Brian T. Sutcliffe and Jonathan Tennyson- A generalized approach to the calculation of...

8/3/2019 Brian T. Sutcliffe and Jonathan Tennyson- A generalized approach to the calculation of ro-vibrational spectra of triat

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MOLECULAR PHYSICS, 1986, VOL. 58, NO. 6, 1053 -1066

A g e n e r a l i z e d a p p r o a c h t o t h e c a l c u l a t i o n o fr o - v i b ra t io n a l s p e c t r a o f t r i a to m i c m o l e c u l e sby B R IA N T . SU T C L IFFE

Chemistry De partmen t, University of York, Heslington,York YO1 5DD, England

and JO N A T H A N T E N N Y S O NDepartment of Physics and Astronomy, University College London,

Gower Street , Lo ndon WC 1E 6BT, England( R e c e i v e d 1 2 M a r c h 1 9 8 6 ; ac c e pt ed 2 8 M a r c h 1 9 8 6)

A generalization of the well known atom-diatom scattering hamiltonian toa coordinate system of two lengths and an angle is derived, another specialcase of which is a previously known bond angle-bond length hamiltonian.Different axis embeddings are also considered. The formalism is applied tothe ro-vibrational levels of D2 H+, CH~ and H DH e (A 1A') and the advan-tage of a judicious choice of coordinates demonstrated. The vibrational bandorigins for HDHe*, the first predictions for this system for which previouscalculations had failed, are obtained using a new geometrically defined coor-dinate system. It is suggested that these coordinates might be used to rep-resent isotopically substituted van der Waals complexes.

1. INTRODUCTIONRecently there has been much interest in the a b i n i t i o calculation of the

vibration-rotation spectra of triatomic systems as experimental interest in suchsystems has turned to those for which extremely good electronic structure calcu-lations are possible. Perhaps foremost amongst such systems are H~ and itsisotopo mers [1]. Naturally m any of the nuclear motion calculations have beenmade using the traditional Eckart hamiltonian [2] in its quantum mechanicalform as given by Watson [ 3 ] . However it has long been realized that this hamilto-nian was inappropriate for large amplitude vibrations. As attention has focused onhigher energy regions where all vibrational motion becomes large, calculationshave been made with using specially constructed hamiltonians which, it wasargued, were particularly appropriate to the problem at hand. This has led to anincreasing number of candidate hamiltonians and co mplement ary solution stra-tegies being suggested for even the relatively simple tria tomic pr oblem.It is of course a relatively straightforward and well understood matter tocompare and to evaluate the results of different calculations using the same hamil-toniap but it is a much more vexed matter to compare calculations using differenthamiltonians. In particular, although one knows that one's choice of coordinates,embedding and consequent hamiltonian should be determined by the potential ofthe problem [4], the best choice is rarely obvious a p r i o r i .


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1054 B. T. Sutcliffe and J. Ten nys onIn a series of papers [5- 10] we have developed a nd applied a solution strategy

based upon the use of body-fixed scattering coordinates. This method has provedsufficiently robust to give reliable results for problems for which these coordi-nates cannot be thought of natural (e.g. [8]). Meanwhile other workers havedeveloped similar procedures based upon the use of different coordinates, forexample the bond length-bond angle approach of Carter and Handy [11-14].This proliferation of hamiltonians and methods is under standa ble b ut not desir-able particularly when one considers polyatomic systems for which the number ofplausible coordinate systems increases alarmingly.

The object of the present paper is to show how the two hamiltonians men-tioned above can be simply related to each other as special cases of a co nti nuu m ofhamiltonians expressed in terms of two lengths and an included angle. We showthat the choice of hamiltonian can be reduced to the definition of a single param-eter to define the coordina te syste m and anothe r to define the embe ddin g of theaxes. T hese parameter s can be part of the data input for a generalized compu terprogram. We also discuss the way in which a third well-known hamiltonian mightalso be related to these two. We present the results of calculations on severalsystems using the various hamiltonians, including another special case, a geo-metrically defined hamiltonian that has not previously been suggested, in order toshow how a comparison and evaluation of them can be effected.

2 . THE C ONSTR UC TI ON OF THE HAMILTON IANSA rather general method for the construction of hamiltonians in body-fixed

coordinates has recently been described [15]. In this method a set of translation-free internal coordinates t i are first constructed in terms of the laboratory fixedcoordinates x i such that

Nt i = ~ x j V ~ i , i = l , 2 , . . . , N - - 1 ;j=l (1)

or t = xV,where N is the total nu mbe r of particles and the elements of V are chosen suchthat

b/V 0= 0 a l l j = l , 2 . . . . . N -1 (2)i=1

to ensure the invariance of the tl unde r u nif orm translations. In terms of thesecoordinat es the translation-free part of the kinetic energ y operator can be writtena s

1 N-1= - - - ~ G 0 . V ( t l ) . V ( t ~ ) ( 3 )/~ 2 i,j=l

where V(ti) is the usual grad ope rat or expressed in the variables t i and where thematrix G has elements

N( ~ 0 = ~ m ~ a V i i I l k (4)

k = l


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G e n e r a l i z e d r o - v i b r a t i o n a l c a lc u l a t io n s 1 0 5 5i n w h i c h t h e m k a r e t h e p a r t i c l e m a s s e s .

T h e t r a n s l a t i o n - f r e e i n t e r n a l c o o r d i n a t e s a r e n o w t r a n s f o r m e d b y a n o r t h o g -o n a l t r a n s f o r m a t i o n C s o t h a t

t i = C z i ( 5 )a n d a m o n g t h e z i t h e r e a r e t h r e e r e l a t io n s

f m ( Z 1 , z 2 , . . . , z N _ 1 ) = 0 , m = 1 , 2 , 3 ( 6 )t h a t d e f in e t h e w a y i n w h i c h t h e c o o r d i n a t e f r a m e i s f i x e d i n t h e s y s t e m . T h em a t r i x C i s p a r a m e t e r i z i b l e i n t e r m s o f t h r e e E u l e r a n g l e s (cr f l, 7 ) a n d t h e c o o r d i -n a t e s z i a r e e x p r e s s i b l e in t e r m s o f 3 N - - 6 i n d e p e n d e n t r o t a t i o n - a n d t r a n s l a t i o n -f r e e i n t e r n a l c o o r d i n a t e s , q k . T h e b o d y - f i x e d h a m i l t o n i a n c a n t h e n b e w r i t t e n i nt e r m s o f t h e a n g u l a r m o m e n t u m o p e r a t o r s ( i n v o l v i n g at, f l , y ) a n d o f o p e r a t o r s a n df u n c t i o n s o f t h e q k . P r e c i s e l y h o w t h i s is t o b e d o n e i s o u t l i n e d i n [- 15 ] a n d s h o w ni n s o m e d e t a i l fo r a p a r t i c u l a r c h o i c e o f t r i a t o m i c c o o r d i n a t e s i n [-5] s o t h a ta l g e b r a i c d e t a i l w i ll n o t b e g i v e n i n t h e p r e s e n t p a p e r .

