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Some aspects of the electronic spectra of small triatomic molecules Hallin, Karl-Eliv Johann 1977

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SOME ASPECTS OF THE ELECTRONIC SPECTRA OF SMALL TRIATOMIC MOLECULES, by KARL-ELIV JOHANN HALLIN B.Sc(Hon), University of Alberta, 1973. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIRMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF CHEMISTRY) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 30, 1977. Karl-Eliv Johann Hallin, 1977 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t o f t h e r e q u i r m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f C h e m i s t r y , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , Vancouver, B.C., agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y Date ( i i ) ABSTRACT 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 o f NO2, SO2, and CS2 have been p h o t o g r a p h e d , and 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 some o f t h e bands have been a n a l y s e d . I n t h e sp e c t r u m o f CS2, l a r g e numbers o f 'hot' bands i n t h e f i r s t e l e c t r o n i c a b s o r p t i o n systems o f 1 2CS 2 and 1 3 C S 2 (34-00-4100 A) have been a n a l y s e d from h i g h - d i s p e r s i o n p l a t e s , and a c c u r a t e r o t a t i o n a l c o n s t a n t s have been o b t a i n e d f o r t h e o v e r t o n e s o f t h e ground s t a t e b e n d i n g v i b r a t i o n up t o V2 = 6 and Si = 3 f o r 1 2CS 2 and v 2 = 4, 1=2 f o r 1 3 C S 2 . The energy d i f f e r e n c e s between t h e v a r i o u s l e v e l s w i t h t h e same Si v a l u e have been d e t e r m i n e d t o an a c c u r a c y o f about ±0.006 cm but (because o f jbhe p a r a l l e l p o l a r i z a t i o n o f t h e e l e c t r o n i c t r a n s i t i o n ) t h e a b s o l u t e e n e r g i e s o f l e v e l s w i t h Si > 0 c o u l d not be o b t a i n e d from t h e s e s p e c t r a . 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 t h e (0,0) band o f t h e 2 2 B 2 - X 2 A x e l e c t r o n i c t r a n s i t i o n o f N0 2, a t 2491 A, has been c a r r i e d o u t . A l t h o u g h t h e l i n e s a r e broadened as a r e s u l t o f p r e d i s s o c i a t i o n , i t has been p o s s i b l e t o d e t e r m i n e t h e f i v e q u a r t i c c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s and t h e s p i n - r o t a t i o n c o u p l i n g c o n s t a n t e f o r t h e upper aa s t a t e . The c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s a l l o w t h e p o s i t i o n o f t h e unseen v i b r a t i o n a l l e v e l v 3 ' t o be e s t i m a t e d : th e r e s u l t s o f f e r no s u p p o r t t o t h e s u g g e s t i o n o f Coon, C e s a n i and Huberman t h a t t h e r e i s a d o u b l e minimum p o t e n t i a l f u n c t i o n i n t h e a n t i s y m m e t r i c s t r e t c h i n g c o o r d i n a t e o f t h e 2 B 2 s t a t e . The g e o m e t r i c a l s t r u c t u r e o f t h e z e r o - p o i l e v e l o f t h e 2 B 2 s t a t e i s r ( N - 0) = 1.314 2 A, Z.0N0 = 1 2 0 . 8 7 ° , and i t s l i f e t i m e (as c a l c u l a t e d from t h e l i n e w i d t h s ) i s 42 ± 5 p i c o s e c o n d s . ( i i i ) About 160 r o t a t i o n a l l i n e s i n t h e r e g i o n 7370 - 74-10 A i n t h e e l e c t r o n i c s p e c t r u m o f NO2 have been a s s i g n e d . The- l i n e s form t h e K = 0, 1, and 2 sub-bands o f a p e r t u r b e d p a r a l l e l band where t h e upper s t a t e A c o n s t a n t i s about 17 cm \ I n a d i a b a t i c r e p r e s e n t a t i o n , t h e band can be c o n s i d e r e d t o be a t r a n s i t i o n w i t h i n the ground s t a t e m a n i f o l d , w h i c h o b t a i n s i t s i n t e n s i t y by v i b r a t i o n a l momentum c o u p l i n g from a nearby band o f t h e A 2 B 2 - X 2 A 1 e l e c t r o n i c t r a n s i t i o n ; i t s d assignment i s 2 13 1-000. Comparison w i t h t h e spec t r u m o f 1 5 N 0 2 shows t h a t t h e nearby A 2 B 2 l e v e l has q u i t e a s m a l l amount o f v i b r a t i o n a l e n e r g y , which i s not i n c o n s i s t e n t w i t h t h e as s i g n m e n t by Brand, Chan-, and Hardwick t h a t t h e (0,0) band o f t h e A - X t r a n s i t i o n i s a t 8350 A. The i m p l i c a t i o n s o f t h e e l e c t r o n s p i n - r o t a t i o n p a r a m e t e r s and t h e i n t e n s i t y o f t h e 7390 A band a r e d i s c u s s e d . R o t a t i o n a l a n a l y s e s have been c a r r i e d o ut f o r t h e (0,0) bands o f t h e a 3 B 1 - X 1 A 1 a b s o r p t i o n systems o f S 1 6 0 2 and S 1 8 0 2 , from h i g h d i s p e r s i o n p l a t e s t a k e n w i t h t h e gases a t d r y i c e t e m p e r a t u r e . The r o t a t i o n a l a n a l y s i s o f t h e (0,0) band o f S 1 6 0 2 g i v e n by Brand, Jones and d i Lauro i s c o n f i r m e d i n - g e n e r a l , but t h e i r v a l u e s f o r t h e a n i s o t r o p i c s p i n f i n e s t r u c t u r e c o n s t a n t s a r e found t o be i n e r r o r . Our new v a l u e s remove t h e d i s c r e p a n c y i n t h e s i g n o f t h e s p i n - s p i n i n t e r a c t i o n parameter g=E between t h e gas phase work and t h e s o l i d s t a t e v a l u e g i v e n by T i n t i . T h i s . d i s c r e p a n c y had been r a t i o n a l i z e d by Brand, Jones and D i Lauro i n terms o f a d i f f e r e n t ^ c h o i c e o f phases f o r t h e a n g u l a r momentum o p e r a t o r s , but t h i s argument i s shown t o be i n c o r r e c t . The spec t r u m o f S 1 8 0 2 c o n f i r m s our new v a l u e s f o r t h e s p i n c o n s t a n t s i n d e t a i l . ( i v ) The C 1 B 2 - X 1 A 1 a b s o r p t i o n s p e c t r a o f S 1 6 0 2 and S 1 8 0 2 between 2350 and 2270 A have been a n a l y s e d i n d e t a i l f rom h i g h d i s p e r s i o n p l a t e s t a k e n w i t h t h e gases a t d r y i c e t e m p e r a t u r e . The c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s f o r t h e (0,0) band a r e found t o d i s a g r e e w i t h t h o s e r e p o r t e d by B r a n d , C h i u , Hoy and B i s t ; r o t a t i o n a l c o n s t a n t s f o r t h e o t h e r members o f t h e v 2 1 p r o g r e s s i o n a r e g i v e n t o h i g h p r e c i s i o n . E f f e c t i v e c o n s t a n t s f o r t h e C 1 B 2 s t a t e o f S 1 8 0 2 a r e r e p o r t e d . I r r e g u l a r i t i e s due t o C o r i o l i s c o u p l i n g o b s e r v e d i n t h e 002 l e v e l s o f both i s o t o p i c s p e c i e s have' been d e p e r t u r b e d t o g i v e n t h e r o t a t i o n a l c o n s t a n t s o f t h e unseen 011 l e v e l s . A s i m i l a r d e p e r t u r b a t i o n o f t h e l e v e l s 012 and 100 o f S 1 6 0 2 has been used t o g i v e t h e r o t a t i o n a l c o n s t a n t s o f t h e unseen 021 l e v e l o f t h e C 1B 2; s t a t e . The s p e c t r o s c o p i c a l l y d e t e r m i n e d C o r i o l i s and anharmonic c o u p l i n g c o n s t a n t s a r e r e p o r t e d . An e s t i m a t e o f t h e energy o f t h e unseen 001 l e v e l from t h e band o r i g i n s o b s e r v e d g i v e s 236 and 234- cm 1 f o r S 1 6 0 2 and S 1 8 0 2 , r e s p e c t i v e l y . The l a r g e a n h a r m o n i c i t y o b s e r v e d i n t h e v 3 ' m a n i f o l d c o n f i r m s t h e d o u b l e minimum p o t e n t i a l i n Q 3 1 s u g g e s t e d by Br a n d , C h i u , Hoy and B i s t , b u t i n d i c a t e s t h a t t h e b a r r i e r h e i g h t i s s m a l l e r t h a n t h e v a l u e o f 100 cm 1 t h a t t h e y r e p o r t . A d e t a i l e d t h e o r e t i c a l a n a l y s i s o f t h e d i r e c t s p i n - o r b i t i n t e r a c t i o n between e l e c t r o n i c s t a t e s o f t h e same s p i n m u l t i p l i c i t y has been c a r r i e d o u t . ( v ) TABLE OF CONTENTS Chapter Page I Theory o f Symmetric T r i a t o m i c M o l e c u l e s i n T h e i r S i n g l e t S t a t e s 1 A) T h e s i s I n t r o d u c t i o n 2 B) The G e n e r a l H a m i l t o n i a n 5 C) The Born-Oppenheimer S e p a r a t i o n o f N u c l e a r and E l e c t r o n i c M o t i o n 7 D) The F a c t o r i n g o f t h e N u c l e a r K i n e t i c Energy and t h e F i e l d - F r e e H a m i l t o n i a n 14 E) E x p a n s i o n o f t h e V i b r a t i o n - R o t a t i o n H a m i l t o n i a n , and t h e Order o f Magnitude C l a s s i f i c a t i o n o f Terms 23 F) A p p l i c a t i o n o f t h e G e n e r a l H a m i l t o n i a n t o Asymmetric Top S p e c t r a 29 G) P e r t u r b a t i o n s 36 H) The R o t a t i o n a l H a m i l t o n i a n 39 I ) C e n t r i f u g a l D i s t o r t i o n i n Asymmetric Top M o l e c u l e s 42 3) V i b r a t i o n a l C o n t r i b u t i o n s t o t h e I n e r t i a l C o n s t a n t s 46 K) The I n e r t i a l D e f e c t 48 L) D e r i v a t i o n o f t h e x ' n n from C e n t r i f u g a l D i s t o r t i o n Data a a ^ ' 5 1 M) D e t e r m i n a t i o n o f t h e M o l e c u l a r F o r c e F i e l d f rom C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s 56 I I M u l t i p l e t S t a t e s o f Bent T r i a t o m i c M o l e c u l e s 60 A) I n t r o d u c t i o n 61 B) Hund's C o u p l i n g Cases 65 C) The O r b i t - R o t a t i o n H a m i l t o n i a n 68 D) D e r i v a t i o n o f t h e S p i n - R o t a t i o n H a m i l t o n i a n 70 ( v i ) C h a p t e r Page I I C o n t i n u e d E) C e n t r i f u g a l C o r r e c t i o n s Due t o S p i n - R o t a t i o n C o u p l i n g 72 F) M a t r i x Elements o f t h e S p i n - R o t a t i o n H a m i l t o n i a n 74 G) E x t e n s i o n b$ t h e E f f e c t i v e H a m i l t o n i a n to. T r i p l e t S t a t e s 79 H) M a t r i x Elements o f t h e S p i n - S p i n H a m i l t o n i a n f o r T r i p l e t S t a t e s o f O r t h o r h o m b i c M o l e c u l e s 80 I I I The D i r e c t S p i n - O r b i t H a m i l t o n i a n 82 A) I n t r o d u c t i o n 83 B) M a t r i x Elements o f t h e D i r e c t S p i n - O r b i t I n t e r a c t i o n 85 IV The Ground S t a t e o f Carbon D i s u l f i d e 92 A) I n t r o d u c t i o n 93 B) Photography o f t h e 3400 t o 4100 A Region o f t h e CS2 Spectrum 95 C) S e l e c t i o n R u l e s f o r t h e R-System o f CS2 96 D) A n a l y s i s o f t h e Bands 98 E) D a t a R e d u c t i o n 101 F) Summary and D i s c u s s i o n 108 V The 2491 A Band o f N0 2 111 A) I n t r o d u c t i o n 112 B) E x p e r i m e n t a l 114 C) Energy L e v e l E x p r e s s i o n s f o r an Asymmetric Top I n a D o u b l e t S t a t e 115 D) D e s c r i p t i o n o f t h e 2491 A Band 118 E) T h e r S p i h s R o t a t i o n C o n s t a n t s o f t h e Upper S t a t e 123 ( v i i ) C h a p t e r Page V C o n t i n u e d F) D e t e r m i n a t i o n o f t h e Upper S t a t e R o t a t i o n a l C o n s t a n t s 130 G) The Q u e s t i o n o f a Double Minimum i n t h e 2 B 2 S t a t e o f N0 2 132 H) L i n e Width and P r e d i s s o c i a t i o n L i f e t i m e 138 I ) D i s c u s s i o n 14-1 V I The 7390 A Band o f N0 2 14-3 A) I n t r o d u c t i o n 144 B) E x p e r i m e n t a l 14-8 C) R o t a t i o n a l A n a l y s i s 14-9 D) V i b r a t i o n a l Assignment o f t h e 7390 A Band 153 E) E l e c t r o n S p i n (fine S t r u c t u r e E f f e c t s 158 F) D i s c u s s i o n 162 V I I E l e c t r o n S p i n F i n e S t r u c t u r e C o n s t a n t s f o r t h e a 3 B i S t a t e o f S 0 2 166 A) I n t r o d u c t i o n 167 B) E x p e r i m e n t a l 169 C) Energy L e v e l s o f a T r i p l e t S t a t e o f S 0 2 170 D) R o t a t i o n a l L i n e Assignments 175 E) D e t e r m i n a t i o n o f t h e R o t a t i o n a l C o n s t a n t s 181 F) D i s c u s s i o n 186 V I I I C o r i o l i s P e r t u r b a t i o n s i n t h e C ^ S t a t e o f S 0 2 191 A) I n t r o d u c t i o n 192 B) E x p e r i m e n t a l 195 ( v i i i ) C h a p t e r Page V I I I C o n t i n u e d C) R o t a t i o n a l A n a l y s i s 196 D) D e t e r m i n a t i o n o f t h e R o t a t i o n a l C o n s t a n t s 200 E) Form o f ^ t h e H a m i l t o n i a n f o r t h e Cou p l e d L e v e l s i n t h e C ^ z S t a t e o f S 0 2 207 F) M a t r i x F o r m u l a t i o n o f t h e Cou p l e d V i b r a t i o n a l L e v e l s 211 G) D e t e r m i n a t i o n o f t h e D e p e r t u r b e d R o t a t i o n a l C o n s t a n t s 219 H) D i s c u s s i o n 225 I ) T h e s i s C o n c l u s i o n 228 B i b l i o g r a p h y 229 A p p e n d i c e s 238 I D e r i v a t i o n o f t h e D i r e c t S p i n - O r b i t H a m i l t o n i a n M a t r i x Elements 239 I I Some R e l e v a n t P o i n t s C o n c e r n i n g S e l e c t i o n R u l e s and and Group Theory f o r C ^ M o l e c u l e s 245 A) C l a s s i f i c a t i o n o f t h e T o t a l W a v e f u n c t i o n 246 B) S e l e c t i o n R u l e s 257 I I I 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 262 ( ix) LIST OF TABLES T a b l e Page 1.1 Terms i n t h e R o v i b r a t i o n a l H a m i l t o n i a n , H/hc, a r r a n g e d by o r d e r o f magnitude and power o f 3 27 1.2 Terms o f t h e ' R o v i b r a t i o n a l H a m i l t o n i a n c l a s s i f i e d by o r d e r o f Magnitude and powers o f ( q ^ , P^) and 3 30 1.3 I d e n t i f i c a t i o n o f t h e M o l e c u l e - f i x e d A x i s System w i t h t h e P r i n c i p a l I n e r t i a l A x i s System 41 1.4- T r a n s f o r m a t i o n s between Watson's "Reduced" and '"Determinable" C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s 53 1.5 The T r a n s f o r m a t i o n from t h e C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s o f a Symmetric AB 2 M o l e c u l e t o t h e I n v e r s e F o r c e C o n s t a n t M a t r i x 58 2.1 C o r r e l a t i o n o f t h e V a r i o u s Non-zero S p i n - R o t a t i o n P a r a m e t e r s f o r an O r t h o r h o m b i c M o l e c u l e 77 2.2 M a t r i x Elements o f t h e S p i n - R o t a t i o n and S p i n - S p i n H a m i l t o n i a n s i n S p h e r i c a l Tensor N o t a t i o n 78 3.1 M a t r i x Elements o f T X ( V ) • T 1 (S) f o r S = \ 86 3.2 M a t r i x Elements o f T ^ V ) • T 1 (S) f o r S = 1 87 3.3 S e l e c t i o n R u l e s i n C_ v f o r t h e D i r e c t S p i n - O r b i t I n t e r a c t i o n 90 2v r 4.1 Summary o f Data f o r t h e A n a l y z e d bands o f CS 2 103 4.2 Comparison o f some Ground S t a t e R o t a t i o n a l C o n s t a n t s f o r 1 2 C S 2 104 4.3 Ground S t a t e R o t a t i o n a l C o n s t a n t s o f 1 2 C S 2 and 1 3 C S 2 r e f e r r e d t o t h o s e o f t h e l o w e r l e v e l o f each Si v a l u e (cm ) 106 4.4 D e r i v e d C o n s t a n t s f o r t h e ground s t a t e o f CS 2 (cm M 107 4.5 Comparison o f Observed and C a l c u l a t e d e n e r g i e s f o r some Z = 0 o v e r t o n e s o f t h e b e n d i n g v i b r a t i o n o f 1 2 C S 2 (cm 1 ) 109 5.1 M a t r i x e l e m e n t s f o r an Asymmetric Top i n a D o u b l e t S t a t e 116 5.2 Comparison o f s p i n s p l i t t i n g s i n c o r r e s p o n d i n g ^P6 l i n e s o f t h e 2491 A (000-000) and 2537 A (000-010|)bands o f N0 2 ( c m - 1 ) 126 (x) T a b l e Page 5.3 R o t a t i o n a l c o n s t a n t s f o r t h e 2491 A band o f N0 2 131 5.4 Comparison o f t h e e x p e r i m e n t a l v i b r a t i o n f r e q u e n c i e s f o r t h e 2 B 2 s t a t e w i t h t h o s e g i v e n by t h e c e n t r i f u g a l d i ' s t o r t - i o n c o n s t a n t s 134 6.1 Subband o r i g i n s and r o t a t i o n a l c o n s t a n t s f o r t h e 7390 A band o f N 0 2 ( c m _ 1 ) 152 6.2 P o s s i b l e v i b r a t i o n a l a s s i g n m e n t s f o r t h e 7390 A band o f N0 2 ( c m - 1 ) 155 7.1 R o t a t i o n a l c o n s t a n t s f o r t h e (0,0) bands o f t h e a - X system o f S 1 6 0 2 ( c m - 1 ) 184 7.2 R o t a t i o n a l c o n s t a n t s f o r t h e (0,0) bands o f t h e a - X system o f S 1 8 0 2 ( c m - 1 ) 185 7.3 CGomparison o f t h e s p i n c o n s t a n t s f o r t h e a 3 B ! s t a t e s o f S 1 6 0 2 and S 1 8 0 2 187 8.1 E f f e c t i v e c o n s t a n t s o f t h e S 1 6 0 2 C 1 B 2 r o t a t i o n a l a n a l y s i s 201 8.2 E f f e c t i v e c o n s t a n t s o f t h e S 1 8 0 2 C1^ r o t a t i o n a l a n a l y s i s 202 833 The r e l a t i o n s h i p between t h e C^ v a x i s s y s t e m and t h e I m o l e c u l e - f i x e d a x i s s ystem 212 8.4 D e p e r t u r b e d r o t a t i o n a l c o n s t a n t s f o r C 1 B 2 002 l e v e l s o f S 1 6 0 2 and S 1 8 0 2 220. 8.5 D e p e r t u r b e d r o t a t i o n a l c o n s t a n t s f o r t h e i n t e r a c t i n g t r i a d 100-021-012 o f S 1 6 0 2 223 8.6 L o w - l y i n g v i b r a t i o n a l l e v e l s o f t h e C 1 B 2 s t a t e o f S 0 2 226 A I I . l C h a r a c t e r t a b l e f o r t h e C p o i n t group 247 A l l . 2 C h a r a c t e r t a b l e f o r t h e D p o i n t g r o u p , i n c l u d i n g o n l y t h t h o s e i r . r e d u c i b l e r e p r e s e n t a t i o n s s y m m e t r i c w i t h r e s p e c t t o i n v e r s i o n 253 A l l . 3 R o v i b r o n i c symmetry s p e c i e s f o r t h e v a r i o u s r o t a t i o n a l s y m m e t r i e s o f t h e asymmetric r o t o r l e v e l s i n each v i b r o n i c s t a t e 256 A l l . 4 V i b r o n i c and r o t a t i o n a l s e l e c t i o n r u l e s f o r AB 2mm6<lecules w i t h C^ y p o i n t group symmetry 261 ( x i ) T a b l e Page A I I I . l A s s i g n e d r o t a t i o n a l l i n e s f o r t h e Si = 0 l e v e l s o f t h e R system o f 1 2 C S 2 264 A I I I . 2 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e £ = 1 l e v e l s o f t h e R system o f 1 2 C S 2 280 A I I I . 3 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e -Si = 2 l e v e l s o f t h e R system o f 1 2 C S 2 301 A I I I . 4 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e Si = 3 l e v e l s o f t h e R system o f 1 2 C S 2 315 A I I I . 5 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e I = 0 l e v e l s o f t h e R sy s t e m o f 1 3CS 2 319 A I I I . 6 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e Si = 1 l e v e l s o f t h e Rssyst-emoof 1 3CS 2 331 A I I I . 7 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e 1 = 2 l e v e l s o f t h e R system o f 1 3 C S 2 341 A I I I . 8 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e 2491 A band o f N 0 2 2 346 A I I I . 9 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e 7390 A band o f N0 2 364 A I I I . 1 0 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e (000-000) F i bands o f t h e a 3B!- X % t r a n s i t i o n o f S 1 6 0 2 368 A I I I . l l a A s s i g n e d r o t a t i o n a l l i n e s f o r t h e (000-000) F 2 band o f t h e a 3 B i - ) ( % t r a n s i t i o n o f S 1 6 ) 2 380 A I I I . 1 2 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e (000-000) F 3 band o f t h e a 3 B i - X % t r a n s i t i o n o f S 1 6 0 2 396 A I I I . 1 3 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e (000-000) F i band o f t h e a 3 B i - X'Aj t r a n s i t i o n o f S 1 8 0 2 415 A I I I . 1 4 A s s i g n e d r o t a t i o n a l l i n e s f o r t h e (000-000) F 2 band o f t h e a 3 B i - X x A i t r a n s i t i o n o f S 1 8 0 2 425 A I I I . 1 5 A s s i g n e d rofcati>onalllinesf,f6r£.the (000-000) F 3 band o f t h e a 3 B i - X ^ i t r a n s i t i o n o f S 1 8 0 2 435 A I I I . 1 6 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 f o r t h e (000-000) band o f t h e C ^ - X % t r a n s i t i o n o f S 1 6 0 2 448 A I I I . 1 7 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 f o r t h e (010-000) band o f t h e C ? B i - X % t r a n s i t i o n o f S 1 6 0 2 453 ( x i i ) T a b l e Page A I I I . 1 8 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 f o r t h e (020-000) band o f t h e X x A i t r a n s i t i o n o f S 1 6 0 2 459 A I I I . 1 9 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 f o r t h e (002-000) band o f tmeeC^Bz- ) ( % t r a n s i t i o n o f S 1 6 0 2 464 A I I I . 2 0 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 f o r t h e (012-000) band o f t h e ClB2- X % t r a n s i t i o n o f S 1 6 0 2 469 A I I I . 2 1 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 f o r t h e (100-000) band o f t h e C X B 2 - X'Ai t r a n s i t i o n o f S 1 6 0 2 473 A I I I . 2 2 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 f o r t h e (000-000) band o f t h e C^Ba- X % t r a n s i t i o n o f S 1 8 0 2 478 A I I I . 2 3 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 f o r t h e (010-000) band o f t h e C 1 B 2 - X J A i t r a n s i t i o n o f S 1 8 0 2 486 A I I I . 2 0 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 f o r t h e (002-000) band o f t h e C 1 B 2 - X ^ A i t . t r a n s i t i o n o d f S 1 8 0 2 492 ( x i i i ) LIST OF FIGURES F i g u r e Page 1.1 Walsh d i a g r a m f o r an AB 2 t y p e m o l e c u l e 11 2.1 Hund's c o u p l i n g c a s e (b) 67 4.1 T y p i c a l s p e c t r a o f CS 2 w i t h t h e i r r o t a t i o n a l a s s i g n m e n t s f o r K' = Z" - 0, 1, 2 and 3 bands; t a k e n from v 2 ' = 5 v 2 " = 4 and 5. 99 5.1 Room t e m p e r a t u r e a b s o r p t i o n s p e c t r u m o f N0 2 between 2491 and 2497 A 119 5.2 (a) A p o r t i o n o f t h e C - X (010-000) band o f S 0 2 (2328 A) a t -78°C (b) A p o r t i o n o f t h e N0 2 a b s o r p t i o n s p e c t r u m (near 2497 A) a t room t e m p e r a t u r e t o t h e same s c a l e as ( a ) ( c ) AK = 2 b r a n c h e s i n t h e 2491 A band o f N0 2 120 5.3 A b s o r p t i o n s p e c t r u m o f N0 2 between 2503 and 2510 A t a k e n a t 200 C, showing t h e l a r g e e s p i n s p l i t t i n g s i n t h e h i g h - K subbands 121 5.4 D i f f e r e n c e i n t h e s p i n s p l i t t i n g s f o r K = 6 between t h e 2537 A (000-010) and 2491 A (000-000) bands o f N0 2 125 5.5 Energy l e v e l d i a g r a m showing how t h e s p i n s p l i t t i n g s i n t h e 2491 A and 2537 A bands o f N0 2. a r e r e l a t e d '128 ! 6.1 E l e c t r o n i c energy as a f u n c t i o n o f bond a n g l e f o r low-l y i n g e l e c t r o n i c s t a t e s o f N0 2 145 6.2 R o t a t i o n a l A s s i g n m e n t s i n t h e l o n g w a v e l e n g t h p a r t o f t h e 7390 A band o f N0 2 150 6.3 Upper s t a t e t e r m v a l u e s from t h e 7390 A band o f N 0 2 l e s s 10K 2 + 0.4 N(N + 1 ) , p l o t t e d as a f u n c t i o n o f N(N + 1) 159 7.1 H a m i l t o n i a n m a t r i c e s f o r t h e r o t a t i o n o f an o r t h o r h o m b i c m o l e c u l e i n a t r i p l e t e l e c t r o n i c s t a t e 171 7.2 (0,0) band o f t h e a 3Bi- X % t r a n s i t i o n o f S 1 6 0 2 a t -78°C; t h e r e g i o n near t h e asymmetry ' s p i k e ' a t 3878 A . 178 7.3 (0,0) band o f t h e a 3B 1-X iX 1A 1 t r a n s i t i o n o f S 1 6 0 2 a t -78°C; t h e t a i l o f t h e band near 3887 A 179 7.4 (0,0) band o f t h e a 3 B j - X M j t r a n s i t i o n o f S 1 6 0 2 a t -78°C; showing t h e h i g h K, AK = +1 s t r u c t u r e o f t h e band 180 ( x i v ) F i g u r e Page 7.5 D i s t r i b u t i o n o f a s s i g n e d term v a l u e s between t h e s p i n components f o r t h e 000 l e v e l o f t h e a 3 B i . s t a t e o f S 1 6 0 2 182 7.6 Observed s p i n s p l i t t i n g , p a t t e r n s f o r K = 0 - 3 f o r t h e 000 l e v e l o f t h e a 3 B i s t a t e o f S 1 6 0 2 189 8.1 The (010-000) band o f t h e C 1 B 2 - X1Ai t r a n s i t i o n o f S 1 6 0 2 a t -78°C; showing t h e main q R band head near 2381 A 197 8,?2 The (010-000) band o f t h e - XlAl s t r a n s i t i o n . o f S 1 6 0 2 a t -78 C; showing t h e t a i l ofi. t h e band near 2335 A 198 8.3 Term v a l u e s o f S 1 6 0 2 r e d u c e d by 0.22 3(3 + 1) and p l o t t e d as a f u n c t i o n o f 3(30+ 1) 203 8.4 Term v a l u e s o f S 1 8 0 2 r e d u c e d by 0.22 3(3 + 1) and p l o t t e d as a f u n c t i o n o f 3(3 + 1) 204 8.5 H a m i l t o n i a n o p e r a t o r s c o u p l i n g t h e v i b r a t i o n a l l e v e l s o f fche i n t e r a c t i n g t r i a d 100-021-012 210 8.6 Term v a l u e s o f t h e i n t e r a c t i n g t r i a d 100-021-012 r e d u c e d by 0.22 3(3 + 1) and p l o t t e d as a f u n c t i o n o f 3 ( 3 + 1) 222 A I I . l A x i s system chosen f o r a C AB 2 m o l e c u l e l y i n g i n t h e yz p l a n e 249 ( x v ) ACKNOWLEDGEMENTS T h i s d i s s e r t a t i o n i s an a c c o u n t o f work c a r r i e d o ut i n t h e Department o f C h e m i s t r y a t UBC under t h e d i r e c t i o n o f Dr. A. 3. Merer. I t g i v e s me g r e a t p l e a s u r e t o thank Dr. Merer f o r h i s a b l e s u p e r v i s i o n and i n s t r u c t i o n d u r i n g t h e c o u r s e o f t h i s r e s e a r c h . I am f u r t h e r g r a t e f u l t o him f o r h e l p i n g t o make my s t a y a t UBC a p l e a s a n t one, t h r o u g h h i s c o n t i n u e d o p t i m i s m and encouragement. I a l s o w i s h t o acknowledge t h e s t i m u l a t i n g e n v ironment p r o v i d e d by Dr. Y. Hamada w h i l e he was w o r k i n g i n our e l e c t r o n i c s p e c t r o s c o p y group a t UBC, e s p e c i a l l y f o r h i s h e l p f u l a s s i s t a n c e and f o r our many u s e f u l d i s c u s s i o n s . I f u r t h e r e x t e n d my s i n c e r e t h a n k s t o t h e N a t i o n a l R e s e a r c h C o u n c i l f o r f i n a n c i a l s u p p o r t d u r i n g t h e c o u r s e o f t h i s r e s e a r c h . 1 CHAPTER I. THEORY OF SYMMETRIC TRIATOMIC MOLECULES IN THEIR SINGLET STATES. (A) T h e s i s I n t r o d u c t i o n . ^ The v a l u e o f s t u d y i n g t h e m o l e c u l a r m o t i o n s o f s i m p l e m o l e c u l e s i n t h e gas phase u s i n g t h e t e c h n i q u e s o f e l e c t r o n i c s p e c t r o s c o p y i s w e l l e s t a b l i s h e d . E l e c t r o n i c s p e c t r o s c o p y i s . i m p o r t a n t because of i t s g e n e r a l a p p l i c a b i l i t y . I t can be used t o s t u d y a l m o s t a l l m o l e c u l e s , i f t h e i r t r a n s i t i o n s a r e o b s e r v a b l e as s h a r p s p e c t r a l f e a t u r e s i n t h e v i s i b l e and u l t r a v i o l e t p o r t i o n s o f t h e e l e c t r o m a g n e t i c s p e c t r u m . I n a d d i t i o n , i n f o r m a t i o n about t h e geometry and e l e c t r o n i c p r o p e r t i e s o f m o l e c u l e s ( o r i o n s ) i n t h e i r e l e c t r o n i c a l l y e x c i t e d s t a t e s can be o b t a i n e d w i t h g r e a t p r e c i s i o n , whacherewealshthehehangeshthatecqceursas a c m o l e c u l e i s a c t i v a t e d towards a c h e m i c a l r e a c t i o n , o r d u r i n g e x c i t a t i o n i n t o d i s s o c i a t i o n f r a g m e n t s . At f i r s t g l a n c e , t h e f a c t t h a t e l e c t r o n i c s p e c t r o s c o p y has been i n a c t i v e use f o r so many y e a r s s u g g e s t s t h a t i n v e s t i g a t i o n s o f " s i m p l e " m o l e c u l e s ought t o have been c o m p l e t e d l o n g ago. I n f a c t t h e r e i s a g r e a t d e a l o f i n t e r e s t i n " c l a s s i c a l " s p e c t r o s c o p i c p r o b l e m s , such as t h o s e c o n t a i n e d i n t h e s p e c t r a o f common and r e l a t i v e l y w i d e l y s t u d i e d m o l e c u l e s s u c h as C S 2 , SO^, and NO^. I n s p i t e o f t h e c o n s i d e r a b l e e f f o r t s t h a t have been expended i n t h e d e t a i l e d s t u d y o f t h e s e m o l e c u l e s , t h e t a s k o f u n r a v e l l i n g t h e i r s p e c t r a i s s t i l l u n f i n i s h e d , and r e m a i n s a t e d i o u s and d i f f i c u l t u n d e r t a k i n g . T h i s i s e s p e c i a l l y t r u e o f t h e s p e c t r a o f NO,,, f o r whi c h an a n a l y s i s o f any v i b r a t i o n a l band can be c o n s i d e r e d t o be a s i g n i -f i c a n t c o n t r i b u t i o n t o t h e u n d e r s t a n d i n g o f i t s p r o p e r t i e s . T h i s m o l e c u l e has been s t u d i e d a t t e m p e r a t u r e s as low as a few de g r e e s K e l v i n (1) u s i n g m o l e c u l a r beam t e c h n i q u e s ; even so i t s s p e c t r u m r e m a i n s u n b e l i e v a b l y com-p l i c a t e d , and c o n t a i n s many more l e v e l s t h a n would be e x p e c t e d on s i m p l e grounds. 