p. bartlett and b. j. howard- the rotational-vibrational spectrum of symmetric non-rigid triatomics...

8/3/2019 P. Bartlett and B. J. Howard- The rotational-vibrational spectrum of symmetric non-rigid triatomics in hyperspherical coordinates: the H^+-3 molecule

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M O L E C U L A R P H Y S I C S , 1990, VOL. 70, NO. 6, 1001-1029

The rotational-vibrational spectrum o f symm etric non-rigid triatom icsin hypersphericai coordinates: the H ~ m oleculeBy P. BARTLETT

School of Chemistry, Cantocks Close, Bristol University, Bristol BS8 1TS, U.K.and B. J. HOWARD

Physical Chemistry Laboratory , South Parks Road, Oxford University, OxfordOX1 3QZ, U.K.

(Received 7 March 1990; accepted 18 April 1990)Theoretical methods are described for the calculation of rovibrational levelsof symmetric non-rigid triatomics using the symmetric hyperspherical coordi-nates of Smith and Whitten. An adiabatic separation of radial and angularmotion is shown to be valid for H~-. The adiabatic angular functions are chosenas a linear combination of products of angular-momentum eigenfunctions andspecially developed hyperspherical basis functions. Corrections to the adiabaticapproximation are included by BriUouin-Wiguer perturbation theory. Resultsare given for the H~ system. The present approach has the advantage that allsymmetric arrangements are treated equivalently.

1 . I n t r o d u c t i o nIn the last few years significant progress has been made in the theoretical treat-ment of large-amplitude motion in small molecules. For near-rigid molecules the

most widely used choice of coordinates originates from the work of Eckart i1]. Inthis approach it is assumed that the potential-energy surface defines an equilibriumnuclear geometry. However, it is widely recognized that the Eckart embedding ofthe body-fixed (BF) axes is unsuitable for molecules with large-amplitude motions.The difficulty is easily seen by considering the case of a triatomic molecule in whicha large-amplitude vibrational motion distorts the molecule into a linear geometry.The instantaneous moment-of-inertia tensor has a zero diagonal element, its inverseis singular, and hence the Eekart Hamiltonian is undefined. This apparent singu-larity arises because it is not possible to define all three Euler angles uniquely for acollinear arrangement of atoms. Consequently, the domain of the Eckart Hamil-tonian is restricted to those functions that vanish sufficiently rapidly close to thelinear geometry so that the matrix elements of the internal Hamiltonian convergeI-2]. Although the singularity in the kinetic-energy operator may always be avoidedby, for example, arbitrarily confining trial functions 1,3], the accuracy of thisapproach is questionable if the molecular potential does not exclude collineargeometries. The treatment of large-amplitude motion in highly symmetric moleculesis further complicated by the Eckart fixing of the body-fixed axes to the equilibriumgeometry. Each symmetry operation requires the setting up of a new Eckart axissystem, with a consequent redefinition of the Euler angles. This leads to consider-able difficulty in constructing trial functions with the full permutational symmetry.

Several new techniques have been developed for the calculation of vibrational-rotat ional levels that avoid some of the problems associated with the Eckart embed-ding of the BF axes. In particular, Tennyson and Suteliffe 14, 5], in calculations on

0026-8976/90 $ 3.00 9 1990 Taylor & Francis Ltd

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1 00 2 P . B a r t l e t t an d B . J . H o w ar ds y s te m s a s d iv e r se a s K C N a n d H e H F , h a v e u s e d J a c o b i (a t om - - d ia t om ) c o o r d i n a t e sw i t h a B F z ax i s a l i g n ed a l o n g t h e co l l i s i o n co o r d i n a t e . I n t h e s e co o r d i n a t e s t h eH a m i l t o n i an i s s u ff ic i en t ly s i m p l e t h a t l a r g e - am p l i t u d e i n t e rn a l m o t i o n m ay b eco m p l e t e l y d e s c r ib ed . A ca r e f ul ch o i ce f o r t h e a s y m p t o t i c f o r m o f b a si s f u n c t i o n sen s u r e s t h e ap p a r en t s i n g u l a r i t y a s s o c i a t ed w i t h t h e l i n ea r g eo m e t r y d o es n o t c au s ean y n u m er i ca l p r o b l em s . T h i s ap p r o ach h a s b een ap p l i ed t o t h e s y m m et r i c o b l a t et o p H ~ [ 6 ] , a l t h o u g h accu r a t e c a l cu l a t i o n s r eq u i r ed v e r y la r g e b a s is s et ex p an s i o ns .Th e s low converg ence o f t hese ca l cu l a t i ons r e f lec t s tw o d i ff icu lt ies wi th t he use o fJ aco b i co o r d i n a t e s . F i r s t, t h e o p e r a t i o n s o f t h e m o l ecu l a r s y m m et r y g r o u p Dahp r o d u ce a co m p l i ca t ed m i x t u r e o f J aco b i co o r d i n a t e s , w h i ch m ak es i t v e r y d if fi cul tt o d e f i n e b a s i s f u n c t io n s w i t h t h e f u l l p e r m u t a t i o n a l s y m m et r y . C o n s eq u en t l y , c a l cu -l a t i ons have been l imi t ed t o u s ing a par t i a l l y symmet r i zed bas i s se t , w i th t he r esu l tt h a t l a r g e n u m b er s o f b a s i s f u n c t i o n s a r e r eq u i r ed f o r a g o o d r ep r e s en t a t i o n . S ec -o n d l y , t h e u s e o f an i n - p l an e B F z ax i s co u p l e s a l l t h e d i f fe r en t B F an g u l a r -m o m e n t a c o m p o n e n t s t o g et h e r, l e a d in g t o a s e t o f s t ro n g l y c o u p l e d r o t a ti o n a ls t a t e s . A n y r ea l i s t i c c a l cu l a t i o n f o r H ~ i n J aco b i co o r d i n a t e s m u s t i n c l u d e a l l o f f -d i ag o n a l C o r i o l i s an d a s y m m et r i c - t o p i n t e r ac t i o n s . N o s i m p l i f y i n g r o t a t i o n a l -d eco u p l i n g ap p r o x i m a t i o n i s ap p r o p r i a t e . T h e s iz e o f t h i s fu ll r o v i b r a t i o n a lca l cu l a t i o n i n c rea s e s li n ea rl y w i t h t h e v a l u e o f t h e t o t a l an g u l a r m o m en t u m d an ds o o n b e co m es i n t rac t ab l e . F u l l y co u p l ed ca l cu l a t i o n s h av e co n s eq u en t l y b eenl imi t ed t o l ow -J s t a t es , t yp i ca l l y J ~< 4 [61 , a l t hou gh Te nn yso n and Su tc l if f e [7"1h av e p r o p o s ed a t w o - s t ep v a r i a ti o n a l p r o ced u r e f o r th e ca l cu l a t i o n o f h i g h ly ex c i tedr o t a t i o n a l s t a t e s o f H ~ an d i ts i s o m er s.I n s y m m et r i c m o l ecu le s , l a r g e - am p l i t u d e m o t i o n i s m o r e e f fi ci ent ly t r e a t ed u s i n ga s y m m e t r i sed s e t o f co o r d i n a t e s i n w h i ch a l l s y m m et r i c a r r an g em en t s a r e t r e a t edeq u i v a l en t l y . S o m e t i m e ag o , S m i t h an d W h i t t en ( S W ) I ' 8 ] d ev e l o p ed a s y s t em o fh y p e r s p h e r i ca l co o r d i n a t e s i n w h i ch a t o m i c p e r m u t a t i o n i s a cco m p l i s h ed s i m p l y b yad d i n g a co n s t an t p h as e an g l e t o o n e o f t h e an g u l a r co o r d i n a t e s . A p a r t f r o m t h i sp h as e - an g l e d i f fe r en ce , t h e i n t e rn a l co o r d i n a t e s a r e u n a f f ec t ed b y p e r m u t a t i o n . I t i st h e r e f o re r e l a ti v e l y ea s y t o g en e r a t e b a s i s s e t s t h a t t r an s f o r m a s o n e o f t h e i r r ed u c -i b l e rep r e s en t a t i o n s o f D 3 h . A v e r y c l ea r d e s c r i p t i o n o f t h e p r o p e r t i e s o f t h i s co o r d i -n a t e s y s t em h as b een g i v en b y J o h n s o n [ 9 ] . F o r c a l cu l a t i o n s o f l a r g e - am p l i t u d em o t i o n s i n s y m m et r i c o b l a t e t o p s s u ch a s H ~ - , t h e S W h y p e r s p h e r i ca l s y s t em h ass ev e r a l ad v an t ag es w h en co m p ar ed w i t h J aco b i co o r d i n a t e s y s t em s . F i r s t , t h e f u l lm o l ecu l a r s y m m et r y m ay b e u s ed t o r ed u ce t h e s i ze o f an y ca l cu l a ti o n . S eco n d l y ,t h e r e i s a u n i q u e r ad i a l co o r d i n a t e p t h a t i s i n v a r i an t u n d e r a l l s y m m et r y o p e r -a t io n s . M o t i o n a l o n g t h e r ad i a l co o r d i n a t e m ay t h e r e f o r e b e f o rm a l l y s ep a r a t edf r o m m o t i o n i n t h e an g u l a r co o r d i n a t e s . W i t h an ad i ab a t i c a s s u m p t i o n , t h e i n t e r n a le i g en s t a te s a r e ex p r e s s ed a s a p r o d u c t o f t h e s o l u t i o n o f a f i v e - d im en s i o n a l eq u a t i o n( t w o in t e r n a l a n d t h r ee r o t a t i o n a l d eg r ee s o f f r e ed o m ) an d a s e t o f co u p l ed r ad i a leq u a t i o n s . T h e u s e fu l n e ss o f t h is ap p r o ac h d ep en d s u p o n t h e s t r en g t h o f t h e n o n -ad i a ba t i c r ad i a l - coup l ing t e rm s . In t he m olec u l a r sys t em s so f a r cons idered [10"1 t her ad i a l co u p l i n g w as s u f fi c ien tl y s m a l l t h a t t h e co r r ec t i o n t e r m s co u l d b e accu r a t e l yi n c l u d ed b y a p e r t u r b a t i o n t r e a t m en t . T h e ad i ab a t i c s ep a r a t i o n , w h i ch i s p o s s i b l e i na h y p e r s p h e r i ca l ( b u t n o t i n a J aco b i ) co o r d i n a t e s y s t em , r ed u ces t h e m ax i m u md i m en s i o n o f t h e m a t r i x eq u a t i o n s t h a t m u s t b e s o lv ed , l e ad i n g t o a co n s i d e r ab l es av i n g i n co m p u t a t i o n a l e f f o r t . T h i r d l y , t h e S W co o r d i n a t e s y s t em i s an i n s t an -t an eo u s p r i n c i p a l - ax i s s y s t em , w i t h t h e B F z ax i s d i r ec t ed a l o n g t h e s y m m et r y ax i so f t h e o b l a t e t o p . T h i s ch o i ce m i n i m i ze s t h e o f f - d iag o n a l co u p l i n g b e t w ee n s t a te s o f

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Rovibrational states o f H ~ 1003d i ff e re n t B F z c o m p o n e n t o f a n g u l a r m o m e n t u m , a n d su g g e s t s t h a t a n a n g u l a r -m o m e n t u m d e c o u p l i n g a p p r o x i m a t i o n , a n a l o g o u s t o t h e c e n t r i f u g a l su d d e n a p p r o x -ima t ion [11] used in ine l a s t i c sca t t e r ing , may be use fu l . Th i s p romises to reduces ig n if ic a n tl y t h e a m o u n t o f c o m p u t e r t im e r e q u i r e d t o c a l c u la t e h i g h r o t a t i o n a lstates.

