jonathan tennyson and brian t. sutcliffe- ab initio vibrational-rotational spectrum of potassium...

8/3/2019 Jonathan Tennyson and Brian T. Sutcliffe- Ab initio vibrational-rotational spectrum of potassium cyanide: KCN

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MOLECULAR PHYSICS, 1982, VOL. 46, NO. 1, 97-109

. 4 b i n i t i o v i b r a t i o n a l -r o t a t i o n a l s p e c t r u m o f p o t a s s i u mc y a n i d e : K C Nby JONAT HAN TENNYSON and BRIAN T. S UT CL IF FE t ,

Institute of Theoretical Chemistry, University of Nijmegen,The Netherlands

(Received 28 September 1981 ; accepted 26 ffanuary 1982)Dynamical calculations are performed on an ab initio potential energy

surface for KCN. Approximate variational solutions of the vibrationalEckart hamihonian are presented for several states. Fundamental vibrationsare found to lie at 302.7 cm -1 and 119.7 cm-1. An effective rotationalhamihonian is solved for several vibrational states allowing vibrationalassignments to be made to the observed rotational spectrum.

1. INTRODUCTIONRecent work, both experimental [I, 2] and theoretical [3, 4], has shown theequilibrium geometry of KC N to be triangular. This, at first, surprising result

makes the dynamics of the K C N molecule of special interest. First, the flatnessof the potential energy surface for the bending motion of the K around the CNmakes the molecule especially floppy in th is coordinate. The low barrier,504 cm -I predicted by [3] for the linear isocyanide structure, should lead tovibrations of very large amplitude after only a few states. The word p o l y t o p i chas been coined [5] to describe projected motions of this type in the LiCNmolecule. The vibrational states lying below this barrier can also be expectedto be of large amplitude.

Secondly, triangular KCN is an asymmetric top. Thi s leads to a complicatedrotational spectrum which has been much studied experimentally [1, 2, 6, 7].At the tempera tures at which these esperiments are conduc ted, 600-900 C,many vibrational states are significantly populated. This fu rthe r complicatesthe observed rotational spectrum. Rotational transitions have been observedfrom at least 11 different vibrations states [2], but as yet no definite assignmentshave been made for anything but the ground vibrational state. For this statealone the spectrum is well characterized [2].The experimental data on the vibrational spectrum of KCN is sketchy andcontradictory. The only direct measurement of the gas phase vibrationalspect rum of KC N [8] observed just one transition (at 207 cm -1) between200 cm -1 and the C- N stretch ing mode at 2158 cm-L This does not agreewith any of the levels reported by Ismail e t a l . [9] from matrix isolation studieswho found the 3 fu ndamental s at 288 cm -1, 139 cm -1 and 2050 cm -1. We

t Permanent address: Department of Chemistry, University. of York, Heslington,York YO1 5DD, England.0026-8976/82/4601 0097 $04"00 1982 Ta ylo r & F r a n c i s L t dM.P. D


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9 8 J . T e n n y s o n a n d B . T . S u t c l if f eno te tha t th e i r (.o1~ 2o~ a n d t h e r e f o r e F e r m i r e s o n a n c e e f f e c ts a r e t o b e e x p e c t e d .A n e s t i m a t e o f t h e i n e r t i a l d e f e c t f r o m t h e g r o u n d s t a t e r o t a t i o n a l s p e c t r u mpla ces o~. a t ab ou t 157 cm -x [2] .

T h e a i m o f t h i s w o r k i s t o c o m p u t e a v i b r a t i o n a l - r o t a t i o n a l s p e c t r u m f o rK C N f r o m f ir s t p r in c i p le s . I t is h o p e d th a t a m o r e d e f i n it e v i b r a ti o n a l s p e c t r u mc a n b e p r o d u c e d a n d s o m e m o r e p o s i t i v e a s s i g n m e n t s m a d e t o t h e r o t a t i o n a ls p e c t r u m o f t h e v i b ra t i o n al e x c i te d s t at e s. K C N a ls o p r o v i d e s a n in t e r e s t i n gt e s t o f t h e th e o r e t i c a l m e t h o d s u s e d t o t a c k le t o - v i b r a t io n a l p r o b l e m s . F i r s t, t h eh i g h a m p l i t u d e b e n d i n g m o t i o n s c a n b e e x p e c t e d t o p r e s e n t s p e c i a l d i f f i c u l t i e sn o t e n c o u n t e r e d in m o r e r i g i d m o l e c u l e s p r e v i o u s l y in v e s t i g a t ed u s i n g v a r i a ti o n a lm e t h o d s , a r e v i e w o f w h i c h i s p r o v i d e d b y C a r n e y e t a l . [ 1 0 ]. S e c o n d l y , t h el a r g e e n e r g y s e p a r a t i o n b e t w e e n t h e l o w e s t t w o f u n d a m e n t a l s a n d t h e C Ns t r e t c h s u g g e s t s t h a t a n a p p r o x i m a t e t h e o r e t i c a l s e p a r a t i o n m i g h t a l s o b eposs ib le .

I n t h i s p a p e r w e e m p l o y a m e t h o d b a s e d o n t h e E c k a r t h a m i l t o n i a n [ 1 1 ]w h i c h w as d e v e l o p e d b y W h i t e h e a d a n d H a n d y [ 1 2] . T h e p o t e n ti a l e n e r g ys u r f a c e u s e d i s t h e o n e c a l c u l a t e d a b i n i t i o a t t h e S C F l ev e l b y W o r m e r a n dT e n n y s o n ( W T ) [3 ].F o r m o l e c u l e s w i t h s m a l l a m p l i t u d e i n t e r n a l m o t i o n , u s e o f t h e E c k a r th a m i l t o n i a n e n s u r e s t h e b e s t p o s s i b l e s e p a r a t i o n o f v i b r a t i o n a l a n d r o t a t i o n a lm o t i o n . T h i s s e p a r a t i o n i n v o lv e s t h e n e g l e c t o f C o r io l is t e r m s b e t w e e n d i f f e r e n tv i b r a ti o n a l st a te s . A l t h o u g h t h e s e te r m s m a y b e si g n i fi c a n t f o r a m o l e c u l eu n d e r g o i n g la r g e a m p l i t u d e v i b ra t io n s , H o l m g r e n e t a l . [ 13 ] f o u n d s i m i l a r C o r i o l ist e r m s t o b e s m a l l f o r t h e v a n d e r W a a l s c o m p l e x A r H C I . K C N i s c o n s i d e r a b l yl e ss f l o p p y t h a n A r H C 1 i n it s l o w l y i n g v i b r a t i o n a l s t a t e s a n d i n 4 w e d e m o n -s t r a t e t h a t t h e s e o f f - d i a g o n a l C o r i o l i s t e r m s a r e i n d e e d s m a l l .I n t h i s c a s e i t i s p o s s i b l e t o c o n s i d e r t h e v i b r a t i o n a l p r o b l e m s e p a r a t e l yf r o m t h e r o t a ti o n a l p r o b l e m , t h u s g r e a t l y r e d u c i n g t h e d i m e n s i o n s o f t h e s e c u l a re q u a t i o n s t o b e s o lv e d . I n th e W h i t e h e a d - H a n d y a p p r o a c h it i s a s s u m e d t h a t t h ev i b r a ti o n a l w a v e f u n c t i o n c an b e e x p a n d e d a s a s u m o f p r o d u c t s o f h a r m o n i c b a s i sf u n c t i o n s a n d t h e r o ta t i o n a l w a v e f u n c t i o n a s a s u m o f s p h e r i c a l h a r m o n i c s . T h eW h i t e h e a d - H a n d y m e t h o d h a s p r e v i o u s l y b e e n a p p l i e d t o s e v e r a l n o n - l i n e a rt i g h t l y b o u n d m o l e c u l e s , w i t h w e l l d e f i n e d e q u i l i b r i u m g e o m e t r i e s , s u c h a sH 2 0 [ 1 2] , S O 2 [ 1 2 ] a n d H 2 C O [ 1 4] w i t h s a t i s f a c t o r y r e s u l t s . R e c e n t l y ,B a r t h o l o m a e e t a l . [1 5] h a v e a p p l ie d t h e W h i t e h e a d - H a n d y m e t h o d t o C H 2 + ,w h i c h h a s a l o w b a r r i e r t o i n v e r si o n . T h e y f o u n d t h a t t h e m e t h o d w o r k e d w e lli f t h e y u s e d a s p e c i a ll y c o n s t r u c t e d b a s i s w h i c h a v o i d e d s i n g u l a r i t i e s in t h e r e g i o no f l in e a r g e o m e t r i e s .I n 2 o f t h i s p a p e r w e g i v e d e t a i l o f t h e m e t h o d u s e d t o s o l v e t h e r o -v i b r a ti o n a l p r o b l e m . S e c t i o n 3 p r e s e n t s so l u t io n s t o t h e v ib r a t io n a l h a m i l t o n i a nw i t h J = 0 . T h e s e a r e u s e d in 4 to s ol v e a n e f f e ct i v e r o t a t i o n a l - v i b r a ti o n a lp r o b l e m .