C o n s i d e r n o w a t r a n s l a t i o n - f r e e s e t o f c o o r d i n a t e s s u c h t h a t t 1 i s t h e b o n dl e n g t h v e c t o r f r o m p a r t i c l e 2 to p a r t i c l e 3 a n d t 2 i s a p r o p e r c o m b i n a t i o n o f a l lt h r e e l a b o r a t o r y - f i x e d c o o r d i n a t e s . T h u s t h e f o r m o f V is

V = - - 1 , 0 ~ < g ~ < l . ( 7)1 g

A m o n g t h e t r a n s l a t i o n - f r e e c o o r d i n a t e s e t s th a t h a v e s u c h a f o r m i s o n e i n w h i c ht 2 i s t h e b o n d l e n g t h v e c t o r f r o m p a r t i c l e 2 to p a r t i c l e 1 , o b t a i n e d b y s e t t i n gg = 1 . A l s o p o s s i b l e i s t h e c o l l i s i o n o r s c a t t e r i n g c o o r d i n a t e s e t i n w h i c h t 2 i s t h ev e c t o r f r o m t h e c e n t r e - o f - m a s s o f t h e d i a t o m i c 2 - 3 t o p a r t i c l e 1 . F o r t h i s c a s e

g = ~ / 2 m d 1 , m d = ~ / 2 "}- ~ / 3 " ( 8 )T h e G m a t r i x g e n e r a t e d b y V a s g i v e n in e q u a t i o n ( 7 ) h a s e l e m e n t s

G l l = # ~ 1 = m 2 1 + m 3 1 , }!G 1 2 # '( 21 = g ( m 2 i + m f l ) - m a l ' t ( 9 )|G22 # 2 1 = m 1 -1 + g 2 m 2 1 + ( 1 - - g ) 2 m a l . ]

A s u i t a b l e s e t o f i n t e r n a l c o o r d i n a t e s i n th i s a p p r o a c h i s r l , t h e l e n g t h o f t l ;r 2 , t h e l e n g t h o f t2 a n d 0 , t h e a n g l e b e t w e e n t l a n d t 2 . T h e m a t r i x C m a y b ed e f i n e d b y c h o o s i n g i t s o th a t t h e e m b e d d e d z - a x i s l ie s e i t h e r a l o n g t l o r t2 a n dt h e r e m a i n i n g c o o r d i n a t e h a s a r a n g e o f k e e p i t i n t h e p o s i t i v e x h a l f o f t h e x zp l a n e , w i t h t h e a d d i t i o n a l r e q u i r e m e n t t h a t th e t h r e e b o d y - f i x e d ax e s f o r m ar i g h t - h a n d e d s e t . T h u s i f t h e c o o r d i n a t e c h o s e n t o d e f in e t h e z - a x i s i s d e n o t e d b ya a n d t h e o t h e r c o o r d i n a t e b y b t h e n

C T a = a , C T b - - b 0 , ( 1 0 )c o s

w h e r e a i s e i t h e r r 1 o r r 2 a n d b i s e i t h e r r 2 o r r 1 a c c o r d i n g t o t h e e m b e d d i n gc h o s e n . W h i c h o f t h e s e e m b e d d i n g s i s u s e d i s u n i m p o r t a n t i n a f u l l c a l c u l a t i o n ,


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1 05 6 B . T . S u t cl if fe a n d J . T e n n y s o nb u t w i ll b e i m p o r t a n t i f a p p r o x i m a t i o n s , s u c h a s t h e n e g l e c t o f o f f -d i a g o n a lC o r i o l i s in t e r a c t i o n s a r e t o b e m a d e [ 7 ] .

I t is n o w p o s s i b le t o w r i te d o w n t h e h a m i l t o n i a n f o r th e s y s t e m i n t e r m s o f th ea n g u l a r m o m e n t u m o p e r a to r s , a n d o p e r a t o r s a n d f u n c t i o n s i n v o lv i n g o n l y th ei n t e r n a l c o o r d i n a t e s b y f o l lo w i n g t h e p r e s c r i p t i o n g i v e n in [ 1 5 ] a n d i l l u s t r a t e d i n[ 5 ]. A s e x p la i n e d i n [ 1 5 ], t h e a n g u l a r m o m e n t u m o p e r a t o r s c a n t h e n b e e l i m i -n a t e d f r o m t h e e x p r e s si o n . T h i s is d o n e b y a l l o w i n g th e o p e r a t o r t o a c t o n am a n i f o l d o f f u n c t i o n s w h i c h a r e p r o d u c t s o f f u n c t i o n s o f th e i n t er n a l c o o r d i n a t e sa n d s t a n d a r d a n g u l a r m o m e n t u m e i g e nf u n c ti o n s a n d t h e n i n t e g r a t in g o v e r th ea n g u l a r v a r ia b l es . B e c a u s e th e e n e r g y d o e s n o t d e p e n d u p o n t h e p r o je c t i o n , M , o ft h e t o ta l a n g u l a r m o m e n t u m , J , a l o n g t h e s p a c e - f ix e d z - a x is , i t is s u ff i ci e n ts i m p l y t o c o n s i d e r t h e p r o j e c t i o n , k , o f J a l o n g t h e b o d y - f i x e d z - a x i s . O f c o u r s e kis n o t i n g e n e r a l a g o o d q u a n t u m n u m b e r a n d t h e r e s u l t i n g o p e r a t o r c o n s i s ts o f2 J + 1 c o u p l e d t e r m s , t h e k i n e ti c e n e r g y p a r t o f w h i c h c a n b e w r i tt e n a s

g: + (11 )w i t h

T 0r---~ + sin-----00 0 s i n 0

+ - - r 2 + - - - - s i n 0 ( 1 2 )/ 22 r ~ ~ s i n 0 O 0 '/~v2, h2 [ c32 c o s 0 ( 1 ~ ~ 0 )= - c o s 0 + - - 0 0 0/212 ~ rlr2 s i T - - s in 0

+ s i n 0 ( 1 ~ 1 ~ 1 ) ~ 0 ]~ r 2 + - - - - +2 ~ r I r l r 2I ~ ( V l R = ( ~ k ' k h2 (J(J + 1 ) - - 2k 2) : o ( , , _+ T c ~ ~ + / 22 r 2 " 1 2 r 1 2 ./)6k,k+l 2#xr 2 Cfk -- + k cot O

( 1 3 )

