UBC Theses and Dissertations

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UBC Theses and Dissertations

Studies in the ro-vibronic spectroscopy of gases McRae, Glenn Aldon 1984

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J STUDIES IN THE RO-VIBRONIC SPECTROSCOPY OF by GLENN ALDON MCRAE A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t 1984 © G l e n n A l d o n McRae, 1984 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requ i rements fo r an advanced degree at the The U n i v e r s i t y of B r i t i s h Co lumbia , I agree tha t the 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 fo r r e f e r e n c e and s tudy . I f u r t h e r agree tha t pe rm i s s i on fo r e x t e n s i v e copy ing of t h i s t h e s i s fo r s c h o l a r l y purposes may be g ran ted by t he . Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s unders tood that copy ing or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l l owed wi thout my w r i t t e n p e r m i s s i o n . DEPARTMENT OF CHEMISTRY The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l ace Vancouver , Canada V6T 1W5 Da te : August 1984 ABSTRACT Microwave s p e c t r a have been i n v e s t i g a t e d fo r G e H „ , A s D 3 , A s D 2 H , AsH 2 D, PD 2H and PH 2D (and P D 3 ) . Fo rb idden d i s t o r t i o n s p e c t r a in the f requency range 8-26 GHz have been measured and ana l yzed fo r GeH„ and A s D 3 . A l lowed r o t a t i o n a l t r a n s i t i o n s , in the cen t ime te r and m i l l i m e t e r wave r e g i o n s , of A s D 2 H , AsH 2 D, PD 2H and PH 2D have been measured and a n a l y z e d . Harmonic f o r ce f i e l d s have been produced f o r ammonia, phosphine and a r s i n e from v a r i o u s e m p i r i c a l da ta i n c l u d i n g the 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 ob t a i ned in t h i s work as we l l as C o r i o l i s c o u p l i n g c o n s t a n t s and v i b r a t i o n a l f r equenc i e s o b t a i n e d from other s t u d i e s . A d i s c u s s i o n of the s tandard " d i a t o m i c " anharmonic i sty c o r r e c t i o n t echn iques to "ha rmon ize " v i b r a t i o n a l f r e q u e n c i e s shows these t echn iques to be u n s a t i s f a c t o r y and so c o r r e c t i o n s of t h i s so r t were not made. From the harmonic f o r ce f i e l d a n a l y s i s average " r z " s t r u c t u r e s have been ob ta ined fo r v a r i o u s i s o t o p i c d e r i v a t i v e s of ammonia, phosphine and a r s i n e . Us ing a l i n e a r e x t r a p o l a t i o n , e q u i l i b r i u m s t r u c t u r e s have been e s t i m a t e d . A compar ison of many of the symmetric top r e d u c t i o n schemes i s a l s o p resen ted in order to make c l e a r a d i s t i n c t i o n between s i m i l a r l y l a b e l e d parameters of d i f f e r e n t r e d u c t i o n s ; of s p e c i a l importance i s the parameter T x x x z * To " f i t " s p e c t r a of l i g h t e a s i l y d i s t o r t e d mo lecu les i t has been suggested by p r e v i ous au thors tha t r a t i o n a l i i f r a c t i o n Pade approx imants might be u s e f u l . A m o t i v a t i o n fo r why these approximants c o u l d prove b e n e f i c i a l i s g i ven a long w i th a review of e a r l i e r a t tempts to i n c o r p o r a t e them i n t o gene ra l f i t t i n g schemes. In t h i s review we f i n d that i nde te rminacy r e l a t i o n s e x i s t between v a r i o u s parameters in these e a r l i e r schemes and so a new method, a method of s u c c e s s i v e s e p a r a t i o n , i s proposed that c i r cumvents these i n d e t e r m i n a c i e s . T h i s method shou ld prove u s e f u l f o r any gene ra l s tudy where power s e r i e s in N v a r i a b l e s are slow to conve rge . Tab le of Conten ts A. I n t r o d u c t i o n 1 B. CHAPTER ONE: THEORY 7 1. The R i g i d Rotor 7 2. C e n t r i f u g a l D i s t o r t i o n 10 3. Hype r f i ne P e r t u r b a t i o n s to the Energy L e ve l s 18 4. E f f e c t s of E x t e r n a l E l e c t r i c F i e l d s 22 5. Force F i e l d s 34 6. De te rm ina t i on of M o l e c u l a r S t r u c t u r e 38 7. Summary and Comments 43 C. CHAPTER 2: EXPERIMENTAL METHODS 46 1. S tark Modu la t ion 47 2. D i s t o r t i o n Spectrometer 48 3. Cent imeter Wave Spect rometer (UBC) 52 4. M i l l i m e t e r Wave Spect rometer (JPL) 54 5. Sundry Items 55 D. CHAPTER 3: GERMANE GeH, 56 1. Germane Gas Sample 58 2. P r e d i c t i o n of the Germane Spectrum 58 3. Measurement of the Germane Spectrum 60 4. Observed Germane Spectrum and A n a l y s i s . . ' 64 5. I sotope S p l i t t i n g in Germane 71 6. F i n a l Comments on Germane 83 E. CHAPTER 4: ARSINE 86 1. P r e p a r a t i o n of the A r s i n e Gas Sample 89 2. P r e d i c t i o n of the AsD 3 Spectrum 90 3. I n i t i a l Searches 94 i v 4. Observed Ars ine-D3 Spectrum 98 5. P r e d i c t i o n and I n i t i a l Search f o r the Asymmetric Top A r s i n e Spec t r a 113 6. Observed AsH 2 D and AsD 2 H Spectrum 117 7. D i s c u s s i o n of Nuc lea r Quadrupole C o u p l i n g in A r s i n e 132 8. D i s c u s s i o n of the Sp in-Ro ta t i on C o u p l i n g Cons tan ts of A r s i n e 134 F. CHAPTER 5: PHOSPHINE 136 1. P r e p a r a t i o n of Phosphine Gas . . . . 1 3 9 2. P r e d i c t i o n of the Asymmetric Top Phosphine Spec t ra 140 3. Observed PH 2D and PD 2H Spec t ra 141 4. A n a l y s i s of the Phosphine Spec t r a 149 G. Chapter 6: FORCE FIELDS, STRUCTURES AND DIPOLE MOMENTS 153 1. An Examinat ion of S tandard Anharmon ic i t y C o r r e c t i o n Techn iques A p p l i e d to V i b r a t i o n a l F r equenc i e s 156 2. Fu r the r C o n s i d e r a t i o n s When Making a Harmonic Force F i e l d C a l c u l a t i o n 172 3. Harmonic Force F i e l d Re f inements : Ammonia, Phosphine and A r s i n e 177 4. E s t i m a t i o n of the D i s t o r t i o n D i p o l e Moment 184 5. S t r u c t u r e of Ammonia, Phosphine and A r s i n e 189 6. D i s c u s s i o n of S t r u c t u r a l Parameters 195 H. CHAPTER 7: INTERESTING EXTRAS 197 1. F a i l u r e s of the Reduc t ions 199 2. On the use of R a t i o n a l F r a c t i o n Approximants in the F i t t i n g of R o t a t i o n a l Spec t ra 206 BIBLIOGRAPHY 227 v LIST OF TABLES Tab l e 1.1 D e f i n i t i o n of r e p r e s e n t a t i o n i n terms of c a r t e s i a n and i n e r t i a l axes 31 66 67 68 Tab le 1.2 Symmetry s e l e c t i o n r u l e s fo r asymmetric top t r a n s i t i o n s Tab le 3.1 Observed germane Q-branch r o t a t i o n a l t r a n s i t i o n s Tab le 3.2 Summary of germane l i n e parameters and expe r imen ta l c o n d i t i o n s \ Tab le 3.3 Tensor 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 of GeH« Tab l e 3.4 Comparison of l i n e a r combina t ions of germane 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 ob ta ined in the p resen t work w i th those of r e f (r68) 69 Tab le 3.5 P r e d i c t i o n of germane IR-MW double resonance t r a n s i t i o n s u s i n g cons t an t s o b t a i n e d s o l e l y from the microwave spectrum 70 Tab l e 3.6 Comparison of f o r c e cons t an t s fo r methane, s i l a n e and germane 75 Tab le 3.7 Comparison of s p e c t r o s c o p i c p a r a m e t e r s : 1 2 C H U , 2 8 S i H a and GeH„ 85 Tab l e 4.1 Fo rb idden