T h e t r ia t o m i c m o l e c u l a r i o n H~ - is o n e o f t h e s i m p le s t k n o w n p o l y a t o m i c m o l -ecu les and has be en the sub jec t o f m an y theo re t i ca l and ex pe r imen ta l i nves tiga tions .Because o f i t s l igh t mass , l a rge -ampl i tude in t e rna l m ot io n i s imp or t an t , even a tre l a tive ly low ene rg ie s . Tha t t h is i s so can be seen f rom the po ten t i a l -ene rgy su r faceof Sch inke , Du pu i s an d Les t e r (SDL ) fo r t he g ro un d e l ec t ron ic s t a t e 1A '~ [12] . Them i n i m u m - e n e r g y c o n f i g u r a t i o n c o r r e sp o n d s t o a n e q u i l a t e r a l t r i a n g l e ( o b l a t e sy m -m e t r i c t op ) . Ye t a t e n e rg i e s o f a b o u t 1 3 0 0 0 c m - 1 a b o v e t h e p o t e n t i a l m i n i m u m , H ~m a y b e c o m e c oUi n e ar ( p ro l a t e sy m m e t r ic t o p ). T h e f la t n es s o f th e p o t e n t i a l a t t h e seene rg ie s r e su l t s i n ve ry- l a rge -ampl i tude in t e rna l mot ion in which the sys t em canaccess a l l poss ib l e tr i angu la r and co l l inea r geom et r ie s so tha t p e rm uta t io n o f t heh y d r o g e n n u c le i is a f a ci le p ro c e s s . W i t h i n c r e as i n g e n e r g y , th e H ~ sy s t e m b e c o m e sloca li sed a t ob tu se i sosce l le s conf igura t ions , w i th h inde red in t e rna l ro t a t ion o f t hed i a t o m i c , a n a l o g o u s t o a v a n d e r W a a l s a t o m - d i a t o m c o m p l e x . A t a n e n e r g y o fa b o u t 3 9 0 0 0 c m - 1 t h e m o l e c u l ar i o n d i ss o c ia te s in t o H 2 ( 1 ~ -) a n d H + . F o r c o m -p a r i so n , t h e c a l c u la t e d z e r o - p o i n t e n e r g y f o r H~ o n t h is su r f a c e i s a b o u t 4 3 3 0 c m - 1 .L a r g e - a m p l i t u d e m o t i o n i s t h e r e f o r e e x p e c t e d t o b e i m p o r t a n t i n H~ , e v e n a t l o wdegrees o f v ib ra t iona l exc i t a tion . A s a r e su l t o f t he s impl i c ity o f t he H ~ ion , seve ra lv e r y a c c u r a t e ab initio poten t i a l -ene rgy su r faces a re ava i l ab le [12-15] . Ca lcu la t ionso n H~ " sh o u l d t h e r e f o r e p r o v i d e a b e n c h m a r k b y wh i c h t o c o m p a r e d i f f e r e n ta p p r o a c h e s t o t h e t r e a t m e n t o f l a r g e - am p l i t u d e m o t i o n s i n n o n -r i g id m o l e c u le s .

T h e f ir s t a n d m o s t e x t e n s iv e ca l c u l a ti o n o f t h e r o t a t i o n a l - v i b r a t i o n a l sp e c t r a o fH~- was made by Carney and Por t e r [13 , 16 , 17] . Vibra t iona l ene rg ie s and wave-f u n c t io n s we r e c al c u l a te d b y e x p a n d i n g W a t so n ' s f o r m o f t h e E c k a r t H a m i l t o n i a nfor non l inea r m olecu les in a bas i s se t o f 220 ha rmon ic -osc i l l a to r p rod uc t func t ions .These ca l cu la t ions concen t ra t ed on the low- ly ing v ib ra t iona l s t a t e s . The h ighe rv i b r a t i o n a l - r o t a t i o n a l s t a t e s h a v e b e e n m o r e a c c u r a t e l y c o n v e r g e d i n t h e c a l c u -l a t ions o f Ten nys on a nd Su tc li f fe [6 ] . In th i s s tudy the bas i s func t ions were no tada p ted to the fu ll sym me t ry g rou p o f D3h , bu t a l a rge se t o f bas i s func t ions (880func t ions fo r d = 0 ca l cu la t ions) was used in o rde r to ensure convergence to a t l ea s t1 cm -1 o r be t t e r . In recen t ca l cu la t ions [15 , 18-20] the ava i l ab i l i ty o f m ore ac cura t ep o t e n t ia l - e n e rg y su r fa c e s h a s i m p r o v e d t h e a g r e e m e n t b e t we e n t h e c a l c u l a te d f u n d a -menta l v ib ra t iona l f r equenc ie s and the expe r imenta l ly de r ived va lues .

P r e v i o u s c a l c u l a ti o n s u s i n g h y p e r sp h e r i c a l c o o r d i n a t e s o f th e b o u n d - s t a t edynam ics o f t r i a tomics h ave been l imi t ed to the pure ly v ib ra t iona l e igens t a te s . Freyand H ow ar d [10 , 21] have ca l cu la t ed the v ib ra t iona l l eve ls o f i ne r t-gas andm o l e c u l a r -h y d r o g e n t r im e r s b y e x p a n d i n g t h e a n g u l a r w a v e f u n c t i o n i n h y p e r -sphe r i ca l ha rmonics . These func t ions a re e igenfunc t ions o f t he k ine t i c -ene rgy op e r -a t o r a n d t h e r e f o r e h a v e t h e c o r r e c t a sy m p t o t i c b e h a v i o u r a t t h e sy m m e t r i c - t o ps i n g u la r it y p r e se n t in t h e J = 0 Ha m i l t o n i a n . Ho we v e r , i n b o u n d s t a te s o f a m o l e c u -l a r sys t em, where the po ten t i a l ene rgy i s comparab le to the k ine t i c ene rgy , a l a rgebas i s se t o f hype rsph e r i ca l ha rm onics i s necessa ry fo r ade qu a te conv ergence . Addi -t ional ly , there are di f f icul t ies in extending this approach to s ta tes wi th non-zero Js in c e th e a n g u l a r b a s is f u n c t io n s m u s t a l so h a v e a d e f i n e d a sy m p t o t i c b e h a v i o u r a tthe co l l inea r ge om et ry wh ere the in t e rna l H am i l ton ian fo r J ~ 0 is unde f ined .

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1004 P. Bartlett and B. J. HowardClosed analytic expressions for the hyperspherical harmonics for non-zero totalangular momentum are unknown.

Hyperspherical coordinates have also been used in a recent calculation of thevibrational eigenstates of H~- [22]. A product basis set of one dimensional functionstogether with successive diagonalisation and truncation was used to generate accu-rate vibrational levels. An alternative body-fixed hyperspherical coordinate system[23] was used. This differs principally from the present hyperspherical system inthat the BF z axis is defined in the plane of the particles. This choice is better forsystems dominated by near-linear configurations. For zero-angular-momentumstates, Whitnall and Light developed a method to handle the singularity in theHamiltonian at the symmetric-top configuration. However, for states with a non-zero angular momentum, the Hamiltonian has an additional singular off-diagonalCoriolis term at the symmetric-top configuration. As a result, the approach ofWhitnall and Light cannot be applied to non-zero-J states.

In this article we present an accurate calculation of the vibrational-rotationaleigenvalues of H~ using the hyperspherical coordinates introduced by Smith andWhitten. In the next section we discuss the coordinate system used in this work, itssymmetry properties, and the form of the molecular Hamiltonian. The orientation ofthe body-fixed axes is labelled by three Euler angles, while the internal radialcoordinate p describes the overall size of the molecular system and the angularcoordinates (3, tp) describe its shape. To aid in visualising the molecular motion inhyperspherical coordinates, we describe a stereographic projection of the H3 poten-tial at fixed p. We assume that motion along the radial coordinate may, as an initialapproximation, be adiabatically separated from motion in the angular coordinates.In section 3 we describe the expansion of the full wavefunction in a set of adiabaticsurface functions which are parametric functions of p. We choose the adiabaticsurface functions as solutions of a fixed-p angular Hamiltonian. In section 4 wediscuss the basis functions used to diagonalise this angular Hamiltonian and theirsymmetry adaption. The full molecular symmetry is used to reduce the size of thepresent calculation. In section 5 we give a description of our method for obtainingthe adiabatic surface functions. Section 6 gives the resulting coupled-channel radialequations and describes their solution, in the limit of weak coupling, by an iterativeform of Brillouin-Wigner perturbation theory. Section 7 contains the results of ourrovibrational calculations on H~- and section 8 gives our conclusions.

2 . H y p e r s p h e r ic a l c o o r d i n a t e s a n d t h e m o l e c u l a r H a m i l to n i a nThe hyperspherical coordinate system of Smith and Whitten [8] has been

described in detail by Johnson [9], so we give here only the details necessary tounders tand our calculations.The hyperspherical coordinates are defined in terms of the mass-scaled [24]Jacobi vectors r k and R k, where r k is the scaled vector from a tom i to a tomj and R kis the scaled vector from the centre of mass of diatomic /j to atom k. The hyper-spherical coordinates divide into two sets: the internal coordinates (p, 3, tp), whichdetermine the size and shape of the triangle formed by the Jacobi vectors, and theexternal coordinates (~t, t , ~,), which specify the orientat ion of the BF axis system.The Euler angles are chosen so that the BF z axis points in the direction of thevector product r k x R k while the BF y axis is parallel to the larger of the twoin-plane instantaneous moments of inertia, with an absolute sense determined by the

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R o v i b r a t i o n a l s t a t e s o f H ~ 1005s e t o f i n t e r n a l c o o r d i n a t e s ( p , 0 , tp ). T h e B F a x e s a r e a s y s t e m o f i n s t a n t a n e o u sp r i n c ip a l a x e s i n w h i c h t h e m o m e n t - o f - i n e r t i a te n s o r i s d i a g o n a l , w i t h t h e e l e m e n t s

Ix x =/ t p 2 sin 2 O, /I . # p 2 c o s 2 0 , ~ (1 )/Izz 12p2, J

s u c h t h a t I . . > / I . / > I x x . O n l y f o r m o l e c u l a r c o n f i g u r a t i o n s w i t h t h e e q u i l i b r i u mg e o m e t r y a r e t h e h y p e r s p h e r i c a l b o d y f i x ed a x e s (x , y , z ) i d e n t i c a l w i t h t h e c o n v e n -t i o n a l l y d e f i n e d E c k a r t p r i n c i p a l a x e s ( a , b , c ). A s t h e m o l e c u l e v i b r a t e s, t h e h y p e r -s p h e ri c al a x e s r o t a t e s m o o t h l y , fo l l o w i n g th e i n t a n t a n e o u s a x e s o f in e r t ia , i nc o n t r a s t w i t h t h e b e h a v i o u r o f th e E c k a r t a x es , w h i c h a r e f ix e d b y t h e e q u i l i b r iu mm o l e c u l a r g e o m e t r y . T h e t h r e e in t e r n a l c o o r d i n a t e s a r e d e f i n e d b y t h e c o m p o n e n t so f th e m a s s - sc a l e d J a c o b i v e c t o r s i n t h e B F a x is s y s t em , n a m e l y

= p co s 0 cos ~o, ]= - p s i n 0 s i n q ~' ( 2 )R k = p c os 0 s in r

R k = p s in 0 co s q~w i t h c o o r d i n a t e r a n g es

0 ~ < p ~ oo, 0 ...< 0 ..< 88 0 ~< q~ ~< 2 n , (3)a n d a v o l u m e e le m e n t

d r ' = 88 sin 40 sin ~ dp dO d~0 d~t d~ d y. ( 4 )T h e r a n g e o f tp s p a n s c o n f i g u r a t i o n s p a c e t w i c e s in c e t h e t w o s e ts o f c o o r d i n a t e s ( p,0 , tp , ~ , p , ~) an d (p , 0 , tp + n , ~t, p , ~, + x) refer t o the sam e p hy s ica l a rr an ge m en t o fp a r t i c le s b u t r e f e r e n c e d t o t h e t w o p o s s i b l e B F a x i s s y s t e m s , w h i c h d i f f e r o n l y i n t h ea b s o l u t e s e n se o f t h e x a n d y a x es . T h i s d u p l i c a t i o n e n s u r e s t h e m o t i o n o f th ei n s t a n t a n e o u s B F a x i s s y s t e m i s a l w a y s c o n t i n u o u s a n d , a s s h o w n b y P a c k a n dP a r k e r [ 2 3 ] , a v o i d s d i s c o n t i n u o u s h a l f - i n te g r a l a n g u l a r - m o m e n t u m f u n c t i o n s o f ~p.T h e H a m i l t o n i a n o p e r a t o r i n h y p e r s p h e r i c a l c o o r d i n a t e s i s [ 2 5 ]