+ 3k,k_ 1 2#1r ~ Cfk + k co t 0 , ( 14 )

- 2) - - Cfk c o s 0 - - k c o t 0 + r 2 ~ r 2 - - k s in 0~ ( ~ [ R = ( ~ k ' k + l 2/212 r l r 2

" ~ - b k , k _ 1 - - Cjk --C O S 0 + k c o t 0 + r2 ~3r---~+ k s in 0 . (15)2/212 rl r2T h e f o r m g i v e n a b o v e i s a p p r o p r i a t e f o r t h e z - a x i s e m b e d d e d a l o n g t 1. T h ee m b e d d i n g a l o n g t 2 is o b t a i n e d s i m p l y b y m a k i n g t h e e x c h a n g e s r 1 ~ -* r2 a n d/21 * -~ /2 2. I n u s i n g t h e o p e r a t o r i n t h i s f o r m a n y i n t e g r a l o v e r t h e i n t e r n a l c o o r d i -n a t e s m u s t b e p e r f o r m e d w i t h t h e v o l u m e e l e m e n t 2 2 s in 0 dr 1 dr 2 dO. T h ei r 2


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G e n e r a l i z e d r o - v i b r a t i o n a l c a l c u l a t i o n s 1057range of r i is (0, oo) and tha t o f 0 (0, 7r). Th e coef ficients C f k are the usual step upand down coefficients

C f k = ( J ( J + 1) -- k ( k - b 1)) 1/2. (16)It is easily seen that if the s catter ing coo rdinates are us ed the n #[21 vanishes and/s reduces to the standard form often used in describing atom-diatom collisions(e.g. [-5]). If the b ond len gth -b on d angle embe ddi ng is used so that r 1 is the 2-3bon d, r 2 the 2-1 bond and the z-axis is placed along tx then

#t =m3 x + m 2 1 , ] 2 2 = m [ x + m r 1, P ~ 2 = m 2 1 (17)and the hamiltonian reduces to the form attributed to Hagstrom and Lai [-16] (seealso [-4] and [,17]). Clearly othe r form s are possible too, for exampl e one in whichg is chosen as 89 so placing the origin for particle 1 at the geometric centre of thediatomic 2-3. In this case (with the z-axis along tl)

m -X = m i x - 1 - - m r 1) ( 1 8 )2 1 = m 2 1 + 3 , ]22 + 88 + m 3 ), ]2 12 -----~ ( m 2 1and naturally this coordinate system is identical with the scattering coordinates inthe case of a homonuclear diatomic 2-3.

The given form of the kinetic energy operator is often called the 'non-hermitian ' form but this is a somewhat misleading appelation. It must be remem-bered that hermiticity is defined only with respect to a specified manifold offunctions and in terms of integration between products of such functions, giventhat the effect of the oper ator on a function is to produ ce an other functio n that isalso in the manifold. If a suitable manifold is chosen and the jacobian in thevolume element for integration is properly included then it is easy to show that agiven operator is hermitian. The choice of a suitable manifold is a matter of somedelicacy in the present case, for inspection of the operator indicates that unlessthe functions are rather carefully chosen then divergent integrals could beencounter ed arising from terms in rf 2 as ri---~ 0 and those in cosec 2 0 as 0- * 0 orzr. Certainly the operator could not be hermitian on any manifold that permittedsuch behaviour. It is easy to accomplish what is necessary to avoid the possibledifficulties with the radial variables because the singularity is of no higher orderthan that which occurs in the central field problem after the angular motion isseparated off. Th us a suitable manifo ld of radial func tions can be construc tedfrom products of central field functions with the form r ; X r 2 1 0 ( r l , r 2 , 0 ), torequ ire that (I) is norma lizab le usi ng the vo lum e element d r 1 d r 2 sin 0 dO and tore-wr ite the kinetic ener gy operato r so that (I) is a suitable trial function . T hisprocess corresponds to incorporating the radial part of the jacobian into theoperator (in the manner described, for example, by Kemble [,18]): The resultingoperato r is often said to be in ' hermitian fo rm '. This form is given below for thepresent operator.The problem with 0 is a little more difficult to deal with. To describe what isto be done it is convenient to imagine that each of the standard angular-mo me nt um funct ions, DSk(0~, fl, 7) [-19], whic h are used above in the derivation of(11) are supplemented for any chosen J, k by a standard associated Legendrefunction Ojk(O as defined by Condon and Shortley [-20]. Clearly, the maximumvalue of [ k [ is J, and j mus t be such that it admits the c hosen k. Th e manifold of


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1 0 58 B . T . S u t c l i f f e a n d J . T e n n y s o nf u n c t io n s o n w h i c h t h e o r i g i n a l h a m i l t o n i a n o p e r a t e s m a y b e t h o u g h t o f a s c o n -s i s ti n g o f t h e p r o d u c t s

r l l r f t s D S k ( O ~ , f l , ( 1 9 )q / = , k j ( r l , r 2 ) O j k ( O ) ~ ) .I f t h e i n t e g r a l s o v e r t h e a n g u l a r v a r i a b l e s t h a t l e a d t o e q u a t i o n s ( 1 2 ) - ( 1 5 ) a r e

n o w e x t e n d e d t o i n v o l v e i n t e g r a t i o n o v e r 0 , i t i s s e e n th a t e a c h o f t h e t e r m s i n / ~ v 1)a n d / ~ v z) t h a t i n v o l v e d e r i v a t i v e s o f 0 g iv e r i se t o t e r m s i n k z c o s e c 2 0 t h a t e x a c t l yc a n c e l t h e t e r m i n k 2 c o s e c 2 0 i n /~V lR. T h u s , p r o v i d i n g t h a t t h e m a n i f o l d c o n s i s t so f f u n c t i o n s l ik e ( 1 9 ), t h e r e i s n o p o s s i b i l i t y o f d i v e r g e n c e a n d t h e o p e r a t o r i sh e r m i t i a n o n s u c h a m a n i f o l d . C h o o s i n g t h e r a d ia l f u n c t i o n s a s i n ( 19 ) o r , e q u i v a -l e n t ly i n c o r p o r a t i n g t h e r a d ia l p a r t o f t h e j a c o b i a n i n t o th e o p e r a t o r , a n d i n t e -g r a t i n g o v e r a ll t h e a n g u l a r v a r i a b l e s l e a d s t o a n e f f e c t i v e o p e r a t o r w h i c h w o r k so n l y o n f u n c t i o n s o f t h e r a d ia l v a r ia b l e s . T h i s o p e r a t o r , w h i c h i s m a n i f e s t l yh e r m i t i a n , h a s th e f o r m :

r _ _ h 2 ~ 2 _ _ ? , /2 ~ 2

/ ~v 2 ) h 2 ( ~ r l ( j + 1 ) . ~ ( . c 1- - - - - d j ~ b i ' j + l 6 k ' kI . l l 2 r l } k & 2- - - - d j _ 1 k ( ~ j ' j - * 6 k ' k + 7