h y p e r f i n e t r a n s i t i o n s of AsD 3 99 Tab le 4.2 A r s i n e l i n e parameters and expe r imen ta l c o n d i t i o n s 1 03 Tab le 4.3 A n a l y s i s of the measured AsD 3 Spectrum (MHz) 1 1 1 v i LIST OF TABLES con t i nued Tab l e 4.4 Observed spectrum of AsH 2 D Tab le 4.5 Observed spectrum of AsD 2 H Tab le 4.6 E m p i r i c a l s p e c t r o s c o p i c parameters of v a r i o u s a r s i n e s Tab le 5.1 Observed PD 2H spectrum in MHz Tab le 5.2 Observed PH 2D spectrum in MHz Tab l e 5.3 A Re-ana l y s i s of the measured PD 3 spectrum in MHz Tab le 5.4 E m p i r i c a l s p e c t r o s c o p i c parameters of v a r i o u s phosph ines 152 1 23 1 28 131 1 45 1 46 147 Tab le 6.1 V a r i o u s f requency r a t i o s of ammonia, phosph ine , a r s i n e and st i b i n e 174 Tab le 6.2 Harmonic f o r c e f i e l d study of ammonia Tab le 6.3 Harmonic f o r c e f i e l d study of phosphine Tab le 6.4 Harmonic f o r c e f i e l d study of a r s i n e Tab le 6.5 Force cons t an t s ( i n t e r n a l symmetry c o o r d i n a t e s ) Tab le 6.6 Force cons t an t s (va lence) Tab l e 6.7 E s t i m a t i o n of i xxxz 179 180 181 182 182 185 v i i LIST OF TABLES con t i nued Tab le 6.8 | 3M/3Q-, | (amu A ) 1 186 Tab l e 6.9 E s t i m a t i o n of the r o t a t i o n a l and v i b r a t i o n a l c o n t r i b u t i o n s to the d i s t o r t i o n d i p o l e moment Tab l e 6.10 r s t r u c t u r e of ammonia ( 1 "N) z Tab le 6.11 r z s t r u c t u r e of phosphine Tab l e 6.12 r s t r u c t u r e of a r s i n e z Tab l e 6.13 E q u i l i b r i u m s t r u c t u r e s of ammonia,phosphine and a r s i n e Tab l e 7.1 E s t i m a t i o n of the c o n t r i b u t i o n of £ to h 3  Tab le 7.2 Comparison of t r unca t ed T a y l o r s e r i e s f T v e r sus Pade f r a c t i o n f p in f i t t i n g data produced from equa t ion 7.12 186 191 1 92 1 93 1 94 204 215 v i i i LIST OF FIGURES F i g u r e 1. The d i s t o r t i o n spec t rometer used between 8 and 26 GHz. I n t e r n a t i o n a l symbols are used fo r the v a r i o u s e lements ( r 57 ) . 48 F i gu re 2. The microwave a b s o r p t i o n l i n e of GeH„ r e s u l t i n g from the d i s t o r t i o n t r a n s i t i o n 18E 2<—18E 1 . I t was ob ta ined by us ing the s i g n a l averager to accumulate 64 sample scans and then s u b t r a c t 64 background scans . The t o t a l scann ing time (sample p l u s background) was 3.64 hou r s . 63 F i g u r e 3. Two of the K = ±1 <— *2 AsD 3 t r a n s i t i o n s . The top r e p r o d u c t i o n i s J=15. I t i s the r e s u l t of 64 sample scans at a S tark f i e l d of 74 v o l t s / c m . The bottom r e p r o d u c t i o n i s J=17 and i s the r e s u l t of 48 scans at a S tark f i e l d of 200 v o l t s / c m . 1 02 F i g u r e 4. The 2 •, -, —2 2 -, r o t a t i o n a l t r a n s i t i o n of A s D 2 H . T h i s was ob ta ined as a s i n g l e scan of 600 seconds d u r a t i o n w i th a t ime cons tan t of 0.1 s e c . and a 100 KHz Stark f i e l d of 2800 v o l t s / c m . 1 18 ix Acknowledgements The i n v e s t i g a t i o n s d e s c r i b e d in t h i s t h e s i s were c a r r i e d out under the p a t i e n t and ab le d i r e c t i o n of M i chae l G e r r y . I shou ld l i k e to thank him fo r h i s i n v a l u a b l e a s s i s t a n c e and encouragement, and fo r the l a r g e degree of independence a l lowed a l l my work. S p e c i a l thanks a l s o goes to I r v i n g O z i e r f o r the e n t h u s i a t i c manner in which he shared many i n s i g h t s and i d e a s . Many of the ideas p resen ted in t h i s t h e s i s were honed in the company of Dav id Spencer and Man Wong. I t i s a p l e a s u r e to acknowledge and thank them. I am indebted to Ed Cohen, Bob D a v i s , Robert Kagann, Herb P i c k e t t , A lan R o b i e t t e and Man Wong fo r t h e i r i n t e r e s t i n , and he lp w i t h , many of the measurments p resen ted in t h i s t h e s i s . I shou ld a l s o wish to thank Zo l Germann fo r h i s p a t i e n t e f f o r t s in keeping the v a r i o u s U .B .C . spec t romete rs a l i v e . F i n a l l y , I shou ld l i k e to d e d i c a t e t h i s t h e s i s to my f a t h e r , mother and s i s t e r in r e c o g n i t i o n of the support and encouragement they have g iven me throughout my s t u d i e s . x 1 A. INTRODUCTION T h i s t h e s i s i s concerned w i th the a b s o r p t i o n of e l e c t r o m a g n e t i c r a d i a t i o n by sma l l mo lecu les in the gas phase . The r a d i a t i o n f r e q u e n c i e s used f a l l in the microwave r e g i o n , between r a d i o and f a r i n f r a r e d , rough ly 1-1000 GHz. A b s o r p t i o n of r a d i a t i o n in t h i s f requency r e g i o n i s u s u a l l y a s s o c i a t e d w i th mo lecu la r r o t a t i o n . A c l a s s i c a l d i a tomic molecu le r o t a t i n g w i th angu la r momentum / p e r p e n d i c u l a r to the bond has a r o t a t i o n a l energy a s s o c i a t e d w i th i t • E - J— ROT 21 where I i s the moment of i n e r t i a about the r o t a t i o n a x i s . If the molecu le possesses an e l e c t r i c d i p o l e moment t h i s d i p o l e moment can r a d i a t e energy or absorb energy from an e x t e r n a l e l e c t r o m a g n e t i c f i e l d . For microwave r a d i a t i o n t h i s r a d i a t e d or absorbed energy a f f e c t s the r o t a t i o n a l energy by, r e s p e c t i v e l y , d e c r e a s i n g or i n c r e a s i n g the r o t a t i o n a l angu la r momentum. For a b s o r p t i o n , the d i p o l e moment becomes a handle by which the microwaves can r o t a t e the m o l e c u l e s . From the above argument we shou ld expect any f requency to be absorbed and g e n e r a l l y f o r heavy mo lecu l a r gases at i n t e rmed i a t e and h igher p r e s s u r e s , we o f t en see r a the r broad f requency a b s o r p t i o n p r o f i l e s c o v e r i n g much of the microwave r e g i o n . As we lower the p r e s su re however, we n o t i c e the broad f requency a b s o r p t i o n f e a t u r e s beg in to sharpen up. At lower p r e s s u r e s (=*0.1mTorr in our s t u d i e s ) gases become 2 q u i t e s e l e c t i v e as to how much of which f r e q u e n c i e s they w i l l absorb and one f i n d s a b s o r p t i o n f e a t u r e s , or l i n e s , whose widths can be one megahertz at one hundred g i g a h e r t z . The l i n e c e n t e r s can u s u a l l y be measured to b e t t e r than one pa r t in 1 0 6 . Th i s p r e c i s i o n of measurement i s the b a s i s of the p r e c i s i o n a s s o c i a t e d wi th microwave s p e c t r o s c o p y . In o rder to r a t i o n a l i z e these seemingly i s o l a t e d s p e c t r a l l i n e s , today we invoke the magic of q u a n t i z e d angu la r momentum and P l a n c k ' s cons tan t ' h ' . Q u a n t i z i n g the angu la r momentum a v a i l a b l e to our mo lecu les q u a n t i z e s the r o t a t i o n a l energy . It i s then t r a n s i t i o n s between these q u a n t i z e d l e v e l s that e x p l a i n microwave s p e c t r a . For i n s t a n c e , f o r a d i a tomic molecu le the angu la r momentum becomes v/JTJ+TT where K equa ls P l a n c k ' s cons t an t d i v i d e d by 27r, and the r o t a t i o n a l energy