3 T h i s d i s s e r t a t i o n p r e s e n t s a n a l y s e s c a r r i e d out f o r a l l t h r e e m o l e c u l e s , and t h e s p e c t r a l i n f o r m a t i o n t h a t t h e s e a n a l y s e s r e v e a l . The r e m a i n d e r o f t h i s c h a p t e r i s d e v o t e d t o a c o n c i s e summary o f t h e t h e o r e t i c a l framework i n t erms o f w h i c h t h e r e s u l t s o f t h i s t h e s i s w i l l be d i s c u s s e d . No a t t e m p t has been made t o a c h i e v e c o m p l e t e n e s s or r i g o r i n t h i s p r e s e n t a t i o n ; i n s t e a d , t h e d i s c u s s i o n has been r e s t r i c t e d t o c o n s i d e r o n l y t h o s e a s p e c t s r e l e v a n t t o t h e s p e c t r a l a n a l y s e s c a r r i e d out f o r t h e s e s m a l l , p l a n a r t r i a t o m i c m o l e c u l e s . The t r e a t m e n t b e g i n s by d e f i n i n g t h e g e n e r a l H a m i l t o n i a n w h i c h d e s c r i b e s t h e t o t a l m o l e c u l a r energy. T h i s g e n e r a l H a m i l t o n i a n i s s i m p l i f i e d i n t h e s e c t i o n s w h i c h f o l l o w , and i s u l t i m a t e l y s e p a r a t e d i n t o s e v e r a l t e r m s , each o f w h i c h may t h e n be t r e a t e d i n d i v i d u a l l y by means o f quantum mechanics. F i n a l l y , t r a n s f o r m a t i o n s a r e d e f i n e d w h i c h d e r i v e an " e f f e c t i v e " H a m i l t o n i a n c o n t a i n i n g o n l y e x p e r i m e n t a l l y d e t e r m i n a b l e p a r a m e t e r s as c o e f f i c i e n t s o f w e l l - d e f i n e d quantum m e c h a n i c a l o p e r a t o r s . A f t e r a g i v e n s p e c t r u m has been a n a l y s e d i n terms o f t h e quantum numbers a s s o c i a t e d w i t h t h e s e o p e r a t o r s , a c o m p a r a t i v e l y t r i v i a l f i t t o t h e e f f e c t i v e H a m i l t o n i a n d e f i n e s t h e s e p a r a m e t e r s , and r e d u c e s a l a r g e number o f s p e c t r a l f r e q u e n c i e s t o a few c o n s t a n t s , from w h i c h m o l e c u l a r p r o p e r t i e s of i n t e r e s t can be o b t a i n e d i n f a v o r a b l e c a s e s . The r e s u l t s o f t h i s f i r s t c h a p t e r a r e extended i n C h a p t e r I I t o i n c l u d e e l e c t r o n i c s t a t e s o f m o l e c u l e s i n w h i c h u n p a i r e d e l e c t r o n s a r e p r e s e n t . The second c h a p t e r t h e r e f o r e c o m p l e t e s t h e b a s i c t h e o r y needed t o f i t t h e s p e c t r a o f s m a l l , s y m m e t r i c AE^-type m o l e c u l e s i n s i n g l e t , d o u b l e t and t r i p l e t s t a t e s . C h a p t e r I I I c o n t a i n s an i n d e p e n d e n t s t u d y o f t h e t h e o r e t i c a l form o f t h e d i r e c t s p i n - o r b i t i n t e r a c t i o n , which o c c u r s between e l e c t r o n i c s t a t e s o f t h e same s p i n m u l t i p l i c i t y and e l e c t r o n i c e n e r g y . T h i s i n v e s t i g a t i o n i s i n c l u d e d w i t h t h e t h e o r e t i c a l s e c t i o n f o r c o m p l e t e n e s s , and was made n e c e s s a r y by t h e o b s e r v a t i o n o f l a r g e numbers o f p e r t u r b a t i o n s i n t h e h i g h e r v i b r a t i o n a l »ley.e-ls;.of' t h e - e i L ' e e t r b n d c r c j B i c s t a t e • o f SO,, d i s c u s s e d i n Cha p t e r V I I . Some o f t h e p e r t u r b a t i o n s found i n t h i s s t a t e can be i n t e r p r e t e d i n terms o o f o d i r e c t s p i n - o r b i t i n t e r a c t i o n w i t h an unseen e l e c t r o n i c s t a t e o f 3 A 2 symmetry, which i s d e g e n e r a t e t o w i t h i n -1 q a few hundred cm w i t h t h e o b s e r v e d B i s t a t e . The r e m a i n i n g c h a p t e r s p r e s e n t t h e e x p e r i m e n t a l r e s u l t s and t h e i r i n t e r p r e t a t i o n s , o r g a n i z e d i n t o c h a p t e r s a c c o r d i n g t o t h e e l e c t r o n i c s t a t e s t o which t h e y r e f e r . C h a p t e r IV c o n c e r n s i t s e l f w i t h t h e ground e l e c t r o n i c s t a t e o f CS^ > <which5 w'als2 therifdrsitiiSjtudyJcun'derttaken ifin=i t h e c o u r s e o f t h i s r e s e a r c h . C h a p t e r s V and VI summarize t h e r e s u l t s o b t a i n e d i n th e s t u d i e s o f HO^',: t h e f i f t h c h a p t e r c o n c e r n s i t s e l f w i t h t h e e l e c t r o n i c t r a n s i t i o n o b s e r v e d near 2491A; w h i l e t h e a n a l y s i s o f a t r a n s i t i o n found near 7390A i s p r e s e n t e d i n c h a p t e r s i x . T h i s l a t t e r t r a n s i t i o n i n v o l v e s a t r a n s i t i o n between v i b r a t i o n a l l e v e l s w i t h i n t h e ground e l e c t r o n i c s t a t e X j A i A i andmristinduGed by y d b r o n i c b i n t e ' E a c i t i o n i b e t w e e n 1 OP. t h i s ground e l e c t r o n i c s t a t e and t h e l o w - l y i n g l e v e l s o f t h e A 2 B 2 s t a t e w hich l i e a t n e a r l y t h e same energy. F i n a l l y , C h a p t e r s V I I and V I I I a r e d e v o t e d t o t h e r e s u l t s o b t a i n e d from t h e a n a l y s i s o f t h e a 3 B j s t a t e , and t h e C 1 B 2 s t a t e o f S02> r e s p e c t i v e l y . he for.-,.' -.'..".•-••'•er t 'Mirc - . s o n l y a , , - t i o n «. t h e q • n- - i s , c o n t a i n - n?.nv o f the t r \ t - h a t e r e ; ..•<••• ";. • r i e d O U t . 5 (g) The G e n e r a l H a m i l t o n i a n The g e n e r a l t r e a t m e n t o f a l l m o l e c u l e s b e g i n s w i t h an e x p r e s s i o n f o r t h e t o t a l energy o p e r a t o r 1 , H, d e f i n e d t o be t h e sum o f t h e t o t a l k i n e t i c e n e r g y , T, and t h e t o t a l p o t e n t i a l e n e r g y , V. Thus H = T + V (1.1) The t o t a l k i n e t i c energy o p e r a t o r i n 1.1 can be r e s o l v e d i n t o a n u c l e a r k i n e t i c energy o p e r a t o r , J^, and an e l e c t r o n i c k i n e t i c energy o p e r a t o r , T e, so t h a t T = T + T (1.2) n e where N T = Y P 2/'2M (1.3) n L , -a da a = l n and - T = I p. 2A2m. (1.4) i = l _ 1 1 1 f o r a m o l e c u l e w i t h n e l e c t r o n s and N n u c l e i . Hefesp. and P denote t - i -a th e l i n e a r momenta o f e l e c t r o n i (mass m.) and n u c l e u s a(mass M ), r e s p e c t i v e l y . S i m i l a r l y , t h e p o t e n t i a l energy o p e r a t o r f o r a m a n y - e l e c t r o n , many n u c l e u s system can be r e s o l v e d i n t o an a l g e b r a i c sum o f many t e r m s , o f w h i c h one s e l e c t s o n l y t h o s e t w o - p a r t i c l e i n t e r a c t i o n s r e l e v a n t t o t h e m o l e c u l a r system under s t u d y . The e l e c t r o s t a t i c -c o u l o m b i c i n t e r a c t i o n s f o r a f i e l d - f r e e m o l e c u l e i n wh i c h no s p i n m u l t i p l i c i t y e x i s t s are«;wfittenaaseequatd:'oni:l55. ^ o j u s t i f i c a t i o n w i l l be made i n t h i s t h e s i s f o r t h e H a m i l t o n i a n form. S p e c i f i c d i s c u s s i o n s c o n c e r n i n g t h e i n d i v i d u a l p a r t s o f t h e H a m i l t o n i a n o p e r a t o r can be found i n r e f e r e n c e s (f2) t h r o u g h (5). 6 V = V + V + V n (1.5) ee en nn i n w h i c h a l l t h r e e terms a r e c o u l o m b i c i n n a t u r e , and r e p r e s e n t t h e e l e c t r o n - e l e c t r o n , t h e e l e c t r o n - n u c l e a r and t h e n u c l e a r - n u c l e a r i n t e r a c t i o n s , r e s p e c t i v e l y . M a t h e m a t i c a l l y , t h e s e terms may be w r i t t e n i n terms o f e l e m e n t a r y c h a r g e s and i n t e r p a r t i c l e d i s t a n c e s as f o l l o w s : V a a - j , J . •''Zl3 <*•«> 1=1 J> 1 J n N V = I l-l e 2 / r . (1.7) en . L , L , a - l a • i = l a = l N N V = I I I Z k e 2 / r . (1.8) nn L , , a b -ab a = l b>a where e i s t h e c h a r g e on an e l e c t r o n , and i s t h e c h a r g e o f n u c l e u s a i n u n i t s o f e. The s e p a r a t i o n between any two p a r t i c l e s i s denoted by a v e c t o r r w i t h a p p r o p r i a t e s u b s c r i p t s . The H a m i l t o n i a n d e f i n e d by e q u a t i o n s 1.1 t h r o u g h 1.8 w i l l be t a k e n t o be t h e z e r o - o r d e r H a m i l t o n i a n f o r m o l e c u l e s i n s i n g l e t s t a t e s ; t h e e f f e c t i o f u n p a i r e d e l e c t r o n s w i l l be c o n s i d e r e d s e p a r a t e l y i n C h a p t e r I I I . 7 (C) The Born-Oppenheimer S e p a r a t i o n of N u c l e a r and E l e c t r o n i c M o t i o n . The z e r o f i e l d H a m i l t o n i a n d e f i n e d i n t h e p r e v i o u s s e c t i o n i s g i v e n by H = T + T + V + V + V (1,9) o n e nn en ee ' The p r e s e n c e o f V g n i n e q u a t i o n 1.9 p r e v e n t s an a n a l y t i c a l s o l u t i o n f.or,ot-fteitdmesimdepemden£-endnP^ f wnGtionSCof- o.^fiec" m o l e c u l e . The complete e i g e n p r o b l e m f o r 1.9 can be w r i t t e n as H ¥ • ( q , Q) = E V ( q , Q) (1.10) o e v r o e v r where ¥ ( q , Q) r e p r e s e n t s t h e t o t a l v -ibr-ohicrwavevfunctiony ' a n d c E i o i s the' corr.espbridingrv.ibr.oriicgernergyt* orTh1© 1'lfetefsc enf' qhandeQtdenot^eat'tn'eQse't-nofeeleetr.omicfnUGl:©a"rOe6ordiiriat'e'g}ear COQ>- . r e s p e c t i v e l y . F o r t u n a t e l y , t h e e l e c t r o s t a t i c f o r c e s t h a t govern t h e motion o f t h e e l e c t r o n s i n a m o l e c u l e a r e o f t h e same o r d e r o f magni-tude as t h o s e which govern t h e motion o f t h e n u c l e i ; as a r e s u l t , t h e l a r g e d i f f e r e n c e between t h e r e s t masses o f t h e eie.Ctrons and n u c l e i allowsJan'vapproximate s e p a r a t i o n o f e l e c t r o n i c and n u c l e a r m o t i o n s . I f one assumes t h a t t h i s s e p a r a t i o n o f n u c l e a r from e l e c t r o n i c m o t i o n i s j u s t i f i a b l e , t h e t o t a l w a v e f u n c t i o n may be s e p a r a t e d i n t o a p r o d u c t o f two w a v e f u n c t i o n s , | e v r ( q , Q) = ^ e ( q , Q) ' V (Q) ( i . l D where ^ e ( q , Q) i s t h e e i g e n f u n c t i o n o f an e l e c t r o n i c H a m i l t o n i a n , H e(Q), whose form i s : H (Q) = T + V + V + V (1,12) e e ee en nn '" w h i c h A l l terms have been p r e v i o u s l y d e f i n e d , w h i l e t h e H a m i l t o n i a n d e f i n i n g (Q) i s d e f i n e d by H Q = T n + H e (Q) (1.13) 8 H e(Q) i s d e f i n e d f o r each i n s t a n t a n e o u s n u c l e a r c o n f i g u r a t i o n , Q, and d e f i n e s t h e e f f e c t i v e p o t e n t i a l i n wh i c h t h e n u c l e i move a c c o r d i n g t o 1.13. A d i r e c t r e s u l t o f e q u a t i o n 1.11 i s t h a t t h e t r u e t o t a l e n e r g y , E q, i s t h e sum o f t h e e l e c t r o n i c e i g e n v a l u e o f H e ( Q ) , denoted by E E ( Q ) , and t h e n u c l e a r e n e r g y , E N, which i s t h e e i g e n v a l u e o f 1.13 above. Thus, i t i s p o s s i b l e t o choose a complete o r t h o n o r m a l b a s i s i n wh i c h t h e r e p r e s e n t a t i o n o f 1.9 i s as b l o c k - d i a g o n a l as p o s s i b l e i n t h e e l e c t r o n i c quantum numbers, q, and e x a c t e i g e n v a l u e s can be o b t a i n e d by d i a g o n a l l z a t i o n o f t h i s m a t r i x r e p r e s e n t a t i o n . T h i s a p p r o x i m a t e s e p a r a t i o n o f t h e t o t a l w a v e f u n c t i o n i n t o a p r o d u c t o f e l e c t r o n i c and n u c l e a r components was f i r s t p r o p o s e d by Born and Oppenheimer (6) and t h e r e f o r e b e a r s t h e i r name. F o l l o w i n g t h e present"at ;xon o f L o n g u e t - H i g g i n s ( 7 ) , one can a p p l y t h e v a r i a t i o n theorem t o '1.13 t o e x t r a c t t h e form o f ^ v r.(Q) > t a k i n g t h e b a s i s f u n c t i o n s t o be t h e s e t d e f i n e d by 1.11. I n D i r a c b r a - k e t notationctoh'ese become |evr;q,Q> == |e;q,Q> . |vr;Q> ( J . 1 4 ) The v a r i a t i o n theorem i m p l i e s t h a t f o r a pro b l e m d e f i n e d i n a complete and o r t h o n o r m a l b a s i s , t h e e s t i m a t e d e i g e n v a l u e , e, w i l l c o r r e s p o n d t o t h e t r u e e i g e n v a l u e when t h e v a r i a t i o n o f t h e w a v e f u n c t i o n s i n t h e b a s i s s e t chosen Una's*, co n v e r g e d t o a f a i t h f u l r e p r e s e n t a t i o n o f t h e t r u e m o l e c u l a r w a v e f u n c t i o n s . M a t h e m a t i c a l l y , t h e same s t a t e m e n t i s w r i t t e n as £ = < 6 ( e v r ) | H Q | e v r > / < 6 ( e v r ) | e v r > (1.15) where "6" denote s lea^small v a r i a t i o n i'm', and H Q i s d e f i n e d i n e q u a t i o n 11.13. P r e s u p p o s i n g t h a t t h e e l e c t r o n i c w a v e f u n c t i o n i s w e l l - d e f i n e d , so t h a t t h e v a r i a t i o n need o n l y be c a r r i e d o u t o v e r t h e n u c l e a r ( v i b r a t i o n -r o t a t i o n ) w a v e f u n c t i o n s , one can show t h a t t h e s u b s t i t u t i o n o f 1.13 i n t o 9 1.15 r e d u c e s t o t h e form N < 6 ( v r ) | e | v r > = < 6 ( v r ) | E (Q) + 7 ( 1 / 2 M J x 1 1 e , a a = l (<e| P 2 |.|e> + 2 <e| P ! |e> P + P 2 ) |vr> (1.16) -a —a -a —a where use has been made o f t h e o r t h o n o r m a l i t y r e l a t i o n f o r 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 s , and t h e e i g e n e q u a t i o n H e(Q) |e> = E e ( Q ) |e> (1.17) has been used t o d e f i n e |e> = ty^iq.Q) w h i c h appears i n 1.11 and 1.14. E q u a t i n g c o e f f i c i e n t s o f t h e < 6 ( v r ) | one f i n d s t h a t t h e w a v e f u n c t i o n s ^ v r ( Q ) a r e t h e e i g e n f u n c t i o n s o f a H a m i l t o n i a n w h i c h can be w r i t t e n as H 0 = T n + < e |. T n | e > + E g ( Q ) + H' (1.18) where N H' = I < e | P | e > • P /M (1.19) . -a - —a a a = l and a l l q u a n t i t i e s a r e as p r e v i o u s l y d e f i n e d . The second term i n 1.18 g i v e s r i s e t o a s m a l l mass-dependent e l e c t r o n i c i s o t o p e e f f e c t . The t h i r d t e rm d e f i n e s t h e e f f e c t i v e p o t e n t i a l energy o f t h e n u c l e i as a f u n c t i o n o f t h e i n s t a n t a n e o u s n u c l e a r c o n f i g u r a t i o n , Q. I t i s The f o u r t h t e r m o f e q u a t i o n 1.18 i s o f most i n t e r e s t , because i t r e p r e s e n t s a c o u p l i n g between t h e e l e c t r o n i c and n u c l e a r a n g u l a r momenta. The s m a l l e r t h e e x p e c t a t i o n v a l u e i n 1.19 becomes, t h e .moreicomplete t h e a s e p a r a t l o m e o f i e l S e c ' ^ w i l l T b e . t T h e B B b r n - G p p e n h e i m e r s e p a r a t i o n i s a l w a y s an a p p r o x i m a t i o n , because t h e i n t e r a c t i o n o f t h e e l e c t r o n s and n u c l e i can never be i n d e n t i c a l l y z e r o f o r a s t a b l e m o l e c u l e . P h y s i c a l l y , t h i s term w i l l be s m a l l i f t h e e l e c t r o n i c m o t i o n i s s u f f i c i e n t l y r a p i d t h a t t h e e l e c t r o n s can compensate i n s t a n t a n e o u s l y f o r s m a l l changes 10 i n t h e n u c l e a r c o n f i g u r a t i o n as t h e y o c c u r . M a t h e m a t i c a l l y , t h e e x p e c t a t i o n v a l u e i s s m a l l f o r an e l e c t r o n i c s t a t e i n which t h e f u n c t i o n a l form o f ip e(q,Q) i s n e a r l y c o n s t a n t f o r t h e v a r i a t i o n s t h a t can o c c u r i n t h e n u c l e a r c o o r d i n a t e s , Q. G e n e r a l l y , t h e s e c o n d i t i o n s w i l l be met f o r s m a l l a m p l i t u d e m o t i o n s w i t h i n non-degenerate e l e c t r o n i c s t a t e s . An e s p e c i a l l y c o n v e n i e n t r e p r e s e n t a t i o n o f t h e e l e c t r o n i c energy l e v e l s w h i c h a r e found i n t r i a t o m i c AB^-type m o l e c u l e s p l o t s t h e energy o f t h e m o l e c u l a r o r b i t a l s as a f u n c t i o n o f t h e B-A-B bond a n g l e . The c o r r e l a t i o n d i a g r a m w h i c h r e s u l t s i s r e f e r r e d t o as a Walsh d i a g r a m ( 8 ) and i s g i v e n i n F i g u r e E i J A I n Whits F i g u r e , t h e I s o r b i t a l s f o r A and B a r e o m i t t e d , but have been i n c l u d e d i n t h e numbering o f t h e o r b i t a l s . S t r i c t l y , t h i s d i a g r a m i s v a l i d o n l y f o r t h e e l e m e n t s o f t h e f i r s t row o f t h e p e r i o d i c t a b l e ; i t can be used t o p r e d i c t t h e m o l e c u l a r geometry and t h e o b s e r v e d e l e c t r o n i c t r a n s i t i o n s i f t h e o r b i t a l s a r e f i l l e d i n sequence, b e g i n n i n g a t t h e bottom, w i t h two e l e c t r o n s a s s i g n e d t o each o r b i t a l f rom t h e v a l e n c e e l e c t r o n s o f t h e m o l e c u l e . H e a v i e r m o l e c u l e s such as S0 2 can be t r e a t e d u s i n g t h e same d i a g r a m , p r o v i d e d one o m i t s t h e numbers w h i c h a r e a t t a c h e d as p r e f i x e s t o t h e m o l e c u l a r o r b i t a l s , and uses o n l y v a l e n c e e l e c t r o n s t o f i l l t h e o c c u p i e d o r b i t a l s . Thus, S0| wi>thal8nvalencetgiiect)pqns isCpfedi ;cted^to°h§ve^§c| r'ound- s t a t e ofcesy'mmet?-^ t h e p r o d u c t ( a 1 ) 2 ( b 2 ) 2 ( a 1 ) 2 ( b 2 ) 2 ( b 1 ) 2 ( a 1 ) 2 ( a 2 ) 2 ( b 2 ) 2 - ( a 1 ) 2 based on t h i s f i g u r e . F o r t r i a t o m i c m o l e c u l e s , t h e s t a b i l i t y o f a g i v e n m o l e c u l e w i t h r e s p e c t t o b e n d i n g has been found t o depend s t r o n g l y on t h e number o f e l e c t r o n s w h i c h p o p u l a t e t h e o r b i t a l s c o r r e l a t i n g w i t h 2TT i n t h e 90'*' 120* 150" 180* F i g u r e 1.1 W a l s h i d i a g r a m f o r a n o n - h y d r i d e AB,, m o l e c u l e . The I s o r b i t a l s o f A aOd B a r e o m i t t e d but have been i n c l u d e d i n t h e numbering o f t h e o r b i t a l s . 12 l i n e a r m o l e c u l e l i m i t . M o l e c u l e s w i t h one e l e c t r o n i n t h e 6a^ o r b i t a l a r e found t o have bond a n g l e s o f t h e o r d e r o f 130°(plus o r minus 10°); w h i l e two e l e c t r o n s i n t h i s o r b i t a l c o r r e s p o n d s t o a bond a n g l e o f t h e o r d e r o f 100°. I f t h e Born-Oppenheimer s e p a r a t i o n i s v a l i d , o n l y s m a l l c o n t r i b u t i o n s t o t h e v i b r i a t i o n - r o t a t i o n H a m i l t o n i a n o f 1.18 w i l l a r i s e due t o t h e n u c l e a r momentum c o u p l i n g o f 1.19. I n t h i s c a s e , t h e energy l e v e l s o f a g i v e n e l e c t r o n i c s t a t e w i l l be d e t e r m i n e d t o w e l l w i t h i n e x p e r i m e n t a l a c c u r a c y w i t h o u t i n t r o d u c i n g w a v e f u n c t i o n s o f t h e o t h e r s t a t e s w i t h w h i c h i t i n t e r a c t s . F o r t h i s r e a s o n , i t i s common p r a c t i c e t o choose t h e Born-Oppenheimer r e p r e s e n t a t i o n as t h e b e s t b a s i s i n w h i c h t o c o n s t r u c t t h e v i b r a t i o n -r o t a t i o n p r o b l e m , and t o e x t e n d t h e b a s i s s e t t o i n c l u d e o t h e r s t a t e f u n c t i o n s o n l y where t h e r e i s c l e a r e v i d e n c e o f v i b r o n i c i n t e r a c t i o n s i n t h e e l e c t r o n i c energy l e v e l s o b s e r v e d . Even when breakdowns o c c u r , t h e d i r e c t p r o d u c t r e p r e s e n t a t i o n o f 1.11 w i l l s t i l l form a c o m p l e t e and o r t h o n o r m a l b a s i s i n which t o c o n s t r u c t t h e m a t r i x r e p r e s e n t a t i o n o f t h e p r o b l e m , f r o m w h i c h e x a c t e i g e n v a l u e s a r e then o b t a i n e d by d i a g o n a l i z a t i o n . The m i x i n g o f e l e c t r o n i c s t a t e s due t o n u c l e a r momentum c o u p l i n g can t r a n s f e r i n t e n s i t y from ' a l l o w e d ' t r a n s i t i o n s t o t h o s e t h a t would be ' f o r b i d d e n ' based on a group t h e o r e t i c a l c l a s s i f i c a t i o n o f e l e c t r o n i c w a v e f u n c t i o n s f o r Q=0. T h i s i s t r u e because t h e n u c l e a r d i s p l a c e m e n t s can change t h e e l e c t r o n i c w a v e f u n c t i o n s s u f f i c i e n t l y t h a t t h e y w i l l no l o n g e r t r a n s f o r m l i k e t h e pure Born-Oppenheimer w a v e f u n c t i o n s under t h e symmetry o p e r a t i o n s o f t h e m o l e c u l a r p o i n t t ? g r o u p ? . W i t h i n t h e s e l i m i t a t i o n s , one may assume t h a t 1.18 r e p r e s e n t s the H a m i l t o n i a n w h i c h d e f i n e s t h e n u c l e a r w a v e f u n c t i o n s (Q). T h i s 13 H a m i l t o n i a n w i l l now be c o n s i d e r e d i n more d e t a i l , t o see how i t may be s e p a r a t e d i n t o r o t a t i o n a l and v i b r a t i o n a l w a v e f u n c t i o n s . 14 (D) The F a c t o r i n g o f t h e N u c l e a r K i n e t i c Energy and t h e F i e l d r F r e e H a m i l t o n i a n . F o r a g i v e n i s o t o p i c s p e c i e s o f a a m o l e c u l e , t h e e l e c t r o n i c i s o t o p e s h i f t i s a c o n s t a n t s h i f t w h i c h may be abso r b e d i n t h e e f f e c t i v e p o t e n t i a l ; i f one assumes t h a t t h e Born-Oppenheimer s e p a r a t i o n i s v a l i d , e q u a t i o n 1.18 s i m p l i f i e s t o t h e form T h i s s e c t i o n o u t l i n e s a p r o c e d u r e f o r s e p a r a t i n g t h e c l a s s i c a l k i n e t i c e n e r g y , T n > i n t o f o u r components, so t h a t one may w r i t e i t as t h e sum o f t h e k i n e t i c e n e r g i e s o f t r a n s l a t i o n , r o t a t i o n , v i b r a t i o n , and v i b r a t i o n -r o t a t i o n ( C o r i o l i s ) i n t e r a c t i o n . Two a x i s systems a r e l o g i c a l c h o i c e s i n t h e c o n s i d e r a t i o n o f t h i s p r o b lem. One sy s t e m i s c o n s i d e r e d t o be f i x e d i n t h e m o l e c u l e , and i s d e f i n e d by t h e E c k a r t c o n d i t i o n s ( s e e b e l o w ) ; t h e o t h e r w h i c h i s f i x e d i n t h e l a b o r a t o r y , i s known as t h e s p a c e - f i x e d a x i s system. These two a x i s systems w i l l be denoted by s m a l l ( x , y , z ) and c a p i t a l (X,Y,Z) l e t t e r s , r e s p e c t i v e l y . The t r a n s f o r m a t i o n between t h e two a x i s systems i s g i v e n by a m a t r i x o f d i r e c t i o n c o s i n e s (9),v.wnich a r e d e f i n e d i n terms o f t h e E u l e r a n g l e s . I n a d d i t i o n , t h e v e c t o r R w i l l be used t o denote t h e c o o r d i n a t e s o f t h e c e n t r e o f mass o f t h e m o l e c u l e , r e l a t i v e t o t h e o r i g i n o f t h e s p a c e - f i x e d a x i s , w h i l e t h e v e c t o r s r . , r . , and d. denote t h e i n s t a n t a n e o u s p o s i t i o n , e q u i l i b r i u m p o s i t i o n , and i n s t a n t a n e o u s d i s p l a c e -ment v e c t o r s o f atom i w i t h r e s p e c t t o t h e o r i g i n o f t h e m o l e c u l e -f i x e d a x i s s ystem, r e s p e c t i v e l y . Thus, by d e f i n i t i o n , H lo = T n + E e W ) (1.20) (1.21) 15 When needed, t h e components o f r^, can be w r i t t e n i n t h e form x^, y^, and Zj,, w h i l e t h e components o f cL can be w r i t t e n as Ax^, Ay^, and Az^, a l l r e f e r r i n g t o atom i . S i n c e t h e r e a r e N atoms i n t h e m o l e c u l e , t h e v e c t o r o f a l l N v a l u e s d^ can be expanded i n t o one c o n t a i n i n g 3N components, and t h i s v e c t o r i s g i v e n t h e s p e c i a l symbol X. By d e f i n i t i o n , t h e e q u i l i b -r i u m c o n f i g u r a t i o n c a n n o t be a f u n c t i o n o f t i m e , so t h a t t h e v e l o c i t y o f atom i i n t h e m o l e c u l e - f i x e d a x i s system can be w r i t t e n r . = d. (1.22) where t h e s u p e r s c r i p t (•) denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e . T n may be w r i t t e n i n terms o f v e l o c i t i e s by s u b s t i t u t i n g t h e d e f i n i t i o n P. = m v. 2" - l i i - l i n t o e q u a t i o n 1.3, from w h i c h T p = I I M. y . 2 (1.23) 1 = 1 The v e l o c i t y o f atom i i n t h e s p a c e - f i x e d c o o r d i n a t e system must be c a l c u l a t e d i n o r d e r t o expand 1.23. T h i s v e l o c i t y i s t h e sum o f t h e v e l o c i t y o f t h e o r i g i n o f t h e m o l e c u l e - f i x e d a x i s system w i t h r e s p e c t t o t h e s p a c e - f i x e d a x i s , t h e a n g u l a r v e l o c i t y o f t h e mo<l>ecule-fixed a x i s w i t h r e s p e c t t o t h e s p a c e - f i x e d a x i s , and t h e v e l o c i t y o f atom i w i t h r e s p e c t t o t h e m o l e c u l e - f i x e d a x i s system. Thus, one may w r i t e : v. = R + co x r . + d. (1.24) - l - - - l - l where to i s t h e a n g u l a r v e l o c i t y v e c t o r , c o n s i s t i n g o f t h e r e s o l v e d components o f t h e v e l o c i t y o f r o t a t i o n o f t h e m o l e c u l e - f i x e d a x i s system w i t h r e s p e c t t o t h e l a b o r a t o r y - f i x e d frame. 16 W i t h o u t l o s s o f g e n e r a l i t y , t h e m o l e c u l a r a x i s system can be d e f i n e d i n a way t h a t g r e a t l y s i m p l i f i e s t h e form o f t h e e q u a t i o n o b t a i n e d on s u b s t i t u t i o n o f e q u a t i o n 1.24 i n t o 1.23. The t r a n s l a t i o n a l k i n e t i c energy can be c o l l e c t e d i n t o one term i n t h e r e s u l t i n g e x p r e s s i o n by r e q u i r i n g t h a t no m o l e c u l a r d i s p l a c e m e n t g i v e s r i s e t o a m o l e c u l a r t r a n s l a t i o n ; t h i s i s e q u i v a l e n t t o t h e r e q u i r e m e n t t h a t t h e c e n t r e o f mass o f t h e m o l e c u l e be t h e o r i g i n o f t h e m o l e c u l e - f i x e d a x i s s y s t e m , so t h a t 0 N 0 = I M. r . (1.25) 1 = 1 1 - 1 The c o n d i t i o n t h a t t h e c e n t r e o f mass re m a i n a t t h e o r i g i n can be w r i t t e n N 0 = I M. d. (1.26) i = 1 1 1 F i n a l l y , a maximum s e p a r a t i o n o f r o t a t i o n can be o b t a i n e d i f one r e q u i r e s t h a t no m o l e c u l a r d i s p l a c e m e n t g e n e r a t e s a r o t a t i o n ; t h i s Cequrtoemeftberscfche'lEck-aast c o n d i t i o n ; v: N 0 = I M. ( r . x d.) (1.27) i = 1 1 _ 1 I n t r o d u c i n g 1.24 i n t o 1.23, and i m p o s i n g c o n d i t i o n s 1.25 t h r o u g h 1.27 a l l o w s t h e n u c l e a r k i n e t i c energy t o be e x p r e s s e d i n t h e f o u r t e r m s : T = T + T . . + T . + T (1.28) n t r a n s v i b r o t Cor N = | I M. { R • R + d. • d. + (a) x r . ) • (OKX.'r') 1 = 1 + 2 oi • d. x d. } (1.29) - - l - l i n which t h e terms i n 1.29 c o r r e s p o n d i n a one-to-one way w i t h t h e terms i n 1.28, and r e p r e s e n t t h e k i n e t i c energy o f t r a n s l a t i o n , v i b r a t i o n , r o t a t i o n , and C o r i o l i s i n t e r a c t i o n , r e s p e c t i v e l y . The 17 t r a n s l a t l o n a l k i n e t i c energy i s not i m p o r t a n t i n t h e f i e l d - f r e e H a m i l t o n i a n , and w i l l not be c o n s i d e r e d f u r t h e r . The t h i r d t e r m , which r e p r e s e n t s t h e k i n e t i c energy o f r o t a t i o n , t a k e s a s i m p l e r form i n terms o f t h e moment o f i n e r t i a t e n s o r , I , whose n i n e components a r e g i v e n by e x p r e s s i o n s o f t h e form N I I rn ( y . 2 + z , 2 ) (1.30) XX . T 1 = 1 N and I = J -m.x.y. (1.31) xy i=t 1 1 1 1 With t h i s d e f i n i t i o n , t h e k i n e t i c energy o f r o t a t i o n t a k e s a q u a d r a t i c f o r m , so t h a t T . = i co + I co (1.32) r o t - = -Here, and e l s e w h e r e , t h e s u p e r s c r i p t (t) w i l l denote t h e H e r m i t i a n a d j o i n t o f a v e c t o r o r m a t r i x . From i t s d e f i n i t i o n , t h e m a t r i x I i s s y m m e t r i c , and can be d i a g o n a l i z e d by a u n i t a r y t r a n s f o r m a t i o n . When t h i s i s done, t h e t h r e e d i a g o n a l e l e m e n t s o f I a r e c a l l e d t h e P r i n c i p a l Moments o f I n e r t i a , d e f i n e d w i t h r e s p e c t t o t h e P r i n c i p a l I n e r t i a l Axes o f t h e m o l e c u l e . The r e m a i n i n g two terms o f 1.29 i n v o l v e v i b r a t i o n a l d i s p l a c e m e n t s o f t h e m o l e c u l e , i n t h e f o r m o f t h e v e c t o r s d^. These can be s i m p l i f i e d as w e l l , by t r a n s f o r m a t i o n from 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 t o normal c o o r d i n a t e s . Normal c o o r d i n a t e s o f v i b r a t i o n c o r r e s p o n d t o t h e o b s e r v e d v i b r a t i o n f r e q u e n c i e s i n a one-to-one f a s h i o n , and have t h e added advantage t h a t t h e r e a r e no c r o s s terms c o u p l i n g v i b r a t i o n a l modes i n t h e m o l e c u l a r v i b r a t i o n a l p o t e n t i a l i n harmonic a p p r o x i m a t i o n . I f one w r i t e s t h e a t o m i c masses i n terms o f a d i a g o n a l m a t r i x M, where each mass appears t h r e e t i m e s (once f o r each c a r t e s i a n c o o r d i n a t e o f t h e atom), one can t r a n s f o r m t o mass-18 w e i g h t e d c a r t e s i a n d i s p l a c e m e n t coordinates-using.-.the t r a n s f o r m a t i o n u = X (1.33) i n w h i c h t h e v e c t o r u c o n t a i n s t h e new c o o r d i n a t e s . Normal c o o r d i n a t e s a r e r e l a t e d t o mass-weighted c a r t e s i a n d i s p l a c e m e n t s by an o r t h o g o n a l t r a n s f o r m a t i o n , £, sucftht)h~at Q = I u (1.34) g e n e r a t e s t h e v e c t o r of' normal c o o r d i n a t e s , Q. The n u l l c o o r d i n a t e s i n Q and £ e'o'BEes^owdiw^ r-otation's'inhave been - 'kept i n thfli»sl d e i f d n ' M M r i p n s o i t h a t I r e m a i n s a square m a t r i x w i t h l f l =-E = | | + (1.35) even though t h e s e n u l l c o o r d i n a t e s v a n ! s'h daceor dimg sto t\' ' "v- " ••' - . - I . c o n d i t i o n s (1.25 t h r o u g h 1.27). Here E i s t h e u n i t m a t r i x o f d i m e n s i o n 3N. U s i n g 1.33 t h r o u g h 1.35 one can'reducerth'ecvibfaati'o'nalt .?. •'• k i n e t i c energy i n terms o f t h e normal c o o r d i n a t e s , Q, t o T v i b = * 9 + " Q ( 1 ' 3 6 ) These same t r a n s f o r m a t i o n s c a s t t h e k i n e t i c eneggy o f v i b r a t i o n - r o t a t i o n i n t e r a c t i o n i n t o a more u s e f u l form based on normal c o o r d i n a t e s . B e g i n n i n g w i t h T_ = 03 • u x u (1.37) Cor -wh i c h f o l l o w s f r o m s u b s t i t u t i o n o f 1.33 i n t o 1.28, one m a y ^ r e a r r a n g e t h i s e x p r e s s i o n as a sum by means o f t h e r e l a t i o n s h i p a x b = I ( a + M ( a ) b ) e (1.38) - - a - = 5 " a where a r u n s o v e r t h e t h r e e c o o r d i n a t e components ( x , y , z ) , and e ^ i s a 19 ( a ) u n i t v e c t o r a l o n g t h e a - a x i s . The m a t r i c e s M were d e f i n e d by Meal and P o l o (10), and s e l e c t components o f t h e c r o s s - p r o d u c t u s i n g m a t r i x n o t a t i o n t h e y a r e n o t t o be c o n f u s e d w i t h t h e u n s u p e r s c r i p t e d m a t r i x o f masses used i n t h e t r a n s f o r m a t i o n 1.33. S u b s t i t u t i n g . u and u f o r a and b i n 1.38, and t a k i n g t h e d o t p r o d u c t w i t h t h e components o f co d e f i n e s ^QQT t o be T r = I co u + M { a ) u (1.39) Cor u a - = a On t r a n s f o r m a t i o n t o normal c o o r d i n a t e s u s i n g 1.34, T^ , becomes T C o r 8 ( 1 ^ 0 ) a ( a ) where t h e d e f i n i t i o n o f t h e C o r i o l i s c, m a t r i c e s g i v e n i n M e a l ' s and P o l o ' s work (10) has been u s e d : which 5s £(a) = I y ( a ) | + (1.41) S u b s t i t u t i n g t h e r e d u c e d forms o f T ... T ., and T„ i n t o 1.29, a v i b ' r o t ' Cor t h e t o t a l c l a s s i c a l n u c l e a r k i n e t i c energy becomes T n = M cof I co + Q + • Q + 2 I co Q f g ( a ) Q } (1.42) oi a T h i s e x p r e s s i o n can be e x p r e s s e d i n terms o f momenta, s i n c e t h e momentum c o n j u g a t e t o Q can be o b t a i n e d from T^ by d i f f e r e n t i a t i o n o f 1.42 w i t h r e s p e c t t o Qi On r e a r r a n g e m e n t , t h i s p a r t i a l d i f f e r e n t i a t i o n g i v e s Q = P - I co C ( a ) + J Q ( l . « ) a S u b s t i t u t i o n o f 1.43 f o r e v e r y o c c u r e n c e o f Q i n 1.42 ogives t h e f i n a l form f o r T , w h i c h i s n' T = P + • P + co+ I ' co (1.44) n - - = -20 The e f f e c t i v e moment o f i n e r t i a t e n s o r , I 1 , whi c h a p p ears i n 1.44, has components d e f i n e d by t h e e q u a t i o n ^ = ^ " « + i l a > i ' m 8 ".