H t = ~2 1 0 5 ~ ~2 F 1 0 A 1 02 ]2 1 t p S - ~ p p O p 2~ -p2 s in 4 0 ~0 s in 4 0 + -o s 2 2 0 ~ 2+ AJ2 x + BJ ~ + CJ2z ih 2 si n 20l t p 2 co s 2 20 Jz ~ + V (p, 0, q~), (5)

w h e r e t h e J i (i = x , y o r z ) a r e t h e B F c o m p o n e n t s o f th e t o t a l - a n g u l a r - m o m e n t u mo p e r a t o r ( in u n i t s o f h),

A = h2/(21up2 sin 2 0), B = h 2 / ( 2 p p 2 co s 2 0), C = h 2 / ( 2 p p 2 cos 2 20),a n d / t i s t h e t h r e e - b o d y r e d u c e d m a s s ( /t 2 = m t m 2 m 3 / (m x + m 2 + m 3 )). W e n o t e ,f o l lo w i n g ( 3 ), t h a t t h e ' r o t a t i o n a l c o n s t a n t s ' A , B a n d C s a ti sf y t h e i n e q u a l it i e sA I> B an d C > i B . I t i s co nv en ien t to rew r i t e (5 ) in the f o rm

H ' = Tp. + T~ + 7"r + T~ + V (p , O, ~o) (6 )

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1006 P . Bar t l e t t and B. J . H ow ardwhe re each consecu t ive line o f (5 ) i s iden t i fi ed w i th the t e rms abo ve : Tp, the rad ia l(p), T~ the an gu lar (~, ~0), T~ the ro tatio na l , T~ the C orio lis k inetic -en erg y op er ato rsand V(p, ~, ~p) the m olec ular poten t ia l . I n the la ter d iscuss ion w e shal l f ind i t usefulto subd iv ide the ro ta t iona l k ine t ic -ene rgy op era to r T~ in to the sy m me t r i c - topp o r t i o n

T~ym = 8 9 + B ) ( J 2 - d ~ ) + C J ~ (7 )a n d t h e (r e m a in i ng ) a sy m m e t r i c - t o p p o r t i o n

Tasy = 8 9 - - B ) (Ja x - - J~ ) . (8 )W e no te th a t T~ i s no t the norm al a sym me t r i c - ro tor k ine t i c -ene rgy ope ra to r s ince ,a l t h o u g h A = h 2 / 2 I = , a n d B = h 2 / 2 I y y , C ~ h 2 / 2 I z z . Simi la r ly , t he ' ro t a t iona lc o n s t a n t s ' d e f in e d a b o v e d o n o t f o l l o w t h e e x p e c t e d o r d e r A i> B > / C . E c k a r t [ 2 6 ]sh o w e d th a t su c h a n o m a l o u s ' r o t a t i o n a l c o n s t a n t s ' a r e a c o n se q u e n c e o f u s i n g a B Fins tan taneous p r inc ipa l -ax i s sys tem. In such a sys tem the Cor io l i s t e rms , desc r ib ingt h e i n te r a c t io n b e t w e e n r o t a t i o n a l a n d v i b r a t io n a l m o t i o n s , a r e o f a la r g e r o r d e r o fm a g n i t u d e t h a n t h o se d u e t o r o t a t i o n a lo n e . I f t h e o f f - d ia g o n a l C o r i o li s te r m s a r einc luded in to a d iagona l e f fec t ive k ine t i c -ene rgy op era to r then , a s Van Vleck [27]showed, the usua l r ig id - ro tor express ion i s ob ta ined fo r the pure ly ro ta t iona l t e rm.I n l at e r w o r k E c k a r t [ 1 ] a b a n d o n e d t h e i n s t a n ta n e o u s - a x i s sy s t e m i n f a v o u r o ff ix ing the mo lecu la r axes by the equ i l ib r ium m om ents o f ine rt ia . Th i s cho ice min i-mises the ma gni tud e of the C or io l i s t e rms .

T h e S c h r 6 d i n g e r e q u a t i o n m a y b e t r a n s f o rm e d i n o r d e r t o s im p l if y t h e d i ff e re n -t ia l t e rms in p . On de f in ing the t r ansfo rm ed wav efunc t ion ~b as~h = p5/2 ~ , (9)

w h e r e ~ i s a n e i g e n fu n c t io n o f t h e u n t r a n s f o r m e d H a m i l t o n i a n o p e r a t o r H ' , ( 5 ), t h eS c h r 6 d i n g e r e q u a t i o n f o r g, b e c o m e sH ~ = E ~ , (1 0)

w h e r e th e t r a n s f o r m e d H a m i l t o n i a n o p e r a t o r i s g i ve n b yH = p S / 2 H ' p - 5/2. (11)

A f te r t h is t r a n s f o rm a t i o n , t h e H a m i l t o n i a n o p e r a t o r m a y b e w r i t t e n a s

w h e r en = ~ + T . + T~ + Tr + V (p, ~, ~0), (12)

T p = 2 / zan d T~, T~ an d T~ are a s in (5) . Th e t rans form ed vo lum e e lem ent i s

dv = 88 in 4~ s in f l dp d~ drp da dp d~ . (13)T h i s H a m i l t o n i a n i s t h e o n e u se d in t h e r e m a i n d e r o f t h is p a p e r .T h e r o t a t i o n a l - v i b r a t i o n a l l ev e ls o f H ~ m a y b e l ab e l le d a c c o r d i n g t o t h e i rr e-duc ib le represen ta t ions o f D3h , which is i somo rphic w i th $3 {E , E*}. The e f fec tso f t h e p e r m u t a t io n - i n v e r s io n o p e r a t o r s o f O3h are easi ly der ived using the def ini -

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Rovibrat ional s ta tes o f H ~ 1007t i o n s a b o v e a n d t h o s e g iv e n b y J o h n s o n [ 9 ] . I n t h e r e p r e s e n t a t i o n o f t h e s y m m e t r i cg r o u p $ 3 a p a s s i v e c o n v e n t i o n i s u s e d i n w h i c h a t o m i c l a b e l s r a t h e r t h a n a t o m i cc o o r d i n a t e s a r e p e r m u t e d .

T h e e x t e r n a l c o o r d i n a t e s ( ~, f l, ~,) a r e u n a f f e c t e d b y t h e o p e r a t o r s E , ( 12 3) a n d( 1 3 2 ) , w h i l e t h e p a i r w i s e p e r m u t a t i o n ( il) i s e q u i v a l e n t t o a r o t a t i o n o f n a b o u t t h eB F y a x i s :

= n - - f l . ( 1 4 )( 0 3 [ ~ _ , JT h e s p a c e- f ix e d i n v e r s io n o p e r a t o r E * i s e q u i v a l e n t t o a r o t a t i o n o f n a b o u t t h e B Fz a x is a n d a c h a n g e i n t h e s e n s e o f a l l a x e s :

E * = B 9 ( 1 5 )

T h e i n t e r n a l c o o r d i n a t e s ( p, ~ ) a r e u n a f f e c t e d b y a n y o f th e o p e r a t o r s o f D 3 h , w h i let h e k i n e m a t i c a n g l e ~o e x h i b i t s t h e f o l l o w i n g b e h a v i o u r .(1 2 )~ 0 = - ~ 0, ]/1 3 ) ~ o = - ~ o - ~ n ,(23)q~ = -~ p + ~ n ,~ (16)

/

I n h y p e r s p h e r i c a l c o o r d i n a t e s 0 = 8 8 d e s c r i b e s e q u i l a t e r a l t r i a n g u l a r c o n f i g u r -a t i o n s , w h i l e f o r 0 = 0 t h e t h r e e a t o m s a r e c o U i n e ar . T h e a n g u l a r d e p e n d e n c e o f t h em o l e c u l a r p o t e n t i a l a t f i x e d p m a y b e v i s u a li s e d b y u s i n g a s t e r e o g r a p h i c p r o j e c t i o no f t h e u p p e r s u r fa c e o f t h e s p h e r e c o v e r e d b y t h e s p h e r i c a l p o l a r c o o r d i n a t e s ~ ' = 89

- 2 0 a n d q~ = 2 g - 2~0. A p o i n t i n t h i s p l o t w i t h C a r t e s i a n c o o r d i n a t e s x a n d yc o r r e s p o n d s t o t h e i n t e r n a l c o n f i g u r a t i o n (0 , ~o),w h e r ex = t a n ( 88 - 0) co s 20 , "~ (17)y - t a n (88 - ~) si n 2q~.J[

T h e r e i s o n l y o n e p o i n t o n t h e p r o j e c t e d p l o t c o r r e s p o n d i n g t o e a c h i n t e r n a l c o n -f i g u r a t i o n s o t h e m a p p i n g i s u n i q u e . F u r t h e r m o r e , e a c h m o l e c u l a r p e r m u t a t i o n i se q u i v a l e n t t o a ro ta t ion o f t h e p l o t a b o u t t h e z a x is w i t h o u t d i s t o r t io n .S t e r e o g r a p h i c p r o j e c ti o n s o f t h e H ~ p o t e n t i a l o f S c h i n k e, D u p u i s a n d L e s t e r a tt h r e e f i x e d v a l u e s o f p a r e s h o w n i n f i g u r e 1 . T h e d a s h e d c i r c le s s h o w c u r v e s o fc o n s t a n t 0 a t 5 ~ i n te r v a ls , w i t h t h e c e n t r a l p o i n t o n t h e p r o j e c t i o n c o r r e s p o n d i n g t ot h e e q u i l a t e r a l t r i a n g l e c o n f i g u r a t i o n ~ = 88 I n f i g u r e 1 ( a) t h e a n g u l a r f o r m o f t h ep o t e n t i a l i s s h o w n a t p = 1 . 2 A , c l o s e t o t h e a b s o l u t e m i n i m u m i n t h e p o t e n t i a l~ = 1-151 A , ~9 = 8 8 M o t i o n a l o n g t h e c o n t o u r s is a p p r o x i m a t e l y p a r a l le l t o ~w h i l e m o t i o n p e r p e n d i c u l a r t o t h e c o n t o u r s i s a p p r o x i m a t e l y p a r a l l e l t o ,9. I t i s cl e a rt h a t a t s m a l l p t h e m o t i o n s a l o n g t h e i n t e r n a l c o o r d i n a t e s ,9 a n d q~ a r e l a r g e l yse pa ra b le . In f a c t , a t 0 = 88 0 a n d q~ a re to t a l ly se pa ra b le , s inc e fo r a n e q u i l a t e ra l

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1 00 8 P . B a r t l e t t a n d B . J . H o w a r d

# I f % ,,; , i ", \ \i i \ 'i i z

( rFigure 1 . S tereographic projec tions of the H ~ poten t ia l of Schinke e t a l . [12] a t p - - 1 .2A(a) , 1 .7A (b) and 2 .0A (c) . Th e sol id conto urs are a t in tervals of 10 00cm -1 in (a) and(b) , and 500 cm -1 in ( c ). The lowes t con tou r co r r e sponds t o po t en t i a l energie s o f2000 crn - 1 (a) , 14 000 cm - 1 (b) and 22 500 er a- 1 (c) abo ve the poten t ia l min imum . T hebr ok en circles depict l ines of co nst ant ~, in 5~ in tervals , wi th the equi la tera l geo m etry

(0 = 45 ~ a t the cent re o f the projec t ion .g e o m e t r y a l l a x e s in t h e p l a n e o f t h e a t o m s a r e p r i n c i p a l a x e s o f i n er ti a . M o t i o n i n tpi s e q u i v a l e n t t o a r o t a t i o n a b o u t t h e B F z a x is , w h i l e m o t i o n i n 3 c o r r e s p o n d s t o a ni n t e r n a l v i b r a t i o n . A t p = 1 -7 A t h e p o t e n t i a l s u r f a c e s h o w s t h r e e s y m m e t r i c a n g u l a rm i n i m a a t o b t u s e i so c el le s g e o m e t r i e s s e p a r a t e d b y s a d d l e p o i n t s a t a c u t e i s oc e ll esg e o m e t r i e s . A t p = 2 . 0 , ~ t h e a n g u l a r m i n i m a h a s s h i f t ed t o s ti ll s m a l l e r v a l u e s o fa n d t h e b a r r i e r t o i n t e r c o n v e r s i o n h a s i n c r e a se d c o n s i d e r a b l y . W i t h i n e a c h o f t h et h r e e c h an n e l s , m o t i o n p a r a l l e l to t h e c o n t o u r s d e s c ri b e s h i n d e r e d r o t a t i o n o f t h ed i a t o m i c i n th e a t o m - d i a t o m c o m p l e x , w h il e m o t i o n p e r p e n d i c u la r t o t h e c o n t o u r si s l a r g e l y v i b r a t i o n o f th e d i a t o m i c . A t l a r g e p , a s t h e p l o t o f f ig u r e 1 ( c) s h o w s ,m o t i o n a l o n g t h e i n t e rn a l a n g u l a r c o o r d i n a t e s 9 a n d tp i s s t r o n g l y c o u p l e d .