becomes „ J ( J + , ) M 2 ROT 21 J i s a quantum number a s s o c i a t e d wi th the angu la r momentum. The observed t r a n s i t i o n s are those where J changes by one. For a b s o r p t i o n one has J—>J+1 and ^ROT I where *> RQ T i s the microwave f r equency . We see from the above equa t ion tha t i f we can measure ^RQT^ 1 "^ w e c a n c a l c u ^ a t e the moment of i n e r t i a I. For a d i a tomic 3 so knowing the v a r i o u s masses a l l ows us to c a l c u l a t e r, the bond l e n g t h . T h i s i s not u n l i k e c a l c u l a t i n g a s imple pendulum's l eng th from i t s " r e s o n a n t " a b s o r p t i o n f r equency . In e f f e c t s h i n i n g microwaves on mo lecu les l e t s us " s e e " what they look l i k e . I t would seem then , in o rde r to absorb microwaves a mo lecu le must have a d i p o l e moment, a handle w i th which to r o t a t e i t . The c l a s s i c a l argument becomes a l i t t l e tenuous now. The "no rma l " d i p o l e moment we a s s o c i a t e w i th a r i g i d mo lecu le i s r e s p o n s i b l e fo r " n o r m a l " or a l l owed r o t a t i o n a l t r a n s i t i o n s . We w i l l be l o o k i n g at a number of these a l lowed t r a n s i t i o n s . We s h a l l a l s o observe a number of s o - c a l l e d " f o r b i d d e n " t r a n s i t i o n s . These f o r b i d d e n t r a n s i t i o n s become "weak ly " a l l owed because of c e n t r i f u g a l d i s t o r t i o n . The d i p o l e moment f o r these s o r t s of t r a n s i t i o n s i s c a l l e d a d i s t o r t i o n d i p o l e moment. As we w i l l see , t h i s d i s t o r t i o n moment i s the sum of two p a r t s , a r o t a t i o n a l borrowing pa r t p l u s a v i b r a t i o n a l borrowing p a r t . The c l a s s i c a l p i c t u r e here has the r o t a t i n g and v i b r a t i n g molecu le d i s t o r t i n g because of c e n t r i f u g a l f o r c e s . In c e r t a i n cases these d i s t o r t i o n s can break the e q u i l i b r i u m symmetry of the mo lecu le thus p roduc ing sma l l d i p o l e moments not a l l owed in the more " s y m m e t r i c a l " r i g i d m o l e c u l e . These f o r b i d d e n s p e c t r a are g e n e r a l l y very weak and we r e q u i r e s p e c i a l i z e d spec t romete r s to see them. S t ud i e s of f o rb idden r o t a t i o n a l s p e c t r a due to c e n t r i f u g a l d i s t o r t i o n date from 1967 wi th Hanson sugges t i ng 4 the p o s s i b i l i t y of s o - c a l l e d AK-±3 t r a n s i t i o n s in C ^ v molecu les ( r l ) . The development of t r a n s i t i o n mechanisms i s a t t r i b u t e d to Watson ( r 2 ) , Fox (r3) and A l i e v (r4) in 1971. The f i r s t exper iment , a mo lecu l a r beam resonance s tudy , was done tha t same year on methane by O z i e r ( r 5 ) . Much of the i n i t i a l expe r imen ta l r e s e a r c h in t h i s f i e l d was c a r r i e d out at the U n i v e r s i t y of B r i t i s h Columbia and at the N a t i o n a l Research C o u n c i l of Canada. The development of d i s t o r t i o n moment spec t ro s copy was rev iewed in 1974 by Oka ( r 6 ) . The l a s t decade has seen a c t i v e expe r imen ta l r e sea r ch at such i n s t i t u t i o n s as Duke U n i v e r s i t y , F y s i s c h Labs at K a t h o l i e k e U n i v e r s i t y H o l l a n d , the Je t P r o p u l s i o n Lab at C a l t e c h , the Russ ian Academy of S c i ences at Gorky , Ha r va rd , and very r e c e n t l y the Swiss F e d e r a l I n s t i t u t e of Technology in Z u r i c h . In t h i s t h e s i s we s h a l l look at three i n d i v i d u a l m o l e c u l e s . The f i r s t s tudy i s on germane, G e H „ . Germane has the same symmetry as methane (Td) f o r which no normal d i p o l e moment e x i s t s due to the t e t r a h e d r a l symmetry. In t h i s s tudy f o r b i d d e n microwave a b s o r p t i o n l i n e s are r epo r t ed in a c o n t i n u a t i o n of e a r l i e r work in t h i s l a b r a t o r y on group IV h y d r i d e s . The second study i s on a r s i n e , s p e c i f i c a l l y A s D 3 , AsD 2 H and AsH 2 D. In the f i r s t case a s e r i e s of AK=±3 f o r b i d d e n t r a n s i t i o n s i s p r e s e n t e d . For the p a r t i a l l y deu te r a t ed forms a l a r g e number of a l l owed "quadrupo le s p l i t " l i n e s in the f requency range 8-303 GHZ i s r e p o r t e d . 5 The t h i r d study i s on phosph ine . T h i s study i s s i m i l a r to the a r s i n e study in tha t we c o n s i d e r P D 3 , PD 2H and PH 2 D. No new t r a n s i t i o n s are r e p o r t e d here fo r P D 3 , i n s t e a d we r e-ana l yze p r e v i ous work in a way c o n s i s t e n t w i th the s t anda rd data r e d u c t i o n t e chn iques rev iewed in t h i s t h e s i s . From ana l y ses of ammonia, phosphine and a r s i n e v i b r a t i o n - r o t a t i o n s p e c t r a , harmonic f o r ce f i e l d s t u d i e s have been p roduced . Here we t r e a t the atoms in a molecu le as i f they were connected to one another by "ha rmon ic " s p r i n g s . We then c o n s i d e r c e n t r i f u g a l d i s t o r t i o n and v i b r a t i o n as d i s p l a c i n g these s p r i n g s from.some e q u i l i b r i u m p o s i t i o n . These s p r i n g s produce a r e s t o r i n g f o r c e o p p o s i t e and p r o p o r t i o n a l to the d i s p l a c e m e n t ; the cons t an t s of p r o p o r t i o n a l i t y we s h a l l l a t e r c a l l f o r ce c o n s t a n t s . For ammonia, a r s i n e and phosphine we have been ab le to es t imate mo lecu l a r atomic p o s i t i o n s by t h i s s imple model . Bes ides the s p e c i f i c mo lecu l a r s t u d i e s the re are some more gene ra l d i s c u s s i o n s p r e sen ted in the t h e s i s that shou ld be of i n t e r e s t to s p e c t r o s c o p i s t s . F i r s t , an attempt has been made to p a i n t a c l e a r p i c t u r e of the d i s t o r t i o n mechanisms working in symmetric mo lecu l a r t o p s . T h i s was deemed necessa ry as many au tho rs use the same l a b e l f o r p h y s i c a l l y d i f f e r e n t q u a n t i t i e s r e s u l t i n g in a c o n f u s i n g l i t e r a t u r e . A l s o of i n t e r e s t shou ld be the q u a s i - t h e o r e t i c a l at tempt to e x p l a i n the i s o t o p i c v a r i a t i o n s of the s p e c t r o s c o p i c parameter D f o r the t e t r a h e d r a l s p h e r i c a l 6 t o p s . T h i s d i s c u s s i o n might prove a s t a r t i n g po in t f o r s i m i l a r e x p l a n a t i o n s in o ther s t u d i e s . T h i r d , the p rocess of anha rmon i c i t y c o r r e c t i o n s in f o r c e f i e l d s t u d i e s i s rev iewed f o r ammonia-l ike m o l e c u l e s . The genera l c o n c l u s i o n s tha t these c o r r e c t i o n s are so f a r u n s a t i s f a c t o r y shou ld be of i n t e r e s t to o ther workers in the f i e l d . F i n a l l y , r e c e n t l y i t has been suggested that r a t he r than the s tandard "power" s e r i e s expans ion method of a c coun t i ng fo r c e n t r i f u g a l d i s t o r t i o n we shou ld c o n s i d e r Pade approx imants . Here we s h a l l review the l i t e r a t u r e on t h i s sub jec t and p o i n t out where these e a r l y a t tempts have gone wrong wh i le at the same t ime g i v i n g a d i r e c t i o n to the d i s c u s s i o n fo r the f u t u r e . 