ts p r e s e n t e d s o f a r w e h a v e f i x e d th e J a c o b i a n g l e t o 9 0 ~ B y d o i n g

s o w e c o n s i d e r o n l y t h e m o t i o n i n o n e o f t h r e e e q u i v a l e n t s y m m e t r y p la n e s d e f i n e db y p e r m u t a t i o n s o f t h e a t o m s . T h i s h a s t h e e f f e c t o f p r e f e r e n t i a l l y s e l e c ti n g E s t a t e so v e r A 1 s ta t e s w h e n t h e y h a v e s i m il a r t y p e s o f w a v e f u n c t i o n l o c a li z a ti o n . A s a ne q u i l a t e r a l t r i a n g l e H ~ h a s a d e g e n e r a t e b e n d m o d e i n t h e n o r m a l m o d e m o d e l ,w h i c h g i v e r is e t o a n l q u a n t u m n u m b e r t o r e p r e s e n t th e v i b r a t i o n a l a n g u l a rm o m e n t u m . A t en e r g ie s b e lo w l i n e a r it y , 9 9 1 2 c m - 1 a b o v e t h e g r o u n d s t at e , l is ar e a s o n a b l y g o o d q u a n t u m n u m b e r , a n d a n a l y si s o f H ~ a n d D ~ a t t h es e e n e rg ie s h a sb e e n p e r f o r m e d h a s b e e n [ 3 9 ] s h o w i n g m a n y o f t h e f ea t u r es p r e d ic t e d b y t o p o l o g i c a lc o n s i d e r a t i o n s . H o w e v e r , f o r th e e n e r g ie s e x a m i n e d h e r e t h e m o d e l is n o t s o s i m p le ,b u t s y m m e t r y e f fe c ts a r e s t il l a p p a r e n t .I f t h e p o w e r s p e c t r u m s h o w n i n f ig u r e 4 ( a) is e x a m i n e d c l os e ly o n e c a n s ee t h a tt h e h o r s e s h o e b e h a v i o u r is c o n t a i n e d i n s e v e ra l st at es . I f t h e h o r s e s h o e H u s i m i p o i n tP rl = - 0 . 3 4 a u , r l = 5 "0 85 a o is c h o s e n b u t w e f o r m a t ri p l e c e n t r e d w a v e p a c k e t w i t he i t h e r E o r A 1 s y m m e t r y , w e c a n o b t a i n p o w e r s p e c t r a fo r th e s y m m e t e r i z e d m o t i o n ,a g a i n u s i n g e q u a t i o n ( 8 ) . T h e s e p o w e r s p e c t r a a r e s h o w n i n fi g u r e 7 ( a ) a n d ( b) ,r e s p ec t iv e l y . I f t h e se p l o t s a r e e x a m i n e d o n e c a n s e e th a t t h e n u m b e r o f c o n t r i b u t i n gs t a te s w i t h a h i g h v a l u e h a s b e e n r e d u c e d . I f w e n o w c o n c e n t r a t e o n t h e p e a k a t~ 3 3 0 0 0 c m - 1 a n d f o r m t h e b a n d w a v e f u n c ti o n s, e q u a t io n (1 1), f o r b o t h p o w e rs p e c t r a , t h e d i f f e re n c e i n l o c a l i z a t i o n c a n b e s t u d i e d . F i g u r e 8 (a ,b) s h o w s h o r s e s h o es t a te s o f d if f er e n t s y m m e t r ie s p l o t t e d i n J a c o b i a n d s y m m e t r y c o o r d i n a t e s . T h es y m m e t r y c o o r d i n a t e s , a r e d e f in e d i n t e rm s o f t h e c h a n g e s i n t h e s e p a r a t i o n o f t h ea t o m s A R ~ , A R 2 a n d A R 3 , w h e r e A R i = R i - R e , f o r i = 1 , 2 , 3 a n d R e i s thee q u i l i b r i u m s e p a r a t i o n o f th e a t o m s , R e = 1 .6 5 a o . T h e s y m m e t r ic s t r e t c h c o o r d i n a t eS~ = ( AR~ + A R 2 + A R3) / x / 3 , t he be nd co o r d i na t e S 2 = (2AR~ - AR 2 - AR 3) / x / 6 ,a n d t h e a s y m m e t r i c s t r e t c h c o o r d i n a t e $3 = ( A R 2 - A R 3 ) / x / 2 . I n t h e J a c o b i p l o t t h eA 1 h o r s e s h o e d o e s n o t h a v e s u c h a w e l l d e f i n e d l o c a l i z a t i o n a s t h e E h o r s e s h o e , d u et o i n t e r f e r e n c e b e t w e e n t h e d i f fe r e n t s y m m e t r y e q u i v a l e n t l o c a l i z a t i o n . I n t h es y m m e t r y c o o r d i n a t e p l o t t h e d e g e n e r a c y o f t h e b e n d S z a n d t h e a s y m m e t r i c s t r e t c hS 3 c a n b e s e e n m o r e c l e a rl y , w i t h t h e l o c a l i z a t i o n h a v i n g d i f f e re n t a m p l i t u d e a n dp h a s e i n th e e q u i v a l e n t c o n f i g u r a ti o n s . F o r t h e E s y m m e t r y w a v e f u n c t i o n th e m a j o r