/ ' / 1 2

+ - - f j ( j + 1 ) + , ( 2 0 )

( j r 2 + ) )+ J ) , ( 2 1 )

l t l 2I~ v ~ )R = 6 t , t / J j , ~ 2 / x , r 2 ( J ( J + 1 ) - 2 k 2 )

~ 2- 6 j, j ~ ( 6 ,,, + l c ; , c j l + 6 , , , _ , c i , c j ; ) ,

^ 2 )I ~ R = 6 k , k + l ~ j, j+ l 2/'/12 C f k aj__kk ( j + 1) C3r 1 r 2

-~ - ( ~ k 'k + l ( ~ j ' j - 1 2#1----22C f k - - +r lO O~k,k_lr a i - k .( j + 1) c9

r i r 2 U T

~ t_ 6 k , k _ l ( ~ j , j - 1 2#1---~2C f k - ~- "r 1

( 2 2 )

( 2 3 )I n t h e a b o v e , t h e e m b e d d i n g i s f o r z a lo n g t 1 a n d t h e a u x i l i a r y q u a n t i t i e s a r ed e f i n e d a s f o l l o w s

d j k = [ ( j + k - - 1 ) ( j + k + 1 ) /( 2 3 ' + 1 ) ( 2 j + 3 ) ] 1 /2 ,a s k = [ ( j + k + 1 ) ( j + k + 2 ) / (2 3 ' + 1 ) ( 2 j + 3 ) ] 1 /2 ,b~ k = [ ( j - - k ) ( j - k - - 1 ) / ( 4 j 2 - 1 ) ] 1 /2

w h i c h a r e s p e c ia l c a s e s o f C l e b s c h - G o r d a n c o e f f ic i e n t s .

(24)(25)(26)


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Genera l i z ed ro -v ibra t iona l ca lcu la t ions 1059^ 2)If scatter ing coo rdina tes are used then #i-~ vanishes. /~v ) and / ~R are thus

absent from the kinetic energy operator and it reduces to the form used by theauthors in some of their previous work (e.g. [-7, 8]).

It can be seen from the form of the kinetic energy operator given above that itwould be possible to take matrix elements once and for all over a given set ofradial functions. These matrix elements could then be combined at will with theappropriate mass factors to give full elements for a problem with any choice ofinternal coordinates and emb edd ing o f the axes. This formal possibility is, unfo r-tunately, of little practical consequence because, as detailed in the next section,the potential appears to be different with each of the internal coordinate choices.This apparent difference determines the details of an optimum expansion func-tion choice and thus in practice the radial matrix elements must be calculatedafresh for each choice. However, the actual computer code necessary to evaluatethese matrix elements is not dependent on these details provided that the sametype of radial functions are used in each case. This means, as asserted in theintroduction, that a single program can be used for a~l cases with the choice ofcoordinates and emb edd ing simply being signalled as part of the input data.

3. MATRIX ELEMENTSTwo considerations need to be addressed in the calculation of matrix ele-

ments. Firstly, in w we integrated the kinetic energy operator over all angularcoordinates, but did not consider the potential. Secondly, for an actual calculationis it necessary to solve for the radial coordinates. We choose to do this by definingradial basis functions and thus need to compute the resulting radial matrix ele-ments. This proc edur e implies the constr uction and diagonalization of a secularmatrix and is fully variational.

Th e electronic potential is purel y a function of the internal coordinates,V ( r x , r 2 , 0 ) . If this potential is expressed as a Legendre expansion in cos 0,

V (r l, rz , O) = ~ V:~(rl , r2)P:~(cos O) (27);t

matrix elements over the potential for the angular basis functions defined abovecan be computed analytically

0 0 0 ] \ - k 0 '(28)

where the integral is over all the angular coordinates and the 3 -j symbols in theGau nt coefficient are convention al [,19]. In the atom -d iat om scattering coordi-nates the potential is often given in a Legendre expansion, but this is unlikely tobe generally the case. However Gauss-Legendre integration has been used suc-cessfully to express a general potential function in the form of equation (27)[-7, 8]. This proce dure, which has been shown to have distinct com putatio naladvantages [-21], is equally applicable to the generalized coordinates consideredhere.

In previous work it has not generally been found necessary to use radialfunction s in (19) that d epen d explicitly on J, j and k (for an exception see [7]) a nd


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1 06 0 B . T . S u t cl if fe a n d J . T e n n y s o ni t h a s p r o v e d s u f fi c ie n t t o e x p r e s s t h e m a s s u m s o f p r o d u c t s o f o n e - v a r i a b l e r ad i alf u n c t i o n s . O f t h e v a r i o u s r a d ia l f u n c t i o n s t h a t h a v e b e e n t e s t e d i n p r e v i o u s c a l c u -l a t io n s [ 5 - 1 0 ] , t h e M o r s e o s c i l l a t o r- l ik e fu n c t i o n s [ 5 ] h a v e p r o v e d t h e m o s td u r a b l e a s t h e y a r e b o t h f l e x ib l e a n d c o m p u t a t i o n a l l y t r a c t a b l e i n m u l t i -d i m e n s i o n a l c a l c u l a t i o n s . T h e n o r m a l i z e d M o r s e o s c i l l a t o r - l i k e f u n c t i o n s a r eg i v e n b y

/ ~ l / 2 N tn ) = H n ( r ) = r _ . ., ~ e x p ( - - y / 2 ) y (~ + x ) / 2 L ~ ( y ) ,y = A e x p [ - - f l ( r - - r e ) ] , (29 )

w h e r e4 D e {" /.t "~1/2

A = f l , f l = coe~ .~ -~ , a = In t e g e r (A ) (30 )a n d t h e p a r a m e t e r s r e , o 9e a n d D e c a n b e a s s o c i a t e d w i t h t h e e q u i l i b r i u m s e p a r a -t io n , f u n d a m e n t a l f r e q u e n c y a n d d i s so c i a ti o n e n e r g y o f t h e c o o r d i n a t e r e s p e c -t i v el y . I n p r a c t i c e t h e y a re u s u a l ly t r e a t e d a s v a r i a t i o n a l p a r a m e t e r s a n d o p t i m i z e da c c o r d i n g l y [ 1 0 ] . N , , ~L ~ is a n o r m a l i z e d L a g u e r r e p o l y n o m i a l [ 2 2 ].

