Jonathan Tennyson and Brian T. Sutcliffe- Highly rotationally excited states of floppy molecules: H2D^+ with J < 20

8/3/2019 Jonathan Tennyson and Brian T. Sutcliffe- Highly rotationally excited states of floppy molecules: H2D^+ with J < 20

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MOLECULAR PHYSICS, 1986, VOL. 58, NO. 6, 1067-1085

H i g h l y r o t a t i o n a l l y e x c i t e d s t a t e s o f f lo p p ym o l e c u l e s : H 2 D + w i t h J ~< 2 0 .by JO N A T H A N T E N N Y S O N

Department of Physics and Astronomy, University College London,Gower Street , Lo ndon WC1 E 6BT, England

and BRIAN T. SUTC LIF FEChemistr y Departme nt, University of York, Heslington,

York YO1 5DD, England(Received 12 March 1986 ; accepted 28 March 1986)

A partitioning of the generalized triatomic hamiltonian of the precedingpaper is developed which allows the calculation of highly-excited rotationalstates, without approximation, in a two-step variational procedure. Iterativediagonalization techniques are found to be particularly useful for the secondvariational step. The rotationally-excited states of HzD + are studied withJ ~< 20, well into the region where the ground and excited state manifoldsoverlap. Comparison of results for two different ab initio potentials and con-vergence considerations suggest that pure rotational transition frequenciesobtained from our results should be accurate to about 1 cm - 1 for J ,~ | 5.

1. INTRODUCTIONThe a priori computation of ro-vibrational spectra of small molecules, mainly

triatomic, has made significant progress over the last decade, but has been largelyconfined to low values of the total angular mome nt um qua ntum number , J. Thereasons for this are twofold, firstly most experimental information has come fromtransitions between modera tely low values of J ( although often muc h high er thanhas been calculated) and secondly, and more crucially, the size of a fully coupledro-vibrational calculation incleases rapidly with J and soon become intractable.Fully co uple d calculations have generally been limited to J ~< 4 [1- 5] .

This lack of work on highly excited rotational states does not mean that thesestates are without interest. They are, for example, crucial for testing and refiningour ideas on model hamiltonians [6] based as they are on perturbation couplingschemes which may cease to be valid in the high J limit.

In such a coupling scheme, it is usual to regard the rotational levels of asystem as a sub-manifold of a particular vibrational state. Although couplingbetween rotational levels of different vibrational manifolds through Coriolis inter-actions is well-known, it is generally regarded as a (small) perturbation. Manytheoretical analyses of large J states have ignored this interaction (e.g. [7]). Wewill call the region where this approximation is valid the low J regime, whilerealizing that for heavy molecules this regime may include J values of several


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1068 J. Te nn ys on and B. T. Sutcliffehundred. A semi-quantitative definition of the range of the regime is given byconsiderin g the ratio of the rotational and vibrational splittings, R j, which can beapproximated by

R j "~ [ A - - ( B + C ) / 2 ] J 2 / ~ o , (1)where A, B and C are the usual rotat ional con stan ts and o~ the lowest vibratio nalfundamental for the molecule. In the low J regime Rj is very much less thanunity. However, as J increases so does the width of the rotational manifold that isthe difference between levels with z = - - J and z = + J for an asy mme tri c top, orK = 0 and K = J for a symme tri c top. Fo r high er J, R 1 will app roa ch un ity andat this point rotational sublevels of the ground vibrational state will begin tooverlap the rotational manifold of the lowest vibrationally-excited state. Th is wecall the intermediate J region.

For high enough J, the situation will arise where the rotational splittings aremuch larger than the spacing between vibrational levels, R s >> 1. In the extre me,this would result in the re-ordering of the quan tum numb ers (or perturbativecoupling scheme) so that levels with low K, in the symmetric top case, will bepreferred and the vibrational parentage of a state will become the small pertur-bation. Th is recoupli ng for large values of the total angular mom en tu m is well-known in nuclei [8] where the study of 'high spin states' is an area ofconsiderable activity. To our knowledge, such recoupling has yet to be observedin molecular systems, but, as we demonstrate, is not outside the scope of curre ntexperiments.In this paper we propose a tractable procedure for the a b i n i t i o calculation ofhighly excited rotational states with full ro-vibrational coupling. This procedure,based on the use of a secondary variational step in the calculation, is similar to themethod used recently by Chen e t a l . [-9] for calculat ing the low-l ying states o fH20 with J ~< 10. Howe ver, it differs in one respec t wh ich we expect to beimpo rtant for the accurate calculation of the highly -excited states consid ered here.

Light systems, with large rotational constants, will tend to display the high Jeffects at lower values of the total angular momentum than heavier systems. H~-,the lightest (and electronically simplest) polyatomic system thus makes a suitablestarting point for such a study. Th e low-lying ro-vibrational states of H~- and itsisotopomers have been the subject of muc h recent study, both experimentally andtheoretically, and a fair measure of understanding has been achieved for thesestates [-4]. In particular, HE D+ is known to be a hig hly -as ymm etr ic top,K - --0"07, whose vibrational levels show strong Coriolis coupling due to thesplitting of levels degenerate in H~ [2]. However, all the calculations on H2D +have concentrated on states with low J(~


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R o t a t i o n a l l y e x c i t e d s ta t es o f H 2 D + 1 0 6 92 . T H E O R Y

I n t h e p r e v i o u s p a p e r [ -1 5] , h e r e a f t e r r e f e r r e d t o a s I , w e d e v e l o p e d a g e n e r -a l i z e d r o - v i b r a t i o n a l h a m i l t o n i a n f o r t r i a t o m i c s in a n y i n t e r n a l c o - o r d i n a t e s y s t e mw h i c h c o n s i s t s o f t w o l e n g t h s a n d a n a n g l e . T h i s h a m i l t o n i a n w a s w r i t t e n

/ - / = t d v ' + ~ v ~ + ~ + ~ + v , (2)w h e r e t h e K v t e r m s a r e p u r e v i b r a t i o n a l k i n e t i c e n e r g y o p e r a t o r s , a n d t h e K vR sa r e v i b r a t i o n - r o t a t i o n 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 z e r o fo r J = 0 . T h et e r m s s u p e r s c r i p t e d 2 a r e t h o s e i n t r o d u c e d b y t h e g e n e r a l i z a t i o n o f t h e c o -o r d i n a t e s f r o m a t o m - d i a t o m s c a t t e r i n g c o - o r d i n a t e s . V i s t h e e l e c t r o n i c p o t e n t i a l .

I n t h i s w o r k w e f i rs t d i a g o n a l i z e a h a m i l t o n i a n f o r w h i c h k , t h e p r o j e c t i o n o f Jo n t o t h e b o d y - f i x e d z ax i s , i s a g o o d q u a n t u m n u m b e r , a n d t h e n w e t a c k le t h e f u l lp r o b l e m . I n t h i s w a y , w e a i m t o r e d u c e t h e s i z e o f t h e f u l l p r o b l e m t h a t n e e d b ec o n s i d e r e d . A s o l u t i o n o f t h e h a m i l t o n i a n

n . , ~ = ~ + ~ v ~ + a~k,/~v~. + V, (3)w i t h e i g e n v a l u e e i ' k c a n b e w r i t t e n

c~Ji, k = ~ ci,-J'm,k'i.H m ( r l ) H , ( r 2 ) O j k ( O ) D ~ k ( O t , f l, ~ ) ( 4 )j , t n , n

i n t e r m s o f th e b a s i s f u r, c t i o n s d e v e l o p e d i n I . W e n o t e t h a t t h e e i g e n v a l u e s a r ei n d e p e n d e n t o f t h e s i gn o f k .

T h e s e e i g e n v e c t o r s c a n t h e n b e u s e d a s a b a s i s s e t fo r th e f u l l h a m i l t o n i a n ( 2) .T h e m a t r i x e l e m e n t s f o r t h i s s e c o n d s t e p a re< k ', i ' [ H l k , i> 6 k , k O i , ie l ' k + f ( k , ' + e j , m , n C j ' m ' n "jmn j ' r a ' n '

, 1 _ _ ! _ _x a j , i a . , . C j < m [ 2 g , r ~ Im >+ a , , , , + , a , , 2#12+km '[ rE 1 m > [ ( j + 1 ) < n ' r-2 1n > + < n ' ] ~ r 2 ]> d ]m

b J '+ - k < m " 1 [ m > [ J n ' ' 1 ] n > - < n ' ] d 3 }+ 3 j ' , , - 1 2 # 1 2 r-7 r '~ ~ i n > , ( 5 )f o r z e m b e d d e d a l o n g r I ( a n d t h e s a m e w i t h r 2 ~ - + r l , ~u1*-~#2 and m ~ - + n i n t e r -c h a n g e d f o r z e m b e d d e d a l o n g r 2 ) . T h e f a c t o r s a j , k a n d b j, k a r e d e f i n e d i n I a n d

f ( k , k ' ) = 1 , k , k ' > O o r k ' = k = O , ~. (6 )= 2 1 / 2 , k = 0 , k ' = 1 o r k = 1 , k ' = 0 ,i s i n c l u d e d t o a c c o u n t f o r t h e s y m m e t r i z a t i o n o f t h e b a s is w i t h r e s p e c t t o t h e t o t a lp a r i t y :

1Ik, i ) = ~ - ~ ( , / , l . ' + ( - 1 ) p , ~ , J . - k ) , p = 0 , 1 ; k > 0 , (7 )d , k= ~ b i , p = O , k = 0 ,

w h e r e , b y c o n v e n t i o n , p = 0 f o r t h e e b l o c k a n d p = 1 f o r t h e f b l o c k . T h e t o t a lp a r i t y i s g i v e n b y ( - 1 ) j + p . W e n o t e t h a t n o m a t r i x e l e m e n t s a p p e a r i n ( 5 ) f o rw h i c h [ k - - k ' l i s g r e a t e r t h a n o n e .


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1070 J. Te nn ys on and B. T. SutcliffeClearly, if all the solutions of hamiltonian (3) are used to diagonalize the full

hamiltonian (2), then the results should be identical to those obtained by lettingthe full hamiltonian act on the untransformed basis functions. However, it ispossible that not all the solutions of H a , k are needed if one is only interested in afew, low-lying levels of H. Furthermore, as the secular equation method is varia-tional, convergence from above can, in general, be demonstrated.

The secondary variational procedure outlined here has one important differ-ence to the method used for their calculations on water by Chen e t a l . [-9]. Thehamiltonian of their first variational step, which was actually expressed in nor maldisplacement co-ordinates, omitted the term equivalent to the diagonal part of/~VlR which we include. This has the advantage that their first variational step isinde pende nt of J and hence the J = 0 vibrational wavefuncti ons (for J = 0, H j , k isthe exact hamiltonian) can be used as the basis for all J states. However, thesebasis functions will not allow for the shifts in geometry caused by the centrifugalterms in the potential, which, as we will show, are large in the high J region.Chen e t a l . ' s calculations, although considering J up to 10, were entirely confinedto the low J , R j ~ 1 region. A similar procedure has also recently been used bySpirko e t a l . on H~ and H2 D+, but on ly for J ~ 4 [5].