3. Th e wa vefun ct ion in hyperspherica i coord inatesW i t h o u t l o ss o f g e n e ra l it y , t h e e i g en f u n c t io n s o f t h e t r a n s f o r m e d H a m i l t o n i a n

(1 2), fo r a g i v e n t o t a l a n g u l a r m o m e n t u m J , M ( s p ac e - fi x e d c o m p o n e n t o f J ) a n d

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R o v i b r a t io n a l s t a t e s o f H ~ 1009s y m m e t ry F , m a y b e e x p a n d e d i n t h e f o r m

Ntg , , M r _ E p ) , ( 1 8 )t

wh ere f l refers to the se t of angu lar coo rdin ates (~, t , y , ~, ~o) an d the fu nct ion sdepend pa ramet r i ca l ly upon the rad ia l coord ina te p . Impl ic i t i n th i s expans ion i s thebe l i e f tha t on ly a few ad iab a t i c s t a t e s t need be cons ide red fo r a go od d esc r ip t ion o fnonl inea r t r i a tomics such as H~- . Th i s a s sum pt ion i s cons ide red in g rea te r de ta i l i nsec t ion 6 . We choose the ad iaba t i c sur face func t ions ~ , used as bas i s func t ions he re ,a s so lu t ions o f the five -d imens iona l Schr6d inger eq ua t io n in the in te rna l co ord in a tes(~, ~o) an d the Eu ler ang les (~, t , y):

[ ], + + + 0 , e / " r ( e ; v ) = (1 9)where the k ine t i c ene rgy and the molecu la r po ten t i a l a re eva lua ted a t a f ixed rad ia lvalue p . H ere T, i s the internal ang ular (~, q~) kinet ic en ergy, T~ is the rota t ion alkinet ic energ y an d T~ the Co riol is term s d ef ined in (12) and (5). As we shal l s ho w insec t ion 4, the so lu t ions to (19) m ay be w r i t ten in the fo rm= Ck,,,,,(p)Yk,,,,,f2), (20)

kvmwhere the symmet r i sed angula r bas i s func t ions a re g iven by

~ . s~ r ( f 2 ) =_ 1 [ 2 ~ + 1 l ' / 2 r s q g 'u oiJ 'x I 'O~k(~, t , y)e - ,v , + ( _ 1) ,+, +,O ff_ ,(~ , t , y)e,V~,]. (21)

I n t h e e x p a n s i o n (2 0) t h e v a l u e s o f t h e a n g u l a r - m o m e n t u m q u a n t u m n u m b e r k a n dt h e k i n e m a t i c q u a n t u m n u m b e r v a r e r e s t r i c t e d b y t h e s t a t e sy m m e t r y F . I n ( 2 1 ) ,J *DMk i s the un norm al i sed W igner D m at r ix , de f ined fo l lowing the con vent ion s o fBr ink and Sa tch le r 1 -28], f svm i s the norm al i sed ~ func t ion in t rodu ced in the nex tse ctio n a nd CkSr~(p) ar e t he ex pa ns io n coefficients.The angu la r sur face func t ion ~( t2 ; p ) t r ansform s as one of the i r reduc ib le repre -

se n t a t io n F o f t h e m o l e c u l a r sy m m e t r y g r o u p D3h.T h i s g r o u p i s g e n e r a t e d b y t h eopera tors fo r space - f ixed inve rs ion E* , pa i rwise pe rmuta t ion (12) and cyc l i c pe rmu-ta t ion (123). F r om the fac t tha t the space - f ixed inve rs ion o pe ra to r E * on ly a f fec t s T,and us ing (15), we imm edia te ly have th a t the ang ula r bas i s func t ion ~(t2 ), de f ined in(21), has the fo l lowing s imple sy mm et ry und er inve rs ion :E* H y.~ (~ ) = ( - 1r $ '~ '~r(~) . (22)

Hence fo r sur face func t ions 9 of def ini te par i ty the k su m in (20) i s rest r ic ted toe i the r a l l -odd or a l l -even va lues o f k . Fur th e rm ore , the two fo ld m app ing of conf igur -a t ion space , d i scussed in sec t ion 2 , leads to the equ a l i ty~-syr(~ , t , y, ~9, q 9 = l~s~r(~, t , ~, + zr, 0, q~ + z , (23)

so t h a t k a n d v i n t h e su m ( 2 0 ) m u s t b e b o t h e v e n o r b o t h o d d , d e p e n d i n g o n t h epar i ty o f the sur face func t ion ~ .To see the e f fec t o f the pa i rwise p e rm uta t ion , we no te f rom (14) tha t (12) t akesJ *Ouk(~, t , y) into

D ~(0 t + n, n -- t , ~ - y) = ( - 1) '+kDff_k(~, t , y). (24)

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1 0 10 P . B a r t l e t t a n d B . J . H o w a r dTab le 1. Th e se t of angular bas is func t ions of angular -m om entum projec t ion k (e /o label sthe par i ty of k and a ) and symmet ry spec ies F . For non-zero k , v i s r es t r i c ted to thevalues _ I 'v] , whi le fo r k = 0, v m ay hav e an y one of the set of values Iv] . Th e

sym m etry of the basis funct ion un der pairwise perm utat io n is labelled by ~r.I r reduciblerepresenta t ion Iv]F Pa r i ty of cr Pa r i ty of k n --- 0, 1, 2 . . .A[ e e 6nA~ e o 3 + 6nA~ o e 6nA[ o o 3 + 6nE' e/o e 2 _ 6nE " e / o o 1 _ 6 n

T o g e t h e r w i t h t h e t r a n s f o r m a t i o n p r o p e r t i e s o f q~, g i v e n b y ( 16 ), p a i r w i s e p e r m u -t a t i o n t r a n s f o r m s t h e p r o d u c t f ~ v m ( ~ ) D ~ ( ~ , fl, ~ ) e - iv , a s

J J:~ -ivcp J+ k J:~ J~( 1 2 ) fk v , ,, (~ ) O M k ( O ~ , f l, ~)e = (- - 1) ~ k , , ,, ,( ~ ) O , ,, _ k ( O ~ , l , ~)e i~ . (25)T h e r a n g e o f k i n (2 0) m a y t h e r e f o r e b e f u r t h e r r e s t r i c te d t o n o n - n e g a t i v e v a l u e s , F o rk I> 1 , t h e b a s i s f u n c t i o n s ~ m a n d ~ _ v,~ a r e d i f f e r e n t a n d i n g e n e r a l b o t h f u n c t i o n sm u s t b e i n c l u d e d i n t h e s u m (2 0). F o r t h e p a r t i c u l a r c a s e o f k = 0 , t h e t w o f u n c t i o n sd i f fe r o n l y b y a p h a s e f a c t o r a n d t h e s u m m a t i o n i n (2 0 ) m a y b e f u r t h e r r e s tr i c te d t op o s it iv e q u a n t u m n u m b e r s . T h e s y m m e t r i s e d a n g u l a r b a si s fu n c t io n s t r a n s f o r m w i t ht h e c h a r a c t e r ( - 1 )" u n d e r p a ir w i s e p e r m u t a t i o n .

U n d e r t h e c y c li c p e r m u t a t i o n ( 12 3), t h e o n l y c o o r d i n a t e a f f e c t ed is t h e k i n e m a t i ca n g l e q~ w h i c h t r a n s f o r m s i n t o q~ + ~ n . H e n c e a n g u l a r f u n c t i o n s ^ s M rk ~ m ( f2) wi th v am u l t ip l e o f t h r e e a r e u n a f f e c t e d b y t h e c y c li c p e r m u t a t i o n ( 12 3 ) a n d t r a n s f o r m a so n e o f th e t w o o n e - d i m e n s i o n a l i r r e d u c i b le r e p r e s e n t a t i o n s o f S 3 , d e p e n d i n g o n t h es ig n o f ( - 1 ) ' . A n g u l a r b a s i s fu n c t i o n s w h e r e v is n o t a m u l t i p le o f t h r e e f o r m ar e p r e s e n t a t io n o f th e t w o - d i m e n s i o n a l i r r e d u ci b le r e p r e s e n t a t i o n o f $ 3 . C o m b i n i n gt h e t r a n s f o r m a t i o n p r o p e r t i e s o f t h e a n g u l a r b a s i s f u n c t io n s , ta b l e 1 s h o w s t h er e s t ri c t io n s o n t h e q u a n t u m n u m b e r s k a n d v i n th e s u m m a t i o n i n ( 20 ) f o r s u r f a cef u n c t i o n s ~ t h a t t r a n s f o r m a s t h e i r re d u c i b l e re p r e s e n t a t i o n F o f D a b .

4. Ad iabat ic surface funct ions: cho ice of bas i s se tT h e a d i a b a t i c s u r f a ce f u n c t i o n s ~ ( Q , p ), w h i c h d e p e n d p a r a m e t r i c a l l y u p o n p ,

a r e s o l u t i o n s t o t h e S c h r 6 d i n g e r e q u a t i o n ( 1 9) w h e r e t h e e f fe c ti v e a n g u l a r p o t e n t i a li s t h e m o l e c u l a r p o t e n t i a l V (p , 2 , q~) e v a l u a t e d a t f i x e d p . A s i s r e a d i l y s e e n f r o mf i g u r e 1 ( a) , t h e H ~ - a n g u l a r p o t e n t i a l a t s m a l l p i s a l m o s t i n d e p e n d e n t o f q~ s o t h a t(1 9) is a p p r o x i m a t e l y s e p a r a b l e . W i t h a n i n c r e a s e i n p , f i g u r e 1 (c ) s h o w s t h e a n g u l a rm i n i m u m s h if ts f r o m t h e e q u i l a te r a l g e o m e t r y a t ~ = 88 t o a n e a r - c o l l i n e a rc o n f i g u r a t i o n o f ~ ,,~ 0 a s t h e a t o m - d i a t o m d i s s o c i a t io n l im i t is a p p r o a c h e d . A tt h e s e l a rg e v a l u e s o f p , t h e m o t i o n i n t h e ~ a n d q~ c o o r d i n a t e s i s s tr o n g l y c o u p l e da n d ( 19 ) i s h i g h l y n o n - s e p a r a b l e . B e c a u s e o f t h e l a r g e c h a n g e i n t h e n a t u r e o f th ea d i a b a t i c s o l u t i o n s ~ ( f 2 ; p ) w i t h p , i t i s v e r y d i ff ic u l t t o c h o o s e a b a s i s s e t t h a t w i l lb e u s e f u l a t a l l v a l u e s o f p . W e s h a ll , h o w e v e r , r e s t r i c t a t t e n t i o n t o t h e l o w - l y i n gs t a te s o f H a , w h e r e t h e n e a r - c o l l i n e a r g e o m e t r y (2 ~ 0 ) i s n o t s i g n if ic a n t. W e s e e k