2. METHODA s s u m i n g , a s u s u a l , t h a t n u c l e a r a n d e l e c t r o n i c m o t i o n s m a y b e s e p a r a t e d ,t h e n t h e h a m i l t o n i a n f o r t h e v i b r a t i o n - r o t a t i o n m o t i o n m a y b e w r i t t e n a s

hef I = ~ ( H = - ~ ) ~ . ~ ( H ~ - ~ ) + ~ P ~ 3- -~ - ~ + V (1 )


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D y n a m ic s o[ K C N 99w h e r e G r e e k l e t t e r s r u n o v e r c a r t e s i a n c o o r d i n a t e s a n d R o m a n o v e r n o r m a lc o o r d in a t es . T h i s f o r m o f t h e h a m i l t o n ia n , w h i c h c o n t a in s o n l y n u c l ea rv a r i a b l e s , w a s d e r i v e d b y W a t s o n [ 1 6 ] f r o m t h e h a m i l t o n i a n D a r l i n g a n dD e n n i s o n [ 1 7] d e v e l o p e d f r o m t h e o r i g in a l f o r m u l a t i o n o f E c k a r t [ 1 1 ].I n ( 1 ) t t is a n e f f e c t iv e i n v e r s e in e r t i a t e n s o r , H = i s a c o m p o n e n t o f t h en e g a ti v e to t a l a n g u l a r m o m e n t u m o p e r a t o r a n d rr= i s a c o m p o n e n t o f t h eC o r i o li s o p e r a t o r

h~== Z k , = 9kP,, P,-- (2)

kt - i ~ Q , "v i s t h e p o t e n t i a l f o r t h e n u c l e a r m o t i o n i n t h e e f f e c t i v e f i e ld o f t h e e l e c t r o n s .

A m o r e d e t a i l e d a c c o u n t o f t h e c o n s t r u c t i o n a n d s i g n i fi c a n c e o f t h e t e r m s i n( 1 ) is g i v e n b y W a t s o n [ 16 ]. A n a c c o u n t o f h o w t o c o n s t r u c t t h e p r o p e r n o r m a lc o o r d i n a t e s Q k f o r a g e n e r a l m o l e c u l e c a n b e f o u n d i n W i l s o n et al. [ 1 8 ] . Fo rp r e s e n t p u r p o s e s , h o w e v e r , it is s u f f ic i e n t t o n o t e t h a t i f t h e i n t e r n a l m o t i o n sr e p r e s e n t e d b y t h e Q k a re v e r y s m a ll , th e n t t b e c o m e s a p p r o x i m a t e l y t h e i n v e r s eo f Io , t h e e q u i l i b r i u m g e o m e t r y in e r t i a t e n s o r , a n d t h e C o r io l i s t e r m b e c o m e sa p p r o x i m a t e l y z e ro .I n t h i s c a s e H b e c o m e s H a n d ( 1 ) s e p a r a te s i n t o t w o p a r t s

w i t h

a n d

H o = Hr o t + H v i b 0H r o t 0 = Z ( I 0 - 1 ) a f l [ I a H f l

( 3 )( 4 )

a n dHro t_v i b = Z/x =p FI =I I p - ~/x=DII=lr p . (9 )

=B

B y s o l v in g t h e p r o b l e m a s s o c ia t e d w i t h H v i b , a m a n i f o l d o f v i b r a t io n a l f u n c t i o n sq ~ c a n b e c a l cu l a te d . A s u i ta b l e m a n i f o l d f o r t h e fu l l p r o b l e m c a n t h u s b eD 2

H vib = l ~k pk2--h-~8 (Io-1)aa+ V . ( 5 )I n t h e a b s e n c e o f m a g n e t i c f ie l d s, H h a s s o l u t i o n s o f t h e f o r m

+ J~ a 0 ( Q , ~ ) = @O( Q ) 2~ d K o [ j g > , (6 )K = - Jw h e r e ] JK > - is a n a n g u l a r m o m e n t u m e i g e n f u n c t i o n i n v o lv i n g t h e E u l e r an g l e ~ ,a n d D is a s o l u t i o n o f t h e p r o b l e m

H v i b 0 ( I ) 0 = E v i b 0 ( I)0 . ( 7 )T h e c o e f f ic i e n t s d K of th e a n g u la r m o m e n t u m e i g en f u n c ti o n s a re d e t e r m i n e d b ys o l v i n g t h e s e c u l a r p r o b l e m s p e c i f ie d b y H r o t a ct in g o n t h e m a n i f o l d o f a n g u l arm o m e n t u m fu n c t io n s of a g i ve n J . T h e c o m p l e te s u m - f u n c t i o n so d e t e rm i n e di s o f t e n c a l le d a r ig i d a s y m m e t r i c - t o p f u n c t i o n .T h i s a n a l y s i s s u g g e s t s th a t , f o r r e a s o n a b l y s m a l l i n t e r n a l m o t i o n s , H c a na p p r o x i m a t e l y b e s e p a r a t e d i n t o t w o p a r t s

h2H v i b = l ~ I x= pc r= ~' p I Z P k 2 - ~ I ~ = = V (8)