The theory given here is for the generalized co-ordinates of I ; in this work weconsider H2 D in scattering co-ordinates. Tha t is a system with r as the H 2bondle ngth, R joinin g the H 2 midp oin t with D a nd 0 the angle between r and R.This co-ordinate system is the best alternative within this formalism for the Hfisotopome rs as shown by the calculations on D2 H in I. F urt her mor e, we willem bed the z axis along R, mak ing r I = R and r 2 = r in the above analysis. Te stcalculations and previous experience [1-3] having shown this to be the bestembedding for this system.

3. PRACTICALCONSIDERATIONSIf the procedur e outlined above is to work three problems must be borne in

mind: convergence of the first variational step, convergence of the second varia-tional step and the computational tractability of the procedure, especially for largevalues of J. T he first variational step is very similar to the pure , J = 0, vibrationalproblem--the centrifugal distortion terms can be considered as extra isotropicterms in the potential. There is considerable experience in converging thesevibrational problems [16] and the only added problem is that of developing abasis appropriate for a range of k values.

Table 1 demonstrates the convergence of the second variational step for thelow-lying states of H2D with J = 4. The sy mme try block is that with even totalparity and even (para) with respect to interchange of the two H atoms. Forcomparison, the 'exact' variational results are also given, that is the resultobtained by directly diagonalizing the full ro-vibrational problem. These resultsare the same as those obtained by including all the eigenfunctions of the first stepin the basis used for the second step, which demonstrates the numerical stabilityof our procedure. However, it is clear from table 1 that, at least for low J, only afraction of these funct ions are needed to converge the second variational step.

The matrices obtained in the second variational step have a structure whichmeans that they become increasingly sparse with J. The only non-zero elementsare the purely diagonal ones in the diagonal block where k = k', and all those in


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Rotation ally excited s tates of H2 D+ 1071

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O0- r , . )

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o

O

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O ~

O ~r

r

', D, 7r

o 0O ~o o,7r'4D

o 0

O ',

6o 0

orO

t" qII


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1 0 72 J . T e n n y s o n a n d B . T . S u t c l if f et h e l e a d i n g o f f - d i a g o n a l b l o c k s w i t h k = k ' _ 1 . I f N s o l u t i o n s o f t h e f i r s t v a r i a -t i o n a l s te p a r e u s e d , t h e s e c u l a r m a t r i x h a s o n l y ( J - p ) N ( N + 1 ) - N 2 u n i q u ee l e m e n t s o u t o f a t o t a l o f ( J - p + 1 ) 2N 2 .

T a b l e 2 c o m p a r e s t h e p e r f o r m a n c e o f 3 d i a g o n a l i z e r s o n a p r o b l e m w i t hJ = 10 e a n d N = 8 2 - - t h e l a r g e s t p r o b l e m f o r w h i c h t h e e n t i r e m a t r i x c o u l d b eh a n d l e d i n c o r e . T h e m o s t n o t a b l e f e a t u r e o f t a b l e 2 i s t h e w i d e v a r i a t i o n i nm e m o r y r e q u i r e d b y t h e th r e e d i a g o n a l i z e r s . T h e f i r s t d i a g o n a l i z e r , w h i c hr e t a i n e d t h e e n t i r e m a t r i x i n c o r e , a s s u m e d o n l y th a t i t w a s s y m m e t r i c . T h es e c o n d d i a g o n a l i z e r t r e a t e d t h e m a t r i x a s b a n d e d a n d s y m m e t r i c , a n d t h u sr e q u i r e d a p p r o x i m a t e l y t w i c e t h e m i n i m u m a m o u n t o f s t o r a g e as t h e b a n d w i d t hi s 2 N . T h e t h i r d d i a g o n a l i z e r u s e s a n i t e r a t iv e a l g o r i t h m d e s i g n e d f o r s p a r s em a t r i c e s . B e c a u s e o f t h e k n o w n s t r u c t u r e o f t h e m a t r i x i t w a s p o s s i b l e n o t o n l y tor e t a i n t h e m i n i m u m n u m b e r o f m a t r i x e l e m e n t s , b u t a l s o t o u s e th i s s t r u c t u r e t os i m p l i f y th e v e c t o r m a t r i x m u l t i p l i c a t i o n r e q u i r e d o n e a c h i t e r a t i o n . C a r e w a st a k e n t o e n s u r e t h a t t h i s m u l t i p l i c a t i o n w a s v e c t o r i z e d . T h e i t e r a t io n s w e r es t a r t e d f r o m u n i t v e c t o r s d e t e r m i n e d b y t h e o r d e r i n g o f t h e d i a g o n a l e l e m e n t s a n dc o n v e r g e d t o a t o l e r a n c e w h i c h g a v e t h e e i g e n v a l u e s a c c u r a t e l y t o 0" 01 c m - 1 . A l lt h e d i a g o n a l i z e r s a g r e e d t o t h i s a c c u r a c y . B o t h t h e f u l l a n d b a n d e d d i a g o n a l i z e r sw e r e t a k e n f r o m a p a c k a g e t h a t h a s b e e n v e c t o r i z e d [ 1 7 ] .

A s c a n b e s e e n f r o m t a b l e 2, t h e b a n d e d m a t r i x d i a g o n a l i z e r w a s t h e s l o w e s t .T h e i t e r a t i v e p r o c e d u r e w a s f o u n d t o b e t h e m o s t e f f i c i e n t , b o t h i n c o r e a n d C P Ut i m e .

F o r J = 1 0, t h e t im e t a k e n f o r t h e s e c o n d d i a g o n a l i z a t i o n i s m o d e s t c o m p a r e dt o s o lv i n g e l e v e n 5 04 d i m e n s i o n a l b a s e p r o b l e m s , w i t h k e q u a l s 0 t o 1 0 i n c l u s i v e ,a n d s i m i l a r to t h a t t a k e n t o c o n s t r u c t t h e f i na l s e c u l a r m a t r i x . H o w e v e r , t h e t i m et a k e n f o r b o t h t h e s e s t e p s i n c r e a s e s o n l y l i n e a r l y w i t h J . O u r b i g g e r c a l c u l a t i o n s ,t h e l a r g e s t o f w h i c h w a s f o r a m a t r i x o f d i m e n s i o n 3 7 80 , d e p e n d e d s t r o n g l y o n t h et i m e t a k e n f o r t h e f i n a l d i a g o n a l i z a t i o n .

U s i n g t h e i t e r a t iv e d i a g o n a i i z a t i o n p r o c e d u r e a n d a n e ig e n v a l u e t o l e r a n c e o f0"0 1 c m - 1 , w e n e x t c o n s i d e r e d t h e c o n v e r g e n c e c h a r a c t e r i s t i c s o f t h e t w o v a r i a -t i o n a l s t e p s f o r h i g h e r v a l u e s o f J . T e s t c a l c u l a t i o n s s h o w e d t h a t t h e b a s i s se to p t i m i z e d b y u s p r e v i o u s l y f o r lo w J c a l c u l a t i o n s [ 1 ] , a n d u s e d i n ta b l e 1 , g a v e ap o o r r e p r e s e n t a t i o n o f s ta t e s w i t h l a r g e k , p a r t i c u l a r l y k > 1 0 . F o r t h e s e s t a t e s ,t h e c e n t r i f u g a l t e r m s a r e la r g e a n d i t w a s f o u n d n e c e s s a r y t o i n c r e a s e th e f l e x -

T a b l e 2 . C o m p a r i s o n o f c o m p u t a t io n a l re q u i r e m e n t s f o r t h r ee d i ag o n a l iz a t io n p r o c e d u r e sf o r t h e 9 0 2 d i m e n s i o n a l s e c u l a r p r o b l e m g i v e n b y J = 1 0 ~ a n d N = 8 2 . 1 7 e i g e n -v a l u e s a n d e i g e n v e c t o r s w e r e o b t a i n e d f o r t h e s a m e s e c u l a r m a t r i x w h i c h t o o k 8 -2 st o c o n s t r u c t . R e q u i r e m e n t s f o r t h e b a s e p r o b l e m s , t h e f i r st v a r i a t io n a l s t e p , a r es h o w n f o r c o m p a r i s o n .D i a g o n a l iz a t io n S t o r a g e T i m em e t h o d ( w o r d s ) ( C r a y - 1 / s )

F u l l m a t r i x [ 1 7 ] 8 4 9 , 4 2 4 2 1" 58B a n d e d m a t r i x [ 1 8 ] 1 8 3 ,7 4 8 5 9" 26S p a r s e m a t r i x 1 1 9] 1 0 3, 97 3 1 1 ' 75 ~B a s e P r o b l e m [ 2 0 ]200 x 200 76 ,6 68 11 x 2"10504 x 504 285 ,979 11 x 15"85T o l e r a n c e 1 0 - 3 , g i v i n g 8 fi g u re (0 .0 1 c m - t ) a c c u r a c y in t h e e i g e n v al u e s .


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Rota t iona l l y exc i t ed s ta t es o f H2 D+ 1 0 7 3T a b l e 3. C o n v e r g e n c e o f 4 t y p ic a l c al c u l a ti o n s w h i c h n e g l e c t C o r i o li s i n t e ra c t i o n s w i t h

i n c r e a s i n g b a s i s s e t s i z e . E n e r g i e s a r e r e l a t i v e t o t h e D + + H 2 d i s s o c i a t i o n l i m i t i n- 1cm

W a v e n u m b e r s / c m - 1J = 0 , k = 0 J = 2 0 , k = 0 J = 2 0 , k = 2 0 J = 3 0 , k = 3 0m maX ~ n m ax ~ j ~ a ~ j ~ax : 1 4 j ~ax = 1 4 j ~ax = 3 0 j ~ax : 4 0

4 4 j ~ x - - 3 2 9 5 3- 0 95 - - 2 3 1 2 3 - 5 9 5 - - 1 9 0 5 6 " 0 1 3 - - 8 2 6 3 . 1 2 74 4 j~ a x + 4 - - 3 2 9 5 3 . 0 9 7 - - 2 3 1 2 3 " 5 9 5 - - 1 9 0 5 6 ' 0 1 3 - - 8 2 6 3 - 1 2 74 6 j~ a~ - - 3 2 9 5 3 . 2 1 8 - - 2 3 1 2 3 " 9 1 0 - - 1 9 0 5 6 ' 1 0 8 - - 8 2 6 4 " 0 0 34 8 j~ a x - - 3 2 9 5 3 - 2 2 2 - - 2 3 1 2 3 " 9 3 3 - - 1 9 0 5 6 . 1 0 9 - - 8 2 6 4 . 0 0 46 4 j~ a x - - 3 2 9 6 1 . 3 2 7 - - 2 3 1 4 0 - 8 5 6 - - 1 9 0 5 6 " 9 0 8 - - 8 2 8 8 . 5 5 48 4 j m a x - - 3 2 9 6 2 . 1 1 1 - - 2 3 1 4 2 - 5 3 8 - -1 9 0 5 6 - 9 3 2 - - 8 2 8 9 - 1 1 9

1 0 4 j ~ - - 3 2 9 6 2 . 1 7 8 - - 2 3 1 4 2 " 7 0 6 - - 1 9 0 5 6 " 9 3 4 - - 8 2 8 9 . 3 7 6O p t i m i z e d8 6 j ~ a x - - 3 2 9 6 2 .2 4 0 - - 2 3 1 4 2 " 8 8 8 -- 1 9 0 5 7 . 0 2 1 - - 8 2 9 1 . 4 9 9

M o r s e - l i k e f u n c t i o n s H,,(r) w i t h m = 0 t o m m ax a n d r e = - 2 . 2 a 0 , ~oe = 0 " 0 0 7 E h a n dD e = 0 . 0 7 E h [ 2 0 ]

3~ M o r s e - l i k e f u n c t i o n s H'.(r) n = 0 t o n m a* an d R e = 1 - 5 5 a0 , 09e = 0 . 0 0 8 E h a n d D e =0 . 2 1 5 E h [ 2 0 ]

w A s s o c i a te L e g e n d r e p o l y n o m i a l s O j ,( 0 ) w i t h j = k to 3 ~ax.

i b i l i t y o f t h e r a d i a l b a s i s f u n c t i o n s t o a l l o w f o r r o t a t i o n a l d i s t o r t i o n s . I t w a s a l sof o u n d n e c e s s a r y t o i n c r e a s e jm ax , t h e o r d e r o f t h e h i g h e s t L e g e n d r e f u n c t i o n i n t h eb a s i s , a l t h o u g h t h e c o n d i t i o n t h a t j > / k m e a n s t h a t t h i s d i d n o t l e a d t o a n i n c r e a s ei n t h e s i z e o f t h e a n g u l a r b a s i s . T a b l e 3 p r e s e n t s r e s u l t s f o r o u r r e - o p t i m i z e d b a s i sa n d d e m o n s t r a t e s c o n v e r g e n c e f o r a s e l e c t i o n o f v i b r a t i o n a l c a l c u l a t i o n s .