T h e k i n e ti c e n e r g y o p e r a t o r d e r i v e d i n w2 c o n t a i n s f o u r r a d i a l o p e r a t o r s , t w oo f w h i c h a r i s e i n t h e s c a t t e r i n g c o o r d i n a t e c a l c u l a t i o n c o n s i d e r e d p r e v i o u s l y , r - 2a n d 0 2 / a r 2 , a n d t w o o f w h i c h a r e n e w , r - 1 a n d a / O r . U s i n g t h e t e c h n i q u e g i v e n[ 5 ] ( i n w h i c h i t i s a s s u m e d t h a t t h e r a n g e o f y i s ( 0 , oo )) i t i s p o s s i b l e t o d e r i v ea n a l y t i c c l o s e d f o r m s f o r m a t r i x e l e m e n t s o f th e d i f f e r e n t ia l o p e r a t o r s

~ 2 j ~ l / 2( n ' l 7 r 2 i n ) = - T - (,~.,.EZn(o~ + n + 1) + o~ + 1] '12

- - 6 , , , , , _ 2 [ ( ~ + n ) ( o ~ + n - - 1 ) n ( n - 1 ) ] 1 / 2

- - ( ~ n , n + 2 [ ( O ~ -- ['- n - ,} - 2)(o~ + n + 1)(n + 2)(n + 1)] 1/1) (31 ), 9 , o< n ' l ~ I n > = ~ ( 6 ,, ,,, + X ( n - - 1 ) (o ~ + n - 1 ) ] 1 / 2

- - 6 . , , , _ 1 [ ( n - - 2 ) ( a + n - - 2 ) ] 1 / 2 ) . (32 )T h e r e is n o s i m p l e c l o se d f o r m f o r t h e m a t r i x e l e m e n t s o f r - 1, b u t t h e se i n t e g r a lsa r e e a s il y e v a l u a te d u s i n g G a u s s - L a g u e r r e i n t e g r a ti o n a s h a s b e e n d o n e f o r th eo p e r a t o r r - 2 p r e v i o u s l y [ 5 ] . I n t e g r a t i o n o v e r th e r a d ia l c o o r d in a t e s i n th e p o t e n -t i a l c a n b e p e r f o r m e d i n a n a n a l o g o u s f a s h i o n .

4 . S Y M M E T R YI n r o - v i b r a t i o n a l c a l c u l a t i o n s i t i s d e s i r a b l e t o m a k e t h e g r e a t e s t p o s s i b l e u s eo f s y m m e t r y , f o r t h i s s i m p l i f ie s b o t h t h e c a l c u l a ti o n a n d t h e i n t e r p r e t a t i o n o fr e s u lt s . I n t w o s p e c ia l c a s es o f t h e g e n e r a l i z e d h a m i l t o n i a n , s c a t t e r i n g c o o r d i -n a t e s , e q u a t i o n ( 8) a n d b o n d l e n g t h - b o n d a n g l e c o o r d i n a t e , e q u a t i o n ( 1 7 ), t h e r e ist h e p o s s i b i l i t y t h a t t h e b a s i s s e t c a n b e s y m m e t r i z e d t o a c c o u n t f o r A B 2 s y s t e m s .A d d i t i o n a l l y t h e a n g u l a r b a s is m a y a l so b e s y m m e t r i z e d t o ta k e a c c o u n t o f t h e


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G e n e r a l i z e d r o - v i b r a t i o n a l c a l c u l a t i o n s 1061t o ta l p a r i t y o f a g i v e n r o - v i b r a t i o n a l s t at e. T h i s is a c h i e v e d b y t r a n s f o r m i n g t h ea n g u l a r f u n c t i o n s t o

I j , k , p ) = 2 - 1 / 2 [ l j , k ) + ( - 1 ) P [ j , - k ) ] X / 2 , k > O , p = O , l , ~ ( 3 3 )= l j , k ) , k = O , p O ,w h e r e I J , k ) = Ojk DJMk i s t h e f u l l a n g u l a r f u n c t i o n d e s c r i b e d i n w2 a b o v e . T h et o t a l p a r i t y o f t h e s t a t e i s g i v e n b y (--1 ) J+p. T h e 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 z e d a n g u l a r b a s is a r e

( j ' , k' , p ' ] I ~ l j , k , p ) = ~ f p f ( k ' , k ) ( j ' , k ' [ I ~ l j , k ) , ( 3 4 )w h e r e

f ( k ' , k ) = l , k ' , k > 0 o r k ' = k = 0 , ( 3 5 )= 2 t/2 , k ' = 0 , k > 0 o r k ' > 0 , k = 0 .I n s c a t t e r i n g c o o r d i n a t e s , t h e s y m m e t r i z a t i o n f o r a n A B 2 s y s t e m i s w e l l k n o w n[ 1 0 ] a n d s i m p l e , a s t h e b a s is s p l it s b e t w e e n a s s o c ia t e d L e g e n d r e f u n c t i o n s w i t h je v e n a n d j o d d . I n t h e b o n d l e n g t h - b o n d a n g le c as e t h is s y m m e t r y is c a r r ie d b yt h e i n t e r c h a n g e o f t h e r a d i a l c o o r d in a t e s . A s w e h a v e s o f a r a s s u m e d a s im p l ep r o d u c t b a s is f o r t h e r a d ia l f u n c t i o n s

JU~m,kj(r1, r2 ) = ]m , n ) = H m ( r l ) H n ( r z ) , (36 )i t i s n e c e s s a r y t o s y m m e t r y a d a p t t h e b a s i s i n t h i s c a s e . T h i s c a n b e a c h i e v e d b yt h e t r a n s f o r m a t i o n[ m , n , q ) = 2 - 1 / 2 [ H m ( r l ) H n ( r 2 ) + ( - - 1 ) q H , ( r l ) H m ( r 2 ) ] , m ~ n , q = 0 , 1 , ~ (37 )

= H , , ( r l ) H , ( r 2 ) , m = n , q O , )w h e r e i t h a s b e e n a s s u m e d t h a t t h e s a m e b a s is s e ts h a v e b e e n u s e d f o r b o t h r a d ia lc o o r d i n a t e s . W i t h t h e s e s y m m e t r i z e d f u n c t i o n s i t is e a s y t o s h o w t h a t t h e m a t r i xe l e m e n t s h a v e t h e f o r m

( m ' , n ' , q ' l / ~ l m , n , q ) = 6 r + 6m,,)-1/2(1 + bin,,,,)-a / 2 [ ( m ' , n ' l I ~ l m , n ) + ( - - 1 )~ ( m , n ' [ R I n , m ) ] , ( 3 8 )

p r o v i d e d t h a t/ ~ ( r ,, r 2) = / ~ ( r 2 , r l ). ( 3 9)