13/19

4 t~

P h a s e s p a ce a n a l ys i s o f H 3 73

(

(

20 22 24 26 28 30 32 34 36 38 40E n e r g y f r o m g r o u n d s t a t e / 10 3 cm "

Figure 7 . Po we r spec t ra fo r wa vepack e ts wi th m om en ta P r, = -0 "3 4 au , P r2 = 7"62 au , Po = 0centre d a t r l = 5 .07 ao, r 2 = 1 0 - 5 a0, 0 = 9 0 ~ ( a ) has even pa r i ty E symmetry , (b ) hasA 1 s y m m e t r y , a n d ( c) h a s o d d p a r i t y E s y m m e t r y . T h e s m o o th e d s p e c t r u m is a ls o a d d e dfor a t ime T = 24 f s. The a r row m arks the d is soc ia t ion energy .

a m p l i t u d e is c o n f i n e d t o o n e r e g io n , b u t i n t h e A 1 w a v e f u n c t i o n t h e r e i s n e a rr o t a t i o n a l s y m m e t r y , w i t h t h e s a m e l o c a l i z a t i o n i n t h e e q u i v a l e n t c o n f i g u r a t i o n s .

T h e s y m m e t r y c o n s i d e r a t i o n s c a n b e ta k e n f u r th e r i f w e n o w l o o k f o r th e o d dp a r i t y E s t a te s . E v e n s t a t e 5 6 7 i s a n E s t a te , a n d h a s t h e s a m e e n e r g y a s o d d s t a t e4 91 . I f t h e H u s i m i f u n c t i o n i s s t u d i e d f o r t h is o d d p a r i t y b u t w i t h t h e a t o m s p e r m u t e dw e o b t a in , f o r t h e s a m e H u s i m i c u t as be fo re , a m a x i m u m c o r r e s p o n d i n g t o o d ds y m m e t r y h o r s e s h o e m o t i o n . I f a s y m m e t r iz e d w a v e p a c k e t is f o r m e d f o r th is p e a kP r, = - 3 . 7 2 a u , r~ = 4 . 9 6 a o , a p o w e r s p e c t r u m c a n b e o b t a i n e d f o r t h e o d d p a r i t yE h o r s e s h o e , s h o w n i n fi g u re 7 ( c ). I f t h e e i g e n f u n c ti o n s u n d e r t h e b a n d a t3 3 0 0 0 c m - 1 a r e c o m b i n e d t h e y c a n b e p l o t t e d i n J a c o b i c o o r d i n a t e s , e x c e p t w i t ha t o m s p e r m u t e d , a n d s y m m e t r y c o o r d i n a t e s . I n f ig u r e 8 ( c) th e i n t e rf e r e n c e i s c l ea ri n t h e J a c o b i p l o t s, a n d i n t h e s y m m e t r y c o o r d i n a t e p l o t t h e o d d p a r i t y is a p p a r e n ta s a r e f l e c ti o n t h o u g h $ 2 = 0 . T h i s t i m e , h o w e v e r , t h e l o c a l i z a t i o n i s r e s t r i c t e d t o t w oe q u i v a l e n t c o n f i g u r a t i o n s b u t w i t h o p p o s i t e p h a s e .