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R o v i b r a t i o n a l s t a t e s o f H ~ 1011a n e x p a n s i o n o f th e a d i a b a t i c s u r f a ce f u n c t i o n s ~ ( t 2 ; p ) in a s et o f s y m m e t r i s e d b a si sf u n c t i o n s ~ ([2 ), w h i c h w e c h o o s e a s e i g e n f u n c t i o n s o f t h e f i xe d -p H a m i l t o n i a n H og i v e n b y

15h 2H o = T, + T~ + T~ + ~ + Vo(P, 0), (26)w h e r e t h e k i n e t i c - e n e r g y t e r m s a r e i d e n t i c a l w i t h t h o s e o f (1 9) a n d t h e r e f e r e n c ep o t e n t i a l Vo h a s t h e s i m p l e f o r m

2/tp 2h2 V0(p, ~) = A + 4~ 2 co s 2 2& (27)T h e a n g u l a r p o t e n t i a l 1/" ( a t f i x e d p ) h a s a m i n i m u m a t ~ = 88 a n d i s i n d e p e n d e n to f q~. W i th inc rea s ing ~ , the w av efu nc t ion i s loca l i s ed a t ~ = 88 T h e s i m i l a r i t y o f t h es m a l l - p H ~ - p o t e n t i a l a n d t h e r e f e re n c e p o t e n t i a l 1Io e n s u r e s t h a t o n l y a f e w t e r m sn e e d b e c o n s i d e r e d i n t h e s u r f a c e w a v e f u n c t i o n e x p a n s i o n ( 2 0 ). A t l a r g e r p , a s th ea m p l i t u d e o f th e w a v e f u n c t i o n a t ~ = 0 i n c r e a se s , t h e c o n v e r g e n c e is e x p e c t e d t o b es lower .W i t h i n c r e a s i n g ~ , t h e b a s i s f u n c t i o n s ~ ( f 2 ) a r e l o c a l i s e d n e a r ~ = 88 T o c o n -s i d e r b e h a v i o u r o f t h e b a s i s f u n c t i o n ~ (f2 ) a t ~ n e a r 8 8 w e r e w r i t e t h e k i n e t i c - e n e r g yp o r t i o n o f (2 6) a s

h 2 1 d s in 4~ + + 89 + B)(J 2 - - j2 )Ti = 2 - ~ s in 43 a9 cos 2 2~ ff~-~2\ ih 2 s in 2~+ C J 2 + 8 9 - - B ) ( J 2 - - J ~ ) - - I t p 2 cos2 29 J~ d--~' (28)

wh ere T~ = T~ + T~ + T~. H ere the coe f f i c ien t o f the a sy rm ne t r i c - ro to r - l ike t e rm i sh2 ( 1 1 ) h2 co s 20 (29)

89 - B) = 4/~p'------ sin~ ~ co s 2 ~ - / z p 2 si n 2 2~ "F o r ~ ~ 88 t h i s la s t t e r m b e c o m e s s m a l l a n d w e s h a l l n e g l e c t i t i n d e r i v i n g th e f o r mo f t h e b a s i s f u n c t i o n s . I n t h i s l i m i t t h e r e is n o c o u p l i n g b e t w e e n t h e d i f f e r e n t c o m -p o n e n t s o f t h e a n g u l a r m o m e n t u m a l o n g t h e B F z ax is . N o t e , h o w e v e r , t h a t a t t h ec o l l in e a r g e o m e t r y (,9 ~ 0 ) t h i s t e r m i s d i v e rg e n t . T h e a n g u l a r m o m e n t u m J i s n o wc o u p l e d t o t h e B F x a x i s , a n d a l l p o s s i b l e c o m p o n e n t s - J ~< k ~< J o f t h e a n g u l a rm o m e n t u m a l o n g t h e B F z a x i s , c o n s i s t e n t w i t h t h e m o l e c u l a r p a r i t y , a r e s t r o n g l ym i x e d . I n s u c h a c a s e t h e p r e s e n t e m b e d d i n g o f th e b o d y - f i x e d a x e s is i n a p p r o p r i a t ea n d i t i s m u c h s i m p l e r t o r e d e f i n e th e B F z a x i s s o i t li e s i n t h e m o l e c u l a r p l a n e .

T h e r e f e re n c e p o t e n t i a l V o (p , ~ ) i s i n d e p e n d e n t o f q ~ a n d c o n s e q u e n t l y t h eS c h r f d i n g e r e q u a t i o n c o r r e s p o n d i n g t o t h e H a m i l t o n i a n o p e r a t o r H o ( n e g le c ti n gt h e a s y m m e t r i c r o t o r t e r m s o f (2 8) a n d ( 29 )) i s se p a r ab l e . T h e u n s y m m e t r i s e d s o lu -t i o n s m a y b e w r i t te n a s

_ / 9-4- 1\1/2

w h e r e v i s t h e e ig e n v a l u e o f t h e k i n e m a t i c a n g u l a r - m o m e n t u m o p e r a t o r - i ~ /~ q~a n d t h e r o t a t io n m a t r i x [ ( 2 J + 1 ) l l 2 / 4 x - I D ~ k is t h e n o r m a l i s e d s y m m e t r i c - t o p w a v e -f u n c t io n f o r t o t a l a n g u l a r - m o m e n t u m q u a n t u m n u m b e r J w i th c o m p o n e n t s k a n dM a l o n g t h e B F a n d S F z a x e s r e sp e c ti v el y . S u b s t i t u t i o n o f th e w a v e f u n c t i o n

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1 01 2 P . B a r t l e t t a n d B . J . H o w a r de x p a n s i o n ( 3 0 ) i n t o t h e S c h r 6 d i n g e r e q u a t i o n f o r t h e H a m i l t o n i a n o p e r a t o r ( 2 6 )g iv e s t h e f o l l o w i n g e q u a t i o n f o r t h e f u n c t i o n s f :

I h 2 1 d s in 40 d h 2- 2 g p 2 s in 4~ d~ d-O + 2 1 z p 2 cos 2 2~ @2 + k 2 _ 2 k v s in 2~)}[ J ( J + 1) -- k a] + Vo (p, 0) - EkSvm(p) fk~vm(/))= 0. (31)/tp z s in 2 2/)T h e e f f e c t iv e a n g u l a r p o t e n t i a l i n (3 1) i s s i n g u l a r a t b o t h t h e e q u i l a t e r a l ( 0 = i n ) a n dc o l l in e a r (0 = 0 ) c o n f i g u r a t io n s . S in c e t h e e i g e n v a lu e s o f th e H a m i l t o n i a n o p e r a t o r( 26 ) m u s t r e m a i n f i n i te f o r w a v e f u n c t i o n s lo c a l i s e d e i t h e r a t 0 = i n o r / ) = 0 , th ea s y m p t o t i c f o r m s o f v a l id s o l u t i o n s a r e d e t e r m i n e d b y t h e s i n g ul a ri ti e s i n t h ee f f ec t iv e - p o t en t ia l t e rm s . F o r 0 n e a r i n (3 1) m a y b e a p p r o x i m a t e d b y

- 2/tp----'~ s in4 / ) cl-/) s in 4 /) - - + nSvm fks ,m(0) = 0, (32)d /) COS2 2 0 _ J J a - , . 1 4 'w h e r e o n l y t h e d i v e r g e n t t e r m s h a v e b e e n r e t a i n e d . E q u a t i o n ( 3 2 ) i s s i m p l y s o l v e da n d t h e / ) = 8 8 a s y m p t o t i c l i m i t o f t h e f u n c t i o n ( 0 ) is g i v e n b y

f ~ , m ( / ) ) " ~ cos v 2/) P ~ ") (c o s 4/)) as / ) --* in , (33)w h e r e t h e i n d e x p = 189 - k) l an d P ~ ' p) is a Jacob8 9 p o l y n o m i a l [2 91 o f d e g r e e m .Th e e igen va lue nSo ,, i s g iven b y

nksom = (4 m + I v - - k l ) ( 4 m + I v - - k l + 4). (34)S i m i l ar l y, i n t h e l i m i t o f s m a l l 0 t h e a s y m p t o t i c f o r m o f t h e f u n c t i o n f ( 0 ) i s th es o l u t i o n o f th e e q u a t i o n

{ h 2 [ 1 d d 2 [ J ( J - I - 1 ) - k 2 ] ] } a _ . o f s ( / ) ) = 0 "-- 2 - ~ si n 40 d/) s in 40 d---~ s in 2 2/) + nks'm (35)E q u a t i o n ( 3 5 ) h a s s o l u t i o n s

fkSom(0) ,-, sin q 20 P ~ ' q)(cos 4/)) a s 0 ~ 0, (36)w h e r e t h e i n d e x q i s g i v e n a s q = { 8 9 + 1 ) - - k 2 3 } / 2 ( 3 7 )T h e c o r r e s p o n d i n g e i g e n v a l u e s a r e g i v e n b y n S~ . = 1 6 ( m + 8 9 + 1 + 89

C o m b i n i n g t h e a sy m p t o t i c f o r m s o f f ( 0 ) , v a l id a t / ) = i n a n d 0 , t h e s o lu t io n o f(3 1) m a y b e w r i t t e n a s

fs~m(/)) = co sv 2 /) sin s 2 /) O ~ m ( O ) , (38)wh ere 9kSom s a w e l l -beha ved fu nc t ion o f / ) . In the l im i t o f l a rge ~ , w i th th e su bs t i -t u t i o n o f (3 8), (3 1) b e c o m e s a n a l y t i c a l l y so l u b l e, a n d t h e n o r m a l i s e d 0 p o r t i o n o f th eu n s y m m e t r i s e d b a s i s f u n c t i o n Y ( t2 ) i s g i v e n b y

f J v m ( / ) ) = 4 r r ( m + 1) 1 ' /2r ( m + - p + 1 ) J c o s " 2 0 s i ns 2 0x exp ( - 8 9 cos 2 2 / )) L~(~ cos 2 20), (39)

w h e r e L ~ i s a n a s s o c i a t e d L a g u e r r e p o l y n o m i a l [ 29 1 o f d e g r e e m . I n f i g u r e 2 t h ea n g u l a r f o r m o f t h i s f u n c t i o n is s k e t c h e d f o r lo w p o l y n o m i a l d eg r ee s . I n t h e l i m i t

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Rovibrational states of H~ 1013

Figure 2.

4o , 1~o ; ~o So 45o1 ( a ) / ~

( c )Angular dependence of the basis functions ~m(,~) for m ---- 0 (a), 3 (b) and 5 (c). Thefunctions correspond to the state J -- 1, k = 1 and v = 3, with ~ = 25.