U s i n g t h e o p t i m i z e d b a s i s f o r t h e f i rs t v a r i a t i o n a l s t e p g i v e n i n t a b l e 3 , t a b l e s4 , 5 a n d 6 s h o w t h e c o n v e r g e n c e o f c a l c u l a t i o n s w i t h J = 10 , 15 a n d 2 0 w i t hi n c r e a s i n g b a s i s . A l l t h e s e c a l c u l a t i o n s a r e e p a r i t y w i t h j e v e n a s t h i s s y m m e t r yg i v e s t h e d e n s e s t s p e c t r u m a n d h e n c e t h e s l o w e s t c o n v e r g e n c e p r o p e r t i e s . T h el e v e l s a n a l y s e d c o r r e s p o n d t o t h o s e t h a t m i g h t b e t h o u g h t o f a s s t e m m i n g f r o mt h e g r o u n d a n d f i rs t t w o v i b r a t i o n a l l y - e x c i t e d s t a t e s ( v 2 = 1 a n d v 3 = 1 ) , d e t e r -m i n e d s i m p l y o n e n e r g y o r d e r i n g . I t c a n b e s e e n t h a t i n a l l c a s e s t h e r e s u l t s a r ec o n v e r g i n g , a l t h o u g h t h i s c o n v e r g e n c e i s s l o w e r as J in c r e a s e s . T h e c o n v e r g e n c e

T a b l e 4 . C o n v e r g e n c e o f s o m e l e v e l s w i t h J = 10 e ( j e v e n ) w i t h i n c r e a s in g v i b r a t i o n a lb a s is , N . E n e r g i e s , i n c m - 1 , a r e a ll r e l a t i v e t o E 0 = - - 3 2 9 6 2 . 2 4 . k , th e p r o j e c t i o n o fJ a l o n g R , v a l u e s a re t a k e n f r o m t h e N o - C o r i o l i s c a l c u la t io n s .

W a v e n u m b e r s / c m - 1L e v e l n u m b e r 1 4 7 1 0 1 3 1 6

k = 0 6 1 2 6 1 0N o C o r i o l i s 2 9 2 8 " 0 4 3 5 7 5 - 0 6 4 9 9 3 " 3 2 5 2 5 5" 8 1 5 7 9 3 - 8 0 6 4 1 1 " 4 9N = 4 0 1 9 8 2 " 2 4 3 1 8 2 " 15 4 1 9 2 " 2 0 4 9 1 4 " 7 0 5 2 7 4 " 4 2 5 7 0 5 " 53

8 0 1 9 7 7 - 1 4 3 1 8 1 " 4 8 4 1 9 1 " 3 7 4 9 0 8 " 6 8 5 2 7 0 " 1 2 5 7 0 2 - 1 51 2 0 1 9 7 6 " 6 6 3 1 8 1 " 4 0 4 1 9 1 " 3 5 4 9 0 7 " 8 6 5 2 6 9 " 1 4 5 7 0 1 " 3 01 4 0 1 9 7 6 "5 3 3 1 8 1 " 3 9 4 1 9 1 " 3 4 4 9 0 7 - 2 6 5 2 6 9 " 0 7 5 7 0 1 " 251 6 0 1 9 7 6 - 5 3 3 1 8 1 " 3 9 4 1 9 1 " 3 4 4 9 0 7 - 1 9 5 2 6 9 " 0 2 5 7 0 1 " 2 11 8 0 1 9 7 6 - 5 2 3 1 8 1 " 3 9 4 1 9 1 " 3 4 4 9 0 7 " 1 1 5 2 6 8 " 9 8 5 7 0 1 " 1 7


8/19

1074Table 5.

J . Tennyson and B. T. SutcliffeConvergence of some J = 15 e (j even) levels with increasi ng N. See legend totable 4 for further explanation.

Wavenumbers/cm- 1Level nu mb er 1 4 7 10 13 16k = 0 6 10 2 4 0No Coriolis 6018-88 6799"41 7861-42 8337-03 8614-36 9072.26N = 40 4154.92 6047-04 6609.22 7173.86 7698"90 8099"8780 4128"18 5932-57 6554.57 7103.85 7660"20 7979"43120 4126-30 5922"71 6549-38 7092-55 7650-89 7973-82160 4123"27 5904"88 6539.20 7084"78 7647-86 7969"43180 4123"13 5904-20 6538.34 7083.89 7647"35 7969"02200 4123"04 5903"39 6537.62 7083"23 7646"19 7968"62

Table 6. Convergence of some J = 20 e (j even) levels with increasing N. See legend totable 4 for further explanation.Wavenumbers/cm-

Level nu mb er 1 2 4 6 8 10k = 0 2 6 1 10 12No Coriolis 9819'4 9927"2 10729-4 11351.3 12049"6 12762-1N = 40 7039'5 8083-5 8896.3 9503-9 10012"6 10189"380 6981"2 8032"4 8825.9 9445"8 9886"0 10132"8120 6968"4 8024"4 8819.8 9439.0 9881"5 10127-1160 6958-5 8012"7 8809-3 9429.5 9874-0180 6957"2 8011"8 8808.6 9428"6 9870"9 10119"8

with the re-optimized basis is also s lower than that observed by us in tes t calcu-lat ions using the smaller basis of table 1. This is because the spectrum of vibra-t ional levels given by the larger re-optimized basis is denser . A feature shown byanaly sing conver gence rates is that the lower levels do not necessari ly convergequicker--for example, compare the lowest level in table 4 with level numbers 4and 7. Th is is because the Coriolis intera ctions for s tates with low k are consider-ably s tronger than those with high k, due to the angular coeff icient Cfk , see I.Ano th er ob vious feature is that neglect ing the Coriolis interaction s is a poorapp ro xi ma ti on for H2 D + . Th e tables show that for N = 180, the J = 10 e levelsare co nv er ge d to a bo ut 0"1 cm -1, J = 15 e to ab ou t I cm -1 an d the J = 20 e levelsto about 5 cm - t . Al tho ugh we wou ld l ike to consider Js highe r th an 20, this mustawait a t ime when this level of convergence can be improved upon.

4. RESULTSCalcula tions were p erfo rmed using the op timi zed basis of table 3 for the f irs t

var ia t ional s tep . T he angular bas is se t parameter j ~x was in creme nted f rom 14 fork = 0 to 34 for k = 20. Th us for each J it was nec ess ary to di ago nal ize J + 1secular pr ob le ms of dim en si on 504 as calc ulati ons wit h p = 0 and p = 1 (e and f)use the same basis. Fo r the seco nd variat ional step N = 180 was used for all J .Thi s was done for consis tency, even tho ugh for lower J values i t gave con-vergence cons iderably bet ter than required . For each J value and symm etry


9/19

R o t a t i o n a l l y e x c i te d s t a t es o f H z D + 1 0 7 5b l o c k , t h e d i a g o n a l i z a t i o n w a s c o n v e r g e d t o 0 -0 1 c m - 1 f o r s u f f i c i e n t l e v e l s toc o v e r a p p r o x i m a t e l y , t h e r o t a t i o n a l m a n i f o l d o f t h e l o w e s t t h r e e v i b r a t i o n a l s t a t e s .

T h r o u g h o u t t h is w o r k t h e 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 [ 2 1 ] h a sb e e n u s e d , f o r p r e v i o u s w o r k h a d s h o w n i t t o p r o v i d e a v e r y g o o d r e p r e s e n t a t i o no f t h e l o w - l y i n g r o t a t io n a l s t a t e s , r e p r o d u c i n g e x p e r i m e n t t o a b o u t 0"1 p e r c e n t[ 1 2 , 2 2 ] . H o w e v e r o u r p r e v i o u s c a l c u l a t i o n s h a v e s h o w n t h a t t h e B V D H p o t e n t i a lf i t t e d b y M a r t i r e a n d B u r t o n [ 2 3 ] t o t h e a b i n i t i o d a t a o f B u r t o n e t a l . [ 2 4 ] g i v e sb e t t e r r e s u l t s f o r t h e v i b r a t i o n a l e x c i t a t i o n e n e r g i e s t h a n d o e s t h e S D L p o t e n t i a l .I t w a s t h e r e f o r e t h o u g h t a p p r o p r i a t e t o c o m p a r e t h e e f f ec t i v e n e s s o f t h e s e t w op o t e n t i a l s i n h i g h J c a l c u l a t i o n s . T h e r e s u l t s o f s u c h a c o m p a r i s o n , f o r J = 1 6 e,a r e s h o w n i n ta b l e 7 . T a b l e s 8 t o 1 1 g i v e th e c a l c u l a t e d l e v e l s f o r H 2 D + w i t h J u pt o 2 0 f o r t h e f o u r s y m m e t r y b l o c k s u s i n g t h e S D L p o t e n t i a l a n d t a b l e 12 c o m -p a r e s o u r l a b e l l in g o f t h e s y m m e t r y b l o c k s w i t h o t h e r l a b e l l i n g s c h e m e s .

I n a l l t h e t a b l e s 7 t o 1 1 , t h e f r e q u e n c i e s o f t h e l e v e l s a r e g i v e n r e l a t i v e to t h eJ = 0 v i b r a t i o n a l g r o u n d s t a te o f H 2 D + . F r o m t h e s e a n d t h e a p p r o p r i a t e s e l e c t i o nr u l e s ( A J = 0 , __+ 1 ; A K a = 0 , 2 . . . . ; A K c = 1 , 3 , . . . ) a l i s t o f p r e d i c t e d t r a n s i t i o nf r e q u e n c i e s h a s b e e n c o n s t r u c t e d , c o p i e s o f w h i c h c a n b e o b t a i n e d f r o m t h ea u t h o r s . I t w a s h o w e v e r n o t p o s s i b l e t o la b e l a l l s t a te s w i t h v a l u e s o f ( K a , K c ) .T h i s i s b e c a u s e o f t h e i n c r e a s i n g o v e r l a p b e t w e e n r o t a t i o n a l m a n i f o l d s o f d i f fe r e n tv i b r a t i o n a l s t a t e s . T h i s b e h a v i o u r i s a l so r e f l ec t e d i n t h e k v a l u e s o f t h e N oC o r i o l i s r e s u l t s p r e s e n t e d i n t a b l e s 4 t o 6.