7 B. CHAPTER ONE; THEORY 1. THE RIGID ROTOR A d i s c u s s i o n of the theory of r o t a t i o n a l spec t roscopy u s u a l l y beg ins wi th a c o n s i d e r a t i o n of a r i g i d l y r o t a t i n g m o l e c u l e . From r o t a t i o n a l s p e c t r a one i n f e r s energy l e v e l s which in the case of a r i g i d r o t o r are c o n s i s t e n t wi th the h a m i l t o n i a n where the c a r t e s i a n a x i s system x , y , z r e f e r s to the p r i n c i p a l i n e r t i a l (mo lecu la r ) axes . The J r ep r e sen t s the y opera to r fo r the component of the angu la r momentum about the g i n e r t i a l a x i s in u n i t s of h7. The B „ are c a l l e d " r o t a t i o n a l c o n s t a n t s " and are p r o p o r t i o n a l to the r e c i p r o c a l s of the moments of i n e r t i a about the g axes . By conven t ion the th ree p r i n c i p a l axes are l a b e l l e d a , b , c so tha t the r o t a t i o n a l cons t an t s obey B >B,>B^. The r o t a t i o n a l a D C c o n s t a n t s are o f t e n renamed A , B ,C such that A>B>C. In a b a s i s where J 2 and J z are both d i a g o n a l one observes the f o l l o w i n g non-zero mat r ix e lements of equa t ion H = B J 2 + B J 2 + B J 2 r x x y y z z [1 .1 ] B g = K V 2 h I g [1 .2 ] ( e 1 . 1 ) , [1 .3 ] < J K ± 2 | H r | J K > = ,{(Bx-B ) { [ J ( J + 1 ) - K ( K ± 1 ) ] y X [ J ( J + 1 ) - ( K ± 1 ) ( K ± 2 ) ] } 1 / 2 [1 .4 ] where J and K are quantum numbers r e p r e s e n t i n g the t o t a l angu la r momentum and i t s mo lecu le f i x e d component r e s p e c t i v e l y , J = 0 , 1 , 2 , 3 , . . . . , K = 0 , ± 1 , . . . ± J . The form of the mat r ix now a l lows c l a s s i f i c a t i o n of mo lecu l a r t y p e s . One has 1 . 1 inear B = B = 0, B x y ' z 2. s p h e r i c a l tops B = B = B r ^ x y z 3. p r o l a t e symmetric tops B x = B^  < B z 4. o b l a t e symmetric tops B = B > B 1 ^ x y z 5. asymmetric tops B x , B^ , B z L i n e a r mo lecu l es r e q u i r e s p e c i a l a t t e n t i o n and w i l l not be d i s c u s s e d f u r t h e r here ( r 7 ) . When B x = B^  the o f f d i a g o n a l terms go to ze ro and one f inds Symmetric Tops ( B x = B y , B z ) E ( J ,K) = B J(J+1) + (B - B ) K 2 [1 .5 ] S p h e r i c a l Tops (B x =B y =B z ) E(J) = B J ( J+1 ) [1 .6 ] A . For asymmetric tops the gene ra l form of the h a m i l t o n i a n i s not d i a g o n a l and one i s f o r c e d to so l ve the s e c u l a r e q u a t i o n . In t h i s case a change of b a s i s known as a Wang T r ans fo rma t i on (r8) b l o c k s the H matr ix i n t o four independent t r i - d i a g o n a l s u b m a t r i c e s . For r i g h t handed c o o r d i n a t e systems K i n g , Ha iner and Cross (r9) i d e n t i f y c a r t e s i a n axes x , y , z , w i th p r i n c i p a l 9 axes a , b , c a c c o r d i n g t o ' t h r e e systems. Tab le 1.1 D e f i n i t i o n of r e p r e s e n t a t i o n in terms of c a r t e s i a n and i n e r t i a l axes r r r I II III x b c a y e a b z a b c The I r e p r e s e n t a t i o n i s o f t e n c a l l e d a p r o l a t e b a s i s and the III r e p r e s e n t a t i o n i s o f t en r e f e r r e d to as an ob l a t e b a s i s . In the case of a p r o l a t e asymmetric top the B r o t a t i o n a l cons tan t i s c l o s e r in magnitude to the C cons tan t than i t i s to the A c o n s t a n t . I d e n t i f y i n g B and C w i th B and By r e s p e c t i v e l y in the p r o l a t e b a s i s means tha t the c o e f f i c i e n t of the o f f - d i a g o n a l terms in H f , namely B x - B^, i s as sma l l as p o s s i b l e . A s i m i l a r argument shows tha t the o f f - d i a g o n a l terms in of an o b l a t e asymmetr ic top are min imized when the III or o b l a t e b a s i s i s used ( fo r an o b l a t e asymmetric top B i s c l o s e r to A than C ) . A l l r e p r e s e n t a t i o n s g i ve the same r e s u l t s so c h o i c e of r e p r e s e n t a t i o n i s based on computa t iona l e f f i c i e n c y a l o n e . For an asymmetric top K i s not a good quantum number. The energy l e v e l s are most o f t en l a b e l l e d J K a K c • Here Ka and Kc are the va lues of |K| one would get by " v a r y i n g " B to the p r o l a t e (A>B=C) and o b l a t e (A=B>C) l i m i t s r e s p e c t i v e l y . Whether Ka and Kc are even or odd can. be r e l a t e d to the symmetries of the asymmetric top wave f u n c t i o n s ( r 9 , r l 0 ) . 1 0 2. CENTRIFUGAL DISTORTION The r i g i d r o t o r h a m i l t o n i a n i s an e x c e l l e n t f i r s t approx ima t ion fo r mo lecu les that are r o t a t i n g " s l o w l y " ( i . e . tha t are at low J ) . However a molecu le i s not a r i g i d r o t o r and i t w i l l d i s t o r t because of c e n t r i f u g a l f o r c e s . The e f f e c t i v e mo lecu l a r cons t an t s w i l l now e x h i b i t a J dependence and as such w i l l not have the same r e l a t i o n to mo lecu l a r geometry as the mo lecu l a r cons t an t s of the r i g i d r o t o r . From a dynamica l study of the c e n t r i f u g a l f o r c e s i n v o l v e d in a r o t a t i n g molecu le ( r11 ,pg 105f f ) i t i s r easonab le to attempt to account fo r c e n t r i f u g a l d i s t o r t i o n w i th h ighe r o rder terms in a gene ra l power s e r i e s expans ion of the h a m i l t o n i a n in components of angu la r momentum o p e r a t o r s 1 . T h i s i s the method of Watson ( r 12 ) , who has shown tha t the gene ra l power s e r i e s can be w r i t t e n in the f o l l o w i n g s o - c a l l e d " s t a n d a r d form" by o m i t t i n g a l l terms not i n v a r i a n t to He rm i t i an c o n j u g a t i o n and time r e v e r s a l , H = Zh ( J p j q j r + J r j q j p ) [ 1 . 7 ] r pqr x y z z y x L • ' J where h are mo lecu l a r c o n s t a n t s ; p , q , r a re i n t e g e r s 0 , 1 , 2 , . . . . For a s p e c i f i c mo lecu l a r type the s tandard form h a m i l t o n i a n i s c o n s t r u c t e d by r e t a i n i n g on l y those terms 1 L a t e r in chapte r 7 we s h a l l c h a l l e n g e t h i s assumpt ion by s u g g e s t i n g tha t a r a t i o n a l f r a c t i o n of gene ra l power s e r i e s can b e t t e r account f o r c e n t r i f u g a l d i s t o r t i o n . 11 which t r ans fo rm as the t o t a l l y symmetric spec i e s of the mo lecu l a r p o i n t g roup . Watson has a l s o shown tha t in gene ra l the s tandard form h a m i l t o n i a n c o n t a i n s more terms than can be ob t a i ned by f i t t i n g r o t a t i o n a l data ob ta ined from ( s t a t i c e l e c t r i c ) f i e l d f r e e measurements. For each mo lecu l a r type he has shown how can be reduced to a " reduced f o r m " , c o n t a i n i n g the maximum number of parameters which can be ob ta ined from r o t a t i o n a l d a t a . Each of these i s d i s c u s s e d in the f o l l o w i n g s e c t i o n s . 