-~ oo the functions f(~) form an orthonormal set in ~ with respect to the volumeelement ~ sin 4& The energy spectrum of the reference Hamiltonian Ho is equi-spaced, with eigenvalues h 2E~vm(P) = 2 - ~ (A + 4~- -- 8~(p + 1)) + 4h---~-2~m. (40)gp2The normalised angular basis functions Y(fl) given by (30) and (39) may be sym-metrised by standard techniques [30]. Specifically, for each irreducible representa-tion F, we construct the projection operator

~ r = ~ ~ o r ( R ) ~ ( 4 1 )R

where R is one of the symmetry operators of D3h, Or(R) is the character of R in Fand the sum extends over all symmetry operators of D3h. By applying the projec-tion operator ~r, for each irreducible representation F of D3h, to theunsymmetrised basis function Y(Q), we generate the symmetry-adapted basis func-tion ~(~) given in (21) together with the symmetry properties summarized in table 1.The symmetrised basis functions ~(Q) constitute nearly ideal functions for theexpansion of the surface functions #. They are orthogonal and complete (in the limitof an infinite set and ~ ~ oo). All matrix elements between these functions are finiteat the equilateral geometry (~----88 and they have simple symmetry properties.However, there are several difficulties, some of which are easily overcome, in usingthese functions. First, for finite values of ~ the set of symmetrised basis functions~(12) is not strictly orthonormal , although, as we shall show in section 5, the appro-priate overlap integrals may be simply evaluated. Secondly, the basis functions donot have the correct limiting behaviour near ~ = 0, which may lead to unpredictableerrors if there is a significant wavefunction amplitude at this point. This problemarises because the basis functions are symmetric-top eigenfunctions with a defined

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1 01 4 P . B a r t l e t t a n d B . J . H o w a r da n g u l a r - m o m e n t u m c o m p o n e n t a l o n g t he B F z ax is . A t 3 = 0 th e E c k a r t s i n g u la r it yi n t h e p r i n c i p a l - a x i s H a m i l t o n i a n ( 1 2) r e q u i r e s a c o u p l i n g o f a l l p o s s i b l e c o m -p o n e n t s o f t h e a n g u l a r m o m e n t u m a l o n g t h e B F z ax is . A n y b a s is f u n c t i o n w i t h ad e f i n e d v a l u e o f k w il l h a v e t h e i n c o r r e c t a s y m p t o t i c b e h a v i o u r a t 3 = 0 . H o w e v e r ,t h e b a s i s f u n c t i o n s d e f i n e d a b o v e d e c a y s u f f i c ie n t l y f a s t f o r s m a l l 3 t o e n s u r e t h a t a l lm a t r i x e l e m e n t s r e m a i n f in it e. T h i s h a s t h e a d v a n t a g e o f e n s u r in g n u m e r i c a l s t a b il -i ty , a l t h o u g h f o r p o t e n t i a l s w i t h a s i g n i f ic a n t w a v e f u n c t i o n a m p l i t u d e a t 3 = 0 c o n -ve rge d e igenv a lues a re d i f f i cu l t to ach ieve .

5 . Adi abat i c s ur f ace f unc t i ons : m a t r i x e l em ent sT h e s u r fa c e f u n c t io n s 9 a r e a d i a b a t i c ; t h a t is , t h e y d e p e n d p a r a m e t r i c a l ly u p o n

p . T h u s , w h e n t h e s u r fa c e f u n c t i o n s ar e e x p a n d e d i n t e r m s o f th e s y m m e t r i s e d b a s isfun c t ion s ~ ( t2 ), the coe f f i c ien t s c a re fun c t ion s o f p ( see (19) an d (20)) . Su bs t i tu t io n o ft h e w a v e f u n c t i o n e x p a n s i o n ( 2 0) in t o t h e S c h r 6 d i n g e r e q u a t i o n g i ve s t h e s t a n d a r dm a t r i x e q u a t i o n s f o r t h e c o e f f i c ie n t s c i n t h e f o r m

E ( (~ ksS r lH a l i>ks~r') - uS r ( p ) (~ sy r l ~ks~r') )cSr;m '(P) = 0 , (42)k'v 'm"

w h e r e , w i t h t e r m s d e f i n e d a s i n (1 2),15h 2= ~ + V (p, 3, tp). (43)a T ~+ T~ + Tr + 8/zp2

T h e m a t r i x e l e m e n t s o f th e H a m i l t o n i a n H a a r e d e s c r i b ed b e l o w a c c o r d i n g t o t h e i ro r ig in .

T h e m a t r i x e l e m e n t s o f t h e a n g u l a r k i n e t i c e n e r g y T , t h e C o r i o l i s te r m s T r a n dt h e s y m m e t r i c - t o p p a r t o f T~ a r e o b t a i n e d f r o m ( 21 ), u s i n g t h e p r o p e r t i e s o f L a g u e r r ep o l y n o m i a l s , a s

g s u r- - k vm I T a " J I- Y s y m " ~ - T~I r{- 2/tp2 6kk'6vv' 4[(m ' + 1)(m' + 2X m' + 2 + p X m ' + 1 + p ) ] l / 2 S m , m , + 2+ 4(2q + 0[ (m ' + 1Xm' + 1 + p ) ' ] l / 2 S m , m , + l+ 4[q 2 -- 2m'(m' + p) + (~ -- 1X2 m' + p + 1) - 1]Sin, m'+ 4(~ -- 2q)[m'(m ' + p )] t / z S . , . ,_ 1+ 4[m ' (m' - 1)(m' + p X m ' + p - - 1 ) ] l /2 S m , m , - 2

2 k v ( f S , m 1 - - s in 23 }c o s2 2 3 I f ~ m ' ) , ( 44 )h e r e t h e 3 - o v e r l a p i n t e g ra l s Sin ,, a r e g i v e n b y ( f ~ m [ f ~ m , ) , a n d t h e i n d ic e s p a n d qa r e d e f i n e d f o l l o w i n g ( 33 ) a n d ( 3 7) . W i t h t h e s u b s t i t u t i o n x = ~ c o s 2 2 3 , t h e o v e r l a pi n t e g r a l s S in ,, m a y b e w r i t t e n i n t h e f o r m

[ F ( m + l ) F ( m ' + l ) 1 '/2 fr ( ~ ) "S .m, = 16~ r ( m + p + 1)r(m ' + p + 1)3 x p 1 - e - X L ~ ( x ) L ~ , ( x )d x . (45)F o r l a r g e v a lu e s o f ~ , c o r r e s p o n d i n g t o a d e e p m i n i m u m a t 3 = i n i n t h e r e fe r en c ep o t e n t i a l V o(p , 3 ) o f (2 7) , t h e u p p e r l i m i t o f t h e i n t e g r a l m a y b e r e p l a c e d b y i n f i n i t yw i t h o u t a p p r e c i ab l e e r r o r . I n t h i s c a se e x p a n s i o n o f t h e t e r m (1 - x / O q i n p o w e r s o f

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R o v i b r a t io n a l s t a te s o f H f 1015x a n d t h e u s e o f t h e r e s u lt d e r iv e d b y T e n n y s o n a n d S u t cl if fe I 4 ] o f

f :' u , = e - ~ x P + ' L ~ , ( x )L ~ ( x ) d x= ( _ l ) . , + m , ~ , ( r ) ~ ) F ( m + p + r - - j + l )i= o * + J F ( m - - j + 1) ' (46)w h e r e m is c h o s e n s o t h a t , = m ' - m i s p o s i t i v e , g i v e s S. , . , , i n t e r m s o f t h e r a p i d l yc o n v e r g e n t s e ri es

[ F ( m + l ) F ( m ' + , ) . ] x / 2 ~ r ~ ( q ) F m , r , , " (47)S m m , = 16~ F ( m + p + 1 ) r (m ' + p + 1 ) J , ~ o \ r /W i t h a s i m i l a r s i m p l i f i c a t i o n , t h e l a s t t e r m w i t h i n t h e b r a c e s i n ( 4 4 ) m a y b e w r i t t e na s

2 k v < f ~ . , I 1 - s i n 23 [ F ( m 1)F (m ' 1) .q x/2c o s 2 2 0 I f S ~ " > = 3 2 kV ~ 2 L + +r i m + t , + 1 ) F (m ' + p + 1 ) J[' | f / 1 - x \ ' ( 1 - x ' ] ' + l n ] e _ . L = L ~ , d x " (48)x J o = " - r j j

T h i s t r a n s f o r m e d i n t e g r a l m a y b e e v a l u a t e d e f f ic i en t ly b y n - p o i n t G a u s s - L a g u e r r eq u a d r a t u r e b a s e d o n t h e z er o s o f t h e p o l y n o m i a l LP. (x ) .T h e a s y m m e t r i c - r o t o r k i n e ti c e n e r g y T ~syh a s m a t r i x e l e m e n t s i n t h e s y m m e t r i s e db a s is s e t t h a t w e m a y w r i te a s

^ s ~ r # s ~ t r \ h 2 / 9 I ~ r COS 2 0 ~,sMr X: --2/tp2 9 k 'rY kvm I T ~ y l - k .~ , . , ,/ , ,- ko m ~ ( j 2 + + j 2 ) l (4 9)w h e r e t h e B F a n g u l a r - m o m e n t u m r a is in g a n d l o w e r in g o p e r a to r s h a v e b e e n d e fi n eda c c o r d i n g t o t h e i d e n t i t y

J = J x -T J , . ( 50 )T h e c h a n g e in s ig n e n s u r e s t h a t t h e B F a n g u l a r - m o m e n t u m o p e r a t o r s s a t i sf y c o m -m u t a t i o n r e l a ti o n s w i t h t h e o p p o s i t e s ig n s t o t h o s e o f t h e S F o p e r a t o r s . W i t h t h em a t r i x e l e m e n t s o f t h e r a is i n g a n d l o w e r i n g o p e r a t o r s g i v e n b y

( O ~ k l J I D SM kT- > = 2 k ) , (51)w h e r e 2 k ) = [ ( J + k + 1 X J " ~ k ) ] x / 2 , o n e r e a d i l y f i n d s t h e a s y m m e t r i c - t o p t e r m so f (4 2) m a y b e w r i t te n a s


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1016 P . Bar t l e t t and B . J . H ow ardwh e r e t h e sy m m e t r y p r o p e r t i e s r e q u i r e t h a t we n e e d o n l y c o n s i d e r e l e m e n t s w i t h kand k ' > /0 . W e no te tha t t he l a s t t e rm in the fma l se t o f squa re b ra cke t s i n (52) i sze ro fo r a l l s t a t e s wi th k o r k ' > /2 . The 0 in t egra l may be eva lua ted us ing the sameGa u ss i a n m e t h o d s a s f o r t h e k i n et ic - e n er g y t e r m s a b o v e . Su b s t i tu t i o n o f t h e a sy m p -t o t i c sm a l l- 0 f o r m f o r t h e f u n c t i o n f , ( 3 6 ), d e m o n s t r a t e s t h a t t h e a sy m m e t r i c - r o t o rt e rms rem a in f in it e nea r ~ = 0 .The angu la r po ten t i a l V(p , ~ , r wi th p fixed , i s i nva r i an t und e r a l l the sym -m e t r y o p e r a t i o n s o f Da b . T h e p o t e n t ia l m a y b e e x p a n d e d i n t h e Fo u r i e r s e ri es

V(p , ~, ~p) = 2 ~ Vr.(~; p) co s 6nq~ (53)n

wh ere ~ '6 . is a Fo ur i e r coe f fi c ien t t ha t depe nds pa ram et r i ca l ly u po n p . Wi th th i sexpans ion , t he po ten t i a l -ma t r ix e l ement s s impl ify to

15h 2/-~sM r I V (p, O, ~o) + - - I -~sMr ,,N a k v r a ! 2 l t p z Jt k ' v ' m ' /

15h 2- 6U . 6~ , .r m m . + {[1 + ( - - 1 ) s + k + ' f k o 6 , o ] [ 1 + ( - - 1 ) J +k ' +ar k , O r v , O ] } 1 /22 l t p 2J + k + ax ~ (fksvmI V r . Ifs'v'm ')( 1 + 6.o)[66., ~ '- , + (- - 1) 66., v'-v 6k, O] (54)

n

T h e i n t e g r a l s a r e e v a l u a t e d u s i n g t h e q u a d r a t u r e m e t h o d s d e sc r i b e d a b o v e , wh i l ethe F ou r i e r coe f f i ci en ts ~ '6 . a re r ap id ly de te rm ined us ing fa s t -Fo ur i e r - t r ansfo rmm e t h o d s .

