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Identification of a ¹Δ [delta] u - ¹Σ [sigma] ⁺g transition of CS₂ in the near ultraviolet Malm, David Nelson 1972-12-31

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IDENTIFICATION  OF A  L  A  -  Z  1  TRANSITION OF C S  +  u  2  g  IN THE NEAR ULTRAVIOLET  by  DAVID NELSON MALM B.Sc,  U n i v e r s i t y of B r i t i s h C o l u m b i a , 1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF SCIENCE In the Department of Chemistry We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October,  1972  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  shall  f u r t h e r agree  in p a r t i a l  the U n i v e r s i t y  make i t  freely  fulfilment of of  British  available  for  the  requirements  C o l u m b i a , I agree  for  that  r e f e r e n c e and s t u d y .  that permission for extensive copying o f  this  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s of  this  representatives. thesis for  It  financial  i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n gain s h a l l  written permission.  Department o f  CHEMISTRY,  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  6  1972.  not be a l l o w e d without my  ABSTRACT  The s t r o n g e s t f e a t u r e s i n the a b s o r p t i o n spectrum o f C S 2 i n o  o  the r e g i o n 2900A t o 3500A a r e i d e n t i f i e d , from t e m p e r a t u r e s t u d i e s , a s a IT -> TT t h e *A  L  A u  I  I  g  t r a n s i t i o n , where t h e R e n n e r - T e l l e r  e f f e c t has s p l i t  s t a t e i n t o *B 2 and *A 2 component s t a t e s o f a bent m o l e c u l e .  A n a l y s i s o f some o f t h e l e a s t s e v e r e l y p e r t u r b e d  bands o f t h e 1 B 2 -  1  E+  8  o  transitions  (3300 - 2900A) shows t h a t t h e y form a p a r a l l e l - p o l a r i z e d  p r o g r e s s i o n i n t h e b e n d i n g v i b r a t i o n , t o an upper s t a t e w i t h r ( C - S) = I.544A,  <SCS = 1 6 3 ° , and a b a r r i e r t o l i n e a r i t y o f ~ 1400 c m - 1 .  Two  h i t h e r t o u n r e c o g n i z e d p r o g r e s s i o n s o f ' h o t ' b a n d s , a weak v i b r o n i c II — n o  o  and a s t r o n g e r v i b r o n i c A - A p r o g r e s s i o n i n the r e g i o n 3300A t o 3500A, are assigned  to the 1 A 2 -  1  E + transiton.  T h i s i s a new t y p e o f t r a n s i t i o n ,  w h i c h does n o t appear i n c o l d a b s o r p t i o n , but whose ' h o t ' bands c a n o b t a i n an amount o f i n t e n s i t y  ( p r o p o r t i o n a l t o K2) through Renner-Teller  mixing.  (ii)  TABLE OF CONTENTS PAGE Abstract  (i)  L i s t o f T a b l e s and F i g u r e s  ( i i i )  Acknowledgment  (v)  I.  INTRODUCTION  1  II.  EXPERIMENTAL  4  III.  THEORY A . Symmetry P r o p e r t i e s o f C S 2 B. V i b r a t i o n a l Energy L e v e l P a t t e r n s C. R o t a t i o n a l Energy L e v e l P a t t e r n s D. R o t a t i o n a l C o n s t a n t s o f C S 2 E. Nuclear Spin S t a t i s t i c s F. S e l e c t i o n Rules  10 13 19 24 25 29  IV.  THE A. B. C. D. E.  V SYSTEM Temperature S t u d i e s and P o l a r i z a t i o n o f t h e V System Rotational Analysis V i b r a t i o n a l Pattern of the V State E l e c t r o n i c Species of the V State Vibronic C o r r e l a t i o n In a 1 A U E l e c t r o n i c State  34 34 38 53 59 61  V.  THE A. B. C.  T STATE The R e n n e r - T e l l e r E f f e c t i n a A E l e c t r o n i c S t a t e G e n e r a l F e a t u r e s o f t h e T System N 2 L a s e r E x c i t e d F l u o r e s c e n c e o f C S 2 a t 3371A  67 68 75 76  D.  The P o t e n t i a l Energy Curves f o r t h e * A S t a t e  VI.  1  U  DISCUSSION  U  77 81  References  84  Appendix I  87  Appendix I I  91  Appendix I I I  93  (iii) L i s t o f T a b l e s and F i g u r e s Table I  Page C h a r a c t e r T a b l e s o f D , and C ooh  II III  C o r r e l a t i o n of Species of D  P o i n t Groups  11  Zy  and C 2 v P o i n t Groups  V i b r a t i o n a l Frequencies of X 1 Z + 8 S t a t e s o f CS 2  (ground) E l e c t r o n i c 19  IV  R o t a t i o n a l L i n e A s s i g n m e n t s o f t h e 3236A and 3322A Bands  V  R o t a t i o n a l L i n e A s s i g n m e n t s o f t h e 3275A and 3365A Bands  o  VI  a.  b.  12  44  o  R o t a t i o n a l Term V a l u e s o f t h e Upper S t a t e s o f t h e 3236A and 3322A Bands .  46 49  R o t a t i o n a l Term V a l u e s o f t h e Upper S t a t e s o f t h e 3275A and 3365A Bands  50  VII  V i b r a t i o n a l Term V a l u e s o f t h e V S t a t e o f CS 2  55  VIII  V i b r a t i o n a l Term V a l u e s o f t h e R S t a t e o f C S 2  66  IX  V i b r a t i o n a l Term V a l u e s o f t h e T S t a t e o f C S 2  70  (iv) Figure 1.  Page 4 Meter White C e l l Experimental Arrangement and Order Separator  6  2.  Ground State V i b r a t i o n a l Energy Level Pattern of CS  18  3.  P r i n c i p a l I n e r t i a l Axes of Non-Linear CS  4.  Rotational Energy Level Patterns  5.  CS  6.  The 0 0 0, K* = 0 - 0 0° 0 and 0 0 0, K' = 0 - 0 2° 0 Bands of the V System  42  7.  The 0 0 0, K' = 1 - 0 l Bands of the V System  43  8.  Rotational Term Values of the 0 0 0, K' = 0 and 0 0 0, K' = 1 Levels of the V State plotted against J ( J + 1)  52  V i b r a t i o n a l Term Values of the V State  54  C o r r e l a t i o n of Vibronic Levels of a A state i n Linear and Bent Limits  62  Behaviour of Vibronic Levels of a A state with a B a r r i e r to Linearity  64  Medium and High Resolution Spectra of CS Wavelength Region (3371A)  69  2  24  2  28 o  9. 10. 11. 12. 13.  2  Absorption Spectrum (3400-2900A)  1  35  0 and 0 0 0, K» = 1 - 0 3  The P o t e n t i a l Energy Curves f o r the *A  2  i n the N  0  1  Laser  2  state of CS  2  79  (v)  ACKNOWLEDGMENT The a u t h o r wishes to thank h i s r e s e a r c h s u p e r v i s o r , Dr. A . J .  Merer, as w e l l as D r . C h r i s t i a n Jungen, f o r t h e i r a d v i c e and  encouragement i n the r e s e a r c h upon which t h i s t h e s i s i s b a s e d .  A  s p e c i a l word o f thanks i s a l s o due E.M. who gave much needed s u p p o r t throughout the  ' d o g d a y s ' o f t h e s i s w r i t i n g and to Rose Chabluk f o r  h e r f i n e j o b i n the t y p i n g o f t h i s  thesis.  - 1 I.  INTRODUCTION  Carbon d i s u l f i d e has a complex e l e c t r o n i c a b s o r p t i o n s p e c t r u m o  i n t h e v i s i b l e and u l t r a v i o l e t r e g i o n s .  o  I n t h e r e g i o n 4300A t o 2900A  t h e r e a r e s e v e r a l r e l a t i v e l y weak systems ( 1 - 6 ) , and i n t h e r e g i o n 2300&. O  t o 1800A (7-8) t h e r e a r e s e v e r a l systems A number o f i n t e n s e bands between  of  1850A and  much s t r o n g e r  absorption.  1375A (7) ( p r o b a b l y  i n g t o s e v e r a l e l e c t r o n i c t r a n s i t i o n s ) have n o t been a n a l y s e d . 1375& t o 600A Rydberg s e r i e s c o n v e r g i n g  t o t h e CS 2 i o n i z a t i o n  belongFrom  limits  have been c h a r a c t e r i z e d ( 9 - 1 0 ) . The e l e c t r o n i c c o n f i g u r a t i o n o f CS 2 i n i t s e l e c t r o n i c ground state i s  ... « y  2  2  <*u>  'V*  l z  g  C S 2 i s l i n e a r i n the ground e l e c t r o n i c s t a t e w i t h a C-S bond l e n g t h o f I.5545A  (11-12).  The shape and b o n d i n g c h a r a c t e r i s t i c s o f t h e f i l l e d  and u n f i l l e d o r b i t a l s a v a i l a b l e t o C S 2 a r e d i s c u s s e d by Walsh ( 1 3 ) . l o w e s t energy e x c i t e d e l e c t r o n c o n f i g u r a t i o n i s . . . ( n g ) TT* i s an a n t i - b o n d i n g o r b i t a l . 1  E , 1 A , 3 E + , 3 E and 3 A u u u u  where  The c o n f i g u r a t i o n g i v e s r i s e t o *E*,  states.  u  The  Only one o f t h e p o s s i b l e TT i r *  t r a n s i t i o n s i s f u l l y a l l o w e d by t h e s p i n and o r b i t a l symmetry s e l e c t i o n r u l e s , and t h i s has been i d e n t i f i e d w i t h t h e s t r o n g 2100A  absorption  e  bands ( 1 3 ) . A r o t a t i o n a l a n a l y s i s o f a few o f the 2100A bands (14) showed t h e t r a n s i t i o n t o be p a r a l l e l - p o l a r i z e d ,  1  B2 -  upper s t a t e i n w h i c h C S 2 i s s l i g h t l y b e n t w i t h an 153° and a C-S bond l e n g t h o f  1.66X. The  1  E + , g o i n g t o an  S-C-S bond a n g l e o f  v i b r a t i o n a l s t r u c t u r e i s not  - 2w e l l u n d e r s t o o d , though t h e r e i s no doubt t h a t t h e upper s t a t e i s the K  TT* 1 B 2  electronic  C*E+) s t a t e . ^ u o  o  I n t h e r e g i o n 3800A t o 3300A Liebermann (4) found s i x v i b r a t i o n a l bands t o have 1 E - 1 E - t y p e r o t a t i o n a l s t r u c t u r e s .  More  recently,  Kleman (15) showed t h a t t h e a b s o r p t i o n i n t h e r e g i o n 4300A t o 3300A c o n s i s t s o f two e l e c t r o n i c s y s t e m s .  I n the l o w e s t e x c i t e d  electronic  s t a t e , w h i c h he c a l l e d t h e R s t a t e , the m o l e c u l e i s b e n t , w i t h an  S-C-S  bond a n g l e o f 135.8° and a C-S bond l e n g t h o f 1.66A.  A long progression  i n the bending frequency, v ^ 3  progression i n the  311 c m - 1 ,  and a s h o r t e r  symmetric s t r e t c h i n g f r e q u e n c y , v j = 691.5 c m - 1 ,  were a s s i g n e d , a s w e l l  as numerous ' h o t ' bands from e x c i t e d v i b r a t i o n a l l e v e l s of t h e ground electronic state.  Kleman was u n a b l e t o i d e n t i f y t h e e l e c t r o n i c s t a t e o f  * the Tr -»• ir e x c i t a t i o n r e s p o n s i b l e f o r the R system e x c e p t t o say t h a t t h e R system was p a r a l l e l - p o l a r i z e d , w i t h a B 2 upper  state.  A l t h o u g h t h e R bands a r e s i n g l e t i n a p p e a r a n c e , t h e m a g n e t i c r o t a t i o n s p e c t r u m (16) and t h e Zeeman e f f e c t state i s actually a t r i p l e t state. as p r e d i c t e d state.  The l o w e s t t r i p l e t e l e c t r o n i c  state,  by t h e o r e t i c a l c o n s i d e r a t i o n s ( 1 3 , 1 8 ) , i s t h e ^A^ (ir -*• IT ) 3  I n t h e bent m o l e c u l e t h e o r b i t a l degeneracy o f t h e A^ 3  state i s l i f t e d , giving r i s e t o A2  and  comprehensive Zeeman e f f e c t s t u d i e s lations  (17) i n d i c a t e t h a t t h e R  3  B2  component s t a t e s .  electronic More  ( 1 9 ) , as w e l l as t h e o r e t i c a l  calcu-  ( 2 0 ) , a s s i g n e d t h e R s t a t e as a Hund's c o u p l i n g c a s e ( c ) s p i n  m u l t i p l e t sub-component ( B 2 ) o f a 3 A 2  electronic state.  The Zeeman  - 3 spectrum o f s o l i d CS 2  a t 4.2°K (21)  a b s o r p t i o n to the o t h e r two the  3  A2  s t a t e , 36 cm - 1  f o r the R s y s t e m . consistent t h a n the  3  (3^u)  B2  3  t o the r e d of the B 2  A2  induced  s p i n m u l t i p l e t sub-components ( A j , B j )  As p r e d i c t e d  w i t h the  showed weak m a g n e t i c f i e l d  (3^u)  of  sub-component r e s p o n s i b l e  by Hougen ( 2 0 ) , t h i s o b s e r v a t i o n i s  component s t a t e l y i n g l o w e r i n energy  component s t a t e .  Kleman i d e n t i f i e d a second e l e c t r o n i c system (which he  called  the S system) i n the r e g i o n 3700A t o 3350A; t h i s c o n s i s t s of a n o t h e r p a r a l l e l - p o l a r i z e d p r o g r e s s i o n i n the upper s t a t e b e n d i n g f r e q u e n c y , v£ = 270  cm-1.  The  r o t a t i o n a l s t r u c t u r e o f the v i b r a t i o n a l bands i s  s i m p l e , l i k e t h a t of the R b a n d s .  The  s t r o n g e s t CS 2  e  i t y of the V system has  was  c a l l e d the V system by Kleman.  The  complex-  so f a r p r e v e n t e d a v i b r a t i o n a l or r o t a t i o n a l  R e c e n t l y , however, the a b s o r p t i o n s p e c t r u m o f the V system  o b s e r v e d i n m a t r i c e s a t 20°K and  Meyer ( 2 2 ) , who structure  the  e  r e g i o n 3400A t o 2900A, was  analysis.  a b s o r p t i o n i n the n e a r - u l t r a v i o l e t , i n  77°K by Bajema, Goutermann  and  found a c o n s i d e r a b l e s i m p l i f i c a t i o n o f the v i b r a t i o n a l  i n the m a t r i x o v e r the gas phase s p e c t r u m .  The  trum c o n s i s t s e s s e n t i a l l y o f a s i n g l e p r o g r e s s i o n o f 580  matrix spec± 30  cm-1,  w h i c h t h e y a s s i g n e d as the symmetric s t r e t c h i n g v i b r a t i o n of a m o l e c u l e l i n e a r i n t h e upper s t a t e . The c o m p l e x i t y o f the V s t a t e was a t t r i b u t e d t o the e f f e c t s o f R e n n e r - T e l l e r i n t e r a c t i o n i n a 1 A e l e c t r o n i c state of u a l i n e a r molecule. t h a t the 580  cm - 1  However, the c o n c l u s i o n r e a c h e d i n t h i s t h e s i s i s i n t e r v a l c o r r e s p o n d s t o the b e n d i n g v i b r a t i o n of a  bent molecule w i t h a 1 B 2  (1A  ) upper s t a t e .  -  II.  4  -  EXPERIMENTAL  Low r e s o l u t i o n s u r v e y s p e c t r a o f C S 2 were t a k e n on a A 10 cm c e l l was f i l l e d w i t h C S 2  Cary 14 r e c o r d i n g s p e c t r o p h o t o m e t e r .  v a p o u r t o 4 cm p r e s s u r e and t h e n opened t o t h e a i r t o g i v e a p r e s s u r e broadened spectrum.^  Without p r e s s u r e - b r o a d e n i n g , a l l a b s o r p t i o n  bands o f w i d t h l e s s t h a n t h e r e s o l u t i o n o f t h e Cary 14 w o u l d be recorded w i t h i n c o r r e c t  intensities.  Medium r e s o l u t i o n gas phase s p e c t r a ( a p p r o x i m a t e l y 150,000 r e s o l v i n g power) were photographed  i n t h e second o r d e r o f a 21 f t  f o c a l l e n g t h Eagle-mounted concave g r a t i n g s p e c t r o g r a p h .  Q u a r t z gas  c e l l s f i t t e d w i t h S u p r a s i l windows were used t o g i v e p a t h l e n g t h s o f 80 cm and 160 cm.  I t was n e c e s s a r y i n a l l e x p e r i m e n t s t o u s e a e  C o m i n g 7-54 f i l t e r t o remove w a v e l e n g t h s  o f l e s s t h a n 2300A from t h e  i n c i d e n t r a d i a t i o n t o p r e v e n t p h o t o d i s s o c i a t i o n o f CS 2  (23) a c c o r d i n g  to the r e a c t i o n cs  h v 2  >  CS(A n) 1  +  S(3P2)  This r e a c t i o n i s e n e r g e t i c a l l y p o s s i b l e a t wavelengths  l e s s than  o  0  2778 ± 10A, b u t because t h e r e i s a l m o s t no a b s o r p t i o n from 2800A t o o  2300A, d e c o m p o s i t i o n i s t r o u b l e s o m e o n l y w i t h i n c i d e n t l i g h t o f wavel e n g t h l e s s t h a n 2300A.  W i t h a 20p s l i t and C S 2 p r e s s u r e r a n g i n g from  ^ A l l p r e s s u r e s w i l l be quoted i n u n i t s o f cm o r mm o f Hg. 1 mm Hg = 133.32  N/m2.  - 50.2 mm  t o 5 mm,  exposure t i m e s were o f the o r d e r o f 10 s e c u s i n g  Kodak SA-1 p l a t e s .  Wavelength c a l i b r a t i o n f o r a l l s p e c t r a  photographed  was w i t h a 120 ma i r o n - n e o n h o l l o w - c a t h o d e lamp o f t h e l a b ' s  own  design.  The t e m p e r a t u r e dependence o f the n e a r - u l t r a v i o l e t bands was i n v e s t i g a t e d by p h o t o g r a p h i n g s p e c t r a o f CS 2 and 200°C.  a t - 7 8 ° , 2 3 ° , '85°, 100°  E l e v a t e d t e m p e r a t u r e s were a c h i e v e d by w r a p p i n g  the c e l l  w i t h e l e c t r i c h e a t i n g tape o v e r w h i c h was wound a s b e s t o s c l o t h tion.  insula-  V a r y i n g t h e a p p l i e d v o l t a g e on t h e h e a t i n g t a p e gave t h e d e s i r e d  temperature.  As measured w i t h a mercury thermometer n e x t t o t h e c e l l  s u r f a c e , a r e g u l a t i o n o f ±2°C was  achieved.  To compare r e l a t i v e i n t e n s i t i e s o f bands as a f u n c t i o n o f t e m p e r a t u r e , t h e number o f a b s o r b i n g m o l e c u l e s i n t h e l i g h t p a t h had t o be k e p t c o n s t a n t a t the d i f f e r e n t t e m p e r a t u r e s .  T h i s p r e s e n t e d no  p r o b l e m e x c e p t a t - 7 8 ° C , f o r , even a t room t e m p e r a t u r e , t h e vapour p r e s s u r e o f CS 2  is  300 mm,  f a r i n e x c e s s o f t h e p r e s s u r e needed t o  o b t a i n s u f f i c i e n t a b s o r p t i o n . However, a t -78° t h e vapour p r e s s u r e o f CS 2  i s o n l y 0.2 mm.  S i n c e the n e a r - u l t r a v i o l e t a b s o r p t i o n i s compar-  a t i v e l y weak, t h i s n e c e s s i t a t e d t h e use o f v e r y l o n g a b s o r p t i o n p a t h lengths.  Using a 4 m White-type m u l t i p l e r e f l e c t i o n c e l l  (24,25) o f  t h e l a b ' s own d e s i g n (as shown i n F i g . 1 ) , p a t h ' l e n g t h s o f up t o 128 m c o u l d be a t t a i n e d .  The c e l l c o n s i s t s of 85 mm  p y r e x t u b i n g , t o the ends  of w h i c h o r d i n a r y p y r e x p i p e f l a n g e s a r e a t t a c h e d .  Steel end-plates sealed  by '0' r i n g s a r e s e c u r e d t o t h e f l a n g e s t o produce a vacuum s e a l .  One  XENON LAMP 25 cm LENS  D MIRRORS 35 cm CYLINDRICAL L E N S ^  ORDER SEPARATOR  25 cm SPHERICAL FIELD LENS CONCAVE MIRROR  FORE SLIT  SPECTROGRAPH  MAIN SLIT FIG. (1) 4 Meter White C e l l and Order Separator  PRISM  - 7 -  e n d - p l a t e c a r r i e s the S u p r a s i l e n t r a n c e and e x i t windows, w h i l e t h e o t h e r e n d - p l a t e has e x t e r n a l c o n t r o l s f o r the concave 'D'-shaped mirrors.  