T a b l e 7 . C o m p a r i s o n o f t h e J = 1 6 e r o t a t i o n a l l e v e l s f o r t w o ab init io p o t e n t i a l s .W a v e n u m b e r s / e m - 1

j e v e n j o d dS t a t en u m b e r S D L B V D H A S D L B V D H A1 4637 .44 4631"65 5 .79 4635 -27 4629"55 5"722 5548-71 554 3.06 5"75 5548 -69 5543 .05 5"743 6225"05 6221"07 3"98 622 4.96 622 1 '01 3-954 6 3 9 6 ' 5 6 6 4 0 7 . 8 7 - - 1 1 - 3 1 6 3 8 4 . 2 5 6 3 9 5 . 7 8 - - 1 1 " 5 35 6708 -13 6705 .12 3 -01 6724 .40 6722 -34 2"066 6 9 7 4 "5 0 6 9 7 3 - 5 0 1 ' 0 0 7 0 4 5 - 3 4 7 0 5 8 - 7 8 - - 1 3 " 4 47 7 04 7" 51 7 0 6 0 . 8 2 - - 1 3 ' 3 1 7 1 5 5 - 4 2 7 1 5 5 - 1 6 0 ' 2 68 7 4 1 0 - 1 7 7 4 1 1 " 04 - - 0 " 7 9 7 4 2 0 - 4 0 7 4 2 4 - 5 5 - - 4 ' 1 59 742 4 '76 7427 .91 - - 3 -15 7597 .19 7590"44 6 '751 0 7 6 0 0 "3 6 7 5 9 3 - 2 5 7 " 1 1 7 7 0 8 -4 1 7 7 0 7 . 8 3 0 - 5 81 1 7 8 8 1 ' 9 3 7 8 9 8 . 1 6 - - 1 6" 77 7 8 8 2 . 1 9 7 8 9 8 "3 9 - - 1 6 ' 2 012 8031 -74 8030 .82 0"92 8217 .33 8218 .21 - -0 -881 3 8 2 1 7 - 4 9 8 2 1 8 . 3 2 - - 0 - 8 3 8 3 0 2 " 88 8 3 2 0 "2 9 - - 1 7 . 4 114 8332"45 8349 .57 - - 17 .12 8373 .36 8372"65 0 .711 5 8 4 3 4 - 3 9 8 4 5 2 . 0 8 - - 1 7 " 6 9 8 4 3 6 - 1 8 8 4 5 3 -4 3 - - 1 7 - 2 51 6 8 5 3 8 ' 2 0 8 5 3 3 "1 0 5 "1 0 8 5 3 8 ' 5 4 8 5 3 3 ' 7 0 4 . 8 4

1 7 8 6 3 7 - 1 2 8 6 6 8 . 0 4 - - 3 0 " 9 2 8 6 2 6 . 6 9 8 6 5 7- 1 1 - - 3 0 - 4 21 8 8 7 2 3 - 2 4 8 7 2 3 - 4 9 - - 0 " 2 5 8 7 8 7 "6 7 8 7 9 9 " 5 4 - - 1 1 - 8 719 8812"47 8813 .61 - - 1 "14 8861 -19 8874 -49 - -1 3 .3 02 0 8 8 6 7 - 55 8 8 9 1 "8 2 - - 1 4 " 2 7 9 0 5 9 ' 7 1 9 0 6 9 ' 8 1 - - 1 0 " 1 02 1 9 1 4 7 . 6 4 9 1 4 2 " 8 4 4 " 8 0 9 0 6 7 . 9 0 9 0 8 1 ' 4 9 - - 1 3 . 5 92 2 9 1 7 1 -1 1 9 1 7 6 - 0 4 - - 4 " 9 3 9 1 5 5 - 0 2 9 1 6 0 " 86 - - 5 . 8 423 9196"02 9196"27 - -0 "2 5 9210 -06 9230 -00 - - 19 .9424 9335"83 9356 .46 - -2 0 -6 3 9323"17 9325"22 - -2 "0 5


10/19

1 0 7 6 J . T e n n y s o n a n d B . T . S u t c l i f f eT a b l e 8 . H z D + r o t a t i o n a l le v e l s w i t h j e v e n a n d e ( p = 0 ) t o t a l p a r i t y r e l a t iv e t o t h e J = 0

g r o u n d s t a te a t - - 3 2 9 6 2 " 2 4 c m - i .J W a v e n u m b e r s / c r n - 11 4 5 " 6 4 2 2 2 5 . 0 9 2 3 7 7 " 4 52 1 3 1 . 4 6 2 2 3 - 6 0 2 2 9 6 " 4 9 2 4 0 5 " 3 7 2 4 6 5 ' 8 83 2 5 1 " 05 3 7 5 . 8 6 2 3 9 3 " 31 2 5 5 8 ' 9 2 2 5 9 3 " 6 8 2 7 9 6 - 7 04 4 0 2 - 2 9 5 8 0 - 7 0 7 7 8 . 1 0 2 5 1 7 ' 7 9 2 7 5 1 ' 0 9 2 7 6 7 - 01

2 9 4 4 ' 0 4 2 9 8 4 . 4 65 5 8 5 ' 4 1 8 3 2 - 0 7 1 0 1 3 " 7 5 2 6 7 1 ' 8 7 2 9 5 0 . 4 0 3 0 0 3 "9 1

3 1 8 1 " 4 6 3 2 1 7 -2 1 3 5 5 6 - 4 76 8 0 0 ' 5 7 1 1 2 2 . 0 2 1 30 0 " 5 3 1 6 4 8 - 6 3 2 8 5 6 " 3 6 3 1 7 8 " 6 9

3 2 7 1 - 5 7 3 4 6 6 . 1 7 3 4 9 9 - 6 6 3 7 8 7 - 9 3 3 7 9 1 ' 0 37 1 0 4 7 ' 6 1 1 4 4 2 . 8 5 1 6 4 0 " 5 4 1 9 6 6 - 5 0 3 0 7 1 " 5 7 3 4 3 7 ' 3 9

3 5 6 4 - 6 6 3 7 8 2 - 3 5 3 8 4 0 - 4 7 4 0 3 4 " 9 5 4 1 0 2 ' 4 8 4 1 4 5 " 0 18 1 3 2 6 ' 2 1 1 7 9 0 . 8 4 2 0 3 1 " 4 7 2 3 27 - 5 1 2 7 9 9 ' 7 5 3 3 1 7 - 6 13 7 2 5 " 9 4 3 8 84 - 2 1 4 1 2 8 - 3 3 4 2 2 9 - 0 3 4 3 1 0 - 4 0 4 4 5 4 " 4 54 5 0 9 ' 1 6 4 7 7 5 . 6 5

9 1 6 3 5 . 9 9 2 1 6 5 - 9 3 2 4 6 6 - 5 0 2 7 3 2 - 01 3 1 9 4 . 6 9 3 5 9 4 - 5 34 0 4 3 " 7 9 4 2 3 1 . 3 9 4 5 0 4 " 4 7 4 6 1 7 "0 8 4 6 5 2 ' 0 5 4 8 4 3 " 4 64 9 2 1 . 3 4 5 1 4 6 . 5 4 5 2 8 3 - 5 5

1 0 1 9 7 6 " 5 2 2 5 6 8 - 7 4 2 9 3 7 - 0 4 3 1 8 1 ' 3 9 3 6 2 6 " 4 4 3 9 0 2 " 3 54 1 9 1 . 3 4 4 3 9 0 . 4 4 4 6 0 6 - 5 8 4 9 0 7 " 1 1 4 9 5 6 - 9 4 5 1 0 0 " 2 35 2 6 8 - 9 8 5 3 8 1 - 0 5 5 5 4 5 ' 9 3 5 7 0 1 - 1 7 5 8 0 2 " 0 2

1 1 2 3 4 7 " 3 2 2 9 9 9 . 3 1 3 4 3 4 - 5 4 3 6 7 7 ' 3 7 4 0 9 4 . 1 7 4 2 4 1 " 0 74 6 5 7 ' 4 6 4 7 6 5 . 3 6 5 0 0 9 " 6 3 5 3 1 1 " 1 3 5 3 5 2 " 8 0 5 5 6 9 - 1 85 7 2 9 " 3 3 5 8 8 4 " 83 5 9 7 3 " 8 8 6 1 4 9 ' 8 8 6 2 1 2 " 8 3 6 2 7 2 " 9 91 2 2 7 4 7 - 9 2 3 4 5 7 - 2 8 3 9 5 3 - 0 6 4 2 1 9 - 2 7 4 5 9 7 - 1 8 4 6 1 0 - 7 65 1 5 2 - 9 0 5 1 6 8 - 0 9 5 4 4 0 " 0 9 5 7 1 2 " 3 4 5 7 8 0 " 4 5 5 8 0 7 - 3 66 0 5 8 " 0 1 6 2 2 1 . 6 8 6 4 2 4 . 9 1 6 4 3 1 " 4 2 65 7 1 - 1 1 6 6 3 0 " 3 06 7 8 0 " 0 3 6 8 5 2 - 2 2

1 3 3 1 7 7 " 7 9 3 9 4 2 . 0 0 4 4 9 0 ' 9 4 4 8 0 1 " 8 5 5 0 1 1 " 2 5 5 1 3 5 " 7 15 5 9 8 -2 1 5 6 7 7 - 0 5 5 8 9 7 ' 3 3 6 1 4 0 - 2 8 6 2 9 0 - 1 4 6 3 1 2 - 9 56 5 6 7 " 1 5 6 7 4 2 . 2 6 6 9 1 4 ' 7 4 6 9 6 3 " 6 3 6 9 9 7 - 2 7 7 1 4 2 ' 2 37 2 5 5 - 4 1 7 3 2 2 . 8 4 7 4 6 3 ' 9 8

1 4 3 6 3 6 -3 8 4 4 5 2 - 7 2 5 0 4 8 ' 7 6 5 4 1 6 - 6 3 5 4 4 2 " 7 6 5 7 1 0 " 1 66 0 5 5 " 1 2 6 2 2 9 . 0 3 6 3 8 0 - 6 0 6 5 9 7 " 2 7 6 7 9 7 - 6 0 6 8 6 5 ' 4 57 0 9 6 " 9 7 7 2 8 7 - 0 7 7 3 8 8 - 3 0 7 4 2 7 " 7 6 7 5 2 4 "1 1 7 5 8 9 . 4 87 6 8 3 - 51 7 6 8 9 . 2 2 7 9 0 1 " 4 4 8 0 1 7 - 1 4 8 1 9 7 " 4 41 5 4 1 2 3 - 1 3 4 9 8 8 . 6 0 5 6 2 6 " 9 4 5 9 0 4 - 2 3 6 0 5 5 - 2 3 6 3 2 2 " 5 86 5 3 8 - 3 5 6 8 0 7 . 9 9 6 8 8 9 " 1 2 7 0 8 3 " 9 0 7 3 2 8 . 5 7 7 4 3 8 ' 4 67 6 4 7 - 3 6 7 8 4 3 . 7 5 7 8 5 3 " 4 8 7 9 6 9 " 0 2 8 1 1 9 - 6 1 8 1 4 7 ' 3 78 1 9 6 - 4 3 8 2 6 0 . 9 0 8 5 1 4 " 2 2 8 5 9 1 - 3 8 8 6 7 6 - 1 6 8 7 5 3 ' 3 9