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1 00 J . T e n n y s o n a n d B . T . S u t c l i ff ec o n s t r u c t e d a s a d ir e c t p r o d u c t o f t h e s e aO a n d t h e c o m p l e t e s e t o f ] J K ) f o r ag iv e n J . H , o t_ vib t h e n o p e r a t e s o n t h e f u l l m a n i f o l d c o u p l i n g i t s m e m b e r s . I ti s c o n v e n i e n t t o t h i n k o f t h e o p e r a t i o n o f H r o t - v i b i n t e r m s o f t h e o p e r a t i o n o f a neflective r o t at i on a l h a m i l to n i a n w h o s e d e m e n t s a re

Hrot~"= Z X ( lO)=t~ ,,8

a n d w h i c h o p e ra t e s o n t h e a n g u la r m o m e n t u m f u n c t i o n s o n l y. I n ( 1 0) t h ein t e g r a l s a r e o v e r t h e v ib r a t i o n a l c o o r d in a t e s Qk o n ly a n d n i s a v ib r a t i o n a lw a v e f u n c t i o n . I f it i s t h e c a s e t h a t , f o r a n y c h o s e n v i b r a ti o n a l s ta t e , t h e o f f -d i a g o n a l (m # n ) c o n t r i b u t i o n o f t h e s e i n t e g r a ls t o H ~ i s s m a l l c o m p a r e d w i t ht h e i r d i a g o n a l ( m = n ) c o n t r i b u t i o n , t h e n t o a g o o d a p p r o x i m a t i o n , t h e c o u p l i n gb e t w e e n d i f fe r e n t v i b r a t i o n a l f u n c t i o n s m a y b e n e g le c t e d . I n t h i s c a s e a s u i t a b l ew a v e f u n c t i o n h a s t h e f o r m

z j ( Q , ) = o d Q) + JZ dKIJK> ( 1 1 )K = - J

wh e r e @,~ i s t h e c h o s e n v ib r a t i o n a l f u n c t i o n . T h e dK a r e d e t e r m i n e d b y s o l v in gt h e s e c u l ar p r o b l e m o v e r th e ] J K ) w i t h H r o t ~n f o r a n a p p r o p r i a t e c h o s e n n .I n t h i s c o n t e x t w e n o t e t h a t t h e o p e r a t o r

Z V'~prrp (1 2 )i s h e r m i t i a n a n d p u r e l y im a g i n a r y . T h u s , a s t h e q b a r e r e al , t h e s e c o n din t e g r a l i n ( 1 0 ) v a n i s h e s f o r m = n . I t i s t h u s r e q u i r e d , f o r ( 1 1 ) t o b e a v a l i da p p r o x i m a t i o n , t h a t t h e m = n c a s e o f t h e f i r s t i n t e g r a l i n ( 1 0 ) d o m i n a t e s bothi n t e g r al s t h a t a r is e w h e n m # n .

I n t h e P r e s e n t w o r k t h e p o t e n ti a l u s e d i n H v i b w a s d e ri v e d f r o m t h e ab initioS C F c a l c u l at i o n s o f W T [ 3] a n d t h e a c t u a l h y p e r s u r f a c e w a s o b t a i n e d b y a na c c u r a t e f i t t o t h e c a l c u l at e d p o i n t s . P r e v i o u s ab initio c a l c u l a t i o n s w i t h t h eW h i t e h e a d - H a n d y m e t h o d [1 2, 15 ] u s e d ab initio p o t e n t i a l p o i n t s p r e c a l c u l a t e d t oa c h o s e n i n t e g r a t i o n s c h e m e . W h i l s t a t t ra c t i v e , a s a n y l os s o f a c c u r a c y d u e t ot h e f it t i n g i s e l i m i n a t e d , t h i s a p p r o a c h h a s t w o p r a c t ic a l d r a w b a c k s . F i r s t , ak n o w l e d g e o f th e s u r f a c e i s r e q u i r e d b e f o r e i t i s c o m p u t e d i n o r d e r t o s e l e c t a ne q u i l i b r i u m g e o m e t r y a n d a s u i t a b l e r a n g e o f i n t e g r a t i o n p o i n ts . S e c o n d l y , ifh i g h o r d e r n u m e r i c a l i n t e g r a t i o n i s r e q u i r e d m a n y m o r e p o t e n t i a l p o i n t s a r en e e d e d , i n o u r c a s e 2 5 6 a s a g a i n s t t h e 2 4 a c t u a l ly u s e d i n t h e s u r f a c e f it . A sc a l c u l a ti n g p o t e n t i a l p o i n t s i s v e r y e x p e n s i v e i n c o m p u t e r t i m e t h i s i s a s e v e r ec a v e a t .T h e s u r fa c e o f W T [ 3] is n o t a fu l l s u r fa c e fo r K e N , b u t h a s o n e in t e rn a lc o o r d i n a t e , t h e C N b o n d le n g t h , f r o z e n . A r e a li s ti c r e p r e s e n t a t i o n o f t h i sc o o r d i n a t e w o u l d n o t b e p o s s i b l e w i t h o u t i n c l u s i o n o f e l e c t r o n c o r r e l a t i o n .H o w e v e r , i t i s n o t p o s s i b l e to s o l v e t h e W a t s o n h a m i l t o n i a n ( 1 ) w i t h a n a r b i t r a r yi n t e rn a l c o o r d i n a t e f r o z e n . T h i s p o i n t w i ll b e d e v e l o p e d i n t h e n e x t s e c t io n .

3. VIBRATIONALPROBLEMT h e p o t e n t i a l e n e r g y s u r fa c e o f W T [ 3] i s e x p a n d e d i n t h r e e i n te r n a l c o -

o r d in a t e s , R , 0 a n d r ( f r o z e n ) , a s d e f in e d in f i g u r e 1 . I n o r d e r t o g e n e r a t e


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D y n a m i c s o f K C N 101

C

K / a

~.~. / /

/ / / R

F ix .r N

Figure 1. Coordinates system, R , 0 and r, for KC N. X denotes the centre of nuclearmass of z~C - Z4N, a an d b schematically represent the principle axes of inertia, w ith cperpendicular to the plane of the paper, and a and R are very nearly coincident.n o r m a l c o o r d i n a t e s f o r th e s o l u t i o n o f t h e v i b r a t i o n a l h a m i h o n i a n ( 8 ) it is n e c e s -s a r y to s a t i sf y t h e E c k a r t c o n d i t i o n s [ 1 1] . I t i s t h e n p o s s i b l e t o s o lv e t h eh a m i l t o n i a n ( 8 ) i n le s s t h a n ( 3 N - 6 ) d i m e n s i o n s b y f r e e z i n g c e r ta i n o f t h e s e n o r m a lc o o r d i n a t e s , as h a s b e e n d o n e r e c e n t l y f o r C O s a n d S O s b y C l a r y [ 19 ] . F o ri n f i n it e s s im a l d i s p l a c e m e n t s f r o m e q u i l i b r i u m o u r i n t e r n a l c o o r d i n a t e s h a v e t h en e c e s s a r y p r o p e r t i e s , b u t n o t f o r r e al d i s p l a c e m e n t s . T h u s i t i s n o t p o s s i b l et o f r e e z e a n a r b i t r a r y i n t e r n a l c o o r d i n a t e . T h i s m e a n s t h a t a l t h o u g h t h e C - Ns t r e t c h i s w e l l s e p a r a t e d i n e n e r g y f r o m t h e o t h e r f u n d a m e n t a l s , i t m u s t b ei n c l u d e d i n t h e s o l u t i o n o f ( 8 ).