6 . Th e rad ia l wav e f unc t ion : inc lusion o f non- ad iaba t i c it yThe ad iaba t i c su r face func t ion 9 d e p e n d s p a r a m e t r i c a ll y u p o n p . T h u s , wh e n t h e

expa ns ion o f (18) i s sub s t i t u t ed in to the fu ll Schr6d inge r equ a t ion (10), t he re su l t ingc o u p l e d - c h a n n e l e q u a t i o n s f o r th e r a d i a l w a v e f u n c t i o n g a r e o f th e f o r m

{ d ~ z + h-~2/~ E - u ~ r( p )] } Z ~ (p ) = ~ I Y f f ( p ) + Xff (p )~p~X ~r(p) , (55)wh e r e t h e n o n - a d i a b a t i c m a t r ix e l e m e n ts X( p ) a n d Y( p ) a r e d e f in e d a s

OXff (p ) = (~ [u r ( t2 ; p ) [ - 2 ~pp [ ~M r( t2 ; p ) ) ,(56)02Y f f ( P ) = ( ~ u r ( ~ 2 ; P) I -- 0p---~ r p ) ) ,

wi th the in t egra t ion re s t r ic t ed to the se t o f angu la r c oord ina te s I2 . W e seek aso l u t i o n t o t h is s e t o f c o u p l e d e q u a t i o n s b y a s su m i n g t h a t m o t i o n i n t h e r ad i a lc o o r d i n a t e p i s ' l o c a l l y s e p a r a b l e ' f r o m m o t i o n i n t h e a n g u l a r c o o r d i n a t e s t 2 . C o r -r e c t i o n s t o t h i s a d i a b a t i c a s su m p t i o n w i l l b e i n c l u d e d b y p e r t u r b a t i o n t h e o r y . T h ecoup led equa t ions (55) a re fo rma l ly iden t i ca l wi th the equa t ions fo r nuc lea r mot ioni n a d i a to m i c m o l e c u l e w i t h in t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n , i f p i s i d e nt i-f i ed wi th the in t e rnuc lea r d i s t ance and I2 re fe r s t o the e l ec t ron ic coord ina te s . Thesmal lness o f t he non -ad iab a t i c coup l ing t e rms a r i se in th is case because o f t he

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Rovibrational states of H ~ 1017di f fe rence in the cha rac te r i s t ic f r equenc ie s o f e l ec t ron ic and nuc lea r mo t ion . Thee q u a t i o n s f o r t h e n u c l e a r m o t i o n , a n a l o g o u s t o ( 5 5 ), a r e o n l y w e a k l y c o u p l ed , soa c c u r a t e so l u t i o n s m a y b e c a lc u l a te d w i t h a p e r t u r b a t i v e t r e a t m e n t o f t h e n o n -a d i a b a t i c t e r m s . Ho w e v e r , a n a d i a b a t i c - t y p e s e p a r a t i o n i s a l so v a l id i f a n g u l a r a n dr a d ia l m o t i o n s a r e o f c o m p a r a b l e f r e q u en c ie s , p r o v i d e d t h a t t h e a n g u l a r wa v e f u n c -t ion 9 d e p e n d s o n l y s l o wl y u p o n t h e r a d i al c o o r d i n a t e . A we a k r a d ia l d e p e n d e n c eo f 9 a r i ses i f t he ra t io o f t he angu la r k ine t i c ene rgy to the angu la r po ten t i a l ene rgyin (19) va r i e s smooth ly wi th the rad ia l coord ina te p . S ign i f i can t non-ad iaba t i c coup-l ing t e rms a re expec ted in reg ions o f p w here , fo r example , t he min im um of theangu la r p o ten t i a l sh i ft s wi th p .

In the i so t rop ic l imi t of no co upl in g (V(p, ~, tp) ~ V(p)) the no n-a diab at ic cou p-l ing t e rms in (55) mus t van i sh , and the coup led rad ia l equa t ions reduce to the se t o fo n e - d i m e n s i o n a l e q u a t i o n s

+ _ v r (p ) ] = ( 5 7 )

wh e r e t~ m a y b e t a k e n a s a z e r o t h - o r d e r a p p r o x i m a t i o n t o t h e r a d ia l wa v e f u n c t io nX in (18) and E ~ i s t he co r re spo nd ing ze ro th -orde r e igenvalue . The rad ia l Schr6d -i n g er e q u a t i o n ( 57 ) c a n b e so l v e d u s i n g a v a r i e t y o f m e t h o d s . W e h a v e u se d t h eNu m e r o v - C o o l e y f i n i t e - e l e m e n t a l g o r i t h m [ 3 1 ] , wh i c h i s b o t h e f f i c i e n t a n d a c c u -r a t e . B y a n a l o g y w i t h t h e e l e c t r o n i c B o r n - Op p e n h e i m e r a p p r o x i m a t i o n , i t c a ne a s il y b e sh o w n [ 3 2 ] t h a t t h e z e r o t h - o r d e r e n e r g y E ~ i s n ec e s sar il y a l o we r b o u n d t othe exac t ene rg y o f t he g rou nd s t a te . I f, how ever , t he d i ago na l coup l ing t e rm Yt,(P) isinc luded in (57) ,

+ ~ _ [E 1 _ U St(p)] _ yS r(p ) tx)ztSr(p) = 0, (58)then the re su l ting e ne rgy E ~, wh ich inc ludes t e rm s up to f ir s t o rde r i n the non-ad iaba t i c coup l ing , is j us t t he exp ec ta t ion va lue o f t he ful l H am i l ton ian fo r t hea p p r o x i m a t e ' l o c a ll y s e p a r a b l e ' w a v e f u n c t io n o f t h e fo r m

~l~osur = t '}~r (p)~ tsur (t~; p). (59)F r o m t h e v a r i a ti o n a l p r in c ip l e, E ~ i s r i g o r o u s l y a n u p p e r b o u n d t o t h e e x a c t e n e rg yo f t h e g r o u n d s t at e . T h e e n e r g y o f t h e g r o u n d s t a te i s t h e r ef o r e b o u n d e d b y t h eva lues E ~ and E 1.

The exac t so lu t ion to the se t o f we ak ly coup led d i f feren t ia l equa t io ns (55) m ayb e f o u n d b y p e r t u r b a t i o n t h e o r y . W e h a v e u se d a n i t e r a t i v e B r i l l o u i n - W i g n e r p r o -e e d u r e , wh i c h h a s t h e a d v a n t a g e t h a t , c o m p a r e d w i t h t h e c o n v e n t i o n a l R a y l e i g h -Sehr6d ing e r t r ea tment , t he e f fec t s o f h ighe r -o rde r t e rms m ay read i ly be inc luded .T h e n u m e r i c a l im p l e m e n t a t io n o f th i s p r o c e d u r e h a s b e e n d e sc r i b e d in d e t a i l els e-where [21] .

C o r r e c t i o n s t o t h e ' l o c a l l y s e p a r a b l e ' z e r o t h - o r d e r wa v e f u n c t i o n r e q u i r e t h ec o m p u t a t i o n o f i n te g r al s o f t h e r a d i a l d e r i v at iv e s o f t h e a n g u l a r wa v e f u n c t i o n ~ .T h e se n o n - a d i a b a t i c t e r m s m a y b e e v a l u a t e d e i t h er i n t e r m s o f m a t ri x e l e m e n t s o fthe de r iva t ive o f t he po ten t i a l [33] o r v i a a fin it e -d i f fe rence app roa ch [34] . He re weu se t h e l a tt e r m e t h o d f o r r e a so n s o f c o m p u t a t i o n a l e ase . I f t h e a n g u l a r su r fa c ef u n c t io n ~ ( t 2 ; p ) i s e v a l u a t e d a t t h e N e q u a l l y sp a c e d r a d ia l g r i d p o i n t s Pk = PO+ k Ap (k = 0 , . . . . N - 1 ) t hen the de r iva t ive a t p = Pro+l/2 m ay be app rox im a ted

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Rovibrational states o f H ~ 1019c u s s i o n t o t h e c o n v e r g e n c e o f t h e v i b r a t i o n a l s t a t e s o f A ~ s y m m e t r y o n t h e C Psur face .

W e s t a r t b y o p t i m i s i n g t h e v a l u e o f ~ . A t e a c h v a l u e o f p , ~ w a s c h o s e n t om i n i m i s e t h e l o w e s t e i g e n v a lu e o f t h e a n g u l a r H a m i l t o n i a n ( 43 ). F o r t h e C a r n e y a n dP o r t e r s u r f ac e , ~ w a s d e t e r m i n e d a s

f3 0 .0 (0.54 A ~< p ~< 0-90 A),= (2 5- 0 (0.90 A < p ~< 2.06 A). (63)

T h e e f fe c t o n t h e A ~ e ig e n v a l u es o f a s m a l l v a r i a t i o n i n t h e p a r a m e t e r ~ fr o m i tso p t i m u m v a l u e i s s h o w n i n t a b l e 2 . T h e a n g u l a r k i n e t i c a n d p o t e n t i a l m a t r i xe l e m e n t s w e r e e v a l u a t e d u s i n g a 3 0 - p o i n t G a u s s i a n - q u a d r a t u r e r u l e . T h e F o u r i e rc o e f f ic i e n ts P6 ,( 0; p ) w e r e c a l c u l a t e d u s i n g a 6 4 - p o i n t f a s t F o u r i e r t r a n s f o r m o f th epo ten t ia l V(p, 0 , tp).O n c e t h e o p t i m u m v a lu e o f ~ h a d b e e n d e t e rm i n e d , th e n u m b e r o f a n g u l a r b a s isf u n c t i o n s Y (I2 ) t o b e i n c l u d e d i n t h e m a t r i x d i a g o n a l i s a t i o n o f th e a n g u l a r H a m i l t o -n i a n H , w a s f o u n d i n a s e ri es o f t r ia l c o n d i t i o n s . F o r e a c h s y m m e t r y sp e ci es t h ea d i a b a t i c s u r fa c e f u n c t i o n r p ) w a s e x p a n d e d in t h e s et o f a n g u l a r b a s is f u n c t i o n s~ m ( g 2 ), w h e r e f o r t h e v i b r a t i o n a l s t a te s k = 0 . T h e v a l u e s o f t h e k i n e m a t i c q u a n t u mn u m b e r v i n t h is s u m a r e d e t e r m i n e d b y t h e s t a t e s y m m e t r y a n d a r e l i st e d i n ta b l e 1as the s e t + Iv ] . T he a ng u la r bas i s fun c t ion s Y0~,~ an d Y0-o~, a re ide n t i ca l ap a r t f ro ma p h a s e f a c t o r (s ee (2 1)) . H e n c e f o r s t a t e s o f k = 0 t h e k i n e m a t i c q u a n t u m n u m b e r vc a n b e f u r t h e r r e s t r i c te d t o t h e s e t I v ] . I n t a b l e 3 t h e o p t i m i s e d a n g u l a r b a s i s s e ts a rel i s t e d f o r t h e t h r e e v i b r a t i o n a l s y m m e t r i e s . C o n v e r g e n c e c h e c k s , d e t a i l e d i n t a b l e 4 ,

Tab le 2. Convergence o f A'I (J = 0) eigenvaluesi" with respect to ~:~.Vibrational state(Vl, v2 , 12) ~ = ~*w ~ = ~* + 2"5w

(0 , 0 , 0 ) 434 5"1 06 43 45" 10 6(1 , 0 , 0 ) 752 9"8 49 752 9"8 49(0 , 2 , 0 ) 913 3"1 60 9133 "171(2, 0 , 0) 1060 3-975 1060 3-974(0, 3 , 3) 11 66 8"6 4 11 66 9-1 4(1 , 2 , 0 ) 121 15-3 0 12 11 5 .3 4

"~ Eigenvalues in cm -1 for C P surface relative to poten tial minim um .Angular basis set described in table 3, 10 non-adiabatic channels.w ~* is g iven b y (63).

Table 3. The ang ular basis functions used in vibrational calculations.Vibrational-state Basissymm etry ( v n , Vm.,)? nr .z ~ s ize

A~ (0, 18) 6 2 8A~ (6, 24) 6 28E ' ( - 2 0, 16) 6 49t The qu an tum num ber v l ies in th e range vmi. ~< v ~< vmz.M axim um degree of Laguerre polynom ial .