2 a . C e n t r i f u g a l D i s t o r t i o n in S p h e r i c a l Tops (Td Symmetry) Under Td symmetry the r e t a i n e d powers of J in our s t anda rd form s e r i e s expans ion of the r o t a t i o n a l h a m i l t o n i a n are much more complex in form than fo r any of the o ther mo l e cu l a r t y p e s . Watson (r13) has shown that the symmetry a l l owed terms (up to but not i n c l u d i n g degree 12 in angu la r momentum) form a minimal s e t ; the h a m i l t o n i a n i s comp le te l y r educed . The r o t a t i o n a l h a m i l t o n i a n i s u s u a l l y w r i t t e n as the sum of two te rms; as s o - c a l l e d s c a l a r and tensor p a r t s ( r 14 ) . To an approx imat ion up to 8th degree in angu la r momentum one has ( r l 5 ) H = H r s c a l a r + H tensor [1 .8 ] where H s c a l a r =B 0 J2 -D J "+ H J 6 +L J s s s [1 .9] 12 and H t e n s o r = [ D t + H 4 t j 2 + L 4 t J < ^ + [ H 6 t + L 6 t j 2 ] " « + L 8 t n 8 [ 1 . 1 0 ] The O n are combina t ions of angu la r momentum o p e r a t o r s , J ± , (= J x ± i J y ) , J z and J of even order up to and i n c l u d i n g order n tha t t r ans fo rm in a t o t a l l y symmetric way under o p e r a t i o n s of the Td p o i n t g roup . For example ( r 1 6 , r l 7 ) 0 , = -1/2 (35 J « +25J 2 ) + (15 J 2 + 3 ) J 2 - 3/2 ( J 2 ) 2 - 5/4( J 2 +J 2 ) [1 .11] Formulae f o r J2S and fi8 a re to be found in r e f e r e n c e s ( r l 3 , r 1 5 , r l 8 ) . The mat r ix e lements of our h a m i l t o n i a n are e va l ua t ed in a b a s i s set of Wang f u n c t i o n s symmetr ized under the Td po in t group u s i n g the method of Fox and O z i e r ( r l 9 ) . The e i g e n f u n c t i o n s are l a b e l e d by J C F C where J i s the t o t a l angu l a r momentum, C i s the symmetry s p e c i e s in the Td p o i n t group ( A , , A 2 , E , F ! , F 2 ) and t i s a number to d i s t i n g u i s h d i f f e r e n t e i g e n f u n c t i o n s of the same J and C, t a k i n g on i n t e g e r va lues 1,2,3... in order of i n c r e a s i n g energy i f D f c i s p o s i t i v e . T h i s i s the n o t a t i o n of Dorney and Watson (rl7) and has been r e l a t e d to other schemes by Ho l t et a l (r20) and by Papousek and A l i e v ( r 14 ) . The s c a l a r pa r t of the ham i l t on i an i s independent of the symmetry s p e c i e s of our r o t a t i o n a l s t a t e and a f f e c t s a l l components of a g i ven va lue of J e q u a l l y . The tensor 1 3 h a m i l t o n i a n g i ves r i s e to a c e n t r i f u g a l s p l i t t i n g of the r o t a t i o n a l l e v e l wi th g i ven J i n t o i t s t e t r a h e d r a l s u b l e v e l s each c h a r a c t e r i z e d by symmetry s p e c i e s C and l e v e l l a b e l t . I t i s t r a n s i t i o n s between these s u b l e v e l s of g i ven J ' s tha t were observed in t h i s work. 2 b . C e n t r i f u g a l D i s t o r t i o n in Asymmetric Tops For asymmetric t o p s , we r e t a i n on l y those terms in our s tandard form power s e r i e s expans ion of H r that t r ans fo rm as the t o t a l l y symmetric s p e c i e s in D,,^ ( r l 2 ) H = LB J 2 + 1 L„T p„J2Jl + J 6 r a a a 4 ap aapp a p a a a a a a*/3 aa/3 a /3 /3 a + * ( J 2 J 2 J 2 + J 2 J 2 J 2 ) [1 .12] xyz x y z z y x where the c o e f f i c i e n t s are r e a l and T NO = T„0 aaPP /3/3aa Equa t ion (e1.12) can be w r i t t e n in c y l i n d r i c a l t ensor form in much the same way as s p h e r i c a l tops are expressed in s p h e r i c a l t ensor form ( r 12 ) . The elements of the or thorhombic h a m i l t o n i a n ( e l . 12 ) a l l have the r e q u i r e d symmetry but t h i s h a m i l t o n i a n i s not s u i t a b l e f o r the f i t t i n g of e m p i r i c a l d a t a . Through a u n i t a r y t r a n s f o r m a t i o n , we can e l i m i n a t e terms from equat ion (e1.12) thereby r educ ing the number of e m p i r i c a l pa ramete rs . There i s no obv ious unique r e d u c t i o n and so the cho i c e i s r a t he r a r b i t r a r y . Two of the p r e f e r r e d c h o i c e s are known as the A and S r e d u c t i o n s - A fo r asymmetr ic t o p , S f o r symmetric t o p . 14 The A r e d u c t i o n , f i r s t proposed by Watson in 1968 ( r 2 l ) , i s the most popu la r and perhaps the e a s i e s t to use . The reason i s the r e l a t i v e ease of i n c o r p o r a t i n g i t s fo rma l i sm i n t o e x i s t i n g asymmetric top r i g i d r o t o r d i a g o n a l i z a t i o n r o u t i n e s . In t h i s case the s t r u c t u r e of the h a m i l t o n i a n mat r ix i s the same as tha t of the r i g i d asymmetric r o t o r <JK|H*|JK> = 1/2[B*+B*]J(J+1)+{B*-l/2[B* +B*]}K 2 L JL z x y - A j J M J + D 2 - A J R J (J+1)K 2 - A K K 4 + <i> J 3 ( j + 1 ) 3 + <i> J 2 (J+1 ) 2 K 2 + <I>KJ '(J+1 )K" + $ K K 6 [1 .13] <JK±2|H^| JK> = 1/4(B*-B*)-5 7J(J+1) - l / 2 8 „ [ ( K ± 2 ) 2 + K 2 ] + x y u j\ 0 J J 2 ( J + 1 ) 2 + 1/2<£ J K J(J+1 ) [ (K±2) 2 +K 2 ] + l ^ ^ n ^ ^ + K ' 1 ] [J(J+1) - K ( K ± 1 ) ] [ J ( J + 1 ) - ( K ± 1 ) ( K ± 2 ) ] 1 / 2 [1 .14] As in the case of a r i g i d r o t o r in a Wang b a s i s the A r e d u c t i o n energy mat r ix b l o c k s i n t o four independent t r i - d i a g o n a l s u b m a t r i c e s . A d i sadvantage of the A r e d u c t i o n i s tha t i t i s i l l - c o n d i t i o n e d in the symmetric top l i m i t ( r 2 l ) (see the d i s c u s s i o n l a t e r on " F a i l u r e of the R e d u c t i o n s " in Chapter 7 ) . For mo lecu les tha t are on l y s l i g h t l y asymmetr ic the p r e f e r r e d c h o i c e i s the S r e d u c t i o n ( r 22 ) . In t h i s case non-zero energy mat r ix e lements are found a l o n g , as w e l l as two, four and s i x o f f , the d i a g o n a l . To s i x t h degree the non-zero mat r ix e lements are 15 [1.15] <JK±2|H^| JK> = 1/4[B^-By]+d,J(J+1)+h, J 2 ( J + 1 ) 2 • F ( J , K ) F ( J , K ± 1 ) [1.16] <JK±4|H^| JK> = d 2 + h 2 j ( j + l ) F ( J , K ) F ( J , K ± 1 ) • F ( J , K ± 2 ) F ( J , K ± 3 ) [1.17] <JK±6|H^| JK> = h 3 F ( J , K ) F ( J , K ± 1 ) F J , K ± 2 ) F ( J , K ± 3 ) • F ( J , K ± 4 ) F ( J , K ± 5 ) [1.18] where F ( J , K ± n ) = { J ( J + 1 ) - ( K ± n ) ( K ± n ± 1 ) } . In a Wang b a s i s one gets four independent hep tad i agona l s u b m a t r i c e s . These two r e d u c t i o n s are j u s t two out of an i n f i n i t e s e t . C h a r a c t e r i s t i c of a l l u s e f u l r e d u c t i o n s i s the number of terms of degree in angu la r momentum, three terms of degree two, f i v e of degree four and seven of degree s i x . A l l the r e d u c t i o n s are e q u i v a l e n t but i t i s not s u r p r i s i n g that some are more " a p p r o p r i a t e " than o the r s depending on l i m i t i n g cases and types of observed t r a n s i t i o n s . The parameters of d i f f e r e n t r e d u c t i o n s are a l l r e l a t e d to each o the r through l i n e a r t r a n s f o r m a t i o n s ( r12 ) . 1 6 2 c . C e n t r i f u g a l D i s t o r t i o n in Symmetric Tops For symmetric t o p s , the s tandard form l eaves the f o l l o w i n g terms through s i x t h degree H = B'J 2+(B' - B ' ) J 2 - D; ( J 2 ) 2 - D' J 2 J 2 - D;J U + r x z x z J J K z K z H J < J 2 > 3 + H ^ ( J 2 ) J 2 + H ^ J ' j J + H^JJ + - " s p l i t [ 1 ' 1 9 ] Equa t ion ( e l . 