I n s e r t e d i n t h e c e l l n e a r t h e q u a r t z windows i s a  'boat'  w h i c h h o l d s i n p o s i t i o n the s h o u l d e r e d c i r c u l a r concave m i r r o r .  The  number of t r a v e r s a l s of t h e c e l l  by  ( m u l t i p l e s o f 4) was  controlled  m a n i p u l a t i o n o f one o f t h e D m i r r o r s .  F o r temperature  s t u d i e s a t - 7 8 ° C , d r y i c e was  t h e W h i t e c e l l and i t s s t y r o f o a m - i n s u l a t e d b o x .  packed between  To a t t a i n e l e v a t e d  t e m p e r a t u r e s , t h e White c e l l was wrapped w i t h h e a t i n g tape and ing  asbestos c l o t h .  Due  t o the l a r g e mass o f p y r e x t u b i n g , a t e m p e r a -  t u r e o f o n l y 85 ± 5°C was  possible.  t r a v e r s a l s 0v48 m p a t h l e n g t h ) , a CS 2 a 30u s l i t w i d t h , exposures m i n u t e s w i t h Kodak SA-1  insulat-  W i t h t h e White c e l l s e t f o r 12 p r e s s u r e a t -78°C o f 0.2 mm  and  on t h e E a g l e s p e c t r o g r a p h were about 10  plates.  H i g h r e s o l u t i o n s p e c t r a ( a p p r o x i m a t e l y 625,000 r e s o l v i n g power) were photographed i n the 1 6 t n * - 19 t * 1 - o r d e r s o f a 7 m l e n g t h Ebert-mounted p l a n e g r a t i n g s p e c t r o g r a p h .  To p r e v e n t an o v e r -  l a p p i n g o f s p e c t r a l o r d e r s , an o r d e r s e p a r a t o r c o n s i s t i n g of  a s m a l l q u a r t z p r i s m monochromator was  g r a p h (main) s l i t and t h e e x t e r n a l o p t i c s S u p r a s i l q u a r t z p r i s m , the w a v e l e n g t h c o u l d be v a r i e d .  focal  essentially  p l a c e d between the s p e c t r o (Fig. 1).  By r o t a t i n g  the  o f l i g h t f a l l i n g on the main  By a d j u s t i n g the f o r e - s l i t  slit  w i d t h between lOOu and o  1500u, t o 300A.  t h e 'bandpass' a d m i t t e d c o u l d be v a r i e d from a p p r o x i m a t e l y W i t h the White c e l l s e t f o r 12 t r a v e r s a l s , a f o r e - s l i t  40A  width  - 8 -  o f 400u and a main s l i t w i d t h o f 20p, e x p o s u r e s were r e q u i r e d w i t h Kodak SA-1 t i o n s p e c t r a was  plates.  of about 10 m i n u t e s  C a l i b r a t i o n o f the h i g h r e s o l u -  a c h i e v e d by i n c r e a s i n g the o r d e r s e p a r a t o r bandpass  so t h a t s e v e r a l o r d e r s a d j a c e n t t o t h e m o l e c u l a r s p e c t r a l o r d e r were photographed.  T h i s i s n e c e s s a r y because a t the h i g h d i s p e r s i o n o f t h e  E b e r t s p e c t r o g r a p h (0.137A/mm i n 1 7 t n * o r d e r ) t h e d e n s i t y of i r o n b r a t i o n l i n e s i n any one o r d e r i s not s u f f i c i e n t f o r a c c u r a t e  cali-  inter-  p o l a t i o n between them.  CS 2  ( F i s h e r RG) was  used w i t h o u t f u r t h e r p u r i f i c a t i o n  f o r d e - g a s s i n g under vacuum a t l i q u i d n i t r o g e n t e m p e r a t u r e . ard  except  The  stand-  a l l g l a s s vacuum l i n e used a d u a l chamber r o t a r y pump and an o i l  d i f f u s i o n pump.  Gas p r e s s u r e s were measured w i t h a thermocouple gauge  and w i t h a manometer f i l l e d w i t h Dow (12.8 mm  s i l i c o n e o i l = 1 mm H g ) .  g r e a s e , used throughout  C o r n i n g . 707 s i l i c o n e  Dow  s i l i c o n e h i g h vacuum s t o p c o c k  t h e vacuum l i n e , was  the o n l y grease t h a t  r e s i s t e d formation of bubbles of d i s s o l v e d CS2. 0.001  mm was  fluid  An u l t i m a t e vacuum o f  r o u t i n e l y achieved w i t h a l l absorption c e l l s .  The  gas  c e l l s were f i l l e d t o t h e d e s i r e d p r e s s u r e s i m p l y by a l l o w i n g t h e f r o z e n , de-gassed CS 2  t o warm s l o w l y , l i q u e f y and  evaporate.  The h i g h r e s o l u t i o n p l a t e s were measured on a G r a n t a u t o m a t i c r e c o r d i n g p h o t o e l e c t r i c comparator i n the P h y s i c s Department, and a l l t h e measurements were reduced t o vacuum wavenumbers by means o f a computer programme t h a t performed a l e a s t s q u a r e s f i t o f the i r o n neon r e f e r e n c e l i n e w a v e l e n g t h s (26) t o a 4-term p o l y n o m i a l .  and  It is  e s t i m a t e d t h a t r e l a t i v e p o s i t i o n s o f unblended l i n e s a r e a c c u r a t e t o b e t t e r than ±0.01  cm-1.  Band head p o s i t i o n s used i n the v i b r a t i o n a l a n a l y s i s were measured from h i g h c o n t r a s t resolution plates.  p h o t o g r a p h i c e n l a r g e m e n t s o f t h e medium  I n t e r p o l a t i o n between two c l o s e r e f e r e n c e  p r o v i d e d a r e l a t i v e band head a c c u r a c y o f ± 1 c m - 1 .  lines  - l O -  IH.  A.  THEORY  Symmetry P r o p e r t i e s o f C S 2  The  p o i n t group o f l i n e a r C S 2 i s  a n a  C 2 v > t h e i r c h a r a c t e r t a b l e s a r e g i v e n as T a b l e I .  that o f bent CS2 i s The l a b e l l i n g o f t h e  symmetry a x i s i s d i f f e r e n t f o r t h e two p o i n t groups b e c a u s e , a c c o r d i n g to Mulliken's convention  (27), the z a x i s i s the molecular  axis for the  l i n e a r molecule, but the two-fold r o t a t i o n a x i s ( C 2 ) f o r the bent molecule.  When l i n e a r C S 2 becomes b e n t , a l l symmetry o p e r a t i o n s o f  D . n o t r e t a i n e d i n C„ a r e no l o n g e r d e f i n e d . °°h 2v symmetry o p e r a t i o n s between remaining  and  The c o r r e l a t i o n o f  i s as f o l l o w s :  C 2 , t h e one  r o t a t i o n about an a x i s p e r p e n d i c u l a r t o t h e l i n e a r a x i s  becomes t h e C 2 ( z ) r o t a t i o n i n t h e  p o i n t group; a ^ , r e f l e c t i o n i n a  p l a n e p e r p e n d i c u l a r t o t h e l i n e a r a x i s becomes c r ^ x z ) ; s i m i l a r l y , t h e one r e m a i n i n g  r e f l e c t i o n , o"^, becomes o " v ( y z ) .  A l l other D  operations  a r e u n d e f i n e d , i n c l u d i n g t h e i n v e r s i o n o p e r a t i o n , so t h a t t h e *g,u' species designations o f the l i n e a r molecule are undefined molecule.  f o r the bent  The r e s u l t i n g c o r r e l a t i o n o f s p e c i e s i s g i v e n a s T a b l e I I . The  axis correlation for D , with C i s as f o l l o w s : «°h 2v  y +-*•' z , z •*-»• y .  x •«-+• x , '  The l i n e a r m o l e c u l e non-degenerate p o i n t group s p e c i e s  c o r r e l a t e i n a one-to-one manner w i t h t h e  p o i n t group s p e c i e s .  However, a l l o t h e r  s p e c i e s a r e d o u b l y d e g e n e r a t e and c o r r e l a t e w i t h  A + B s p e c i e s o f C2v«  That i s , t h e c o r r e l a t i o n o f symmetry s p e c i e s i s  u n i q u e o n l y when t h e symmetry i s l o w e r e d upon b e n d i n g t h e l i n e a r m o l e c u l e ; t h e c o r r e l a t i o n i s ambiguous when b e n t C S 2 becomes l i n e a r .  TABLE D ,  coh  I  z  +  2C*  2C *  CO  OO  2  2C * 3  I  and C Point 2v  Groups  »  V  00  2S* OO  2S * 2  •••  s  2  =i  OO  1  1  1  1  1  1  1  1  1  1  1  1  1  1  -1  -1  1  -1  -1  -1  z  E~  1  1  1  1  -1  1  -1  g Z~ u  1  1  R  1  1  1  1  1  -1  -1  -1  -1  n  2  2cos(j)  2cos2(j)  2cos3<j>  0  -2  0  -2cos<f>  -2cos2cf>  ...  2  n  2  2cos<J>  2cos2<}>  2cos3cj>  0  +2  0  2cos<j>  2cos2<f>  ...  -2  2  2cos2<}>  2cos4cf>  2cos6<{>  0  +2  0  2cos2<J>  2cos4<J>  2  2  2cos2<}>  2cos4(j)  2cos6<|)  0  -2  0  -2cos2<f>  -2cos4<j>  -2  2  2cos3<{>  2cos6<}>  2cos9<{)  0  -2  0  -2cos3<}>  -2cos6(j)  2  2  2cos3<}>  2cos6<j>  2cos9(|)  0  +2  0  2cos3<J>  2cos6<j>  -2  g  u  g  u A g A u g u • * •  • • •  • ••  • •  • • •  1. -1  • • •  * •*  •• <  • • «  z R  ,R x  y  TABLE I ( c o n t i n u e d )  C  I  2V  CT  c2  (x_)  a  (y*) V  V  1  1  z  1  -1  -1  R  1  -1  1  -1  x,R  1  - l  -1  1  Al  1  l  A2  1  Bl B2  X  y  p o i n t group a x i s system  y,R  p o i n t group a x i s s y s t e m  The x a x i s i s o u t o f t h e p l a n e o f t h e p a p e r .  TABLE I I C o r r e l a t i o n o f S p e c i e s o f D , and C * °°h 2 v  D , °°h C  2v  I  + S  if u  g  Ai  A2  Bl  E  + u  B2  n  g  A 2 +B 2  n  u  Al+Bi  A g Al+B!  P o i n t Groups  A u A 2 +B 2  g A 2 +B 2  u A 2 +B 2  • • o  • o o  - 13 B.  Vibrational  Energy L e v e l P a t t e r n s  In order t o gain f a m i l i a r i t y w i t h the conventional spectroscopic  n o m e n c l a t u r e used t h r o u g h o u t t h i s t h e s i s , a c u r s o r y d e v e l o p -  ment o f t h e t h e o r y o f v i b r a t i o n and r o t a t i o n o f a symmetric m o l e c u l e w i l l be o u t l i n e d .  triatomic  We b e g i n w i t h a c o n s i d e r a t i o n o f t h e t h e o r y  of Normal Modes o f v i b r a t i o n .  The t h r e e t r a n s l a t i o n a l degrees o f freedom p o s s e s s e d by each of t h e N c o n s t i t u e n t atoms become t h e t r a n s l a t i o n a l , v i b r a t i o n a l and r o t a t i o n a l degrees o f freedom o f t h e m o l e c u l e formed from t h e s e a t o m s . The number o f v i b r a t i o n a l degrees o f freedom i s t h u s 3N - 6, ( o r 3N - 5 f o r a l i n e a r m o l e c u l e w h i c h has o n l y two r o t a t i o n a l degrees o f f r e e d o m ) . It i s very d i f f i c u l t  t o consider the v i b r a t i o n s  of a polyatomic mole-  c u l e u s i n g C a r t e s i a n d i s p l a c e m e n t c o o r d i n a t e s because t h e v i b r a t i o n a l m o t i o n s a r e q u i t e c l o s e l y i d e n t i f i a b l e as s t r e t c h i n g s angle changing motions, o r t w i s t i n g s .  o f b o n d s , bond  F o r t h i s r e a s o n i t i s customary  t o i n t r o d u c e Normal c o o r d i n a t e s o f v i b r a t i o n , w h i c h c o r r e s p o n d t o t h e a c t u a l m o t i o n s o f t h e atoms, w i t h s u i t a b l e m a s s - w e i g h t i n g .  The d e t a i l e d  m o t i o n s o f t h e atoms a r e governed by t h e F o r c e F i e l d , a s e t o f p o t e n t i a l energy e x p r e s s i o n s c o n t a i n i n g t h e F o r c e C o n s t a n t s and t h e Normal coordinates:  t h e s e a r e c o n v e n i e n t l y w r i t t e n as a T a y l o r s e r i e s i n t h e  3N - 5 ( 6 ) Normal c o o r d i n a t e s  - 14 -  W i t h the c h o i c e o f o r g i n as the ' E q u i l i b r i u m C o n f i g u r a t i o n ' the two terms may  first  be removed, and the q u a d r a t i c c r o s s terms v a n i s h because  t h e Normal c o o r d i n a t e s a r e d e f i n e d t o cause them t o be z e r o .  The  p o t e n t i a l energy i s thus  V = f [ f k  \  Q  a Q  2  +  h i  g h e r terms  (2)  k/o  I f the h i g h e r terms a r e n e g l e c t e d , t h e P o t e n t i a l Energy i s j u s t a o f q u a d r a t i c t e r m s , one f o r each v i b r a t i o n , k .  sum  T h i s i s analogous t o  t h e s i t u a t i o n o f a c l a s s i c a l system u n d e r g o i n g s i m p l e harmonic m o t i o n s , and i s known as the 'Harmonic  Approximation'.  The v i b r a t i o n a l h a m i l t o n i a n f a c t o r i z e s , i n harmonic a p p r o x i m a t i o n , i n t o a sum  o f o n e - d i m e n s i o n a l h a m i l t o n i a n s , one f o r each v i b r a -  t i o n a l motion  k where the * k ' s a r e the F o r c e C o n s t a n t s , i . e . ( 3 2 V / 9 Q 2 ) Q and momentum o p e r a t o r s c o n j u g a t e  t o the Q^'s.  The  the P k ' s  eigenvalues of t h i s  a r e the  hamil-  t o n i a n , the 'Harmonic O s c i l l a t o r ' h a m i l t o n i a n , a r e w e l l - k n o w n (28,29) t o be E = I  (v f c + |) h v fc  joules  (4)  k  where t h e v^'s a r e n o n - n e g a t i v e i n t e g e r s , and  the  v k  's  are r e l a t e d to  t h e F o r c e C o n s t a n t s by X k = 4*2 v 2  (5)  I t i s customary  to define the quantity  = vic/c»  s o  t n a t  »  *-n c  m  u n i t s , t h e v i b r a t i o n a l energy l e v e l s a r e  E/hc  -I  (v + | ) cok  cm-1  (6)  k  The e f f e c t s o f t h e c u b i c and h i g h e r terms o m i t t e d from t h e h a m i l t o n i a n are  t o add c o r r e c t i o n s t o t h i s e x p r e s s i o n , c a l l e d A n h a r m o n i c i t y Terms.  A l i n e a r symmetric  t r i a t o m i c m o l e c u l e has 3N-5 = 4 v i b r a t i o n s ,  b u t two o f t h e s e c o r r e s p o n d t o i d e n t i c a l b e n d i n g motions e x e c u t e d a t r i g h t a n g l e s , and t h e r e f o r e have t h e same f r e q u e n c y v ; t h e s e a r e s a i d t o be ' d e g e n e r a t e ' and one speaks l o o s e l y o f t h e m o l e c u l e a s " h a v i n g t h r e e v i b r a t i o n s , one o f w h i c h i s d e g e n e r a t e " .  The o t h e r two v i b r a t i o n s  c o r r e s p o n d t o symmetric and a n t i s y m m e t r i c c o m b i n a t i o n s o f t h e s t r e t c h i n g s o f t h e two b o n d s .  The v i b r a t i o n a l energy l e v e l p a t t e r n i s s t i l l g i v e n by e q u a t i o n ( 6 ) , w i t h t h e summation r u n n i n g o v e r b o t h components o f t h e d e g e n e r a t e bending v i b r a t i o n .  There i s , however, a n a d d i t i o n a l p r o p e r t y a s s o c i a t e d  w i t h t h e b e n d i n g v i b r a t i o n , w h i c h i s t h e v i b r a t i o n a l a n g u l a r momentum. P h y s i c a l l y i t s o r i g i n may be v i s u a l i s e d as f o l l o w s .  I n a normal v i b r a -  t i o n a l l t h e atoms move w i t h t h e same f r e q u e n c y so t h a t t h e C a r t e s i a n components o f t h e d i s p l a c e m e n t s change a c c o r d i n g t o s i n e c u r v e s .  The  s u p e r p o s i t i o n o f two i d e n t i c a l b e n d i n g m o t i o n s o f e q u a l a m p l i t u d e a t r i g h t a n g l e s and w i t h a 90° phase s h i f t r e s u l t s i n t h e m o t i o n s o f t h e i n d i v i d u a l atoms d e s c r i b i n g a c i r c l e about t h e l i n e a r a x i s as i l l u s t r a t e d below.  - 16 -  Such an a d d i t i o n of two harmonic motions at r i g h t a n g l e s r e s u l t i n g r o t a t i o n about the l i n e a r  axis i s equivalent  t o i m p a r t i n g an a n g u l a r  momentum to the n u c l e i w i t h the v e c t o r of the v i b r a t i o n a l momentum a l o n g the l i n e a r  axis.  r o t a t i o n about the l i n e a r  a x i s at  in  angular  The i n d i v i d u a l atoms execute a  full  the f r e q u e n c y of the degenerate  bend-  ing v i b r a t i o n . In m a t h e m a t i c a l terms the v i b r a t i o n a l because i t  is  convenient  angular momentum a r i s e s  to t r a n s f o r m from Normal c o o r d i n a t e s to  cylin-  d r i c a l p o l a r c o o r d i n a t e s , when the p r o d u c t of two o n e - d i m e n s i o n a l harmonic o s c i l l a t o r  e i g e n f u n c t i o n s becomes an A s s o c i a t e d  f u n c t i o n m u l t i p l i e d by an a n g u l a r f a c t o r , (l//2 n )e ^^. ;  factor,  as i s w e l l known ( 3 1 ) ,  system w i t h a x i a l symmetry. is  £ is  This angular  c o r r e s p o n d s to an a n g u l a r momentum i n a  The n o t a t i o n used f o r a bending v i b r a t i o n  2  is  the sum of one d i m e n s i o n a l o s c i l l a t o r  the v i b r a t i o n a l  v  2~2»«.•,-v . 2  As we show below,  function) it  2  and I,  quantum numbers, and  a n g u l a r momentum quantum number, which  p r o p e r t i e s of the A s s o c i a t e d Laguerre 2»  1  t h a t the e i g e n f u n c t i o n s are d e s c r i b e d by quantum numbers v  where v  v  r  Laguerre  (from  may take the  i s convenient,  for  the  values  rotational  energy l e v e l c a l c u l a t i o n s , t o d e f i n e t h e v i b r a t i o n a l a n g u l a r momentum o p e r a t o r , G.  The e i g e n v a l u e o f i t s z-component i s , o f c o u r s e , £(tl),  i.e.  Anharmonic terms cause components o f a v i b r a t i o n a l l e v e l v 2 w i t h d i f f e r e n t values of  to l i e at s l i g h t l y different energies.  The  v i b r a t i o n a l l e v e l s a r e c l a s s i f i e d as E , II, A, . . . a c c o r d i n g t o whether |£| = 0 , 1, 2 , . . . .  I n f a c t , any a n g u l a r momentum s t a t e s a r e c l a s s i -  f i a b l e i n t h i s manner.  The l o w e s t v i b r a t i o n a l l e v e l s o f t h e ground s t a t e o f C S 2 a r e g i v e n i n T a b l e I I I and i l l u s t r a t e d i n F i g . 2 . symmetric s t r e t c h i n g Normal v i b r a t i o n s  The symmetric and a n t i -  a r e d e s i g n a t e d V j and v 3 > w i t h  quantum numbers v 1 and v 3 , r e s p e c t i v e l y ;  the doubly degenerate bending  v i b r a t i o n i s l a b e l l e d by t h e v a l u e s o f v 2 and £ . t i o n a l wavefunctions  i n the  The s p e c i e s o f t h e v i b r a -  p o i n t group a r e a l s o g i v e n i n F i g . 2 . The  £  n o t a t i o n ( j V 2 v 3 ) s p e c i f i e s t h e v i b r a t i o n a l l e v e l o f a l i n e a r symmetric v  triatomic molecule. the 'overtones' are not g i v e n .  