«> E q u a t i o n 1.45 means t h a t even i f t h e e f f e c t i v e moments o f i n e r t i a f o r a m o l e c u l e can be d e t e r m i n e d s p e c t r o s c o p i c a l l y , an a c c o u n t o f t h e v i b r a t i o n a l c o n t r i b u t i o n s t o t h e s e moments must be made b e f o r e one can c a l c u l a t e a c c u r a t e m o l e c u l a r g e o m e t r i e s . A f i n a l t r a n s f o r m a t i o n o f 1.44 produces an e q u a t i o n i n v o l v i n g o n l y a n g u l a r momenta and l i n e a r momenta c o n j u g a t e t o t h e normal c o o r d i n a t e s , By d e f i n i t i o n , t he a n g u l a r momentum about t h e a - a x i s , 3 , i s d e f i n e d as t h e p a r t i a l d e r i v a t i v e o f T R w i t h r e s p e c t t o t h e a - t h component o f t h e a n g u l a r v e l o c i t y v e c t o r , to, from which 3 = 1 o ) D I ' + G (1.46) p where an i n t e r n a l a n g u l a r momentum, G^, has been i n t r o d u c e d , d e f i n e d as G q = Q/+ £ ( a ) P (1.47) I f e q u a t i o n 1.46 i s r e w r i t t e n as t h e m a t r i x e q u a t i o n 3 = I 'a) + G (1.48) one may r e a r r a n g e e q u a t i o n 1.48 t o d e f i n e t h e a n g u l a r v e l o c i t y as a f u n c t i o n o f t h e a n g u l a r momenta, g i v e n by a) = u (3 - G) (1.49) i n which t h e i n v e r s e o f t h e m a t r i x I'has been g i v e n t h e l a b e l y. S u b s t i t u t i o n o f 1.49 i n t o 1.44 g i v e s t h e d e s i r e d r e s u l t , which i s Tn = * (9 " Q ) + U (5 " -} + * V- ' - ( 1 , 5 0 ) 21 Thus, t h e c l a s s i c a l H a m i l t o n i a n w h i c h can be d e r i v e d from 1.20 i s H = \ (3 - G ) + u (3 - G) + 1 P + • P + E (Q) (1.51) o - - - - - e A c l a r i f i c a t i o n o f t h e s i g n i f i c a n c e o f t h e i n t e r n a l a n g u l a r momentum, G^, must be made. I f d e f i n i t i o n 1.47 i s expanded by t h e e x p r e s s i o n f o r P o b t a i n a b l e from 1.43, one f i n d s t h a t G = Q + S ( a ) Q + l«a Q + £ ( a ) S ( 3 ) t Q d . 5 2 ) E c k a r t (11) p o i n t e d out t h a t G a i s not s t r i c t l y a v i b r a t i o n a l a n g u l a r momentum, s i n c e t h i s would be g i v e n by t h e f i r s t t erm o f 1.52 a l o n e . However, t h r o u g h c o n v e n t i o n a l u s e , G^ i s o f t e n r e f e r r e d t o as t h e v i b r a t i o n a l a n g u l a r momentum o f t h e m o l e c u l e , and t h i s use i s adopted i n t h i s t h e s i s as w e l l . The u s u a H t r a n s f o r m a t i o n o f t h e c l a s s i c a l H a m i l t o n i a n t o t h e quantum m e c h a n i c a l H a m i l t o n i a n i s a s i m p l e m a t t e r o f r e p l a c e m e n t o f t h e l i n e a r momenta w i t h d e r i v a t i v e s w i t h r e s p e c t t o t h e normal c o o r d i n a t e s . T h i s r e p l a c e m e n t i s n o t v a l i d , however, s i n c e e q u a t i o n 1.43 shows t h a t P i s not j u s t e q u a l t o Q but c o n t a i n s a r o t a t i o n a l c o n t r i b u t i o n as w e l l . I n a d d i t i o n , G ; i s a l s o riot c o n j u g a t e t o any c o o r d i n a t e because i t a l s o c o n t a i n s a c o n t r i b u t i o n from t h e r o t a t i o n a l m o t i o n . A r i g o r o u s t r a n s f o r m a t i o n from t h e c l a s s i c a l H a m i l t o n i a n i n g e n e r a l i z e d c o o r d i n a t e s t o t h e a p p r o p r i a t e quantum m e c h a n i c a l H a m i l t o n i a n must be used. The P o d o l s k y t r a n s f o r m a t i o n (12) o u t l i n e d i n s e c t i o n 35 o f r e f e r e n c e (5) c a n be used t o t r a n s f o r m 1.51 t o t h e quantum m e c h a n i c a l o p e r a t o r f o r m , w h i c h g i v e s H = i u* (3 - G ) + u wf*(>KA G) y* + * u* P y J *P u* + E (Q) (1.53) 22 where u i s t h e d e t e r m i n a n t o f t h e m a t r i x y. Watson (13) showed t h a t a a and as a r e s u l t , t h e o r d e r o f t h e o p e r a t o r s i n t h e f i r s t t erm o f 1.53 does not m a t t e r . T h i s a l l o w e d Watson t o r e a r r a n g e t h e f i e l d - f r e e H a m i l t o n i a n t o t h e form H Q = 1 ( 3 - G ) + y (3 - G) + I P + • P + U + E G(Q) (1.55) T h i s r e a r r a n g e m e n t g i v e s r i s e t o an e x t r a t e r m , U, w h i c h Watson showed t o be (13) U = - (h/4Tr ) 2 U y a a (1.56) a S i n c e 1.56 shows t h a t U i s a f u n c t i o n o f t h e masses and c o o r d i n a t e s o n l y , but n o t o f t h e momenta, i t can be a b s o r b e d as a s m a l l i s o t o p e - d e p e n d e n t s h i f t i n t h e p o t e n t i a l t e r m , E E ( Q ) , so t h a t t h e f i n a l form o f t h e quantum m e c h a n i c a l H a m i l t o n i a n i s i d e n t i c a l t o 1.51, w i t h t h e r e d e f i n i t i o n o f t h e e l e c t r o n i c p o t e n t i a l , E G ( Q ) . Thus, 1.51 i s t a k e n t o be t h e f i e l d - f r e e H a m i l t o n i a n f o r a m o l e c u l e i n i t s s i n g l e t e l e c t r o n i c s t a t e s . T h i s form o f t h e v i b r a t i o n - r o t a t i o n H a m i l t o n i a n can be expanded i n t o a sum o f s e v e r a l terms o f s u c c e s s i v e l y s m a l l e r o r d e r s o f magnitude. T h i s e x p a n s i o n i s e f f e c t e d by c a l c u l a t i n g t h e b i n o m i a l e x p a n s i o n o f t h e i n v e r s e e f f e c t i v e i n e r t i a t e n s o r , : y, and i s t h e s u b j e c t o f t h e n e x t s e c t i o n . 23 (E) E x p a n s i o n o f t h e V i b r a t i o n - R o t a t i o n H a m i l t o n i a n , and t h e Order o f Magnitude C l a s s i f i c a t i o n o f Terms. I n t h i s s e c t i o n , t h e v i b r a t i o n - r o t a t i o n H a m i l t o n i a n o f 1.51 w i l l be expanded i n t o a sum o f t e r m s , and t h e s e terms w i l l be c l a s s i f i e d a c c o r d i n g t o t h e i r o r d e r o f magnitude. The i n d i v i d u a l terms c o r r e s p o n d t o v a r i o u s p h y s i c a l i n t e r a c t i o n s , and a r e t h e r e f o r e u s u a l l y t r e a t e d s e p a r a t e l y when i t becomes n e c e s s a r y t o b r i n g them i n t o t h e energy l e v e l c a l c u l a t i o n o f a s p e c i f i c m o l e c u l e . T h i s approach i s n e c e s s a r y because t h e i n d i v i d u a l terms i n t h e expanded H a m i l t o n i a n b e a r q u i t e a complex r e l a t i o n s h i p t o energy l e v e l s o b s e r v e d i n t h e sp e c t r u m . N i e l s e n (14,15) used a Van V l e c k transformation(5,20,21) t o g i v e a v e r y g e n e r a l H a m i l t o n i a n and d i s c u s s e d t h e v a r i o u s terms t h a t r e s u l t . An a l t e r n a t i v e method o u t l i n e d here was f i r s t s u g g e s t e d by Oka (16), and i s based on t h e e x p a n s i o n o f t h e y t e n s o r ( s u c h as t h a t g i v e n i n Watson(13)) t o a s s i g n o r d e r s o f magnitude t o t h e v a r i o u s terms t h a t r e s u l t . F o l l o w i n g t h i s p r o c e d u r e , i n d i v i d u a l c o n t r i b u t i o n s t o t h e o b s e r v e d energy l e v e l s o f a m o l e c u l e can be o b t a i n e d u s i n g s t a n d a r d p e r t u r b a t i o n t h e o r y . The t e n s o r y i s expanded as a ^ a y l o r s e r i e s i n t h e normal c o o r d i n a t e s . In ; i t h e s p r o e e s s v d e r i \ v a t i y e s h o f i i t h e t i n e r t i a l o t e n s o r h o f f i b h e form ^ 3 ) = ^ a ^ V e " ( 1 ' 5 7 ) must be e v a l u a t e d . ^ a t - n a r i d S f l e i ^ i ; f r f R e ' d e f i n i t i o n il?21»risnSubsti>tuted M t o ^ e q ~o t h a t one may e q u a l l y w e i i d e f i n e 1 5/ i n terms o^ a d i f f . a e r a t o r on I 1 .. A.-jt and Henrv ' .17; showed t h a t i f t h e d~ a 3 -. s u b s t i t u t e d :hto e q u a t i o n s 1,30 and lo31 w h i c h de .' 24 t h e moment o f i n e r t i a t e n s o r , t h e n t h e m a t r i x V d e f i n e d i n e q u a t i o n 1.45 has e lements I ' = 1 R e + I a < a 3 ) ( L + i I a < a Y ) ( I a < 6 0 ) (Ml , Y,'<5 (1:58) where t h e s u p e r s c r i p t "e" d e n o t e s t h a t e q u i l i b r i u m v a l u e s a r e t o be t a k e n . A change o f n o t a t i o n t o t h a t o f 3 x 3 m a t r i c e s a l l o w s one t o c a s t t h i s e x p r e s s i o n i n a more s i m p l e form. L e t t h e 3 x 3 m a t r i x denoted by § k be composed o f t h e e l e m e n t s a ^ a ^ . "Tilifien a s q u a r e m a t r i x o f o r d e r 3N-6 can be d e f i n e d by t h e e x p r e s s i o n a = I a Q k (1.59) k U s i n g s i m i l a r r e a s o n i n g , i t i s c o n v e n i e n t t o d e f i n e two more m a t r i c e s , w h i c h a r e b k = ( i V ^ a k ? ( I e f * (1.60) and f i n a l l y , t h e m a t r i x b o f o r d e r 3N-6 by a n a l o g y t o 1.59. F o l l o w i n g Watson ( 1 3 ) , t h e d e f i n i t i o n o f a n o t h e r i n e r t i a l m a t r i x , I " by t h e e x p r e s s i o n • I " = I 6 + \ | (1.61) a l l o w s t h e r e - e x p r . e s s i o n o f e q u a t i o n 1.58 i n t h e form I ' = I " ( I 6 ) " 1 I " (1.62) T h i s h can be v e r i f i e d by d i r e c t s u b s t i t u t i o n o f 1.61 i n t o 1.62 and e x p a n s i o n u s i n g 1.59. The i m p o r t a n t f e a t u r e w h i c h makes 1.62 u s e f u l i s t h a t i t can be t r i v i a l l y i n v e r t e d , from which one f i n d s i m m e d i a t e l y t h a t u 25 H = (I')"1 = (I")"1 f (I")'1 (1.63) Rearrangement o f 1.61 g i v e s l" = ( | e ) * (| + i | ) ( I 6 ) 1 (1.64) and a t r i v i a l i n v e r s i o n o f t h i s e x p r e s s i o n can be used t o s u b s t i t u t e f o r t h e i n v e r s e o f I " i n 1.63, wh i c h f i n a l l y g i v e s y = ( I e ) _ i (| + i b j " 2 ( I e ) ~ * (1.65) B i n o m i a l e x p a n s i o n o f t h e c e n t r a l term i n 1.65 y i e l d s t h e f i n a l f orm f o r y, which i s o b t a i n e d by s u b s t i t u t i n g (E + | b ) " 2 = | - b + 3b 2/4 - | b 3 + ... (1.66) f o r t h e m i d d l e term i n 1.65. R e t u r n i n g t o t h e H a m i l t o n i a n , one may r e w r i t e 1.51 i n t h e form H =.H + H (1.67) o v v r where t h e v i b r a t i o n a l p a r t i s g i v e n by H y = k P + • P + E e ( Q ) (1.68) and t h e v i b f a t i o n p r d t a t i o n p a r t by Hv r = i (3 - G ) + y (3 - G) (1.69) I t i s t h e form o f w h i c h must be expanded, s i n c e t h e f i r s t t erm i n H i s t f a c t a b l e r a l r e a d y .alByaa;nalbgya,wd.thg;l.'49t;haln'eWs operator., Q = ( I e ) ~ * (3 - G) (1.70) i s d e f i n e d , so t h a t t h e v i b r a t i o n - r o t a t i o n H a m i l t o n i a n t a k e s t h e p a r t i c u l a r l y s i m p l e form "or :h- • • v : H v r = ^ 0 + ( l + ^ ) _ 2 ^ (1.71) S u b s t i t u t i o n o f 1.66 l e a d s t o t h e m a t r i x e x p r e s s i o n f o r t h e v i b r a t i o n -r o t a t i o n H a m i l t o n i a n e x p a n s i o n : 26 H v r = * fif § - i fi+ b fl +(3/8) ft+ b 2 - ... (1.72) The f i r s t term i n t h i s e x p r e s s i o n d e f i n e s a z e r o - o r d e r v i b r a t i o n -r o t a t i o n H a m i l t o n i a n , and subsequent h i g h e r - o r d e r terms a r e r e a l l y c o r r e c -t i o n s t o t h i s z e r o t h o r d e r e x p r e s s i o n . The most u s e f u l f e a t u r e o f t h i s form i s t h a t t h e c o n t r i b u t i o n o f each term i s l a b e l l e d by t h e exponent o f b. Oka (16) p o i n t e d out t h a t t h e d e f i n i t i o n o f t h e m a t r i x b r e q u i r e s i t t o be o f t h e same o r d e r o f magnitude as t h e Born-Oppenheimer e x p a n s i o n p a r a m e t e r , i K , w h i c h i s n u m e r i c a l l y e q u a l t o t h e r a t i o (m e/M n)* o r a p p r o x i m a t e l y 0.1. I f a t a b u l a t i o n i s made o f t h e terms w h i c h a r i s e from t h e e x p a n s i o n 1.72 t o g e t h e r w i t h t h o s e o f 1.68 a c c o r d i n g t o t h e i r o r d e r s o f magnitude, one o b t a i n s t h e r e s u l t s p r e s e n t e d i n T a b l e 1.1. A l l o p e r a t o r s t h a t appear i n T a b l e 1.1 a r e d i m e n s i o n l e s s , and t h e r e f o r e a l l c o e f f i c i e n t s a r e i n energy u n i t s . A l l summations i n t h i s T a b l e a r e u n r e s t r i c t e d . Thus, T a b l e 1.1 c o m p l e t e l y s p e c i f i e s t h e terms which a r i s e on e x p a n s i o n o f t h e H a m i l t o n i a n g o v e r n i n g n u c l e a r m o t i o n f i r s t g i v e n i n 1.20. I n o r d e r t o make t h e o p e r a t o r s d i m e n s i o n l e s s , i t i s c o n v e n i e n t t o i n t r o d u c e t h e s c a l i n g f a c t o r X = 4Tt.2cco /h = X */(h/2Tr) (1.73a) r n r k r where co^ and X a r e , r r e s p e c t i v e l y , t h e harmonic wavenumber and f o r c e c o n s t a n t a s s o c i a t e d w i t h t h e r t h normal mode. I n a d d i t i o n , t h e v i b r a t i o n a l o p e r a t o r s P f and Q r have been r e p l a c e d by t h e i r d i m e n s i o n l e s s e q u i v a l e n t s , q r = QrVr* and p r = P r /(hY r*/2Tr) (1.73b) F o r c o n s i s t e n c y , t h e o p e r a t o r G has been r e d e f i n e d i n u n i t s o f (h/2iT), so t h a t G a = I ^ q r P s K > / U' 7*) V • s T a b l e 1.1 Terms i n t h e R o v i b r a t i o n a l H a m i l t o n i a n , H / ( h c ) , A r r a n g e d by Order o f Magnitude and Power o f 3. Order o f Magnitude C o e f f i c i e n t o f 3° 3 1 3 2 r v i b L. r r s t ^ r ^ s x r s t K 2V .. .+1/24 y <J) . q q q.q v i b L. r r s t u T ^ S T ^ U r s t u +y (h/8TT 2c) y 6 (G 2 -2G 3 + 3 2 ) ^ K a a a a a a a K 3V ., + q u i n t i c anharmonic v i b ^ (a3) a3r | _ e e U * a 3 3 a a 3 Y r a a 33 K^V ., + s e x t i c anharmonic v i b or (G G„ -(G 3 n + G n3 ) +3 3„) a3?rs i a 3 a 3 3 c r a 3 ( Y Y ) 2 I I e l 6 ^ y rV a a L& B3 vv i b i s a t y p i c a l ( o r average) v i b r a t i o n a l f r e q u e n c y -of t h e - m o l e c u l e . Adapted from r e f ( 3 5 ) . 28 F i n a l l y , t h e $ s t and a r e c u b i c and q u a r t i c anharmonic p o t e n t i a l f o r c e c o n s t a n t s , d e f i n e d from t h e p o t e n t i a l energy e x p r e s s i o n E (Q) = V/(hc) = | I a) q 2 + (1/6) £ <j> q q q r r s t + ( 1 / 2 4 ) { Kstu % % \ % + ( 1 ' 7 3 ) r s t u A t r a n s f o r m a t i o n d e s c r i b e d i n more d e t a i l i n t h e n e x t s e c t i o n s has a l s o been used i n t h i s tafcibe, i n o r d e r t h a t t h e o f f - d i a g o n a l e l e m e n t s o f t h e u t e n s o r v a n i s h , t o l e a v e o n l y 3 p r i n c i p a l i n v e r s e moments o f i n e r t i a . One must now c o n s i d e r t h e p r o c e d u r e s r e q u i r e d t o s p e c i a l i z e t h i s g e n e r a l form f o r t h e H a m i l t o n i a n t o t h a t which can be used t o p r e d i c t t h e m o l e c u l a r t r a n s i t i o n s o b s e r v e d i n a g i v e n s p e c t r u m . 29 (F) A p p l i c a t i o n o f t h e G e n e r a l H a m i l t o n i a n t o Asymmetric Top S p e c t r a . The p r e c e d i n g s e c t i o n s d e s c r i b e d how t h e n u c l e a r H a m i l t o n i a n c o u l d be expanded, and t h e r e s u l t i n g terms a r r a n g e d i n an o r d e r o f magnitude c l a s s i f i c a t i o n , f o l l o w i n g Oka ( 1 6 ) . A s i m p l e way o f w r i t i n g Oka's scheme i s t o r e p l a c e a term i n t h e H a m i l t o n i a n s u c h as ( q , p ) n 3 m by t h e s h o r t form h . T a b l e 1.1 i s t h e r e b y r e d u c e d t o t h e form ofi. T a b l e 1.2. The nm J f u l l H a m i l t o n i a n H = H + H + H (1.76) v r v r i s now t h e sum o f a p u r e l y v i b r a t i o n a l p a r t H H V = 7 h . (1.77) v L _ nO n = 2 a r o t a t i o n a l p a r t H r = h Q 2 (1.78) and v . i b r a t i o m o ? . - r o t a t i o n c r o s s t e r m s , w h i c h a r e e v e r y t h i n g e l s e : oo H v r = h 1 2 + I 7 ( h n l + hn2> ( 1 ' 7 9 ) n = 2 The f i r s t t erm o f e q u a t i o n 1.77, h ^, i s t h e H a m i l t o n i a n f o r t h e harmonic o s c i l l a t o r . Normal c o o r d i n a t e a n a l y s i s i n terms o f t h e harmonic a p p r o x i m a t i o n assumes t h a t t h e n u c l e a r o s c i l l a t i o n s a r e o f s u f f i c i e n t l y s m a l l a m p l i t u d e t h a t t h e p o t e n t i a l energy i s w e l l d e s c r i b e d by t h e q u a d r a t i c terms i n 1.75 a l o n e . I n r e a l m o l e c u l e s , t h e n u c l e a r o s c i l l a t i o n s a r e not i n f i n i t e s i m a l , even though t h e i r a m p l i t u d e s a r e s m a l l . As a r e s u l t , r e a l m o l e c u l e s behave as anharmonic o s c i l l a t o r s , so t h a t a s t r i c t s e p a r a t i o n o f t h e m o l e c u l a r o s c i l l a t i o n s i n t o a s u p e r p o s i t i o n o f normal v i b r a t i o n s i s not p o s s i b l e . S i n c e t h e p o t e n t i a l energy o p e r a t o r must re m a i n i n v a r i a n t under a l l symmetry 30 T a b l e 1.2 Terms i n t h e R o v i b r a t i o n a l H a m i l t o n i a n C l a s s i f i e d by Order o f Magnitude and Powers o f (q , p ) and 3. Order o f Magnitude Power o f 3° 3 1 3 2 -v i b h 2 0 " l b v i b feo J 1 K \ i b \o h 2 1 H 0 2 K 3V .. v i b h 5 0 h 3 1 h 1 2 v i b h 6 0 \ l h 2 2 Taken from r e f e r e n c e ( 3 5 ) . 31 o p e r a t i o n s o f t h e m o l e c u l a r p o i n t group (see Appendix I I ) , terms i n 1.75 t h a t i n v o l v e a n t i s y m m e t r i c v i b r a t i o n a l modes t o odd powers must v a n i s h . E m p i r i c a l l y , t he v i b r a t i o n a l term v a l u e s o f an anharmonic o s c i l l a t o r can be g i v e n by G(v) = I co (v + ?) + 1 1 x (v + i ) ( v +!) + ... (1.80) r r r r S>B R S R S where t h e summations r u n over a l l 3N-6 normal modes o f t h e m o l e c u l e . Here, t h e x a r e a n h a r m o n i c i t y c o n s t a n t s , and t h e co have been d e f i n e d ' r s . r i n e q u a t i o n 1.73. E q u a t i o n 1.80 i s v a l i d o n l y f o r m o l e c u l e s i n wh i c h a l l v i b r a t i o n a l modes t r a n s f o r m as non-degenerate i r r e d u c i b l e r e p r e s e n t a t i o n s (such as C, ). 2v From T a b l e 1.2 one can see t h a t t h e l e a d i n g v i b r a t i o n a l t e r m , h,,Q,r i n 1.77, s h o u l d be about two o r d e r s c o f magnitude l a r g e r t h a n t h e l e a d i n g r o t a t i o n a l term h^,,. T h i s f a c t was used by W i l s o n and Howard (18) and N i e l s e n (4,15) t o o b t a i n an e f f e c t i v e r o t a t i o n a l H a m i l t o n i a n , v a l i d f o r a p a r t i c u l a r v i b r a t i o n a l s t a t e , u s i n g a c o n t a c t t r a n s f o r m a t i o n ( 5 , 2 0 , 2 1 ) . The e f f e c t i v e H a m i l t o n i a n t h a t t h e y d e r i v e d was i n t e n d e d f o r t h e c a s e where no l a r g e m a t r i x e l e m e n t s remain w h i c h c o n n e c t t h e v i b r a t i o n a l s t a t e o f i n t e r e s t t o o t h e r v i b r a t i o n a l s t a t e s o f n e a r l y t h e same e n e r g y , and i s not v a l i d o t h e r w i s e . The use o f a c o n t a c t t r a n s f o r m a t i o n u n f o r t u n a t e l y t e n d s t o o b s c u r e t h e meanings o f t h e par a m e t e r s o f t h e e f f e c t i v e H a m i l t o n i a n and t h e s e a r e l e s s easy t o i n t e r p r e t i n terms o f m b i e c u l a r p a r a m e t e r s t h a n t h e o r i g i n a l p a r a m e t e r s . An e q u i v a l e n t p r o c e d u r e uses p e r t u r b a t i o n t h e o r y i n s t e a d . The e f f e c t i v e r o t a t i o n a l H a m i l t o n i a n i s chosen t o be t h e symmetric t o p r o t a t i o n a l H a m i l t o n i a n n 2 Q , p l u s t h e e f f e c t s o f t h e v i b r a t i o n - r o t a t i o n terms t r e a t e d (as f a r as p o s s i b l e ) by means o f p e r t u r b a t i o n t h e o r y . 32 Sometimes p e r t u r b a t i o n t h e o r y cannot be used t o t r e a t a l l t h e terms o f 1.79 because t h e v i b r a t i o n a l s t a t e o f i n t e r e s t l i e s v e r y c l o s e t o a n o t h e r one, t o w h i c h i t i s c o n n e c t e d by l a r g e m a t r i x e l e m e n t s ; i n t h i s c a s e t h e i n t e r a c t i n g s t a t e s must be t a k e n t o g e t h e r , and t h e e nergy l e v e l s o b t a i n e d as Roots o f a l a r g e r m a t r i x . I n m o l e c u l a r s p e c t r a , t h i s s i t u a t i o n shows up e i t h e r i n t h e form o f l o c a l r o t a t i o n a l p e r t u r b a t i o n s o r as p e r t u r b a t i o n s a f f e c t i n g a l l l e v e l s , and i s d i s c u s s e d i n s e c t i o n (G) below. Not a l l t h e v i b r a t i o n - r o t a t i o n c r o s s terms o f 1.79 need n e c e s s a r i l y be c o n s i d e r e d , because o f symmetry r e s t r i c t i o n s . Any termti i n t h e H a m i l t o n i a n has t o be t o t a l l y s y m m e t r i c under a l l t h e symmetry o p e r a t i o n s o f t h e p o i n t g r o u p , so t h a t i n m o l e c u l e s w i t h e l e m e n t s o f symmetry, many o f t h e terms i n 1.79 v a n i s h because t h e y a r e n o t a t o t a l l y s y m m e t r i c . Symmetry r e s t r i c t i o n s t h a t a p p l y t o t h e C m o l e c u l e s s t u d i e d i n t h i s t h e s i s can be o b t a i n e d from t h e group t h e o r e t i c a l arguments o f Appendix I I . One b e g i n s by c h o o s i n g a complete and o r t h o n o r m a l b a s i s which spans t h e space o f m o l e c u l a r v i b r a t i o n s . F o r c o n v e n i e n c e , t h e e i g e n f u n c t i o n s o f t h e harmonic o s c i i L l a t o r H a m i l t o n i a n , h^Q? were chosen f o r t h i s p u r p o s e , denoted by t h e v i b r a t i o n a l s p e c i f i c a t i o n s { v } . A s i m i l a r c h o i c e o f a s e t o f r o t a t i o n a l s p e c i f i c a t i o n s i s not n e c e s s a r y a t t h i s s t a g e , s i n c e t h e c o r r e c t i o n s t o t h e e f f e c t i v e r o t a t i o n a l H a m i l t o n i a n o f 1.78 d e r i v e d from 1.79 a r e most u s e f u l l y e x p r e s s e d as c o e f f i c i e n t s o f v a r i o u s p r o d u c t s o f t h e o p e r a t o r s 3a. I n t h e language o f p e r t u r b a t i o n t h e o r y , t h e z e r o t h -o r d e r H a m i l t o n i a n w h i c h d e f i n e s t h e v i b r a t i o n a l b a s i s i f i i i n c t i o n s i s t h e r e f o r e . 33 H° = .h = 'I <o <p 2 + q/) (1.81) r and the Hamiltonian to be treated by perturbation theory i s given by o o o o H1 = y H'- ' = H •+ I h . (1.82) • L n m vr L -> nO m = 0 n = 3 equivalent to the sum of 1.77 and 1.79, excluding h,>Q> the f i r s t term of 1.77. In defining 1.82, each term in 1.79 has been relabelled using an arbitrary running index m. When perturbation theory i s applied to the sum of 1.81 and 1.82, many contributions to the energy levels appear, of which the f i r s t few are E ( 0 ) = < v | H° | v > (1.83) E ( D = ^ E ( 1 ) = y < v I H ' |:v > (1.84) m m m m -(2) _ y E (2) _ y y <v| Hm' |v'><y'| H ' |v> mn mn v' E 0 - E , 0 v v (1.85) The total energy i s obtained from the sum of a l l such expressions, so that 00 E = I E ( k ) (1.86) k = 0 gives the vibrational corrections to the effective rotational Hamiltonian of equation 1.78. Equation 1.86 w i l l contain the vibrational term value expression for the anharmonic oscillator given in 1.80, as well as expressions in which matrix elements of vibrational operators appear as coefficients of various products of the rotational angular momentum operators 3^. 34 The l a t t e r terms a re t h e d e s i r e d c o r r e c t i o n s t o t h e r o t a t i o n a l H a m i l t o n i a n o f 1.78, w h i l e e q u a t i o n 1.80 c o n t r i b u t e s t o t h e band o r i g i n o f t h e r r o t a t i o n a l energy l e v e l s . Thus t h e e i g e n v a l u e s o f t h e f u l l v i b r a t i o n -r o t a t i o n H a m i l t o n i a n o f 1.76 a r e g i v e n by t h e sum o f t h e anharmonic term v a l u e s , G ( v ) , and t h e e i g e n v a l u e s o f t h e r o t a t i o n a l H a m i l t o n i a n , h ^ j c o r r e c t e d by t h e o p e r a t o r e x p r e s s i o n s o f 1.86. The advantage o f t h e c l a s s i f i c a t i o n scheme o f T a b l e 1.2 i s t h a t one can e a s i l y e v a l u a t e t h e r e l a t i v e i m p o r t a n c e o f t h e v a r i o u s terms t h a t appear i n t h e t r a n s f o r m e d H a m i l t o n i a n . Because t h e commutation r e l a t i o n q p - p q = i (h/2TT) (1.87) r e l a t e s a q u a d r a t i c e x p r e s s i o n i n q and p t o t h e c o n s t a n t i ( h / 2 f r ) , t h e p e r t u r b a t i o n t r e a t m e n t w i l l r e d u c e t h e o r d e r o f t h e v i b r a t i o n a l o p e r a t o r s by 2 i n t h e t r a n s f o r m e d H a m i l t o n i a n . On t h e o t h e r hand, t h e r o t a t i o n a l a n g u l a r momenta obey t h e commutation r e l a t i o n 3 x 3 = 1 (h/2TT) 3 (1.88) w h i c h means t h a t q u a d r a t i c terms i n t h e components o f 3 w i l l be r e d u c e d by o n l y one power i n 3. Thus, i t i s p o s s i b l e t o drop e i t h e r one power i n 3 o r two powers i n q and p by t h e p e r t u r b a t i o n t r e a t m e n t . F o r example, i f t h e two terms h, „ ( o f o r d e r o f magnitude K 3V ., ) and h ( o f maqnitude kl 3 v i b mn K^V^^) a r e t r e a t e d by means o f second o r d e r p e r t u r b a t i o n t h e o r y , t h e y g i v e r i s e t o d i a g o n a l terms ( o f magnitude K ^ a + ' 3 ) °f t n e form a n d < v I hVm,£+n-J I v > ( 1 ' 9 0 ) i n t h e t r a n s f o r m e d H a m i l t o n i a n . Even though t h e h i g h e s t o r d e r terms i n T a b l e s 1.1 and 1.2 a r e o f o r d e r K 4V , terms o f h i q h e r o r d e r i n < a r e v i b ' - 3 i m p l i e d by t h e b i n o m i a l e x p a n s i o n o f e q u a t i o n s 1.66 and 1.72. The 35 t r u n c a t i o n o f b o t h t a b l e s a t terms o f o r d e r K 1* i s c o m p l e t e l y a r b i t r a r y , and f o r t h i s r e a s o n t h e summations i n e q u a t i o n s 1.77, 1.79 and 1.82. t h r o u g h 1.86 a r e g i v e n w i t h i n f i n i t e upper l i m i t s . Terms of. h i g h e r o r d e r c o u l d be added i f needed, but t h e terms g i v e n i n T a b l e s 1.1 and 1.2 a r e s u f f i c i e n t t o f i t p r e s e n t l y o b t a i n a b l e d a t a t o w i t h i n e x p e r i m e n t a l u n c e r t a i n t y . F o r t h i s r e a s o n , a t r u n c a t i o n c r i t e r i o n must be adopted. One may compute t h e o r d e r o f magnitude o f a g i v e n c o n t r i b u t i o n t o t h e energy e x p r e s s i o n o f 1.86 from t h e r e l a t i o n s h i p 0 ( E ( k ) ) = n 0 (H"-)/0(Ae ) (1.91) mn.. • m mn... mn. .. where Ae i s t h e denominator o f t h e a p p r o p r i a t e p e r t u r b a t i o n t h e o r y e x p r e s s i o n from e q u a t i o n s l i k e 1.85, d e f i n e d f o r each v a l u e o f m and n. F o l l o w i n g Oka ( 1 6 ) , terms a r e n e g l e c t e d i f t h e i r o r d e r o f magnitude f r o m 1.91 i s s m a l l compared t o t h e e x p e r i m e n t a l u n c e r t a i n t y o f t h e d a t a o b s e r v e d . 36 (G) P e r t u r b a t i o n s . In terms o f t h e e n t r i e s o f T a b l e 1.1, t h e r e a r e two main s o u r c e s from w h i c h p e r t u r b a t i o n s can a r i s e . I f t h e p e r t u r b a t i o n a r i s e s from terms i n t he f i r s t column ( o t h e r t h a n h , ^ ) , t h e i n t e r a c t i o n i s termed an a anharmonic r e s o n a n c e , s i n c e i t a r i s e s from t h e e f f e c t s o f c u b i c and h i g h e r o r d e r c o n t r i b u t i o n s t o t h e p o t e n t i a l e n e r g y . C r o s s terms between 3 and G g i v e r i s e t o what a r e c a l l e d C o r i o l i s p e r t u r b a t i o n s . A v i b r a t i o n a l p e r t u r b a t i o n c o n s i s t s o f a s h i f t i n t h e p o s i t i o n o f a v i b r a t i o n a l energy l e v e l , accompanied by a change i n i t s r o t a t i o n a l c o n s t a n t s . A t t h e same t i m e , a m i x i n g o f t h e e i g e n f u n c t i o n s o f t h e two v i b r a t i o n a l s t a t e s . o c c u r s . The s p e c i a l c a s e o f anharmonic r e s o n a n c e between t h e v i b r a t i o n a l l e v e l s (v ,v ) and (v + 2, v . - .1) i s due t o a c u b i c anharmonic term o f r s r s th e f orm d> q 2 q a n d ' i s c a l l e d a F e r m i r e s o n a n c e . (Only t h e quantum r r s ^ r s 1 M numbers which a r e d i f f e r e n t i n t h e i n t e r a c t i n g s t a t e s have .been s e l e c t e d i n t h i s l a b e l l i n g c o n v e n t i o n ) . F e r m i r e s o n a n c e i s p o s s i b l e whenever t h e f u n d a m e n t a l has t h e same symmetry s p e c i e s as one o f t h e symmetry s u b l e v e l s o f 2v^. I n g e n e r a l , a l l v i b r a t i o n a l l e v e l s a s s i g n e d t o a m u l t i p l y e x c i t e d f u n d a m e n t a l w i l l be a f f e c t e d by F e r m i r e s o n a n c e , which i s o f t e n t h e l i m i t i n g f a c t o r i n t h e d e t e r m i n a t i o n o f a t r u e m o l e c u l a r f o r c e f i e l d f rom o b s e r v e d v i b r a t i o n a l f r e q u e n c i e s . In m o l e c u l e s f o r which v r and v g have d i f f e r e n t symmetry s p e c i e s , t h e s t a t e ( v r , v g ) can s t i l l be p e r t u r b e d by t h e s t a t e (v - 2, \^ + 2) f o r a l l v i b r a t i o n a l l e v e l s f o r which'v > 2. T h i s anharmonic r e s o n a n c e r i s termed a D a r l i n g - D e n n i s o n r e s o n a n c e , and i s caused by terms i n <t> q Z q 2 . r r s s r s 37 V i b r a t i o n - r o t a t i o n ( C o r i o l i s ) p e r t u r b a t i o n s o c c u r between r o t a t i o n a l l e v e l s o f v i b r a t i o n a l s t a t e s b e l o n g i n g t o d i f f e r e n t i r r e d u c i b l e r e p r e s e n t -a t i o n s . The symmetry r e q u i r e m e n t i s t h a t t h e d i r e c t p r o d u c t o f t h e i r i r r e d u c i b l e r e p r e s e n t a t i o n s c o n t a i n s t h e r e p r e s e n t a t i o n o f a r o t a t i o n , because G i s an a n g u l a r momentum which t r a n s f o r m s l i k e a r o t a t i o n . The C o r i o l i s o p e r a t o r -2 G • y • 3 i s l i n e a r i n 3, so t h a t i t l e a d s t o a mutual r e p u l s i o n o f two " u n p e r t u r b e d " v i b r a t i o n a l l e v e l s w hich i n c r e a s e s w i t h r o t a t i o n a l quantum number. As a r e s u l t , t h e v i b r a t i o n a l l e v e l h i g h e r i n energy w i l l d i s p l a y l a r g e r r o t a t i o n a l c o n s t a n t s , and t h e l o w e r energy l e v e l w i l l d i s p l a y s m a l l e r r o t a t i o n a l c o n s t a n t s t h a n t h o s e i t would have i f t h e r e were n o . C o r i o l i s i n t e r a c t i o n . The magnitude o f t h i s i n t e r a c t i o n i s i n v e r s e l y p r o p o r t i o n a l t o t h e v i b r a t i o n a l f r e q u e n c y d i f f e r e n c e between t h e two i n t e r a c t i n g s t a t e s , and i t s dependence on th e r o t a t i o n a l quantum numbers s e r v e s t o i d e n t i f y i t ; a n h a r m o n i c i t y p e r t u r b a t i o n s have no such dependence. Only l e v e l s w i t h t h e same r o v i b r o n i c symmetry s p e c i e s and t h e same v a l u e o f 3 may i n t e r a c t i n any p e r t u r b a t i o n . I f t h e r o t a t i o n a l c o n s t a n t s o f two i n t e r a c t i n g v i b r a t i o n a l s t a t e s a l l o w t h e u n p e r t u r b e d t e r m v a l u e s , p l o t t e d as a f u n c t i o n o f 3, t o c r o s s , t h e n t h e r o t a t i o n a l l e v e l s i n t h e n e i g h b o r h o o d o f t h e c r o s s i n g w i l l be s h i f t e d most. To r e p r o d u c e t h e energy l e v e l p a t t e r n o b s e r v e d i n such c a s e s , t h e i n t e r a c t i n g s t a t e s must be c o n s i d e r e d t o g e t h e r , and t h e energy l e v e l s o b t a i n e d as e i g e n v a l u e s o f a l a r g e r m a t r i x . j T h i s . p r o c e d u r e changes t h e meaning o f t h e m o l e c u l a r c o n s t a n t s o b t a i n e d , so t h a t t h e g e n e r a l e x p r e s s i o n s f o r t h e s e c o n s t a n t s d e r i v e d from T a b l e s 1.1 and 1.2 must be m o d i f i e d t o e x c l u d e c o n t r i b u t i o n s t o t h e p a r a m e t e r s t h a t have been removed i n t h e d i a g o n a l i z a t i o n . I t i s common p r a c t i c e t o i n d i c a t e r e d e f i n i t i o n by a d d i n g 38 a s u p e r s c r i p t a s t e r i s k t o t h e m o l e c u l a r c o n s t a n t s a f f e c t e d . P a r a m e t e r s o b t a i n e d i n t h i s way mean v e r y l i t t l e u n l e s s a c o m p l e t e d e s c r i p t i o n o f t h e methods used t o d e t e r m i n e them accompanies t h e i r t a b u l a t i o n . 