T o t h e s u r f ac e o f W T w e t h u s a d d e d a h a r m o n i c t e r m i n re N w i t h f o r c ec o n s t a n t k = 0 . 1 7 2 / ~ N A - z w h i c h i s t y p i c a l o f o b s e r v e d C N v a l u e s [ 1 8 ], p . 1 7 5.T h i s f o r c e c o n s t a n t l e a d s to a f u n d a m e n t a l f r e q u e n c y o f 2 1 2 9 . 7 c m - 1 f o r t h e C Ns t r e t c h , i n t e r m e d i a t e t o t h o s e o b s e r v e d f o r K C N [8 , 9 ] . W e a i m t o s h o w t h a t t h ef i na l s o l u t i o n o f t h e v i b r a t i o n a l p r o b l e m f o r t h e o t h e r t w o f u n d a m e n t a l s i si n s e n s it i v e t o t h i s c h o i c e . T h i s l e a d s t o p o t e n ti a l s w h i c h c a n b e e x p r e s s e d a s

6V ( R , O , r ) = Y ~ V l ( R ) P l ( co s O ) + k ( r - r e ) ~ ( 1 3 )1= 0

w h e r e V t ( R ) i s g iv e n i n r e f e r e n c e [ 3 ] a n d r e wa s t a k e n a s 1 .1 5 7 A , t h e v a lu e o fW T .V i b r a t i o n a l b a s i s f u n c t i o n s w e r e ta k e n a s p r o d u c t s o f g au s s i a n w e i g h t e d

H e r m i t e p o l y n o m i a l s f o r e a c h n o r m a l c o o r d i n a t e~ ( n l , n s, n 3) = H . , ( ~ z ) H . , ( ~ s ) H . 3 ( ~ 3 ) ex p [ - (~)1 + ~)ss + ~85)] (14)

Qk (15)w h e r e o k a r e f u n d a m e n t a l f r e q u e n c i e s g i ve n b y s o l u ti o n o f t h e h a r m o n i c p r o b l e m .T h e n o r m a l c o o r d i n a te s Q k w e r e o b t a i n e d i n c a r te s i an f o r m u s i n g p r o g r a m sd e v e l o p e d b y S e h a c h t s c h n e i d e r [ 20 ].


6/13

1 02 J . T e n n y s o n a n d B . T . S u t c l i f feA s in [ 1 2 ] m a t r i x e l e m e n t s w e r e e v a l u a t e d u s i n g a p r o d u c t o f o n e d i m e n s i o n a l

G a u s s - H e r m i t e i n t e g r a ti o n s c h e m e s w i t h th e p o i n t s d i s t ri b u t i o n w e i g h t e d b yt h e h a r m o n i c f r e q u e n c y o f t h e f u n d a m e n t a l s . E a c h n u m e r i c a l i n t eg r a t io n p o i n ti s t h u s d e f i n e d i n t e r m s o f w e i g h t e d n o r m a l c o o r d i n a t e s O k . O n e g e ts t h ec u r r e n t c a r te s ia n g e o m e t r y f ro m t h e s e b y a p p l y in g t h e t r a n s f o r m a t i o n g e n e r a te db y t h e S c h a c h t s c h n e i d e r p r o g r a m s [ 20 ] ; t h i s g e o m e t r y is u s e d t o g i v e a n i n v e r sei n e r ti a te n s o r f o r t h e c u r r e n t p o i n t . T h e p o t e n t i a l w a s e v a l u a t e d a t e a c hi n t e g r a t i o n p o i n t i n t h e b o d y - f i x e d c o o r d i n a t e s R , r a n d c o s 0 , w h i c h c a n b eo b t a i n e d s i m p l y f r o m t h e c a r t es i an g e o m e t r y .A s c o m p a r a t i v e l y l a r g e b a s i s s e t s ( u p t o 3 8 5 f u n c t i o n s ) w e r e u s e d a n d , s i n c eo n l y t h e l o w e s t v i b ra t i o n a l s t a t e s w e r e o f i n t e r e s t, w e u s e d t h e m e t h o d o f S h a v i t te t a l . [ 2 1] t o g iv e t h e fi r s t f e w v i b r a t i o n a l s ta t e s . T h i s p r o v e d m o r e e c o n o m i ci n b o t h s t o r a g e a n d c o m p u t e r t i m e t h a n d i a g o n a li z in g t h e w h o l e m a t r i x , e s p e c i al l ya s v e c t o r s f r o m a p r e v i o u s r u n o f t e n p r o v i d e a g o o d s t a r t i n g p o i n t f o r t h e c y c li cd i a g o n a l i z a t i o n .I n o r d e r t o s o l v e t h e v i b r a t i o n a l p r o b l e m i t i s n e c e s s a r y t o o b t a i n c o n v e r g e n c ew i t h b o t h b a s i s s e t s i ze a n d i n t e g r a t i o n p o i n t g r id . T o t h i s e n d w e d i v i d e d o u rn o r m a l c o o r d i n a te s i n to t w o t y p e s : A c o m p r i s i n g ( ~ 1 a n d ~)2 wh ic h c a n a p -p r o x i m a t e l y b e i d e n t if i e d w i t h t h e K - C N s t r e tc h a n d b e n d r e s p e c t i v e l y ; Bc o m p r i s i n g ~ o r t h e C - N s t re t c h . W h a t w e r e q u i r e i s c o n v e r g e n c e f o r t h ef u n d a m e n t a l s a n d lo w - l y i n g s t a te s o f t y p e A . B a s is f u n c t i o n s f o r t y p e B w i t hn 3 > 0 a r e i n c l u d e d s o l e ly to r e m o v e t h e e f f e c t s o f t h e C N s t r e t c h i n c l u d e d i n t h eb a s i s f u n c t i o n s o f t y p e A . C o n v e r g e n c e w i t h M A a n d M B w a s t e s t e d se p a r a t e l y

8and /Origin/crn-1

8 0 0

500

4 0 0

200

--------_ ~ ~ . 1002

01M A

I I I I J I4 5 6 7 8 9( a )


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D y n a m i c s o I K C N 103

B a n d /O r i g i n / c r n - 1I000

~00

Z O 0 ~ MBOIJ L i i J0 1 2 3 L , 5 6

(b )Figu re 2 . Co nverg ence of ban d or igins with increas ing basis se t s ize . (a ) Increas ingM A with M B = 4. (b) Increas ing MB with MA = 7. S ta tes a re ident if ied by the irma jo r ha rm onic c om pone n ts w i th n3 = 0 .ze ro.w h e r e

M a > n 1 + n 2 , M B > n 3 .T h i s l ea d s to a b a si s of ( M a + 1 ) ( M n + 2 ) ( M B + 1 ) /2 fu n c t i o n s .