The energy l e v e l s o f t h e ' f u n d a m e n t a l s ' ( v ^ = 1) and  ( v ^ = 2 , 3 , . . . ) a r e shown, b u t t h e ' c o m b i n a t i o n  levels'  E  - 18  v,  / C M "  1600  4  Ag u  i  1200  3  n  -  U  u  0 E  800  2  A  8  1  1 n.  4 0 0  u  +  FIG.  (2)  0  -Q__  +  g  Fundamental and L o w e s t - l y i n g Overtone L e v e l s o f t h e X 1 ^ Ground S t a t e o f (15,31).  1 2  C  3 2  S 2 Referred to the v i = v  2  = v  =  3 0  Level  8  - 19 TABLE I I I  Fundamental and L o w e s t - l y i n g Overtone L e v e l s ( c m - 1 ) o f t h e X X E Ground S t a t e o f } C S 2  32  R e f e r r e d t o t h e v : = v 2 = v 3 = 0 L e v e l (15,31)  2  Level  ( 1 2 3) V  V  (1 0° 0)  657.98  (0 0° 1)  1532.35  (0 l  C.  V  1  0)  395.9  (0 2 2 0)  791.9  (0 2° 0)  802.6  (0 3 3 0)  1187.8  (0 3 1 0)  1207.2  R o t a t i o n a l Energy L e v e l P a t t e r n s  The g e n e r a l e x p r e s s i o n f o r t h e r o t a t i o n - v i b r a t i o n of a polyatomic molecule i n the r i g i d approximation  .  I  rotator-harmonic  Hamiltonian  oscillator  (32) i s  V a = x,y,z  (J - P )  .  2  X  a  1 U 2 k  .  .  _. I V i n2 k  The l a s t two terms r e p r e s e n t t h e Normal v i b r a t i o n s o f t h e m o l e c u l e and have been d i s c u s s e d .  The f i r s t term r e f e r s t o t h e k i n e t i c energy f o r  - 20 n u c l e a r r o t a t i o n about t h e t h r e e p r i n c i p a l i n e r t i a l a x e s , a s w e l l a s t h e i n t e r a c t i o n o f r o t a t i o n a l and i n t e r n a l a n g u l a r momenta.  We now c o n s i d e r t h e n u c l e a r r o t a t i o n a l k i n e t i c e n e r g y . n u c l e a r r o t a t i o n a l a n g u l a r momentum R i s n o t q u a n t i z e d : quantities  t h e conserved  a r e J , t h e t o t a l a n g u l a r momentum, and P , t h e i n t e r n a l  a n g u l a r momentum,which c o n s i s t s electron  The  of the v i b r a t i o n a l , electron  s p i n and  o r b i t a l a n g u l a r momenta, i . e .  P = G + S + L .  (2)  The n u c l e a r r o t a t i o n a l a n g u l a r momentum i s thus o b t a i n e d by v e c t o r subtraction.  R = J - P  (3)  There i s a t once a d i f f e r e n c e a non-linear molecule.  between a l i n e a r m o l e c u l e and  S i n c e t h e mass o f t h e l i n e a r m o l e c u l e i s c o n -  c e n t r a t e d a l o n g t h e a x i s , 1^ i s u n d e f i n e d , and 1^ = I .  For a non-  l i n e a r t r i a t o m i c m o l e c u l e t h e t h r e e moments o f i n e r t i a a r e a l l d i f f e r e n t and non-zero, and t h e p l a n a r i t y c o n d i t i o n ,  I = 1 + I , applies. x y z  We assume t h a t t h e m o l e c u l e i s i n an e l e c t r o n i c  singlet  o r b i t a l l y non-degenerate s t a t e , so t h a t S and L may be o m i t t e d :  the  o n l y r e m a i n i n g i n t e r n a l a n g u l a r momentum a r i s e s from v i b r a t i o n .  I t can  be shown (33,34) t h a t t h e v i b r a t i o n a l a n g u l a r momentum o n l y makes important c o n t r i b u t i o n s  t o t h e r o t a t i o n a l energy l e v e l s f o r a l i n e a r  m o l e c u l e , s o t h a t t h e two t y p e s o f m o l e c u l a r g e o m e t r i e s may be c o n s i d ered s e p a r a t e l y .  i)  Linear molecules  Expanding t h e s q u a r e s , t h e K.E. o p e r a t o r f o r a l i n e a r molecule i s  T =  (J2+ J2) J G + J G X. + ^ L * v_JL  I, 2  1  1  , (G2 + G2) l _ _ x y_  2  . .  1  The l a s t term (34) has t h e form o f c o r r e c t i o n s t o t h e a n h a r m o n i c i t y t e r m s , and we n e g l e c t i t .  T=  The f i r s t term may be r e - w r i t t e n  tl <?-S>  P)  2J  whose e i g e n v a l u e s a r e  E = | j  [ J ( J + 1) - A 2 ]  joules  (6)  where t h e component o f J a l o n g t h e a x i s o f t h e m o l e c u l e i s t h e z-cbmponent o f t h e v i b r a t i o n a l a n g u l a r momentum, £(ft).  I t i s customary  t o use wave number u n i t s ( c m - 1 ) i n s t e a d o f t r u e energy u n i t s  (joules),  so t h a t we d i v i d e b o t h s i d e s by h e :  E/hc = — [ J ( J 8TT c l  2  + 1) - £ ]  cm-  1  (7)  A g a i n , i t i s customary t o a b b r e v i a t e h / 8 i r 2 c l by t h e symbol 'B' ( c a l l e d the r o t a t i o n a l c o n s t a n t ) , g i v i n g E/hc = B [ J ( J + 1) - £ 2 ]  cm - 1  (8)  The e f f e c t s o f t h e second term i n (4) a r e beyond t h e scope o f t h i s t h e s i s , b u t g i v e r i s e t o t h e e f f e c t c a l l e d £.-type d o u b l i n g (33)  i n d e g e n e r a t e v i b r a t i o n a l s t a t e s where \z\ >  0:  g i v e n J a r e s p l i t by an amount p r o p o r t i o n a l t o  the l e v e l s of a B[^- J ( J + 1 ) ] ^  s i n c e i n g e n e r a l B << u>, t h e s p l i t t i n g i s l a r g e s t f o r decreases  rapidly for larger I values.  expressed  by t h e energy l e v e l term  The magnitude o f t h e  q [ J ( J + 1 ) ] ^ may  u n i t s at the h i g h e s t observable J v a l u e s . p r o p o r t i o n a l i t y c o n s t a n t , q , i s o n l y 5.27 ii)  Non-linear  =1  I n CS 2 x 10~5  and  and splitting, cm - 1  r e a c h a few  t h e £-type d o u b l i n g cm - 1  (35).  molecules  Energy l e v e l e x p r e s s i o n s f o r n o n - l i n e a r t r i a t o m i c  molecules  cannot be o b t a i n e d i n c l o s e d form e x c e p t f o r the l o w e s t J v a l u e s because t h e m o l e c u l e s b e l o n g t o t h e c l a s s known as Asymmetric t o p s , where a l l t h r e e moments o f i n e r t i a a r e d i f f e r e n t .  However a p p r o x i m a t e  forms a r e r e a d i l y o b t a i n e d by o m i t t i n g t h e o f f - d i a g o n a l m a t r i x elements o f t h e h a m i l t o n i a n t h a t a r i s e when Symmetric top b a s i s f u n c t i o n s a r e used ( t h e Near-Symmetric top  The  approximation).  i n t e r n a l a n g u l a r momentum components can be  neglected  s i n c e t h e i r e f f e c t i s m e r e l y t o m o d i f y the r o t a t i o n a l c o n s t a n t s of molecule i n a p a r t i c u l a r v i b r a t i o n a l l e v e l  (33).  the  The h a m i l t o n i a n f o r  n u c l e a r r o t a t i o n i s then .  J  3  2  21  x  2  21  5 y  2  21  v  z  W i t h t h e d e f i n i t i o n o f the ' l a d d e r ' o p e r a t o r s  - 23 -  t h i s becomes  « " I <2Tx  +  2 1y -  +  tIT " z  +  Converting  t o cm - 1  4  (  \ <2T  +  2 I ~ - 2I->  ( J  +  2T>y ^  x  i  + J 2 )  (10)  u n i t s , and i n t r o d u c i n g t h e t h r e e  rotational  constants . A =  h  _ B =  2  8ir cl  h  „ C =  2  8u cl  z  h 8TT CI  - l cm 1  2  y  x  such t h a t A > B > C, we have  (ti2/hc) H = \  (B + C) J  2  + [A - | ( B + C ) ] J 2 + 1 (C - B) ( J 2 + J * )  (11)  U s i n g symmetric top b a s i s f u n c t i o n s , |JK>, w r i t t e n i n m o l e c u l e - f i x e d c o o r d i n a t e s , t h e m a t r i x e l e m e n t s o f t h e s e o p e r a t o r s a r e (36,37) <JK|J 2 |JK> = J ( J + l ) f t 2  <JK|J z |JK> = Kfi  <J,K + 1|J + |JK> -  (12)  *\J(J + 1) - K(K + 1)  h  The f i r s t two terms i n t h e h a m i l t o n i a n a r e d i a g o n a l , and  provided  •^•(C - B) i s s m a l l enough t o a l l o w t h e o f f - d i a g o n a l e l e m e n t s t o be n e g l e c t e d , g i v e t h e energy l e v e l s a s E/hc = [A - -|(B + C ) ] K 2 + | ( B + C) J (J + 1)  cm"1  (13)  - 24 -  The l a s t term i n t h e h a m i l t o n i a n , w h i c h i s o f f - d i a g o n a l i n K, g i v e s r i s e t o what i s c a l l e d t h e 'asymmetry d o u b l i n g ' o f l e v e l s w i t h K t 0 . The asymmetry d o u b l i n g has t h e same form as t h e £-type d o u b l i n g , being p r o p o r t i o n a l to [ J ( J + 1 ) ] :  however, t h e p r o p o r t i o n a l i t y  c o n s t a n t i s v e r y much l a r g e r , so t h a t t h e asymmetry d o u b l i n g o f a n asymmetric t o p m o l e c u l e i s an o r d e r o f magnitude g r e a t e r t h a n t h e A-type doubling of a l i n e a r  D.  molecule.  R o t a t i o n a l Constants  The t h r e e d i f f e r e n t moments o f i n e r t i a o f a n o n - l i n e a r symmetrical  t r i a t o m i c m o l e c u l e a r e c a l c u l a t e d i n terms o f t h e bond  l e n g t h and t h e i n t e r i o r a n g l e . e v a l u a t e d f o r1 2 C 3 2 S 2 .  *s  The t h r e e r o t a t i o n a l c o n s t a n t s w i l l be  s n o w n  i n  a  n o n - l i n e a r geometry i n F i g . 3 ,  where t h e C-S bond l e n g t h i s I and t h e S-C-S bond a n g l e i s 2<}>. The o r i e n t a t i o n o f t h e t h r e e p r i n c i p a l axes o f i n e r t i a i s d e f i n e d by t h e C,  symmetry and t h e i r o r i g i n by t h e c e n t e r o f mass c o n d i t i o n .  z(b) C  y(a) S  FIG. (3)  T  I s i n <t>  I c o s <|>-r  - 25 The moments of i n e r t i a I  I  = 2m.g &  sin cj>  2  2  2m m„  = -^-l  a  I  2m +  a I  are  mc.  s  = 2m £ c S c  where m  cosH  2  2m„ [1 - •= 2irig  2  +  = 12 amu and m  expressed  in A units,  cos *] 2  m  c  = 31.97207 amu.  c  W i t h the C-S bond l e n g t h  the r o t a t i o n a l c o n s t a n t s i n cm  °  -1  units  are  1.6684  A = Z.2 cos 2<f>, „ .  0.26363 £2 sin2(j)  c  0.26363  =  [1 - 0.84199 cos tj)J  I  2  2  To c a l c u l a t e  the s t r u c t u r e  of C S  2  (& and 2<J))  requires  knowledge of o n l y two of the t h r e e r o t a t i o n a l c o n s t a n t s because I  + I  = 1  x E.  y  .  z  Nuclear Spin S t a t i s t i c s The t o t a l e i g e n f u n c t i o n of a m o l e c u l e may be w r i t t e n as a  product of the e l e c t r o n i c , eigenfunctions.  total  e  v  r  s  v i b r a t i o n a l , r o t a t i o n a l and n u c l e a r  spin  - 26 The  t o t a l e i g e n f u n c t i o n can o n l y be symmetric o r  anti-  symmetric under t h e o p e r a t i o n s o f the p o i n t group c o r r e s p o n d i n g e x c h a n g i n g p a i r s of i d e n t i c a l n u c l e i .  I f t h e sum  of the  to  individual  n u c l e a r a n g u l a r momenta o f the i d e n t i c a l n u c l e i , c a l l e d I ( f i ) , i s i n t e g r a l , t h e t o t a l e i g e n f u n c t i o n has t o be symmetric w i t h r e s p e c t t o exchange o f i d e n t i c a l n u c l e i ( B o s e - E i n s t e i n s t a t i s t i c s ) ; a n d , i f I i s h a l f - i n t e g r a l , t h e t o t a l e i g e n f u n c t i o n has t o be a n t i s y m m e t r i c  with  r e s p e c t t o exchange of i d e n t i c a l n u c l e i ( F e r m i - D i r a c s t a t i s t i c s ) . 12  In  C 3 2 S 2 , t h e two i d e n t i c a l s u l f u r n u c l e i each have z e r o n u c l e a r s p i n so  that Bose-Einstein s t a t i s t i c s  apply.  F o r a l i n e a r m o l e c u l e (29) one need c o n s i d e r o n l y the symmetry p r o p e r t i e s of the e i g e n f u n c t i o n s under i n v e r s i o n o f a l l the p a r t i c l e s i n t h e c e n t r e o f mass, and r e f l e c t i o n i n any p l a n e p a s s i n g t h r o u g h the a x i s . Depending on whether t h e f u n c t i o n does or does not change s i g n under  these  o p e r a t i o n s i t i s c l a s s i f i e d as u o r g (ungerade o r g e r a d e ) , and - o r  +,  respectively.  I f t h e m o l e c u l e has a c e n t r e o f symmetry i t s H a m i l t o n i a n  i s a l s o i n v a r i a n t w i t h r e s p e c t to an i n t e r c h a n g e o f the c o o r d i n a t e s o f t h e n u c l e i , and a s t a t e i s c l a s s i f i e d as ' s y m m e t r i c ' o r ' a n t i s y m m e t r i c  with  r e s p e c t t o t h e n u c l e i ' depending on t h e b e h a v i o u r o f i t s e i g e n f u n c t i o n with respect to t h i s operation.  An i n t e r c h a n g e of the c o o r d i n a t e s o f  the  n u c l e i i s e q u i v a l e n t t o a change i n the s i g n o f the c o o r d i n a t e s o f a l l t h e particles  ( e l e c t r o n s and n u c l e i ) f o l l o w e d by a change i n s i g n o f the  c o o r d i n a t e s o f t h e e l e c t r o n s o n l y , i . e . an i n v e r s i o n f o l l o w e d by r e f l e c t i o n described above.  the  Hence i t f o l l o w s t h a t , i f the s t a t e i s gerade  and p o s i t i v e o r ungerade and n e g a t i v e , i t i s symmetric w i t h r e s p e c t t o the nuclei.  - 27 -  S i n c e B o s e - E i n s t e i n s t a t i s t i c s a p p l y f o r t h e ground v i b r a t i o n a l s t a t e o f C S 2 (where \\>e,  and ^ g a r e s y m m e t r i c ) , t h e r o t a t i o n a l  f u n c t i o n must a l s o be s y m m e t r i c . related and  eigen-  Now a r o t a t i o n a l e i g e n f u n c t i o n |j£> i s  t o t h e A s s o c i a t e d Legendre p o l y n o m i a l  c h a r a c t e r i s e d by t h e t o t a l  component a n g u l a r momenta J and I r e s p e c t i v e l y ,  and i t s b e h a v i o u r under  the i n v e r s i o n o p e r a t i o n i s  i|J£>  = ( - 1 ) J | J , -£>  (2)  T h e r e f o r e , s i n c e % = 0, o n l y even J f u n c t i o n s a r e s y m m e t r i c , and a r e p e r m i t t e d as r o t a t i o n a l e i g e n f u n c t i o n s . (including  A more d e t a i l e d  consideration  t h e e f f e c t o f t h e r e f l e c t i o n o p e r a t i o n ) shows t h a t when  o n l y one member o f t h e i s permitted.  £-type d o u b l e t  > 0,  f o r each J v a l u e i s symmetric and  The o t h e r r o t a t i o n a l l e v e l s a r e a b s e n t .  These p a t t e r n s a r e  i l l u s t r a t e d i n F i g . 4.  The n u c l e a r s p i n s t a t i s t i c s o f t h e n o n - l i n e a r C S 2 m o l e c u l e produce s i m i l a r r o t a t i o n a l p a t t e r n s . properties classify  The d e r i v a t i o n  o f t h e symmetry  (28,29,38) i s more d i f f i c u l t , f o r i t t u r n s o u t t h a t one must  t h e r o t a t i o n a l e i g e n f u n c t i o n s under t h e o p e r a t i o n s o f t h e  r o t a t i o n a l sub-group C 2 :  according  to the P a u l i P r i n c i p l e o n l y those  s p e c i e s c o r r e l a t i n g w i t h t h e t o t a l l y symmetric r e p r e s e n t a t i o n o f C 2 ( i . e . Ax and A 2 o f C 2 v ^ a r e p e r m i s s i b l e as r o t a t i o n a l s p e c i e s .  Thus,  as shown i n F i g . 4 , a l t e r n a t e J l e v e l s a r e absent i n t h e K = 0 r o t a t i o n a l s t a c k s , and one asymmetry d o u b l i n g component f o r each J v a l u e o f t h e K > 0 stacks i s absent.  - 28  J  -  Ai(Bi)  +  _.  -  2u(A2)  3  •+  1  •- +  0  + +  (K=0  u of  /IN  ?U(B2)  +  2  u  (  A  2>  Bl(Ai) Ai(Bx)  .  K=l  -+  2-CBi)  •++  ^c  A l  )  Eu(B2)  Z  8  C B  1>  B2(A2)  /IN  +  E~(A ) 2  £J(B2) (  +  of  "  E-(A2)  E ; B  "  0  -  Al(Bi) + _ Bl(Ai)  B2)  1  +  3  E >  2  2  +—  )  )  n u  (K=0  FIG.  (4)  of  Ax)  (K=l o f  Ax)  Asymmetric top s p e c i e s (+-) and r o v i b r o n i c ( o v e r - a l l ) s p e c i e s of the l o w e s t r o t a t i o n a l l e v e l s of a l i n e a r m o l e c u l e i n E g' _and u u I I v i b r o n i c s t a t e s w i t h the c o r r e s p o n d i n g s p e c i e s of a E,, u bent m o l e c u l e i n b r a c k e t s . A l l r o t a t i o n a l l e v e l s of B s p e c i e s ( d o t t e d l i n e s ) a r e absent i n 1 2 C 3 2 S 2 and f o r K=l o f A 2 d o t t e d and s o l i d l i n e s are exchanged. P a r a l l e l and p e r p e n d i c u l a r l y p o l a r i z e d t r a n s i t i o n s between r o t a t i o n a l s t a c k s a r e shown by s o l i d and d o t t e d a r r o w s , r e s p e c t i v e l y . For a l l t r a n s i t i o n s the r o v i b r o n i c s e l e c t i o n r u l e i s E + ( A i )-<-»-E~(A2). U  8  - 29 F.  S e l e c t i o n Rules  The  i n t e n s i t y o f an a b s o r p t i o n o r e m i s s i o n t r a n s i t i o n f o r  a gaseous m o l e c u l e i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e t r a n s i t i o n moment i n t e g r a l  R  where:  , ,, evr , e v r "  = <eVr'11.F |u_|e"v"r"> 1  v  (1) '  t h e symbols ' e v r ' r e f e r t o t h e e l e c t r o n i c , v i b r a t i o n a l , and  r o t a t i o n a l eigenfunctions o f the molecule;  t h e p r i m e and d o u b l e p r i m e  s u p e r s c r i p t s r e f e r t o t h e upper and l o w e r s t a t e s ; and, u  i s the tran-  s i t i o n moment o p e r a t o r r e f e r r e d t o space f i x e d c o o r d i n a t e s  (F) w h i c h ,  f o r an e l e c t r o n i c t r a n s i t i o n , i s t h e e l e c t r i c d i p o l e moment o p e r a t o r o f t h e molecule.  The e l e c t r i c d i p o l e moment o p e r a t o r i s a f u n c t i o n o f i n t e r -  n u c l e a r d i s t a n c e s , and i s t h e r e f o r e u s u a l l y expanded a s a T a y l o r i n t h e Normal c o o r d i n a t e s o f t h e m o l e c u l e .  series  The t r a n s i t i o n moment  i n t e g r a l then becomes  R  , „ = <e'v,r*|yn + \ ( % \ evr',evr" '_0 k \ ^k/  Q, |e"v"r"> ,  (2)  where t h e *F* s u b s c r i p t and t h e h i g h e r terms a r e o m i t t e d .  By n e g l e c t i n g t h e s m a l l i n f l u e n c e o f t h e n u c l e a r k i n e t i c energy upon t h e e l e c t r o n i c e i g e n f u n c t i o n (Born-Oppenheimer a p p r o x i m a t i o n ) , and  the (small) i n t e r a c t i o n of r o t a t i o n w i t h other  internal  m o t i o n s , we have  |evr> = |e>|v>|r>  (3)  - 30 Thus, equation  (2) becomes  (4)  The second term i s t h e t r a n s i t i o n moment o f a  vibrational-rotational  ( i n f r a r e d ) t r a n s i t i o n and w i l l n o t be d i s c u s s e d f u r t h e r . t e r m , however, i s t h e t r a n s i t i o n moment o f a n e l e c t r o n i c t i o n a l (microwave) t r a n s i t i o n .  