39 (H) The R o t a t i o n a l H a m i l t o n i a n . The l e a d i n g t e r m i n t h e r o t a t i o n a l H a m i l t o n i a n i s d e f i n e d by t h e e x p a n s i o n o f h^,,, from w h i c h H = I ( h / 8 ^ 2 c ) y 6 0 2 (1.92) r L aa a a T h i s e q u a t i o n i s u s u a l l y w r i t t e n i n t h e more f a m i l i a r form H = X 3 2 + Y J 2 + Z 3 2 (1.93) r x y z ' where ( i n cm u n i t s ) X = (h/87T 2c) y x x = h / ( 8 T T 2 d x x ) (1.94) w i t h s i m i l a r e q u a t i o n s h o l d i n g f o r Y and Z. E q u a t i o n s - 1 . 9 3 and 1.94 d e f i n e t h e H a m i l t o n i a n f o r t h e r i g i d r o t a t o r . Because t h e symmetric t o p e i g e n f u n c t i o n s o f 32 and 3^ form a complete and o r t h o n o r m a l b a s i s o f t h e r o t a t i o n a l s p e c i f i c a t i o n s , t h e y a r e chosen t o be t h e b a s i s i n whi c h t h e m a t r i x e l e m e n t s f o r H 0 a r e e v a l u a t e d . . One cannot d e f i n e r an a x i s about which an asymmetric t o p m o l e c u l e w i l l r o t a t e w i t h o u t n u t a t i o n , and t h i s means t h a t no component o f 3 i s a c o n s t a n t o f t h e m o t i o n . F o r a g i v e n v a l u e o f 3, t h e m a t r i x e l e m e n t s o f H 0 can be a ' r computed from t h e m a t r i x e l e m e n t s i n symmetric t o p b a s i s from t h e s e t < J,K | H^ 0 | 3,K' > . D i a g o n a l i z a t i o n o f t h i s m a t r i x d e f i n e s t h e e i g e n f u n c t i o n s o f H r ° , r e l a t e d by t h e l i n e a r t r a n s f o r m a t i o n | 3,T > = I a T | 3,K > (1.95) K x t o t h e symmetric r o t o r b a s i s f u n c t i o n s . The c o e f f i c i e n t s a^ a r e e i g e n v e c t o r s o f t h e d i a g o n a l i z i n g t r a n s f o r m a t i o n , and T i s s i m p l y an 40 i n t e g e r l a b e l used t o i d e n t i f y t h e (23 + 1) b a s i s f u n c t i o n s . I t s v a l u e s r u n from -3 t h r o u g h +3 i n i n t e g r a l s t e p s . B e f o r e m a t r i x e l e m e n t s a r e c a l c u l a t e d i n t h e | 3,K > b a s i s s e t , i t i s c o n v e n i e n t t o r e a r r a n g e t h e r o t a t i o n a l H a m i l t o n i a n t o t h e form H r° = M X + Y) 3 2 + {Z - |(X + Y)} 3 z 2 + MX - Y ) { 3 + 2 + 3_ 2} (1.96) The o p e r a t o r s 3 + and 3_ a r e s h i f t down and s h i f t up o p e r a t o r s i n K, r e s p e c t i v e l y , whose m a t r i x e lements a r e ( i n u n i t s o f h/2TT) 3_ | 3,K,M > = | / 3 ( 3 + 1) - K(K ± 1) | 3,K±1,M > (1.97) The phase c h o i c e c o r r e s p o n d s t o t h e i d e n t i f i c a t i o n 3, = (3 ± i 3 ) as K ± x y h a v i n g r e a l and p o s i t i v e m a t r i x e l e m e n t s , which i s t h e Condon and S h o r t l e y (22) phase c o n v e n t i o n . I t i s n o t t h e same c h o i c e as t h a t ,made by K i n g , H a i n e r and C r o s s ( 2 3 ) , who chose t h e m a t r i x element o f (3 ± i 3 ) y x t o be r e a l and p o s i t i v e . The l a s t t erm i n 1.96 g i v e s o f f - d i a g o n a l m a t r i x e l e m e n t s and t h e r e f o r e removes t h e symmetric t o p form o f H^ 0; t h e s i z e o f t h e l a s t t e r m r e l a t i v e t o t h e f i r s t i s a measure o f j u s t how f a r t h e m o l e c u l e d e v i a t e s from symmetric t o p b e h a v i o r . One n o r m a l l y chooses t o i d e n t i f y t h e m o l e c u l a r i n e r t i a l axes a, b, and c, w i t h t h e m o l e c u l e - f i x e d axes x, y, and z, i n such a way t h a t t h e c o e f f i c i e n t \ (X - Y) i s as s m a l l as p o s s i b l e i n 1.96. There a r e s i x p o s s i b l e ways o f d o i n g t h i s , and t h e s e a r e g i v e n i n T a b l e 1.3 u s i n g l a b e l s due t o K i n g , H a i n e r and C r o s s ( 2 3 ) . F o r n e a r - p r o l a t e symmetric t o p s such as t h e A B 2 - t y p e m o l e c u l e s s t u d i e d i n t h i s t h e s i s , t h e I r e p r e s e n t a t i o n i s used, f o r which x, y, and z become b, c, and a, r e s p e c t i v e l y . T a b l e 1.3. I d e n t i f i c a t i o n o f t h e M o l e c u l e - F i x e d A x i s System w i t h t h e P r i n c i p a l I n e r t i a l A x i s System. M o l e c u l e - F i x e d I n e r t i a l A x i s System A x i s R e p r e s e n t a t i o n I r I I r I I I r I * 0 i r i n * x b c a c a b y c a b b c a z a b c a b c -p-42 ( I ) C e n t r i f u g a l D i s t o r t i o n i n Asymmetric Top M o l e c u l e s . The energy l e v e l s f o r t h e r i g i d r o t o r g i v e n i n e q u a t i o n 1.92 must be m o d i f i e d t o a l l o w f o r t h e e f f e c t s o f v i b r a t i o n , w hich appear as terms i n 0 4 i n t h e e f f e c t i v e H a m i l t o n i a n . These c o r r e c t i o n s would have t h e c l a s s i f i c a t i o n h_," i n t h e t r a n s f o r m e d H a m i l t o n i a n , and a r i s e f rom t h e s e c o n d - o r d e r v i ' b r a t i o n i r b t a t i 6 n o i n ( t e r a f e i t i 6 n r ; o f ( i t h e form (ft" x h )/ E ... 12 12 12v i b 12 v i b Frbme-Tablesaildlland 1.2 • (h / 4 T r 2c) a r ( Q l 3 ) q r hi2 = -* { — —p—~ V B (1-98) ' r aa 33 i n w h i c h a l l terms have been p r e v i o u s l y d e f i n e d . U s i n g 1.85, t h e second o r d e r p e r t u r b a t i o n c o r r e c t i o n due t o h^^ c a n be c a l c u l a t e d , and has t h e form (h/4TT 2c) a ( a B ) , a ( y 6 ) it h 0 4 = * I I °a 3 B \ °6 X r a B Y 6 a) I 6 I R R e I 6 I « r aa BB YY <S6 I v' <v q v ' x v ' q v> r r E v " E V (1.99) E q u a t i o n 1.73 has been used t o o b t a i n t h e form f o r h ^ g i v e n i n 1.99. Because t h e v i b r a t i o n a l i n t e r v a l s a r e much l a r g e r t h a n r t h e r o t a t i o n a l e nergy d i f f e r e n c e s , t h e denominator o f t h e l a s t summation o f 1.99 i s a p p r o x i m a t e l y co . The numerator i s j u s t -\, from which h" = i I x 0 , 3 3D 3 3X (1.100a) 04 ag s a$y& a B Y <5 a (aB) a ( f 6 ) a a u i v r r where x „ - = -? ) i r i r i . \ aBY<5 £ n — (1.100b) XrTaa J33 """YY I S 6 43 E q u a t i o n 1.73 has been used a g a i n t o o b t a i n t h e f i n a l form g i v e n i n 1.100b. These e q u a t i o n s a r e i d e n t i c a l t o t h o s e f i r s t g i v e n by W i l s o n and Howard ( 1 8 ) , e x c e p t f o r t h e f a c t t h a t xa^y^ has been n o r m a l i z e d t o energy u n i t s by removing a f a c t o r o f (h/2TT) from 1.100. T h e 7 a p p l i c a t i o n o f t h e c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n s g i v e n by E q u a t i o n s 1.100 t o t h e f i t t i n g o f a c t u a l m o l e c u l a r s p e c t r a t u r n s o u t t o be an e x t r a o r d i n a r i l y c o m p l i c a t e d p r o c e d u r e . W i t h o u t i m p o s i n g t h e r e s t r i c t i o n s o f symmetry, e q u a t i o n 1.100a i m p l i e s 81 terms i n t h e H a m i l t o n i a n ; c o n s i d e r a t i o n o f t h e symmetry p r o p e r t i e s o f t h e v a r i o u s T'S i n t h e D,, p o i n t group ( t o wh i c h t h e a symmetric t o p wa v e f u n c t i o n s ' ' b e i o n g ) c a u s e s a l l but 21 o f t h e s e terms t o v a n i s h i d e n t i c a l l y . The commutation r e l a t i o n s w h i c h a p p l y t o t h e v a r i o u s p o s s i b l e a n g u l a r momentum o p e r a t o r s o f 1.100a a l l o w s a r e a r r a n g e m e n t o f t h e r o t a t i o n a l H a m i l t o n i a n t o t h e form H = 1 £ U 1 3 2 + i y T' 0 2 3 2 (1.101a) r L Kaa a LD aa33 a 3 a a3 where Ka = ( yaa " \ Ta3a3 " \ V Y " 3 / 2 £ ( 1 ' 1 0 1 B ) and x' = T - (1.101c) aaaa aaaa T' FLE = T Q 0 + 2x (a ± 3) ( l . l O l d ) aa33 aa33 a3a3 T h i s s e e m i n g l y " s i m p l e " r e a r r a n g e m e n t i n j e c t s c o n t r i b u t i o n s due t o t h e T'S i n t o t h e " r i g i d r o t o r " p a r t o f H r. A f t e r t h i s has been done, s i x T'S remain a p p a r e n t l y i n d e p e n d e n t o f each o t h e r ; t h e s e s i x a r e g i v e n by e q u a t i o n s 1.101c and l . l O l d by a l l o w i n g a,3 t o r u n o v e r t h e m o l e c u l e -f i x e d axes ( x , y , z ) . Watson (24) showed t h a t a g e n e r a l r u l e a p p l i e s t o t h e H a m i l t o n i a n o f any n o n - p l a n a r asymmetric r o t o r : i f t h e H a m i l t o n i a n i s p a r t i t i o n e d i n t o a sum o f terms so t h a t H = H ( 3 2 ) + H(3 1*) + H ( 3 6 ) + ... 44 t h e n t h e number o f p a r a m e t e r s on w h i c h t h e energy l e v e l s depend i s 3 + 5 + 7 + 9 + r e s p e c t i v e l y . T h i s means t h a t t h e r e a r e , i n f a c t , o n l y 5 d e t e r m i n a b l e c o e f f i c i e n t s o f terms i n 3h, not s i x as i m p l i e d by e q u a t i o n s 1.101. Watson a t t a c h e d t h e l a b e l " r e d u c e d " t o t h e form o f t h e e f f e c t i v e H a m i l t o n i a n i n w h i c h o n l y as many p a r a m e t e r s appear as a r e d e t e r m i n a b l e . There i s no u n i q u e way t o r e d u c e t h e form o f 1.101, but an e s p e c i a l l y u s e f u l form f o r g e n e r a t i n g th e m o l e c u l a r H a m i l t o n i a n f o r r o t a t i o n i n a sy m m e t r i c t o p b a s i s s e t was f o u n d t o be H r = I (X + Y) g 2 + {Z - I (X + Y)} J z 2 - A K 3 z " - A 3 K 3Z 3/ - A 3 g1* + ( 3 x 2 - 3 y 2 ) U ( X - Y) - 6 3 J 2 - 6 K 3 Z 2 } + {£ (X - Y) - 6. 3 2 - 6.. 5z 2 } (3 2 - 3 2 ) (1.102) j - i \ z x y The form o f 1.102 was found by Watson(24) by c h o o s i n g a u n i t a r y t r a n s f o r m a t i o n i n which t h e m a t r i x e l e m e n t s v a n i s h between symmetric to p b a s i s f u n c t i o n s whose quantum numbers d i f f e r by 4. The m a t r i x r e p r e s e n t a t i o n o f 1.102 c o n t a i n s AK = 0, ±2 e l e m e n t s o n l y . F o r some m o l e c u l e s w h i c h behave as v e r y n e a r l y - p r o l a t e symmetric t o p s , t h i s r e d u c t i o n sometimes f a i l s , and an a l t e r n a t i v e r e d u c t i o n must be used. T h i s l a t t e r r e d u c t i o n i s d i s c u s s e d a t l e n g t h by Watson ( 2 4 ) , Yamada (25, 26, 2 7 ) , and a l s o by W i n n e w i s s e r ( 2 8 ) . R e l a t i o n s h i p s between t h e c o n s t a n t s o b t a i n e d u s i n g e i t h e r r e d u c t i o n i s a l s o found i n Yamada's work. The a p p l i c a t i o n o f t h e s e r e s u l t s t o t h e f i t t i n g o f e x p e r i m e n t a l d a t a has been d i s c u s s e d i n d e t a i l by Watson (24) and KdrjChh^fif ;C29). I n p r i n c i p l e , a s i m p l e r e v e r s a l o f t h e p e r t u r b a t i o n p r o c e d u r e o f t h i s s e c t i o n ought t o d e f i n e m o l e c u l a r p a r a m e t e r s w i t h p h y s i c a l meaning, w i t h 45 r e f e r e n c e t o T a b l e 1.1. U n f o r t u n a t e l y , even though t h e t r a n s f o r m a t i o n p r o c e d u r e s o u t l i n e d e x t r a c t t h e c o r r e c t o p e r a t o r f o r m f r o m t h e t r a n s f o r m e d H a m i l t o n i a n , the C o e f f i c i e n t s which appear i n t h e H a m i l t o n i a n used t o f i t m o l e c u l a r s p e c t r a r e p r e s e n t t h e sum o f a l l c o n t r i b u t i o n s o f h i g h e r - o r d e r terms t h a t g e n e r a t e c o e f f i c i e n t s o f t h e same r o t a t i o n a l o p e r a t o r s . T h i s means t h a t s p e c t r o s c o p i c c o n s t a n t s can be r e l a t e d t o t h e m o l e c u l a r p r o -p e r t i e s o n l y i n s p e c i a l c a s e s where a p p r o x i m a t i o n s can be a p p l i e d w i t h some c e r t a i n t y . 46 (3) V i b r a t i o n a l C o n t r i b u t i o n s t o t h e I n e r t i a l C o n s t a n t s . To a good a p p r o x i m a t i o n , t h e energy l e v e l s o f a v i b r a t i n g and r o t a t i n g a symmetric t o p a r e g i v e n by t h e sum o f the..eigenvalues' o f 1.91 and 1.102. R o t a t i o n a l c o n s t a n t s o b t a i n e d " f r o m - a s . f i t - t b ^ . s u c h V a a ".expression' have an" ^mpir.iical-evi'br.at-ionalbdeperidence., which can be w r i t t e n A v = A e - I a r A (v + i ) + ... (1.103) r i n w h i c h t h e summation ru n s over a l l t h e normal c o o r d i n a t e s . Analogous e x p r e s s i o n s may be w r i t t e n f o r B and C . A , B , and C a r e t h e v a l u e s ' v v e e e t h a t t h e r o t a t i o n a l c o n s t a n t s would have when t h e m o l e c u l e assumes t h e e q u i l i b r i u m c o n f i g u r a t i o n , and a r e r e l a t e d t o t h e r e c i p r o c a l moments o f i n e r t i a by t h e r e l a t i o n s h i p 1.94. U s u a l l y one assumes t h a t t h e i d e n t i f i c a t i o n o f t h e m o l e c u l e - f i x e d a x i s system w i t h t h e p r i n c i p a l i n e r t i a l a x i s system has been chosen ( u s i n g T a b l e 1.3) i n such a way t h a t A g > B g > C e < The c o n s t a n t s a f a r e s m a l l v i b r a t i o n a l c o r r e c t i o n s t o t h e r o t a t i o n a l c o n s t a n t s , and t h e p r e s e n c e o f such a dependence i s i n d i c a t e d by t h e s u b s c r i p t v f o r t h e o b s e r v e d r o t a t i o n a l c o n s t a n t i n 1.103. In a t r i a t o m i c asymmetric r o t o r , t h e r e can be no d e g e n e r a t e normal modes o f v i b r a t i o n . From 1.93 and 1.103, t h e a's a r e c o e f f i c i e n t s o f (v + j) 3 2 i n t h e t r a n s f o r m e d H a m i l t o n i a n . T h i s term would be c l a s s i f i e d II as h^2 i n t h e n o t a t i o n o f T a b l e 1.2, s i n c e a v i b r a t i o n a l dependence on (v + j) a r i s e s from t h e average v a l u e o f t h e v i b r a t i o n a l o p e r a t o r p r' 2 + q r 2 . Such a term w i l l be g e n e r a t e d i n t h r e e ways from t h e e n t r i e s o f Ta b l e 1.2;, from h ^ by f i r s t o r d e r p e r t u r b a t i o n t h e o r y ; from (h.,^ x. h ^ ) / ^ ^ by s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y ; and from ( h ^ x h - ^ ) / by s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y . A l l t h r e e terms g i v e r i s e t o c o n t r i b u t i o n s o f o r d e r o f magnitude K ^ y - j ^ * U s i n g t h e terms o f Ta b l e 1.1 and t h e v i b r a t i o n a l m a t r i x e l e m e n t s , 47 one f i n d s ^ ( A 3 ) . 2 (3 a r v ' , p / ) 2 ( 3 u r z + u ) s 2 ) - a A = (2A2/u> ) { I + I (?rs(A))2 — — 3 4T 6 S t , 2 - . 2 3B r s + u ( c / h ) ^ I $ a ( M ) U3 (co ) " 3 / 2 } (1.104) Y r r s s r s s i n w h i c h t h e terms have been a r r a n g e d t o c o r r e s p o n d t o t h e s o u r c e s o u t l i n e d above. The e x p r e s s i o n w i t h i n t h e b r a c e s i s d i m e n s i o n l e s s and o f o r d e r u n i t y ; t h e whole e x p r e s s i o n has magnitude and u n i t s g i v e n by 2A 2 / t o r . The p h y s i c a l i n t e r p r e t a t i o n o f t h e t h r e e terms i s c l a r i f i e d by c o n s i d e r -a t i o n s o f t h e i r o r i g i n . The f i r s t t erm r e p r e s e n t s t h e q u a d r a t i c dependence o f u.. on normal c o o r d i n a t e q ; i t r e f l e c t s t h e f a c t t h a t a mean sq u a r e A A r d i s p l a c e m e n t o f q r c a u s e s a change i n t h e e f f e c t i v e r o t a t i o n a l c o n s t a n t . A s e c o n d - o r d e r C o r i o l i s c o u p l i n g between to'r and to causes t h e l e v e l s o f t h e • l o w e r energy v i b r a t i o n a l s t a t e t o c l o s e up, w h i l e t h o s e r o t a t i o n a l l e v e l s b e l o n g i n g t o t h e h i g h e r energy v i b r a t i o n a l s t a t e open up - t h i s i s g i v e n as t h e second term. F i n a l l y , t h e l a s t t e r m a r i s e s from t h e f a c t t h a t a mean squ a r e d i s p l a c e m e n t o f q r can g e n e r a t e a l i n e a r d i s p l a c e m e n t i n q s , which i n t u r n changes t h e r o t a t i o n a l c o n s t a n t s . In many m o l e c u l e s , i m p o r t a n t n e g a t i v e a n h a r m o n i c i t y terms such as ^ r r r a r e a s s o c l a t e d w i t h bond s t r e t c h i n g , due t o t h e a n h a r m o n i c i t y i n h e r e n t i n t h e d i s s o c i a t i o n p r o c e s s f o r l a r g e d i s p l a c e m e n t s . O f t e n , t h e s e a r e t h e dominant c o n t r i b u t i o n s t o a , which makes t h e r i g h t - h a n d s i d e l a r g e and p o s i t i v e . T h i s i s t h e r e a s o n f o r t h e n e g a t i v e s i g n i n t h e d e f i n i t i o n o f a . E q u a t i o n 1.104 can be w r i t t e n more s i m p l y as t h e sum o f t h r e e t e r m s : (harm.) ( C o r . ) (anharm.) 1 r > I- N a = a + a + a (1.105) r r r r i n w h i c h terms on t h e r i g h t a r e d e f i n e d by one-to-one c o r r e s p o n d e n c e w i t h 1.104. 48 (K) The I n e r t i a l D e f e c t The r o t a t i o n a l energy l e v e l s o f a m o l e c u l e a r e d e t e r m i n e d by t h e e x p e c t a t i o n v a l u e s o f A y , B^, and C^, and not by t h e e f f e c t i v e moment o f i n e r t i a t e n s o r g i v e n i n 1.45. Even i f t h e C o r i o l i s and anharmonic c o n t r i b u t i o n s t o a g i v e n i n t h e p r e v i o u s s e c t i o n were t o v a n i s h i d e n t i c a l l y , t h e r e would s t i l l r emain a harmonic c o n t r i b u t i o n . A f t e r c o r r e c t i o n f o r t h e v i b r a t i o n a l c o n t r i b u t i o n s , t h e e q u i l i b r i u m r o t a t i o n a l c o n s t a n t s can be r e l a t e d t o t h e e f f e c t i v e moment o f i n e r t i a t e n s o r , from w h i c h t h e e q u i l i b r i u m moments o f i n e r t i a can be o b t a i n e d u s i n g 1 . 4 5 s t o remove r o t a t i o n a l c o n t r i b u t i o n s . From d e f i n i t i o n 1.30 we have N ? I - I - I = -2 J M. y. (1.105) yy xx z z i - 1 1 where y^ i s t h e d i s t a n c e o f t h e i - t h atom from t h e xz p l a n e . ( S t r i c t l y s p e a k i n g , t h e r e l a t i o n s h i p a l s o ought t o sum ov e r t h e e l e c t r o n i c d i s t a n c e s from t h e xz p l a n e as w e l l ) . I f a l l p a r t i c l e s a r e c o n s t r a i n e d t o t h e xz p l a n e , t h e m o l e c u l e w i l l be c o m p l e t e l y p l a n a r , and t h e r i g h t hand s i d e o f 1.105 v a n i s h e s i d e n t i c a l l y . T h i s s i t u a t i o n ought t o be o b s e r v e d i n t h e symmetric t r i a t o m i c m o l e c u l e s s t u d i e d , w h i c h a r e p l a n a r by d e f i n i t i o n . When v i b r a t i o n a l motion i s t a k e n i n t o a c c o u n t , t h e r i g h t hand summation does n o t v a n i s h , even f o r t h e p l a n a r m o l e c u l e c a s e . Thus, t h e i n e r t i a l d e f e c t o f a p l a n a r m o l e c u l e i n i t s ground v i b r a t i o n a l s t a t e i s d e f i n e d by t h e e x p r e s s i o n A° = I ..° -. I ° - I n- n° (1.106) c c bb aa 1 A° i n w e l l - b e h a v e d p l a n a r m o l e c u l e s i s a s m a l l p o s i t i v e number, o f t h e o r d e r o f 0.05 t o 0.5 amu A 2. F o r a n o n - p l a n a r m o l e c u l e , 1.105 i n d i c a t e s t h a t A 0 must be n e g a t i v e . D a r l i n g and Dennison (30) dem o n s t r a t e d t h a t A was in d e p e n d e n t o f 49 c u b i c anharmonic f o r c e c o n s t a n t s , so t h a t t h e v i b r a t i o n a l c o n t r i b u t i o n t o A c o u l d be c a l c u l a t e d d i r e c t l y from t h e harmonic f o r c e f i e l d a l o n e . Oka and Morino (31) have g i v e n a g e n e r a l e x p r e s s i o n f o r A, i n c l u d i n g t h e c e n t r i f u g a l and e l e c t r o n i c c o n t r i b u t i o n s . The i n e r t i a l d e f e c t can be w r i t t e n i n w h i c h A v ° , A e ° , and A^0 a r e t h e v i b r a t i o n a l , e l e c t r o n i c and c e n t r i f u g a l c o n t r i b u t i o n s t o t h e i n e r t i a l d e f e c t , r e s p e c t i v e l y . The f i r s t two terms were shown by Oka and M o r i n o (3*1) t o be A ° = ( h / 4 u 2 c ) I (3/co. ) - I 2oi 2/{u) (o> 2 - co 2 ) } x • V t ^ r s S r r S { ( C r s ( d ) ) 2 - ( ? r s ( B ) ) 2 - a r s ( a ) ) 2 ) (1.108) 3 n d A e ° = " K ( I c c 9 C c ' hV>W ' ^ a W U - 1 0 9 ) i n w h i c h K i s t h e Born-oppenheimer e x p a n s i o n p a r a m e t e r , and g - d e f i n e t h e g t e n s o r f o r t h e r o t a t i o n a l magnetic moment. The s m a l l c e n t r i f u g a l c o r r e c t i o n i s an a r t i f a c t o f t h e r e a r r a n g e m e n t s used t o o b t a i n e q u a t i o n 1.101. The E c k a r t c o n d i t i o n s were s p e c i f i c a l l y c hosen so t h a t no s e t o f m o l e c u l a r d i s p l a c e m e n t s gave r i s e t o a r o t a t i o n ; hence x,, , and x must be i d e n t i c a l l y z e r o . T h e r e f o r e t h e u ' r e l a t i o n s bebc ac a c ' aa g i v e n i n 1.101b c o n t a i n c o n t r i b u t i o n s from T a ^ a ^ o n l y > ar>d when t h e s e a r e s u b s t i t u t e d i n t o 1.106, one f i n d s t h a t A c d ° = " T a b a b { ( 3 I c c ^ C ) + % a / 2 A ) + ( W 2 B ) } ( 1 - 1 1 0 ) W i t h t h e s e s u b s t i t u t i o n s , e q u a t i o n 1.107 i s found t o p r e d i c t t h e o b s e r v e d i n e r t i a l d e f e c t s o f s m a l l t r i a t o m i c s q u i t e w e l l (see Gordy and Cook (32) or Oka and Mor i n o ( 3 3 ) ) . U s u a l l y t h e c o n t r i b u t i o n o f t h e v i b r a t i o n a l p a r t t o A i s an o r d e r o f magnitude l a r g e r t h a n t h e o t h e r t e r m s , though f o r m o l e c u l e s l i k e SO,, i n wh i c h a l a r g e o u t - o f - p l a n e e l e c t r o n d e n s i t y e x i s t s , 50 t h e e l e c t r o n i c c o r r e c t i o n can be s i g n i f i c a n t . F i n a l l y , a p o s i t i v e i n e r t i a l d e f e c t i s o n l y i n d i c a t i v e o f m o l e c u l a r p l a n a r i t y i f t h e c a l c u l a t e d and o b s e r v e d d e f e c t s a r e i n good agremment. Fo r p l a n a r t r i a t o m i c s , a l a r g e and n e g a t i v e i n e r t i a l d e f e c t i s s u g g e s t i v e (though n o t c o n c l u s i v e ) e v i d e n c e f o r s t r o n g v i b r a t i o n - r o t a t i o n i n t e r a c t i o n , w h i c h ought t o be c o r r o b o r a t e d w i t h the o b s e r v a t i o n o f p e r t u r b a t i o n s . 51 (L) D e r i v a t i o n o f t h e T ^ a g g from C e n t r i f u g a l D i s t o r t i o n D a t a . S i n c e Watson's method was f i r s t p u b l i s h e d ( 2 4 ) , many m o l e c u l e s have been c h a r a c t e r i z e d u s i n g h i s e f f e c t i v e r o t a t i o n a l H a m i l t o n i a n . The e f f e c t i v e c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s o b t a i n e d f r o m 1.102 c o n t a i n a g r e a t d e a l o f i n f o r m a t i o n about t h e m o l e c u l a r f o r c e f i e l d , though t h e e x t r a c t i o n o f m e a n i n g f u l f o r c e c o n s t a n t s from them i s by no means a t r i v i a l t a s k . T h i s s e c t i o n and t h e next a r e d e v o t e d t o t h e d e f i n i t i o n o f a c o n s i s t e n t method f o r o b t a i n i n g f o r c e f i e l d i n f o r m a t i o n from s p e c t r o s c o p i c c o n s t a n t s . An advantage o f Watson's method i s t h a t t h e d e t e r m i n a b l e c o n s t a n t s a r e i n v a r i a n t t o any u n i t a r y t r a n s f o r m a t i o n o f t h e e f f e c t i v e r o t a t i o n a l H a m i l t o n i a n . Watson's d e t e r m i n a b l e r o t a t i o n a l c o n s t a n t s (K,V,2) can be c a l c u l a t e d from t h e r e d u c e d ( s p e c t r o s c o p i c ) c o n s t a n t s (X,Y,Z) from t h e r e l a t i o n s h i p X X 2 1 0 -2 -2 V Y - 2 1 0 2 2 z z 2 0 0 0 0 (1.111) where D i s t h e v e c t o r o f r e d u c e d c e n t r i f u g a l c o n s t a n t s , d e f i n e d by ° + = ( A 3 ' A 3 K ' V &3> V (1.112) Watson's d e t e r m i n a b l e c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s a r e g i v e n by t h e v e c t o r r t T = (T , T , T , T,, T 0 ) xx' y y ' z z ' 1' 2 (1.113a) The e l e m e n t s i n t h e v e c t o r T a r e g i v e n i n terms o f t h e T'S by t h e r e l a t i o n s h i p s T = T / M a = x, y, o r z) (1.113b) 52 T. = ( x ' + T ' + T * )/4 ' (1.113c) 1 xxyy y y z z z z x x and L , = ( X x ' + Y x ' + Z x ' )/4 (1.113d) 2 y y z z z z x x xxyy W i t h t h e s e d e f i n i t i o n s , t r a n s f o r m a t i o n s can be d e f i n e d r e l a t i n g t h e d e t e r m i n a b l e c o n s t a n t s t o t h e reduced ones, so t h a t b o t h T = Q • D (1.114) and t h e i n v e r s e t r a n s f o r m a t i o n D = ! • T (1.115) a r e p o s s i b l e . The m a t r i c e s Q and B needed i n t h e s e e q u a t i o n s a r e g i v e n i n T a b l e 1.4, and may be s p e c i a l i z e d t o a p p l y f o r any c h o i c e o f a x i s r e p r e s e n t a t i o n by s e l e c t i o n o f t h e a p p r o p r i a t e columns o f T a b l e 1.3. S t r i c t l y s p e a k i n g , t h e c o n s t a n t s X , Y , and Z o f T a b l e 1.4 r e f e r t o t h e e q u i l i b r i u m r o t a t i o n a l c o n s t a n t s . However, e q u i l i b r i u m c o n s t a n t s a r e known o n l y r a r e l y , and so one n o r m a l l y s u b s t i t u t e s t h e d e t e r m i n a b l e c o n s t a n t s from 1.111 i n t o t h e m a t r i c e s o f T a b l e 1.4. T h i s means t h e t r a n s f o r m a t i o n g i v e n i n 1.114 i s not u n i q u e , so t h a t t h e c a l c u l a t i o n o f Watson's " d e t e r m i n a b l e " c o n s t a n t s becomes an i t e r a t i v e p r o c e d u r e . One p r o c e e d s by c a l c u l a t i n g an i n i t i a l s e t o f X , y, and Z from 1.111, and us'e's t h e s e c o n s t a n t s t o d e f i n e t h e m a t r i c e s Q and B. The v e c t o r T i s t h e n e s t i m a t e d from 1.114, a f t e r which t h e i n v e r s e t r a n s f o r m a t i o n o f 1.115 g i v e s a c o r r e c t e d v e c t o r , D, f o r s u b m i s s i o n t o 1.111. T h i s p r o c e s s i s i t e r a t e d t i l l a s t a b l e s e t o f d e t e r m i n a b l e c o n s t a n t s i s o b t a i n e d . I f one a d o p t s t h e more compact n o t a t i o n T a B = i W <a,S = x f - y , o r z> (1.116a) a' n d T a b — T a b a b ( 1 ' 1 1 6 b > th e seven possi'bl!escentrifugal :distortion r ,comstantss(..;,-.some.-t o f which 53 T a b l e 1.4 T r a n s f o r m a t i o n s Between Watson's "Reduced" and " D e t e r m i n a b l e " C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s . -1 0 0 -2 0 -1 0 0 2 0 Q •y -1 -1 -1 0 0 -3 -1 0 0 0 -(X+Y+Z) -(X+Y)/2 0 X-Y X-Y -1/2 -1/2 0 0 0 3/2 3/2 -1 0 -1 -1 -1 1 0 -1/4 1/4 0 0 0 x - z Y-Z 0 X+Y 1 2(X^-Y) 2(X-Y) 2(Y-X) X-Y 54 w i l l v a n i s h by symmetry) b e l o n g t o t h e s e t T , T,, , T aa' bb' c c ' T a b ' T b c ' T , and T* a c ' ab Watson's d e t e r m i n a b l e c o n s t a n t s , and a r e l i n e a r c o m b i n a t i o n s o f t h e l a s t f o u r o f t h e s e c o n s t a n t s , so t h a t 1 ab T. + T + T* be ca ab (1.117) and TT = AT, + 2 be BT + C(T . ac ab + 2T* ) ab (1.118) One must complete t h e s e t o f a l l seven c o n s t a n t s u s i n g t h e f i v e d e t e r m i n a b l e ones , and t o do so i t i s customary t o i n v o k e t h e s o - c a l l e d p l a n a r i t y c o n d i t i o n s when one i s d e a l i n g w i t h p l a n a r m o l e c u l e s . These a r e g i v e n by T a b - < A B> 2 !