T h e g round 00 s tate i s t a ke n a s the e ne rgy

( 1 6 )F i g u r e s 2 ( a )

a n d ( b ) sh o w t h e c o n v e r g e n c e o f t h e b a n d o r ig i n s w i t h i n c r ea s i n g M 4 a n d M Br e s p e c t i v e l y . E x c i t e d st a t es a r e i d e n t i f i e d w i t h t h e h a r m o n i c b a s i s f u n c t i o nh a v i n g t h e l a r g e st c o m p o n e n t (n 3 = 0 f o r al l l o w ly i n g st a t es ) . F o r s ta t e s w i t hn 2 > 2 t h i s l a b e l li n g i s n e c e s s a r i ly r a t h e r a r b i t r a r y a s t h e a n h a r m o n i c i t y i n t h ea n g u l a r p o t e n t i a l s t r o n g l y m i x e s t h e h a r m o r / i c b a si s. F i g u r e 2 s h o w s s e v e ra ls t a te c r o s si n g s . A s a ll v i b r a t i o n a l s t a t es a re o f A ' s y m m e t r y , t h e s e s h o u l ds t r i c tl y b e d e p i c t e d a s a v o i d e d c r o s si n g s . H o w e v e r a s 3 // .4 a n d M B a r e b o t hi n t e g e r v a r i a b l e s i t i s p o i n t l e s s t o s p e c u l a t e w h a t f o r m t h e c u r v e s t a k e f o r n o n -i n t e g e r v a l u es , h e n c e w e h a v e l i n k e d d i r e c t l y p o i n t s w i t h s i m i l a r h a r m o n i cc o m p o n e n t s . F i g u r e 2 s u g g e s t s t h a t o u r b a si s s e t i s w e l l c o n v e r g e d f o r t h eg r o u n d a n d f i r s t s e v e n e x c i t e d s t a t e s f o r M A = 9 a n d M B = 6 . N e x t w e t e s t t h ec o n v e r g e n c e o f o u r n u m e r i c a l i n t e g r al s . W e c o m p a r e d c a l c u l a t io n s f o r 21//4 = 8 ,M B = 6 a n d a 1 6, 1 6, 8 p o i n t g a u s s i a n i n t e g r a t i o n s c h e m e f o r m o d e s 1 , 2 a n d 3r e s p e c t i v e l y w i t h o n e s u s i n g 1 8 , 18 , 6 a n d 1 6 , 1 6 , 1 0 s c h e m e s . N o n e o f t h e


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1 04 J . T e n n y s o n a n d B . T . S u t c l i ff eb a n d o r i gi n s w e r e c h a n g e d b y m o r e t h a n 0 .0 2 c m - 1. T h i s s h o w s t h a t w e h a v en o n e o f t h e p r o b l e m s w i t h n e a r l in e a r g e o m e t r i e s e n c o u n t e r e d b y B a r t h o l o m a ee t a l . [ 15 ]. I n th e i r c as e t h e p r o b l e m is p r o b a b l y e x a c e r b a t e d b y s t u d y i n g at r i a t o m i c c o n t a in i n g t w o h y d r o g e n a t o m s . A s a f in a l c h e c k o n t h e c o n v e r g e n c eo f t h e b a s i s , w e t e s t e d t h e e f f e c t o f a l t e r i n g t h e e m p i r i c a l C N f o r c e c o n s t a n ti n c l u d e d i n o u r p o t e n t i a l . A n i n c r e a s e o f 1 0 p e r c e n t l if ts t h e h a r m o n i c C Ns t r e t c h f r e q u e n c y b y 1 0 4 c m - x ; f o r a c a l c u l a t i o n w i t h M . 4 = 8, M B = 6 i t o n l yr a is e s th e o t h e r t w o f u n d a m e n t a l s b y a b o u t 0"5 c m - t . W e t h u s f ee l t h a t o u rc a l c u l a t i o n s a r e w e l l c o n v e r g e d . T a b l e 1 p r e s e n t s r e s u l ts f o r a n M a = 9, M n = 6c a l c u l a t i o n .Tabl e I . Low est 9 v ibra t iona l s ta tes for ca lcula t ion wi th Ma = 9 , MB = 6 . Energies a rer e la t iv e to th e g r o u n d s t a t e ; s t a t es a r e lab e l led b y th e h a r m o n ic b as i s f u n c t io n swhic h have a coef f ic ien t g rea ter than 0 .40 in abso lu te va lue , n3 =0 for a l l com -p o n en t s g iv en .

H a r m o n i c H a r m o n i cSta te En ergy /cm -1 basis funct ion Coef f ic ien t basis funct ion Coef f ic ien tn 1 n~ n I nu

1 0.0 0 0 0.9 82 119-7 0 1 0.873 241.9 0 2 0.684 302-7 t 0 0.925 382-9 0 3 0.46 1 1 0-416 438:6 1 1 0.747 524.0 1 2 0.578 588.3 2 0 0-699 616.7 2 0 -0 .4 5 1 2 0 -45

T h e m i n i m u m o f t h e p o t e n ti a l o f W T l ie s a t - 3 9 0 8 6 . 1 c m - 1 r e la t iv e to az e r o a t f u l l y s e p a r a t e d K + a n d C N - i o n s. O u r g r o u n d v i b r a t i o n a l s t a te h a s a ne n e r g y o f - 3 7 7 9 8 - 3 c m -1 i n c l u d i n g 1 06 4. 9 c m -1 z e r o p o i n t e n e r g y f r o m t h eh a r m o n i c C N - p o t e n ti a l . T h e z e ro p o i n t e n e r g y f o r t h e o t h e r t w o f u n d a m e n t a l si s t h u s 2 2 2 .9 c m - 1 w h i c h i s h i g h e r t h a n t h a t g i v e n b y h a l v i n g t h e s u m o f t h et w o f u n d a m e n t a l e n e r g i e s , 2 11 . 2 c m - a . H o w e v e r , a s t h e r e i s s i g n i f i c a n t i n t e r -a c t i o n b e t w e e n t h e 0 2 a n d 1 0 s t a te s , t h e l a tt e r m e t h o d o f d e r i v i n g z e r o p o i n te n e r g i e s i s n o t v e r y r e l i a b l e .