I n a pure r o t a t i o n a l  The f i r s t o r pure  rota-  t r a n s i t i o n , e ' = e",  so t h a t t h e i n n e r i n t e g r a l i s s i m p l y t h e permanent d i p o l e moment o f t h e molecule.  I n an e l e c t r o n i c  t r a n s i t i o n , t h i s i n t e g r a l i s the e l e c t r o n i c  t r a n s i t i o n moment o p e r a t o r ; t h e o t h e r i n t e g r a l i n t h e f i r s t <v'|v">, i s t h e Franck-Condon v i b r a t i o n a l  overlap  The components o f t h e e l e c t r o n i c  integral.  t r a n s i t i o n moment e x p r e s s e d  i n a m o l e c u l e - f i x e d c o o r d i n a t e system (g) a r e r e l a t e d referred to space-fixed coordinates  ^F=  term,  t o t h e components  (F) by t h e d i r e c t i o n c o s i n e s , X_  I g=x,y.z  C5)  U s i n g symmetric t o p e i g e n f u n c t i o n s r e f e r r e d t o t h e m o l e c u l e - f i x e d c o o r d i n a t e s , we have  e v r ' e v r .II = <v'|v"> I <J'K'|A F g y e , e , ( >  = <v'|v"> £ u g  <J'K'  |j"K">  "K">  (6)  - 31 -  The d e t e r m i n a t i o n o f r o t a t i o n a l l i n e s t r e n g t h s and s e l e c t i o n r u l e s i s done by c a l c u l a t i n g t h e ' d i r e c t i o n c o s i n e m a t r i x e l e m e n t ' , t h e i n t e g r a l w i t h i n t h e sum. The r e l a t i v e i n t e n s i t i e s o f r o t a t i o n a l l i n e s i n a v i b r a t i o n a l band o f an e l e c t r o n i c  transition.are  g i v e n by t h e s q u a r e o f t h e  l i n e s t r e n g t h ( e q u a t i o n (6) w i t h t h e v i b r a t i o n a l o v e r l a p i n t e g r a l o m i t t e d ) summed o v e r t h e s p a c e - f i x e d components, J and m u l t i p l i e d by t h e B o l t z maim p o p u l a t i o n f a c t o r o f t h e l o w e r r o t a t i o n a l s t a t e .  The s e l e c t i o n  r u l e s on J and K a r e d e t e r m i n e d from t h e n o n - v a n i s h i n g  direction  m a t r i x elements (40) as summarized below f o r a l i n e a r  cosine  molecule.  Non-zero component o f e l e c t r o n i c t r a n s i t i o n moment  ^e'e"  z  P  a r a  liei  polarization  Selection  AK=K'-K"=0  Rules  AJ=±1 I f K=0 AJ=0,±1 i f K+0  AK=±1  Ve",x or  perpendicular  AJ=0,±1  polarization  Ve",y  The a x i s l a b e l l i n g must be changed f o r t h e n o n - l i n e a r m o l e c u l e b u t t h e selection rules are unaffected.  - 32 -  The p r o j e c t i o n projection  of J on t h e a x i s o f a - l i n e a r m o l e c u l e i s P;  of J on the p r i n c i p a l a x i s o f l e a s t i n e r t i a  l i n e a r m o l e c u l e i s K.  the  (a a x i s ) o f a n o n -  A change i n geometry o f the m o l e c u l e does not  a f f e c t the s e l e c t i o n r u l e s a l t h o u g h  the o r i g i n of the a n g u l a r momenta  i s d i f f e r e n t i n t h e b e n t and l i n e a r m o l e c u l e s .  With p a r a l l e l - p o l a r i -  z a t i o n , a K* = 0 - K" = 0 band (E - E) has a P b r a n c h (AJ = -1)  and  an R b r a n c h (AJ = +1) w i t h l i n e s t r e n g t h s t h a t i n c r e a s e w i t h J ; o t h e r p a r a l l e l b a n d s , K =f= 0 (II-II, A-A,  e t c . ) have Q b r a n c h e s (AJ = 0) t h a t  a r e weak i n comparison t o the R and P b r a n c h e s and whose l i n e decrease w i t h J . t h e P,Q,  W i t h p e r p e n d i c u l a r p o l a r i z a t i o n (E-II, n-E,  strengths etc.),  and R b r a n c h e s a r e o f comparable i n t e n s i t y , w i t h l i n e  increasing  strengths  with J .  The  electronic transition vibrational selection rules for a  l i n e a r molecule  are a l r e a d y p a r t i a l l y i n c l u d e d i n the  rotational  s e l e c t i o n r u l e s s i n c e P i s e q u i v a l e n t t o the v i b r a t i o n a l a n g u l a r momentum  I of a l i n e a r m o l e c u l e .  The  only other v i b r a t i o n a l s e l e c t i o n  rule  a r i s e s from the r e q u i r e m e n t t h a t t h e Franck-Condon o v e r l a p i n t e g r a l , <v'|v">, be t o t a l l y s y m m e t r i c . as v 2  (^JJ)  ANC  *  V3  s e l e c t i o n r u l e Av  (0^)  k  T h u s , any a n t i s y m m e t r i c  v i b r a t i o n , such  o f a l i n e a r t r i a t o m i c m o l e c u l e must f o l l o w  the  = 0, ±2, ±4, e t c . s i n c e e i g e n f u n c t i o n s b e l o n g i n g  to  even v i b r a t i o n a l quantum numbers c o n t a i n a t o t a l l y symmetric component. I n a n o n - l i n e a r m o l e c u l e , the d o u b l y d e g e n e r a t e v i b r a t i o n v t o t a l l y symmetric v i b r a t i o n and one  o f the r o t a t i o n s  r e s t r i c t i o n no l o n g e r a p p l i e s and a l l v a l u e s o f Av  2  becomes a  so t h a t t h i s are  allowed.  - 33 -  One f u r t h e r  p o i n t t o n o t e i s t h a t t h e Franck-Condon o v e r l a p  i n t e g r a l , w h i c h d e t e r m i n e s t h e e x t e n t and i n t e n s i t y d i s t r i b u t i o n o f a v i b r a t i o n a l p r o g r e s s i o n , depends s t r o n g l y on t h e change i n geometry o f the m o l e c u l e upon e l e c t r o n i c e x c i t a t i o n . excited  The v i b r a t i o n  predominantly  i s t h a t one w h i c h most n e a r l y d e s c r i b e s t h e shape change upon  excitation.  I f t h e shape change i s l a r g e enough, t h e maximum i n t e n s i t y  i n t h e v i b r a t i o n a l p r o g r e s s i o n may l i e w e l l away from t h e e l e c t r o n i c o r i g i n band.  - 34  IV.  The  low  ANALYSIS OF  THE  -  V AND  T SYSTEMS  r e s o l u t i o n p r e s s u r e broadened s p e c t r u m of CS 2  o  in  o  the r e g i o n 2900A to 3400A i s shown as F i g . 5 .  At l e a s t t h r e e  t r a n s i t i o n s are responsible f o r absorption i n t h i s r e g i o n .  electronic  The  higher o  v i b r a t i o n a l members of the R and S systems l i e i n the r e g i o n 3400-3300A are v e r y weak ( e m a x % 2 l i t e r s / m o l e cm ( 4 1 ) ) . The s t r o n g e s t a b s o r p t i o n o f C Szo i n the n e a r u l t r a v i o l e t i s due mole cm 1  B2  (41)).  *•  tion.  max  % 40  However, i t i s s t i l l a weak system compared t o  liters/ the  system a t 2100A, i n d i c a t i n g t h a t i t i s a ' f o r b i d d e n ' t r a n s i U n l i k e the R and  p a t t e r n s and any  to the V system (e J  but  S systems, which e x h i b i t r e g u l a r  simple r o t a t i o n a l s t r u c t u r e ,  the V system does not  r e a d i l y a p p a r e n t v i b r a t i o n a l p r o g r e s s i o n s , n o r i s the  structure  simple.  have  rotational  Under h i g h r e s o l u t i o n , the r o t a t i o n a l s t r u c t u r e  the v i b r a t i o n a l bands o f the V system i s dense and obvious simple b r a n c h - l i k e s t r u c t u r e . i s not  vibrational  irregular with  of no  Without temperature s t u d i e s , i t  even p o s s i b l e t o e s t a b l i s h the ground s t a t e v i b r a t i o n a l numbering  of the V b a n d s .  A.  Temperature S t u d i e s and  P o l a r i z a t i o n of the V System  P a r t o f the c o m p l e x i t y o f the V system i s because bands a r i s i n g f r o m v a r i o u s v i b r a t i o n a l l e v e l s of the ground s t a t e a r e p r e s e n t , o f t e n with very considerable i n t e n s i t y . s p e c t r u m of CS 2 100° and  200°C:  between 2900 and  To  i d e n t i f y t h e s e 'hot' bands  3500A was  the bands found t o be  the Cary s p e c t r o p h o t o m e t e r t r a c i n g  the  photographed a t - 7 8 ° , 2 3 ° ,  'hot'  (Fig. 5 ) .  are marked w i t h s h a d i n g  on  E /cirri 32COO  34000 ' I  |  2900  :  1  I  1  3000  30000  I  «  1  I  :  1  1  3200  I  1  1  '  o  F I G . (5)  Wavelength / A  Room temperature pressure-broadened gas phase a b s o r p t i o n spectrum o f 1 2 C 3 2 S 2 i n the r e g i o n 2900A t o 3A00A r e c o r d e d by a Cary 14 s p e c t r o p h o t o m e t e r . The shaded a r e a s under the a b s o r p t i o n c u r v e i n d i c a t e a b s o r p t i o n from e x c i t e d v i b r a t i o n a l l e v e l s of t h e X 1 E + ground s t a t e of C S 2 .  !  3400  - 36 -  A ' h o t ' band may be r e c o g n i s e d  by t h e f a c t t h a t i t s i n t e n s i t y  i n c r e a s e s on r a i s i n g t h e t e m p e r a t u r e o f t h e g a s .  The i n t e n s i t y o f an  a l l o w e d v i b r a t i o n a l band i n a b s o r p t i o n i s p r o p o r t i o n a l t o t h e square o f the Franck-Condon o v e r l a p i n t e g r a l t i m e s t h e number o f m o l e c u l e s i n t h e l o w e r v i b r a t i o n a l l e v e l , w h i c h i s g i v e n by t h e Boltzmann d i s t r i b u t i o n N  - exp (-E v hc/RT) In equation  (1)  ( 1 ) , N^, t h e number o f m o l e c u l e s i n l e v e l v , a t energy E ^  r e l a t i v e t o t h e z e r o - p o i n t l e v e l , and a t t e m p e r a t u r e T} i s g i v e n a s a f r a c t i o n o f t h e number, N Q , i n t h e z e r o - p o i n t l e v e l . temperature T i s increased  Obviously  as t h e  i n c r e a s e s r e l a t i v e t o N Q , and NQ a c t u a l l y  d e c r e a s e s because o f t h e number o f m o l e c u l e s promoted t o t h e e x c i t e d vibrational levels.  'Hot' bands a r e t h e r e f o r e r e a d i l y i d e n t i f i e d by  comparing s p e c t r a t a k e n a t d i f f e r e n t t e m p e r a t u r e s , and a rough  estimate  o f t h e energy o f t h e l o w e r l e v e l may be o b t a i n e d by comparing s p e c t r a " taken a t s e v e r a l d i f f e r e n t  temperatures.  As soon a s t h e l a r g e number o f ' h o t ' bands i n t h e V system had been r e c o g n i s e d , a s y s t e m a t i c s e a r c h was made f o r ground s t a t e v i b r a t i o n a l c o m b i n a t i o n d i f f e r e n c e s between bands w i t h common upper s t a t e s b u t d i f f e r e n t lower s t a t e s . 0 l  1  0 (respectively  v a l s corresponding  O n l y t h e i n t e r v a l s 0 2° 0 - 0 0° 0 and 0 3 * 0 -  802 and 811 cm ') c o u l d be i d e n t i f i e d , and no i n t e r -  t o 1 0° 0 - 0 0° 0 ( t h e ground s t a t e s t r e t c h i n g  t i o n , 658 c m - 1 ) were found d e s p i t e c o n s i d e r a b l e e f f o r t .  vibra-  As i s shown l a t e r ,  the shape change i n t h e V system i s a l m o s t p u r e l y a change i n t h e bond a n g l e on e x c i t a t i o n , w h i c h i s c o n s i s t e n t w i t h t h e o b s e r v e d ground s t a t e intervals.  - 37 -  The o b s e r v e d ground s t a t e i n t e r v a l s o n l y o c c u r a t t h e l o n g w a v e l e n g t h end o f t h e s p e c t r u m , where t h e ' h o t ' bands a r e v e r y i n t e n s e . However g i v e n t h e i d e n t i f i c a t i o n of t h e l o w e r s t a t e s of t h e s e b a n d s , t o g e t h e r w i t h t h e known l o w e r s t a t e assignments of t h e R s y s t e m , i t i s p o s s i b l e t o compare t h e degree o f t e m p e r a t u r e s e n s i t i v i t y o f a l l t h e bands o f t h e V system a g a i n s t t h e known b a n d s , and hence r o u g h l y e s t a b l i s h the energy o f t h e i r l o w e r s t a t e s r e l a t i v e t o the z e r o - p o i n t l e v e l .  It is  a l s o p o s s i b l e t o make a l o w e r s t a t e v i b r a t i o n a l quantum number a s s i g n m e n t , b e c a u s e , assuming t h a t o n l y t h e b e n d i n g v i b r a t i o n i s i n v o l v e d , the d i f f e r ences i n t e m p e r a t u r e - s e n s i t i v i t y a r e s u f f i c i e n t l y l a r g e t o d i s t i n g u i s h bands w i t h v'2' = 0, 1 and 2.  I t i s p r o b a b l e t h a t 'sequence b a n d s ' i n t h e  symmetric s t r e t c h i n g v i b r a t i o n , o f the t y p e l v O - l v O , may  o c c u r on t h e  h i g h e r t e m p e r a t u r e p l a t e s , but t h e y do n o t have such f a v o u r a b l e F r a n c k Condon f a c t o r s as t h e ' h o t ' bands i n the b e n d i n g v i b r a t i o n , and none o f them have been r e c o g n i s e d . The o b s e r v e d c o m b i n a t i o n d i f f e r e n c e s a l s o d e t e r m i n e t h a t t h e transition i s parallel-polarized.  The ground s t a t e i n t e r v a l 0 2° 0 -  0 0° 0 o c c u r s between two v e r y s t r o n g b a n d s , the ' c o l d ' band a t 3236A (10V i n our n o t a t i o n ) and t h e ' h o t ' band a t 3322A ( I V ) . I f the t r a n s i t i o n was p e r p e n d i c u l a r l y - p o l a r i z e d  (K' -  = ± 1 ) , the p r e s e n c e o f t h e  ' c o l d ' band would show t h a t t K e upper s t a t e had K' = 1; t h e n t h e ' h o t '  - 38 -  band would be s p l i t by t h e c h a r a c t e r i s t i c 10 cm t h e 0 2° 0 and 0 2 2 0 components o f t h e splitting  1  s e p a r a t i o n between  = 2 level.  However no such  (which would be r e a d i l y o b s e r v a b l e ) a p p e a r s , so t h a t t h e  transition i sparallel-polarized.  T h i s d e d u c t i o n i s c o n f i r m e d by t h e  absence o f t h e c o r r e s p o n d i n g s p l i t t i n g s i n o t h e r i n s t a n c e s where they would be e x p e c t e d on t h e b a s i s o f p e r p e n d i c u l a r p o l a r i z a t i o n , and a l s o (see below) by t h e r e s u l t s o f t h e r o t a t i o n a l a n a l y s i s .  B.  Rotational Analysis  The c o m p l e x i t y o f t h e V bands makes r o t a t i o n a l a n a l y s i s o f most o f t h e bands q u i t e i m p o s s i b l e .  However two o b v i o u s p a i r s o f bands  w i t h common upper s t a t e s (as i n d i c a t e d by t h e l o w e r s t a t e v i b r a t i o n a l c o m b i n a t i o n d i f f e r e n c e s d e s c r i b e d above) were s u f f i c i e n t l y t o i n v i t e an a t t e m p t a t a n a l y s i s .  unperturbed  These a r e t h e K' = 0 p a i r a t 3236&  and 3322A, and t h e K* = 1 p a i r a t 3275A and 3365A.  The 3365A band h a s  n o t p r e v i o u s l y been r e c o g n i s e d as b e l o n g i n g t o t h e V s y s t e m .  R o t a t i o n a l a n a l y s e s o f t h e s e bands were p o s s i b l e f o r t h r e e reasons.  F i r s t , t h e e x t r e m e l y h i g h r e s o l u t i o n a t t a i n e d on o u r p l a t e s  ( t h e measured f u l l w i d t h a t h a l f maximum o f a n unblended l i n e b e i n g <0.05 cm 1 ) reduced  t h e b l e n d i n g o f l i n e s t o a minimum.  The second  f a c t o r w h i c h h e l p e d g r e a t l y was t h e c o o l i n g o f t h e gas t o -78°C:  this  - 39 -  has  the e f f e c t o f r e d u c i n g the D o p p l e r w i d t h o f the l i n e s and  removing  t h e a b s o r p t i o n l i n e s a r i s i n g from h i g h e r r o t a t i o n a l l e v e l s of the lower s t a t e s .  F i n a l l y , a recent high r e s o l u t i o n i n f r a r e d analysis  (35)  o f the b e n d i n g v i b r a t i o n l e v e l s of the ground s t a t e has g i v e n a c c u r a t e c o n s t a n t s from w h i c h i t was  possible  f o r the a n a l y s i s of the e l e c t r o n i c  to c a l c u l a t e the data  (Appendix I )  spectra.  I n o r d e r t o check t h e a c c u r a c y o f the i n f r a r e d  combination  o  d i f f e r e n c e s , an a n a l y s i s of a s i m p l e E - E band a t 3326A (Kleman's band) was  undertaken.  17U  I n t h i s b a n d , w h i c h was measured f r o m the same  p l a t e as t h e 3322A band ( I V ) , the r o t a t i o n a l c o m b i n a t i o n  differences  of 18 b r a n c h l i n e s d e v i a t e d by no more t h a n ± 0.006 cm - 1  from t h e i r  calculated values.  Hence, t h e i n f r a r e d c o n s t a n t s g i v e c a l c u l a t e d com-  b i n a t i o n d i f f e r e n c e s as l e a s t as a c c u r a t e as the measured r e l a t i v e l i n e positions  ( ± 0.01  cm  1  ) .  The w e l l known method o f c o m b i n a t i o n d i f f e r e n c e s used t o a n a l y s e t h e b a n d s .  (42)  was  I f t h e p o s i t i o n o f a r o t a t i o n a l l e v e l above  t h e v i b r a t i o n a l o r i g i n of t h e l e v e l i s c a l l e d F ( J ) , the r o t a t i o n a l l e v e l e x p r e s s i o n f o r a v i b r a t i o n a l l e v e l o f t h e ground s t a t e  (see  section  I I I . C ( i ) ) becomes  F ( J ) = T 0 + B [ J ( J + 1) - I } 2  - D J2(J+D  2  + |(-1)V(J+D  (1)  - 40 where Tg i s t h e e l e c t r o n i c term v a l u e , B and D a r e t h e r o t a t i o n a l and c e n t r i f u g a l d i s t o r t i o n constants doubling constant.  r e s p e c t i v e l y , a n d q i s t h e £-type  T h e r e f o r e , the r o t a t i o n a l l i n e p o s i t i o n s i n a  p a r a l l e l band a r e g i v e n by t h e e x p r e s s i o n s  R ( J ) = F 1 ( J + 1) - F " ( J )  = v 0 + 2B 1 + (3B 1 - B " ) J + (B* - B " ) J 2  (2)  P ( J ) = F ' ( J - 1) - F " ( J ) = VQ - ( B ' + B " ) J + ( B ' - B " ) J 2  (3)  where v Q i s t h e band o r i g i n , and t h e e f f e c t o f t h e D and q terms has been neglected.  However c o m p l i c a t e d  t h e r o t a t i o n a l s t r u c t u r e s o f t h e upper  s t a t e may b e , an a n a l y s i s i s i n p r i n c i p l e a l w a y s p o s s i b l e i f t h e R and P b r a n c h l i n e s t e r m i n a t i n g on t h e same upper l e v e l c a n be i d e n t i f i e d . T h e i r s e p a r a t i o n , known as t h e second c o m b i n a t i o n  difference A2F"(J),  i s g i v e n by  A 2 F " ( J ) = F " ( J + 1) - F " ( J - 1) = R ( J - 1) - P ( J + 1)  .  (4)  I n v i e w o f t h e f a n t a s t i c c o m p l e x i t y o f the s p e c t r u m , we have o n l y a t t e m p t e d t o a n a l y s e p a i r s o f bands w i t h a common upper l e v e l b u t d i f f e r e n t l o w e r v i b r a t i o n a l l e v e l s , because a d d i t i o n a l  combination  d i f f e r e n c e s become a v a i l a b l e : the same J i n the two  t h e s e are the. s e p a r a t i o n  of l e v e l s w i t h  lower l e v e l s , w h i c h can of c o u r s e be  c a l c u l a t e d from the h i g h r e s o l u t i o n i n f r a r e d d a t a (35).  accurately  There i s a  s m a l l d i f f e r e n c e of B w i t h v, r e s u l t i n g from a n h a r m o n i c i t y and e f f e c t s , w h i c h i s s u f f i c i e n t to make, a d i f f e r e n c e o f ^ 0.5 the v i b r a t i o n a l separations bands of i n t e r e s t .  of l e v e l s w i t h J " = 0 and  T h i s t u r n e d out  cm  Coriolis  -1  between  J " = 40 i n the  t o be p a r t i c u l a r l y u s e f u l because  the J dependence of the v i b r a t i o i t a l c o m b i n a t i o n d i f f e r e n c e s a l l o w e d assignment ( f o r J > 20)  The  J  to w i t h i n ± 2 quanta.  r o t a t i o n a l a n a l y s i s was  s e a r c h f o r v i b r a t i o n a l and  c a r r i e d out by a t r i a l and  r o t a t i o n a l combination d i f f e r e n c e s . o  h e l p w i t h the a n a l y s i s of the 3275A and  error To  o  3365A bands we w r o t e a computer  programme t h a t gave a l l the p o s s i b l e s e t s of f o u r l i n e s (an R and l i n e from each band) w h i c h were i n agreement w i t h the ground combination d i f f e r e n c e s . t o agree w i t h i n 0.04  a  cm . -1  The  a P  state  f o u r c a l c u l a t e d term v a l u e s were r e q u i r e d  V - e r e were of c o u r s e l a r g e numbers of  p o s s i b l e s e t s of l i n e s p i c k e d  out by the computer, b u t , by c a r e f u l l y  c h e c k i n g e v e r y p o s s i b l e s e t to ensure t h a t the r e l a t i v e i n t e n s i t i e s of the f o u r l i n e s , were r o u g h l y the same, we managed t o e l i m i n a t e so many p o s s i b l e s e t s t h a t , i n v i r t u a l l y a l l c a s e s , o n l y one f o r each  rotational line.  T a b l e s IV and V; The found i n any  The  r o t a t i o n a l assignments are g i v e n  the computer programme i s g i v e n  r o t a t i o n a l a n a l y s e s show the o f the f o u r bands.  assignment remained  as Appendix I I .  following.  No Q b r a n c h i s  Hence, the two bands o f F i g . 6  2 - £ t y p e , and t h o s e o f F i g . 7 are II - II t y p e .  as  This  are  r e s u l t confirms  2973420 cm-'  29712.45 cm-  1  - 44 TABLE IV  Rotational Line Assignments of the 3236 A (10V) and _l 12 32 3322 A (IV) Bands of C S (cm ). 2  ooo,K' J  II  0  = 0 - 00°0  (10V)  P(J+2)  R(J) 30 902.24  ooo,K'=  R(J)  30 901.62  30 100.42*  0 - 02°0 (IV) PCJ+2) 30 099.77  2  901.23* 901.62 906.48 908.38  899.71 900.09 904.96 906.84  099.38 099.77 104.63 106.54*  097.83 098.24 103.08* 105.03  4  900.93 902.10 911.29  898.53 899.71 908.87*  099.07 100.27 109.40  096.66 097.83 107.01  6  900.78 903.02 903.51 904.19 904.39 906.53*  897.51 899.71 900.26 900.93 901.12 903.29*  098.93 101.14 101.63 102.37 102.56 104.66  095.63 097.83 098.36 099.07 099.25 101.40  8  903.84 904.96 907.26 908.12  899.71 900.78 903.14 903.96  102.01 103.03 105.41 106.26  097.83 098.93 101.28 102.10  10  898.12 905.01* 906.15 908.08  893.10 899.96* 901.12 903.02  096.24 103.10* 104.27* 106.19  091.21 098.09 099.25 101.14  12  908.60 908.88* 911.13 911.36*  902.71 902.97 905.22* 905.46  106.70 106.93 109.19 109.45*  100.79 101.04 103.30 103.57  14  900.26 9.05.72 905.97 909.30  893.49 898.95 899.20 902.57  098.36 103.77 104.02 107.37  091.57 097.00* 097.25 100.61  •  - 45 TABLE  16  30 905.24 905.88 906.33 909.30 910.23 910.48  30 897.58 ' 898.24 898.69 901.67* 902.57 902.82  IV ( C o n t i n u e d ) 30 103.30 103.92 104.39 107.34 108.28 108.53*  30 095.63 096.24 096.70* 099.68 100.61 100.87*  18  907.63 907.76* 908.21 909.52  899.08 899.25 899.71 900.93  105.64* 105.81 106.24* 107.50  097.10 097.25 097.68 098.93  20  907.26 911.01 911.87  897.88 901.62 902.53*  105.25 109.03 109.87  095.86* 099.62 100.45  22  908.93 909.62 910.91 911.59  898.69 899.40 900.65 901.32  106.93 107.60 108.90 109.50  096.60* 097.33 098.61 099.25  24  909.88 910.07 913.07  898.79 898.95 901.93  107.85 108.02* 111.00  096.66 096.88 099.85  26  910.70 910.80 911.76 913.91 915.00  898.69 898.79 899.81 901.93 903.02  108.58* 108.72 109.64 111.80 112.93  096.53 096.66* 097.60 099.77 100.89*  28  910.53* 914.26 914.66  897.58* 901.40* 901.77  108.38* 112.14 112.54  095.46 099.25 099.62  30  913.66 916.11  899.92 902.37  111.46* 113.94  097.68 100.10  32  912.73* 914.94*  898.12 900.28*  110.50 112.68  095.86* 098.01  34  914.23*  898.75*  111.92  096.40  36  916.31  899.95*  113.97*  097.59*  38  916.31 917.04  899.08 899.80  113.94 114.66  096.60* 097.35*  *  measured from h i g h r e s o l u t i o n  print  -  46  -  TABLE V  R o t a t i o n a l L i n e Assignments o f t h e 3 2 7 5 A ( 6 V ) and 12 32 -1 3 3 6 5 A Bands o f C S2(cm ) . 0  000,K'  R(J)  = 1 -  01*0  (6V)  P(J+2)  000,K'=  03*0  R(J)  1  30 5 2 9 . 9 6  30 5 2 8 . 9 0  29 7 1 9 . 1 0  2  30.80  29.30  19.93  3  1 -  (3365  P ( J + 2)  29  717.98 18.40  30.56  28.63  19.66  17.74  29.96  28.02  19.10  17.09  30.96  28.55  20.06  17.63  6  32.24  29.00  21.34  18.07  7  32.64  28.90  21.74  17.98  33.07  28.93  22.16  17.98  33.69  29.50  22.75  18.58  33.58 33.79  29.00 29.19  22.67 22.87  18.07 18.26  4 5  10  '  34.12  29.09  23.17  18.12  34.49  29.43  23.56  18.48  11  34.49 34.89 35.55  29.00 29.43 30.09  23.56 23.97 24.60  18.07 18.48 19.10  12  33.85  27.96  22.87  16.97  34.81  28.93  23.88  17.98  35.41  29.50  24.48  18.58  13  35.79  29.43  24.85  18.48  14  35.72  28.93  24.76  17.98  36.43  29.67  25.47  18.67  36.99  30.22  26.00  19.21  15  35.55  28.35  24.60  17.36  35.95  28.72  25.00  17.74  A band)  - 47 -  TABLE V  (Continued)  16  30 536.81 38.16  30 529.14 30.47  29 725.82 27.14  29 718.12 19.49  17  35.23 36.75 38.90 39.48  27.16 28.63 30.80 31.40  24.24 25.76 27.88 28.46  16.12 17.63 19.77 20.35  18  34.55 34.66 36.54 37.11  26.02 26.13 28.02 28.55  23.51 23.65 25.51 26.09  14.97 15.11 16.97 17.51  34.66 36.00 38.16  25.71 27.07 29.19  23.65 25.00 27.14  14.64 16.02 18.19  37.44  28.02  26.40  16.97  34.00 35.26 35.41 36.43  24.17 25.43 25.54 26.58  22.99 24.24 24.36 25.39  13.13 14.34 14.51 15.52  37.78 38.22 38.50 39.36  27.51 27.96 28.23 29.09  26.70 27.14 27.39 28.27  16.42 16.85 17.09 17.98  36.43  25.71  25.39  14.64  39.89 41.73 42.19 42.67  28.72 30.56 31.04 31.52  28.78 30.64 31.08 31.56  17.60 19.43 19.93 20.35  37.69 38.16 38.50 39.89  26.11 26.58 26.91 28.27  26.56 27.09 27.39 28.78  14.97 15.48 15.79 17.14  39.29 39.48 41.78 42.70  27.27 27.43 29.74 30.68  28.16 28.34 30.64 31.56  16.12 16.29 18.58 19.49  42.33 42.94  29.84 30.47  31.18 31.80  18.67 19.31  41.55  28.63  30.38  17.43  - 48 TABLE V  (Continued)  29  30 541.14  30 527.82  29 529.98  29 516.62  30  40.59 43.49  26.79 29.74  29.35 32.27  15.52 18.48  31  41.85 44.28  27.64 30.09  30.64 33.08  16.42 18.82  32  40.75 45.48  26.11 30.80  29.48 34.20  14.80 19.49  33  43.44 44.91  28.40 29.84  32.22 33.66  17.09 18.58  - 49 TABLE V I ( a )  A s s i g n e d R o t a t i o n a l Term V a l u e s o f t h e (000),K 1 = 0 1 1 12 32 -1 * Level o f the B 2 ( Au) V State o f C S 2 (cm ) . Term V a l u e s  J* 1 30 902.27 3 901.88 5 903.11 7 905.37 9 911.72 11 910.13 13 925.62 15 923.18 934.91 17 19 944.92 21 953.08 23 964.14 25 975.36 27 987.27 999.10 29 31 31 015.11 33 027.93 35 044.03 061.61 37 077.97 39  30 902.28 904.29 907.58 912.80 917.00 925.88 928.62 935.55 945.07 956.84 964.83 975.54 988.33 31 002.86 017.56 030.11  30 907.13 913.46 908.09 915.14 918.16 928.13 928.87 936.00 945.52 957.70 966.12 978.52 988.39 31 003.24  30 909.04 908.79 916.98 920.07 928.38 932.22 938.97 946.79  30 908.99  30 911.12  939.89  940.15  966.77 990.50  991.60  078.70  *Term v a l u e s a r e averages from 4 r o t a t i o n a l level.  l i n e s w i t h a common u p p e r  - 50 -  TABLE VI (b)  A s s i g n e d R o t a t i o n a l Term V a l u e s o f the ( 0 0 0 ) , Kf = 1 1  Level o f the  1  12  B2 ( A ) V S t a t e o f  32  C  S2  -1 J.  (cm j .  Term V a l u e s  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33  i*  30 926.18 927.46 927.87 929.12 930.64 932.83 934.73 936.93 939.41 942.12 944 .89 946.88 951.66 955.38 957.77 962.52 964.65 967.91 972.17 979.34 980.49 989.09 992.74 31 001.47 004.70 012.04 020.91 026.31 032.20 038.23 046.24 052.19 062.05  Term v a l u e s  30 937.52 939.60 942.48 945.30 947.87  30 945.95 948.46  955.93 958.16 963.87 966.16 968.04 973.53  968.30 969.92 975.67  30 968.89 970.47  981.73 989.53  981.87 989.79  982.90 990.66  31 003.31 005.20 012.21 021.53  31 003.78 005.51 014.51  31 004.24 006.89 015.44  959.00  041.16 048.67 056.89 063.51  9  °  (averaged) from t h e 3275 A (6V) and 3365 A bands  - 51 -  beyond doubt t h e p a r a l l e l p o l a r i z a t i o n o f the V s y s t e m .  The upper  s t a t e term v a l u e s a r e g i v e n as T a b l e V I ; t h e s e term v a l u e s , s u i t a b l y s c a l e d by s u b t r a c t i o n o f B"  J ( J + 1 ) , a r e shown p l o t t e d a g a i n s t  000 J ( J + 1) i n F i g . 8.  The two E - E bands have o n l y even J " b r a n c h  l i n e s , as e x p l a i n e d i n s e c t i o n I I I . E . a l l values of J " present.  The II - II b a n d s , however, have  The o b s e r v a t i o n o f s e v e r a l (up t o 6) a s s i g n -  ments f o r a s i n g l e b r a n c h l i n e i s e v i d e n c e o f t h e p e r t u r b a t i o n s o f t h e upper s t a t e w h i c h p l a g u e d any p r e v i o u s a t t e m p t s a t r o t a t i o n a l a n a l y s i s of the V bands.  The s i z e o f t h e p e r t u r b a t i o n s i s i l l u s t r a t e d i n F i g . 8  by t h e s p r e a d o f p o i n t s f o r each J ' v a l u e w h i c h i s on t h e o r d e r o f 5 cm-1.  D e s p i t e these p e r t u r b a t i o n s  slope of the l i n e s  there i s a d e f i n i t e p o s i t i v e  w h i c h r e p r e s e n t s t h e c o u r s e o f t h e 'mean' upper  s t a t e energy l e v e l s , i n d i c a t i n g t h a t B' > B". The v i o l e t o f t h e s e bands s u p p o r t s s u c h a r e s u l t .  degradation  From the s l o p e s and i n t e r c e p t s  o f t h e l i n e s a r e o b t a i n e d the v i b r a t i o n a l band o r i g i n s and r o t a t i o n a l constants: Bands  To  B'  (E)  3236A, 3322A"  30902.8 + 0.5 c m - 1  0.1129 ± 0.0010  (n)  3275A, 3365A  30926.5 ± 0.5  0.1144 ± 0.0010  I n t h e II v i b r o n i c b a n d s , one might e x p e c t t o f i n d a ' s t a g g e r i n g ' o f even and odd J v a l u e b r a n c h l i n e s , caused by the A o r £- d o u b l i n g o f a l i n e a r m o l e c u l e , o r , t h e asymmetry m o l e c u l e as i l l u s t r a t e d i n F i g . 4.  effects i n a non-linear  The ' s t a g g e r i n g ' w i l l appear a s a s l i g h t  309110  'E +  "3  9 0 9 0 9 0 7 0 9 0 5 0 903.0-  J>3090I.O  [a]  liJ30899.0-|  97.0-1  3 0 8 9 5 . 0 _  F I G . (8)  3 0 0  6 0 0  9 0 0  1200  I500  1800  2 0 0  4 0 0  6 0 0  8 0 0  IOOO  1200  3 0 9 3 8 . 0  R o t a t i o n a l Term V a l u e s o f the 000,K' = 0 L e v e l (a) and t h e 000,K' = 1 L e v e l (b) of t h e V S t a t e o f 1 2 C S 2 p l o t t e d a g a i n s t J ( J + 1 ) . The s l o p e o f t h e l i n e - l e a s t squares f i t t e d t o t h e average term v a l u e s - i s reduced by s u b t r a c t i n g BQQOJP+I) where BQOO ° 0.1090917 c m - 1 . 32  J[j + l]  - 53 -  d i f f e r e n c e i n t h e r o t a t i o n a l c o n s t a n t f o r even and odd J. upper s t a t e term v a l u e s a r e p l o t t e d s e p a r a t e l y  When t h e  f o r even and odd J ,  however, t h e B' v a l u e s were w i t h i n one s t a n d a r d d e v i a t i o n o f each o t h e r . The  r o t a t i o n a l perturbations  bands make t h e o b s e r v a t i o n  i n t h e upper s t a t e o f t h e 3275A and 3365A  of such a ' s t a g g e r i n g '  impossible.  Thus,  the r o t a t i o n a l a n a l y s i s i n d i c a t e s t h a t C S 2 i n t h e V s t a t e i s e i t h e r l i n e a r or not s i g n i f i c a n t l y bent.  As shown i n s e c t i o n s D and F , C S 2 i s  s l i g h t l y bent ( 1 6 3 ° ) i n t h e V s t a t e so t h a t t h e c a l c u l a t e d asymmetry 'staggering'  i s much s m a l l e r  t h a n t h e s t a n d a r d d e v i a t i o n , ± 0.0010 cm  1  ,  i n t h e r o t a t i o n a l c o n s t a n t o f t h e K' = 1 upper s t a t e .  A few s p u r i o u s r o t a t i o n a l a s s i g n m e n t s a r e i n e v i t a b l e i n s u c h a complicated a n a l y s i s .  N e v e r t h e l e s s , a t l e a s t 90% o f t h e a s s i g n m e n t s  a r e beyond d o u b t , w h i c h i s c e r t a i n l y s u f f i c i e n t t o draw r e g a r d i n g t h e n a t u r e o f t h e upper s t a t e s  C.  conclusions  involved.  V i b r a t i o n a l Pattern of the V State  W i t h t h e ground s t a t e v i b r a t i o n a l a s s i g n m e n t s o f t h e V bands e s t a b l i s h e d , t h e upper s t a t e term v a l u e s were c a l c u l a t e d by a d d i n g t h e ground s t a t e v i b r a t i o n a l energy t o t h e band head p o s i t i o n s : t h e s e a r e shown i n F i g . 9 and i n T a b l e V I I . H a v i n g d e t e r m i n e d t h e p a r a l l e l p o l a r i z a t i o n , the V l e v e l s were grouped a c c o r d i n g t o t h e i r v a l u e s o f t h e p r o j e c t i o n o f the t o t a l a n g u l a r momentum on t h e m o l e c u l a r a a x i s , w h i c h i s c a l l e d £ o r K  (depending on whether t h e m o l e c u l e i s l i n e a r o r b e n t ) .  The V l e v e l s  a r e l a b e l l e d as £ , II, A, . . . c o r r e s p o n d i n g t o K' o r £.' = 0 , 1, 2 , . . . .  - 54 -  34000i 13  E/cm-i  )5 12 11  33000 -  10  r  }3,o  )3c 3, 8  32000 -  }  }o  31000-  Q.  3  } ° 3  F I G . (9)  n  A  The V S t a t e v i b r a t i o n a l term v a l u e s c l a s s i f i e d by t h e v a l u e o f £" = 0,1,2 (E,II,A) o f the c o r r e s p o n d i n g V b a n d s . Shaded r e g i o n s i n d i c a t e bands w i t h no h e a d - l i k e o r i g i n ; term v a l u e s o f o v e r l a p p e d bands a r e shown as d o t t e d l i n e s .  - 55 -  TABLE  A s s i g n e d Term V a l u e s o f the B  (bent)  1  VII  B ( A ) - X*E* (V)  System o f  Term V a l u e s and v i  (linear)  2  E  1  u  n  30 756 30 904  (3)  30 928  (O  31 066  31 529 (6)  31 519  31 496  31 703  (5) 31 594  31 625  (31 712)  31 787  32 373  32 140 (8)  2  (6)  31 974  (7)  32 499  32 253  32 074  32 338  32 240  32 610  32 906 (10)  32 322 32 445 3  (8) 32 529 32 703 32 747  (9)  32 772 32 974  S (cm 2  - 1  ),  30 965  (31 372)  31 339 (O  3 2  30 757  (31 057)  1  C  A  (2) 0  1 2  33 058  (5)  31 178  - 56 TABLE V I I ( c o n t i n u e d )  Term V a l u e s and v 2  (linear)  (bent)  32 907  33 269 (ll)  32 945 (10)  33 455  33 010 33 133 32 218  33 389  (13)  33 952  33 481 33 605 (12)  33 703 33 780 33 865  The v 2 ( l i n e a r ) quantum number i s t h e b r a c k e t e d number t o t h e l e f t o f each group o f term v a l u e s .  - 57  The  p a t t e r n of the  E levels  bands) i s v e r y i r r e g u l a r , but  m a t r i x spectrum a t 77°K ( 2 2 ) . r a t i c i n the p r o j e c t i o n £' = 0 ,  1 and  structure  can be  ± 30 cm" 1 ,  p r o g r e s s i o n of ^ 580  the  ( c o r r e s p o n d i n g t o the interpreted  as a v e r y p e r t u r b e d  same i n t e r v a l as seen i n  quantum number, the  2 can be a s s i g n e d as the  the  l o w e s t group, w i t h K'  asymmetric top  o f a n o n - l i n e a r m o l e c u l e (see e q u a t i o n (13)  II l e v e l s w i t h i n  'cold'  S i n c e t h e i r e n e r g i e s are r o u g h l y quad-  w i t h A - -|- (B + C) Z. 33 + 5 cm" 1 , and  -  and,  rotational of s e c t i o n  from r o t a t i o n a l a n a l y s i s cm - 1 .  t h e s e groups ( s e c t i o n B ) , B - C — 0.11  s t r i c t , we  s h o u l d use  the b e n t m o l e c u l e quantum number K'  projection  of the a n g u l a r momentum a l o n g the n e a r - s y m m e t r i c top a  r a t h e r t h a n the l i n e a r m o l e c u l e a n a l o g u e , K-type r o t a t i o n a l s t r u c t u r e as l i t , I5  an<  * i6 (as  The  explained i n section  The  E and  III.C) of  £  To  be  the axis, This  = 1 levels, labelled  E).  p a t t e r n o f the V l e v e l s above 32,000 cm - 1  d i f f e r e n t , however.  for  i . e . K' - l" = 0.  i s a l s o seen i n the v 2  or  i s quite  A l e v e l s i n t h i s r e g i o n c o i n c i d e and  II l e v e l s l i e h a l f w a y between the  E  and  A levels.  Such a p a t t e r n  the (see  F i g . 