• II II ;iaa bb + O/ 2 c c (1.119a) T b c = (tic)2 II II aaa bb + 0 / z c c (1.119b) T a c - ( A C ) 2 (+ T + T, , aa bb + T )/2 c c (1.119c) where T = T /A w i t h s i m i l a r e x p r e s s i o n s h o l d i n g f o r T,, and T . aa aa ' r M bb c c J u s t as i n t h e d e f i n i t i o n o f t h e t r a n s f o r m a t i o n s o f T a b l e 1.4, e q u a t i o n s 1.117 t h r o u g h 1.119 a r e s t r i c t l y v a l i d o n l y f o r t h e e q u i l i b r i u m geometry o f p l a n a r m o l e c u l e s . N e v e r t h e l e s s , i t i s customary t o assume t h a t t h e s e r e l a t i o n s h i p s h o l d i n t h e v i b r a t i o n a l z e r o - p o i n t l e v e l s as w e l l . Thus, t h e d e t e r m i n a b l e c o n s t a n t s o b t a i n e d by i t e r a t i o n w i l l be used i n t h e s e e q u a t i o n s as w e l l . a s i n T a b l e 1.4. The l a s t f i v e e q u a t i o n s form a system o f f i v e s i m u l t a n e o u s e q u a t i o n s i n w h i c h t h e r e a r e but f o u r unknown p a r a m e t e r s , and t h i s f a c t has been remarked on by s e v e r a l a u t h o r s ( 2 5, 26, 27, 2 8 ) . A c l e a r a c c o u n t o f t h e 55 problems t o be c o n s i d e r e d as w e l l as t h e v a r i o u s c h o i c e s which can be made has been p r e s e n t e d i n t h e work o f Yamada and W i n n e w i s s e r ( 2 5 ) . One c o n s i s t e n t c h o i c e made i n t h i s work was as f o l l o w s . The c o n s t a n t s T > T, , , T , and T, can be t a k e n as t h e y a r e d e t e r m i n e d from 1.114. Then 3.3. DD C C X one p l a n a r i t y c o n d i t i o n i s used t o c a l c u l a t e T 2 ° = (A + B) T c c + CT 1 (1.120) which c o r r e s p o n d s t o t h e v a l u e 1 s h o u l d have i f t h e p l a n a r i t y c o n d i t i o n s were e x a c t l y f u l f i l l e d , and t h e i n e r t i a l d e f e c t were i d e n t i c a l l y z e r o , From t h e s e c o n s t a n t s , e q u a t i o n s 1.119 can be used t o d e t e r m i n e T ^ , , and T ; f i n a l l y , t h e s u b s t i t u t i o n o f t h e n e c e s s a r y c o n s t a n t s i n t o 1.117 3C d e f i n e s T*^. The advantage o f t h i s c h o i c e i s t h a t a l l t h e p l a n a r i t y c o n d i t i o n s e n t e r i n t o t h e c a l c u l a t i o n o n l y once, and t h e v a l u e o f T^ (which i s o f t e n t h e most p o o r l y d e t e r m i n e d p a r a meter o f t h e v e c t o r T) i s not used. Yamada's and W i n n e w i s s e r ' s work shows t h a t t h i s c h o i c e i s n o t as a r b i t r a r y as i t might seem, s i n c e a l l o t h e r c h o i c e s d i f f e r o n l y i n t h e way i n which t h e T - d e f e c t A T c c = T c c " ( T 2 " C T 1 ) / ( A + B ) = T 2 " T 2 ° ( 1 ' 1 2 1 ) i s i n c l u d e d i n the c a l c u l a t i o n . S i n c e t h i s means t h a t o t h e r methods o f e x t r a c t i n g t h e T c o n s t a n t s c o n t a i n no a d d i t i o n a l i n f o r m a t i o n c o n c e r n i n g t h e f o r c e c o n s t a n t s o f t h e m o l e c u l e , t h e y w i l l n ot be c o n s i d e r e d f u r t h e r . The e x t e n t t o which t h e a p p r o x i m a t i o n s i n t r o d u c e d by r e p l a c i n g t h e e q u i l i b r i u m c o n s t a n t s w i t h t h o s e o f t h e z e r o - p o i n t l e v e l do not h o l d i s g i v e n by AT and can be o b t a i n e d by s u b s t i t u t i n g t h e l a s t t h r e e e l e m e n t s o f T i n t o C C — ( 1 . 1 2 1 ) . F o r p l a n a r m o l e c u l e s , t h i s d e f e c t ought t o v a n i s h , and t h e f a c t t h a t i t does not g i v e s r i s e t o t h e ambiguous n a t u r e o f t h e t r a n s f o r m a t i o n s d i s c u s s e d i n t h i s s e c t i o n . 56 (M) D e t e r m i n a t i o n o f the M o l e c u l a r F o r c e F i e l d from C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s . C e n t r i f u g a l d i s t o r t i o n c o n s t a n t s a r e o f c o n s i d e r a b l e i n t e r e s t t o s p e c t r o s c o p i s t s , because t h e y a r e among t h e few w e l l - d e t e r m i n e d s p e c t r o s c o p i c c o n s t a n t s t h a t a r e s e n s i t i v e t o the m o l e c u l a r f o r c e f i e l d . They a r e i m p o r t a n t because t h e number o f q u a d r a t i c v a l e n c e f o r c e c o n s t a n t s exceeds 3Ne6, t h e number o f normal modes o f v i b r a t i o n f o r a g e n e r a l m o l e c u l e , which p r e v e n t s a d e t e r m i n a t i o n o f t h e f o r c e f i e l d f rom e x p e r i m e n t a l v i b r a t i o n a l f r e q u e n c i e s f o r o n l y one i s o t o p i c s p e c i e s . A d d i t i o n a l i n f o r m a t i o n needed t o r e s o l v e t h e a m b i g u i t i e s i n t h e f o r c e f i e l d must t h e r e f o r e be sought i n o t h e r d a t a , s u c h as c e n t r i f u g a l d i s t o r t i o n , i n e r t i a l d e f e c t s ( 3 1 ) , i s o t o p e s h i f t s , and C o r i o l i s c o u p l i n g c o e f f i c i e n t s ( 1 0 ) , whenever t h e l a t t e r a r e a v a i l a b l e . The u s e f u l n e s s o f d i s t o r t i o n c o n s t a n t s i n such; f o r c e f i e l d c a l c u l a t i o n s i s l i m i t e d t o t h e number o f i n d e p e n d e n t l y d e t e r m i n a b l e c o m b i n a t i o n s o f t h e x p a r a m e t e r s , as d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n s . T r i a t o m i c m o l e c u l e s w i t h 0, p o i n t group symmetry form a s p e c i a l c a s e , s i n c e symmetry a l l o w s t h e d e t e r m i n a t i o n o f f o u r x c o n s t a n t s 1 and t h e s e m o l e c u l e s have a v a l e n c e f o r c e f i e l d g i v e n by t h e p o t e n t i a l 2V = f ( 6 r ! 2 + 6 r 2 2 ) + f 6 a 2 + 2 f " ( 6 r i + 6 r 2 ) 6 a r a r a + 2 f 6 r i 6 r 2 (1.122) r r 1 which i n v o l v e s o n l y f o u r q u a d r a t i c f o r c e c o n s t a n t s . T h i s p o t e n t i a l Watson's red u c e d H a m i l t o n i a n i s s t i l l used t o f i t t h e s p e c t r a o f C m o l e c u l e s , as t h e f i f t h c o n s t a n t i s r e q u i r e d because t h e p l a n a r i t y v r e l a t i o n s b reak down due t o t h e x-defect. 57 d e f i n e s a f o r c e c o n s t a n t m a t r i x F o f t h e form ( i n dynes/cm) f + f r r r 22 f / r r a 0 F 0 (1.123) 0 0 f - f r r r E x p l i c i t r e l a t i o n s h i p s between t h e x c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s f o r t h e e q u i l i b r i u m c o n f i g u r a t i o n and t h e i n v e r s e o f F have been g i v e n i n t h e f o r m u l a e o f H e r b e r i c h e t . a l . (34) r e p r o d u c e d i n Gordy and Cook ( 3 2 ) . These r e l a t i o n s h i p s a r e g i v e n by t h e t r a n s f o r m a t i o n o f T a b l e 1.5. In t h i s t a b l e , t h e v a l u e s o f t h e r o t a t i o n a l c o n s t a n t s and t h e T'S ought t o be i n cm \ so t h a t t h e e l e m e n t s o f t h e i n v e r s e o f F w i l l be i n cm/dyne. R i s g i v e n by where r i s t h e A - B bond l e n g t h f o r t h e AB 2 m o l e c u l e , i n angstrom u n i t s . The a n g l e 6 i s h a l f t h e B - A - B bond a n g l e and m^, n^, and M a r e t h e masses ( i n a t o m i c mass u n i t s ) o f n u c l e u s A, n u c l e u s B, and t h e m o l e c u l e , r e s p e c t i v e l y . The e l e m e n t s o f F 1 not d e f i n e d i n T a b l e 1.5 a r e i d e n t i c a l l y t h e e l e m e n t s o f t h e i n v e r s e o f F - 1 w i t h t h e m a t r i x g i v e n i n 1.123. In f a v o r a b l e c a s e s , f o r c e c o n s t a n t s o b t a i n e d i n t h i s way agree w i t h i n 4% w i t h t h e b e s t v a l u e s o b t a i n e d from v i b r a t i o n a l a n a l y s i s ( 3 2 , 3 4 ) . For t r i a t o m i c m o l e c u l e s , t h e s e d e t e r m i n a t i o n s a r e i m p o r t a n t as an i n d e p e n d e n t c o n f i r m a t i o n o f t h e m o l e c u l a r v i b r a t i o n f r e q u e n c y a s s i g n m e n t . T h i s c o m p l e t e s t h e t h e o r y n e c e s s a r y t o u n d e r s t a n d t h e r o t a t i o n a l s p e c t r a o f t r i a t o m i c m o l e c u l e s i n s i n g l e t e l e c t r o n i c s t a t e s . , Many o f t h e R = r 2 • 10 / ( 2 h c ) (1.124) z e r o ; t h e c o m p l e t e q u a d r a t i c v a l e n c e f o r c e f i e l d i s o b t a i n e d by i d e n t i f y i n g 58 Ta b l e 1.5 The T r a n s f o r m a t i o n from t h e C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s o f a Symmetric AB^, m o l e c u l e t o t h e I n v e r s e F o r c e C o n s t a n t M a t r i x . e 2 d 2 2 0 T /A 2 aaaa = 1 R a 2 b 2 • -e/2 d/2 (e - d)^2 0 bbbb F" 1 22 2 2 -4 0 T ,./AB aabb F 3 3 0 0 0 c / ( a2 b 2) T . ./AB abab D e f i n i t i o n s : a = s i n 9 b = cos 0 c = (m A - a 2m B) 2/(m AM) d = t a n 6 e = c o t 0 59 s y m m e t r i c t r i a t o m i c m o l e c u l e s i n t h i s t h e s i s have been a n a l y z e d i n m u l t i p l e t e l e c t r o n i c s t a t e s ; t h e t h e o r y p r e s e n t e d i n t h i s c h a p t e r must be g e n e r a l i z e d t o i n c l u d e t h e e f f e c t s o f e l e c t r o n s p i n i n o r d e r t o f i t t h e r o t a t i o n a l s p e c t r a o b s e r v e d f o r t h e s e m o l e c u l e s . I t i s t h i s t o p i c w h i c h i s t h e c o n c e r n o f t h e n e x t C h a p t e r . CHAPTER I I MULTIPLET STATES OF BENT TRIATOMIC MOLECULES. 61 (A) I n t r o d u c t i o n . The p r e c e d i n g c h a p t e r d i s c u s s e d t h e H a m i l t o n i a n f o r m o l e c u l e s i n s i n g l e t e l e c t r o n i c s t a t e s , from w h i c h an e f f e c t i v e r o t a t i o n a l H a m i l t o n i a n was d e r i v e d . In t h a t t r e a t m e n t , i t was s u f f i c i e n t t o c o n s i d e r o n l y t h e c o u l o m b i c c o n t r i b u t i o n s t o t h e m o l e c u l a r p o t e n t i a l e n e r g y , because t h e s p e c t r a l s t r u c t u r e can be e x p l a i n e d i n terms o f t h e n u c l e a r k i n e t i c e nergy a l o n e . However, whenever one o r more u n p a i r e d e l e c t r o n s a r e p r e s e n t , c o n t r i b u t i o n s t o t h e energy l e v e l s t r u c t u r e i n a r o t a t i o n a l s p e c t r u m a r i s e w h i c h r e q u i r e more e x p l i c i t c o n s i d e r a t i o n s o f t h e p o t e n t i a l i n whi c h t h e e l e c t r o n s move. T h i s p o t e n t i a l c o n s i s t s o f a l l n o n - s p h e r i c a l charge d i s t r i b u t i o n s w i t h w h i c h t h e e l e c t r o n s can i n t e r a c t . The f i r s t s e c t i o n s o f t h i s c h a p t e r w i l l d e a l w i t h t h e m o d i f i c a t i o n s o f t h e H a m i l t o n i a n g i v e n i n t h e p r e v i o u s c h a p t e r w h i c h a r e n e c e s s a r y t o e x t e n d t h e d e s c r i p t i o n o f r o t a t i o n a l s p e c t r a t o d o u b l e t s t a t e s ; t h i s w i l l be f o l l o w e d w i t h a f u r t h e r e x t e n s i o n t o m o l e c u l e s i n . t r i p l e t , s t a t e s ' (*i.?e: p o s s e s s i n g : t w o ' u n p a i r e d eleqtr.q.ns.).i^ - v . t^o -.< .-'.red e ' e : n n s f o r t r i p l e t s t r u c t u ^ I n m o l e c u l e s h a v i n g o n l y one u n p a i r e d e l e c t r o n , t h e m o d i f i e d form o f the p o t e n t i a l energy i s k e s t h e farm V = V + V + V + V + V (2.1) ee en nn so soo The f i r s t t h r e e terms on the r i g h t hand s i d e o f 2.1 were - d e f i n e d and : ©'©"iifii,d,^red'.dirtoGh"i^t€pdIVn-Pand f». v arentjhe' t w o - p a r t i e l e h i ' n t e f a c t i o n s 1 SO SOO s" Soo 1 df'tth"e,cspi>R o'fPa^seigetrdn'ewithtiit-sc-own Lo"rbital , imoti-bnf-and'-with t h e o r b i t a l m o ^ i d f l ^ o f n t ^ e a ^ t h e f €feeWoim^ai.s§^e§U\el^ s p i n w i t h t h e o r M t ^ the o t h e r e l e c t r o r ^ r e s n e c t 4 v e l y . I n terms o f e l e m e n t a r y c h a r g e s , i n t e r p a r t i c l e d i s t a n c e s and v e l o c i t i e s , t h e s e two terms can be w r i t t e n "'~ V o^r^. .-;.t a -•• 62 n N V = (g3/c) I J Z e ( r . x ( i v . - v ) • s . ) / r . (2.2) so 3 ' i L-, a - i a - l -a -I i a i = l a = l n n V 3 (2.3) = (g3/c) y J e ( r . . x ( v . - i v . ) • s . ) / r . . soo y H i ^ 1 £ L - i J " J - i - i 1 J f o r a m o l e c u l e i n w h i c h t h e r e a r e n e l e c t r o n s and N n u c l e i . I n t h e s e e q u a t i o n s , s^ i s t h e e l e c t r o n s p i n o p e r a t o r f o r e l e c t r o n i , g i s the g - f a c t o r f o r an e l e c t r o n , e q u a l t o 2.002319, and 3 i s t h e Bohr magneton, e q u a l t o (eh/#irmc). As a l w a y s , c i s t h e speed o f l i g h t . The f a c t o r s o f \ whi c h appear b e f o r e v^ i n e q u a t i o n s 2.2 and 2.3 ar e a c o r r e c t i o n c a l l e d t h e Thomas p r e c e s s i o n , which r e f l e c t s t h e d i f f e r e n c e between t h e d e s c r i p t i o n o f t h e e l e c t r o n ' s s p i n w i t h i n i t s own frame and t h a t r e l a t i v e t o a frame f i x e d on t h e n u c l e i o r on a n o t h e r e l e c t r o n . E q u a t i o n 2.2 i s a c o l l e c t i o n o f v e c t o r p r o d u c t s , s c a l e d by t h e f a c t o r Z e (g3/c)/r. ^. The s i z e o f t h i s t erm can be e s t i m a t e d from t h e e x p e c t a t i o n a i a v a l u e o f r ^ a 3 o v e r h y d r o g e n - l i k e w a v e f u n c t i o n s |nl>: < n l | r j L 3 a |nl> = Z 3 / { a 3 n 3 l ( l + i ) ( l + l ) } (2.4) where a Q i s t h e Bohr r a d i u s and has t h e v a l u e 5.292 x 10 1 Jm. T h i s e q u a t i o n 2 3 33 g i v e s t h e r e s u l t t h a t <-v. i - s p p r . p p o r t i o n a l t t o Z Z , ,\%whichrrmeans t h a t t h e * ii a ' 3a c o n t r i b u t i o n o f a g i v e n atom t o t h e s p i n - o r b i n c o u p l i n g w i l l v a r y i n p r o -4 p o r t i o n t o Z . T h i s f o u r t h power dependence means t h a t s p i n - o r b i t e f f e c t s 3 w i l l be most i m p o r t a n t i n m o l e c u l e s composed o f heavy n u c l e i . ^ s o o > t n e i n t e r a c t i o n o f t h e e l e c t r o n w i t h t h e o r b i t s o f t h e o t h e r e l e c t r o n s w i l l be p r o p o r t i o n a l t o Z 3 by t h e same argument, and w i l l t h e r e f o r e be a s m a l l e r a c o n t r i b u t i o n t h a n 2.2. S i n c e t h e c r o s s p r o d u c t o f two v e c t o r s d e f i n e s a new v e c t o r , t h e e x p r e s s i o n s i n V and V can be combined i n t o a dot p r o d u c t o f two r so soo r v e c t o r s so t h a t 63 n V + V = y a.I. • s. (2.5) SO SOO .L, 1 - 1 - 1 1=1 where t h e summation r u n s o v e r a l l e l e c t r o n s , and t h e components o f a^JL a r e d e f i n e d by t h e e q u a t i o n N a. A. = (gB/c) I Z e r " J ( r x (|v. - v )) ± —X _1 a -La " i d —J- — J B—X - (gB/c) I e r " 3 ( r . x (|v - v )) (2.6) j > i J J J Where I i s t h e o r b i t a l a n g u l a r momentum v e c t o r o p e r a t o r and a^ i s t h e s p i n - o r b i t c o u p l i n g parameter f o r e l e c t r o n i . F o r any m o l e c u l e , one can form a v e c t o r sum o f a l l e l e c t r o n i c s p i n o p e r a t o r s , s^, w h i c h i s denoted by S. One can efeosfe t o d e f i n e a m o l e c u l a r s p i n o r b i t c o u p l i n g c o n s t a n t , A s * ° * , so t h a t • ™-tr , . , t l . - •- . V = A S ° L • S (2.7) so - -has d i a g o n a l m a t r i x e l e m e n t s e q u a l t o t h o s e o f 2.5. E q u a t i o n 2.7 g i v e s a c o n v e n i e n t and s i m p l e form f o r e v a l u a t i n g m a t r i x e l e m e n t s o f t h e s p i n -o r b i t i n t e r a c t i o n when i t i s d i a g o n a l i n S. The o p e r a t o r L i s never w e l l - d e f i n e d e x c e p t f o r m o l e c u l e s w i t h s p h e r i c a l symmetry. F o r t h i s r e a s o n , L i s never s t r i c t l y a c o n s t a n t o f t h e m o t i o n . N e v e r t h e l e s s has a s h a r p e i g e n v a l u e i n l i n e a r m o l e c u l e s , and i t i s customary t o a s s o c i a t e t h e quantum number A w i t h t h e e i g e n v a l u e o f o p e r a t i n g on an e l e c t r o n i c w a v e f u n c t i o n . Large s p i n - o r b i t e f f e c t s a r e n o t e x p e c t e d i n bent m o l e c u l e s , b"e ucauVe-tne'tbt"al Jwa^efunctions o f bent moleculehSitates--thattGor,r*elateiwithJa ,ra6r.bi o f a l i n e a r i m o l e c u l e have o r b i t a l f a c t o r s g i v e n by 2- 2 {|A>±|-A>} . As a r e s u l t , t h e e x p e c t a t i o n v a l u e o f L^ i s 0 even i n s t a t e s c o r r e l a t i n g w i t h o r b i t a l l y d e g e n e r a t e l i n e a r m o l e c u l e g s t a t e s . Bent t r i a t o m i c m o l e c u l e s b e l o n g t o 64 t h e o r t h r h o m b i c p o i n t g r o u p s , w h i c h do n o t have d e g e n e r a t e i r r e d u c i b l e r e p r e s e n t a t i o n s . S i n c e e v e r y p o s s i b l e e l e c t r o n i c s t a t e o f a m o l e c u l e must t r a n s f o r m as one o f t h e i r r e d u c i b l e r e p r e s e n t a t i o n s o f t h e m o l e c u l a r p o i n t group i n t h a t s t a t e , s p i n - o r b i t e f f e c t s must be second o r d e r i n bent m o l e c u l e s , and not f i r s t o r d e r as one o b s e r v e s i n l i n e a r m o l e c u l e s . 65 (B) Hund's C o u p l i n g Cases The two c o u p l i n g schemes n o r m a l l y e n c o u n t e r e d i n t r i a t o m i c s ymmetric m o l e c u l e s a r e termed Hund's c a s e (a) and Hund's case (b) ( 1 ) . Hund's c o u p l i n g c a s e s a r e s i m p l y c h o i c e s o f b a s i s s e t , and a r e u s u a l l y s e l e c t e d t o c onform as c l o s e l y as p o s s i b l e t o t h e a n g u l a r momentum c o u p l i n g w h i c h a p p l i e s most e x a c t l y t o t h e m o l e c u l e under c o n s i d e r a t i o n . Hund's c a s e (a) a p p l i e s t o t h e s i t u a t i o n i n w h i c h t h e dominant f i e l d f r o m the p o i n t o f view o f t h e e l e c t r o n s i s t h e f i e l d g e n e r a t e d by t h e i r own o r b i t a l m o t i o n . T h i s c o u p l i n g scheme i s a p p r o p r i a t e t o t h e s i t u a t i o n f o u n d i n a l i n e a r m o l e c u l e f o r w h i c h l _ z and a r e w e l l - d e f i n e d o p e r a t o r s , w i t h e i g e n v a l u e s A and E, r e s p e c t i v e l y . I n t h i s c a s e , one would use t h e k e t s d e f i n e d by |0fi;LA;SZ> as m o l e c u l a r b a s i s f u n c t i o n s , where fi=A+E. I n 'molecules c o n t a i n i n g v e r y heavy atoms, t h e s p i n - o r b i t . i n t e r a c t i o n can g i v e r i s e t o v e r y l a r g e s p l i t t i n g s . T h i s s i t u a t i o n i s an'extreme c a s e (a) and i s c l a s s i f i e d s e p a r a t e l y as Hunds c a s e ( c ) . In t h i s c a s e , th e s p i n - o r b i i t i n t e r a c t i o n s have become so l a r g e t h a t L and S l o s e t h e i r meanings, and a r e r e p l a c e d by 3a=L+S w h i c h has--1, thje p r o j e c t i o n quantum number 0, = A + Z. F o r t h i s s i t u a t i o n , one would use k e t s d e f i n e d by I Ofi; 3 £l> as m o l e c u l a r b a s i s f u n c t i o n s . 3 I f f t h e s p i n - o r b i t c o u p l i n g i s weak, or i f t h e f i e l d g e n e r a t e d by th e r o t a t i o n o f t h e n u c l e a r c h a r g e s i s : s t r o n g , t h e n t h e g r e a t e s t c o u p l i n g may e x i s t between t h e t o t a l s p i n and t h e magnetic f i e l d g e n e r a t e d by t h e n u c l e i r o t a t i n g as a whole. Even i f t h e m o l e c u l e were t o conform t o Hund's c a s e (a) f o r t h e l o w e s t r o t a t i o n a l l e v e l s , t h e f i e l d cause by t h e n u c l e a r r o t a t i o n w i l l a l w a y s i n c r e a s e w i t h r o t a t i o n a l a n g u l a r momentum, so t h a t t h e m o l e c u l a r c o u p l i n g scheme t e n d s t o w a r d Hund's c a s e ( b ) . 66 I n t h e r o t a t i o n a l s p e c t r a c o n s i d e r e d i n t h i s work, t h e a n g u l a r momentum c o u p l i n g scheme was c l o s e t o Hund's cas e ( b ) , w i t h s p l i t t i n g s m a l l e r t h a n t h e r o t a t i o n a l energy l e v e l s p a c i n g s . F o r t h i s c o u p l i n g c a s e , a r e d e f i n i t i o n o f t h e a n g u l a r momenta due t o n u c l e a r m o t i o n becomes n e c e s s a r y . One r e f e r s t o t h e t o t a l a n g u l a r momentum o f t h e m o l e c u l e as 3, and names t h e t o t a l a n g u l a r momentum e x c l u d i n g s p i n as N, such t h a t 3 = N + § (2.8) T h i s s i t u a t i o n 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 i n F i g u r e 2.1. The good quantum numbers d e f i n e d by such a scheme a r e 3, N, K, and S, so t h a t the b a s i s f u n c t i o n s w h i c h w i l l be used a r e l a b e l l e d | 0,N,K,S > . K i s d e f i n e d t o be t h e quantum number a s s o c i a t e d w i t h t h e p r o j e c t i o n o f N onto t h e m o l e c u l e - f i x e d z - a x i s , and i s t h e r e f o r e t h e e i g e n v a l u e o f N . K gure 2 = 1 HuncTs •» ">', J ~ " V D ) „ F i g u r e 2.1 Hund's c o u p l i n g c a s e ( b ) . 68 (C) The O r b i t - R o t a t i o n H a m i l t o n i a n The p a r a m e t e r s a p p e a r i n g i n e q u a t i o n s 2.2 t h r o u g h 2.6 a r e n o t o b s e r v a b l e s , i n t h e sense t h a t t h e m o l e c u l a r s p e c t r u m cannot be a s s i g n e d i n terms o f t h o s e e q u a t i o n s . I n g e n e r a l , one must c o n s i d e r f i v e s o u r c e s o f a n g u l a r momentum p r e s e n t i n t h e m o l e c u l e . These f i v e a r e t h e a n g u l a r momenta c r e a t e d by (a) The n u c l e a r r o t a t i o n ( which was c a l l e d 3 i n Ch a p t e r I ) (b) t h e o r b i t a l m o t i o n o f t h e e l e c t r o n s , L ( c ) t h e i n t r i n s i c s p i n o f t h e u n p a i r e d e l e c t r o n s , S (d) t h e n u c l e a r s p i n s ( I ) (e) and t h e d e g e n e r a t e v i b r a t i o n s (G) Of t h e s e , o n l y t h e a n g u l a r momenta due t o terms (a) and (e) were c o n s i d e r e d i n C h a p t e r I . I n t e r a c t i o n s o f t h e n u c l e a r s p i n s o f te r m (d) w i t h t h e e l e c t r o n i c a n g u l a r momenta and w i t h t h e r o t a t i o n o f t h e m o l e c u l e w i l l not be c o n s i d e r e d i n t h i s t r e a t m e n t , because t h e h y p e r f i n e s t r u c t u r e f o r which t h e y a r e r e s p o n s i b l e i s not u s u a l l y r e s o l v e d i n c l a s s i c a l u l t r a v i o l e t , v i s i b l e o r i n f r a r e d s p e c t r o s c o p y , though t h e y a r e i m p o r t a n t i n microwave and i n m i c r o -w a v e - o p t i c a l d o u b l e r e s o n a n c e e x p e r i m e n t s . E f f e c t s due t o t h e v i b r a t i o n a l a n g u l a r momentum o f te r m (e) have been d i s c u s s e d a t l e n g t h i n t h e p r e c e d i n g c h a p t e r and need n o t be c o n s i d e r e d f u r t h e r h e r e . Because N i s d e f i n e d i n terms o f 0 and S ( e q u a t i o n 2 . 8 ) , i t c o n t a i n s c o n t r i b u t i o n s due t o t h e e l e c t r o n i c o r b i t a l a n g u l a r momentum, L, as w e l l as t h e a n g u l a r momentum due t o n u c l e a r r o t a t i o n . Even though t h e e x p e c t a t i o n v a l u e o f l _ z becomes s m a l l , t h e i d e n t i t y L 2 = L 2 + L 2 + L 2 (2.9) - x y z must s t i l l h o l d , and t h i s means t h a t L and l _ v must grow t o compensate 69 f o r t h e q u e n c h i n g o f as- the m o l e c u l e d e p a r t s from l i n e a r i t y . The norm o f L rem a i n s d e f i n e d by t h e v a l u e i t had i n t h e s p h e r i c a l l i m i t . Thus, even though t h e c o n t r i b u t i o n t o N i n a bent m o l e c u l e may be s m a l l , i t s e f f e c t s a r e not n e g l i g i b l e , and one must s t r i c t l y w r i t e N - L f o r t h e a n g u l a r momentum due t o t h e n u c l e a r r o t a t i o n a l o n e . In s i n g l e t s t a t e s , t h e r e f o r e , t h e q u a n t i t y 3 i n e q u a t i o n 1.51 s h o u l d s t r i c t l y be w r i t t e n N - L . On e x p a n d i n g t h e f i r s t t erm o f th e e q u a t i o n t h a t r e s u l t s , one o b t a i n s H = i N p N + | P + - P + E ( Q ) + i L u L - L u N (2.10) O - E - - - e - E - - E -I n t h i s e x p r e s s i o n , t h e term i n \ L y L i s a s m a l l c o r r e c t i o n t o the e l e c t r o n i c p o t e n t i a l e n e r g y , and w i l l be ab s o r b e d i n t o E g ( Q ) . The e q u a t i o n t h a t remains i s i d e n t i c a l t o 1.51, e x c e p t f o r t h e p e r t u r b a t i o n H = - L y N (2.11) or - = -i n t h e new n o t a t i o n . I t i s t h e p r e s e n c e o f t h i s t e r m i n t h e H a m i l t o n i a n which c o u p l e s t h e e l e c t r o n s p i n ( v i a t h e S p i n - O r b i t i n t e r a c t i o n ) t o t h e n u c l e a r r o t a t i o n , and t h i s c o u p l i n g i s t h e t o p i c o f the n e x t s e c t i o n . ( I n l i n e a r m o l e c u l e s , t h i s i s t h e term t h a t a l l o w s t h e t w o f o l d o r b i t a l degeneracy f o r A > 0 t o be l i f t e d by r o t a t i o n , g i v i n g r i s e t o t h e s m a l l s p l i t t i n g s o f t h e r o t a t i o n a l energy l e v e l s c a l l e d A - d o u b l i n g . ) 70 (D) D e r i v a t i o n o f t h e S p i n - R o t a t i o n H a m i l t o n i a n I n s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y , H ( d e f i n e d as V + V r '' so so soo i n e q u a t i o n 2.5) and H Q r g i v e r i s e t o terms i n t h e e f f e c t i v e t r a n s f o r m e d ^ H a m i l t o n i a n o f t h e form <v| H |v'> = I I <5 0v| H |.gv"><£v"| H |5oV>/(E - Ep.) s r v „ so o r t , o (2.12) i n w h i c h E,0 i s t h e l a b e l t h a t i d e n t i f i e s t h e e l e c t r o n i c s t a t e under c o n s i d e r a t i o n , and t h e summation o v e r £ r u n s over a l l o t h e r e l e c t r o n i c s t a t e s o f t h e m o l e c u l e . * S u b s t i t u t i n g f o r H s q and H Q R u s i n g 2.5 and 2.11, r e s p e c t i v e l y , one f i n d s <v| H |v'> = 1 1 <CoH I'a.Ji. • s |5v"><5v"| -L.y N |?0v.!:> S r v" 5>?o { i 1 - 1 _ 1 * ( E r - E F ) (2.13) Co C T h i s e e a n e s b e c . i s i m p l i f i e d s , because many o p e r a t o r s may be e x t r a c t e d from t h i s e x p r e s s i o n . A v i r t u a l e l e c t r o n i c e x c i t a t i o n i s f o r m a l l y e q u i v a l e n t t o t h i s p e r t u r b a t i o n e x p r e s s i o n , and by t h e Frarick-Condon p r i n c i p l e , t h e n u c l e i w i l l not move a p p r e c i a b l y d u r i n g s s u c h an - e x c i t i a t i o n . ^ A s - a " r e s u l t , t h e y t e n s o r may be e x t r a c t e d from t h e m a t r i x e l e m e n t s , as can N which must be a c o n s t a n t o f t h e m o t i o n . By t h e W i g n e r - E c k a r t theorem, i t has been shown ( 2 , 3) t h a t <C 0 | s. |£>.= <£o| k. |?> § (2.14) so t h a t S may a l s o be e x t r a c t e d from t h e f i r s t t e r m . I n 2.14, k^ i s 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 t h a t r e l a t e s t h e s p i n o p e r a t o r f o r e l e c t r o n i t o t h e t o t a l s p i n o p e r a t o r f o r t h e m o l e c u l e . Thus, 2.13 r e d u c e s t o t h e f o r m 71 H s r = S • e • N (2.15) where | = M • I I -2 <to| k . a . ^ |?><5| L |? 0> e > ? 0 i ^ (2.16) Co 5 Two i m p o r t a n t p o i n t s must be made. E q u a t i o n 2.15 r e l a t e s two non-^commuting o p e r a t o r s , N and S, t h r o u g h t h e t e n s o r e . S t r i c t l y s p e a k i n g , a h e r m i t i a n average must be employed t o p r e s e r v e t h e h e r m i t i a n form o f t h e m o l e c u l a r H a m i l t o n i a n , so t h a t H s r = ^ ( - * i * - + - ' l ' - } ) (2.17) Such an average i s not r e q u i r e d f o r symm e t r i c t r i a t o m i c m o l e c u l e s because t h e e t e n s o r i s made d i a g o n a l by t h e same t r a n s f o r m a t i o n t h a t d i a g o n a l i z e s t h e y_ t e n s o r t o w h i c h i t i s p r o p o r t i o n a l . A n o t h e r i n t e r e s t i n g consequence o f 2.16 i s t h a t t h e s p i n - r o t a t i o n c o n s t a n t s o f t h e same v i b r a t i o n a l l e v e l o f t h e c o r r e s p o n d i n g e l e c t r o n i c s t a t e s o f two i s o t o p i c m o l e c u l e s s h o u l d be r e l a t e d by t h e r a t i o s o f the a p p r o p r i a t e y_ t e n s o r s e i e m e n t s . T h i s means t h a t a c o n v e n i e n t check on t h e ass i g n m e n t o f t h e r o t a t i o n a l l i n e s o f t h e two s p e c i e s can be o b t a i n e d from t h i s r a t i o , as was found i n t h e a n a l y s i s o f t h e t r i p l e t s t a t e o f SO- d i s c u s s e d i n C h a p t e r V I I . 72 (E) C e n t r i f u g a l C o r r e c t i o n s t o t h e S p i n - R o t a t i o n C o u p l i n g . Because t h e e t e n s o r a r i s e s from a s e c o n d - o r d e r c o u p l i n g of H - - so w i t h H Q r , a c e n t r i f u g a l c o r r e c t i o n i s p o s s i b l e . a s - w a s , d e m o n s t r a t e d by Bli-xtfnr.aadaD.tixfeu.Eyiu.(*4) . > ) S u b s t i t u t i " b h - s d f N --b fiorMO-ih I r l O O g e n e r a t e s t,e'r lmsairasthe:reififectiv.e Ha~mi»it6.roiah'ao.fItheifiormf t h e *orm H" = i I x D . (N - L )(N_ - L 0)(N - L )(N. - U ) (2.18) E x p a n s i o n o f t h e H a m i l t o n i a n producessmany.yterms? • E x p r e s s i o n s i n L L„L L_, N L.L L R, and N N~L L~ can be r e a r r a n g e d t o g i v e n s m a i i l ' c o r r e c t i o n s t o t h e e f f e c t i v e e l e c t r o n i c e n e r g y , E g ( Q ) , t h e s p i n - r o t a t i o n c o n s t a n t s e n d e f i n e d by 2.16 above, and t h e r o t a t i o n a l c o n s t a n t s u „, a 3 a 3 r e s p e c t i v e l y . D i x o n and Duxbury (4) have shown t h a t t h e r e m a i n i n g e x p r e s s i o n s i n N^ N^ N c o u p l e w i t h H s q t o g i v e a a e e n t r i f l i i g a l a c o r r e c r t s i o n o t o ^ t h e s p i n -r o t a t i o n , o f t h e form HC r . ( i f ) = I ri „ J N J S, (2.19) s r £ 'OI3YS a 3 Y <5 Thus t h e g e n e r a l i z a t i o n o f t h e e f f e c t i v e H a m i l t o n i a n t o i n c l u d e o r b i t a l l y non-degenerate w M i p l e t t s t a t e s j l e a d t t - o - t h e ^ e x p r e s s i o n H * = I N ( ^ 0G N D + e 0 S D ) r o t a L R a a 3 3 a 3 3 + J i T „ . N N.N N. + n _ . N N_N S_ (2.20) aSy6 Y a Y Y a Y Many o f t h e c o e f f i c i e n t s o f 2.20 a r e l i n e a r l y r e l a t e d o r v a n i s h i d e n t i c a l l y , beGausespi%symmet£y.orequirementsts. These c e n t r i f u g a l c o r r e c t i o n s a r i s e from two i n d e p e n d e n t mechanisms. F i r s t , t h e " m a g n e t i c f i e l d g e n e r a t e d by t h e r o t a t i o n a l a n g u l a r momentum o f 73 t h e n u c l e a r framework depends on t h e r o t a t i o n a l v e l o c i t y , and t h e r e f o r e on t h e moments o f i n e r t i a . As a r e s u l t , c e n t r i f u g a l d i s t o r t i o n w i l l m o d i f y t h e s p i n - r o t a t i o n i n t e r a c t i o n . Second, t h e c e n t r i f u g a l d i s t o r t i o n o f t h e n u c l e a r framework w i l l a l t e r t h e e l e c t r o n i c s t r u c t u r e o f t h e m o l e c u l e , and t h e r e b y m o d i f y both t h e s p i n - o r b i t c o u p l i n g and t h e form o f E g ( Q ) . U s u a l l y t h e second c o n t r i b u t i o n i s c o m p l e t e l y n e g l i g i b l e w i t h i n t h e p r e c i s i o n o f e x p e r i m e n t a l d a t a . Because t h e t r a n s f o r m a t i o n t h a t d i a g o n a l i z e s y d i a g o n a l i z e s e a l s o ( s i n c e e i s p r o p o r t i o n a l t o y ) , one can s t i l l w r i t e a r e d u c e d form f o r t h e e f f e c t i v e H a m i l t o n i a n H 4- = I N d-U™eti, + e s ) + y. N 2 ( £T 0 0 N 0 2 + n D 0 N 0 S D ) r o t £ a a ^aa a aa a a% a a a a 3 3 3 3 (2.21) which a p p l i e s t o o r t h o r h o m b i c m o l e c u l e s and i s complete up t o terms i n N 4. R e l a t i o n s h i p s between t h e T'S and t h e n's e x i s t , w hich have been g i v e n by Di x o n and Duxbury ( 4 ) . 