14/19

374 G. G. de Polavieja et a l .

=r

6 -

5 -

4 -

3 -

2 -

1 -

O "

5 -

4 -

3 "

2 -

1 -

0 -

5 -

4 -

3 -

2 -

1 -

i I 1 I J I J I J I J I1 2 3 4 5 6

r 2 / a o

( a / 1

o

- 2 - 1

-2---I

-ti i n n n I n I I I I I I L l I l l l n I

- 2 - 1 0 1

s , l a oFigure 8. Parts (a), (b) and (c) show ban d wavefunctions for the peaks centred at ~ 33 300 cm - 1in figure 7 (a), (b) and (c), respectively. The plot on the left is in Jacobi coordinates withr I versus r 2 and 0 = 90~ however, for (c) due to a node at 0 = 90 ~ the atoms were

permuted. The plot on the right in each case is in symmetry coordinates with S2 versusS3 and $1 = 0. Note that in the Jacobi plots the wavefunct ions look very simi lar exceptfor some interference at the equilibrium geometry, rl = 1-65 %, r 2 = 1-43 ao, 0 = 90~whereas in symmetry coordinates the wavefunctions differ considerably.

3.5. E x c i t e d h o r s e s h o e sIn their classical analysis of H+3, Go me z Llorente and Pollak [5] explain the

coarse-grained regular i ty in the Car r i ngt on- Ke nne dy spectrum [2] as ar ising fromrovi brat iona l trans itions from the horseshoe states to similar States with anti-symm etri c excitatio n, i.e., mo ti on out of the 0 = 90 ~ plane. To con fir m the exist enceof these states we have assumed that if the excitation is only one quantum, thestrongest localization wil l be in the odd symmetry calculat ions with maximumlocalization near 0 = 90 ~ w ith a node on the 0 = 90 ~ plane. The power spec trum ofsuch a motio n can be obtained by considering a wavepacket moving in the horseshoemoti on, just away from the 0 = 90 ~ plane. The initial Hus imi p oint chosen for thewa vep ac ket was r 1 = 5.085 a o , r z = 10 - s ao, 0 = 78'5 ~ Pn = 0'0, Pr2 = 9.1906 au,


15/19

Phase space analysis of H ~ 3750 . 3

0 . 2 '

0 . 1 '

0.0

1.0

0.75

~ 0 . 5

0.25

0 . 020

I I 25

( a )

~ , I , i i , , , i , , i i i , , , L - I t ~ r - i ~ l " l ' ~5 0 7 5 1 O 0 1 2 5 1 5 0T i m e / f s

2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0E n e r g y f r o m g r o u n d s t a te / 1 0 3 c m a

Figure 9 . (a) Co rre la t ion funct ion for an H~ odd par i ty wavepa cket wi th m om en ta p , , = 0"0 ,P2~ = 9"190 6 au , Po = 0"0 cen tred at r 1 = 5'085 ao, r 2 = 10 -5 a 0, 0 = 78"5~ showing as t rong rec urrence a t 20 fs. (b) Pow er spect rum for the sam e wavepa cket . The sm oothedspect rum is for a t ime T = 24 fs. The peaks in th is spect rum do not coincide wi th thosein the h orsesho e p owe r spect rum in f igure 4 (a) .