2) i s the v i b r a t i o n a l energy l e v e l p a t t e r n of a l i n e a r m o l e c u l e . I n o t h e r w o r d s , the energy l e v e l p a t t e r n below 32,000 cm - 1 i s t i c of the K r o t a t i o n a l s t r u c t u r e above 32,000 cm - 1 ,  i s character-  of a s l i g h t l y bent m o l e c u l e , w h i c h ,  becomes the v i b r a t i o n a l p a t t e r n o f a l i n e a r m o l e c u l e  with successive excitations  of the d e g e n e r a t e b e n d i n g v i b r a t i o n .  The  v i b r a t i o n a l energy o f the n o n - l i n e a r m o l e c u l e t h e r e f o r e exceeds a p o t e n t i a l b a r r i e r to l i n e a r i t y a t ^ 31,900 ± 100  cm-1.  The  barrier  to  - 58  -  l i n e a r i t y , H, r e f e r r e d to the z e r o p o i n t energy i s g i v e n  H=  (31,900 ± 100) - E ( v 2  cm - 1  = 0) + -| v 2  where the l o w e s t - l y i n g V l e v e l i s t a k e n as the v2  = 0 (K' = 0) group a t 30,830 cm - 1 .  c a l c u l a t e d as 1350 gression  ± 150  cm-1.  The  by  The  'mean' p o s i t i o n o f  b a r r i e r to l i n e a r i t y i s  assignment o f the upper s t a t e p r o -  to the b e n d i n g v i b r a t i o n r a t h e r t h a n the symmetric s t r e t c h i n g  f r e q u e n c y (22) i s c o n f i r m e d by the v i b r a t i o n a l p a t t e r n of the V The  the  state.  m a n i f o l d s of v i b r a t i o n a l l e v e l s , as grouped i n F i g . 9, show t h e  b e n d i n g v i b r a t i o n and  K-type r o t a t i o n a l s t r u c t u r e e x p e c t e d f o r a n o n -  l i n e a r m o l e c u l e t h a t overcomes a b a r r i e r t o l i n e a r i t y on e x c i t a t i o n o f s e v e r a l quanta of v 2  The  (43).  geometry o f CS 2  s t a t e can be d e t e r m i n e d . s o l v e s e q u a t i o n s (8) and and  Knowing the A and  and  r 0 ( C - S ) = 1.544  =  The  163  ±  can be  one  length,  B v a l u e s o b t a i n e d from a n a l y s i s o f  6V bands a r e averaged t o g i v e B' = 0.1130 ± 0.0005 cm"1  t h a t the geometry of CS 2  <SCS  B r o t a t i o n a l constants,  (9) o f s e c t i o n I I I . D f o r £ , the C - S bond  2<[>, the S - C - S bond a n g l e .  t h e 10V  The  i n the lowest v i b r a t i o n a l l e v e l of the V  so  calculated:  ± 0.006A,  2°.  shape change upon e x c i t a t i o n of CS 2  m a i n l y a bond a n g l e change and not  from X 1 E +  t o the V s t a t e i s  a bond l e n g t h change.  r e a s o n a b l e i n l i g h t of the o b s e r v a t i o n  This i s  of e x t e n s i v e ' h o t ' band s t r u c t u r e  - 59 -  i n t h e b e n d i n g v i b r a t i o n and t h e absence o f t h e a b s o r p t i o n ground s t a t e symmetric s t r e t c h i n g v i b r a t i o n a l l e v e l .  from t h e  According  t o t h e Franck-Condon p r i n c i p l e , t h e c a l c u l a t e d geometry o f t h e V s t a t e l e n d s f u r t h e r s u p p o r t t o t h e assignment o f t h e V s t a t e v i b r a t i o n a l progression  t o the bending v i b r a t i o n , v ,because t h i s i s t h e o n l y 2  v i b r a t i o n one e x p e c t s t o see g i v e n t h i s shape c h a n g e .  D.  E l e c t r o n i c Species of the V State  The  lowest e x c i t e d e l e c t r o n c o n f i g u r a t i o n of CS2 i n v o l v e s  p r o m o t i o n o f an e l e c t r o n from t h e h i g h e s t o c c u p i e d o r b i t a l , o f symmetry Tr , t o an a n t i b o n d i n g o r b i t a l o f T T ^ symmetry. a  (TT ) 3 ( 7 r u ) 1  The e l e c t r o n i c s t a t e s o f  c o n f i g u r a t i o n a r e t h e s i n g l e t and t r i p l e t s t a t e s  obtained  * by group t h e o r e t i c a l m u l t i p l i c a t i o n , T T ^ X TT .  The -rr -»• IT  electronic  states o f CS2 are therefore  ^ ( ^ z ) 3  ,  I+(3B2) ,  ^ ( ^ z ) , 3  Z"(3A2) ,  where t h e e l e c t r o n i c s p e c i e s  \( A  1  3  1  2  *U(3A2  2  +  3fi  2>  of the non-linear  As i n d i c a t e d i n s e c t i o n I I I . D , an the integrand  + B )  1  molecule a r e bracketed.  e l e c t r o n i c t r a n s i t i o n i s allowed i f  o f <e'| u |e">, where g = x , y o r z , i s t o t a l l y s y m m e t r i c .  8 A p a r a l l e l - p o l a r i z e d t r a n s i t i o n i n v o l v e s an o s c i l l a t i n g d i p o l e moment p a r a l l e l t o t h e S-S d i r e c t i o n , i . e . o f s p e c i e s V s t a t e , where t h e m o l e c u l e i s n o n - l i n e a r ,  (or B 2 ) .  i s a B2 s t a t e .  Hence, t h e  - 60 -  To d e c i d e whether t h e V s t a t e i s s i n g l e t o r  triplet  (S = 0 or 1) we must c o n s i d e r t h e r e s u l t s o b t a i n e d from t h e m a g n e t i c r o t a t i o n spectrum (16,44).  As shown i n Appendix I I I , a m a g n e t i c  r o t a t i o n spectrum (m.r.s.) a r i s e s from the p r e s e n c e o f e i t h e r an e l e c t r o n o r b i t a l o r s p i n a n g u l a r momentum i n one o f the two Now,  t h e 1 E + ground e l e c t r o n i c s t a t e has no i n t r i n s i c  8  states.  a n g u l a r momentum  so t h a t any observed m.r.s. comes from t h e upper e l e c t r o n i c s t a t e .  The R and S systems of CS 2 e x h i b i t a s t r o n g m.r.s. ( 1 6 ) ; t h e V system, which i s considerably  stronger i n absorption  t h a n the R and  S s y s t e m s , has a much w e a k e r , though o b s e r v a b l e , m.r.s. the V s t a t e i s n o n - l i n e a r  S i n c e CS 2 i n  so t h a t the o r b i t a l a n g u l a r momentum i s  e s s e n t i a l l y quenched, o n l y the s p i n m u l t i p l i c i t y o f the V s t a t e i s i n question.  S i n c e the R s y s t e m , known t o have a t r i p l e t upper s t a t e ( 2 1 ) ,  shows a much s t r o n g e r m.r.s. t h a n t h e V s t a t e , t h e m.r.s. t h e r e f o r e i n d i c a t e s t h a t t h e V s t a t e i s e s s e n t i a l l y s i n g l e t but t h a t  perturbations  by t r i p l e t l e v e l s (see s e c t i o n V I ) l e n d some t r i p l e t c h a r a c t e r  t o the  V s t a t e , and h e n c e , a weak m.r.s.  O n l y two *B 2 s t a t e s can a r i s e from the ( U g ) 3 ('"'u)1 c o n f i g u r a t i o n , w h i c h a r e the 1 E ^ ( 1 B 2 ) and  1  A u ( 1 B 2 component) s t a t e s .  The  1  E^(1B2)  state  o  i s the upper s t a t e o f the i n t e n s e  2100A s y s t e m , so t h a t the V s t a t e has  t o be the 1 B 2 component o f t h e 1 A  state.  of t h e V system o f CS 2  T h u s , t h e e l e c t r o n i c assignment  i s TT -* I T * 1 B 2 ( 1 A ) - X  1  E+.  - 61 -  E.  i n a *A E l e c t r o n i c  Vibronic Correlations  State  F i g u r e 10 shows how t h e v i b r a t i o n a l l e v e l s o f a ' i s t a t e o f u a l i n e a r molecule c o r r e l a t e w i t h the v i b r a t i o n - r o t a t i o n two  electronic  s t a t e s w h i c h i t becomes when t h e m o l e c u l e i s b e n t .  the n e x t s e t i s t h a t f o r a Renner-Teller i n t e r a c t i o n .  1  1  Where t h e R e n n e r - T e l l e r e f f e c t i s i m p o r t a n t  ( s e c . V.A.) t h e o n l y good quantum number i s the v i b r o n i c quantum number K = | A + z\  t  a n g u l a r momentum  w h i c h becomes t h e r o t a t i o n a l quantum number K  of t h e s t a t e s o f a b e n t m o l e c u l e . t h e r e f o r e draw t h e c o r r e l a t i o n s  bent s t a t e s d i r e c t l y .  Obeying t h e ' n o n - c r o s s i n g r u l e ' one between t h e l e v e l s o f t h e l i n e a r and  These,are shown as t h e t i e l i n e s between t h e  second and t h i r d s e t s o f l e v e l s . The column headed ' v i b r o n i c g i v e s t h e v a l u e (v, } f o r each K - r o t a t i o n a l l e v e l . ° bent v , . lin To  The  E + state; and, g A state of a l i n e a r molecule with small u  l e f t - h a n d s e t o f v i b r a t i o n a l energy l e v e l s i s t h a t f o r a  can  levels of the  correlation'  i l l u s t r a t e the effect of a p o t e n t i a l b a r r i e r t o l i n e a r i t y  upon t h e energy l e v e l s o f a m o l e c u l e i n a *A state i t i suseful  (1A2 + 1B2) electronic  t o represent the c o r r e l a t i o n s  i n a d i f f e r e n t manner.  F i g u r e 11 i l l u s t r a t e s s c h e m a t i c a l l y t h e c o u r s e o f t h e K-type r o t a t i o n a l l e v e l s o f t h e v i b r a t i o n a l l e v e l s o f b o t h t h e upper and l o w e r component s t a t e s of the n o n - l i n e a r molecule; r a r i l y assumed t o l i e a t about t h e  The  the b a r r i e r to l i n e a r i t y i s a r b i t = 7 level.  r e a l l y i n t e r e s t i n g r e s u l t o f F i g . 11 i s t h e b e h a v i o u r o f  the K-type r o t a t i o n a l l e v e l s o f t h e upper and l o w e r components n e a r the b a r r i e r t o l i n e a r i t y .  I f t h e r e i s no o r b i t a l a n g u l a r momentum,  - 62 Vibronic Correlation  1  F I G . (10)  C o r r e l a t i o n o f v i b r o n i c l e v e l s (shown as t i e l i n e s ) between l i n e a r and bent g e o m e t r i e s . The b e n d i n g v i b r a t i o n quantum numbers of a C S 2 - t y p e m o l e c u l e i n t h e l i n e a r and bent l i m i t s a r e v l i n a n d vbent' respectively. The l e v e l s w i t h t h e same v a l u e o f K - t h e R e n n e r - T e l l e r quantum number o f a *A U s t a t e of a l i n e a r m o l e c u l e and t h e r o t a t i o n a l quantum numbers o f a bent m o l e c u l e - obey a ' n o n - c r o s s i n g r u l e ' .  - 63 -  i . e . i f A = 0, t h e symmetric t o p energy l e v e l f o r m u l a a p p l i e s ( e q u a t i o n (13) o f s e c t i o n I I I . C ) and t h e p o s i t i o n s o f t h e K-type t i o n a l l e v e l s a r e q u a d r a t i c i n K.  rota-  T h i s ' n o r m a l ' p a t t e r n changes near  the b a r r i e r t o l i n e a r i t y t o t h e energy l e v e l p a t t e r n o f a l i n e a r molecule.  W i t h i n t h e upper component, t h e v a l u e o f t h e A r o t a t i o n a l  c o n s t a n t ( d e f i n e d h e r e as t h e energy s e p a r a t i o n o f t h e K = 1 l e v e l above t h e K = 0 l e v e l w i t h i n a v i b r a t i o n a l l e v e l ) s t e a d i l y i n c r e a s e s a s v. i n c r e a s e s , u n t i l , near the b a r r i e r t o l i n e a r i t y , the v a l u e o f A bent becomes t h e l i n e a r m o l e c u l e b e n d i n g f r e q u e n c y .  W i t h i n the lower  component, however, t h e b e h a v i o u r o f t h e v a l u e o f A i s u n e x p e c t e d . W e l l below t h e b a r r i e r t o l i n e a r i t y , t h e v a l u e s o f A i n b o t h components a r e s i m i l a r though n o t e q u a l s i n c e t h e g e o m e t r i e s o f t h e m o l e c u l e i n the upper and l o w e r component s t a t e s a r e n o t i d e n t i c a l .  However, i n  the h i g h e r v i b r a t i o n a l members o f t h e l o w e r component s t a t e t h e v a l u e of A d e c r e a s e s and a c t u a l l y becomes n e g a t i v e n e a r t h e b a r r i e r t o l i n e a r i t y , r a p i d l y approaching the l i m i t i n g value of the negative of the l i n e a r m o l e c u l e b e n d i n g f r e q u e n c y .  A 'negative r o t a t i o n a l constant'  i s an a r t i f a c t o f t h e n e c e s s a r y c o r r e l a t i o n o f energy l e v e l s w i t h i n a degenerate e l e c t r o n i c  s t a t e , s i n c e a symmetric t o p A r o t a t i o n a l  con-  s t a n t cannot be d e f i n e d f o r a l i n e a r m o l e c u l e .  T h i s b e h a v i o u r o f t h e A r o t a t i o n a l c o n s t a n t i n a A s t a t e has n o t been r e c o g n i z e d b e f o r e .  As can be seen i n T a b l e V I I I , i n the R s t a t e  the v a l u e o f A d e c r e a s e s and becomes n e g a t i v e n e a r t h e v 2 = 10 l e v e l (15).  S i n c e t h i s e f f e c t was n o t u n d e r s t o o d by Kleman, he was u n a b l e t o  a s s i g n h i g h e r v i b r a t i o n a l members o f t h e p r o g r e s s i o n .  However, we c a n  - 64 -  upper lower  F I G . (11)  upper lower  upper lower  upper lower  B e h a v i o u r of v i b r o n i c l e v e l s of the two components o f a A s t a t e w i t h a b a r r i e r to l i n e a r i t y a t about v 'Upper' and  l i n  = 7.  ' l o w e r ' r e f e r t o the two components o f a A  s t a t e i n the bent molecule  limit.  - 65  now  say  t h a t the R s t a t e i s d e r i v e d  -  from t h e l o w e r component of what  would be an o r b i t a l l y d e g e n e r a t e s t a t e of the l i n e a r m o l e c u l e . p o s s i b l e d e g e n e r a t e IT -*• if and  Wiersma (21), and  t r i p l e t s t a t e i s the  a l s o Douglas and M i l t o n  The  s t a t e : Hochstrasser  (19) and  Hougen (20), had  previously  suggested t h i s a s s i g n m e n t from Zeeman e f f e c t s t u d i e s ,  we  c o n f i r m t h i s from the v i b r a t i o n a l s t r u c t u r e .  can now  On  the o t h e r h a n d , the v a l u e of A i n t h e V s t a t e  r a p i d l y near 32,000 c m - 1 , be  the b a r r i e r t o l i n e a r i t y .  the upper component s t a t e .  The  subscripted  The  increases  V s t a t e must  t h e n be a p p l i e d  I t s h o u l d be noted t h a t , above  b a r r i e r t o l i n e a r i t y , t h o s e l e v e l s w i t h the same v a l u e o f the c o r r e l a t i o n number (v . ) now  but  c o r r e l a t i o n l a b e l s of  the upper component s t a t e , as g i v e n i n F i g . 10, can the V s t a t e term v a l u e s i n F i g . 9.  c o i n c i d e , as  only  expected.  to the  subscripted  TABLE V I I I Term Values of the R A 3  1 2  V  2  CS  u  2  1 3 2  K = 1  K = 0  ( ^ ) State of CS 3  2  K = 2  K = 0  CS  2  K = 2  K = 1  3  27101.1  7.3  27108.4  14.8  27123.2  27100.3  7.1  27107.4  14.0  27121.4  4  27401.9  7.7  27409.6  16.9  27426.5  27395.9  7.3  27403.2  15.3  27418.5  5  27697.3  8.5  27705.8  18.8  27724.6  27685.6  8.6  27694.2  16.9  27711.1  6  27986.4  9.0  27995.4  21.2  28016.6  27970.0  8.2  27978.2  19.5  27997.7  7  28269.8  6.5  28276.3  23.2  28299.6  28249.4  8.1  28257.5  20.7  28278.2  8  28542.8  7.3  28550.1  16.8  28566.9  28523.0  4.2  28527.2  1.6.3  28543.5  9  28810.0  -2.7  28807.4  28786.4  -7.0  28779.4  10  29081.0  -37.1  29043.9  29044.0  -38.2  29005.8  "Values ( i n cm" ) 1  from B. Kleman, Can. J . Phys. 41, 2034 (1963).  - 67 -  V.  THE T STATE  I t has been shown t h a t t h e V s t a t e i s t h e upper R e n n e r - T e l l e r component o f a *A  state.  One m i g h t t h e r e f o r e expect a weak t r a n s i t i o n  t o t h e l o w e r component, 1 A 2 ( 1 A u ) l y i n g t o l o n g e r w a v e l e n g t h s .  We  have  found such an a b s o r p t i o n (which we c a l l the T s y s t e m , i n k e e p i n g w i t h Kleman's n o m e n c l a t u r e ) , l y i n g among t h e h i g h e r bands o f t h e R and S systems.  A l t h o u g h a t r a n s i t i o n from a t o t a l l y symmetric l o w e r s t a t e t o a *A 2  e l e c t r o n i c s t a t e i s e l e c t r i c d i p o l e f o r b i d d e n , a v a r i e t y o f mech-  anisms can g i v e a non-zero  t r a n s i t i o n moment.  T h u s , a *A2  -  1  Z+  t r a n s i t i o n i s a l l o w e d by e l e c t r i c quadrupole and m a g n e t i c d i p o l e tion rules.  These t y p e s of t r a n s i t i o n s a r e g e n e r a l l y l e s s i n t e n s e t h a n  an e l e c t r i c d i p o l e t r a n s i t i o n by f a c t o r s o f ^ 1 0 5 tively  (45).  selec-  and ^ 1 0 8 ,  respec-  M o r e o v e r , i n any p o l y a t o m i c m o l e c u l e , most t r a n s i t i o n s  ( e x c e p t t h e e l e c t r o n i c o r i g i n band) a l l o w e d i n t h e s e ways c a n appear more s t r o n g l y as  v i b r o n i c a l l y allowed e l e c t r i c d i p o l e t r a n s i t i o n s v i a  Herzberg-Teller mixing.  However, a n o t h e r t y p e o f v i b r o n i c i n t e r a c t i o n , t h e RennerT e l l e r e f f e c t , can cause a t r a n s i t i o n o f the t y p e 1 A 2  -  to appear,  8 when the 1 A 2  s t a t e i s one component of a d e g e n e r a t e s t a t e whose o t h e r  component does appear i n a b s o r p t i o n .  Such a t r a n s i t i o n i s p a r a l l e l -  p o l a r i z e d , K' - £" = 0 , and w i t h i n t e n s i t y p r o p o r t i o n a l w i t h K 2 , t h a t o n l y a b s o r p t i o n from I" in section A). bands.  =1,  so  2, 3, ... l e v e l s o c c u r s (as d i s c u s s e d  That i s , one e x p e c t s a t r a n s i t i o n c o n s i s t i n g o f ' h o t '  - 68 -  o  o  I n t h e r e g i o n 3300A t o 3400A t h e r e a r e a number o f bands.  unassigned  Only t h o s e bands t h a t show no ( s t r o n g ) m a g n e t i c r o t a t i o n  spec-  trum o r Zeeman e f f e c t c a n be c o n s i d e r e d c a n d i d a t e s f o r a s i n g l e t electronic  system.  Numerous ' h o t ' bands o f s i n g l e t n a t u r e a r e p r e s e n t  i n t h i s r e g i o n a n d , i n f a c t , two v i b r a t i o n a l p r o g r e s s i o n s were f o u n d . The T s t a t e  term v a l u e s a r e g i v e n i n T a b l e  I X and a medium  p h o t o g r a p h o f a few o f t h e T bands i s p r e s e n t e d  resolution  i n F i g . 12.  Since the  T system i s ' h o t ' o n l y , t h e R e n n e r - T e l l e r e f f e c t i s p r o b a b l y c a u s i n g t h e t r a n s i t i o n t o a p p e a r , as d e s c r i b e d i n t h e f o l l o w i n g A.  !  