74 (F) M a t r i x Elements o f t h e S p i n - R o t a t i o n H a m i l t o n i a n M a t r i x e l e m e n t s f o r t h e e f f e c t i v e H a m i l t o n i a n i n c l u d i n g s p i n - , r o t a t i o n i n t e r a c t i o n have been t r e a t e d by van V l e c k ( 2 ) , Raynes ( 5 ) , and d i Lauro ( 6 ) . Raynes' f o r m u l a e a r e t h e e a s i e s t t o a p p l y , but th e y c o n t a i n an u n f o r t u n a t e m i x t u r e o f phases t h a t has caused some c o n f u s i o n i n t h e l i t e r a t u r e . The pro b l e m a r i s e s because Raynes combined van V l e c k ' s s p i n f o r m u l a e w i t h P o s e n e r , ..and S t r a n d b e r g ' s (7) f o r m u l a f o r t h e c e n t r i f u g a l d i s t o r t i o n d i r e c t l y . Van V l e c k used t h e Condon and S h o r t l e y (8) phase c o n v e n t i o n , i n which m a t r i x e l e m e n t s o f (3 ± t 3 ) a r e r e a l and p o s i t i v e , w h i l e Posener and x y S t r a n d b e r g f o l l o w e d K i n g , H a i n e r and C r o s s (9) and chose t h o s e o f ( 3 y ± ^ x ) t o be r e a l and p o s i t i v e . The r e s u l t was t h a t Raynes gave t h e phases f o r t h e asymmetry and o f f - d i a g o n a l c e n t r i f u g a l d i s t o r t i o n r e v e r s e d from t h o s e o f t h e s p i n i n t e r a c t i o n s t h r o u g h o u t h i s t a b l e . B i r s s e t . a l . c o r r e c t e d t h e e r r o r i n s i g n o f Raynes' asymmetry t e r m , but t h i s change was c o n t e s t e d by Brand e t . a l . (11) who s t a t e d t h a t Raynes was u s i n g t h e "phase c o n v e n t i o n t h a t t h e el e m e n t s o f ( 3 y ±^3 x) a r e p o s i t i v e and r e a l " , and t h e r e f o r e m a i n t a i n e d t h a t t h e r e was no e r r o r . N e v e r t h e l e s s , Raynes t r a n s c r i b e d van V l e c k ' s s p i n m a t r i x e l ements d i r e c t l y , so he was i m p l i c i t l y u s i n g t h e Condon and S h o r t l e y phase c o n v e n t i o n f o r them. The s p e c i f i c phase c h o i c e made i s not i m p o r t a n t , b u t t h e phase c h o i c e must be c o n s i s t e n t f o r a l l m a t r i x e l e m e n t s which a r e used t o p r e d i c t m o l e c u l a r s p e c t r a . I f a m i x t u r e o f phases i s used f o r o p e r a t o r s t h a t appear as sums o r d i f f e r e n c e s i n o f f - d i a g o n a l e l e m e n t s , t h e r e s u l t s o f t h e d i a g o n a l i z a t i o n w i l l be i n c o r r e c t . Bowater, Brown and C a r r i n g t o n (12) have g i v e n t h e m a t r i x e l e m e n t s 75 o f t h e e f f e c t i v e s p i n - r o t a t i o n H a m i l t o n i a n i n a c o n s i s t e n t phase c h o i c e , w h i c h i s t h a t o f Condon and S h o r t l e y ( 8 ) . T h e i r m a t r i x e l e m e n t s a r e w r i t t e n as c o r r e c t i o n s t o Watson's " r e d u c e d " H a m i l t o n i a n ( e q u a t i o n 1.102) u s i n g s p h e r i c a l t e n s o r n o t a t i o n . F o r t h e s e r e a s o n s , t h e f o r m a l i s m o f Bowater e t . a l . (12) as e x t e n d e d by Brown and Howard (13) has been used t h r o u g h o u t t h i s t h e s i s . T h e i r p r o c e d u r e c o n s i d e r s N as a s p a c e - f i x e d , r a t h e r t h a n a m o l e c u l e - f i x e d a n g u l a r momentum o p e r a t o r . From a p h y s i c a l p o i n t o f v i e w t h i s c h o i c e i s l o g i c a l , s i n c e t h e o p e r a t o r N i s d e f i n e d i n terms o f an i n f i n i t e s i m a l r o t a t i o n o f t h e m o l e c u l e - f i x e d a x i s w i t h i n t h e l a b o r a t o r y frame. T h e i r approach i s t o expand t h e i r r e d u c i b l e t e n s o r components o f t h e e f f e c t i v e H a m i l t o n i a n as s p a c e - f i x e d o p e r a t o r s which t h e y l a b e l by t h e i n d e x p. These o p e r a t o r s can t h e n be r e l a t e d t o t h e m o l e c u l e - f i x e d a x i s system ( i n d e x e d by q) by means o f t h e r o t a t i o n o p e r a t o r , so t h a t T k (A) = 7 D ( k )*(co) • T k (A) (2.22) p - Jj pq q -i n w h i c h k i s t h e r a n k o f t h e s p h e r i c a l t e n s o r , and t h e form o f t h e r o t a t i o n o p e r a t o r i s i m p l i c i t i n t h e c h o i c e o f phases f o r t h e symmetric t o p b a s i s f u n c t i o n s , w h i c h i s : | N,K;,,M > = {(2N + D / S u 2 } * (u>) (2.23) where co s t a n d s f o r t h e t h r e e E u l e r a n g l e s . I n i r r e d u c i b l e t e n s o r f o r m ^ i e q u a t i o n (2.15) t a k e s t h e form I k k H = I T K ( e ) • T K(N,S) (2.24) S k = 0 76 where T (N,S) i s an i r r e d u c i b l e t e n s o r o p e r a t o r d e f i n e d by c o u p l i n g t h e two f i r s t - r a n k t e n s o r s T J ( N ) and T1(S) u s i n g t h e r e l a t i o n s h i p T k (N,S) = ( - 1 ) P ( 2 k + l ) * I T 1 (N) T 1 (S) P " " _ " pi - P2 P 1 J P 2 r I l k I P 2 P2 P J (2.25) k T (e) i s an i r r e d u c i b l e t e n s o r (k = 0, 1, 2 ) , w h i c h would have n i n e components f o r a m o l e c u l e w i t h no symmetry. F o r an o r t h o r h o m b i c m o l e c u l e t h e r e a r e o n l y t h r e e n on-zero components. These t h r e e p a r a m e t e r s a r e r e l a t e d t o t h e and t h e van V l e c k / Raynes p a r a m e t e r s as i n T a b l e 2.1. The m a t r i x e l e m e n t s o f t h e s p i n - r o t a t i o n H a m i l t o n i a n a r e g i v e n i n T a b l e 2.2. I t can be shown t h a t t h e e x p r e s s i o n s a r e i d e n t i c a l t o t h o s e g i v e n by Raynes and van V l e c k e x c e p t f o r t h e s i g n s o f t h e AN = ±1 e l e m e n t s , w h i c h a r e r e v e r s e d because o f t h e c h o i c e o f N as a s p a c e - f i x e d r a t h e r t h a n as a m o l e c u l e - f i x e d o p e r a t o r . T a b l e 2.2 i n c l u d e s t h e s p i n - s p i n m a t r i x e l e m e n t s , w h i c h a r e d e r i v e d i n t h e n e x t s e c t i o n . 77 T a b l e 2.1 C o r r e l a t i o n o f t h e V a r i o u s Non-Zero S p i n - R o t a t i o n P a r a m e t e r s f o r an O r t h o r h o m b i c M o l e c u l e . T k (e) q = van Vleek/Raynes p a r a m e t e r . T° o ( | ) _ i - 3 - 2 ( e + e + e )-\ xx yy z z 3*a 0 T 2 ± 2 ( = ) He - e ) xx yy -b T 2 0 ( | ) ! 6"*(2e - e - £ ) z z xx yy -6^ 3 78 T a b l e 2.2 M a t r i x Elements o f t h e S p i n - R o t a t i o n and S p i n - S p i n H a m i l t o n i a n s I n S p h e r i c a l Tensor N o t a t i o n . < N', K ' , S , 3' | H r | N/'K, S, 3 > = 6 £ (2k + 1) S J J k = 0 { S ( S + 1 ) ( 2 S + 1)(2N + 1 ) ( 2 N ' + l ) } 2 x (-1) x ( - l ) k {N(N + 1 ) ( 2 N + 1 ) } * f J , J 3+S+N' fN S 3 S N' 1 X I (-D q N'-K' N' k N i -K' q K J TVl> < N' ; K' , S' , 3' | H | N, K, S, 3 > = -| /30 6 g 2 6 2 ( - 1 ) N + : I + S S3 J J } x I ( - 1 ) N ' " K ' {(2N + 1)(2N' + l ) } 4 J n 3 S N ' I v / , x ' -  2 N S q N' -K' 2 q N K T M C ) 79 (G) E x t e n s i o n of. t h e E f f e c t i v e H a m i l t o n i a n t o T r i p l e t S t a t e s . T r i p l e t e l e c t r o n i c s t a t e s a r i s e i f t h e r e a r e two u n p a i r e d e l e c t r o n s i n a m o l e c u l e , f o r w h i c h t h e quantum number S = 1. Oust as f o r d o u b l e t s t a t e s , each u n p a i r e d e l e c t r o n can i n t e r a c t w i t h t h e f i e l d o f t h e n u c l e i r o t a t i n g as a whole and so g i v e r i s e t o t h e s p i n - r o t a t i o n i n t e r a c t i o n a l r e a d y d i s c u s s e d . The r e s u l t s o f t h e p r e v i o u s s e c t i o n and T a b l e 2.2 can be used d i r e c t l y f o r m o l e c u l e s i n t h e t r i p l e t s t a t e i f t h e s u b s t i t u t i o n S = 1 i s made i n s t e a d o f S = \. Howewerada^taoifiu^thebecom^lbi'cation,pt-fte.?-fcwd e l e c t f b n i s p i ' n s ^ c a n i n t e r a c t s w i t h eachrotherithThishadGlseandtheEtterm:, r hdenol3^a b y o i n'"r V t o t h e p o t e n t i a l , w h i c h i s p t h e e o p e r a t - o r o f o r a t f i e (di'poiie-dipol'e;) A e l e c t r o n s p i n - e l e c t r o n s p i n i n t e r a c t i o n . F o r m o l e c u l e s i n o r b i t a l l y n on-degenerate t r i p l e t s t a t e s , t h e p o t e n t i a l o p e r a t o r w h i c h c o r r e s p o n d s t o 2.1 becomes V = V + V + V + V + V + V (2.26) nn en ee so soo s s where V = g 2B 2 I I ( ( s . • s . ) r . . 2 - 3 ( s . • r . . ) ( s . • r . . ) ) / r . . 3 s s y i ^ 1 £ ± - l - j i j v - i - i j - j - i j i j (2.27) where, j and i r u n o v e r t h e n e l e c t r o n s i n t h e m o l e c u l e , van V l e c k (2) showed t h a t f o r o r t h o r h o m b i c i m o l e c u l e s ? tke e f f e c t i v e spin^spTn*"' ' i i n . t e f ' a c t i w a d p e r a t o ' r has t h e form H ' = a ( 2 S 2 - S 2 - S 2 ) + 6(S 2 - S 2 ) (2.28) s s z x y x y f o r o r t h o r h r ^ b t c - - ^ ( j 1 ^ 80 (H) M a t r i x Elements o f t h e S p i n - S p i n H a m i l t o n i a n f o r T r i p l e t S t a t e s o f O r t h o r h o m b i c M o l e c u l e s . I n s p h e r i c a l t e n s o r n o t a t i o n , 2.27 can be w r i t t e n H = - /6 g 2 B 2 I T 2 ( C ) • T 2 ( s . , s ) (2.29) S j > i _ 1 " J Herev t h e components o f T 2 ( C ) a r e r e l a t e d t o t h e a v e r a g e s o f s p h e r i c a l h a r monics o v e r v i b r a t i o n a l c o o r d i n a t e s . The t r u e s p i n - s p i n p a r t (see below) can be d e f i n e d by t h e e x p r e s s i o n T 2 (C) = I (4TT/5) 1 Y 0 (9,d>)/r. . 3 (2.30) In a c a s e (b) b a s i s s e t , f o r a I r a x i s c h o i c e , t h e m a t r i x e l e m e n t s o f 2.29 a r e g i v e n by t h e e x p r e s s i o n o f T a b l e 2.2. The s p h e r i c a l t e n s o r m a t r i x e l e m e n t s r e d u c e t o t h e form g i v e n by Raynes on s u b s t i t u t i n g a = - i g 2 B 2 T 2 0 ( C ) (2.31a) and g = - 1 U g 2 3 2 T 2 (C) (2.32b) ' ± 2 Only t h e f i e l d - f r e e H a m i l t o n i a n was c o n s i d e r e d i n t h i s work. S i n c e t h e s p i n - s p i n i n t e r a c t i o n ' s d i a g o n a l i n t h e n u c l e a r s p i n quantum numbers, t h e m a t r i x e l e m e n t s g i v e n i n T a b l e 2.2 can be combined w i t h e q u a t i o n s (27) t h r o u g h (37) o f r e f e r e n c e (12) t o d e s c r i b e t h e e nergy l e v e l s o f a t r i p l e t s t a t e c o m p l e t e l y , i n c l u d i n g t h e S t a r k and Zeeman e f f e c t s . T a b l e 2.2 g i v e s t h e e l e m e n t s r e q u i r e d 'to s e t up t h e s p i n p a r t o f t h e c o m p lete v i b r a t i o n - r o t a t i o n H a m i l t o n i a n f o r m o l e c u l e s i n any s t a t e where S < 3/2. The H a m i l t o n i a n m a t r i x must al w a y s f a c t o r i z e i n t o f o u r 81 s u b m a t r i c e s f o r t h e f i e l d - f r e e p r o b l e m , and each o f t h e s e can o n l y c o n t a i n A i , A 2 , B i o r B 2 r o t a t i o n a l l e v e l s f o r a m o l e c u l e w i t h C p o i n t group symmetry. The f a c t o r i z a t i o n o f t h e c o m p l e t e H a m i l t o n i a n m a t r i x i n t o even and odd K s u b m a t r i c e s f o r a g i v e n v a l u e o f 3 f o l l o w s from t h e f a c t t h a t non-zero m a t r i x e l e m e n t s o n l y o c c u r f o r AK = 0, ±2. F u r t h e r f a c t o r i z a t i o n i n t o f o u r s u b m a t r i c e s must o c c u r because of p a r i t y c o n s i d e r a t i o n s . I n p r a c t i c e , e x p r e s s i o n 2.29 g i v e s o n l y p a r t o f what i s u s u a l l y c a l l e d t h e s p i n - s p i n i n t e r a c t i o n . I n a d d i t i o n , t h e r e a r e second o r d e r s p i n - o r b i t e f f e c t s w i t h t h e same quantum number dependence t h a t c o n t r i b u t e t o t h e o b s e r v e d a and 8. U s u a l l y t h e t r u e s p i n - s p i n i n t e r a c t i o n i s q u i t e s m a l l , so t h a t t h e o b s e r v e d p a r a m e t e r s a and 8 are m a i n l y s e c o n d - o r d e r s p i n - o r b i t e f f e c t s . I n t h i s t h e s i s , no a t t e m p t has been made t o s e p a r a t e t h e two c o n t r i b u t i o n s t o a and 8. I t would seem t h a t t h i s development o f t h e e f f e c t i v e H a m i l t o n i a n and i t s m a t r i x e l e m e n t s ought t o be s u f f i c i e n t t o u n d e r s t a n d t h e d o u b l e t s t a t e s o f N0 2 and t h e s i n g l e t and t r i p l e t s t a t e s o f S 0 2 . However, d e t a i l e d a n a l y s i s o f t h e 3 B i J A i t r a n s i t i o n i n S 0 2 as w e l l as t h e t r a n s i t i o n o b s e r v e d i n N0 2 a t 7390 A r e v e a l e d a n o m a l i e s c o n s i s t e n t w i t h a c o n s i d e r a b l e s p i n - o r b i t i n t e r a c t i o n between n e a r l y d e g e n e r a t e s t a t e s o f t h e same m u l t i p l i c i t y . I n o r d e r t o u n d e r s t a n d t h e s e l o c a l p e r t u r b a t i o n s , i t was n e c e s s a r y t o e v a l u a t e m a t r i x e l e m e n t s f o r t h e d i r e c t s p i n - o r b i t i n t e r a c t i o n between m u l t i p l e t s t a t e s o f a ^ m o l e c u l e , and t h e n e x t c h a p t e r c o n s i d e r s t h i s p r o b l e m i n d e t a i l . CHAPTER I I I . THE DIRECT SPIN-ORBIT INTERACTION. 83 (A) I n t r o d u c t i o n . S p i n - o r b i t i n t e r a c t i o n s between d i f f e r e n t e l e c t r o n i c s t a t e s have been e x t e n s i v e l y s t u d i e d i n d i a t o m i c m o l e c u l e s , but have not been c o n s i d e r e d t o the same e x t e n t i n t h e 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 f o r s e v e r a l r e a s o n s . F i r s t , t h e a t o m i c number dependence i n t h e s p i n - o r b i t c o u p l i n g ( d i s c u s s e d i n C h a p t e r I I ) means t h a t f o r t h e l a r g e m a j o r i t y o f p o l y a t o m i c m o l e c u l e s composed o f l i g h t atoms, t h e s p i n - o r b i t i n t e r a c t i o n i s l i k e l y t o be s m a l l , though not n e g l i g i b l e . Second, o t h e r s o u r c e s o f p e r t u r b a t i o n s b b e c o m e T f l a r m m o r e i i m p o r t a h t t i t h a n t i t h e s s p d ^ s o r b i t c c o p p l i n g i n t h e s p e c t r a o f n o n - l i n e a r p o l y a t o m i c m o l e c u l e s , ~ - ? - f o r - i n s t a n c e , n u c l e a r momentum c o u p l i n g . T h i r d , a r e q u i r e m e n t f o r t h e p r e s e n c e o f t h i s i n t e r a c t i o n i s t h a t one o f t h e i n t e r a c t i n g e l e c t r o n i c s t a t e s have .•§5i>! 0t:lSinceccbj^af.aifed\velvyniiPe.wamdiltiplet- ' s t a t e s o d f t p o l y & t o m i e " m o l e c u l e s have been a n a l y s e d i n d e t a i l a t a r e s o l u t i o n h i g h enough t o see r o t a t i o n a l p e r t u r b a t i o n s , i t had notMbeen h e c e s s a r y t d ^ d e v e l o p f o r m u l a e f o r d e s c r i b i n g d i r e c t s p i n - o r b i t i n t e r a c t i o n s i n t h e s e m o l e c u l e s ; A s i m i l a r s i t u a t i o n e x i s t s r e g a r d i n g s i n g l e t - t r i p l e t i n t e r s y s t e m c r o s s i n g p e r t u r b a t i o n s , w h i c h a r e v e r y i m p o r t a n t i n t h e t h e o r y , o f r a d i a t i o n l e s s p r o c e s s e s . U s u a l l y h i g h l e v e l d e n s i t i e s i n t h e i n t e r a c t i n g s t a t e s , and t h e c o m p l e x i t y o f t h e competing p e r t u r b a t i o n mechanisms means t h a t even i f t h e spec t r u m o b t a i n e d c o u l d be a n a l y s e d i n d e t a i l , v e r y l i t t l e c o u l d be s a i d about t h e p e r t u r b a t i o n s o b s e r v e d . F o r a . m o l e c u l e w i t h C^ v p o i n t group symmetry, s i n g l e t and t r i p l e t s t a t e s o f t h e same o r b i t a l symmetry cannot i n t e r a c t d i r e c t l y . Any i n t e r a c t i o n w i l l t h e r e f o r e be a second o r d e r p r o c e s s ( 1 ) , and w i l l appear as a s p i n - r o t a t i o n i n t e r a c t i o n by t h e mechanism o u t l i n e d i n t h e p r e v i o u s C h a p t e r . 84 F o r a bent m o l e c u l e , i n which t h e a n g u l a r momentum c o u p l i n g scheme i s c l o s e t o Hund's c a s e ( b ) , t h e e x p e c t a t i o n y a l u e o f l _ z i s s m a l l . However, a s m a l l v a l u e o f <L z> need not s i g n i f y a h s m a l l o f f - d i a g o n a l s p i n - o r b i t i n t e r a c t i o n , because t h e o r b i t a l a n g u l a r momentum L 2 i s c o n s e r v e d . T h i s means t h a t t h e s m a l l v a l u e o f < L z > i i m m e d i a t e l y i m p l i e s c o r r e s p o n d i n g l y l a r g e r v a l u e s f o r <l-x> and < L y > • Large e f f e c t s may r e s u l t i f t h e d i f f e r e n c e i n energy between e l e c t r o n i c s t a t e s t h a t a r e a l l o w e d t o i n t e r a c t v i a L and L i s s u f f i c i e n t l y s m a l l , x y J T h i s c h a p t e r d e v e l o p s e x p l i c i t m a t r i x e l e m e n t s f o r t h e d i r e c t s p i n -o r b i t i n t e r a c t i o n between e l e c t r o n i c s t a t e s o f p o l y a t o m i c m o l e c u l e s o f t h e same s p i n m u l t i p l i c i t y . The 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 p r e s e n t e d h e r e were m o t i v a t e d by an a t t e m p t t o u n d e r s t a n d t h e numerous r o t a t i o n a l p e r t u r b a t i o n s t h a t were o b s e r v e d i n t h e h i g h e r v i b r a t i o n a l l e v e l s o f t h e a 3 B i s t a t e o f S 0 2 , d i s c u s s e d i n C h a p t e r V I I . T h i s s t a t e i s p e r t u r b e d by t h e 3 A 2 s t a t e w i t h which i t i s d e g e n e r a t e t o w i t h i n a few hundred cm \ and some o f t h e p e r t u r b a t i o n s o b s e r v e d (though not a l l o f them) can be a s c r i b e d t o d i r e c t s p i n - o r b i t i n t e r a c t i o n between t h e s e two s t a t e s . As i n t h e p r e c e d i n g c h a p t e r , t h e n o t a t i o n and phase c h o i c e u adopted i s t h a t o f Bowater, Brown and C a r r i n g t o n (2) as a m p l i f i e d i n t h e work o f Brown and Howard ( 3 ) , i n t h e phase c o n v e n t i o n o f Condon and S h o r t l e y ( 4 ) . T h e i r d e f i n i t i o n o f N i n terms o f a s p a c e - f i x e d , r a t h e r t h a n as a m o l e c u l e -f i x e d q u a n t i t y i s a l s o a d o p t e d . M a t r i x e l e m e n t s a r e d e v e l o p e d f o r a m o l e c u l e i n c a s e (b) c o u p l i n g , which i s t h e Hund's c o u p l i n g case a p p r o p r i a t e t o t h e s m a l l t r i a t o m i c m o l e c u l e s s t u d i e d i n t h i s work. The r e s u l t s p r e s e n t e d here can t h e r e f o r e be c o n s i d e r e d t o be an e x t e n s i o n o f t h e r e s u l t s d e r i v e d f o r d i a t o m i c and l i n e a r p o l y a t o m i c m o l e c u l e s by Kovacs (5) and Chow C h i u ( 6 ) . 85 (B) The Matrix Elements for Direct Spin-Orbit Interaction. The spin-orbit coupling Hamiltonian for a non-linear molecule was given in equation 2.7. This equation can be recast into a form H = L + • A s o • S (3.1) so , - = in which L and S are the orbital and spin angular momentum vectors, SO respectively, and A i s a 3 x 3 matrix containing the spin-orbit coupling coefficients. Since the molecules in question do not have spherical symmetry, L w i l l be far from being a good quantum number. *t* SO This makes i t necessary to consider the product L • A as a vector, V, whose components are given by the three relations V = (L + • A s o) =-L A S 0 + L A S 0 + L A s° (3.2) a - = a x xa y ya z za where a runs over the set of molecular axes (x, y, z ) . The interaction Hamiltonian i s therefore a scalar product of this vector with S.^ , Converting 3.2 to spherical tensor notation, the interaction Hamiltonian of equation 2.5 becomes H = T^V) • Jl(S) (3.3) so -Standard angular momentum techniques (7, 8, 9) wer.ebusedet:odevaluate the matrix elements of 3.3 in a I case (b) basis. As shown in 'Appendix I? the matrix elements are < n ' , N', S , 3 , K' | T X(V) • T l(S) | n, N,3S,,3, K > = (_1)N+S+3 x { s ( s + 1 ) ( 2 S + 1 ) ( 2 N + 1 ) ( 2 N , + 1 ) } 1 N S I X I N' 1 N [ -K* q K J -< n' F CV) I r, > (3.4) v q -Table 3.1 * Matrix elements of T X(V) • T X(S) for S = |. VO oo AK=0 |3,K,N=3-i> |3,K,N=3+i> <3,K,N=3-i| K/(23+l) -{(3+|)2-K2}i/(23i1-l) -{(3+i)2-K2}*/(23+l) -K/(23+l) AK=±1 |3,K,N=3-i> |3,K,N=3+1> <3,K±l,N=3-i| <3,K±l,N=3+i| i 2 (23-l)(23+l)-4K(K+l)U [(3+K+§)(3+K-|)l* 2(23+1)2 J r(3±K+i)(3±M)l* + if 2(23*1)2 > i (23+l)(23+3)-4K(K±l)l2 <• 2(23+l) 2 J 2(23+l) 2 ^ T a b l e 3.2 M a t r i x e lements o f T ^ V ) • T*(S) f o r S = 1. AK=0 |3,K,N=3-1> |J,K,N=3> |3,K,N=3+1> <3,K,N=3-l| K/3 - { ( 3 2 - K 2 ) ( 3 + l ) / ( 2 3 + l ) } * / 3 0 <3,K,N=3| - K/{ 3 ( 3 + l ) } - ( { ( 3 + l ) 2 - K 2 } a / ( 2 a + l ) } i / ( 3 + l ) <3,K,N=3+l| f' -V S y m m e t r i c a l - K /,(3+l) AK=±1 <3,K±1,N=3-1 <3,K±T,N=3 <3,K+1,N=3+1 3,K,N=3-1> 3(3-l)-K(K±l) 23' J (3+l)(3±K)(3±K+l) 2(23+1).3 2 3,K,N=3> 3,K,N=3+1> (3+l)(3+"K)(3+K-l) 2 ( 2 3 + l ) 3 2 3(3+1)-K(K±1) 2 3 l ( ; 3 + l j 2 ' 3(3±K+l)(3±K+2)> 2 2 ( 2 3 + l ) ( 3 + l ) 2 ', 3(3+K)(3+K+l) 2 ( 2 3 + l ) ( 3 + l ) 2 J (3+l)(3+2)-K(K±l)' 2 ( 3 + l ) : 88 The r e d u c e d m a t r i x element i s t h e e x p e r i m e n t a l l y d e t e r m i n a b l e parameter i n t h i s e q u a t i o n . F o r d i a t o m i c m o l e c u l e s , t h e s e r e d u c e d m a t r i x e l e m e n t s would be w r i t t e n < n1 I T 1 (V) I n > = < A | A S 0 L I A > (3.5a) o - z < n' I T J + 1(y) | n > = + 2~* < A ± l | A S O L + | A > (3.5b) S O where A i s t h e d i a t o m i c s p i n - o r b i t c o u p l i n g c o n s t a n t . I n e q u a t i o n 3.5b, L + d e n o t e s t h e o p e r a t o r s ( l _ x ± i L ^ ) , and t h e p r o j e c t i o n quantum number K has been r e p l a c e d by A . E x p l i c i t r e l a t i o n s f o r e q u a t i o n 3.4 i n d o u b l e t s t a t e s a r e g i v e n i n T a b l e 3.1, and f o r t r i p l e t e l e c t r o n i c s t a t e s i n T a b l e 3.2 ( see a l s o r e f e r e n c e s 5 and 6 ) . The form o f t h e m a t r i x e lements g i v e n i n 3.4 d e f i n e s * r o t a t i o n a l s e l e c t i o n rruibes: :these a r e AK = 0 f o r terms i n v o l v i n g T 1 o r AK = ±1 ' o f o r terms i n T x + 1 , s u b j e c t t o t h e o v e r a l l s e l e c t i o n r u l e AN = 0, ±1. Which o f t h e components o f T g i v e s t h e i n t e r a c t i o n between two s p e c i f i c v i b r o n i c s t a t e s , ri and n ' , f o l l o w s from a c o n s i d e r a t i o n o f theirotat<i6ma!l!asymmet-ries<?s„ F o r m o l e c u l e s w i t h C^ v symmetry i n which t h e b i n e r t i a l a x i s i s t h e z - a x i s o f t h e p o i n t group, ( f o r example, NO,, and SO,,), t h e r u l e s a r e summarized by t h e e n t r i e s o f T a b l e 3.3. . ' M l t h e l m o l e c u l e t b e l b n g e d g t d totlhee'C p o i n t g r o u p , any two o f i t s v i b r o n i c l e v e l s c o u l d i n t e r a c t a c c o r d i n g t o both AK*= 0 and ±1. Because)r;(cry) (x F(^';c)ntransf6rmS'3likeoaar-otati6h^ th'e~AN =^-0 m a t r i x e l e m e n t s must a l s o i n c l u d e a c o n t r i b u t i o n from t h e C o r i o l i s o p e r a t o r f o r t h e o r b i t - r o t a t i o n i n t e r a c t i o n , w hich was g i v e n by e q u a t i o n 2.11 o f Ch a p t e r I I . F o r n o n - l i n e a r m o l e c u l e s , t h e el e m e n t s w i l l be r ass-v'eighted ~vpect<.ition v a l u e s o f L, m u l t i p l i e d by t h e c j p z : J-• * t.+ . •".erits due t o N. these a r e w r i t t e n as 89 < n1 NSOK | H Q r | nNSOK > = -2K < n' I zi_z I n > (3.6a) < n'NS3K±l I H I nNSGK > = - {N(N + 1) - K(K ± 1)}* x ' or 1 < n' | XL X + i Y L y | n > (3.6b) i n w h i c h t h e s p e c i f i c i d e n t i f i c a t i o n o f t h e r o t a t i o n a l c o n s t a n t s X, Y, and Z has n o t been made. I n a I r e p r e s e n t a t i o n , t h e o p e r a t o r s c o r r e s p o n d t o AL and (BL + i C L ) r e s p e c t i v e l y . The A i n AL i s t h e r o t a t i o n a l A z x y' f J z so c o n s t a n t h e r e , and i s not t o be c o n f u s e d w i t h A used p r e v i o u s l y as a c o e f f i c i e n t o f t h e same o p e r a t o r . I f t h e m o l e c u l e u n d e r ^ c o n s i d e r a t i o n i s one t h a t may become l i n e a r t h r o u g h v i b r a t i o n , t h e n t h e e x p r e s s i o n ; g i v e n i n 3.6a r e p r e s e n t s t h e R & i n e r - T e l l e r e f f e c t }J and w i l l p r o b a b l y be t h e dominant c o n t r i b u t i o n t o t h e m a t r i x element c o n n e c t i n g t h e v i b r o n i c s t a t e s ? - U s u a l l y , t h e s p i n - o r b i t c o n t r i b u t i o n t o t h e m a t r i x element w i l l be t h e more i m p o r t a n t c o n t r i b u t i o n . Franck-Condon o v e r l a p i n t e g r a l s a r e i m p l i c i t i n t h e p a r a m e t e r s o f e q u a t i o n s 3.6 and T a b l e s 3.1 and 3.2, but have been s u p p r e s s e d t o s i m p l i f y t h e n o t a t i o n . I t t u r n s o u t t h a t f a v o r a b l e c o n d i t i o n s f o r t h e i d e n t i f i c a t i o n a d i r e c t s p i n - o r b i t i n t e r a c t i o n a r i s e o n l y r a r e l y . The m o l e c u l e must be c o m p a r a t i v e l y l i g h t , so t h a t t h e r o t a t i o n a l s t r u c t u r e o f t h e band i s not t o o crowded; a t t h e same ti m e i t must be heavy enough t h a t t h e s p i n - o r b i t c o u p l i n g c o n s t a n t i s a p p r e c i a b l e . I n a d d i t i o n , t h e v i b r a t i o n a l l e v e l d e n s i t i e s i n t h e i n t e r a c t i n g s t a t e s ought not t o be u n d u l y h i g h , f o r o t h e r w i s e p e r t u r b a t i o n s due t o o t h e r mechanisms such as n u c l e a r momentum c o u p l i n g and v i b r a t i o n a l C o r i o l i s e f f e c t s w i l l c o n f u s e t h e s p e c t r u m beyond a l l r e c o g n i t i o n . ' 90 T a b l e 3.3 S e l e c t i o n r u l e s i n C f o r t h e d i r e c t s p i n - o r b i t i n t e r a c t i o n . r(n) x r(n') AK AN ±1 0, ±1 B i 0 0 ( e x c e p t K = 0 ) , ±1 B 2 ±1 0, ±1 To o b t a i n t h e s e r e s u l t s , t h e a x i s c h o i c e C 2 = z = b , . x = c has been made (see Appendix I I I ) . 91 E v i d e n c e f o r a d i r e c t s p i n - o r b i t i n t e r a c t i o n has been found i n t h e t r i p l e t m a n i f o l d o f SO2, w h i c h w i l l be d i s c u s s e d i n C h a p t e r V I I . Here, s m a l l p e r t u r b a t i o n s i n some o f t h e low v i b r a t i o n a l l e v e l s o f t h e a 3 B i s t a t e appear t o be s p i n - o r b i t p e r t u r b a t i o n s caused by t h e unseen b A 2 s t a t e , w i t h w h i c h i t i s d e g e n e r a t e t o w i t h i n a few hundred cm ( 1 0 ) . The p e r t u r b a t i o n s a r e q u i t e s m a l l , w h i c h i s c o n s i s t e n t w i t h t h e u n f a v o r a b l e Franck-Condon f a c t o r s . The A - r o t a t i o n a l c o n s t a n t s o f t h e two s t a t e s a r e v e r y d i f f e r e n t , so t h a t t h e AK = +1 and AK = -1 p e r t u r b a t i o n s p r e d i c t e d by T a b l e 3.3 w i l l f o l l o w i n q u i c k s u c c e s s i o n i n t h e s e c a s e s . The t h e o r e t i c a l b a s i s o f t h e work p r e s e n t e d i n t h i s t h e s i s i s c o m p l e t e d w i t h t h i s c h a p t e r , and t h e r e m a i n i n g c h a p t e r s w i l l p r e s e n t t h e t r a n s i t i o n s a n a l y s e d i n terms o f t h i s t h e o r y , t o g e t h e r w i t h t h e r e s u l t s o b t a i n e d . CHAPTER IV. THE GROUND STATE OF CARBON DISULPHIDE. 93 (A) I n t r o d u c t i o n . The near u l t r a v i o l e t s p e c t r u m o f CS2 has been s t u d i e d a t v a r i o u s r e s o l u t i o n s by many w o r k e r s (1 - 7 ) , who found a p e r p l e x i n g c o l l e c t i o n o f d a t a w h i c h seemed a t f i r s t t o be i n c o n s i s t e n t . The f i r s t e x t e n s i v e v i b r a t i o n - r o t a t i o n a n a l y s i s was c a r r i e d out by Kleman ( 1 ) , who o b s e r v e d s i m p l e r o t a t i o n a l s t r u c t u r e i n a t r a n s i t i o n w h i c h he c a l l e d t h e R-system. The a s s i g n m e n t o f t h e r o t a t i o n a l s t r u c t u r e was c o n s i s t e n t w i t h a t r a n s i t i o n o r i g i n a t i n g i n t h e l o w e s t few v i b r a t i o n a l l e v e l s o f t h e * ^ g + l i n e a r ground s t a t e and f o r m i n g a p r o g r e s s i o n i n t h e b e n d i n g f r e q u e n c y o f a n o n - l i n e a r B2 e l e c t r o n i c upper s t a t e , as M u l l i k e n had o r i g i n a l l y p r e d i c t e d ( 8 , 9 ) . S u b s e q u e n t l y , a pronounced Zeeman e f f e c t was o b s e r v e d f o r t h i s s y s t e m , w h i c h caused Douglas and M i l t o n (2) t o r e a s s i g n t h i s s p e c t r u m as a t r a n s i t i o n t o t h e B 2 s p i n component o f a 3 A 2 e l e c t r o n i c s t a t e . Temperature s t u d i e s o f Oungen, Malm, and Merer (4,5) c l a r i f i e d t h e s h o r t w a v e l e n g t h a b s o r p t i o n i n t h e r e g i o n 2900-3400 A. These w o r k e r s f o u n d t h a t t h e i n t e n s e ' c o l d ' band s t r u c t u r e ( K l e m a n i s V system) i n t h i s r e g i o n was a I T - I T * t r a n s i t i o n from t h e ground s t a t e t o t h e *B 2 component •,of- a A u tipper s t a t e . Some weak 'hot' bands near 3300 A, f o r w h i c h no c o r r e s p o n d i n g c o l d bands c o u l d be f o u n d , were t h e n i n t e r p r e t e d as t r a n s i t i o n s t o t h e o t h e r component o f 1 A U , *A 2, a s s o c i a t e d w i t h t h e V system. Even though t h e *A 2 <- *£g + t r a n s i t i o n i s f o r b i d d e n f o r e l e c t r i c d i p o l e r a d i a t i o n , t h e r e i s R e n n e r - T e l l e r m i x i n g between t h e *A 2 and *B 2 components o f a x A u s t a t e , w h i c h g i v e s r i s e t o t h e p a r a l l e l - p o l a r i z e d 'hot' bands o b s e r v e d . 