W T p r e d i c t s a b a r r i e r t o i n v e r s i o n o f 50 3 -9 c m - 1 f o r a n i s o c y a n i d e g e o m e t r y .T h i s p l a c e s t h e v i b r a t i o n a l s t a t e 4"38.6 c m - a a b o v e t h e g r o u n d s t a t e a t a b o u t t h es a m e h e i g h t a s t h e b a r r i e r , w h e n a l l o w a n c e i s m a d e f o r th e z e r o p o i n t e n e r g y o ft h e b e n d i n g m o d e . B e n d i n g v i b ra t i o n s a b o v e t h is s h o u l d s h o w la r ge a m p l i t u d e sa n d w o u l d n o t n e c e s s ar i ly b e w e ll d e s c r ib e d b y a n i n c o m p l e t e h a r m o n i c b a s iss e t s i t e d a t o n e p o i n t . I t i s t h u s n o t s u r p r i s i n g t h a t s o m e o f t h e v i b r a t i o n a ls t a t e s a b o v e t h e b a r r i e r a r e n o t w e l l c o n v e r g e d . C a l c u l a t i o n s , e m p l o y i n g t h ec l o s e - c o u p l i n g f o r m a l i s m ( f o r r e v i e w s e e [ 2 2] ) a r e c u r r e n t l y i n p r o g r e s s [ 2 3] .T h e s e m i g h t b e e x p e c t e d t o p r o d u c e b e t t e r r e s u l ts f o r t h e s e p o l y t o p i c s t at e s as th ep o t e n t i a l i s a n a l y t i c a l l y i n t e g r a t e d o v e r a f u l l b a s i s in t h e t h e t a i n t e r n a l c o -o r d i n a t e .


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Dynamics o f KC N 105Table 1 shows that we find the fundamental frequencies of KCN to lie at

302.7 cm -1 and 119.7 cm -1 for the stre tch and bend respectively. These disagreeby 10-20 cm -1 with those observed by Ismail et al. [9] in a mat rix isolation study ;as .~ve are calculating a gas phase spect rum, bett er agreement cannot be expected.We find no transit ion directly comparabie to the one at 207 cm -I observed byLeroi and Klemperer [8], however this could possibly be the 02 overtone. Itwould be interesting to see if further lines could be observed in a repeat of thisexperiment.

4. ROTATIONAL PROBLEMUsing the vibrational functions calculated in 3, we calculated the coupling

integrals needed for the effective rotational hamilton ian (10). This was done bythe same methods used to calculate the integrals needed in the vibrationalproblem. In a typical case it was foun d that both types of integral for m # nwere a hundred or so times smaller than the non-vanishing integral in the casem = n . In view of the inaccuracies of the potential of WT, particularly theequilibrium geomet ry, it was decided that a wavefunction like that given byequation (11) was a sufficiently good approximation. Even though such awavefunction has the same form as that of a rigid asymmetric top, obtained bysolving equation (6), there is an impor tant difference. In the rigid rotor modelit is necessary to assume a rigid structure , normal ly the equilibrium geometry ; thepresent approach allows us to take a correct vibrational average of the fullinverse effective inertia tensor for each vibrational state.

For a given value of J the set of angular momentum eigenfunctions [ J K ) iscomplete. The matrix elements can be evaluated analytically and the results arewell known (see for example [12]). We chose to work with a basis of realharmonics which give a real symmetric rather than hermitian secular problem.The matrix elements of the real harmonics, [JK cos) and [JK sin), can be ob-tained from those of the spherical harmonics I J K ) using the relationships

I J K c o s ) = ( ( - - 1 ) K ] J K ) + [ J - K ) ) /V '2 , }[JK sin) = (( - 1 )K I J K ) _ [ j _ K ) ) / 2 i . (17)

Table 2 compares some of the ab initio rotational spect rum with the experimentallyassigned lines for the vibrational ground state. Rotational spectra are verysensitive to the geometric parameters of a molecule. The ab initio surface ofWT has a minimum at R= 2. 67 5A and 0=105.7 , r =l .1 57 A being heldconstant ; we shall label this geomet ry I. Thus their surface has a minim umat larger R than observed experimentally [2] ; this they attribute to lack of electroncorrelation in their calculations. Moreover, they worked with an SCF optimizedbond length for CN-, which is shorter than that found experimentally [2] orwhen extensive configuration interaction is used [24].

A rigid rotor calculation at the experimental geometry [2] gave a spectrumin much better agreement with experiment, especially for the a-type transitionswhich were now reproduced to within 5 per cent, confirming that it is inaccuraciesin the SCF equilibrium geometry of WT which is the major source of dis-agreement in table 2.


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106 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 2. C o m p a r i s o n w i th t h e r o t a t io n a l s p e c t r a o f [ 2 ] f o r g r o u n d s ta t e K C N . I w a sc a l c u l a te d f o r t h e e q u i l i b r i u m g e o m e t r y o f W T [ 3 ] a n d I I c o r r e s p o n d s t o t h e b e s te x p e r i m e n t a l g e o m e t r y [2 6 ]. T h e s p e c t r u m o f [ 1 ] i s c o m p a r e d w i t h t h a t c a l c u l a te du s i n g g e o m e t r y I I i n ta b l e 3 .

C a l c u l a t e d / M H zJ ' K ' - l K ' + I J K - I K + I T y p e I I IE x p e r i -m e n t a l / M H z[ 2 ]

3 1 2 3 1 3 a 1798 2366 24234 1 3 4 1 4 a 2996 3943 40386 1 5 6 1 6 a 6291 8279 84787 1 6 7 1 7 a 8388 11038 113038 1 7 8 1 8 a 1078 4 14191 145299 1 8 9 1 9 a 13480 17736 18157

10 1 9 10 1 10 a 16474 21673 221851 0 1 0 0 0 a 863 6 9481 94752 0 2 1 0 1 a 17272 18961 189493 1 3 4 0 4 b 259 17 16041 144437 0 7 6 1 6 b 213 4 15156 16796

12 2 11 13 1 12 b 245 70 193 603 1 3 2 1 2 a 25459 27852 278173 0 3 2 0 2 a 25905 28436 284173 2 1 2 2 0 a 25913 2844 4 284263 1 2 2 1 1 a 26358 29035 290284 1 4 3 1 3 a 33944 37133 370864 0 4 3 0 3 a 34534 37905 )4 3 2 3 3 1 a~ 34548 37930 / 378784 3 1 3 3 0 a J4 2 3 3 2 2 a 34545 37924 378874 2 2 3 2 1 a 34556 37945 379104 1 3 3 1 2 a 35142 38710 38701

W e t h u s d e c i d e d t o p e r f o r m c a l c u l a t i o n s f o r g e o m e t r i e s b a s e d o n t h e e x p e r i -m e n t a l m i n i m u m . I n o r d e r t o d o t h i s w i t h i n t h e E c k a r t f r a m e w o r k , i t i sn e c e s s a r y t o s h i f t t h e p o t e n t i a l e n e r g y s u r f a c e , c o m p u t e a n e w s e t o f n o r m a l c o -o r d i n a t e s a n d r e - s o l v e t he v i b r a t i o n a l p r o b l e m a t t h e n e w g e o m e t r y w i t h t h en e w n o r m a l c o o r d i n a te s . T h e s e v i b r a t i o n a l s o l u t io n s w i l l t h e n h a v e t h e c o r r e c tp r o p e r t i e s f o r t h e e f f e c t i v e h a m i l t o n i a n o f e q u a t i o n ( 1 0 ).