The R e n n e r - T e l l e r E f f e c t i n a A  u  Electronic  In o r b i t a l l y degenerate e l e c t r o n i c l i n e a r molecules  t h e v i b r a t i o n a l and e l e c t r o n  section.  State  states  ( I I , A, $ , ...) o f  o r b i t a l angular  momenta  a r e c o u p l e d , as f i r s t shown by Renner (46) w h i l e w o r k i n g w i t h E . T e l l e r . Because o f t h e R e n n e r - T e l l e r e f f e c t , t h e u s u a l Born-Oppenheimer  separa-  t i o n o f m o l e c u l a r m o t i o n i n t o e l e c t r o n i c , v i b r a t i o n a l , and r o t a t i o n a l p a r t s b r e a k s down.  The R e n n e r - T e l l e r e f f e c t causes t h e e l e c t r o n i c and  v i b r a t i o n a l wavefunctions and r o t a t i o n a l m o t i o n s .  t o be mixed so t h a t one c o n s i d e r s  vibronic  The c o u p l i n g between v i b r a t i o n a l and  electron  o r b i t a l a n g u l a r momenta i s d e s c r i b e d by an e f f e c t i v e h a m i l t o n i a n , d e r i v e d by Renner  IL = Jj 6 2 A cos2A(x-<t>) Renner where x i s t h e a n g l e between t h e r a d i u s v e c t o r t o a ' v a l e n c e ' and  (1)  electron  an a r b i t r a r y r e f e r e n c e p l a n e , <J> i s t h e a n g l e between t h e p l a n e o f  3390A  3349A 12U11U  10U  s+  9U  4  8U  7U  6U  s+3  I H S H H I BBS  V system 000-030(n)  If ft! \ lis  m l  :  1:  N X  a j r  = 3370.0755A  7 f f l n fMS 1 i 1 I f •:  ' ;  2  0 :;  n  f  u,  8U  laser line positions 3371.4289 A  F I G . (12) CS 2 a b s o r p t i o n a t medium r e s o l u t i o n : -78°C ( a ) , 23°C (b) and 100°C ( c ) ; a n d , t h e 9U r e g i o n a t h i g h r e s o l u t i o n : -78°C ( d ) , 23°C (e) and 100°C ( f ) . The (t+2)A and t i l 'hot' bands o f t h e T system a r e i n t h e r e g i o n o f N 2 l a s e r e m i s s i o n . The r o t a t i o n a l a n a l y s i s o f t h e ' c o l d ' 9U band i s shown as a F o r t r a t - l i k e d i a g r a m .  - 70 -  TABLE I X  A s s i g n e d Term V a l u e s o f t h e T  K' = 1  *A ( 2  X  A ^ State of K* = 2  1 2  C32S2  (cm-1) K' = 3  30 031 30 055  30 024 30 035  30 252 t+1  (30 246) 30 259  30 438  30 454  30 444  30 465  t+2  30 662 t+3 30 670  30 844 t+4 30 851  - 71 -  t h e bent m o l e c u l e and t h e same r e f e r e n c e p l a n e , 6 i s t h e supplement t o the i n s t a n t a n e o u s a n g l e o f bend and j i s a parameter d e s c r i b i n g t h e Renner-Teller  interaction.  The R e n n e r - T e l l e r p e r t u r b a t i o n i s b e s t w r i t t e n i n e x p o n e n t i a l form making use o f t h e d e f i n i t i o n  I  (2)  - e.* **-*) 1  giving L  - 1 j 2A 21A( -#) e  Renner  2  ( e  X  ^ K x - ^ j  +  J  = 2" J  ( q + + q_ )  I n t h e case o f a *A  state  (3)  (A=2),  the Renner-Teller h a m i l t o n i a n i s  u \enner = 1 * K  +  &  >  <> 4  the R e n n e r - T e l l e r h a m i l t o n i a n c o n s i s t s of ' l a d d e r ' operators  mixing  e l e c t r o n i c and v i b r a t i o n a l s t a t e s a c c o r d i n g t o t h e r u l e AA = ±2, A£, = + 2. The R e n n e r - T e l l e r term c o u p l e s t h e p u r e l y e l e c t r o n i c wavefunction  |A> = ( l / / 2 T ) e 1 A x  (5)  w i t h the doubly degenerate o s c i l l a t o r wavefunction,  |v£>, which  c o n t a i n s a v i b r a t i o n a l a n g u l a r momentum t e r m , (l//2y)e*~ '^. !l  turbed  The u n p e r -  ' v i b r o n i c ' b a s i s f u n c t i o n i s s i m p l y t h e p r o d u c t o f t h e s e wave-  functions \hvl> = |A>|v£>  .  (6)  - 72 -  Only the t o t a l Renner-Teller  ( v i b r o n i c ) a n g u l a r momentum, A + £,is c o n s e r v e d by t h e interaction.  Because o f t h e ' l a d d e r i n g ' p r o p e r t y o f  the o p e r a t o r , t h e m a t r i x elements o f t h e R e n n e r - T e l l e r evaluated  hamiltonian  i n t h e |Av£> b a s i s f u n c t i o n s a r e e n t i r e l y o f f - d i a g o n a l .  This  s u g g e s t s t h a t symmetrized l i n e a r c o m b i n a t i o n s o f b a s i s f u n c t i o n s , a s g e n e r a t e d by a Wang t r a n s f o r m a t i o n , w i l l t r a n s f e r t h e i n t e r a c t i o n m o s t l y t o t h e d i a g o n a l m a t r i x e l e m e n t s , and a l s o r e d u c e t h e s i z e o f t h e o f f - d i a g o n a l elements.  Then, second o r d e r p e r t u r b a t i o n t h e o r y , r a t h e r  t h a n e x a c t d i a g o n a l i z a t i o n , c a n be a p p l i e d .  W i t h t h e d e f i n i t i o n K = |A +  where A and I a r e s i g n e d  q u a n t i t i e s , t h e 'sum' and ' d i f f e r e n c e ' b a s i s f u n c t i o n s a r e  |A*> = ( l / / 2 ) { | 2 , v , K - 2 > ± |-2,v,K+2>} The  (7)  form o f t h e Renner m a t r i x i s i l l u s t r a t e d s c h e m a t i c a l l y b e l o w .  |A+> <A+|  reord  <A"|  ^R  |A"> ^R  H  reord  The d i a g o n a l s u b - m a t r i c e s c o n t a i n t h e ' r e o r d e r i n g m a t r i x e l e m e n t s ' , s o designated  because t h e s e p a r t s o f t h e i n t e r a c t i o n c o u p l e b a s i s f u n c t i o n s  o f t h e same symmetry.  A t t h i s s t a g e , one c a n a s s o c i a t e t h e 'sum' m a t r i x  w i t h t h e upper R e n n e r - T e l l e r the l o w e r component.  component and t h e ' d i f f e r e n c e ' m a t r i x w i t h  The r e o r d e r i n g m a t r i x elements s c r a m b l e a l l t h e  v i b r a t i o n a l l e v e l s w i t h i n a component b u t do n o t connect t h e upper and l o w e r component s t a t e s .  - 73 -  The m a t r i x elements c o n n e c t i n g t h e upper and l o w e r components t a k e t h e form  \ j <A  4  | ^  +  + q>_ | A >  (8)  w h i c h we c a l l t h e 'Fermi r e s o n a n c e m a t r i x e l e m e n t s ' . m a t r i x e l e m e n t s i n t h e |Av£> b a s i s f u n c t i o n s , we  |  ±  j  (1//2~){<2,V,K-2| <-2,V,K+2|}  Expressing  have  ( l / / 2 ) { | 2,v' ,K-2>+| - 2 , v ' ,K+2>}  |  \ 3 {<-2,v,K+2|q^|2,v',K-2> - <2,v,K-2|q^|-2,v',K+2>}  =  these  .  (9)  I t c a n be shown t h a t f o r K = 0 t h e F e r m i r e s o n a n c e m a t r i x e l e m e n t s vanish i d e n t i c a l l y .  I n o t h e r w o r d s , t h e E + and E  be i d e n t i f i e d w i t h t h e B 2  (upper) and A 2  v i b r o n i c s t a t e s can  ( l o w e r ) component s t a t e s ,  r e s p e c t i v e l y , w i t h no i n t e r a c t i o n between them.  The K == { 0 matrix  elements  can be c a l c u l a t e d e a s i l y i n harmonic a p p r o x i m a t i o n w i t h the d e f i n i t i o n  ^  ±  =  e  **X  ( 1 0  because m a t r i x elements o f t h e r e c t i l i n e a r d i s p l a c e m e n t o p e r a t o r s have been e v a l u a t e d  )  x+  ( 4 7 ) . M a t r i x m u l t i p l i c a t i o n g i v e s the m a t r i x .A.  e l e m e n t s o f t h e o p e r a t o r x ^ , and n o t i c i n g t h a t 2TT  :±2|e : p t i X|+2> = ^  e ± 2 ± * e** 1 * e ± 2 i * d X  }  0 = 1  we have  ,  (11)  - 74 -  <+2,v,K±2|q^_|±2,v,K+2> = Jtt  <+2,v+4,K±2|q^|±2,v,K+2>  /(V+K)(V-K)(V+K+2)(V-K+2)  = -|N /(V±K)(V±K+2)(V±K+4)(V±K+6)  <+2,v+2,K±2|q^_|±2,v,K+2> =  N /(v+K+2)(v±K)(v±K+2)(v±K+4)  <+2,v-2,K±2|q^|±2,v,K+2> =  N /(v+K+2)(v+K)(v±K)(v+K-2)  <+2,v-4,K±2|q'j|±2,v,K+2> = | w  (12)  /(v+K+2) (v+K) (v+K-2) (v+K-4)  where N i s a n o r m a l i z a t i o n f a c t o r .  The n o n - v a n i s h i n g F e r m i r e s o n a n c e  m a t r i x elements obey t h e s e l e c t i o n r u l e Av = ± 2 , ± 4 . 2 V *0? rK '  Av = ± 2 ; —————  = N[/(v-K+2) (v+K) (v+K+2) (v+K+4) - /(v+K+2) (v-K)  = N /(v+2)2-K2  For  [/(v+K+2) 2 -2 2 - / ( v - K + 2 ) 2 - 2 2 ]  v f a i r l y l a r g e , so t h a t ( v + 2 ) 2 »  HJ^ ' 2  V  (v-K+2) (v-K+4)]  K 2 and ( v ± K + 2 ) 2  (13)  >>4  y N(v+2) [(v+K+2) - (v-K+2)] = 2N(v+2)K  (14)  Even f o r v = 3 , K = 1, t h i s a p p r o x i m a t i o n i s a c c u r a t e w i t h i n  7%.  S i m i l a r l y , t h e Av = ±4 F e r m i r e s o n a n c e m a t r i x elements a r e a l s o p r o p o r t i o n a l t o K, w h i c h i s a 'good' quantum number.  A p p l y i n g t h e s e r e s u l t s to C S 2 , the 1 B 2 and *A 2 *  s t a t e s o f t h e TT ->- TT K.  component  1 A  A  s t a t e i n t e r a c t w i t h a m a t r i x element p r o p o r t i o n a l t o  The f i r s t o r d e r w a v e f u n c t i o n o f t h e A 1  2  state i s  - 75 -  l ^ . v I O  -  (0)  M°\v*>+1  1  \ 4°\v>K>  ( 0 ) 2  ,vK|^'hBf>,v.K>  ^  l  V  The  < A  '  E l  .  A ,vK" 2  El  ( 1 5  )  B ,v'K 2  v i b r o n i c m i x i n g produces a n o n - z e r o e l e c t r i c d i p o l e t r a n s i t i o n  moment  < 1 A| 1 ) ,vK|y IX !^ 8 1  proportional of t h e  t o K such t h a t t h e t r a n s i t i o n t a k e s t h e p a r a l l e l p o l a r i z a t i o n (XA ) -  L  Z  transition.  Since the i n t e n s i t y of a t r a n s i t i o n  i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e t r a n s i t i o n moment, t h e 1 A 2 ( 1 A ^ ) -  Xl l  B.  +  g  t r a n s i t i o n has a R e n n e r - T e l l e r i n d u c e d i n t e n s i t y p r o p o r t i o n a l t o K 2 . G e n e r a l F e a t u r e s o f t h e T System I t i s i n t e r e s t i n g t o n o t e how, a t 100°C, t h e A - A bands o f  the T system a r e s t r o n g e r t h a n t h e II - II b a n d s .  The K 2 i n t e n s i t y term  i s r e s p o n s i b l e , s i n c e t h e a d d i t i o n a l f a c t o r o f 4 f o r t h e A - A bands (K" = 2) o v e r t h e IT — II bands (K" = 1) o u t w e i g h s t h e l e s s f a v o u r a b l e Boltzmann f a c t o r .  Only t h e R e n n e r - T e l l e r e f f e c t , a s e x p l a i n e d  accounts f o r the v i b r o n i c  The  vibronic  above,  intensities.  II - II and A - A p r o g r e s s i o n s show a v i b r a t i o n a l  i n t e r v a l o f ~ 220 cm" 1 w h i c h must be a s s i g n e d a s t h e T s t a t e b e n d i n g frequency.  Because t h e T s t a t e b e n d i n g f r e q u e n c y i s about h a l f t h e ground  s t a t e f r e q u e n c y o f 396 c m - 1 , II and A bands d i f f e r i n g by two upper s t a t e quanta n e a r l y c o i n c i d e .  The T bands a r e o f t e n d o u b l e d and i n a l l c a s e s  e x h i b i t r o t a t i o n a l s t r u c t u r e a t l e a s t a s s e v e r e l y p e r t u r b e d as that o f t h e V bands so t h a t no r o t a t i o n a l a n a l y s i s has been p o s s i b l e .  As i n t h e  V . s t a t e , t h e term v a l u e s o f t h e T s t a t e show t h e r o t a t i o n a l s t r u c t u r e  - 76 -  o f a (near) symmetric top m o l e c u l e .  By p l o t t i n g the a v e r a g e p o s i t i o n s  o f the p e r t u r b e d l e v e l s o f the l o w e s t v i b r a t i o n a l l e v e l a g a i n s t K 2 , A r o t a t i o n a l c o n s t a n t o f 14.7  ± 1.7  cm"1  an  i s o b t a i n e d . U s i n g the a v e r a g e  o f t h e B r o t a t i o n a l c o n s t a n t s found f o r the V s t a t e , the geometry o f CS 2  i n the T s t a t e was  r ( C - S ) = 1.54it  calculated:  ± 0.006A  (assumed)  <SCS = 155 ± 2 ° . I n p r i n c i p l e , a d e t a i l e d r o t a t i o n a l a n a l y s i s o f a II - II band o f t h e T system would a l l o w t h e magnitude and s i g n of t h e asymmetry d o u b l i n g c o n s t a n t to be d e t e r m i n e d . i s opposite i n the  1  B2  and  ^  2  The  sense o f the asymmetry d o u b l i n g  component s t a t e s so t h a t a c o n f i r m a t i o n  o f t h e assignment of t h e T s t a t e as  1  A 2 ( 1 A ) would be p r o v i d e d .  Further-  more, w i t h o u t i s o t o p i c s t u d i e s , the a b s o l u t e v i b r a t i o n a l numbering o f t h e T bands cannot be o b t a i n e d .  As y e t , the a s s i g n m e n t o f t h e T system  must n o t be c o n s i d e r e d c o n c l u s i v e .  C.  N2  Laser E x c i t e d Fluorescence of C S 2 a t Brus (49) has r e p o r t e d t h a t C S 2  pulsed N2  337ll  f l u o r e s c e n c e , e x c i t e d by a  l a s e r , decays e x p o n e n t i a l l y i n the manner expected  l a s e r i n i t i a l l y populates  two f l u o r e s c i n g s t a t e .  when t h e  F i g . 12 shows the o  a b s o r p t i o n spectrum near t h e N 2  l a s e r w a v e l e n g t h (3371A) w i t h the l a s e r  * l i n e s marked below the h i g h r e s o l u t i o n s p e c t r a .  The  laser l i n e s overlap  & I n comparing the CS 2 spectrum w i t h the N 2 l a s e r l i n e s , Brus o m i t t e d the vacuum c o r r e c t i o n to the a i r w a v e l e n g t h s o f the N 2 e m i s s i o n , as g i v e n by P a r k , Rao and J a v a n ( 5 0 ) ; the l a s e r l i n e s do n o t , t h e r e f o r e , f a l l i n the 9U b a n d , as Brus m i s t a k e n l y t h o u g h t .  - 77  a II - II and  a A - A band of the  -  ( s i n g l e t ) T s y s t e m , and  a weak ' c o l d '  band (8U), w h i c h i s known to be a t r i p l e t from i t s c o m p a r a t i v e l y s t r o n g m.r.s. ( 1 6 ) .  Thus i t i s p o s s i b l e  n e n t i a l decays of the l a s e r e x c i t e d by  t h a t B r u s ' o b s e r v a t i o n of two f l u o r e s c e n c e o f CS 2  may  be  explained  the f a c t t h a t the l a s e r l i n e s o v e r l a p bands b e l o n g i n g to two  t r o n i c t r a n s i t i o n s , w h i c h have s i n g l e t and  expo-  elec-  t r i p l e t upper s t a t e s , r e s p e c -  tively.  D.  The  P o t e n t i a l Energy Diagram of the  *A  State  I n a *A s t a t e the p o t e n t i a l energy can be w r i t t e n  v±(e) = v0  +  i  as  ke + -a—. + (I ± I) ne1* 2  (D  b+8 2 - r e f e r to the upper ( 1 B 2 )  where + and  states, respectively. and  The  and  the l o w e r ( 1 A 2 )  component  f i r s t t e r m , V Q , i s a c o n s t a n t ; the  second  t h i r d terms r e p r e s e n t the p o t e n t i a l energy of a b e n t m o l e c u l e  w r i t t e n as a q u a d r a t i c term p e r t u r b e d by a L o r e n t z i a n ; term d e s c r i b e s the s p l i t t i n g of the  1  A  and  the  fourth  s t a t e as the l i n e a r m o l e c u l e u  bends.  A l l the R e n n e r - T e l l e r q u a r t i c s p l i t t i n g  i s included i n  the  upper s t a t e p o t e n t i a l c u r v e f o r c o n v e n i e n c e i n c a l c u l a t i o n . The calculated  f o u r p o t e n t i a l energy p a r a m e t e r s , k , a , b and  n can  from the bond a n g l e s , v i b r a t i o n a l f r e q u e n c i e s and  to l i n e a r i t y  of the component s t a t e s .  minima a t the upper and  The  be  barriers  p o t e n t i a l c u r v e s must have  l o w e r s t a t e e q u i l i b r i u m a n g l e s , 9 + and  the second d e r i v a t i v e s of e q u a t i o n (1) e v a l u a t e d a t 6  and  6  6 , and . are  - 78 -  required  to reproduce the bending f r e q u e n c i e s ,  3V (G)  co , i . e .  ±  = 0  (2)  G.  3 V (6) 302 2  ±  =  A  ±  .  (3)  The two f o r c e c o n s t a n t s , A , must be c a l c u l a t e d f o r t h e g e o m e t r i e s o f CS2  i n t h e two s t a t e s . A = (4Tr c/h)  {m (m + 2m s i n s c s  2  Aforementioned u n c e r t a i n t i e s value of 6  2  ^ 0)/2M) I  r Lb 2  np  u  (4)  2  i n t h e a n a l y s i s o f t h e T system make t h e  so d e t e r m i n e d l e s s u s e f u l i n t h e d e t e r m i n a t i o n o f t h e  p a r a m e t e r s than t h e b a r r i e r t o l i n e a r i t y f o r t h e upper s t a t e , = V (0) - V+(0 +  H  ) = I - \ k6  2  Four s i m u l t a n e o u s e q u a t i o n s ( o b t a i n e d and  - -JL_ - nej  (5)  b+07 by e v a l u a t i n g  t h e '+' component o f ( 2 ) ) were s o l v e d n u m e r i c a l l y  a computer programme.  .  equations ( 3 ) ,  (5)  with the a i d of  The p o t e n t i a l c o n s t a n t s g i v e t h e p o t e n t i a l  diagram shown as F i g . 13. The bond a n g l e i n t h e T s t a t e i s w e l l p r e d i c t e d (1),  as i s t h e b a r r i e r t o l i n e a r i t y .  by e q u a t i o n  I f t h e l o w e s t energy T band (see  T a b l e I X ) b e l o n g s t o t h e v^ = 0 l e v e l , the T s t a t e b a r r i e r t o l i n e a r i t y i s 2000 ± 100 c m , b r a c k e t i n g - 1  t h e c a l c u l a t e d v a l u e o f 1960 c m . - 1  - 79 -  F I G . (13)  The p o t e n t i a l energy c u r v e s of the V and T s t a t e s lower c u r v e s , r e s p e c t i v e l y ) . +  V (8)  <| I>  (upper and  *  3 3 9 0 0B  = 29,000 + 1730 6 + 0 > Q524+e2 U s i n g t h e d a t a 6 = 1 8 ° , u + = 560 c m - 1 , u)_ = 220 cm - 1 and H = 1350 cm - 1 gave t h e above s o l u t i o n w i t h 9 _ = 2 8 ° , H = 1960 c m - 1 . C f . t h e o b s e r v e d v a l u e s o f 6 ~ = 25 ± 2° and -1 t e n t a t i v e l y , H = 2000 ± 100 c m . ±  2  +  ±  - 80 -  A l t h o u g h t h e p o t e n t i a l energy e x p r e s s i o n can r e p r o d u c e t h e a p p r o x i m a t e f e a t u r e s o f t h e s p e c t r u m , i t s h o u l d n o t be r e g a r d e d as a c o n f i r m a t i o n of the v i b r a t i o n a l a n a l y s i s of the V and T s y s t e m s .  - 81 -  VI.  DISCUSSION  A n a l y s i s o f the e l e c t r o n i c a b s o r p t i o n spectrum o f CS 2  i n the  o.  r e g i o n 2900 - 3500A has shown t h a t the s t r o n g e s t a b s o r p t i o n i n t h i s 1  r e g i o n goes t o an upper s t a t e of bent a t e q u i l i b r i u m .  where t h e .  