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 f o r t h e s e 'hot' bands were needed i n o r d e r t o r o t a t i o n a l l y a n a l y s e t h e V s y s t e m . These were o b t a i n e d by r e a n a l y s i n g Kleman's R system. I t was found t h a t t h e 94 measured c o m b i n a t i o n d i f f e r e n c e s o b t a i n e d showed o b v i o u s and s y s t e m a t i c d e v i a t i o n s from t h e v a l u e s one would c a l c u l a t e from t h e b e s t i n f r a r e d c o n s t a n t s . T h i s c h a p t e r r e p o r t s t h e work done i n an a t t e m p t t o remove t h e s e d i s c r e p a n c i e s from t h e ground s t a t e d a t a f o r CS2• The r e s o l v i n g power of our s p e c t r o g r a p h i s l i m i t e d by t h e D o p p l e r w i d t h s o f t h e l i n e s , so t h a t our r e s o l u t i o n approaches t h a t o f a c o n v e n t i o n a l i n f r a r e d g r a t i n g s p e c t r o m e t e r . I n t h e s p e c t r a o b t a i n e d , l i n e s b e l o n g i n g t o h i g h 3 v a l u e s were o b s e r v e d i n t r a n s i t i o n s i n v o l v i n g t h e h i g h e r o v e r t o n e s o f t h e ground s t a t e b e n d i n g f r e q u e n c y . S i n c e s u e h ^ t r a n s i t i o n s a r e e i t h e r weak or symmetry f o r b i d d e n i n an i n f r a r e d s p e c t r u m , t h e p r e s e n t a n a l y s i s complements t h e i n f r a r e d d a t a f o r CS2, and w i l l be of c o n s i d e r a b l e i n t e r e s t i n t h e d e t e r m i n a t i o n o f an a c c u r a t e f o r c e f i e l d f o r t h e ground s t a t e o f t h i s m o l e c u l e . 95 (B) Photography of the 3400 to 4100A region of the CS2 Spectrum. The absorption spectrum of CS2 was photographed at both room and dry ice temperatures between 3400 and 4100A, using a 7 meter focal length Ebert-mounted plane grating spectrograph. Pressure-paths of up to 40 m-atm. were obtained for the 1 2CS 2 isotopic species in a White-type multiple reflection c e l l 4 meters long. The smaller sample size available for the 1 3CS 2 isotopic species limited the pressure path to 3 m-atm for this molecule. An average f u l l width at half-maximum (FWHM) line width of about 0.045cm-"'" was observed on these plates, which corresponds to a resolving power of the order of 620,000. A l l line measurements were punched onto computer cards using a Grant automatic recordihgccomparator. Atomic iron reference lines from an iron-neon hollow cathode lamp were used to calibrate molecular line positions. The assignable iron reference wavelengths were used to define a fourth order polynomial in the plate positions by means of a least squares f i t , and from the function obtained, , the molecular line positions were interpolated and corrected to yield vacuum wavenumbers for the observed spectral lines. Relative positions of lines that are unblended w i l l be accurate to considerably better than 0.010 cm "*" using this technique. A sample of 1 3CS 2 was purchased from Merke, Sharp and Dohme of Canada, and was stated to be of 95% purity; commercially available CS 2 of spectroscopic quality was used without purification to obtain the spectrum of 1 2CS ?. 96 (C) S e l e c t i o n R u l e s f o r t h e R s y s t e m o f C S 2 . The R system o f CS 2 i s an e l e c t r o n i c t r a n s i t i o n from a l i n e a r ground s t a t e t o an upper s t a t e i n which t h e S-C-S bond a n g l e i s 137°. The upper e l e c t r o n i c s t a t e has been c h a r a c t e r i z e d as t h e 3 A 2 Renner-T e l l e r component o f what becomes a 3 A u s t a t e i n t h e l i n e a r l i m i t (2 - 9 ) . Only t h e I = 0 s p i n component a p p e a r s i n t h e a b s o r p t i o n t r a n s i t i o n f r o m th e s i n g l e t ground s t a t e . For a m o l e c u l e w i t h an e q u i l i b r i u m c o n f i g u r a t i o n t h a t t r a n s f o r m s as t h e C^ v p o i n t g r o u p , t h i s s p i n f u n c t i o n t r a n s f o r m s as R ^ B i ) , w i t h t h e r e s u l t t h a t t h e upper s t a t e l e v e l s t r a n s f o r m as t h e i r r e d u c i b l e . .. s p i n n o r b i t . s p i n - o r b i t , , . ,, r e p r e s e n t a t i o n ^ B i x A 2 = ^ B 2. As a r e s u l t , t h e o b s e r v e d t r a n s i t i o n f r o m a t o t a l l y symmetric ground s t a t e t o a B 2 upper s t a t e i s a l l o w e d by t h e z-component o f t h e e l e c t r i c d i p o l e moment 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 p r i n c i p a l i n e r t i a l a x i s s y s t e m (10) . (see Appendix I I ) . Thus, the r o t a t i o n a l s e l e c t i o n r u l e s a r e g i v e n by AO = 0, ±1 and AK = 0, e x c e p t f o r K = 0, f o r w h i c h A3 = ±1 o n l y . T h i s g i v e s r i s e t o p a r a l l e l - t y p e bands. These c o n s i d e r a t i o n s f o l l o w from t h e group m u l t i p l i c a t i o n t a b l e when one i d e n t i f i e s t h e p o i n t group axes ( x , y , z ) w i t h t h e m o l e c u l a r i n e r t i a l axes i n t h e o r d e r ( y , z , x ) as i s u s u a l f o r s y m m e t r i c a l t r i a t o m i c m o l e c u l e s . In t h e l i n e a r ground s t a t e , o n l y 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 a n g u l a r momenta c o n t r i b u t e t o K, so t h a t K" = (V ± A " ) . S i n c e t h e l o w e r s t a t e i s an e l e c t r o n i c E s t a t e , A = 0, w h i c h means t h a t K" i s g i v e n by I". From t h e s e l e c t i o n r u l e AK = 0, i t f o l l o w s t h a t K' = K" = Si" where K' i s t h e n e a r - p r o l a t e symmetric t o p quantum number f o r t h e upper e l e c t r o n i c s t a t e , and i s t h e ground s t a t e v i b r a t i o n a l a n g u l a r momentum quantum number. 9 7 A d i r e c t l i m i t a t i o n t h a t r e s u l t s from t h e s e l e c t i o n r u l e s above i s t h a t one i s u n a b l e t o d e t e r m i n e t h e p o s i t i o n s o f energy l e v e l s w i t h Z > 0 w i t h r e s p e c t t o t h e l e v e l s f o r w h i c h Z" = 0 from t h e e l e c t r o n i c s p e c t r u m , even though t h e r e l a t i v e p o s i t i o n s and s e p a r a t i o n s o f l e v e l s w i t h i n t h e same v a l u e o f Z" can be d e t e r m i n e d t o h i g h p r e c i s i o n . A c c o r d i n g t o t h e Franck-Condon p r i n c i p l e , t h e 'hot' bands i n t h e ground s t a t e b e n d i n g v i b r a t i o n w i l l be enhanced i n a l i n e a r t o bent t r a n s i t i o n , s i n c e e x c i t a t i o n o f t h e ground s t a t e b e n d i n g mode i n c r e a s e s t h e o v e r l a p i n t e g r a l between t h e ground s t a t e w a v e f u n c t i o n and t h e upper s t a t e w a v e f u n c t i o n . An a d d i t i o n a l enhancement e x i s t s i n CS2 because t h e 'hot' bands f o r t h e R system i n v o l v e t r a n s i t i o n s t o an upper s t a t e w h i c h i s an e l e c t r o n i c A s t a t e , f o r w h i c h Z = 2. I n such a c a s e , t h e t r a n s i t i o n moments become i n t e g r a l s o v e r t h e s q u a r e o f t h e supplement t o t h e bond a n g l e , and a r e n o t s i m p l e o v e r l a p i n t e g r a l s . As a r e s u l t , t h e e l e c t r o n i c t r a n s i t i o n moment i s found t o i n c r e a s e as t h e m o l e c u l e bends. The d a t a o f Barrow and D i x o n (6) show t h a t t h e e x p e r i m e n t a l i n t e n s i t y o f t h e E bands g i v e s a t r a n s i t i o n moment f o r t h e 040-02°0 band w h i c h i s l a r g e r t h a n t h a t o f t h e 040-00°0 band by a f a c t o r o f 100. Bands a r i s i n g f rom 060, w h i c h l i e s a t 2400 cm ^ above t h e upper s t a t e o r i g i n , a r e e a s i l y o b s e r v e d a t room t e m p e r a t u r e as a r e s u l t o f t h i s enhancement, even though t h e i r Boltzmann f a c t o r s p r e d i c t an e x t r e m e l y s m a l l p o p u l a t i o n o f t h e s e l e v e l s . 98 (D) A n a l y s i s o f t h e Bands. Some t y p i c a l s p e c t r a and t h e i r r o t a t i o n a l a s s i g n m e n t s a r e shown i n F i g u r e 4.1. As i l l u s t r a t e d , t h e Z, II, A, and $ subbands c o r r e s p o n d t o t h e quantum a s s i g n m e n t s K 1 = = 0, 1, 2, and 3, r e s p e c t i v e l y . The bands shown were chosen from t h e 050 - 04^0 and 050 ?X05£0 bands o f 1 2 C S 2 - H a l f t h e r o t a t i o n a l l i n e s a r e m i s s i n g from t h e s p e c t r u m , because t h e e q u i v a l e n t s u l p h u r atoms have z e r o n u c l e a r s p i n s ; o n l y even 3" l i n e s o c c u r f o r t h e Z bands as a r e s u l t . The pronounced s t a g g e r i n g o b s e r v e d i n t h e II bands i s due t o t h e l a r g e asymmetry s p l i t t i n g i n t h e upper s t a t e K' = 1 r o t a t i o n a l l e v e l s , i n wh i c h CS^ i s a bent asymmetric t o p . These bands c o n s i s t o f an undegraded odd 3" component, and a s t r o n g l y v i o l e t degraded 3" even component. One a l s o f i n d s asymmetry e f f e c t s as a' double;head i n t h e P b r a n c h o f t h e I = 2 (A) bands, but t h e $ bands a r e u n a f f e c t e d by asymmetry. Abband head i s seen between 3" = 30 and 3" = 50 i n a l l t h e A and $ bands; u n f o r t u n a t e l y , t h i s a l s o t e n d s t o l i m i t t h e A2F" c o m b i n a t i o n d i f f e r e n c e s t o wh i c h one can f i t t h e ground s t a t e c o n s t a n t s . N e v e r t h e l e s s , one can i d e n t i f y a few o f t h e r e t u r n i n g high-3 components o f t h e P bra n c h e s i n f a v o r a b l e bands, p a r t i c u l a r l y i n t h e A bands and i n t h e even 3" components o f t h e n bands. F i g u r e 4.1 a l s o i l l u s t r a t e s t h e p r e s e n c e o f numerous upper s t a t e r o t a t i o n a l p e r t u r b a t i o n s w h i c h appear t h r o u g h o u t t h e s p e c t r u m ; a l l f o u r subbands c o n t a i n a t l e a s t one p e r t u r b a t i o n . When t h e s e p e r t u r b a t i o n s a r e s m a l l , t h e y a i d t h e r o t a t i o n a l a n a l y s i s by i d e n t i f y i n g l i n e s i n t h e P-and R- bra n c h e s w h i c h b e l o n g t o t h e same upper s t a t e 3' assignment.' The main d i f f i c u l t y i n t h e a n a l y s i s o f t h e s e bands a r o s e from t h e o v e r l a p p i n g o f subbands, w h i c h made l i n e . b l e n d i n g , a s e r i o u s problem t o co n t e n d w i t h . The l o n g e s t w a v e l e n g t h bands o f t h e system y i e l d e d t h e b e s t d a t a , s i n c e t h e 261 0 6 . 0 cm-i 2 6 0 7 6 . 3 cm- 1 F i g u r e 4.1 T y p i c a l s p e c t r a o f C S 2 w i t h t h e i r 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 f o r K' = I" = 0, 1, 2, and 3 bands; t a k e n f r o m v 2 ' = 5 and v 2 " = 4 and 5. 100 p e r t u r b a t i o n s d e s c r i b e d d e c r e a s e d markedly i n magnitude f o r t h e l o w e r v' v a l u e s . U n f o r t u n a t e l y , t h e s e bands were a l s o t h e weakest, and t h i s made l o n g p a t h a b s o r p t i o n t e c h n i q u e s mandatory. From t h e photographed s p e c t r a , bands were a n a l y s e d up t o V 2 " = 6 and £" = 3 f o r 1 2 C S 2 , and v 2 " = 4, £" = 2 f o r 1 3 C S 2 - Subbands f o r h i g h e r i i v a l u e s were e a s i l y found on t h e p l a t e s t a k e n , b u t were not a n a l y z e d , s i n c e f o r t h e most p a r t t h e y were weak and h e a v i l y o v e r l a p p e d . One o f t h e main problems e n c o u n t e r e d was t h a t o f a b s o l u t e c a l i b r a t i o n . The p o s s i b i l i t y o f an a b s o l u t e c a l i b r a t i o n s h i f t was made more l i k e l y by t h e f a c t t h a t t h e h i g h d i s p e r s i o n made i t n e c e s s a r y t o measure bands s e v e r a l hundred cm x a p a r t from d i f f e r e n t p l a t e s . To m i n i m i z e t h e s e c a l i b r a t i o n e r r o r s , r e p l i c a t e d e t e r m i n a t i o n s o f t h e same ground s t a t e i n t e r v a l were used wherever p o s s i b l e . I n some c a s e s , up t o 10 d i f f e r e n t and i n d e p e n d e n t measurements o f a c o m b i n a t i o n d i f f e r e n c e were o b t a i n e d ; a l l d e t e r m i n a t i o n s c l u s t e r e d about a c e n t r a l v a l u e and f e l l w i t h i n 0.030 cm x o f t h e mean v a l u e . To t h e e x t e n t t h a t a G a u s s i a n d i s t r i b u t i o n i s d e f i n e d f o r 10 p o i n t s , t h e d i s t r i b u t i o n o f t h e s e v a l u e s can be s a i d t o be G a u s s i a n . I n t h e f i n a l a n a l y s i s , a l l a v a i l a b l e v a l u e s were a v e r a g e d , and t h e number o f d a t a p o i n t s c o n t r i b u t i n g t o t h i s a verage was made as e x t e n s i v e as p o s s i b l e t h r o u g h e x h a u s t i v e a n a l y s i s o f t h e bands o b s e r v e d O v e r 170 subbands were a n a l y s e d i n a l l : f o r 1 2 C S 2 47 £ bands, 33 I I ' s and 22 A ' s were a s s i g n e d ; f o r 1 3 C S P , 36 E ' s , 22 TPs and 8 A ' s were o b t a i n e d . 101 (E) Data R e d u c t i o n . The method chosen t o reduce t h e d a t a was as f o l l o w s . F i r s t : , a l l " p o s s i b l e ground s t a t e c o m b i n a t i o n d i f f e r e n c e s t h a t c o u l d be formed u s i n g o n l y unblended l i n e p o s i t i o n s were c a l c u l a t e d . These c o m b i n a t i o n d i f f e r e n c e s were o f two t y p e s ; t h e r e were t h e normal A2F"(3) v a l u e s f o r each (v,£) l e v e l c a l c u l a t e d from t h e l i n e p o s i t i o n s a c c o r d i n g ; : tbng A 2 F " ( 0 ) = R (3 '• + 1) - P ( 3 ' -.1) = (4B" - 6 D " ) ( 3 ^ + i ) - 8D"(3' + | ) 2 (4.01) and i n a d d i t i o n , t h e r e were separaitionsmbfneorresp6ndingn'3eievels'»t d n t s u c c e s s i v e r v i b r a t i o n a l l e v e l s w i t h o d i i f f e ' r e n t n k y a l u e s v o f s v J b u t u t h e ; s a m e a v a l u e l o f e f e i i T h e s e h l a t t e r ' e o m b i n a t d ' o n o d i f f k r e n c e s areesimpOiy c"" s;epaEationsdbeit;weenathenR((3i) and"B(03)aMnesroihvtwopbandstwithea'.»common u p p e r ( s t a t e i j e s o i i b h a t o L-^nds. w.th a common uppe: ^ t a t e , so t h a t -A E y , v ( 3 ) = R v , ( 3 ) - R v ( 3 ) = P y , ( 3 ) - P y ( 3 ) (4.02) where v and v' a r e l o w e r s t a t e v i b r a t i o n a l quantum numbers. As a r e s u l t o f t h e p a r a l l e l s e l e c t i o n r u l e s , t h e v a r i o u s c o m b i n a t i o n d i f f e r e n c e s f o r each v a l u e o f £ form i n d e p e n d e n t d a t a s e t s . The n e x t s t a g e i n v o l v e d a l e a s t s q u a r e s f i t o f t h e s e c o m b i n a t i o n d i f f e r e n c e s t o e x p r e s s i o n s d e r i v e d from t h e u s u a l energy l e v e l f o r m u l a + \ (-1)° q 3(3 + U ) 6 £ (4.03) Because each band c o u l d be measured from a s i n g l e p l a t e , i t was assumed t h a t t h e r e were n o n s i g n i f i c a n t c a l i b r a t i o n e r r o r s a l o n g t h e 30 cm o r so wcbut-ainrngfet^ A b s o l u t e c a l i b r a t i o n e r r o r s o f t h e k i n d a l r e a d y mentioned were a l l o w e d f o r by i n t r o d u c i n g one parameter AG f o r each p a i r o f bands used i n t h e 102 d e t e r m i n a t i o n o f a v i b r a t i o n a l l e v e l i n t e r v a l . T h i s s o r t o f a n a l y s i s p e r m i t s t h e use o f a s i n g l e AB, AD, and Aq t o r e l a t e t h e r o t a t i o n a l c o n s t a n t s o f a g i v e n band t o t h o s e o f t h e v = Z l e v e l . F o r example, t h e c o m p l e t e d a t a s e t f o r t h e Z = 0 ( I ) bands o f CS,, r e q u i r e d 29 par a m e t e r s i n t h e f i n a l f i t : t h e s e were B(00°0) and D(00°0), ( which were a l l o w e d t o v a r y f r e e l y o r f i x e d t o some s u i t a b l e v a l u e s ), t h e t h r e e AB's and t h r e e AD's c o r r e s p o n d i n g t o t h e v i b r a t i o n a l i n t e r v a l s 2-0, 4-2, and 6-4, and n i n e v a l u e s o f AG _ = G(02°0) -, U G(00°0), n i n e v a l u e s o f AG. ~ and t h r e e o f AG. .. F i n a l l y , t h e v a r i o u s AG's 4 , 2 6 ,4 were averaged t o o b t a i n t h e ' t r u e ' v i b r a t i o n a l i n t e r v a l s . T a b l e 4.1 summarizes t h e d a t a o b t a i n e d from a l l t h e CS^ a n a l y s e s . Appendix I I I . > c b n t a i n s h t h e s a s s i g r i e e L r o t a t i o n a l l i n e s , I c l a s s i f i e d : by >Z v a l u e l e s s o b t a i n e d t'~ hsi! ; - ab l e ,'. n l a a s I f i r - 4 K As wasi'aindricc'atferd"' a>bwe ;, t h ^ s r b i t . a i i t i i b l w a l 'corrs>t'ate'si-" f o r t h e l o w e s t l e v e l o f each Z v a l u e c o u l d be f i x e d , o r a l l o w e d t o f l o a t i n t h e l e a s t s q u a r e s c a l c u l a t i o n s . Some o f t h e r e s u l t s t h a t were o b t a i n e d a l l o w i n g t h e s e c o n s t a n t s t o v a r y fireely a r e compared w i t h t h e most r e c e n t v a l u e s o b t a i n e d i n i n f r a r e d work i n T a b l e 4.2. I t i s s a t i s f y i n g t o n o t e t h a t t h e v a l u e o f B(00°0) f o r 1 2 C S ^ o b t a i n e d from t h e e l e c t r o n i c d a t a a l o n e a g r e e s w i t h t h a t g i v e n by Maki and Sams (11) t o w i t h i n one s t a n d a r d d e v i a t i o n , w h i c h was 1.0 x 10 6 cm The s t a n d a r d d e v i a t i o n s found from t h e e l e c t r o n i c d a t a a r e l a r g e r t h a n t h o s e o f Maki and Sams, and t h e c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s a r e l e s s w e l l d e t e r m i n e d , jpcesymab/Iyy because t h e e l e c t r o n i c d a t a do n o t e x t e n d t o as h i g h ; 3 , v a l u e s as t h e i n f r a r e d d a t a . F o r t h e 01*0 l e v e l , & -r^ai. f l i f f erfeh.Gerfimatthe, B j v a l G i e s was. f oundf,oisiinpe; jtto.ee o y e r all- s t a n d a r d d e v i a t i o n o f t h e f i t was i n c r e a s e d s l i g h t l y when 6 ( 0 1 ^ ) was f i x e d t o t h e i n f r a r e d v a l u e . 103 T a b l e 4.1 Summary o f d a t a f o r t h e a n a l y z e d bands o f CS Band t y p e .3 . max No. o f band i n t e r v a l s a No. o f AE's No. o f A 2F's T o t a l d a t a p o i n t s 1 2 c s 2 E 76 21 928 601 1529 n 76 14 951 531 ' 1482 A 69 10 522 322 844 $ 63 3 238 157 395 1 3 c s 2 z 70 14 523 477 1000 n 81 7 462 230 892 A 59 2 140 111 251 i . e . t o t a l number o f AG p a r a m e t e r s u s e d . 104 T a b l e 4.2 Comparison o f some Ground S t a t e R o t a t i o n a l C o n s t a n t s f o r L e v e l E l e c t r o n i c S p e c t r u m 3 'Best' IR Spectrum* 3 B V 00°0 0.1091176±72 0.1091165±33 01*0 0.1093374±68 0.1093294±50 10 8D V 00°0 1.100 ±130 1.134 ±23 01 '0 1.136 ±108 . 1.180 ±38 1 0 5 q O'^O 7.56 ±77, 7.81 ±49 a T h i s work; no c o n s t r a i n t s i n l e a s t s q u a r e s ^ t a k e n from r e f e r e n c e (11) e r r o r l i m i t s q u oted a r e 3a; a l l e n t r i e s a r e g i v e n i n cm-"*". 105 A l l t h e same, t h e i n f r a r e d d a t a must be o f h i g h e r a c c u r a c y t h a n t h e e l e c t r o n i c c o n s t a n t s whenever t h e y a r e a v a i l a b l e , s i m p l y because t h e i n f r a r e d d a t a have been d e t e r m i n e d from a more e x t e n s i v e d a t a base. F i n a l c a l c u l a t i o n s were made w i t h t h e c o n s t a n t s f o r t h e l o w e r l e v e l o f each £ v a l u e f i x e d a t t h e i n f r a r e d v a l u e where p o s s i b l e . The i m p o r t a n t p o i n t i s t h a t an a t t e m p t t o d e t e r m i n e t h e a b s o l u t e v a l u e s o f t h e r o t a t i o n a l c o n s t a n t s was not made, but r a t h e r t h e d i f f e r e n c e s between t h e s e c o n s t a n t s and t h o s e o f t h e h i g h e r v i b r a t i o n a l l e v e l s w i t h i n each v a l u e , w h i c h can be e x t r a c t e d from t h e e l e c t r o n i c s p e c t r u m w i t h f a r h i g h e r a c c u r a c y . The v a l i d i t y o f t h i s approach has been d i s c u s s e d by P l i v a ( 1 2 ) , who showed t h a t by c a l c u l a t i n g a l l r e l a t e d band c o n s t a n t s as i n c r e m e n t s t o t h o s e o f t h e most p r e c i s e l y d e t e r m i n e d l e v e l , one not o n l y e n s u r e s t h a t t h e f i n a l p a r a m e t e r s w i l l be s t a t i s t i c a l l y u n c o r r e l a t e d , but a l s o t h a t subsequent improvements i n t h e d e t e r m i n a t i o n o f t h e base c o n s t a n t s w i l l i n no way r e n d e r t h e i n c r e m e n t s d e t e r m i n e d o b s o l e t e . T h i s i s so because t h e i n c r e m e n t s depend on t h e s e p a r a t i o n o f t h e s p e c t r a l l i n e p o s i t i o n s , and n o t on t h e a b s o l u t e v a l u e s o f t h e l i n e f r e q u e n c i e s . The agreement seen i n T a b l e 4.2 t h e r e f o r e v a l i d a t e s t h e r e s u l t s o b t a i n e d from t h e e l e c t r o n i c s p e c t r u m , and a l s o shows t h a t t h e raw p a r a m e t e r s a r e i n s u f f i c i e n t l y good agreement t h a t f u t u r e r e f i n e m e n t o f t h e "base" c o n s t a n t s w i l l n o t i n v a l i d a t e t h e d i f f e r e n c e s t h a t have been d e t e r m i n e d i n t h e e l e c t r o n i c work. I n a d d i t i o n , i t a l s o v e r i f i e s t h e s t a t i s t i c a l e q u i v a l e n c e o f t h e e l e c t r o n i c and i n f r a r e d d a t a s e t s assumed i n t h i s work. The p r i n c i p a l r e s u l t s o f t h i s work a r e summarised i n t h e form o f T a b l e 4.3, w h i c h g i v e s t h e s e d i f f e r e n c e s f o r b o t h 1 2CS,, and 1 3 C S 2 - As one might e x p e c t , t h e p r e c i s i o n o f t h e AB's c a l c u l a t e d i s f a r h i g h e r t h a n t h e a b s o l u t e p r e c i s i o n o f t h e B's t h e m s e l v e s , as g i v e n i n r e f e r e n c e ( 1 1 ) . T a b l e 4.3 Ground s t a t e r o t a t i o n a l c o n s t a n t s o f C S 2 and C S 2 r e f e r r e d t o t h o s e o f t h e l o w e s t l e v e l o f each % v a l u e (cm x ) . I n t e r v a l AG lO^AB 108 10 5Aq 1 2 c s 2 z 02° 0 - 00° 0 801.863 ± 6 3.714 ± 16 0.003 ± 34 -04° 0 - 00°0 ' 1619.777 7 6.773 20 -0.125 43 -06°0 - 00° 0 2450.053 7 9.502 33 y0.302 82 -n 03*0 - Ol 1*) 810.883 6 3.261 12 -0.035 25 7.01 ± 5 05*0 - 01*0 1635.535 7 6.109 21 -0.057 44 13.69 8 A 0420 - 0220 818.993 6 2.940 21 -0.001 60 -0620 - 0220 1650.045 7 5.484 64 -0.438 290 -$ 0530 - 0330 826.400 6 2.677 26 -0.089 74 -1 3 c s 2 E 02°0 - 00°0 776.542 6 3.374 16 -0.048 47 -04°0 - 00°0 1569.221 7 6.076 49 -0.471 202 -n 03*0 - 01*0 785.864 6 2.907 13 -0.081 27 6.95 5 A 0420 - 0220 794.060 10 2.589 43 -0.037 128 -E r r o r l i m i t s q uoted a r e 2a, i n t h e u n i t s o f t h e l a s t s i g n i f i c a n t f i g u r e . The AB£ 2 terms have been t a k e n i n t o a c c o u n t so t h a t t h e Ta b l e g i v e s t h e " t r u e " AG's/* T a b l e 4.4 D e r i v e d c o n s t a n t s f o r t h e ground s t a t e o f CS,, (cm" ), L e v e l 10 8D 1 0 \ 1 2 C S 2 : 00°0 0.000 01*0 396.090 02°0 801.863 0 2 2 0 792.7 03*0 1206.973 0 3 3 0 $ . 04°0 1619.777 0 4 2 0 1611.7 05*0 2031.625 05 3 0 $ + 826.400 06°0 2450.053 0 6 2 0 2442.7 1 3 c s 2 e 00°0 0.000 01*0 n 02°0 776.542 0 2 2 0 A 03*0 n + 785.861 04°0 1569.221 0 4 2 0 A + 794.060 0.006 0 . 1 C 0.007 0.1 0.007 0.006 0.007 0.1091165 ± 3 3 a 1.134 ± 2 3 a 0.1093294 5 0 a 1.180 3 8 a 7.81 + 4 9 a 0.1094879 49 1.137 57 0.1095457 1 5 7 a 1.310 3 7 0 a 0.1096554 62 1 ^ 5 . 63 14.82 + 54 0.1097599 109 1.17 d 0.1097938 53 1.009 65 0.1098397 178 1.310 430 0.1099403 71 1.123 82 21.50 + 57 0.1100276 83 1.08 0.1100667 66 0.832 105 0.1100941 221 0.872 660 0.1091316 0.1093368 0.1094690 0.1095454 0.1096275 0.1097392 0.1098403 32 46 48 122 59 81 165 (1.134)1 ( 1 U 8 0 ) T ( 1 . 0 8 6 ) ' (1.31 ) T (1.099) (0.661) (1.27 ) 7.81 ± 49 14.76 ± 54 d T a k e n from r e f e r e n c e ( 1 1 ) . ^Taken from r e f e r e n c e ( 1 3 ) ; t h i s i s the " t r u e " v 0 , not v 0 - B. Q E l e c t r o n i c s p e c t r u m - see t e x t ; t h i s i s t h e " t r u e " V o , n o t v 0 - 4B. ^Assumed. e A l l B v a l u e s g i v e n f o r C S ? a r e c a l c u l a t e d e n t i r e l y from e l e c t r o n i c d a t a . f V 12 Assumed by an a l o g y w i t h CS 2- U n c e r t a i n t i e s quoted a r e 2a i n u n i t s o f t h e l a s t s i g n i f i c a n t d i g i t . 108 (F) Summary and D i s c u s s i o n . The i n c r e m e n t s o f T a b l e 4.3 can be combined w i t h t h e a v a i l a b l e i n f r a r e d c o n s t a n t s t o produce d e r i v e d 1 c o n s t a n t s , which a r e summarized i n T a b l e 4.4. A b s o l u t e e n e r g i e s a r e not g i v e n f o r l e v e l s w i t h Jc > 0, w i t h t h e e x c e p t i o n o f t h e I = 1 l e v e l s o f 1 2 C S 2 , where t h e o r i g i n o f t h e 01 10-00°0 band has been d e t e r m i n e d by S m i t h and 0 v e r e n d ( 1 3 ) . From t h e e l e c t r o n i c s p e c t r u m , an e s t i m a t e o f t h e energy o f ( 0 2 2 0 ) can be o b t a i n e d , i f one assumes t h a t D^ f o r t h e l o w e s t o b s e r v e d v i b r a t i o n a l l e v e l o f upper s t a t e 010 can be n e g l e c t e d . The v a l u e o b t a i n e d i n t h i s way f o r G ( 0 2 2 0 ) agreed w e l l w i t h t h e v a l u e g i v e n by Smith and Overend (13) based on t h e i r a s s i gnment o f t h e u n r e s o l v e d Q - br a n c h o f t h e 02 20-010 band. The e l e c t r o n i c d a t a o b t a i n e d f o r t h e l e v e l (04°0) c o m p l e t e s t h e l o w e s t F e r m i t r i a d o f i n t e r a c t i n g l e v e l s , s i n c e d a t a f o r t h e l e v e l s 20°0 and 12°0 a r e a v a i l a b l e from r e f e r e n c e ( 1 1 ) , and t h e r e f o r e s h o u l d p r o v i d e much u s e f u l i n f o r m a t i o n c o n c e r n i n g t h e f o r c e f i e l d o f C S 2 . S i n c e t h e p r i m a r y p u r p o s e o f t h i s i n v e s t i g a t i o n d i d n o t i n c l u d e a r e f i n e m e n t o f t h e f o r c e f i e l d , s u c h f u r t h e r c o r r e c t i o n s were n o t u n d e r t a k e n . A c o m p a r i s o n o f t h e d a t a o b t a i n e d from t h i s work f o r t h e I l e v e l s w i t h what one would c a l c u l a t e from t h e most r e c e n t l y p u b l i s h e d s p e c t r o s c o p i c c o n s t a n t s o f r e f e r e n c e s (13,14,15,16) was made, and t h e r e s u l t s a r e p r e s e n t e d i n T a b l e 4.5. C l e a r l y , s m a l l c o r r e c t i o n s w i l l be n e c e s s a r y t o b r i n g t h e i n f r a r e d d a t a i n t o agreement w i t h t h e e l e c t r o n i c c o n s t a n t s . A c o m p a r i s o n o f r e c e n t l y d e t e r m i n e d v a l u e s f o r r o t a t i o n a l c o n s t a n t s o f t h e ground s t a t e o f 1 2 C S 2 has been g i v e n by Maki ( 1 7 ) , t o which t h e i n t e r e s t e d r e a d e r i s r e f e r r e d . 1 3 C S 2 d a t a a r e s c a r c e r ; t h e v a l u e o f T a b l e 4.5 Comparison o f o b s e r v e d and c a l c u l a t e d e n e r g i e s f o r some & = 0 o v e r t o n e s o f t h e be n d i n g v i b r a t i o n o f 1 2 C S ~ (cm-''"). C a l c u l a t e d from t h e f o r c e f i e l d s o f L e v e l Observed • r e f 1 4 r e f 15 r e f 13 r e f 16 62° o 801.863 801.865 801.872 801.680 801.784 04°0 16,19.777 1619.850 1619.858 1619.743 1619.954 12°0 1447.083 a 1447.082 1447.469 1446.716 1446.745 20°0 1313.696 a 1313.711 1314.479 1311.671 1312.316 06°0 2450.053 2450.424 2450.200 2451.559 2452.102 Data from r e f e r e n c e ( 1 1 ) . 110 f o r B(00°0) = 0.109132 cm x o b t a i n e d i n t h i s work a g r e e s w e l l w i t h B l a n q u e t and C o u r t o y ' s v a l u e (18) o f 0.109128 cm" 1 and M a k i ' s and Sam's (11) v a l u e o f 0.109135 cm N e i t h e r F e r m i r e s o n a n c e nor ^ - u n c o u p l i n g were t a k e n i n t o a c c o u n t i n t h e c o n s t a n t s p r e s e n t e d i n T a b l e 4.3. W i t h i n t h e I l e v e l s , a l a r g e and n e g a t i v e d e c r e a s e i n t h e c e n t r i f u g a l d i s t o r t i o n parameter D w i t h i n c r e a s i n g v i s o b s e r v e d , which i s p r o b a b l y due t o J t - u n c o u p l i n g . CHAPTER V THE 2491 A BAND OF N 0 2 112 (A) I n t r o d u c t i o n . The v i s i b l e and u l t r a v i o l e t a b s o r p t i o n s p e c t r a o f NO^ have become a c h a l l e n g e t o many s p e c t r o s c o p i s t s , many o f whom have expended a g r e a t d e a l o f e f f o r t i n t h e i r a t t e m p t s t o u n d e r s t a n d t h e s t r u c t u r e o f t h e s e e l e c t r o n i c t r a n s i t i o n s . From th e l i m i t e d r o t a t i o n a l a n a l y s e s t h a t have been a c h i e v e d , i t i s p r e s e n t l y b e l i e v e d t h a t t h e i l l - b e h a v e d n a t u r e o f t h e s e s p e c t r a a r i s e •fromtP v i b r o n i c i n t e r a c t i o n s between th e e l e c t r o n i c a l l y e x c i t e d s t a t e l e v e l s and t h e h i g h e r 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 . I n c o n t r a s t t o t h e v i s i b l e a b s o r p t i o n o f NO,,, the 2491A e l e c t r o n i c t r a n s i t i o n has a c o m p a r a t i v e l y s i m p l e v i b r a t i o n a l s t r u c t u r e , and seems t o be r o t a t i o n a l l y u n p e r t u r b e d . The v i b r a t i o n a l s t r u c t u r e o f t h i s system i s o f i n t e r e s t f o r two r e a s o n s . F i r s t , t h e bands a r e p r e d i s s o c i a t e d , and t h e d egree o f d i f f u s e n e s s i n c r e a s e s r a p i d l y w i t h d e c r e a s i n g w a v e l e n g t h , as can be seen i n t h e medium r e s o l u t i o n p h o t o g r a p h t a k e n by Huber ( u n p u b l i s h e d ) and r e p r o d u c e d i n H e r z b e r g ( 1 ) . The (0,0) band a t 2491A appears t o be c o m p l e t e l y s h a r p , w h i l e no r o t a t i o n a l s t r u c t u r e can be seen t o t h e s h o r t w a v e l e n g t h s i d e o f 2450A. S e c o n d j v t h e a n t i s y m m e t r i c s t r e t c h i n g v i b r a t i o n i s t h e l o w e s t o f t h e t h r e e upper s t a t e f u n d a m e n t a l f r e q u e n c i e s . The V 3 1 f u n d a m e n t a l i t s e l f c annot be o b s e r v e d f o r r e a s o n s o f symmetry, but-Coon e t . a l . (2) a s s i g n e d t h e band a t 2447 A as 002 - 000, which g i v e s a v a l u e o f 713 cm" 1 f o r 2 v 3 1 . The u n e x pected s t r e n g t h o f t h i s band was i n t e r p r e t e d by Coon e t . a l . (2) as e v i d e n c e f o r a d o u b l e minimum i n the a n t i s y m m e t r i c c o o r d i n a t e Q^', where t h e p o t e n t i a l e nergy b a r r i e r a t t h e s y m m e t r i c a l c o n f i g u r a t i o n i s about 722 cm 1 . T h i s c h a p t e r g i v e s 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 t h e (0,0) band o f t h e 2 2 B 2 - X 2 A j e l e c t r o n i c t r a n s i t i o n o f W<d^. A p a r t i a l r o t a t i o n a l a n a l y s i s had been c a r r i e d out by R i t c h i e e t . a l . ( 3 ) , f o l l o w i n g an e a r l i e r 113 u n s u c c e s s f u l a t t e m p t by H a r r i s and K i n g ( 4 ) . R i t c h i e e t . a l . (3) found t h a t t h e t r a n s i t i o n was p a r a l l e l - p o l a r i z e d , and t h a t i n t h e 2E>2 s t a t e , t h e 0 - N - 0 bond a n g l e i s 13° s m a l l e r and t h e N - 0 bonds 0.12A l o n g e r t h a n i n t h e ground s t a t e . T h e i r a n a l y s i s had o n l y d e t e r m i n e d v a l u e s f o r t h e r o t a t i o n a l c o n s t a n t s A - B, B, and o f t h e upper s t a t e ; t h i s work p r e s e n t s a c o m p l e t e d e t e r m i n a t i o n o f t h e m o l e c u l a r c o n s t a n t s as f a r as t h e q u a r t i c c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s . From t h e s p e c t r o s c o p i c c o n s t a n t s o b t a i n e d , t h e energy o f t h e v 3 ' f u n d a m e n t a l i n t h e e x c i t e d s t a t e can be e s t i m a t e d , and l e a d s t o t h e c o n c l u s i o n t h a t t h e p o t e n t i a l energy b a r r i e r i n t h e Q^' c o o r d i n a t e i s v e r y much s m a l l e r t h a n t h a t p r o p o s e d by Coon e t . a l . ( 2 ) , and i n f a c t may not even e x i s t . A d o u b l i n g o f t h e r o t a t i o n a l l i n e s due t o t h e s p i n o f t h e u n p a i r e d e l e c t r o n can be seen i n t h e s p e c t r a o f a l l subbands e x c e p t K 1 = 0 , though a i t i s o n l y r e s o l v e d f o r t h e low N v a l u e s up t o K g' = 3 . From t h e known v a r i a t i o n o f t h e ground s t a t e s p i n c o n s t a n t s w i t h V2" ( 5 ) , t h e energy o r d e r o f t h e upper s t a t e s p i n components may be deduced by comparing t h e s p l i t t i n g s o f c o r r e s p o n d i n g l i n e s i n t h e 2491A (000 - 000) band w i t h t h o s e i n t h e 2537A (000 - 010) band. The s p i n s p l i t t i n g i n t h e upper s t a t e was found t o be i n a sense o p p o s i t e t o t h a t o f t h e ground s t a t e , so t h a t t h e s p e c t r a l s p l i t t i n g s o b s e r v e d r e p r e s e n t t h e sums o f t h e s p i n s p l i t t i n g s i n t h e ground and e x c i t e d e l e c t r o n i c s t a t e s . F i n a l l y , t h e s p e c t r a show t h a t t h e l i n e s o f t h e 2491A system a r e not c o m p l e t e l y s h a r p , b u t i n s t e a d d i s p l a y l i n e w i d t h s o f t h e o r d e r o f t w i c e t h e D o p p l e r w i d t h . T h i s b r o a d e n i n g can be i n t e r p r e t e d i n terms o f a weak p r e d i s s o c i a t i o n w h i c h i s i n d e p e n d e n t o f t h e r o t a t i o n a l quantum numbers. From t h i s , t h e l i f e t i m e o f t h e z e r o - p o i n t l e v e l o f t h e e x c i t e d s t a t e was found t o be 42 ± 5 p i c o s e c o n d s . 