19 ) i s q u i t e gene ra l fo r a l l symmetric t o p s , the s p e c i f i c po in t group dependence be ing c o n t a i n e d in " s p l i t " T ^ e s P- ' -^ t t i n 9 term fo r v a r i o u s p o i n t groups i s g i ven by Watson ( r 12 ) . Us ing the |J,K> b a s i s i n t r o d u c e d fo r r i g i d r o t o r s , f o r C 3 V mo lecu les H < l p i i t through s i x t h degree i n t r o d u c e s non-zero mat r ix e lements of the form <J ,K|H R | J ,K±3> and < J , K | H | J , K ± 6 > . The o f f - d i a g o n a l terms due to H gp]_^ t a r e sma l l in magnitude (order of a s e x t i c c o n t r i b u t i o n ) and are t r e a t e d as p e r t u r b a t i o n s . Two e n t i r e l y e q u i v a l e n t (r23) approximate d i a g o n a l i z a t i o n schemes, s t andard p e r t u r b a t i o n theory and con tac t t r a n s f o r m a t i o n , have been used by O lson (examples: r 2 4 , r 2 5 , r 2 6 ) and Watson ( r12 ) , r e s p e c t i v e l y . U n f o r t u n a t e l y , the. r e s u l t s of Olson have g e n e r a l l y l e d to a host of new r e d u c t i o n s tha t unduly comp l i c a t e the l i t e r a t u r e . (See the l a t e r s e c t i o n e n t i t l e d " F a i l u r e s of the Reduc t i ons " ) F o l l o w i n g the con tac t t r a n s f o r m a t i o n method of Watson one ge ts a f i n a l reduced h a m i l t o n i a n i d e n t i c a l w i th equa t ion ( e l . 1 9 ) in form, but wi th the terms in < J , K | | J , K ± 3 > removed. Pr imes are removed from the parameters in the f i n a l 17 reduced h a m i l t o n i a n . The H S p ] _ ^ t term i s now g iven by " s p l i t = h 3 ^ + J 6 > [1-20] To a very good a p p r o x i m a t i o n , w i th the excep t i on of the K=3 l e v e l s , H g can be ignored and the energy l e v e l s are g i ven by the d i a g o n a l pa r t of an unprimed equa t ion e 1 . 1 9 . H s p l i t r e c e ^ v e s ^ t s name from.what i t does to the J K|= 3 l e v e l s ( a c t u a l l y to the |K|=3n where n=1 ,2 ,3 . . .when h ighe r degrees are c o n s i d e r e d ) . The j K j = 3 l e v e l s a re s p l i t by the h 3 te rm. The K=3 l e v e l s can be t r e a t e d by a p e r t u r b a t i o n e x p r e s s i o n f o r the h 3 c o n t r i b u t i o n (r27) AE(K=±3) =2h 3 J ( J+1)[ J ( J+1)-2][ J ( J+1)-6] [1 .21] or more g e n e r a l l y , as w i l l be the case in t h i s s tudy , by s o l u t i o n of the s e c u l a r e q u a t i o n . The energy mat r i x e lements are i d e n t i c a l to those of the asymmetr ic top S r e d u c t i o n wi th d !=d 2 =h,=h 2 =0. The problem of the f a i l u r e of the symmetric top r e d u c t i o n in the s p h e r i c a l top l i m i t a long w i th a more in-depth look at the mechanics of the r e d u c t i o n i s to be found l a t e r in a s e c t i o n e n t i t l e d " F a i l u r e s of the Reduct i o n s " . 18 3. HYPERFINE PERTURBATIONS TO THE ENERGY LEVELS In t h i s s e c t i o n we s h a l l look at p e r t u r b a t i o n s to the r o t a t i o n a l energy l e v e l s of the l a s t s e c t i o n . The p e r t u r b a t i o n s tha t tu rn out to be important in the s t u d i e s p resen ted here are due to nuc l ea r quadrupole e f f e c t s and sp in r o t a t i o n e f f e c t s . These e f f e c t s w i l l be c o n s i d e r e d in the low i n c i d e n t e l e c t r i c f i e l d l i m i t o n l y . The s p e c i f i c a p p l i c a t i o n of these methods w i l l be to a r s i n e (AsH 3 ) where i t w i l l be found that we need on l y c o n s i d e r the nuc l ea r h y p e r f i n e e f f e c t s of the a r s e n i c nuc leus and can ignore the sp ins of the hydrogens (or d e u t e r i u m s ) . 3a .Nuc l ea r Quadrupole Moments A nuc leus w i th an i n t r i n s i c asymmetry possesses an e l e c t r i c quadrupo le moment that w i l l t r y to a l i g n with an i n t r a m o l e c u l a r e l e c t r i c f i e l d g r a d i e n t . For an a x i a l l y symmetric nuc l eus w i th un i fo rm charge d e n s i t y a measure of the d e v i a t i o n of the nuc l ea r charge d i s t r i b u t i o n from s p h e r i c a l symmetry i s the i n t r i n s i c quadrupole moment. Depending on whether the net charge .dens i ty i s g rea te r or l e s s a long the nuc l ea r symmetry a x i s t h i s moment i s p o s i t i v e ( p r o l a t e ) or nega t i ve (ob la te ) r e s p e c t i v e l y . I f the nuc l ea r sp in I i s g r e a t e r than or equa l to one the quadrupole moment i s "measurab le " and the e x p e r i m e n t a l l y de te rminab le quadrupo le moment Q i s r e l a t e d to the i n t r i n s i c moment by a s imple f u n c t i o n of I tha t r e f l e c t s the ave rag ing of the charge d i s t r i b u t e d by the necessa ry motion of I ( r 28 ) , about the f i e l d g r a d i e n t symmetry a x i s . 19 If our nuc leus i s in a r o t a t i n g mo l e cu l e , in a weak e x t e r n a l f i e l d , we can get a c o u p l i n g of / and / to form a r e s u l t a n t F F=I+J [1.22] 1 /2 wi th e i genva l ues [F(F+1)] ' where F i s r e s t r i c t e d to the va lues J + I , J + I - 1 | J - I | . The i n t e r a c t i o n energy of our quadrupo le n u c l e i in the f i e l d g r a d i e n t of the molecu le i s g iven as the s c a l a r p roduc t of the quadrupole and f i e l d g r ad i en t t e n s o r s ( r29 ) . S ince the pr imary c o n t r i b u t i o n to the e l e c t r i c f i e l d g r a d i e n t w i l l be due to the e l e c t r o n s neares t the n u c l e u s , the s p l i t t i n g s u l t i m a t e l y produced by our i n t e r a c t i o n g ive us some idea as to how the e l e c t r o n s a r e . d i s t r i b u t e d in bond ing . A u s e f u l model to have in mind d u r i n g d i s c u s s i o n s of quadrupo le c o u p l i n g i s the s e m i - c l a s s i c a l " v e c t o r " model . In t h i s p i c t u r e , s i n ce the e l e c t r o n o r b i t a l s a re f i x e d in the m o l e c u l e , r o t a t i o n of the molecu le w i l l r e s u l t in an ave rag ing of the f i e l d g r a d i e n t about the r o t a t i o n a x i s . Rap id r o t a t i o n averages the components of the f i e l d g r ad i en t p e r p e n d i c u l a r to the r o t a t i o n a x i s to zero and makes the r o t a t i o n a x i s a symmetry a x i s f o r the f i e l d g r a d i e n t e x p e r i e n c e d by the quadrupo la r n u c l e i . To a v o i d t ime dependent f i e l d g r a d i e n t s the f i e l d g r ad i en t t ensor i s r o t a t e d from the nuc l ea r f i x e d frame to the p r i n c i p a l i n e r t i a l axes in the molecu le f i x e d frame. The p r i n c i p a l 20 i n e r t i a l axes are the axes a long which the angu la r momentum vec to r / i s decomposed. The r o t a t i o n a l angu la r momentum vec to r / , t hen , p r e ces ses about the t o t a l angu la r momentum a x i s d e f i n e d by F. In a f i r s t order theory the p r e c e s s i o n of J about F i s assumed slow and the f i e l d g r a d i e n t i s averaged about the / a x i s . I t i s an e x c e l l e n t approx imat ion fo r moderate quadrupo le moments in the case of no r o t a t i o n a l near degenerac i es to use the f i r s t o rder c o r r e c t i o n to the r o t a t i o n a l energy l e v e l s f i r s t ob ta ined by Cas im i r ( r 30 ) . (3/4)C(C+1 )-I (1 + 1 )J(J+1 )" Wg = eQ 2J12J-1) I (21-1) \VF/J H . 