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Ta ble 4. Co nverg ence ofA 'l (J = 0) eigenvaluesl" with respect to an gula r basis size:~.m ~ , vm x w

2 8 '

2 4

Vibrat ional s tate(V 1 , V 2 , 1 2 ) (6, 12) (5, 18) (6, 18)

(0 ,~ 0 ) 4345.107 4345.107 4345.106(1 , 0 , 0 ) 7529 '853 7 5 2 9 -8 5 1 7529 .849( 0 ,2 , 0 ) 9 1 3 3 - 7 8 8 9 1 3 3 - 2 1 7 91 33 .16 0(2 , 0 , 0 ) 10603 .984 10 6 03 . 9 7 5 1060 3 , 976( 0, 3, 3) 1 1 6 7 6 .1 3 1 1 6 7 1 . 7 7 1 1 6 6 8 . 6 4(1, 2 ,0 ) 1 2 1 1 6 . 3 0 1 2 1 1 5 . 5 2 1 2 1 1 5 . 3 01" Eigenvalues in c m - 1 for the C P surface.:~ ~ fixed at values giv en by (63), 10 no n-a dia ba tic channels.w Maxi m um degree o f Lague r r e po l ynomi a l and max i mum va l ue o f v inc l uded i n t hean gu lar basis set. T he basis set includes all states 0 ~< v ~< v~ x.

s h o w t h a t t h e l o w - l y i n g s t a te s a r e v e r y c l o se l y c o n v e r g e d ( < 0 -0 1 c m - 1 ) , w i t h h i g h e re i ge n v a lu e s b e in g c o n v e r g e d t o b e t t e r t h a n 1 c m - 1 .

I n t h e f i n a l s t e p o f t h e c a l c u l a t i o n t h e n o n - a d i a b a t i c c o r r e c t i o n t e r m s w e r ei n c l u d e d v i a B r i l l o u i n - W i g n e r p e r t u r b a t i o n t h e o r y . T h e a n g u l a r H a m i l t o n i a n w a sd i a g o n a l i s e d f o r 7 7 e q u a l l y s p a c e d v a l u e s o f t h e r a d i a l c o o r d i n a t e p i n t h e r a n g e0 - 5 4 A ~< p ~< 2 . 0 6 A . F i g u r e 3 d e p i c t s t h e r a d i a l d e p e n d e n c e o f t h e e i g e n v a l u e s U ( p )o f t h e a n g u l a r H a m i l t o n i a n f o r s o lu t i o n s o f A ~ s y m m e t r y . T o a c h i e v e e n e r g ie sa c c u r a t e t o 0 .0 1 c m - 1 , t h e e ig e n v a l u e s U ( p) a n d t h e n o n - a d i a b a t i c m a t r i x e l e m e n t sX ( p ) a n d Y (p ) w e r e i n t e r p o l a t e d , u s i n g t h e m e t h o d o f c u b i c s p li n es , o n t o a n e q u a l l ys p a c e d f i ne g r id o f 75 1 p o i n t s . T o c o m p l e t e t h e c a l c u l a t i o n a n d c o n v e r g e t h e p e r t u r -!

"7 2Ou% -

~ 1 2

8 '

4 ,

O

1 0 20 P . B a r t l e t t a n d B . J . H o w a r d

0 , 6 0 .8 1 0 1 2 1 4 1 .6 1 8 2 . 0

Figu re 3. Rad ial depe nden ce of the eigenvalues u~r(p) o f the fixed-p ang ular Ha m il tonianfor the v ibrat iona l s tates (J = 0) of A'~ symm etry. Calcu lat ions a re for the surface o fC arney and Por t e r .

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Table 5.Rovibrational states o f H~ 1021

Convergence of A'~ (J = 0) eigenvalues1" with respect to num ber of no n-adiab aticchannels included in iterative Brillouin-Wigner solution:~.N,w Rayleigh-Schr6dinger tlVibrational s tate

(V l , V , 12 ) 6 10 E 0 E 1 E 2(0, 0, 0) 4 3 4 5 .1 0 6 4 3 4 5 . 1 0 6 4 3 4 1 . 1 5 6 4 3 4 7 - 5 0 3 4 3 45 -1 1 3(1, 0, 0) 7 5 29 -8 4 9 7 5 2 9 . 8 4 9 7 5 3 6 . 1 8 5 7 5 4 5 . 8 4 0 7 5 2 9 .7 6 9(0, 2, 0) 9 1 33 - 17 6 9 1 3 3 - 1 6 0 9 1 0 9 . 4 3 9 9 1 4 8 - 7 5 9 9 1 33 -5 1 2(2, 0 , 0) 1 0 6 0 3 - 9 7 6 1 0 6 0 3 . 9 7 5 1 0 6 1 3 . 4 4 5 1 0 6 3 2 - 2 8 0 1 06 0 3.5 3 8(0, 3, 3) 1 1 6 6 9 - 19 1 1 6 6 8 - 6 4 1 1 7 3 5 - 4 3 1 1 8 1 1 . 0 8 1 1 6 44 -0 8(1, 2, 0) 1 2 1 15 .6 2 1 2 1 1 5 - 3 0 1 2 0 2 6 - 9 3 1 2 1 0 0 . 9 8 1 2 1 31 .9 6

t Eigenvalues in cm - 1 for CP surface.:~ Ca lcula tion w ith ~ fixed at v alues given by (63), an d the ang ular b asis described intable 3.wNu m ber of non-adiab atic channels included in i terative Bril louin-W igner solution.II Results of zeroth-, first- an d second-order Rayleigh-Schr6dinger per turb atio n theo ry.

b a t i o n s o l u t i o n o f t h e c o u p l e d r a d i a l e q u a t i o n s , a n i te r a ti v e f o r m o f B r i l l o u i n -W i g n e r p e r t u r b a t i o n t h e o r y w a s u s e d . T a b l e 5 i l l u s t r a t e s t h e r a p i d c o n v e r g e n c e o ft h e p e r t u r b a t i o n c a l c u l a t io n w i t h t h e n u m b e r o f r a d i a l c h a n n e l s N t, in c l u d e d i n t h ei t e r a t i v e s o l u t i o n . T e n r a d i a l c h a n n e l s e n s u r e c o n v e r g e n c e t o w i t h i n 0 .0 1 e r a - ~. T h ez e r o t h - , f i r s t - a n d s e c o n d - o r d e r R a y l e i g h - S c h r 6 d i n g e r e n e r g i e s a r e a l s o g i v e n f o rc o m p a r i s o n .T h e z e r o t h - a n d f i r s t - o r d e r e n e r g i e s , a s d i s c u s s e d i n s e c t i o n 6 , b r a c k e t t h e e x a c tg r o u n d - s t a t e e n e r g y . T h e s m a l l s h if t o f a b o u t 4 e m - 1 b e t w e e n t h e z e r o t h - o r d e r a n de x a c t e n e r g ie s s h o w s t h a t t h e a s s u m p t i o n o f a n a d i a b a t i c s e p a r a t i o n b e t w e e n r a d i a la n d a n g u l a r m o t i o n is j u st if i ed . T a b l e 5 s h o w s t h a t t h e n o n - a d i a b a t i c c o r r e c t i o nt e r m s b e c o m e i n c r e a s i n g l y s i g n i f i c a n t a s t h e v i b r a t i o n a l e n e r g y i n c r e a se s . T h er e a s o n s f o r t h i s a r e i l l u s t r a t e d i n f i g u r e 4 , w h e r e t h e n o n - a d i a b a t i c m a t r i x e l e m e n t sb e t w e e n t h e g r o u n d a n d f ir s t e x c i te d A ~ c h a n n e l s a re p l o t t e d a s a f u n c t i o n o f p . T h en o n - a d i a b a t i c m a t r i x e l em e n t s s h o w a s h a r p p e a k a t p ~ 1 .8 A , w h e r e t h e c h a r a c t e ro f t h e a n g u l a r e i g e n f u n c ti o n s c h a n g e s r a p i d ly . S t e r e o g r a p h i c p r o j e c t io n s o f t h ep o t e n t i a l , s i m i l a r t o f i g u r es 1 (b ) a n d (c ), s h o w t h a t a t t h i s v a l u e o f p t h e a n g u l a rm i n i m u m s h if ts f r o m t h e e q u i l a t e r a l g e o m e t r y t o w a r d s a n e a r - c o ll i n e a r g e o m e t r y .T h e h i g h - e n e r g y v i b r a t io n a l s t a te s s a m p l e t h is l a rg e -p p o r t i o n o f t h e p o t e n t i a l-e n e r g y s u r f a c e , a n d h e n c e t h e n o n - a d i a b a t i c c o r r e c t i o n t e r m s b e c o m e m o r e s i g n i f i -c a n t .

7.3. Rotationa l-basis-set selectionI n H ~ - t h e b o d y - f i x e d z p r o j e c ti o n k o f t h e t o t a l a n g u l a r m o m e n t u m i s n o t

s t ri c tl y a c o n s e r v e d q u a n t i t y . T h u s a t e a c h J a n d F t h e m a t r i x r e p r e s e n t a t i o n o f t h ea n g u l a r H a m i l t o n i a n h a s a b l o c k e d s t r u c t u r e w i t h s u b m a t r i c e s ( J F I k [ ), d i a g o n a l i nt h e B F p r o j ec t io n q u a n t u m n u m b e r I k l , c o u p l e d b y a s y m m e t r ic - t o p m a t r i xe l em e n t s t h a t a r e o f f -d i a g on a l i n I k l . E a c h s u b m a t r i x ( J F I k l ) is c o m p o s e d o fm a t r i x e l e m e n t s o f t h e a n g u l a r H a m i l t o n i a n i n t h e s et o f s y m m e t r i s e d a n g u l a r b a s isfunctions Ylktvm(t2)w i t h k fi x ed a n d t h e k i n e m a t i c q u a n t u m n u m b e r v l i m i t e d t o t h es y m m e t r y s e ts + _[ v] l i s te d i n t a b l e 1. T h e f u n c t i o n s Ylklvrnan d i~l~l_vm are ine qu iva -l e n t , a p a r t f r o m t h e s p e c i a l c a s e o f k = 0 , w h e r e t h e y d i f f e r b y a t r i v i a l p h a s e f a c t o r .

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1022 P. Bartlett and B. J. Howard

"rE 15~ o x ( # )

> - 1 0 .o_

~, 5 '

0 !; -

~ - 5 -&7~ -10-

- 1 5 -

Figure 4. Radial dependence of the non-adiabatic matrix elements xIr(p) and y~r(p) forJ = 0 and symmetry F = A~, between the ground-state adiabatic surface function~l~ur(t2;p) and the first excited state qr~2ur(g2;p). Calculations are for the surface ofCarney and Porter.

For a complete rovibrational calculation both functions must be included in thematrix representation ( JF I k I) for non-zero k.For states of E symmetry the size of the angular basis set may be reduced by theuse of an approximate symmetry classification. Equations (44), (52) and (54) demon-strate that the matrix elements of the angular Hamiltonian obey the selection ruleAv = 0 (mod 6), with the single exception of the asymmetric-top kinetic-energy oper-ator, which connects the functions ~lklv,~ and YIk'l-v- if k and k' ~< 2. In states of A1and A2 symmetry v is a multiple of three, so matrix elements between Ylkl~m andgk'l -m satisfy the selection rule Av = 0 (mod6) and are generally non-zero.However, for states of E symmetry, where o is not a multiple of three, onlyasymmetric-top terms connect these two functions. In the conventional normal-mode description of H~- [35] the two angular basis functions ~lklmn and Ylkl-omcorrespond to rota tional levels of different vibrational 12 manifolds, where 12 is thevibrational angular momentum quantum number associated with the degeneratebending mode v2. Perturbation arguments suggest that the effect of these vibra-tional off-diagonal terms will be small consequently we shall ignore them here. Withneglect of these terms, the submatrix (JF I k t ) may be partitioned into two, with thekinematic quantum number v limited to the set + [v] in the upper partition and to- I v ] in the lower partition.

For the general rovibrational problem the optimum size of the angular basis isfound by methods similar to those described in section 7.2. Table 6 lists the angularbasis sets used in the rotational calculations of H~" described below. All other

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p. bartlett and b. j. howard- the rotational-vibrational spectrum of symmetric non-rigid triatomics...

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