T h e p o t e n t i a l w a s s h i f t e d b y r e f i t t in g i t f o r g e o m e t r y I w i th H e r m i t e p o l y -n o m i a l s o f t h e d i m e n s i o n l e s s n o r m a l c o o r d i n a t e s b e l o n g i n g t o t h e n e w g e o m e t r y( O ' i ) . T h e p o t e n t i a l o f e q u a t i o n ( 1 3 ) is t h u s r e w r i t t e n

L M NV (R , O, Y) = V( Q tl , 0 ',,>, Ors) = E E E atmnHt(O'l)Hm(O'~)Hn(O'a)"( 1 8 )l = 0 m = 0 n = 0t h e c o e f f i c ie n t s a im ~ b e i n g o b t a i n e d b y G a u s s - H e r m i t e i n t e g r a t i o n . L = 1 4,M = 9 , N = 2 w a s f o u n d t o g i ve a g o o d r e p r e s e n t a t i o n o f t h e p o t e n t i a l . I f t h er e f i t t e d p o t e n t i a l w a s u s e d t o p e r f o r m t h e c a l c u l a t i o n o f t a b l e 1 , t h e t w o f u n d a -m e n t a l s , o1 a n d coz, w e r e c h a n g e d b y o n l y 0 - 4 c m - t a n d 1 -4 c m - 1 r e s p e c t i v e l y .


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Dynamics o[ K C N 10 7T a b le 3 . Com par i son o f ca lcu la ted and exper im en ta l spec t r a in the r ange 85 -107 GH z fo r the g roundand f i r s t v ib ra t iona l ly exc i t ed s ta te . T he exper im en ta l l ines a re f rom [1 ] w i th as s ignm en ts due to[2]. All t rans i t ions are of the a- ty pe . All calculat ions are for geo me try I I .

J ' K ' _ I K ' + I J K - 1G r o u n d s ta t eE x p e r i -K + I C a l c u la t e d /M H z m e n t a l / M H z

Fi r s t e xc i t ed s ta teE x p e r i -C a l c u l a t e d / M H z m e n t a l / M H z9 1 8 8 1 7 87044 87005 86120 86080

10 1 10 9 1 9 92762 92621 91707 9143910 0 10 9 0 9 94468 94360 93439 9353810 5 5 9 5 4 94833 94567 93788 9342110 4 6 9 4 5 94842 94643 93797 9351411 0 11 10 0 10 103835 103706 102708 10247911 2 9 10 2 8 104684 104587 103519 10337911 1 10 10 1 9 106347 106285 105219 105152

T h e r e f i t te d p o t e n t i a l w a s t h e n s h i f te d t o g e o m e t r y I I . T h i s w a s t h ea p p r o pr ia t e e x p e r i m e n ta l g e om e t r y , R = 2 - 5 5 5 A , 0 = 9 8 . 8 a n d r = l . 1 7 1 Ao b t a i n e d b y i s o t o p i c s u b s t i t u t i o n [ 7 ]. A r e p e a t o f t h e v i b r a t i o n a l c a l c u l a t i o n o ft a b l e 1 w i t h t h e n o r m a l c o o r d i n a t e s a p p r o p r i a t e t o t h is g e o m e t r y s h o w e d t h a tn o n e o f t h e f i r st 8 b a n d o r i g in s c h a n g e d b y m o r e t h a n 2 c m - 1 a n d t h e f u n d a -m e n t a l s b y 0 . 4 c m - 1 a n d 0 .1 c m - 1 r e s p e c t i v e l y .

P a r t o f t h e c a l c u l a t e d r o t a t i o n a l s p e c t r u m f o r g e o m e t r y I I is g i v e n in t a b l e2 . I t i s c l e a r t h a t , f o r t h e a - t y p e tr a n s i t i o n s , g o o d a g r e e m e n t is n o w o b t a i n e d .F i g u r e 1 s h o w s a p p r o x i m a t e d i r e c t io n s o f t h e p r i n c i p l e a x e s of i n e r ti a f o r K C N .W e n o t e t h a t , a l t h o u g h K C N i s a n a s y m m e t r i c t o p , i t i s v e r y n e a r l y a p r o l a t es y m m e t r i c t o p a s I a a -l > > I b b- 1 -- Ice 1 . T h e s m a l l m o m e n t o f i n e r t i a l a a i sd e p e n d e n t a l m o s t e n t i r e l y o n t h e p r o p e r t i e s o f t h e C N b o n d . I t is t h i s m o m e n to f i n e r t i a w h i c h is s e n s i t i v e l y m e a s u r e d b y t h e b - t y p e t r a n s i t i o n s [ 2 5 ] . A s w eh a v e n o t a t t e m p t e d a f u ll so l u t i o n o f t h e v i b r a t i o n a l p r o b l e m a n d h a v e o n l yr e p r e s e n t e d t h e C N c o o r d i n a t e b y an e m p i r ic a l h a r m o n i c p o t e n t ia l , o n e c a n n o th o p e t o r e p r o d u c e t h a t p a r t o f t h e r o t a t io n a l s p e c t r u m d e p e n d e n t o n i t. T a b l e 2s h o w s th a t th e r e is p o o r a g r e e m e n t fo r t h e s e b - t y p e t r a n s it i o n s . H o w e v e r , b - t y p et r a n s i ti o n s h a v e l o w i n t e n s it y a n d a s y e t h a v e n o t b e e n o b s e r v e d f o r e x c i t e dv i b r a t i o n a l s ta t e s [2 6 ] . T h e a - t y p e tr a n s i t i o n s a r e r e l a t i v e l y i n s e n s i t i v e t o t h eC N g e o m e t r y a n d h e n c e t h e r e is g o o d a g r e e m e n t b e t w e e n c a lc u l at e d a n d o b s e r v e dv a l u e s .

A l t h o u g h m a n y r o t a t i o n a l t ra n s i t i o n s f r o m e x c i t e d v ib r a t i o n a l s t a te s h a v e b e e no b s e r v e d a n d a s s ig n e d r o t a ti o n a l q u a n t u m n u m b e r s , n o d e f in i t e a s s i g n m e n t s t ov i b r a t io n a l s t a te s h a s y e t b e e n p o s s i b le [ 7 ]. H o w e v e r , o u r m e t h o d a l lo w s t h er o t a t io n a l s p e c t r u m o f e x c i t e d v i b ra t i o n a l s t a te s t o b e c a l c u l a t e d a n d h e n c ea s s ig n m e n t s to b e m a d e b y c o m p a r is o n s w i th o b s e r v e d s p e c t r u m . T a b l e 3p r e s e n t s o n e s u c h c o m p a r i s o n f o r t h e f i r s t e x c i t e d st a te f o r t h e r a n g e 8 5 -1 0 6 G H z o f [ 1 ]. A s i m i l a r c o m p a r i s o n f o r t h i s s t a t e w i t h t h e r e s u l t s o f [ 7 ]i n t h e ra n g e 1 0 - 4 0 G H z a ls o g i v es g o o d a g r e e m e n t . T h i s c o n f i r m s t h a t t h e s er o t a t i o n a l t r a n s i t i o n s b e l o n g t o t h e f i r s t e x c i t e d s t a t e .

S i m i l a r c o m p a r i s o n s h a v e l e d t o a t e n t a t i v e a s s i g n m e n t o f t h e s p e c t r u mb e l o n g i n g t o t h e s e c o n d v i b r a t i o n a l l y e x c i t e d s t a te . T a b l e 4 g i v e s c a l c u l a t e d


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1 08T a b l e 4 .