2 3 J where C = F(F+1)-I (1+1)-J ( J+1), e i s the charge on the e l e c t r o n , <j> i s the p o t e n t i a l f i e l d of the e x t r a - n u c l e a r cha rges , and 2 i s the nuc l ea r symmetry a x i s . R o t a t i n g from the nuc l ea r f i x e d frame to the mo lecu le f i x e d frame y i e l d s f o r l i n e a r m o l e c u l e s , /3 2«A _ _ J /d 2<j>\ r , \WTZ/J ~ 2J+3 YaFV u . ^ 4 j symmetric tops (nuc leus on a x i s ) ,K 2 JTJ+TT 1 /a2<A = J \J¥7/J 2J + 3 and fo r asymmetric tops X B i 7 / [ 1 - 2 5 ] / 9 2 0 \ _ 2J v / 9 2 A <J 2 > [1.26] \JIZR/J ~ (2J+3) (2J+1 ) g Xag 7/ ug where the <J 2> are c a l c u l a t e d as in Gordy and Cook ( r 3 l ) . 9 21 When h ighe r order e f f e c t s are impor tan t , i t i s o f t e n necessa ry to employ an exact s o l u t i o n to the p rob lem. In t h i s case the vec to r model i n t e r p r e t a t i o n i s that the p r e c e s s i o n of / about F cannot be i gnored and the f i e l d g r a d i e n t i s averaged about the F a x i s . The a p p r o p r i a t e mat r ix e lements are g iven in r e f e r e n c e ( r 32 ) . The s e l e c t i o n r u l e s fo r t r a n s i t i o n s are AF = 0 , ± 1 , AI=0. [1 .27] The t r a n s i t i o n s where AJ=AF are found to be the s t r o n g e s t . The quadrupo le s p l i t t i n g con f i rms the r o t a t i o n a l a s s ignments . The e x p e r i m e n t a l l y de te rm inab le parameters are the c o u p l i n g cons t an t s i y where i and j r e f e r to the p r i n c i p a l i n e r t i a l , g , a x i s . The t r a c e of the c o u p l i n g tensor i s c o n s t r a i n e d to zero by L a p l a c e ' s e q u a t i o n . 3 b . S p i n - r o t a t i o n Coup l i ng (/•/) The nuc l ea r magnetic moment of a nuc leus wi th s p i n I can i n t e r a c t w i th the e f f e c t i v e magnet ic f i e l d produced at the nuc l eus by mo lecu la r r o t a t i o n to g i ve f u r t h e r c o n t r i b u t i o n s to the s p l i t t i n g of r o t a t i o n a l l e v e l s . D e f i n i n g magnetic c o u p l i n g c o n s t a n t s C^g wi th the s i gn conven t i on of Gordy and Cook (r31) one can wr i t e the f i r s t 22 order c o r r e c t i o n to the r o t a t i o n a l energy l e v e l s of asymmetric tops as . F(F+1)-I(I+1)-J(J+1) m 2J(J+1) g gg g In a symmetric top two of the c o u p l i n g c o n s t a n t s above are equa l because of a x i a l symmetry. Choos ing C =C„ and Z Z K C =C =CXT y i e l d s the f o l l o w i n g xx yy N J 3 m 2 K 2 C +(C-C V T )-'N K U N ; J ( J+1 ) [F (F+1)- I (1+1)-j ( j+1) ] [1.30] 4. EFFECTS OF EXTERNAL ELECTRIC FIELDS In the presence of an a p p l i e d e x t e r n a l e l e c t r i c f i e l d a new term must be added to our r o t a t i o n a l energy l e v e l h a m i l t o n i a n H E = -M-E [1 .31] where M i s the d i p o l e moment ope ra to r and E i s the e x t e r n a l e l e c t r i c f i e l d v e c t o r . D i a g o n a l i z a t i o n of t h i s f i n a l energy mat r ix r e s u l t s in e i g e n f u n c t i o n s that are l i n e a r combina t ions (mix ings) of the o r i g i n a l r o t o r wave func t i ons . For s t a t i c e l e c t r i c f i e l d s t h i s e f f e c t i s known as the S tark s h i f t . For t y p i c a l l a b o r a t o r y e l e c t r i c f i e l d s of 2000 v o l t s / c m . one sees s h i f t s of energy l e v e l d i f f e r e n c e s on the o rder of tens to hundreds of megahertz . For these f i e l d s the mix ing i s g e n e r a l l y so s l i g h t that the ze ro f i e l d quantum numbers J , K , are r e t a i n e d and one t a l k s of the S tark e f f e c t of a c e r t a i n l e v e l . 23 For o s c i l l a t i n g e x t e r n a l f i e l d s r o t a t i o n a l l e v e l s are coup l ed in a time dependent way to g i ve a t r a n s i t i o n s t a t e ( r 33 ) . 4 a . D i p o l e Moment ( R o t a t i o n a l and V i b r a t i o n a l C o n t r i b u t i o n s ) Rotat i o n a l The e l e c t r i c f i e l d i s coup led to our r o t a t i n g molecu le through the d i p o l e moment o p e r a t o r . C l a s s i c a l l y one p i c t u r e s a d i p o l e moment as due to a l i n e a r s e p a r a t i o n of o p p o s i t e c h a r g e s . One shou ld expect tha t a molecu le in d i f f e r e n t v i b r a t i o n a l and r o t a t i o n a l s t a t e s shou ld have d i f f e r e n t d i p o l e moments r e f l e c t i n g the ave rag ing of the charge d i s t r i b u t i o n over the d i f f e r i n g s t a t e s . The example of a C ^ v symmetric top w i l l perhaps i l l u s t r a t e the p o i n t . The d i p o l e moments a re l a b e l e d by the s o r t of mat r ix e lements they are a s s o c i a t e d w i t h . For an a l lowed R branch (AJ=1,AK=0) t r a n s i t i o n the " e f f e c t i v e " d i p o l e moment p a r a l l e l to the symmetry a x i s of the m o l e c u l e , i s g i ven by (r34) u r ( J , K ) = Mo + MJ(J+1) 2 + M R K 2 . . . [1 .32] where MO i s the e q u i l i b r i u m d i p o l e moment and M j and M k the c o r r e c t i o n s due to c e n t r i f u g a l d i s t o r t i o n . The d i s t o r t i o n c o r r e c t i o n s a re t y p i c a l l y four to s i x o rde r s of magnitude sma l l e r than Mo ( r 35 ) . A s i m i l a r e x p r e s s i o n e x i s t s f o r a l l owed Q branches ( r 35 ) . 24 R e c a l l tha t for C ^ v symmetric tops the l a b e l i n g of r o t a t i o n a l energy l e v e l s by J and K i s not q u i t e c o r r e c t . E l i m i n a t i o n of the <JK |H |JK±3> terms to a r r i v e at the reduced h a m i l t o n i a n meant tha t r o t a t i o n a l s t a t e s had to be d e s c r i b e d as l i n e a r combina t ions of | J K > s t a t e s . T h i s m i x i n g , a l t hough i n s i g n i f i c a n t in the above a l l owed t r a n s i t i o n s i s r e s p o n s i b l e fo r the s o - c a l l e d d i s t o r t i o n d i p o l e moment tha t d r i v e s the f o rb idden AK=±3 t r a n s i t i o n s . The magnitude of the mix ing i s set by the q u a r t i c d i s t o r t i o n cons tan t i n t r o d u c e d by the above mat r ix e lement , r , 1.. x x x z ' ( r 2 , r 1 2 , r 1 4 ) and the mechanics of the u n i t a r y t r a n s f o r m a t i o n g i v i n g the reduced h a m i l t o n i a n . S ince each s t a t e <J,K| has a sma l l admixture <JK±3| and each s t a t e |J,K±3> has a sma l l admixture of |J,K> we get a non-zero va lue fo r the Q branch mat r ix element T V Y V 7 M ° <JK|M1JK±3> = 2 ( B -B ) [1 .33] X z where the above equa t ion co r responds to our conven t ion of r e t a i n i n g the J K l a b e l i n g , because the mix ing i s sma l l ( i . e . f i r s t o rder p e r t u r b a t i o n ) . O v e r l o o k i n g the mix ing of s t a t e s i m p l i c i t in equa t ion ( e l . 33 ) above, t r a n s i t i o n s d r i v e n by t h i s d i p o l e moment would appear to change the component of the angu la r momentum a long the symmetry a x i s . C l a s s i c a l l y a change in r o t a t i o n speed about the symmetry a x i s would r e q u i r e a d i p o l e moment p e r p e n d i c u l a r to the a x i s . For t h i s reason the d i s t o r t i o n d i p o l e moment r e s p o n s i b l e fo r AK=±3 t r a n s i t i o n s i s o f t en c a l l e d a p e r p e n d i c u l a r moment. 