F o r / ~ v t h i s c o n d i t i o n i s s a ti s fi e d , b u t f o r / ~ v R i t i s n o t . A s / ~ V R i s n u l l f o r J = 0 ,t h is m e a n s t h a t w i t h t h e p r e s e n t f o r m a l i s m t h is s y m m e t r i z a t i o n c a n o n l y b ea c h i e v e d f o r t h e r o t a t i o n a l g r o u n d s t a t e ( p u r e v i b r a t i o n a l s ta t e s) . P h y s i c a l l y t h i ss i t u a t i o n a r i s e s b e c a u s e , w h i l e t h e r e i s n o t h i n g i n h e r e n t l y a s y m m e t r i c a b o u t t h ec o o r d i n a t e s y s t e m , t h e r e is a b o u t t h e e m b e d d i n g o f t h e b o d y - f i x e d z - a x is a l o n go n e o f th e b o n d s .A s y m m e t r i c e m b e d d i n g c a n b e a c h i e v e d b y c h o o s i n g t h e b o d y - f i x e d a xis toli e a l o n g t h e b i s e c t o r o f t h e i n t e r n a l a n g l e, 0 . T h i s w a s t h e p r o p o s a l m a d e b yC a r t e r e t a l . [ 1 2, 1 3 ] fo l l o w i n g B h a t i a a n d T e m k i n [ 2 3 ] . N o t s u r p r i s i n g l y th i sl e a d s t o k i n e t i c e n e r g y o p e r a t o r s w h i c h a r e i d e n t i c a l w i t h g t~ ) ) a n d K~v ) ( ( 1 2 ) a n d( 1 3) ) b u t t o d i s t i n c t l y d i f f e r e n t f o r m s f o r th e K v R t e r m s ( 1 4 ) a n d ( 1 5 ) , w h o s et r e a t m e n t r e q u i r e s s o m e w h a t d i f f e r e n t c o n s i d e r a t i o n s f r o m t h o s e u s e d a b o v e . T h e


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1062 B. T. Sutcliffe and J . Te nn ys onterms are such that we have not so far been able to show the analytic cancellat ionof potent ia l ly d ivergent terms . In consequence, the hamil t onian in such a formcanno t easily be incorpora ted into the present scheme and mor e detailed consider-ation of i t is deferred u nti l a later publicati on. We note, however, that nu meri calwork with these kinetic energy terms would seem to indicate that they cause notrouble [12, 13].

5 . S P E C I M E N C A L C U L A T IO N SIn order to test the theory given above we perf orme d tr ial calculations on

three mol ecula r systems for which calculations in scatter ing coordinates had bee nprevious ly a t tempted . Results for the molecules D2H +, CHf and HD He* aregiven in tables 1 to 3 respectively. I n these tables , the en ergy of the rotational andvibra tiona l gro und s tates are given relative to dissociation and the vibrat ionalba nd origin s are given relative to the (0, 0, 0) level. In table 1 the rotati onalfrequenc ies are relative to the J = 0 level with the same vi brati onal qua nt umnumb ers . The calcula t ions were per formed with program TRI AT OM [24] , agenera l i za t ion o f ATOM DIAT /ATO M DIA T2 [9] .

Co lu mn (a) of table 1 reprodu ces a scatter ing coordinate calcula tion from table6 of [25] . T he potent ia l used was Mar t i re and B ur to n ' s BVD H potent ia l [26] .

Co lu mn (b) is for a coordinat e system for which r I was the H D separation andr 2 l inked the geometric mid po in t of HD with the other D atom. T his coordinat esystem is not a sensible choice as the calculation does not take advantage of theper mut ati on sym met ry of the l ike D nucl ei . However, i t gives a nume ric al tes t ofour general ized hamil ton ian . The energies obta ined f rom th is ca lcula t ion are ingood agreement with those given in column (a) al though, s ince the calculationsare varia tional , they are inferior. Th is is to be expected as the coor dina tes are

Table 1. Energy levels of D2H +, in cm -1, for calculations using (a) scattering coordi-nates, equation (8), (b) midpoint coordinates, equation (18) with H-D as thediatomic and (c) DH D bon d l eng th- bon d angle coordinates, equation (17). T hemax imu m num be r of basis functions used in each coordinate is shown for compari-son.

Wavenumbers/cm- 1(v 1, v2, v3) J k (a) (b) (c)f

0 0 0

0 1 0

0 0 1

0 --71477"482 --71477.371 --71477"3851 0 e 49"151 49.1401 1e 34"820 34.8290 1964"943 1964-849 1965" 1041 0 e 45 "886 46.0671 1e 30'344 30.4770 2075"080 2076-167 2077-0531 0 e 50'191 50.1321 I e 40.290 40.023

j ~ 14 14 28m~x 4 4 4n max 4 4 4

I- Basis functi ons paramete rs: r e = 1-89a0, coe = 0-0116Eu, D e = 0.115Eh .


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Generalized ro-vibrational calculations 1 0 6 3T a b l e 2 . V i b r a t i o n a l b a n d o r i g i n s o f C H ~ , i n c m - 1 , c a l c u la t e d u s i n g (a ) s c a t te r i n g c o o r -

d i n a te s , e q u a t i o n ( 8) a n d ( b) H C H + b o n d l e n g t h - b o n d a n g l e c o o r d i n a te s , e q u a t i o n( 17 ). T h e m a x i m u m n u m b e r o f b a s is f u n c t i o n s u s e d i n e a c h c o o r d i n a t e is s h o w n f o rc o m p a r i s o n .

W a v e n u m b e r s / c m - 1(v 1, v 2 , v3) j~" (a ) (b)~

0 0 0 e - 7 0 4 0 0 " 4 8 - 7 0 4 0 0 " 6 10 2 0 e 7 1 8 -6 7 1 8 - 40 4 0 e 1 6 1 2 '1 1 6 1 1 . 40 6 0 e 2 7 7 4 . 7 2 7 7 2 " 71 0 0 e 2 9 9 9 . 0 2 9 9 8 . 90 0 1 o 3 2 7 1 - 5 3 2 7 0 ' 70 8 0 e 3 6 8 0 " 5 3 6 7 8 . 60 2 1 o 3 9 6 9 . 1 3 9 6 6 " 9

jmax 1 7 1 4m m a x 7 4n max 5 4

P a r i t y w i t h r e s p e c t t o i n t e r c h a n g e o f t h e H a t o m s : e = e v e n , o = o d d .B a s i s f u n c t i o n s p a r a m e t e r s : r e = 2 - 1 8 a 0 , c oe = 0 . 0 1 4 E h , D e = 0 ' 2 5 E h .