Po = 0 " 0 , a n d t h e r e c u r r e n c e f u n c t i o n a n d p o w e r s p e c t r u m a r e s h o w n i n fi g u re 9 ( a )a n d ( b ), r e s p e c t i v e l y . C l e a r l y t h e r e i s a s t r o n g r e c u r r e n c e w i t h a p e r i o d o f 1 9 f s a n da t r a c e o f a w e a k e r r e c u r r e n c e o f 2 2 fs.O n t h e p o w e r s p e c t r u m a c le a r s t r u c tu r e s h o w s , a n d i n te r e st in g l y t h e p e a k s d on o t c o i n c i d e w i t h e i t h e r t h e h o r s e s h o e , f i g u r e 4 ( a ), o r t h e e l e p h a n t f o o t , fi g u r e 4 ( b ),p o w e r s p e c t r a . U s i n g e q u a t i o n ( 1 1 ) w e c a n c o m b i n e t h e s t a t e s a g a i n u n d e r t h es m o o t h e d p o w e r s p e c t r u m t o r e v e a l l o c a l iz a t i o n . I n f ig u r e 1 0, 3 s t a te s a r e p l o t t e d i nJ a c o b i a n d s y m m e t r y c o o r d i n a t e s , c o r r e s p o n d i n g t o t h e p e a k s l a b e ll e d i n fi gu r e 9 (b ).D u e t o t h e n o d e a t 0 = 9 0 ~ t h e J a c o b i ' s l i c e ' i s w i t h a n a n g l e 0 = 7 8.5 ~ h o w e v e r , t h es a m e h o r s e s h o e s h a p e c a n b e s e en . I n t h e s y m m e t r y p l o t s i t i s c l e a r t h a t t h e n o d e i sn o t s i m p l y a s y m m e t r y r e l a t e d e f fe c t, b u t s p li ts t h e l o c a l i z a t i o n d o w n t h e m i d d l e.H i g h a m p l i t u d e i s r e s t r ic t e d t o o n l y o n e o f t h e 3 s y m m e t r y r e l a t e d c o n f i g u r a t i o n s ,b u t i t is l i k e ly t h a t b o t h E a n d A 2 l o c a li z e d w a v e f u n c t i o n s c a n b e r e s o l v e d e m p l o y i n gt h e m e t h o d d e s c r i b e d i n s e c t i o n 3 . 4 . P r e l i m i n a r y r e s u l t s i n d i c a t e t h a t t h e r e a r e t w oe x c i te d h o r s e s h o e m o t i o n s w i t h d i s ti n c t ly d i f f e re n t b a n d c e n t r es , a n d s l ig h t ly d i f fe r in gl o c a l i z a t io n , a n d i t i s t h e i n t er f e re n c e o f t h e s e t w o m o d e s t h a t m a k e t h e J a c o b iw a v e f u n c t i o n p l o t s a p p e a r ' m e s s y ' . T o c o m p a r e t h e r el a ti v e e n er g ie s o f t h e m o t i o n s ,t a b l e 1 s h o w s t h e b a n d c e n t r e s f o r h o r s e s h o e , e l e p h a n t f o o t , a n d t h e e x c it e d h o r s e s h o em o d e . T h e e n e r g i e s i n c m - 1 a r e r e f e r r e d t o t h e v i b r a t i o n a l g r o u n d s t a t e . T h e b a n d s


16/19

376 G. G. de Polavieja e t a l .

t, .

6 ~

5 -

4 -

3 -

2 -

1 -

5 -

4 -

3 -

2 -

i i i i J l l l t l l l1 2 3 4 5 6 -2 -1 0 1 2

r~ / a . s3 / a o

Figure 10. Band wavefunctions under the three numbered peaks in figure 9 (b) in Jacobi andsymmetry coordinates. The Jacobi plots on the left with r I versus r 2 and 0 = 78.5~ showlocalization with horseshoe shape; however, the symmetry plots on the right with Szversus Sa and $1 = show that there is a node along the symmetry plane 0 = 90~ andso the wavefunctions are not simple horseshoes.

are ordered in bending nod e nu mber order, and energy differences are quoted betweenhorseshoe and excited horseshoe, horseshoe and elephant foot, and the horseshoeand the horseshoe with one more node.

4 . C o n c l u s i o n

In this article we have applied Husimi analysis to full 3D calculations on H~-and, by studying repeated phase space features, recovered several types of localizationonto classical periodic orbits. The method not only recovers localization whichis apparent in eigenvectors, but also motion that does not appear in a singleeigenstate. Here we have presented localization on horseshoe, elephant foot and 0excited horseshoe periodic orbits, but have also found strong localization on thesymmetric stretch, inverted hyperspherical, symmetric stretch excited horseshoes and


17/19

Phase space analysis of H ~ 377Table 1. Energies of band centres in the smoothed power spectra for the horseshoe E h,elephant foot E e and the excited horseshoe E exh motions. All energies are quoted fromthe ground state . The number of nodes n is obatined from examination of the

wavefunctions. As the elephant foot does not exist below 30 000 cm ~ the band centresare quoted only in the valid energy window; other peaks in the power spectrum arisefrom the horseshoe. The difference between the nth horseshoe and excited horseshoe,and the nth horseshoe and elephant foot are also quoted. In the final column the peakseparation of the horseshoe is given to show the approximate harmonic progression.Energies marked with a * are above the dissociation energy and should be consideredwith caution.h h~ e ~ ~ x h ~ ; x h _ n ~ ~.~ - e . ~ E , , _ I - ~ .

cm 1 cm- 1 cm- 1 cm - 1 cm - 1 cm - 111 20160 22480 2320 207012 22230 24220 1990 184013 24070 25970 1900 190014 25970 27780 1810 192015 27890 29 650 1760 180016 29 690 31100 31400 1710 1410 181017 31 500 32840 33060 1560 1340 183018 33 330 34670 34690 1360 1340 173019 35060 36730* 36310 1250 1670 165020 36710* 38810* 37900* 1190 2100 161021 38 320* 40230, 39490* 1170 1910 160022 39 920* 41345* 1425

several un- nam ed types of motion, all of which persist up to and above thedissociation energy. The m etho d employed is not classically guided, al though classicalcalculations were performed to demonstra te types of motion. Due to the number ofdimensions in phase space one of the momen ta has been fixed, partially restrictingthe allowed motions. As a result of this work, a new method, based on wavepacke trecurrence, has now been developed and more complicated orbits have been isolated,including free rotor type motions [40].