1 6 4 6 3 7 " 4 4 5 5 4 8 . 7 1 6 2 2 5 - 0 5 6 3 9 6 " 5 6 6 7 0 8 " 1 3 6 9 7 4 - 5 07 0 4 7 - 51 7 4 1 0 . 1 7 7 4 2 4 - 7 6 7 6 0 0 " 3 6 7 8 8 1 " 9 3 8 0 3 1 . 7 48 2 1 7 "4 9 8 3 3 2 .4 5 8 4 3 4 " 39 8 5 3 8 ' 2 0 8 6 3 7 " 1 2 8 7 2 3 ' 2 48 8 1 2 . 4 7 8 8 6 7 . 5 5 9 1 4 7 - 6 4 9 1 7 1" 1 1 9 1 9 6 " 0 2 9 3 3 5 -8 39 3 6 5 - 6 9 9 3 8 1 . 6 0

1 7 5 1 7 8 " 7 0 6 1 3 2 - 1 2 6 8 4 1 " 4 1 6 9 2 0 " 1 0 7 3 7 1 " 0 6 7 5 8 1 " 6 37 6 6 4 . 7 4 7 9 7 4 - 9 2 8 0 4 7 " 2 9 8 1 4 6 " 5 0 8 4 5 7 - 01 8 6 4 4 - 7 38 8 0 6 "2 6 8 8 5 0 . 9 9 9 0 3 1 "4 7 9 1 3 3 ' 1 5 9 1 5 7 . 4 3 9 3 3 8 " 3 99 4 3 8 " 9 3 9 5 0 1 - 6 1 9 6 8 9 ' 9 8 9 7 8 5 - 6 9 9 8 4 5 . 2 2 9 8 9 4 " 6 79 9 4 4 " 3 4 9 9 6 0 " 9 9 1 0 0 3 8 - 6 5

1 8 5 7 4 6 " 2 3 6 7 3 7 - 7 8 7 4 5 7 " 0 7 7 4 9 2 - 1 4 8 0 4 3 - 5 0 8 1 4 1 " 6 68 3 8 4 - 5 7 8 5 5 2 - 4 7 8 7 0 6 ' 6 4 8 7 2 1 " 7 6 9 0 5 2 " 5 4 9 2 7 7 - 0 39 3 9 1 . 3 6 9 4 2 0 . 7 6 9 6 4 1 . 6 0 9 7 0 6 - 7 4 9 7 5 3 " 5 4 9 9 6 6 " 2 6

1 0 0 7 9 " 3 5 1 0 1 5 6 - 2 9 1 0 2 4 3 ' 7 9 1 0 4 1 7 - 1 2 1 0 4 4 0 - 8 5 1 0 5 3 0 - 0 71 0 5 5 0 - 0 9 1 0 5 8 0 . 5 9 1 0 6 8 8 . 4 1 1 0 7 7 5 - 1 5


11/19

Rota t iona l l y exc i t ed s ta tes o f H 2 D+Table 8 (continued).

1077

Wavenumbers/cm -119 6339.34 7364.71 8044.26 8138.90 8713"57 8739-889122-25 9154-34 9325-20 9396.01 9667.73 9927"839966-84 10046-28 10254-04 10300-84 10392'70 10606"7710734-13 10825.77 10827-83 11014.84 11065.06 11150"2111228.35 11253.45 11340"51 11369"18 11543'21 11589.2920 6957-24 8011-79 8652'67 8808"63 9328.80 9428.639770.92 9875"63 9954.54 10119.79 10301"07 10566-9010597-10 10691-81 10864.73 10940.45 11049-83 11260"3511402.01 11438.62 11507"76 11614'44 11725-79 11789-8811908.55 11963-04 11982-20 11998.39 12206"70 12252"5412328.81 12426.94

Table 9. H2 D+ rotational levels wi th j even a nd f ( p = 1) total parity relative to the J = 0ground state at - - 3 2962 - 24cm-1J Wavenumbe rs/cm- 11 2383.712 218.41 2393.45 2486"643 354-33 2517.44 2638.16 2796.634 530.63 777.76 2676'26 2834-14 2942-59 2984.865 744.08 1011-06 2867-33 3068-50 3173"44 3220"10

3556.476 991-93 1289-47 1648-62 3089.30 3334"84 3447"423504.52 3788.01 3827.717 1272.33 1610-42 1966.36 3341'67 3629-60 3759-423837.81 4102.48 4143 '55 4259"008 1584-12 1970.44 2326.68 2799.75 3624.25 3952'184105.06 4215.47 4455.16 4503-51 4568.39 4895'379 1926-56 2365-77 2728.58 3194.69 3936.88 4302.734481-31 4630.56 4846'96 4902.14 4912.20 5282"405342.8310 2299.06 2792-97 3170-49 3626'41 4191-34 4279-444681.14 4886.06 5075-04 5268.76 5292"10 5345"775698.25 5770.96 5802-3311 2701.06 3249.26 3650.10 4093'99 4651-69 4657-495087.06 5317.79 5541"46 5676-58 5764'15 5820.346144.60 6228.35 6271-51 6606.4412 3131.98 3732.61 4164.29 4596-56 5053'53 5152"925520-00 5775.39 5780-47 6025"05 6110.76 6279.686329.34 6622'45 6713.07 6775-47 7018'64 7118"9313 3591-21 4241.56 4709.46 5133-13 5484'76 5677"055979.48 6257.82 6312.93 6520-82 6581'69 6819'906876.02 7132-49 7224.63 7311.67 7432-88 7643.677701-4714 4078.13 4774.99 5281"85 5702.53 5945"15 6229"036464.96 6764.27 6865.42 7024.72 7094"24 7376"097457.30 7524.11 7676-13 7763-37 7872"05 7883"668175.28 8181.28 8268"6815 4592.07 5331'91 5878-02 6303.09 6434"67 6808"046975.96 7293"95 7438-40 7545.42 7639-60 7949-208060.37 8119.64 8253-43 8331.38 8354'38 8471"478670.13 8730.89 8862-27 8976.35


12/19

1 0 7 8 J . T e n n y s o n a n d B . T . S u t c l i f f eT a b l e 9 (continued).

W a v e n u m b e r s / c m - 11 6 5 1 3 2 - 3 3 5 9 1 1 - 3 2 6 4 9 5 . 0 2 6 9 3 0 ' 7 2 6 9 5 4 . 6 8 7 4 1 3 ' 2 5

7 5 1 1 ' 9 5 7 8 4 5 - 8 2 8 0 3 1 " 5 5 8 0 8 9 . 9 9 8 2 0 9 - 6 6 8 5 3 9 ' 6 78 6 7 5 . 2 6 8 7 2 3 " 4 4 8 8 5 2 ' 7 5 8 8 6 4 " 2 5 8 9 3 8 ' 9 0 9 0 8 8 " 7 29 1 8 8 " 2 7 9 3 0 8 . 4 4 9 3 8 1 . 6 0 9 4 7 5 . 5 2 9 5 8 0 " 8 0 9 6 6 9 " 2 4

1 7 5 6 9 8 ' 2 2 6 5 1 2 ' 2 7 7 1 3 0 ' 6 2 7 4 9 7 ' 9 6 7 5 8 9 ' 5 7 8 0 4 2 " 6 88 0 7 4 - 0 8 8 4 1 9 . 5 0 8 6 4 3 ' 7 4 8 6 6 0 " 5 7 8 8 0 1 - 5 5 9 1 4 7 " 3 69 2 9 9 "3 1 9 3 3 9 ' 1 4 9 3 9 6 - 3 8 9 4 7 5 " 3 9 9 5 8 9 - 6 5 9 7 2 6 - 8 49 7 3 4 " 9 6 9 9 1 1 ' 0 3 1 0 0 3 8 . 6 4 1 0 1 0 5 - 8 9 1 0 2 0 8 " 5 6 1 0 2 2 1 - 4 1

1 0 3 5 0 . 4 41 8 6 2 8 8 " 9 8 7 1 3 3 ' 7 3 7 7 8 3 ' 1 2 8 0 7 2 " 7 0 8 2 6 7 - 3 5 8 6 5 3 ' 7 1

8 7 0 3 . 7 8 9 0 1 4 ' 5 5 9 2 5 3 " 3 6 9 2 7 7 " 9 1 9 4 1 3 " 6 6 9 7 7 1 " 9 09 9 3 0 " 2 8 9 9 6 5 . 9 8 9 9 7 0 - 2 6 1 0 1 0 6 " 3 2 1 0 2 7 3 . 4 1 1 0 3 0 5 ' 4 01 0 3 8 7 " 4 2 1 0 5 4 0 ' 6 0 1 0 6 8 8 " 4 1 1 0 7 5 1 -7 3 1 0 7 9 5 ' 5 3 1 0 8 6 2 . 9 7

1 0 9 2 7 - 9 4 1 1 0 6 2 " 8 0 1 1 1 0 0 - 6 01 9 6 9 0 3 ' 8 6 7 7 7 4 ' 7 5 8 4 5 1 " 1 8 8 6 7 4 "8 8 8 9 6 3 ' 7 2 9 2 6 2 ' 2 0

9 3 8 4 . 3 0 9 6 3 0 " 7 6 9 8 7 2 - 4 6 9 9 2 8 . 3 4 1 0 0 4 5 " 1 6 1 0 4 1 2 . 7 71 0 5 4 6 - 0 7 1 0 5 9 4 . 7 6 1 0 6 1 2 - 9 4 1 0 7 5 3 ' 9 0 1 0 9 0 3 . 2 4 1 0 9 7 3 ' 5 11 1 0 6 5 " 1 2 1 1 1 9 9 ' 4 7 1 1 3 4 0 " 5 1 1 1 3 9 1 ' 8 9 1 1 4 2 1 ' 1 4 1 1 5 1 2 " 1 61 1 5 5 0 - 8 1 1 1 6 8 0 " 9 5 1 1 7 3 2 - 7 8 1 1 8 3 4 . 4 2

2 0 7 5 4 2 " 0 6 8 4 3 4 ' 3 2 9 1 3 3 " 7 0 9 3 0 3 ' 9 8 9 6 7 4 . 6 8 9 8 9 2 ' 7 21 0 0 8 9 - 4 4 1 0 2 6 7 . 7 9 1 0 5 1 3 - 5 3 1 0 5 9 7 . 5 0 1 0 6 9 4 " 9 3 1 1 0 6 9 ' 7 61 1 1 6 9 " 6 9 1 1 2 5 1 ' 8 7 1 1 2 7 4 " 7 3 1 1 4 1 8 ' 3 4 1 1 5 2 5 " 7 6 1 1 6 7 7 ' 7 81 1 7 5 7 - 2 8 1 1 8 9 4 " 2 2 1 1 9 9 8 " 1 7 1 2 0 2 1 . 4 7 1 2 0 9 8 ' 0 1 1 2 1 3 8 " 6 81 2 2 3 5 " 8 7 1 2 3 2 1 ' 0 2 1 2 3 8 3 " 1 9 1 2 4 6 5" 1 3 1 2 6 2 9 . 1 2 1 2 7 0 4 . 6 2

T a b l e 1 0. H 2 D + r o t a t i o n a l l e v e ls w i t h j o d d a n d e ( p = 0) to t a l p a r i t y r e l a t i v e t o t h eJ = 0 g r o u n d s t a t e a t - - 3 2 9 6 2 - 2 4 c m - 1 .