erate  A  u  B2  1  and  A2  symmetry where t h e m o l e c u l e i s  A weaker t r a n s i t i o n a t l o n g e r w a v e l e n g t h s , w h i c h  had not been r e c o g n i z e d 1  B2  p r e v i o u s l y , i s assigned  s t a t e s are Renner-Teller  as g o i n g t o a *A 2  components o f t h e degen-  * s t a t e (n -»• IT ) o f the l i n e a r m o l e c u l e .  t u r e i n t h e ^B 2  state,  s t a t e has been d e t e r m i n e d , and  The m o l e c u l a r  struc-  the a p p r o x i m a t e form o f  t h e p o t e n t i a l energy c u r v e s f o r the two s t a t e s has been d e s c r i b e d . The above the  1  A2  o b s e r v a t i o n t h a t the  1  B2  component of the ^  component when the m o l e c u l e i s bent g i v e s  information  about the p o s i t i o n o f the s o - f a r unobserved TT -> TT  A  TT -> i r * 1 B 2 z  above the  u  s t a t e (19): the 1  1  A2  B2 C^ 1  s t a t e i s known to l i e 15000 cm"1  A2  state l i e s  ( E ) state. X  A  The  u  by e l e c t r o s t a t i c i n t e r a c t i o n i t must l o w e r the energy o f  component.  However t h e  1  A2  state interacts similarly with  the  ) s t a t e , and t o account f o r the energy o r d e r of the components  one can c o n c l u d e t h a t the 1 Ao  (1Z  ) s t a t e l i e s above the 1 A state, i n ^ u u c o n t r a d i c t i o n t o the c o n s i d e r a t i o n s of McGlynn e t a l . ( 4 1 ) . The v i b r a t i o n a l and r o t a t i o n a l s t r u c t u r e s o f the components o f the V1B2  1  A^  s t a t e are very severely perturbed.  T h i s i s e s p e c i a l l y so i n the  component, where o n l y two upper s t a t e l e v e l s c o u l d be  r o t a t i o n a l l y , and  the v i b r a t i o n a l s t r u c t u r e  analysed  a t the s h o r t w a v e l e n g t h  end  - 82  is chaotic.  -  E v i d e n c e from t h e m a g n e t i c r o t a t i o n spectrum (16)  t h a t the p e r t u r b i n g  s t a t e s a r e p r o b a b l y t r i p l e t , because t h e  component l e v e l s below the b a r r i e r t o l i n e a r i t y  (where t h e  shows ^2  orbital  a n g u l a r momentum i s quenched) a r e s e n s i t i v e t o a m a g n e t i c f i e l d . c l a s s i f y i n g the p o s s i b l e TT -> TT n o t a t i o n one  can show t h a t t h e  t o be Kleman's R s t a t e , and  states according  t o the c a s e  (strongly) perturbing  (c)  s t a t e s are  the ft = 0 component o f the  3  Eu  By  likely  state  (which  i s presumably K l e m a n ' s S s t a t e ) : -  n = 0  1  l  +  1  u  Z  u  1  Au  3  Z^  3  Au  3  ^  The  B2  (2100A bands)  A2 A2(T) B2(S)  B2(V)  Ai + Bi Ai + B j  A2  Ai + B j  + B2(R)  B  Ai +  A2  s i n g l e t and  +  X  t r i p l e t s t a t e s i n t e r a c t by s p i n - o r b i t c o u p l i n g .  l i n e a r molecule l i m i t t h i s i s diagonal  In  the  i n the quantum number ft, and  w i l l a l l o w s t r o n g c o u p l i n g between the ^  s t a t e and 2u  the  3  A„  state. 2u  I n the bent m o l e c u l e t h e case (c) s p i n - e l e c t r o n i c s t a t e s b e l o n g i n g t o the same r e p r e s e n t a t i o n  are c o u p l e d .  Thus the V 1 B 2 ( ^ A u ) s t a t e w i l l  c o u p l e d s t r o n g l y to the R ( 3 A 2 y ) s t a t e a t a l l v a l u e s o f the SCS but more w e a k l y t o t h e S ( 3 Z Q + u ) s t a t e .  The  be  bond a n g l e ,  observed p e r t u r b a t i o n s  i n the  T ( * A u ) s t a t e a r e t h e r e f o r e not u n e x p e c t e d , s i n c e s i m i l a r s p i n - o r b i t i n t e r a c t i o n s w i t h l o w e r - l y i n g s t a t e s are  possible.  - 83 -  The  d e n s i t y o f p e r t u r b a t i o n s i n t h e V system i s i n t e r e s t i n g  because one i s s e e i n g t h e o n s e t o f t h e s p e c t r o s c o p i c m a n i f e s t a t i o n o f the p r o c e s s t h a t p e r m i t s t r i p l e t manifold  i n t e r s y s t e m c r o s s i n g from t h e s i n g l e t t o t h e  of a molecule.  The l e v e l s we have a n a l y s e d  b e l o n g t o t h e l o w e s t v i b r a t i o n a l l e v e l o f the V s t a t e ;  rotationally  even s o ,  there  a r e o f t e n s e v e r a l a s s i g n e d upper s t a t e l e v e l s w i t h t h e same J v a l u e showing t h a t t h e d e n s i t y o f p e r t u r b i n g t r i p l e t l e v e l s i s a l r e a d y erable.  At higher energies  consid-  t h e d e n s i t y o f l i n e s i s much g r e a t e r and  a c c o r d i n g l y the d e n s i t y of p e r t u r b i n g t r i p l e t l e v e l s i s o b v i o u s l y a l s o greater. although  T h u s , t h e s p e c t r u m appears t o be c o n t i n u o u s a t l o w d i s p e r s i o n , at high resolution  so d e n s e l y width.  i t appears a s a m u l t i t u d e o f o v e r l a p p i n g  lines  packed t h a t t h e i r a v e r a g e s e p a r a t i o n s a r e l e s s t h a n t h e l i n e  The d i f f u s e a b s o r p t i o n i s n o t a t r u e continuum spectrum because  t h e l e v e l s s t i l l l i e below the ground s t a t e d i s s o c i a t i o n l i m i t s i n g l e t r o t a t i o n a l l e v e l i s perturbed  (23).  A  by so many t r i p l e t l e v e l s i n t h e  r e g i o n o f d i f f u s e a b s o r p t i o n t h a t t h e m o l e c u l e c o u l d have a g r e a t e r p r o b a b i l i t y of being i n the t r i p l e t manifold than i n the s i n g l e t  manifold.  References  1.  E.D. W i l s o n , A s t r o p h y s . J . 6 9 , 34 ( 1 9 2 9 ) .  2.  F.A. J e n k i n s , A s t r o p h y s . J . 7 0 , 191 ( 1 9 2 9 ) .  3.  W.W. Watson and A.E. P a r k e r , P h y s . R e v . 3 7 , 1013 ( 1 9 3 1 ) .  4.  L.N. L i e b e r m a n n , P h y s . R e v . 6 0 , 496 ( 1 9 4 1 ) .  5.  C. R a m a s a s t r y , P r o c . N a t l . I n s t . S c i . I n d i a A18, 177 ( 1 9 5 2 ) .  6.  C. R a m a s a s t r y , P r o c . N a t l . I n s t . S c i . I n d i a A18, 621 ( 1 9 5 2 ) .  7.  W.C. P r i c e and D.M. Simpson, P r o c . R o y . S o c . 165A, 272 ( 1 9 3 8 ) .  8.  C. Ramasastry and K.R. Rao, I n d i a n J . P h y s . 2 1 , 313 ( 1 9 4 7 ) .  9.  Y . T a n a k a , A.S. J u r s a , and F . 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P h y s . 5 4 , 4165 ( 1 9 7 1 ) .  22.  L . Bajema, M. Gouterman, and B. M e y e r , J . P h y s . Chem. 75_, 2204 ( 1 9 7 1 ) .  23.  H. Okabe, J . Chem. P h y s . 5 6 , 4381 ( 1 9 7 2 ) .  24.  J . U . W h i t e , J . O p t . S o c . Am. 3 2 , 285 ( 1 9 4 2 ) .  25.  H . J . B e r n s t e i n and G. H e r z b e r g , J . Chem. P h y s . 16_, 30 ( 1 9 4 7 ) .  (1963).  - 85 -  26.  H.M. C r o s s w h i t e , Fe-Ne H o l l o w Cathode T a b l e s , John Hopkins U n i v e r s i t y (1965).  27.  R.S. M u l l i k e n , J . Chem. P h y s . 2 3 , 1997 ( 1 9 5 5 ) .  28.  G. H e r z b e r g , I n f r a r e d and Raman S p e c t r a o f P o l y a t o m i c M o l e c u l e s , Van N o s t r a n d , P r i n c e t o n , New J e r s e y , U.S.A. C h . I I 2 . ( 1 9 4 5 ) .  29.  L.D. Landau and E.M. L i f s h i t z , Quantum M e c h a n i c s , 2 n d E d i t i o n , Pergamon P r e s s and Addison-Wesley P u b l i s h i n g Company, I n c . , R e a d i n g , M a s s a c h u s e t t s , U.S.A. § 1 0 1 , 5 104, § 1 0 5 , ( 1 9 6 5 ) .  30.  D. A g a r , E.K. P l y l e r , and E.D. T i d w e l l , J . R e s . N a t l . B u r . S t a n d . 66A, 259 ( 1 9 6 2 ) .  31.  H.C. A l l e n , J r . and P.C. C r o s s , M o l e c u l a r V i b - R o t o r s , John W i l e y and S o n s , I n c . , New Y o r k and London, 7 ( 1 9 6 3 ) .  32.  H. N i e l s e n , R e v . Mod. P h y s . 2 3 , 90 ( 1 9 5 1 ) .  33.  I.M. M i l l s , M o l . P h y s . ]_, 549 ( 1 9 6 4 ) .  34.  J . T . Hougen, J . Chem. P h y s . 3 6 , 519 ( 1 9 6 2 ) .  35.  D.F. S m i t h , J r . and J . Overend, J . Chem. P h y s . 5 4 , 3632 ( 1 9 7 1 ) .  36.  E.U. Condon and G.H. S h o r t e l y , The Theory o f A t o m i c S p e c t r a , The U n i v e r s i t y P r e s s , Cambridge, C h . I l l ( 1 9 6 3 ) .  37.  J . H . Van V l e c k , R e v . Mod. P h y s . 2 3 , 213 ( 1 9 5 1 ) .  38.  G. H e r z b e r g , E l e c t r o n i c Spectra- o f P o l y a t o m i c M o l e c u l e s , Van N o s t r a n d , P r i n c e t o n , New J e r s e y , U.S.A. 109-114, 223 ( 1 9 6 6 ) .  39.  J . T . Hougen, N a t . B u r . S t a n d . (U.S.) Monogr. 1 1 5 , Ch. 3 ( 1 9 7 0 ) .  40.  H.C. A l l e n , J r . and P.C. C r o s s , M o l e c u l a r V i b - R o t o r s , J o h n W i l e y and Sons, I n c . , New Y o r k and London, 102 ( 1 9 6 3 ) .  41.  J.W. R a b e l a i s , J.M. McDonald, V. Scher and S.P. M c G l y n n , Chem. R e v . 7 1 , 73 ( 1 9 7 1 ) .  42.  G. H e r z b e r g , I n f r a r e d and Raman S p e c t r a o f P o l y a t o m i c M o l e c u l e s , Van N o s t r a n d , P r i n c e t o n , New J e r s e y , U.S.A. 390 ( 1 9 4 5 ) .  43.  R.N. D i x o n , T r a n s . Faraday Soc. 6 0 , 1363 ( 1 9 6 4 ) .  44.  W.H. E b e r h a r d t , p r i v a t e  45.  G. H e r z b e r g , E l e c t r o n i c S p e c t r a o f P o l y a t o m i c M o l e c u l e s , Van N o s t r a n d , P r i n c e t o n , New J e r s e y , U.S.A. 134-136, 2 2 2 , 265-271 ( 1 9 6 6 ) .  communication.  - 86 -  46.  E. Renner, Z. P h y s . 9 2 , 172  (1934).  47.  W. M o f f i t and A.D. L i e h r , P h y s . Rev. 106, 1195  48.  A . J . Merer and D.N. T r a v i s , Canad. J . P h y s . 44, 353  49.  L . E . B r u s , Chem. P h y s . L e t t e r s 1 2 , 116  50.  J . H . P a r k s , D.R.  51.  T. C a r r o l l , P h y s . Rev. 52, 822  52.  W.H.  (1957). (1966).  (1971).  Rao and A. J a v a n , A p p l . P h y s . 13_, 142  (1968).  (1937).  E b e r h a r d t and H. R e n n e r , J . M o l . S p e c . 6_, 483  (1961).  APPENDIX I V i b r a t i o n a l and R o t a t i o n a l C o m b i n a t i o n D i f f e r e n c e s of t h e X*E* Ground E l e c t r o n i c S t a t e o f  1 2  C  3 2  S2  The r o t a t i o n a l energy f o r m u l a e o f C S 2 i n i t s ground e l e c tronic state F(J) = v  used were 2  0  2  + B v [ J ( J + 1) - £ ] - D y J ( J + l )  2  + |(-l) qJ(J+D J  (1)  A2Fv(J) = Fv(J+l) - F v ( J - l )  (2)  A2G(J) = F v , ( J ) - F v „ ( J )  (3)  The c o n s t a n t s used (35) a r e summarized below i n cm  Level  v u0  B  v  1  units.  D *108 v  q*10 5  0 0° 0  0  0.1090917  0.993  0  0 2° 0  801.849  0.1094604  0.810  0  0 l  0  396.092  0.1093146  0.953  5.27  0 31 0  1206.980  0.1096572  1.49  8.62  1  The v a l u e s o f e q u a t i o n s  (1 - 3) were c a l c u l a t e d w i t h t h e a i d o f a  computer programme and a r e g i v e n h e r e .  - 88 -  (0 2° 0) Level  (0 0° 0) Level J  Ffj(J)  0  0  2  A F (J-l) 2  0  F (J) 2  A G(J) = 2  A F (J-1) 2  2  F ( J ) -Fo(J) 2  -  801.849  -  0.655  0.655  802.506  0.657  1.851  4  2.182  1.527  804.038  1.532  1.856  6  4.582  2.400  806.446  2.408  1.864  8  7.855  3.273  809.730  3.284  1.876  10  12.000  4.145  813.890  4.159  1.890  12  17.018  5.018  818.925  5.035  1.907  14  22.909  5.891  824.835  5.911  1.927  16  29.672  6.763  831.622  6.786  1.949  18  37.308  7.636  839.284  7.662  1.975  20  45.817  8.509  847.821  8.537  2.004  22  55.198  9.381  857.234  9.413  2.036  24  65.451  10.254  867.522  10.288  2.071  26  76.577  11.126  878.686  11.164  2.109  28  88.576  11.998  890.726  12*. 039  2.150  30  101.447  12.871  903.640  12.915  2.193  32  115.190  13.743  917.430  13.790  2.240  34  129.805  14.615  932.095  14.665  2.290  36  145.293  15.487  947.636  15.540  2.343  38  161.652  16.360  964.052  16.416  2.399  40  178.884  17.232  981.342  17.291  2.459  801.849  - 89 -  (0 l  1  (0 3  0) L e v e l  1  0) L e v e l  J  Fi(J)  A2Fi(J)  F3(J)  A2F3(J)  A2G(J)  1  396.201  -  1207.090  -  2  96.639  1.093  07.529  1.096  0.890  3  97.294  1.531  08.186  1.536  0.892  4  98.170  1.967  09.064  1.973  0.895  5  99.261  2.405  10.159  2.413  0.897  6  400.575  2.841  11.478  2.850  0.903  7  02.103  3.280  13.009  3.291  0.906  8  03.855  3.716  14.769  3.727  0.914  9  05.819  4.155  16.735  4.169  0.917  10  08.010  4.590  18.937  4.604  0.927  11  10.409  5.030  21.339  5.046  0.931  12  13.040  5.464  23.983  5.480  0.944  13  15.873  5.904  26.820  5.924  0.947  14  18.944  6.338  29.907  6.357  0.963  15  22.211  6.779  33.177  6.801  0.966  16  25.723  7.213  36.708  7.234  0.985  17  29.424  7.653  40.411  7.678  10.987  18  33.376  8.087  44.386  8.111  811.010  19  37.511  8.528  48.522  8.556  1.011  20  41.904  8.961  52.942  8.987  1.038  21  46.472  9.403  57.509  9.433  1.037  22  51.307  9.835  62.375  9.864  1.068  810.888  - 90 -  J  Fi(J)  A2Fi(J)  F3(J)  A2F3(J)  A2G(J)  23  456.307  10.277  1267.373  10.310  811.066  24  61.584  10.709  72.685  10.740  1.101  25  67.016  11.152  78.113  11.187  1.097  26  72.735  11.583  83.873  11.617  1.137  27  78.599  12.026  89.730  12.065  1.131  28  84.761  12.457  95.937  12.493  1.176  29  91.056  12.900  1302.223  12.942  1.167  30  97.662  13.331  08.879  13.370  1.217  31  504.387  13.775  15.593  13.819  1.206  32  11.436  14.205  22.697  14.246  1.261  33  18.592  14.649  29.839  14.695  1.247  34  26.085  15.079  37.393  15.122  1.308  35  33.671  15.523  44.960  15.572  1.290  36  41.608  15.952  52.965  15.998  1.357  37  49.623  16.397  60.958  16.449  1.335  38  58.005  16.826  69.413  16.874  1.408  39  66.449  17.271  77.832  17.235  1.383  40  75.276  17.700  86.739  17.750  1.463  - 91 -  APPENDIX II CS2 COMBINATION  DIFFERENCE PRCGRAMME  LINES OF TWO EANDS WITH COMMON UEPER STATES ARE STORED IN ' V ( N I ) AND •HV(NJ) . •PREC IS THE REQUIRED PRECISION (CM-1) OF THE FOUR CALCULATED TERM VALUES GENERATED BY THE R AND P ERANCH ASSIGNMENTS FROM EACH BAND. COMEINATION DIFFERENCES OF CS2 LOWER LEVELS ARE STORED IN ARRAY THE CONSTANTS SHOWN ARE FOR THE V2=1 L=1 AND V2=3,L=1 LEVELS. 1  1  f  22 1 20  DIMENSION P (3 7,5),V (250) ,HV (250) ,SH (10),WR (4,6) CO 1 J=1,37 X^FLOAT ( J * J + J) P ( j , 1) =-14.017+ (0. 10928 83-0. OC COO 000953 *X) *X P (J,2)=6.870+ (0. 1096141-0.0000000119*X)*X IF (MOD (J,2) .NE.O) GO TO 22 P(J,1)=P ( J , 1)+0.0000527*X P(J,2)=P (J,2) +0.0000862*X P (J,3)=P (J,2)~P (J,1) CONTINUE DO 20 J=2,36 P(J,4)=P (J + 1, 1 ) - P ( J - 1 , 1) P ( J , 5) =P (J+1 ,2)-P ( J - 1 , 1) REAE (5,100)NI f NJ PREC FORMAT (214,F5.3) REAE(5,101) (V (K) , K= 1 ,NI) READ (5,101) (H V (K) , K= 1 , NJ) FORMAT (16F5.3) ZL=F (1,3)-PREC ZU=P (36,3) + PREC LZ= 1 QZ=1.3*PREC DO 2 N=2,NI Z=V(H) WRITE(6,49)Z FORMAT (/ LINE«,F10.3) KK=0 IF ( (Z-HV (LZ) ) .LT.ZL) GO TO 2 IF ( (Z-HV (LZ)).LT.ZU) GO TO 5 LZ=IZ+1 IF (LZ.GT.NJ) GO TO 2 GO 10 1 I=LZ KK=KK+1 SM (KK)=HV (I) 1=1 + 1 IF (I.GT.NJ) GO TO 6 IF ( (Z-fiV (I) ) .GT.ZL) GO TO 7 KN=N-1 LN-LZ- 1 DO 8 J=1,35 ZA=P(J,3) KL=0 KR=0 r  100 101  49 4 3 5 7  6 10  1  - 92 -  KS=0 DO 9 1=1,KK IF (ABS (Z-SM (I)-ZA). GT. PR EC) GO TO 9 KL=KL•1 WR (KL.1)=SM (I) 9 CCNTINUE IF(KL.EQ.O) GO TO 8 I=KM ZA=P(J*1,4) 11 IF(I.EQ.0) GO TO 13 IF ( (Z-V (I)+PREC) .LT.ZA) GOTO 12 IF (ABS (Z-V (I) -ZA) .GT.PREC) GO TO 13 KN=I KR=KR +1 WR (KR,2) = V (I) 12 1=1-1 GO TO 11 13 IF (KR.FQ.O) GO TO 8 I=LN ZA=P (J + 1 ,5) 14 IF (I.EQ.O) GO TO 16 IF ( (Z-HV (I) +PREC) .LT.ZA) GO TO 15 IF (ABS (Z-HV (I)-ZA) . GT. PREC) GO TO 16 LN=I KS=KS+1 WR (KS,3)-HV (I) 15 1=1-1 GO TO 14 16 IF (KS.EQ.O) GO TO 8 T=Z*P(J,1) DO 17 1=1,KL 17 HR (I,4)=WR (I, 1) +P(J 2) DO 18 1=1,KR 18 WR (I,5)=WR ( 1 , 2 ) + P(J + 2,1) DO 19 1=1,KS 19 WR (I,6)=WR (1,3) *P(J + 2,2) IF ( (KL+KR + KS).NE.3) GO TO 21 IF( (ABAX1 (WR (1,4) , WR (1,5) , WR (1,6) )-ABINl (WR (1,4) ,WR (1,5) ,WR (1,6) ) ) 1.GT.QZ) GO TO 8 2 1 WRITE (6, 50) J,T, (WR(I,1) , WR (1,4) ,I=1,KL) 50 FORMAT (' R (',12,•)•,F10.3,45X4(F7.3,F8„3)) WRITE(6,51) (WR ( 1 , 2 ) ,WR (1,5) ,1=1 ,KR) 51 FORMAT (1X4 (F7.3,F8.3)) WRITE(6,52) (WR (1,3) ,WR (1,6) ,1=1 ,KS) 52 FORMAT (•+ •,60X4 (F7.3,F8.3)) 8 CONTINUE 2 CONTINUE STOP END f  - 93 -  APPENDIX I I I *  The magnetic trum of  the  light  transmitted  and w i t h a m a g n e t i c the is  propagation of proportional  path  length  rotation  to  times  field  netic  the  product of  the  components,  2J+1  and n e g a t i v e line  of  u  light,  ,and,  in a direction  of  magnetic  the  the  plane  field  spatial  Hence,  of  This  - J . lines  in  the  is  d e t e r m i n e d by  is  the wings  spectrum.  (51,52).  of  times  the  the  total  the  plane  of  a  of  give  as  a  spectrum. by  positive  Zeeman b r o a d e n e d polarization  the  and e l e c t r o n  moments, o n l y  of  scattered  the magnitude o f  orbital  magnetic states  polarization  manifested  through crossed p o l a r i z e r s .  electronic  to  into  observed  significant  parallel  direction  field  an e l e c t r o n i c  polarized light  electron  polarizers  mag-  as  only  spec-  associated  rotation  is  a  strength  degeneracy  J and  the magnetic  J,J-1,  circularly  is  gas.  removes  rotational  gas  negative  since  spin multiplet  References  the  a  or  effect  momenta g e n e r a t e or  Mj. =  gas  Mj. c o m p o n e n t s , r e s p e c t i v e l y ,  the m . r . s .  moment,  the  the m o l e c u l e .  and l e f t  show p o s i t i v e  transmitted  of  the  of  p l a c e d between c r o s s e d  upon the  quantized along  i.e.  Zeeman b r o a d e n i n g o f Because r i g h t  field  gas  (m.r.s.)  The r o t a t i o n  density  of  are  u  acting  light.  momentum, J  moment,  by a  the  The m a g n e t i c angular  spectrum  The  of  strength  magnetic  spin  orbitally  angular degenerate  an o b s e r v a b l e m a g n e t i c  rotation  

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