114 (B) E x p e r i m e n t a l . The a b s o r p t i o n s p e c t r u m o f NO,, between 2490 A and 2560 A was photographed a t room t e m p e r a t u r e and a t v a r i o u s t e m p e r a t u r e s up t o 200°C u s i n g a 7 meter f o c a l 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 i n 2 3 r d o r d e r . The q u a r t z a b s o r p t i o n c e l l was 1.85 m l o n g , and t h e gas p r e s s u r e s were v a r i e d o ver t h e range from 3 mm Hg t o 56 mm Hg. Good p l a t e s were o b t a i n e d f o r t h e head o f t h e 2491 A band u s i n g 3 mm Hg a t room t e m p e r a t u r e ; t h e e x p o s u r e t i m e on Kodak SA-1 p l a t e s w i t h t h e s p e c t r o g r a p h s l i t s e t t o 20 urn was 75 m i n u t e s . A background continuum was p r o v i d e d by a 1000W xenon a r c lamp, and w a v e l e n g t h s t a n d a r d s were s u p p l i e d by an i r o n - n e o n h o l l o w cathode lamp. The p h o t o g r a p h i c p l a t e s were measured u s i n g a G r a n t a u t o m a t i c r e c o r d i n g comparator. P r e v i o u s l y photographed s p e c t r a o f CS,, and SO2 v e r i f y t h a t t h e s p e c t r o g r a p h i s c a p a b l e o f a f u l l w i d t h a t half-maximum r e s o l v i n g power i n e x c e s s o f 600,000 a t t h e s e w a v e l e n g t h s . F o r N0 2 a t room t e m p e r a t u r e , t h e r e s o l v i n g power s h o u l d be l i m i t e d by t h e Do p p i e r w i d t h s o f the l i n e s t o t h e o r d e r o f 550,000. I n s t e a d , t h e r e s o l v i n g power o b s e r v e d was o n l y 250,000. Comparison s p e c t r a were t a k e n o f t h e 2328 A band o f SO2 c h a n g i n g n o t h i n g but t h e gas f i l l t o prove t h a t t h i s r e s u l t was not s i m p l y an e f f e c t o f o p t i c a l m i s a l i g n m e n t o f t h e a p p a r a t u s . As' a f u r t h e r c h e c k , t h e o p t i c a l system was dismounted and r e a s s e m b l e d ; i n both c a s e s , t h e NO,, spec t r u m was f d u n d i t o l b e unchanged. The SO,, s p e c t r a t a k e n w i t h t h e same o p t i c a l arrangement were found t o have a r e s o l v i n g power o f 700,000, which i s o n l y s l i g h t l y l e s s t h a n t h e l i m i t of:800','000: imposed by t h e Do p p l e r l i n e w i d t h f o r t h i s m o l e c u l e . 115 (C) Energy L e v e l E x p r e s s i o n s f o r an Asymmetric Top i n a D o u b l e t S t a t e . I n C h a p t e r I I , t h e t h e o r y o f t h e e l e c t r o n , s p i n f i n e s t r u c t u r e o f an asymmetric t o p m o l e c u l e i n a d o u b l e t e l e c t r o n i c s t a t e was d i s c u s s e d a t l e n g t h . T a b l e 2.2 g i v e s t h e m a t r i x e l e m e n t s f o r t h e s p i n - r o t a t i o n i n t e r a c t i o n i n s p h e r i c a l t e n s o r n o t a t i o n , u s i n g t h e s p i n -r o t a t i o n p a r a m e t e r s d e f i n e d i n T a b l e 2.1. I f t h e t y p e I r Hund's case (b) b a s i s s e t |N,S,3,K> i s a b b r e v i a t e d t o |N,K> t h e f o u r e x p r e s s i o n s g i v e n i n T a b l e 5.1 a r e t h e m a t r i x e l e m e n t s r e q u i r e d t o f i t t h e 2491A band o f NO,,. The m a t r i x e l e m e n t s o f t h i s t a b l e combine a p p r o p r i a t e terms from t h e redu c e d c e n t r i f u g a l d i s t o r t i o n H a m i l t o n i a n o f 1.102 w i t h t h e s p i n - r o t a t i o n m a t r i x e l e m e n t s o f Ta b l e 2.2, m a i n t a i n i n g t h e phase c h o i c e o f Condon and S h o r t l e y (6)r.throughout. To r e p r o d u c e t h e ground s t a t e energy l e v e l s o f •SO^, a c e n t r i f u g a l c o r r e c t i o n was n e c e s s a r y t o f i t t h e o b s e r v e d K-dependence o f t h e s p i n c o n s t a n t e . T h i s was made by r e p l a c i n g e by e +;>n.K2, aa aa aa where n(=n ) has been d e f i n e d i n e q u a t i o n 2.21. aaaa One may t r a n s f o r m t h e m a t r i x e l e m e n t s o f T a b l e 5.1 i n t o t h o s e o f t h e Wang-symmetrized sym m e t r i c t o p b a s i s u s i n g t h e t r a n s f o r m a t i o n |N,K> ' = 2 - 2"{|N,K> ± IN,-K >} (5.01) The m a t r i x r e p r e s e n t a t i o n g i v e n by T a b l e 5.1 w i l l have m a t r i c e s o f the o r d e r 3 + 1, o r t w i c e as l a r g e as t h o s e e n c o u n t e r e d i n s i n g l e t asymmetric t o p m o l e c u l e s . Because o f t h e g r e a t e r c o m p u t a t i o n ' t i m e s i n v o l v e d , t h e most comprehensive d a t a p r e s e n t l y a v a i l a b l e f o r t h e ground s t a t e o f NO,, (5) have been re d u c e d t o m o l e c u l a r p a r a m e t e r s u s i n g an a p p r o x i m a t e method t o t r e a t t h e s p i n s p l i t t i n g s . Cabana e t . a l . (5) used t h e a p p r o x i m a t i o n t h a t t h e a n i s o t r o p i c p a r t o f t h e s p i n - r o t a t i o n t e n s o r ( i v e . r t h e s terms i n b i n Ta b l e 5.1) may be n e g l e c t e d e x c e p t f o r |K| = 1, and t h a t t h e r e m a i n i n g s p i n - r o t a t i o n m a t r i x e l e m e n t s o f f - d i a g o n a l ; i n N can be t r e a t e d t o s u f f i c i e n t T a b l e 5.1 M a t r i x Elements f o r an Asymmetric Top M o l e c u l e i n a D o u b l e t S t a t e . < N,K | H | N,K > = | ( B + C)N(N + 1) + {A - £(B + C ) } K 2 - A K* - A N R N ( N + 1 ) K 2 - A N N 2 ( N + l ) 2 + H K K 6 + H K N K 4 N ( N + 1) + H N N 3 ( N + 1 ) 3 - L ^ 8 • H 3 ( 3 + 1) - N(N +• 1) - 3/4} a + a o 3K2 N(N + 1) l - n K N(N + 1) < N,K ± 2 | H | N,K > < N - 1 , K | H | N,K > < N - 1 , K ± 2 |. H I N,K > ( i ( B > C) - i b { 3 ( 3 . + 1) - N(N + 1) - 3/4 } / {N(N + N(N + 1) , '^ - i < y K 2 + (K ± 2 ) 2 } ) x ( { ¥ ( N + 1) - K(K ± 1 ) } {N (N + 1) - (K ± 1 ) ( K > ± 2)}} H3a - n ) K ( N 2 - K 2 ) * / N ±£b ( {N(N + 1) - K(K ± 1 ) } ( N '+, K i " l ) ( N + K - 2 ) ) * 117 a c c u r a c y u s i n g s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y . In t h i s a p p r o x i m a t i o n , t h e energy l e v e l s become t h o s e o f a s i n g l e t asymmetric top w i t h c o r r e c t i o n s t o t h e energy f o r t h e e l e c t r o n ; s p i n e f f e c t s . The c o r r e c t i o n s a r e g i v e n by E S p i n ( 3 = N ± i ) = ± H ( a - a ± i b f i . ,.)N(N + 1) - 3aK 2 + r]Kk 0 X j K 9 a 2 K 2 ( l - K 2 / ( N + 1 ± i ) 2 ) / 4 B } / ( N + \ ± i ) (5.02) f o r t h e two s p i n components. The ± b e f o r e |b i n 5.02 i s d i f f e r e n t from t h e o t h e r terms p r e c e d e d by ±, s i n c e t h i s term i n d i c a t e s ' t h e two;>?-': asymmetry componentsoof t o K = 1; B i s t h e r o t a t i o n a l c o n s t a n t i ( B + C ) . Because t h e e q u i v a l e n t oxygen atoms o f N0 2 have z e r o n u c l e a r s p i n s , t h e a l l o w e d r o t a t i o n a l l e v e l s a r e t h o s e b e l o n g i n g t o t h e r o v i b r o n i c s p e c i e s and A.,. E x c e p t f o r t h e l e v e l N = 0, t h e s e l e v e l s w i l l a l l be d o u b l e d by t h e u n p a i r e d e l e c t r o n s p i n , a c c o r d i n g t o t h e r e l a t i o n g i v e n i n 5.02. I t i s n e c e s s a r y t o v e r i f y t h e a s s u m p t i o n s m a d e s i n r t h e n d e r i v a t i o n o f 5.02 i n t h e s p e c t r u m o f NO.,. Energy l e v e l s c a l c u l a t e d u s i n g 1.102 and 5.02 and t h o s e c a l c u l a t e d u s i n g t h e same m o l e c u l a r c o n s t a n t s i n t h e m a t r i x e l e m e n t s o f T a b l e 5.1, f o l l o w e d by d i a g o n a l i z a t i o n , were compared. The d i f f e r e n c e s f o und were n e v e r l a r g e r t h a n 0.01 cm 1 i n t h e range o f N and K v a l u e s o f i n t e r e s t . T h i s r e s u l t i s r e a s o n a b l e f o r two r e a s o n s : N0 2 i s v e r y n e a r l y a' p r o l a t e symmetric t o p , so t h a t asymmetry e f f e c t s a r e s m a l l , and t h e r o t a t i o n a l c o n s t a n t B i s l a r g e compared w i t h t h e s p i n c o n s t a n t = -a - 2 a Q . The d a t a o b t a i n e d were r e d u c e d u s i n g 5.02 as c o r r e c t i o n s t o t h e e i g e n v a l u e s o f 1.102, s i n c e the 0.01 cm 1 p r e c i s i o n i s adequate i n t h e p r e s e n c e o f t h e l a r g e l i n e w i d t h s due t o p r e d i s s o c i a t i o n (see s e c t i o n H ) . T h i s c h o i c e means t h a t t h e upper s t a t e c o n s t a n t s - c a n be t r e a t e d as i n c r e m e n t s t o t h e i n f r a r e d ground s t a t e c o n s t a n t s : f u t u r e r e f i n e m e n t o f t h e l a t t e r c a n t h e r e f o r e be d i r e c t l y t r a n s f e r r e d t o t h e upper s t a t e c o n s t a n t s d e r i v e d from t h i s work. 118 (D) D e s c r i p t i o n o f t h e 2491 A Band. The 2491 A band o f NO^ , i s a ' t e x t b o o k ' example o f a t y p e A ' , t r a n s i t i o n o f a d o u b l e t asymmetric t o p m o l e c u l e i n which t h e s p i n c o u p l i n g i s c l o s e t o Hund's case ( b ) . F i g u r e 5.1 shows t h e l i n e a s s i g n m e n t s f o r t h e s h o r t w a v e l e n g t h p a r t o f t h e band. The K— s t r u c t u r e o f t h e band i s s t r o n g l y r e d - d e g r a d e d , and t h e N - s t r u c t u r e o f t h e subbands i s a l s o r e d - d e g r a d e d , w i t h t h e r e s u l t t h a t t h e subbands form a c l e a r s e r i e s v o f heads. I n t h e low K r e g i o n o f t h e band shown i n F i g u r e 5.1, t h e s p i n s p l i t t i n g s a r e q u i t e s m a l l and a r e not p a r t i c u l a r l y o b v i o u s because o f t h e crowded nature: o f t h e s p e c t r u m . L i n e s c o r r e s p o n d i n g t o even and odd N" a s s i g n e m n t s a r e marked w i t h d o t s above and below t h e t i e - l i n e s , r e s p e c t i v e l y , so t h a t t h e two s p i n components o f a l i n e (where t h e y a r e r e s o l v e d ) a r e i n d i c a t e d i n F i g u r e 5.1 by two c l o s e l y spaced d o t s on t h e same s i d e o f t h e t i e - l i n e . F i g u r e 5.1 shows t h e spectrum a t room t e m p e r a t u r e , where t h e b r a n c h e s can be f o l l o w e d t o about N = 35 and K = 6. The l i n e a s s i g n m e n t s c o u l d be e x t e n d e d t o about N = 5 0 i n J a l l t h e subbands up t o K = 10 u s i n g p l a t e s t a k e n a t h i g h e r t e m p e r a t u r e s and p r e s s u r e s . As e x p e c t e d from t h e asymmetric t o p l i n e s t r e n g t h e x p r e s s i o n s , t h e b r a n c h e s a r e q u i t e weak f o r low K, but t h e y become t h e most p r o m i n e n t f e a t u r e s o f each subband f o r t h e h i g h e r K v a l u e s . At h i g h e r p r e s s u r e s , t h e weak asymmetry-induced AK = +2 b ranches ( RQ and R-^ ) can be seen beyond t h e main head o f t h e band: t h e s e b r a n c h e s a r e i l l u s t r a t e d i n F i g u r e 5.2c. Some l i n e s due t o t h e A 3Z + - X 3E u g ( H e r z b e r g system o f 0-, a l s o appear i n F i g u r e 5.2c; t h e s e a r e due t o t h e a i r i n the s p e c t r o g r a p h . The o b s e r v e d s p i n s p l i t t i n g s i n c r e a s e w i t h K, and become v e r y o b v i o u s i n t h e q P b ranches f o r K > 5,('a'seshowneinaFigure~5.30*, where i n c r e a s e d s e p a r a t i o n o f s u c c e s s i v e K-sabbands'has IfcedQced'""line cblending. F i g u r e 5.1 Room t e m p e r a t u r e a b s o r p t i o n s p e c t r u m o f N0 2 between 2491 and 2497 A. 4 0 0 5 1 . 5 8 c m " ^ 4 0 0 4 1 . 0 0 c m " 1 1 3 4 i i < > 5 6 3 4 II . 3 3 3 6 3 5 —' 2 > 9 : 3 2 31 3 4 3 3 2 3 < 11 11 J , •* ^ 4 0 1 4 6 . 9 8 c m - 1 - j 4 0 1 3 3 . 5 2 c m -1 (0 2) SR o I R 14 3 4 3 6 head of 3 4 0 2 I 4 F i g u r e 5.2 (a) A p o r t i o n o f t h e C-X (010-000) band o f S 0 2 (2328 A) a t -78 C. (b) A p o r t i o n o f t h e N0 2 a b s o r p t i o n s p e c t r u m n e a r 2497 A a t room t e m p e r a t t o t h e same s c a l e as ( a ) (c) AK = 2 b r a n c h e s i n t h e 2491 A band o f N 0 2 . F i g u r e 5.3 A b s o r p t i o n s p e c t r u m o f N0 2 between 2503 and 2510 A t a k e n a t 200°C, s h o w i n g t h e l a r g e d o u b l e t s p i n s p l i t t i n g s i n t h e h i g h - K s u b b a n d s . S p i n components o f t h e same v a l u e o f N a r e c o n n e c t e d by s o l i d b a r s . 122 I t i s not p o s s i b l e t o d e t e r m i n e t h e 3 numbering o f t h e s p i n components d i r e c t l y , s i n c e t h e A3 = AN s e l e c t i o n r u l e w h i c h a p p l i e s t o Hund's case (b) c o u p l i n g does not p e r m i t s a t e l l i t e b r a n c h e s w i t h AN = A3 ± 1. The l a t t e r a r e n o r m a l l y used t o a s s i g n t h e 3 v a l u e s i n m o l e c u l e s where case (a) c o u p l i n g o c c u r s . F o r t u n a t e l y , i t was not n e c e s s a r y t o know t h e energy o r d e r o f the+.two s p i n components i n t h e upper s t a t e i n o r d e r t o a s s i g n t h e N and K quantum numbers c o r r e c t l y . Even though e q u a t i o n 5.02 shows t h a t t h e s p i n c o n t r i b u t i o n s t o t h e energy l e v e l s a r e n o t e q u a l and o p p o s i t e f o r t h e two s p i n components, t h e means o f t h e s p i n - d o u b l e d l i n e s p o s i t i o n s w i t h i n a g i v e n v a l u e o f K were f o u n d t o be s h i f t e d by an amount which remained a p p r o x i m a t e l y c o n s t a n t f o r each sub-band, r e l a t i v e t o where t h e l i n e s would be i f t h e s p i n c o n s t a n t s were i d e n t i c a l l y z e r o . As a r e s u l t , c o n v e n 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 t e c h n i q u e s based on t h e means o f each d o u b l e t c o u l d be used t o make t h e N and K a s s i g n m e n t s i n t h e s p e c t r u m . 123 (E) The S p i n - R o t a t i o n C o n s t a n t s o f t h e Upper S t a t e . The d i f f e r e n c e s between t h e s p i n c o n s t a n t s f o r t h e ground s t a t e l e v e l s 000 and 010 can be used t o e s t a b l i s h t h e energy o r d e r i n g o f t h e s p i n components i n t h e 2491 A band. These a r e known, from t h e work o f Cabana e t ^ a l . ( 5 ) , b u t i t s h o u l d be emphasized t h a t even i f t h e y had n o t been measured, i t would s t i l l have been p o s s i b l e t o p r e d i c t t h e s i g n o f t h e d i f f e r e n c e i n the' p r i n c i p a l s p i n c o n s t a n t e f f r o m t h e t h e o r y o f t h e aa I R e n n e r - T e l l e r e f f e c t , and t o come t o t h e same s o l u t i o n . The p r e d i c t i o n can be made as f o l l o w s . The ground s t a t e o f hKj^j X 2 A i , c o r r e l a t e s w i t h a 2II u s t a t e o f t h e l i n e a r m o l e c u l e , f o r w h i c h t h e s p i n -o r b i t c o u p l i n g c o n s t a n t s A S " ° ' i s p o s i t i v e because t h e r e i s one e l e c t r o n i n a rr m o l e c u l a r o r b i t a l . I n t h e bent m o l e c u l e , t h e 2 n s t a t e s p l i t s u ' u r i n t o a 2B 1 s t a t e and a 2r\1 s t a t e , t h e l a t t e r b e i n g t h e ground s t a t e . The s p i n - r o t a t i o n i n t e r a c t i o n i n t h e ground s t a t e r e p r e s e n t s t h e r e s i d u a l s p i n - o r b i t c o u p l i n g when t h e m o l e c u l e i s no l o n g e r l i n e a r and t h e e x p e c t a t i o n v a l u e o f t h e o r b i t a l , a n g u l a r momentum, <L Z> , becomes v e r y s m a l l . The s o r e l a t i o n s h i p between e and A ' ' has been g i v e n i n e q u a t i o n s 2.6, 2.7 and aa 2.16 o f c h a p t e r I I . As a p p l i e d t o t h e ground s t a t e o f NO,,, t h e s e e q u a t i o n s -• become e = 4 A A S , ° ' /AE( 2Bi - 2 A X ) (5.03) aa where t h e c o n s t a n t A(=u ) i s t h e a - a x i s r o t a t i o n a l c o n s t a n t . A l l q u a n t i t i e s aa on t h e r i g h t - h a n d s i d e o f 5.03 a r e p o s i t i v e , so t h a t t h e v a l u e o f e must aa a l s o be p o s i t i v e . S i m i l a r e x p r e s s i o n s a p p l y e t d h t h e r c o h s t a n t s ^ E ^ g and £ c c t h e s e c o n s t a n t s w i l l be some 20 t i m e s s m a l l e r s i n c e t h e y i n v o l v e t h e r o t a t i o n a l c o n s t a n t s B and C. On e x c i t a t i o n o f t h e b e n d i n g mode, one o f t h e c l a s s i c a l t u r n i n g p o i n t s w i l l c o r r e s p o n d t o a more l i n e a r c o n f i g u r a t i o n . T h i s means t h a t <L > i n c r e a s e s , and t h e r e f o r e e „ does as w e l l , z aa 124 An e x a c t d e t e r m i n a t i o n o f how much e ought t o i n c r e a s e w i t h v 0 aa d. c a n n o t be made w i t h o u t making d e t a i l e d c a l c u l a t i o n s o f < L Z > , but t h i s q u a n t i t y has been d e t e r m i n e d e x p e r i m e n t a l l y by Cabana e t . a l . ( 5 ) . The change i n t h e v a l u e o f e i s q u i t e s m a l l , so t h a t i n t h e p r e s e n c e o f aa w i de l i n e s due t o p r e d i s s o c i a t i o n t h e d i f f e r e n c e i n s p l i t t i n g s becomes l a r g e enough^to be d e f i n i t e l y m easureable o n l y a t h i g h e r v a l u e s o f K. The s p l i t t i n g s i n t h e q P . b r a n c h e s o f t h e 2491 A (000 - 000) band 6 a r e compared t o t h o s e o f t h e 2537 A (000 - 010) band i n T a b l e 5.2 and F i g u r e 5.4. The measurements a r e n o t as p r e c i s e as would be d e s i r e d , because t h e 2537 A band had t o be photographed a t a t e m p e r a t u r e o f 200°C i n o r d e r t o a c h i e v e s u f f i c i e n t i n t e n s i t y , and t h i s c a u s e d i n c r e a s e d b l e n d i n g o f t h e l i n e s . Even s o , T a b l e 5.2 and F i g u r e 5.4 show t h a t t h e d i f f e r e n c e s j i i n t h e s p l i t t i n g s measured i n t h e two bands a r e i n s a t i s f a c t o r y agreement w i t h t h o s e c a l c u l a t e d from t h e i n f r a r e d d a t a o f r e f e r e n c e ( 5 ) . T a b l e 5.2 shows t h a t t h e s p l i t t i n g s o f t h e l i n e s i n t h e 2491 A band a r e l a r g e r t h a n t h o s e o f t h e l e v e l s o f t h e ground s t a t e , w h i c h i s i n d i c a t e d g r a p h i c a l l y by t h e f a c t t h a t a l l t h e p o i n t s p l o t t e d l i e above t h e c a l c u l a t e d change i n t h e ground s t a t e - s p l i t t i n g o f F i g u r e 5.4. The f a c t t h a t t h e o b s e r v e d s p l i t t i n g s i n t h e e l e c t r o n i c t r a n s i t i o n i n c r e a s e when t h o s e o f t h e ground s t a t e i n c r e a s e p r o v e s t h a t t h e o b s e r v e d s p l i t t i n g s a r e sums o f t h e s p l i t t i n g s i n t h e two s t a t e s . S i n c e t h e Hund's case (b) s e l e c t i o n r u l e s p e r m i t o n l y F^ - and F^ - F^ t r a n s i t i o n s , t h i s r e s u l t i m p l i e s t h a t t h e energy o r d e r o f t h e s p i n components i n t h e upper s t a t e must be o p p o s i t e t o t h a t o f t h e ground s t a t e . From t h e microwave d a t a o f Lees e t . a l . ( 7 ) , i t i s known t h a t t h e v a l u e o f e i s p o s i t i v e i n t h e ground s t a t e ( as was a l s o p r e d i c t e d by t h e R e n n e r - T e l l e r t h e o r y ), and t h a t t h e F 1 (0 = N + \) l e v e l s l i e above 125 0.1 Difference in the spin splittings for K = 6 between the 2537 A (000-010) and 2491 A (000-000) bands of NO2 cm -1 0 .05 0.0 calculated change in the ground ^ state F | -F 2 splitting: 0 1 0 - 0 0 0 16 18 20 22 N 24 F i g u r e 5.4: D i f f e r e n c e i n t h e s p i n s p l i t t i n g s f o r K = 6 between t h e 2537 A (000 - 010) and 2491 A (000 - 000) bands o f N 0 2 > T a b l e 5.2 Comparison o f s p i r p s p l i t t i n g s i n c o r r e s p o n d i n g l i n e s o f t h e 2491 A (000-000) and 2537 A (000-010) bands o f N0 2 ( c m - 1 ) . Observed s p l i t t i n g s C a l c u l a t e d ground s p l i t t i n g s (F-^ -s t a t e F2> N" 000-010 000-000 D i f f e r e n c e 010 000 D i f f e r e n c e 16. 0.825 a 0.795* 0.030 a 0.355 0.317 0.038 17 0.890 3 0.745 0.145 a 0.330 0.295 0.035 18 0.727 3 0.684 a 0.043 a 0.309 0.275 0.034 19 0.737 0.649 0.088 0.289 0.259 0.030 20 0.681 0.652 0.029 0.271 0.242 0.029 21 0.762 a 0.579 0.183 a 0.255 0.229 0.026 22 0.641 0.597 0.044 0.240 0.215 0.025 23 0.626^ 0.562 0.064 0.225 0.203 0.022 24 0.551 0.528 0.023 0.213 0.192 .0.021 I n c l u d e s d a t a from a b l e n d e d l i n e . 127 t h e F 2 (0'-= N - |) l e v e l s . From t h i s f a c t , one may c o n c l u d e t h a t t h e F 2 l e v e l s l i e above t h e F^ l e v e l s i n t h e upper s t a t e , as i l l u s t r a t e d by t h e e n ergy l e v e l d i a g r a m o f F i g u r e 5.5; The a s s i g n m e n t o f t h e c'p^(2) l i n e c o n f i r m s t h i s energy o r d e r , s i n c e f o r t h i s l i n e , t h e two s p i n components have v e r y d i f f e r e n t i n t e n s i t i e s , as can be seen from F i g u r e 5.1. S i n c e t h e r e l a t i v e i n t e n s i t i e s f o r t h e s p i n components o f t h i s l i n e w i l l be r o u g h l y p r o p o r t i o n a l t o t h e 3" v a l u e s , t h e weaker ( s h o r t w a v e l e n g t h ) component can be a s s i g n e d as 3'°= 1 -<- 3" = l i and t h e s t r o n g e r as;'3! = 1| 3" = 2|. T h e r e f o r e , where s p i n s p l i t t i n g s a r e r e s o l v e d i n t h e s p e c t r u m , t h e F 2 ( 3 = N - 1) s p i n component w i l l l i e t o t h e s h o r t w a v e l e n g t h s i d e o f t h e F^ component*.'-D i r e c t measurement o f t h e ^ s p l i t t i n g s i n the upper s t a t e N = K l e v e l s ( f o r example, i n t h e l i n e ^P/^(2) and t h e f i r s t l i n e s o f t h e h i g h e r K subbands) c o n f i r m s t h e d e d u c t i o n t h a t e has a n e g a t i v e s i g n i n t h e upper aa s t a t e . I t does not seem t o have been p r e v i o u s l y r e c o g n i z e d t h a t an a m b i g u i t y i n t h e s i g n o f e can a r i s e i f t h e low N s p l i t t i n g s a r e n o t r e s o l v e d , a a The problem i s s i m i l a r t o t h a t a r i s i n g i n f t h e zJl and 2 A s t a t e s o f l i n e a r m o l e c u l e s c l o s e t o c a s e (b) c o u p l i n g , s i n c e t h e energy l e v e l p a t t e r n s g i v e n by t h e H i l l and Van V l e c k e q u a t i o n (see H e r z b e r g ( 8 ) ) a r e s y m m e t r i c a l about s o t h e p o i n t where A " " = 2B. U n l e s s t h e l e v e l w i t h 3 = A - \ can be i d e n t i f i e d , i t i s not p o s s i b l e t o d e c i d e w h i c h o f t h e two s o l u t i o n s t o t h e H i l l and Van V l e c k e x p r e s s i o n i s t h e c o r r e c t one ( see a l s o Merer e t . a l . ( 9 ) ) . An e n t i r e l y a n a l o g o u s s i t u a t i o n e x i s t s i n n o n - l i n e a r m o l e c u l e s , though the n o t a t i o n i s d i f f e r e n t i n t h i s c a s e : f o r A, A, and B, one must now r e a d e , K, and B. a a ' ' The a m b i g u i t y a r i s e s because t h e upper s t a t e s p l i t t i n g s , w i t h t h e e x c e p t i o n o f t h o s e f o r which N = K, can be e q u a l l y w e l l f i t t e d w i t h e '/B CO (NJ B, 000 Fa F, 2491 A 2537 A ft 010 000 Fa Fi F 2 F i g u r e 5.5 Energy l e v e l d iagram showing how t h e s p i n s p l i t t i n g s i n t h e 2491 A and 2537 A bands o f N0\, a r e r e l a t e d . 129 e q u a l t o -0.484 o r +4.484, and t h a t t h e second f i g u r e i s not u n r e a s o n a b l e , s i n c e i t c o r r e s p o n d s t o a v a l u e o f e 1 = -1.72 cm \ However, t h e second aa f i g u r e i s e x c l u d e d i n t h i s c a s e by t h e p o s i t i o n s o f t h e 3 = K - i l i n e s . The upper s t a t e c o n s t a n t s a n c ' e c c a r e z e r o w i t h i n t h e a c c u r a c y o f t h e l i n e p o s i t i o n s o b s e r v e d . These c o n s t a n t s a r e g i v e n most c o n v e n i e n t l y by t h e h i g h N s p l i t t i n g s f o r K g = 0 and 1, w h i c h g i v e a - a Q = ^ ( e ^ + £ c c ^ and a - a Q ± i b = + e c C ) + ^ e b b ~ £ c c ^ ' r e s P e c t ^ - v e ^ y • E x p e r i m e n t a l l y , an a l t e r n a t i o n o f l i n e w i d t h w i t h N i s found i n t h e ^P^ b r a n c h o f F i g u r e 5.2c. T h i s a l t e r n a t i o n a r i s e s because h a l f t h e l e v e l s a r e m i s s i n g i n NO,,, so t h a t t h e l i n e s c o r r e s p o n d s t o t h e upper and l o w e r asymmetry components, a l t e r n a t e l y , t o w h i c h t h e a n i s o t r o p i c p a r t o f t h e s p i n - r o t a t i o n t e n s o r c o n t r i b u t e s a q u a n t i t y t h a t a l t e r n a t e s w i t h N. The o b s e r v e d l i n e w i d t h s can be e x p l a i n e d e n t i r e l y i n terms o f u n r e s o l v e d ground s t a t e s p l i t t i n g s , and t h e r e f o r e one may c o n c l u d e t h a t t h e upper s t a t e c o n s t a n t s l e ^ l - a n d |c | a r e l e s s t h a n 0.001 cm 1 i n magnitude. 130 (F) D e t e r m i n a t i o n o f t h e Upper S t a t e R o t a t i o n a l C o n s t a n t s . The unblended l i n e f r e q u e n c i e s were c o n v e r t e d t o upper s t a t e term v a l u e s by a d d i n g t h e a p p r o p r i a t e ground s t a t e t e r m v a l u e s c a l c u l a t e d from t h e d a t a o f Cabana e t . a l . ( 5 ) . A H a m i l t o n i a n i d e n t i c a l t o t h a t used by Cabana e t . a l . (5) was used i n t h e l e a s t s q u a r e s f i t ; m a t r i x e l e m e n t s from which t h e s i n g l e t e i g e n v a l u e s were c a l c u l a t e d can be o b t a i n e d f r o m : T a b l e 5.1 by o m i t t i n g t h e s p i n - r o t a t i o n i n t e r a c t i o n , and i n c l u d i n g s p i n e f f e c t s l a t e r by means o f 5.02. A t o t a l o f 513 upper s t a t e term v a l u e s were o b t a i n e d from t h e unblended l i n e p o s i t i o n s , w h i c h correspond*_to i n d i v i d u a l s p i n components o f a r o t a t i o n a l l e v e l e x c e p t f o r t h e l o w e s t K v a l u e s , where t h e y r e p r e s e n t e d t h e means o f t h e u n r e s o l v e d s p i n components. With t h e e x c e p t i o n s o f H^ and L^, t h e upper s t a t e c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s o f T a b l e 5.1 were a l l d e t e r m i n a b l e . S u r p r i s i n g l y , t h e c o n s t a n t H ^ was w e l l - d e t e r m i n e d and had t o be i n c l u d e d i n t h e f i n a l f i t t o a c c o u n t f o r t h e o b s e r v e d d a t a , even though H^ was p o o r l y d e t e r m i n e d and was t h e r e f o r e dropped from t h e f i n a l f i t . The f i n a l q u a r t i c c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s found were q u i t e s i m i l a r t o t h o s e o f t h e ground s t a t e , w i t h t h e e x c e p t i o n o f A ^ w h i c h had a p o s i t i v e , r a t h e r t h a n a n e g a t i v e v a l u e . The r o t a t i o n a l c o n s t a n t s o b t a i n e d a r e g i v e n i n T a b l e 5.3; t h e a s s i g n e d r o t a t i o n a l l i n e s have been p u b l i s h e d as T a b l e k o f r e f e r e n c e ( 1 0 ) , and i n Appendix I I I . 131 T a b l e 5.3. R o t a t i o n a l c o n s t a n t s f o r t h e 2491 A band o f NO 2 B 2 upper s t a t e 2 A i ground s t a t e T 0 40,125.849 t 10 0.00 -1 cm A 4.11572 53 8.002383 ± 21 B 0.403340 54 0.4337088 41 q . 0.365445 60 0.4104469 22 A K ANK A N 6 K 6.331 x10" 1.208 x l O " 5.03 x l O ' 3.26 x l O " -4 -5 -7 -5 56 74 13 60 2.700 x l O - 3 -1.969 x 1 0 " 3 3.013 x 1 0 " 7 4.13 x l O " 6 13 29 30 173 6 N 9.7 x l O " •8 17 3.103 x l O - 8 96 H K HKN H N -4.27 x l O " •8 75 3.26 x l O &