J W a v e n u m b e r s / c m - 11 5 9 " 9 6 2 2 3 6 - 9 8 2 3 5 8 - 3 12 1 3 8 ' 6 7 2 3 0 0 - 6 9 2 4 5 2 - 1 9 2 5 4 4 - 513 2 5 3 " 6 9 4 5 7 - 8 2 2 3 9 4 - 3 9 2 5 8 5 " 2 8 2 6 2 9 . 2 4 2 6 9 1 ' 6 54 4 0 3 . 0 8 6 4 4 " 7 7 2 5 1 7 " 9 5 2 7 5 1 - 9 9 2 8 0 9 ' 0 6 2 8 8 9 . 4 4

3 1 3 5 " 5 65 5 8 5 . 61 8 7 5 "4 1 1 1 7 5 " 9 0 2 6 7 1 " 7 2 2 9 4 9 ' 9 8 3 0 2 8 . 1 5

3 1 3 6 " 1 9 3 3 3 0 - 5 0 3 3 6 5 - 2 16 8 0 0 " 5 9 1 1 4 6 - 3 4 1 4 5 2 - 4 8 2 8 5 5 " 9 9 3 1 7 8 . 5 4 3 2 8 2 ' 3 9

3 4 2 7 ' 2 1 3 6 0 6 - 0 8 3 6 3 9 - 3 2 3 7 9 1 - 0 07 1 0 4 7 ' 5 7 1 4 5 3 ' 9 4 1 7 7 3 " 5 9 2 1 9 1 " 5 2 3 0 7 0 ' 9 3 3 4 3 7 . 3 1

3 5 6 8 ' 8 3 3 7 5 6 ' 0 7 3 9 2 5 "3 7 3 9 5 8 "0 6 4 0 3 5 .1 2 4 3 1 0 ' 9 98 1 3 2 6 ' 1 2 1 7 9 5 " 0 6 2 1 3 7 " 6 6 2 5 4 8 " 6 7 3 3 1 6 ' 5 8 3 7 2 5 " 8 3

3 8 8 5 "6 7 4 1 1 5 ' 5 7 4 2 8 7 ' 4 0 4 3 0 8 " 7 8 4 3 2 3 " 1 7 4 6 6 2 . 2 54 7 2 2 ' 2 6

9 1 6 3 5 . 8 4 2 1 6 7 .3 1 2 5 4 2 - 1 5 2 9 4 5 - 7 2 3 4 6 8 . 1 4 3 5 9 2 " 9 64 0 4 3 . 6 2 4 2 3 1 - 8 3 4 5 0 0 - 1 8 4 6 1 6 " 6 5 4 6 8 8 . 1 2 4 7 3 0 " 1 35 0 4 6 " 9 1 5 1 2 0 - 4 5 5 1 4 9 " 0 3

1 0 1 9 7 6 " 2 8 2 5 6 9 - 1 2 2 9 8 3 " 7 2 3 3 8 1 - 8 0 3 8 9 9 " 3 1 3 9 0 0 - 0 84 3 9 0 . 1 8 4 6 0 6 - 6 6 4 9 0 6 - 4 8 4 9 5 6 -4 1 5 1 1 9 " 4 3 5 1 8 3 - 0 05 4 6 5 . 8 2 5 5 4 7 - 7 2 5 5 6 0 " 9 1 5 8 7 9 - 2 6

1 1 2 3 4 6 ' 9 6 2 9 9 9 " 4 0 3 4 5 8 - 6 3 3 8 5 5 " 8 5 4 2 3 7 " 7 5 4 3 6 3 ' 8 44 7 6 4 ' 9 9 4 9 6 3 " 9 2 5 0 0 9 " 5 9 5 3 1 0 " 7 4 5 3 5 2 . 9 1 5 5 7 7 ' 6 75 6 7 4 ' 1 0 5 9 1 9 " 4 4 5 98 0 " 9 9 6 0 34 " 5 0 6 2 0 1 .2 5 6 3 9 9 ' 2 6


13/19

R o t a t i o n a l l y e x c i t ed s ta t e s o f H 2 D +Table 10 (continued).

1079

Wavenumbers /cm - 112 2747.36 3457.29 3963-24 4366-47 4606"07 4860-725167.54 5439.97 5463.75 5711"47 5807.69 6061"01

6194-16 6403-97 6450-12 6542-44 6556'56 6846-876875"0813 3176.98 3942.00 4494.40 4911.62 5005-25 5389-115597-53 5897-26 5988.50 6138"98 6290-43 6567"936635.58 6734.11 6905.21 6945.76 6972-82 7081-497249.26 7377.90 7476"28

14 3635.22 4452.72 5049"62 5434"35 5488.90 5948"166054-17 6380-39 6537-95 6595-41 6797'77 7096"997199-88 7287.94 7366.52 7424"14 7546-68 7647'797679-56 7908-98 8031.38 8124-0915 4121.53 4988-59 5627-02 5894.20 6094"27 6533"646540.55 6888.53 7081"40 7111'53 7328-70 7647-237778.62 7818.85 7855"32 7968"17 8138.39 8156-688241.00 8441.02 8469.52 8607-79 8667"05 8711'5716 4635.28 5548-69 6224.96 6384"25 6724-40 7045"347155.42 7420-39 7597.19 7708.41 7882"19 8217"338302.88 8373.35 8436-18 8538.55 8626.68 8787-688861-19 9059.70 9067-90 9155-02 9210.06 9323"179365.69

17 5175.86 6132-05 6840.80 6905-81 7375"41 7579"507801.25 7975.00 8142.80 8327"91 8457-43 8802.408821.08 8984.26 9032-15 9132.57 9146-86 9427"589507.61 9671.93 9688.68 9696'51 9828.40 9888-509950-41 9971.34 10187.5018 5742-59 6737.65 7443.71 7488-02 8043.48 8138.908472.77 8552.04 8717.70 8969'11 9053.61 9355'979418.26 9610.52 9640.56 9696.05 9753-83 10074"85

10169"24 10226.54 10327-68 10348.70 10438-50 10468"1010537.74 10625.72 10773.48 10803.9219 6334-76 7364.43 8024"76 8138-26 8711-04 8738.149143.50 9175-45 9320.77 9629"94 9671.41 9926.18

10044.98 10241.11 10255.07 10298-79 10392"69 10732.5010807.10 10837.80 10971.71 11011.10 11035-84 11125.2511158.35 11303.02 11365-94 11441.80 11584-51 11688"4220 6951"61 8011'27 8630-18 8808"35 9324'47 9428'299766.61 9888.78 9950.05 10291.41 10328.33 10522.8810690.35 10848.33 10909.84 10944-91 11049'56 11399.6111419.47 11512.24 11609.74 11625.58 11723.10 11780.6211848-46 11976.66 11996.96 12103.59 12202.07 12365-3912373.48

Tabl e 11. H2D rotational levels with j odd a nd f (p = 1) total parity relative to the J ~ 0ground state at -- 32962'24 cm - 1.

J Wavenumbers /em- 11 72-37 2256.862 175-71 2357.94 2543.573 325.72 459.30 2498.78 2634.23 2685-244 515.44 653.68 2668 '90 2831'92 2869' 64 3135"63


14/19

1 0 8 0 J . T e n n y s o n a n d B . T . S u t c l i f f e

56789

1 0

11

1 2

1 3

1 4

1 5

1 6

1 7

1 8

1 9

2 0

T a b l e 1 1 (continued).W a v e n u m b e r s / c m - 1

7 3 7 " 7 4 9 0 2 " 8 6 1 1 7 5 " 9 7 2 8 6 4 " 8 9 3 0 7 6 ' 2 1 3 1 0 1 - 3 03 3 3 0 - 7 3 3 3 6 5 ' 6 6