T a b l e 3. V i b r a t i o n a l b a n d o r i g i n s f o r t h e t w o is o m e r s o f H e H D ( A : A ' ) , i n c m - 1 , f o r ac a l c u la t i o n u s i n g d i a t o m m i d p o i n t c o o r d i n a t e s , e q u a t i o n ( 1 8) . (a ) s h o r t H e - D , l o n gH e - H i s o m e r a n d (b ) s h o r t H e - H , l o n g H e - D i s o m e r . B a s is s e t p a r a m e t e r s f o r r l :r e = 4 " 2 4 5 a 0 , coe = 0 " 0 0 7 E u , D e = 0 - 4 E h ; f o r r 2 : r e = 2 ' 0 a 0 , cO = 0 " 0 0 4 E h , D e =0 " 0 2 E h . C o l u m n (c ) s h o w s t h e c o r r e s p o n d i n g l e v el s o f H e H 2 ( A 1 A ') 1 -3 0] f o r c o m -p a r i so n . T h e a b s o lu t e m i n i m u m i n t h e p o t en t ia l is - 1 2 , 7 5 6 c m - 1 .

W a v e n u m b e r s / c m - 1(v 1, v2 , va) (a) (b ) (c)

0 0 0 - 9 9 7 8 - 4 - 9 5 5 9 . 9 - 9 3 9 4 - 40 1 0 1 1 1 6 9 2 0 1 1 6 20 0 1 1 2 5 3 1 3 6 1 1 4 3 80 2 0 2 2 0 7 1 8 2 5 2 2 9 20 1 1 2 3 3 9 2 2 8 2 2 5 7 50 0 2 2 5 0 3 2 6 5 7 2 8 1 41 0 0 3 0 3 5 3 8 6 90 3 0 3 3 0 0 2 7 2 4 3 4 1 0

j ~ ' ~ 3 4 3 3m max 4 5n m a x 4 5

n o n - o p t i m a l a n d w e d i d n o t s p e c i f i c a l l y o p t i m i z e t h e r a d i a l b a s i s s e t f o r t h i s c a se ,b u t u s e d f u n c t i o n s o p t i m i z e d p r e v i o u s l y [ 1 ] f o r t h e D ~ i o n .

C o l u m n ( c ) g i v e s r e s u l t s o b t a i n e d u s i n g t h e b o n d l e n g t h - b o n d a n g l e c o o r d i -n a t e s y s t e m , b u t o n l y f o r t h e v i b r a t i o n a l b a n d o r i g i n s b e c a u s e o f t h e d i f f i c u l t yw i t h s y m m e t r i z a t i o n i n t h is c o o r d i n a t e s y s t e m . A g a i n t h e r e s u l t s a r e i n g o o da g r e e m e n t w i t h , b u t v a r i a t i o n a l l y i n f e r i o r to , t h e s c a t t e r i n g c o o r d i n a t e c a l c u l a t i o no f c o l u m n ( a ) . F u r t h e r m o r e , w h i l e t h e c a l c u l a t i o n s o f c o l u m n s ( a ) a n d ( b) u s e dt h e s a m e n u m b e r o f b a s is f u n c t i o n s , i t w a s n e c e s s a r y t o u s e t w i c e a s m a n y a n g u l a rf u n c t i o n s t o o b t a i n r e a s o n a b l e c o n v e r g e n c e i n t h i s c o o r d i n a t e r e p r e s e n t a t i o n .


12/14

1064 B. T. Sutcliffe and J. Ten nys onThe scattering coordinate calculation on CH~- given in column (a) of table 2 is

taken from table 6 of [8] which used Carter and Handy's [-27] representation ofthe potential of Bartholomae et al. [28]. By contrast, the bond length-bond anglecoordinate calculation of column (b), which gives results variationally superior tocolumn (a), used a basis set less than half as big as the one used for the scatteringcoordina te calculation.

Table 3 gives results for a vibrational calculation on the first electronicallyexcited state of HDHe. Although the electronic ground state of H2He shows onlya weak Van der Waals minimum, the electronically excited state shows a deepmin imu m for a highly asymmetric, bent geometry where the He is muc h closer toone H [29]. Previous ro-vibrational calculations on H/He* showed small split-tings between the two symmetry related minima due to tunnelling [30], but failedfor HDH e* because of the difficulty of obtainin g a suitable radial basis set whichgave a good representation of both minima simultaneously [31]. The previouscalculations used scattering coordinates, but test calculations by us using bondlength-bond angle coordinates failed for the same reason. We note that the vibra-tional assignments given in table 3 must be regarded as tentative, but are sup-ported by scattering-coordinate calculations which only stabilized states for oneminimum.

6. DISCUSSIONAND CONCLUSIONThe three sets of calculations presented in the previous section provide auseful insight into the behaviour of our hamiltonian in different coordinate

systems. F or D2H + the scattering c oordinate calculation proved easily the mo stefficient of those tested. D2H + is a highly be nt molecule with a bond angle of 60 ~In contrast CH~ is quasi-linear and was represented much more efficiently in thebond length-bond angle coordinates. This behaviour was noted by Carter andHan dy [11] who foun d that their metho dol ogy was more suited to (near) linearmolecules than strongly bent ones.

The calculations on HDH e* demonstrate the usefulness of our hamiltonianfor a problem where other procedures fail. Of course, one could perform separatecalculations for each minimum using an Eckart hamiltonian but that would meanneglecting the tunnelling interaction of the states from the two minima.

The coordinates system in which r: measures the distance of He from the HDgeometric midpoint does not suffer from this problem as the two minima have thesame (rl, r2) coordinates. We were thus able to stabilize vibrational levels for themolecules with a short He-D bond and a long He-H bond (the more stableisomer) and a short He-H bond and a long He-D bond simultaneously. Physi-cally, the isotopic substitution leaves the potential unaltered, but creates extraoff-diagonal elements in the kinetic energy operator. Our calculation recognizesthis by not transforming to a new set of coordinates which breaks the symmetryof the potential.A situation where this commonly arises is for the van der Waals complexesformed between homonuclear diatomics (e.g. H2) and rare gas atoms. The poten-tials of these systems are usually represented as Legendre expansions. We suggestthat use of our hamiltonian in diatomic midpoin t coordinates would provide moreinsight into the behaviour of such complexes upon asymmetric isotopic substitu-tion than does the usual pro cedu re [-32] of transform ing the potential.