The theoretical model currently used to describe the Carrington-Kennedycoarse-grained spectrum of H~ [2] was first proposed by Gomez-Llore nte and Pollak[5], based on classical analysis. The main features are explained by transitions fromthe hor seshoe mo tion, to a similar mot ion with some excitation out of the 0 = 90 ~plane. Rotational excitation keeps the dissociation states temporarily bound behinda centrifugal barrier. We have found strong localization on 0 excited horseshoe stateswhich preliminary analysis seems to give non-rotational excitation transition fre-quencies from the corresponding horseshoe state in a range 1200 -200 0cm -1However, we have found many other motions in the dissociating energy range, allwith strong power spectra, which cannot be ignored in the explanation of thedissociation mechanism.The methods we have employed to analyse the motions could be applied to anytriatomic species for which eigenvectors can be calculated. The accuracy of thepotential is important for quantitative analysis, but for qualitative studies for findingimpor tant motions, topologically correct potentials may be adequate.

The authors would like to thank James Henderson for the H~- vectors, Ruth Le


18/19

3 78 G . G . d e P o l a v ie j a et al .S u e u r f o r t h e t ra n s i t io n s p e c tr u m a n d T i n o B o r o n d o f o r s u p p l y in g t h e ' G e a r ' r o u t in eu s e d i n t h e c l a s si c a l t r a j e c t o r y c a lc u l a t io n s . G . G . P . a c k n o w l e d g e s a f e l lo w s h i p f r o mM i n i s t e r i o d e E d u c a c i o n y C i e n c i a ( S p a i n ) a n d w o u l d l i k e t o t h a n k U n i v e r s i t yC o l l e g e L o n d o n f o r t h e ir h o s p i ta l i ty d u r i n g s u m m e r 1 99 3. N . G . F . w o u l d l i ke to t h a n kt h e S c i e n ce a n d E n g i n e e r i n g C o u n c i l ( S E R C ) f o r a s t u d e n t s h ip . T h i s w o r k h a s b e e ns u p p o r t e d b y g ra n t s u n d e r t he B ri ti sh C o u n c i l A c c i o n e I n t e g ra d e s c h e m e - - a n dS E R C g r a n ts f o r t im e o n th e U L C C C o n v e x e s a n d a n I B M R S / 6 0 0 0 w o r k s t a ti o n .

References[1] CARRINGTON,A., BUTTENSHAW,J., and KENNEDY,R. A., 1982, Molec. Phys., 45, 753.[2] CARRINGTON,A., and KENNEDY, R . A ., 1984, J. chem . Phys., 81, 91.[3] CARRINGTON,A ., MCN AB, I. R ., an d WEST, Y. D., 1993, J. chem . Phys., 98, 1073.[4] BERBLINGER,M., POLLAK,E., and SCHEIER, CH ., 1988, J. chem . Phys., 88, 5643.[5 ] GOMEZ-LLORENTE, . M ., and POLLAK,E., 1987, Chem. Phys. Lett ., 138 , 125.[6 ] GOMEZ-LLORENTE, . M ., ZAKRZEWSKI, ., TAYLOR, H . S., an d KULANDER,K. C., 1988,J. chem. Phys., 89, 5959,[7] CHILD, M . S., 1986, J. chem. Phys., 90, 3595.[8] PFEIFFER,R., an d CHILD, M . S., 1987, Molec. Phys., 60, 1367.[9] HENDERSON,J. R ., a nd TENNYSON, J., 199 0, Chem. Phys. Lett ., 173, 133.[10] HENDERSON,J. R., TENNYSON,J., and SUTCLIFFE,B. T. , 1993, J. chem. Phys., 98, 7191.[ l 1] LE SUEUR, C. R., HENDERSON, . R., an d TENNYSON,J., 1993, Chem . Phys. Lett ., 206, 429.[12] SHNmELMAN,A. L., 1974, Ups. Mat . Nauk , 29, 1181; see also ZELDITCH,S., 1987, DukeMath . J . , 55, 91 9; C OLIN DE VERDIERE, Y., 198 5, Commun. Math. Phys. , 102, 497.[13] HELTON,J. W ., a nd TABOR, M., 1985, Physiea D, 14, 409,