J . T e n n y s o n a n d B . T . S u t c l if f eRo ta t iona l cons tan t s fo r t h e l o w e s t f ive v ibrat ional s ta tes calculated us ingg e o m e t r y I I .

S ta te A / M H z B / M H z C / M H z1 59 860-59 4 937.87 4 543.602 63 234.24 4 887.05 4 490.033 64 752.44 4 889.99 4 477.144 59 702.58 4 927-55 4 528.185 64 645-54 4 900.17 4 475.90

Exp er ime ntaFf 58 265-81 4 940.055 4 536-214~- E xper im en ta l g ro und s ta te va lue f rom [2 ].

r o t a t i o n a l c o n s t a n t s f o r t h e l o w e s t 5 v i b r a t i o n a l st a t e s. I t is c l e a r t h a t t h ec a l c u l a te d r o t a t i o n a l c o n s t a n t s , a n d h e n c e s p e c t r a , a r e v e r y s im i l a r f o r th e s e c o n da n d f o u r t h v i b r a t i o n a l l y e x c i t e d s t a te s , st a t e s 3 a n d 5 i n t a b l e 4 . T h i s m e a n st h a t a n y a s s i g n m e n t o f th e s e s t a te s c a n o n l y b e t e n t a t i v e .

5 . CONCLUSIONSIn this paper we have successfullyapplied the Whitehead-Handy method to

the low lyingvibrational states of KCN and hence obtained a rotational spectrumfor these states. This method relies on being able to define an equilibriumstructure for the molecule and thus is not appropriate for the higher vibrationalstates where polytopic [5] motion is to be expected. However,for states oflarge amplitude, separation of vibrational and rotational problems can no longerbe such a good approximation. In a future paper [23] we hope to explore thehigh amplitude vibrations in the bending coordinatemore fully by using a method[22] which integrates analyticallyover this coordinate with no assumptions aboutequilibrium structures or separability. However, or the low-lying states ourvibrational calculations are well converged and when a realistic equilibriumgeometry is used good agreement is obtained with the experimental rotationalspectrum. This has allowedvibrational assignments o be made o the rotationalspectrum of vibrationally excited states.The main problem in this ab {n{t{ocalculation of vibrational and rotationalspectra has been the accuracy of the ab in{riosurface and more particularly thelocation of the ab initio minimum. Wormerand Tennyson [3] give the time tocalculate one potential point as 6 hours on an IBM 370/158. By comparison thecalculation with M~ = 9, M B= 6 of table 1 took 70 minutes, the integration overvibrational states of equation (10) took 10 minutes per vibrational state andcalculation of a rotational spectrum with J =


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D y n a m i c s o f K C NT h e y a n d D r . P a u l W o r m e r a r e t h a n k e d f o r s u g g e st i ng t h e p r o b l e m .a c k n o w l e d g e t h e a d v i c e a n d i n t e r e s t o f P r o f e s s o r A d v a n d e r A v o i r d .

1 0 9F i n a l l y w e

REFERENCES[1] KUIJPERS,P. , TRRING, T . , a n d DYMANUS,A., 1976, Chem . Phys. Let t . , 42, 423.[2] T6RRING , T . , BEKOOY,J. P., MEERTS, W . L ., HOE FT, J., TIEMANN, E., an d DYMANUS,A., 1980, J . chem. Phys., 73, 4875.[3] WORMER, P. E. S., an d TENNYSON, J., 1981, J. chem. Phys., 75, 1245.[4] KLEIN, M . L. , GODDARD, J. D ., a nd BOUNDS,D. G. , 1981, J. chem. Phys., 75, 3909.[5] CLEMENTI,E ., KISTENMACHER,H. , and POPKIE, H. , 1973, J. chem. Phys., 58, 2460.[6] REINARTZ,J . M. L . J . , 1976 , T hes i s , U n iver s i ty o f N i jm egen .[7] VAN VAALS, J . J . , MEERTS, W . L. , an d TORRING, T . ( to b e pub lishe d).[8] LEROI, G. E . , and KLEMPERER, W. , 1961, J. chem. Phys., 35, 774.[9] ISMAIL, Z. K ., HAUGE, R. H ., an d MARGRAVE, J. L ., 1972, J. chem. Phys., 57, 5137 ;1973, 3 . molec. Spectrosc., 45, 304; 1975, Ibid., 54, 402.

[10] CARNEY, G . D ., SPRANDEL, L . L ., a nd KER N, C. W ., 1978, Ad v. Chem. Phys ., 37, 305.[11] ECKART, C., 1935, Phys. Rev., 47, 552.[12] WHITEHEAD, R. J., an d HANDY , N . C., 1975, if- molec. Spectrosc., 55, 356 ; 1976,aT. mo lec. Sp ectrosc ., 59, 459.[13] HOLM GgeN, S. L ., WALDMAN, M ., a nd KLEMPERSa,W . , 1978 , J. chem. Phys., 69, 1661.[14] HANDY, N . C. , and CARTER, S. , 1981, Chem . Phys . Le t t . , 79, 118.[15] BARTHOLOMAE,R. , MARTIN, D. , an d SUTCLIFFE,B. T . , 1981, .7. molec. Spectrosc., 87,367.[16] WATSON, J . K . G. , 1968, Molec. Phys., 15, 479.[17] DARLING,B. T. , a nd DENNISON, D . M ., 1940, Phys . Rev. , 57, 128.[18] WILSON , E. B. , JR . , DEcIUS, J . C. , an d CROSS, P. C. , 1955, Molecular Vibrations( M c G r a w - H i l l) , C h a p . 4 .[19] CLARY, D . C. , 1981, Molec. Phys., 43, 469.[2 0] SCHACHTSCHNEIDER, . H ., 19 62 , Vibrational Analysis of Polyatomic Molecules (She l lDeve lopm en t Com pany) , P rog . No . 31450 , T ech . Rep . 231-64 .[21] SHAVITT,I ., BENDER, C. F., PIPANO, A., an d HOSTENY,R. P . , 1973,J . comp ut. Phys., 11,90. RAFFENETTI,R. C . , 1979, J . comput. Phys., 32, 403.[22 ] LEROY, R. J ., a nd CARLEY, J. S., 1980, Ad v. Chem . Phys. , 42, 353.[23] TENNYSON, J., an d VAN DER AVOIRD, A., 19 82 , J . chem. Phys. ( in the press ) .[24] TAYLOR,P. R., BACSKAY, G . B., H USH , N . S., a nd HURLEY,A. C. , 1979, .~. chem. Phys.,70, 4481.[25] GORDY, W . , an d CooK , R. L . , 1970, Microwave and Molecular Spectra, 2nd ed i t ion( In te r s c ience-W i ley ) , Chap . 7 .[26 ] VAN VAALS, J. J., an d MEERTS, W . L. , 1981 (p riva te com m unica t ion ) .

jonathan tennyson and brian t. sutcliffe- ab initio vibrational-rotational spectrum of potassium...

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