9 8 9 - 7 4 1 2 0 4 " 3 7 1 4 5 3 . 1 4 3 0 8 8 - 5 8 3 3 4 0 . 3 5 3 3 9 4 - 3 63 6 0 7 " 1 2 3 6 4 1 " 4 9 3 9 7 9 " 9 41 2 7 1 - 6 6 1 5 5 1 " 43 1 7 7 6 - 9 3 2 1 9 1 " 5 3 3 3 4 1 " 5 0 3 6 3 1 " 4 93 7 2 8 " 3 2 3 9 2 5 - 8 5 3 9 6 7 " 2 9 4 2 5 8 - 4 7 4 3 1 1 - 0 41 5 8 3 - 9 4 1 9 3 5 " 6 1 2 1 4 9 - 2 8 2 5 4 8 " 7 0 3 6 2 4 " 2 7 3 9 5 2 " 5 34 0 8 9 ' 7 9 4 2 8 1 - 8 6 4 3 4 7 " 1 3 4 5 6 7 ' 5 6 4 6 6 2 - 5 8 4 7 2 3 " 1 51 9 2 6 - 5 3 2 3 4 8 " 7 3 2 5 7 1 " 9 7 2 9 4 5 " 9 0 3 4 6 8 - 1 4 3 9 3 7 " 0 04 3 0 2 " 6 6 4 47 4 " 81 4 6 7 2 " 5 4 4 7 7 6 " 9 7 4 9 0 7 ' 2 5 5 0 4 7 ' 7 75 1 2 3 " 0 7 5 3 3 5 - 4 52 2 9 9 . 0 6 2 7 8 6 ' 0 5 3 0 4 3 " 3 6 33 8 2 " 6 3 3 8 9 9 "3 6 4 2 7 9 ' 6 34 6 8 1 ' 0 4 4 8 8 3 " 6 2 5 0 9 5 - 7 7 5 2 4 7 "7 3 5 2 7 7 " 5 3 5 4 6 7 " 3 45 5 6 5 " 1 8 5 7 6 6 . 6 3 5 9 5 9 ' 9 32 7 0 1" 0 8 3 2 4 6 .9 0 3 5 5 7 . 1 8 3 8 5 8 -9 5 4 3 6 3 ' 8 4 4 6 5 2 ' 0 04 9 6 3 - 9 2 5 0 8 7 - 0 2 5 3 1 7 " 0 2 5 5 4 8 - 2 7 5 6 7 8 - 5 25 9 2 1 "6 1 6 0 4 9 - 0 7 6 2 2 2 - 6 0 6 4 0 8 - 9 5 6 5 3 8 " 8 7 6 5 5 4 ' 0 13 1 3 2 " 0 1 3 7 3 1 - 9 5 4 1 0 4 - 9 2 4 3 7 5 - 8 5 4 8 6 0 " 7 4 5 0 5 3 " 9 35 4 6 3 - 7 3 5 5 2 0 " 0 0 5 7 7 5 " 21 6 0 2 5 " 4 9 6 1 1 1 - 6 8 6 2 7 5 - 6 96 4 1 0 " 1 9 6 5 74 " 4 6 6 7 0 4 - 0 2 6 8 8 5 - 63 7 0 2 0 ' 1 4 7 0 3 7 , 0 33 5 9 1 " 2 7 4 2 4 1 . 4 4 4 6 7 7 " 6 7 4 9 3 5 " 1 2 5 3 8 9 - 2 2 5 4 8 5 " 4 25 9 7 9 - 4 1 5 9 8 8 " 6 3 6 2 5 7 " 9 1 6 5 2 0 " 0 5 6 5 8 1 " 7 1 6 6 3 5 - 5 86 8 1 8 ' 7 9 6 9 3 2 - 1 9 7 1 3 9 " 0 7 7 2 1 1 ' 1 7 7 3 9 1 ' 0 3 7 4 3 5 ' 6 17 5 6 5 " 7 5 7 6 6 7 . 2 84 0 7 8 . 2 1 4 7 7 4 . 9 9 5 2 6 8 - 2 4 5 5 3 7 - 21 5 9 4 3 . 6 0 5 9 5 1 . 0 96 4 6 4 . 9 7 6 5 3 7 ' 9 9 6 7 6 4 . 4 1 7 0 2 4 . 4 9 7 0 9 3 " 6 2 7 1 9 9 . 8 97 3 7 7 " 0 2 7 4 8 5 -4 1 7 7 3 0 " 9 0 7 7 5 0 - 7 3 7 8 8 1 - 1 3 7 9 2 5 . 0 58 1 2 5 - 3 4 8 1 8 0 - 2 6 8 2 4 9 - 5 84 5 9 2 - 1 8 5 3 3 1 - 9 2 5 8 7 3 - 6 7 6 1 7 8 ' 3 1 6 4 3 5 - 2 5 6 5 3 9 - 3 96 9 7 5 ' 8 9 7 1 1 1 "5 5 7 2 9 4 " 1 4 7 5 4 5 ' 4 7 7 6 3 9 " 1 2 7 7 7 8 .6 77 9 5 0 ' 8 9 8 0 6 6 ' 4 6 8 2 9 5 " 6 7 8 3 5 4 .7 8 8 3 7 3 ' 0 9 8 4 4 1 . 0 38 4 8 7 . 7 5 8 6 7 1 . 2 6 8 7 1 5 . 7 0 8 8 4 7 - 0 6 9 0 4 2 " 0 75 1 3 2 " 4 8 5 9 1 1 " 3 4 6 4 9 4 " 2 6 6 8 5 0 " 4 3 6 9 5 3 " 7 3 7 1 6 1 . 2 87 5 1 1 - 6 7 7 7 0 8 - 4 4 7 8 4 6 - 2 7 8 0 9 0 - 0 4 8 2 0 9 - 4 3 8 3 7 3 - 5 58 5 4 1 - 2 2 8 6 7 0 - 71 8 8 6 1 - 3 2 8 8 7 9 " 4 8 9 0 1 8 - 6 7 9 0 6 7 . 0 79 0 7 8 . 35 9 1 8 9 . 5 8 9 3 3 6 - 9 9 9 4 5 9 ' 4 6 9 6 2 0 - 1 4 9 6 7 0 '2 15 6 9 8 "4 1 6 5 1 2 ' 2 8 7 1 3 0 " 8 0 7 4 9 9 "0 1 7 5 4 5 ' 7 7 7 8 1 6 - 9 38 0 7 1 . 9 7 8 3 2 7 . 9 6 8 4 2 0 " 3 7 8 6 5 9 - 9 1 8 8 0 1 . 2 8 8 9 8 4 . 8 99 1 4 8 " 5 4 9 2 9 3" 9 3 9 4 0 0 ' 9 8 9 4 8 1 - 2 2 9 6 7 9 " 2 5 9 6 9 2 ' 9 39 6 9 9 . 2 0 9 7 3 5 - 5 0 9 9 8 8 - 6 7 1 0 0 8 6 " 0 9 1 0 2 1 9 - 7 9 1 0 2 2 4 ' 8 7

1 0 3 4 1 ' 3 0 1 0 3 4 9 " 1 06 2 8 9 " 2 3 7 1 3 3 " 7 6 7 7 8 3 . 3 4 8 0 7 4 - 4 6 8 2 5 0 . 6 8 8 5 0 9 - 1 88 6 5 6 . 2 9 8 9 6 8 . 9 2 9 0 1 6 . 7 1 9 2 5 4 - 7 4 9 4 1 3 . 2 0 9 6 1 2 . 3 19 7 7 2 . 7 6 9 9 2 9 . 5 6 9 9 7 2 " 1 6 1 0 1 0 6 " 7 5 1 0 3 0 6 - 71 1 0 3 2 5 " 2 0

1 0 3 3 7 ' 1 5 1 0 3 5 9 " 7 9 1 0 6 6 5 " 2 0 1 0 7 2 8 -9 5 1 0 7 9 5 " 9 0 1 0 8 5 5 " 0 61 0 9 2 5 ' 0 8 1 1 0 2 9 - 0 9

6 9 0 4 " 1 8 7 7 7 4" 8 1 8 4 5 1 " 3 2 8 6 7 6 " 9 7 8 9 6 1 ' 3 4 9 2 3 5 - 9 99 2 6 6 " 3 4 9 6 2 2 " 1 4 9 6 4 4 ' 4 0 9 8 7 3 - 5 0 1 0 0 4 4 " 2 8 1 0 2 5 5 " 6 6

1 0 4 1 3 " 7 3 1 0 5 5 1 ' 6 1 1 0 6 0 0 - 0 0 1 0 7 5 2" 8 2 1 0 90 4 - 8 1 1 0 9 7 0 ' 1 41 1 0 0 0 . 4 2 1 1 0 4 2 - 4 9 1 1 3 4 5 " 2 5 1 1 3 9 6 - 9 2 1 1 4 0 5 " 8 5 1 1 5 1 1 - 0 51 1 5 2 4 . 8 5 1 1 6 8 2 - 2 2 1 1 6 9 9 "7 5 1 1 7 6 5 . 6 2 1 1 8 3 5 " 0 6

7 5 4 2 " 4 7 8 4 3 4" 4 1 9 1 3 3 " 8 7 9 3 0 6 ' 3 7 9 6 7 6 . 7 4 9 8 9 2 " 8 19 9 9 9 ' 5 2 1 0 2 6 5 ' 7 3 1 0 3 2 6 " 3 2 1 0 51 4 " 9 1 1 0 6 9 3 " 4 4 1 0 9 1 5 ' 1 7

1 1 0 7 1 - 3 8 1 1 1 7 6. 5 6 1 1 2 6 6 ' 1 7 1 1 4 1 7 - 4 9 1 1 5 2 7 ' 5 3 1 1 6 2 4 . 2 01 1 6 8 4 . 2 9 1 1 7 3 6 - 4 5 1 2 0 1 7 - 4 3 1 2 0 2 7 - 1 3 1 2 1 1 0 . 5 3 1 2 1 5 2 - 5 5


15/19

Rota t io na l ly exc i t ed s tate s o f HzD + 1081Table 12. Symmet ry labels and sizes of the rotational manifolds (My) for HzD +.

v = 0w M~I I

Point Parity Totalgroup [7] of j r parity~ K a K c v = 0 v2 = 1 v3 = 1A 1 e e 2n - 2 J + 2 - 2n J/ 2 + 1 J/2 + 1 (J + 1)/2A 2 o e 2n - 1 J + 2 -- 2n (J + 1)/2 (J + 1)/2 J/2 + 1BI e f 2n J + 1 -- 2n J/2 J/2 (J + 1)/2B 2 o f 2n -- 1 J + 1 -- 2n (J + 1)/2 (J + 1)/2 J/ 2

e = even, o = odd.e is p = 0, f i s p = 1. Total parity is (--1) a+v.w 2 . .. . M~= 0 .II Mani fold size using integer arithmetic .

5. DiscussioNFrom the results shown in table 7 i t is possible to make an assessment of the

likely error in our calculations ar is ing from the chosen potential . In fact theagreement between the resul ts obta ined f rom the S DL and BVD H potent ia ls i sfair ly good, the largest difference being a bout 30 cm -1 for a few of the highers tates . A detailed comparison of the levels shows a number of systematic featureswhich i t is possible to understand from a knowledge of the results obtained at lowJ with each of the potentials.

Th e frequenci es of the lowest 3 s tates with j even or odd an d a num be r of thehigher ones (for example, bot h s tates 16) are foun d to be 2-6 cm -1 hig her for theSDL than the BVD H potent ia l . Examina t ion of the wavefunct ion shows thatsuch s tates can, at least approxim ately, be associated with the vib rati onal gro un dstate. Th e previous calculations [3] for low J showed that a precisely s imilardiscrepan cy was observed in the gro und vibrat ional s tate results . The discrep ancymay be thought of as due to the discrepancy between the calculated rotationalconstants in each potential and because the terms in the energy proportional tothese constant s are mul tip lie d by j2, the difference will be magnif ied whendeali ng with hi gh J states.

On the other hand for certain s tates (for example both s tates number 11) theSDL potent ia l g ives f requencies that are 10- 20c m -1 lower than the BVDHpotential . E xam ina tio n of the wave func tions show that such s tates are in factassociated, approxi mately, with the v2 and v3 vibrat ional ly excited s tates . T hi sdiscrep ancy is again to be expected from pr evious calcula tions where i t was shownthat the SD L potential gives v 2 and v a fun dam ent al freque ncies which are abou t20 cm -t too low. T ha t the disc repancy is generally less than this is due to acancellat i on of errors. Th er e are a few levels, both s tates nu mb er 17 for example,for which the SD L frequencies are up to 30c m -1 lower than the BV DH ones.These levels can be associated with states with 2 quanta in the v2 and v3 modes,probably 2v2, for which d iscrepancies of th is magni tude between the potent ia lsare to be expected [3].

Not all the states fall exactly into the three categories given above, which canbe explained by the mixed vi brati onal parentage of some s tates and the s implis t ictreatment of the rotational constants in the above discussion. However, i t is clearthat the comparat ive behav iour of the two potent ia ls can be unde rs tood f rom thei rproperties at low J . I t is worth noti ng that , al tho ugh the absolute shif ts for J = 16


16/19

1 0 82 J . T e n n y s o n a n d B . T . S u t c l i f f el e ve l s a r e a s m u c h a s 3 0 c m -1 , t r a n s i t i o n s i n s i m u l a t e d s p e c t r a , w h i c h o f c o u r s eo n l y i n v o l v e A J = 0 , ~ 1 , w i l l g e n e r a l l y a g r e e t o a b o u t 1 c m - 1 p r o v i d e d t h e t r a n -s i t i o n s i n v o l v e n o v i b r a t i o n a l e x c i t a t i o n .