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G e n e r a l i z e d r o - v i b r a t i o n a l c a lc u la t io n s 1065T h e s t r e n g t h o f t h e C o r i o l is f o r c e s i n a s y s t e m i s a n a r t i f a c t o f t h e c o o r d i n a t e

c h o i c e a n d t h e e m b e d d i n g o f t h e a x e s in t h e i n t e rn a l c o o r d i n a t e f r a m e . I t isd e s i r a b l e t o m i n i m i z e t h e s e C o r i o l i s i n t e r a c t i o n s a s t h i s c a n l e a d t o u s e f u l s i m p l i -f i c a ti o n s . I n d e e d t h i s is t h e b a s i s o f t h e s e c o n d E c k a r t c o n d i t i o n I-2 ]. I n d e r i v i n go u r g e n e r a l i z e d h a m i l t o n i a n w e h a v e a l l o w e d f o r s o m e f le x i b i li t y i n t h e c h o i c e o fe m b e d d i n g a s t h is a l lo w s o n e t o c h o o se t h e a p p r o p r i a t e b o d y - f i x e d a x es fo r ap a r t i c u l a r p r o b l e m . T h i s i s p a r t i c u l a r l y i m p o r t a n t i f t h e r e i s a p a r t i t i o n i n gb e t w e e n t h e v i b r a t i o n a l a n d r o t a t i o n a l s t e p s i n th e c a l c u l a t i o n 1 -33 ].

I n s u m m a r y w e h a v e d e r i v e d a h a m i l t o n i a n e x p r e s s e d i n t e r m s o f a g e n e r -a l i z ed i n te r n a l c o o r d i n a t e s y s t e m a n d w i t h o p t i o n s a b o u t th e e m b e d d i n g u s e d.T h i s h a m i l t o n i a n is a u n if ic a t i o n o f s ev e r al h a m i l t o n i a n s w h i c h h a v e p r e v i o u s l yf o u n d f a v o u r . B y p e r f o r m i n g t e s t c a l c u la t io n s o n t h r e e t r i a t o m i c p r o b l e m s o fi n t e r e s t w e s h o w h o w t h e c h o i c e o f c o o r d i n a t e s s t r o n g l y e f f e c ts t h e e f f o r t i n v o l v e din o b t a i n i n g r e s u l t s - - o r e v e n th e p o s s i b i li t y o f d o i n g s o . T h e m e t h o d o l o g y w eh a v e d e v e l o p e d , w h i c h g i v e s o n e f l e x ib i l it y i n c h o o s i n g t h e a p p r o p r i a t e c o o r d i -n a te s , h as b e e n i m p l e m e n t e d a s g e n e r a l c o m p u t e r p r o g r a m w h i c h w il l b ep u b l i s h e d 1 - 2 4 ] .

REFERENCES[1] TENNYSON,J., an d SUTCLIFFE, B. T ., 1986, f t . chem. Soc . Farada y 1I ( in the press) .[2 ] ECKART, C., 1935, Phys . Rev . , 47, 552.[3] WATSON, J . K . G ., 1968, Molec . Phys . , 15, 476.[4] SUTCLIFFE,B. T. , 1983, Molee . Phys . , 48, 561.[5 ] TENNYSON, J., an d SUTCLIFFE, B. T ., 1982, ff. chem. Phys. , 76, 5710.[6] TENNYSON,J. , a nd SUTCLIFFE, B. T . , 1983 ,ff . chem. Phys. , 79, 43.[7] TENNYSON,J., an d St~TCLIFFE, B. T ., 1983, ft. molec. Spectrose., 101, 71.[8] TENNYSON,J., a nd SUTCLIFFE, B. T ., 1984, Molec . Phys . , 5 1 , 8 8 7 .[9] TENNYSON,J. , 1983, Comput . Phys . Commun. , 29, 307; 1984, Ibid. , 32, 109.[10] TENNYSON,J. , 1986, Comput. Phys. Rep. ( in the press) .[11] CARTER, S. , an d HANDY, N . C . , 1982, Molec . Phys . , 47, 1445; 1986, Ibid. , 52, 175.[12] CARTER, S. , HA NnY, N . C. , an d SUTCLIFFE, B. T . , 1983, Molec . Phys . , 49, 745.[13] CARTER, S. , an d HANDY, N . C. , 1984, Molec . Phys . , 52, 1367.[14 ] CARTER, S . , 1984, Re por t CC P1/84 /2 , D ares bu ry La bora to ry .[15 ] SUTCLIFFE, B. T ., 1982, Current Aspects o f Quantum Chemis try , e d i t e d b y R . C a r b o(S tud ie s i n Theore t i ca l Ch em is t ry , Vo l . 21 ), p . 99 .[16 ] LAI, E . K . C . , 1975, M as t e r s The s i s , De pa r tm en t o f Ch em is t ry , I nd i ana Un ive r s i t y .[17] BURDEN, F. R. , an d QUINEY, H . M ., 1984, Molec . Phys . , 53, 917.[18] KEMBLE, E. C . , 1937, The Fundamental Pr inciples o f Quantum Mechanics ( M c G r a w -Hi l l ) .[19] BRINK, D . M . , and SATCHLER, G . R. , 1968, A n g u l a r M o m e n t u m , 2nd ed i t i on(Cla rendon P res s ) .[20] CONDON, E. U. , and SHORTLEY, G . H . , 1935, The Theory o f A tom ic Spec t ra( C a m b r i d g e U n i v e r s i t y P r e ss ).[21] TENNYSON,J. , 1985, Comput . Phys . Commun. , 38, 39.

[22] GRADSHTEYN, . S . , and RYZHIK, I . H . , 1980, Tables of Integrals, Series an d Products(Academic ) .[23] BHATm, A. K . , and TEMKIN, A. , 1965, Phys . Rev. A, 137, 1335.[24] TENNYSON,J ., Comput . Phys . Commun. ( in the p ress) .[25] TENNYSON,J., anc l SUTCLIFFE, B. T ., 1985, Molec . Phys . , 56 , 1175.[26] MARTIRE, B. , an d BURTON, P. G . , 1985, Chem. Phys . Let t . , 121 ,479 .[27] CARTER, S. , an d HANDY , N . C. , 1982, ft . molec. Speetrosc., 95, 9.[28] BARTHOLOMAE,R., MART1N, D ., and SUTCLIFFE, B. T . , 1981, ft . molec. Spectrosc., 87 ,367.


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1 06 6 B . T . S u t cl if fe a n d J . T e n n y s o n[29] FARANTOS,S. C., THEODORAKOPOULOS,G. , and NICOLAIDES,C. A., 1983, C h e m . P h y s .L e t t . , 100, 263.r30] FARANTOS,S. C., and TENNYSON,J. , 1985,ff . c h e m . P h y s . , 82, 2163.[31] FARANTOS, S. C. , and TENNYSON, J. (u np ub lish ed wo rk).[32] KREEI~, H. , and LE R oy , R. J . , 1975, f t . c h e m . P h y s . , 63 , 338 . L Iu , W . -K. ,GRABENSTETTEa, J . E. , LE RoY, R . J . , an d M cC ouB T, F. R. , 1978, f t . c h e m . P h y s . ,68, 5028.[33] TENNYSON,J., an d SUTCLIFFE, B. T ., 1986, M o l e c . P h y s . , 58, 1067.

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