[14] VOROS, A., 19 77, Stochastic Behaviour in Classical and Quantum Hamiltonian Systems,edited b y G. C asati an d J. F or d (Springer-Verlag); BERRY, M. V., 1977, J. Phys. A, 10,2083.[15 ] MACDONALD, S. W ., a nd KAUFMAN, A. N ., 1979 , Phys. Rev. Lett., 42, 1189.[16] MACDONALD, S. W., 1983 , Lawrence Berkeley Lab. Rep. LBL, 1483 7; TAYLOR, R. D .,an d BRUMER, P., 1983 , Faraday Discuss. chem. Soc., 75, 170; HELLER, E. J., 1984 , Phys.Rev. Lett ., 53, 1515.[17] BOGO~OLNY,E., 1988, Physica D, 31, 169.[18] BERRY, M V., 1 989, Proc . R. Soc . Lond. A, 423, 219.[19] DELANDE, D., 199 1, Cha os and Quantum Phys ics , edi ted by M.-J. Giannoni , A. Voros,and J . Zinn-Just in, Session LII , 1-31, L es Hou ches (Elsevier).[20] GOMEZ-LLORENTE, . M. , and POLLAK,E., 1992, Ann. Rev. Phys. Chem., 43, 91.[2 1] DE POLAVlEJA, G . G ., BORONDO, F. an d BENITO, R . M ., 199 4, Phys. Rev. Lett., in press.[22] HEELER,E. J. , 1991, Chaos and Quantum Phys., edi ted by M .-J. Giann oni , A. Voros, andJ. Zinn-Justin, Session LII, 1-31, Les Houches (Elsevier),[23] CHO , S.-W., a nd CHILD, M. S., 1994, Molec. Phys., 81, 447.[2 4] WA LTON, A . R. , a nd MANOLOVOLOUS,D. E., unpublished.[25] WIGNER,E., 1932, Phys. Rev., 40, 749.[26 ] O'CONNELL, R. F., and WIGNER, E. P., 1981, Phys. Let t . A, 83, 145.[27] WIGNER, E., 193 8, Trans. Faraday Soc., 34, 29.[28] HUSIMI, K., 1940, Proc. phys. m ath. Soc. Jap., 22, 264.[29] TAKAHASHI,K. , 1989, K. Pro gr. theoret. Phys. Suppl., 98, 109.[30 ] HARRIMAN, J. E., a n d CASIDA, M . E ., 199 3, Int . J . Quantum Chem., 45, 263.[31] VILELA,MENDES, R., 1991 , J. Phys. A, 24, 4349.[32] HENDERSON,J. R., TENNYSON,J., LE SUEUR, C. R. an d MILLER, S., 1993 , Com p. Phys.Commun., 75, 379.[33 ] MEY ER, M ., BOTSCHWINA,P., an d BURTON, P. G ., 1986, J. chem. Phys., 84, 891.[34] FARANTOS,S. C., 1993, Laser Chem., 13, 87.[35] FULTON,N. G., POLAVIEJA,G. G . an d TENNYSON, J., unp ublish ed.[36] BERBL1N~ER,M ., SCHLIER, C., TENNYSON, J., an d MILL ER, S., i992, J. chem. Phys., 96,6842.[37] TAYLOR,H . S., ZAKRZEWSKI, ., a nd SAINI, S., 198 8, Chem . Phys. Lett ., 14 5, 5 55; TAYLOR,


19/19

Phase space ana ly s i s o f H ~ 379H. S., and ZAKRZEWSKI,J., (1988), Phys. Rev. A, 38, 3732; GOMEZ-LLoRENTE, . M.,ZAKRZEWSKI, J., TAYLOR, H . S., an d KULANDER, K . C., 198 9, J. chem. Phys., 90, 1505.1-38] SCHLIER,CrI. , and VIx, U., 1985, Chem . Phys., 95, 401.

[3 9] SADOVSKII, D . A ., FULTON , N . G ., HENDERSON, J. R., TZNNVSON, J., an d Zr~ILINSICn,B. I., 1993, J. chem. Phys. , 99, 906.[40] POLAVIEJA,G . G ., FULTON, N . G ., an d TZNNYSON, J., unpub lished.

Documents

Gonzalo Garcia De Polavieja et al- Coarse grained spectra, dynamics and quantum phase space structures of H3^+