F i g u r e s 1 a n d 2 i l l u s t r a t e t h e q u a l i t a t i v e b e h a v i o u r o f t h e r e s u l t s o f t a b l e s 8 t o1 1. F i g u r e 1 s h o w s t h e b r o a d e n i n g o f t h e r o t a t i o n a l m a n i f o l d s f o r o n e s y m m e t r yb l o c k ( t h e t o t a l s y m m e t r i c e e b l o c k ) a s J i n c r e as e s 9 F o r c o m p a r i s o n , t h e p o s i t i o no f t h e l o w e s t 1 0 b a n d o r i g i n s a r e g i v e n f o r J = 0 . W h a t f i g u r e | s h o w s i s t h ep r o g r e s s i v e o v e r l a p o f t h e r o t a t i o n a l m a n i f o l d s b e l o n g i n g t o d i f f e r e n t v i b r a t i o n a ls t a te s . T h u s f o r J = 4 a n d a b o v e , t h e v 2 a n d v a s t a t e s b e c o m e s i g n i f i c a n t l y m i x e d ,s o m e t h i n g t h a t is k n o w n f r o m b o t h t h e o r y [ 2 ] a n d e x p e r i m e n t [ 1 0 , 1 2 ] . A b o v eJ = 11 t h e r o t a t i o n a l l ev e l s o f t h e g r o u n d s t a te o v e r l a p t h o s e o f t h e v i b r a t i o n a l l ye x c i t e d s t a te s . B y J = 2 0 t h e l o w e s t v 2 le v e l i s t h e t h i r d e x c i t e d s t a te 9 I t i s t h u sp o s s i b l e t o e x t r a p o l a t e t o J ~ 25 w h e r e o n e w i l l f i n d th a t t h e l o w e s t v i b r a t i o n a le x c i t a t i o n i s l e ss t h a n t h e l o w e s t r o t a t i o n a l e x c i t a t i o n !

I n m o s t o f t h e p r e v i o u s w o r k o n h i g h l y - e x c i t e d r o t a t i o n a l s ta t e s t h e m o l e c u l eh a s b e e n t r e a t e d a s a r i g i d a s y m m e t r i c t o p . I t i s w e l l - k n o w n [ 7 , 2 5 ] t h a t t h i sm o d e l p r e d i c t s a la r g e n u m b e r o f a c c i d e n t a l d e g e n e r a c i e s , w i t h s p l i t t i n g s a s s m a l la s 1 0 - 1 3 c m - 1 w h e n ~: = 0 a n d J = 2 0 [ 7 ] . I t i s i n t e r e s t i n g t o s e e h o w f a r t h i ss t r u c t u r e i s p r e s e r v e d w h e n t h e c o n s t r a i n t s o f r i g i d i t y a n d n o r o - v i b r a t i o n a l i n t e r -a c t i o n a r e l if t e d . F i g u r e 2 g i v e s t h e l e v e l s o f J - - 1 5 p l o t t e d t o e m p h a s i z e t h e

i EU

. , 0Er

>o

1 0

8

6

4

2

- - - " : = -

H 2 D + e ~ _ , . . "_a r i t v = = - = -j e v e n ___ - __ - - - . -

-_- _- _ - - :

- - _- - _-

_ - - - - = __ - __ . .- _ ~ _

= - -

= _ - _

= _I

I 1(} - 5 10 j 15 2LOF i g u r e 1 . R o t a t i o n a l le v e l s a s s o c i a t e d w i t h t h e l o w e s t t h r e e v i b r a t i o n a l s t a t e s o f H 2 D + a sa f u n c t io n o f J . R e s u l ts a r e f o r t h e s y m m e t r y b l o c k w i t h j e v e n ( p a r a H 2 ) a n d e( p = 0 ) t o t a l p a r i t y .


17/19

Rotationally excited states of HzD + 1083.

.

U

~ o 7 ,

.0EDr

> 6.0

.

-Z 1.1 -" == - = 0.002 =O k _ "_ - - _= = ( 1 0 3 - _ =- 0 . 8 - - 1 . 7 - -- 1 .8 - -~ 5 -- 0 . 1 - - 0 . 5 - -

- 0 . 0 6 -- 0 . 1 - - 0 2 - -

= 2 5 - - 0 . 0 2 =- 0 . 6 - - 0 1 1 7 - _

- 0 1 1 5 -

- 2 . 2 - - 1 . 1 -

- 0 . 0 8 -

- 0 . 0 0 7 -

- 0 . 6 -

- - 0 , 0 1 - -

- 0 . 1 -

- 1 . 6 -

4 o e e e e f o f o eFigure 2. Comparison between symmetry species for rotational levels with J = 15. Thefigures give the splittings, in cm-1, of near degenerate levels between neighbouringsymmetries. The symmetries are labelled by parity of j (e for even, 0 for odd) andtotal parity (e for p = 0 and f for p = 1), see table 12.degeneracies . Th e separa tion o f all levels closer than 5 cm -1 are given explicitly.Splittings of less tha n 0"01 cm -1 can have no significance because of the c on-vergence criteria of the diagonalization and it should be remem ber ed that theabsolute co nverge nce of the J = 15 calculation is only about 1 c m - 1, although thesplittings between systematically degenerate levels are probably better converged.

Inspection of figure 2 shows that the structure of the nearly degenerate levelscorresponds, for the lowest levels, with that found by Harter and Patterson for arigid asymme tric top with ~: = 0 [7]. T he lowest states are three nearly degen eratepairs fo r each total parit y : each pair is compo sed of one state with j even and onewith j odd, the so-called Cz(x)-Type Clusters [7].

At higher energy there are another 3 interleaved pairs, this time with thememb ers of each pair having a co mm on j parity and differing in the total parity,the so-called C2(z)-Type Clusters [7]. Overlapping the higher degeneracies are alarge num be r of near degeneracies with c om mo n j parity and differing in totalparity. These come from rotational states associated with vibrationally excitedstates. However, because of the overlap between the levels associated with vibra-tional states v2 and v3, and no doubt higher vibrational states, there is no easilydiscernible structure in this energy region. One would expect that for J ~ 15 thestructure of the rotational manifold of the vibrational ground state would also bedestroyed.


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1084 J. Te nn ys on and B. T. Sutcliffe6. CONCLUSION

In this paper we propose a method of performing calculations on the highlyrotationally excited state of triatomic molecules. The method is in principle exactwithin the constraints of the variation theorem, which we apply twice. We use theme thod to calculate the rotational levels with J ~< 20 of the lowest three vibra -tional state of H 2 D +, a molecule chosen because of the likely strength of Coriolisand centrifugal distortion effects. Qualitatively, we see the progressive overlap ofrotational manifolds belonging to different vibrational states. We thus probe bothlow and intermedia te J regions and expect the high J region, where the width of arotational manifold is much greater than the vibrational spacing, to be reached atJ ~ 25 for HED+.

We hope that our results mi ght aid the observ ation of high J states of H2D +.There are two main sources of error in our calculations caused by inaccuracies inthe und erly ing electronic potential and trunc ation o f the vibrational basis sets. Bycomparing results for different potentials we see that the absolute energy of thelowest states vary by about 5 cm -1 for J = 16, where the variational calculationsare converged to about lc m -1. Transition frequencies, which do not involvevibrational excitation, will allow for a cancellation of much of this absolute errorand s hould be accurate to about I c m - 1 for J ~ 15 and 5 c m - 1 for J ~ 20, wherethe conver gence of the basis set is poorer.

We are grateful to Professor J.-L. Destombes for bringing this problem to ourattention and for helpful correspondence.

REFERENCES[1] TENNYSON,J., and SUTCLIFFE,B. T., 1984, Molec . Phys . , 51,887.[2] TENNYSON,J., and SUTCLIFEE,B. T., 1985, Molec . Phys . , 54, 141.[3] TENNYSON,J., and S~TCLIFFE, B. T., 1985, Molec . Phys . , 56, 1175.[4] TENNYSON,J., and SUTCLIFFE, B. T., 1986, J. chem. Soc . Faraday II (in the press)[5] SPIRKO, V., JENSEN, P., BUNKER, P. R., and CEJCHAN, A., 1985, J. molec. Spectrosc.,112, 183. JENSON, P., SPIRKO, V., and BUNKER, P. R., 1986, J. molec. Spectrosc., 115,269.[6] LEMOINE, B., BOG/~Y, M., and DESTOMBES, J. L., 1985, Chem. Phys . Le t t . , 117, 532.[7] HARTER,W. G., and PATTERSON,C. W., 1984, J. chem. Phys. , 80 , 4241.[8] BOHR, A., and MOTTELSON, B. R., 1975, Nuclear S truc ture , Vol. II (Benjamin),Chap. 4.[9] CHEN , C .-L ., MAESSEN,B., and WOLFSBERG,M., 1985, J. chem. Phys. , 83, 175.[10] SHY,J.-Z., FARLEY,J. W., and WING, W. H., 1981, Phy s . Re v . A, 24, 1146.[11] AMANO,T., and WATSON,J. K. G., 1984, J. chem. Phys. , 81, 2869.[12] FOSTER, S. C., McKELLAR,A. R. W., PETERKIN,I. R., WATSON, J. K. G., PAN, F. S.,CROFTON, M. W., ALTMAN, R. S., and OKA, T., 1986, ff. chem. Phys. , 84, 91.[13] BOGEY, M., DEMUYNCK, C., DENIS, M., DESTOMBES, J. L., and LEMOINE, B., 1984,Astron . As trophys . , 137, L15. WARNER, H. E., CONNER, W. T. , PETRMICHL, R. H.,and WooDs, R. C., 1984, ] . chem. Phys. , 81, 2514. SAITO, S., KAWAGUCHI,K., andHIROTA, E., 1985,J. chem. Phys. , 82, 45.[14] DESTOMBES,J. L. (private communication).1-15] SUTCLIFFE,B. T., and TENNYSON,J., 1986, Mote c . Phy s . , 58 , 1053.[16] TENNYSON,J., 1986, Comput . Phys . Rep . (in the press).[17] Subroutine EIGSFM, GARBOW, B. S., BOYLE, J. M., DONCARRA,J. J., and MOLER,C. B., 1977, M a t r i x E i g e n s y s t e m R o u t i n e s - - E I S P A C K G u i d e E x t e n s i o n (LectureNotes in Computer Science, Vol. 51) (Springer).[18] Subroutines BANDR and BISE CT [17].


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R o t a t i o n a l l y e x c i t e d s t a te s o f H 2 D + 1085[ 1 9 ] F O 2 F J F , 1 98 3, N A G F o r t r a n L i b r a r y M a n u a l , M a r k 1 1, V o l . 4 . NIKOLAI, P. J . ,1979, A C M T ra ns . M a t h . S o ft w a re , S , 118.[20] TENNYSON,J. , 1983, Comput. phys. Commun. , 29, 307.[21] SCHINKE,R., DUPUIS, M ., and LESTER, W . A., JR. , 1980, J . chem. Phys. , 72, 3909.[22] WATSON, J . K . G ., FOSTER, S. C . , M cKELLA R, A. R. W ., BERNATH, P. , AMANO, T .,PAN, F. S. , CROFTON, M . W ., ALTMAN, R. S. , a nd OICA, T . , 1984, Can. J . Phys . , 62,1875.[23 ] MARTIRE, B., an d BURTON, P. G ., 1985, Chem. Phys . Le t t . , 121 ,479 .[24] BURTON, P. G . , VON NAGY-FELSOBUKI,E., DOHERTY, G ., a n d HAMILTON, M ., 1985,Molec . Phys . , 5 5 , 5 2 7 .[25] GORDY, W . , and CooK , R. L. , 1970, Mic r owav e Mo le c u la r Spe c t r a , 2nd ed i t i on ,( In tersc ience -W iley) , A ppe nd ix 1 . TOWNES, C. H. , and SCHAWLOW, A. L. , 1955,M icrow ave Spec troscopy ( M c G r a w - H i l l) , A p p e n d i x I V .

Documents

Jonathan Tennyson and Brian T. Sutcliffe- Highly rotationally excited states of floppy molecules: H2D^+ with J < 20