2 5 L a b e l i n g our symmetric top r o t a t i o n a l energy l e v e l s by J and K i s then j u s t an a p p r o x i m a t i o n . I t i s the breakdown of t h i s approx imat ion and subsequent d e s c r i p t i o n of s t a t e s as s u p e r p o s i t i o n s of |JK> s t a t e s tha t e x p l a i n s our " f o r b i d d e n " t r a n s i t i o n s . V i b r a t i o n a l In the l a s t s e c t i o n we saw the breakdown of the r i g i d r o t o r J , K d e s c r i p t i o n of r o t a t i o n a l s t a t e s . C e n t r i f u g a l d i s t o r t i o n coup led s t a t e s J , K w i th s t a t e s J , K ± 3 . T r a n s i t i o n s t a t e s where the e x t e r n a l o s c i l l a t i n g e l e c t r i c f i e l d was coup l ed to a r o t a t i o n a l s t a t e v i a the d i p o l e moment ope ra to r were i n t r o d u c e d . A p e r p e n d i c u l a r d i s t o r t i o n d i p o l e moment was i d e n t i f i e d g i v i n g AK=±3 t r a n s i t i o n s . In t h i s s e c t i o n we c o n s i d e r the e f f e c t of r o t a t i 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 on the 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 c l u d i n g c e n t r i f u g a l d i s t o r t i o n a l l ows new terms in the 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 that can coup le the harmonic o s c i l l a t o r v i b r a t i o n s t a t e s ( r 14 ) . In t h i s case a more a c cu ra t e d e s c r i p t i o n of a v i b r a t i o n a l s t a t e i s as a L i nea r combina t ion of harmonic o s c i l l a t o r s t a t e s . T h i s means the ground v i b r a t i o n a l s t a t e shou ld i n c l ude sma l l admixtures of " h i g h e r " harmonic o s c i l l a t o r s t a t e s u s u a l l y a t t r i b u t e d to h ighe r v i b r a t i o n a l s t a t e s . Dur ing t r a n s i t i o n the r e s u l t of t h i s v i b r a t i o n a l c o u p l i n g in ammonia-l ike mo lecu les i s to aga in a l l ow ground r o t a t i o n a l s t a t e t r a n s i t i o n s between s t a t e s J , K and J , K ± 3 . Here the d i p o l e moment ope ra to r i s expanded in a T a y l o r s e r i e s about the e q u i l i b r i u m 26 c o n f i g u r a t i o n as a f u n c t i o n of the normal c o o r d i n a t e s £K , M = M e + I i ( 3 M 6 / 3 Q i ) Q i [1 .34] The f i r s t term i s r e s p o n s i b l e f o r the r o t a t i o n a l t r a n s i t i o n s and e s p e c i a l l y the f o r b i d d e n r o t a t i o n a l t r a n s i t i o n s when jue i s e va l ua t ed in the b a s i s of "m ixed " r i g i d r o t o r b a s i s states— a mix ing due to the T x x x z > three o f f the 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 terms d i s c u s s e d in the l a s t s e c t i o n . The second term i s the s t anda rd t r a n s i t i o n ope ra to r connec t i ng v i b r a t i o n a l s t a t e s . The o v e r a l l d i p o l e mat r ix element w i l l then have a sma l l c o n t r i b u t i o n from the non-zero e lements of t h i s term because of the s l i g h t " admix tu re " of h ighe r harmonic o s c i l l a t o r s t a t e s i n t o the ground v i b r a t i o n a l s t a t e . In t h i s case the AK=±3 t r a n s i t i o n s are a l lowed by a 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 produced by c e n t r i f u g a l d i s t o r t i o n . We have seen then tha t c e n t r i f u g a l d i s t o r t i o n induced by r o t a t i o n causes a breakdown of both a r i g i d r o t o r and harmonic o s c i l l a t o r d e s c r i p t i o n of r o t a t i o n and v i b r a t i o n and that a f i n a l d e s c r i p t i o n of " r o - v i b r o n i c " s t a t e s i s as a l i n e a r combina t ion of r i g i d ro tor-harmonic o s c i l l a t o r b a s i s s t a t e s . The harmonic o s c i l l a t o r d e s c r i p t i o n of v i b r a t i o n a l s o breaks down when anha rmon i c i t y i s taken i n t o a c coun t . T h i s e f f e c t r e s u l t s in v i b r a t i o n a l s t a t e s as l i n e a r combina t ions of harmonic o s c i l l a t o r s t a t e s and aga in i s a l s o r e s p o n s i b l e fo r f o r b i d d e n t r a n s i t i o n s . These t r a n s i t i o n s are o f t e n 27 c a l l e d M.V. t r a n s i t i o n s a f t e r the au tho rs Mizushima and Venkateswar lu (r36) who f i r s t proposed them. The e f f e c t of anha rmon i c i t y i s to a l l ow p r e v i o u s l y f o rb idden r o t a t i o n a l t r a n s i t i o n s in degenerate v i b r a t i o n a l s t a t e s (r14 p214) . For the purposes of t h i s t h e s i s we s h a l l on l y be concerned wi th v i b r a t i o n a l mix ing caused by c e n t r i f u g a l d i s t o r t i o n . For a d i s c u s s i o n of o ther mechanisms l e a d i n g to o ther f o rb idden t r a n s i t i o n s see the e x c e l l e n t book by Papousek and A l i e v ( r 14 ) . For ammonia the v i b r a t i o n a l s t a t e s are of two symmetr ies . There are the t o t a l l y symmetric p a r a l l e l v i b r a t i o n s c o r r e s p o n d i n g to the symmetric bond s t r e t c h cu, and symmetric ang le bend co2r p l u s the an t i s ymmet r i c p e r p e n d i c u l a r doubly degenerate v i b r a t i o n s c o r r e s p o n d i n g to the an t i symmet r i c bond s t r e t c h o>3 and ang le bend CJ^ ( f o r p i c t o r i a l r e p r e s e n t a t i o n see (r37 p l O O ) ) . M ix ing the p a r a l l e l v i b r a t i o n s i n t o our ground v i b r a t i o n a l s t a t e w i l l not change the mo lecu l a r symmetry. The net e f f e c t of mix ing i n t o the ground v i b r a t i o n a l s t a t e sma l l amounts of p a r a l l e l v i b r a t i o n s w i l l be to a l t e r the a long a x i s 'permanent ' d i p o l e moment r e s p o n s i b l e f o r the a l l owed t r a n s i t i o n s . If we mix in sma l l amounts of an e x c i t e d doubly degenerate p e r p e n d i c u l a r v i b r a t i o n , however, the e q u i l i b r i u m symmetry w i l l be d e s t r o y e d . D e s t r o y i n g the e q u i l i b r i u m symmetry means the r i g i d r o t o r , harmonic o s c i l l a t o r d e s c r i p t i o n of our molecu le i s no longer adequate . T h i s i s of course what we are say ing when we wr i t e 28 the r o - v i b r o n i c energy l e v e l s as l i n e a r combina t ions of the r i g i d r o t o r harmonic o s c i l l a t o r b a s i s s t a t e s . In 1969 Watson ( r 2 , r6 ) gave the f o l l o w i n g c o n t r i b u t i o n to the d i s t o r t i o n d i p o l e moment in the ground v i b r a t i o n a l s t a t e , due to c e n t r i f u g a l l y induced v i b r a t i o n a l m i x i n g , 0 X X = 2B 2 x • xx 9M xx du 1 3 3 x _^  a ii x u'j 9Q 3 w« 9Q« [1 .35] where the sma l l a ' s are d e r i v a t i v e s at e q u i l i b r i u m of moments of i n e r t i a I1-' w i th r espec t to normal c o o r d i n a t e s Q T , ; a * - ' = 9 I 1 - ' /9Q„ . The 9 M / 9 Q ' S a re the v i b r a t i o n ana logues of the r i g i d r o t o r d i p o l e moment ope ra to r in r o t a t i o n a l t r a n s i t i o n s ; the e l e c t r i c f i e l d i s coup l ed to our v i b r a t i n g molecu le through the ope ra to r 9 M / 9 Q . Here we have the change in the p e r p e n d i c u l a r d i p o l e moment genera ted by a change i n the normal c o o r d i n a t e s of the p e r p e n d i c u l a r v i b r a t i o n s C J 3 and c j i , . The net d i s t o r t i o n d i p o l e moment fo r a C ^ v symmetric top i s the sum of the v i b r a t i o n a l and r o t a t i o n a l c o n t r i b u t i o n s , = 0 X X + D x T xxxz 2(B -B ) x z Mo [1 .36] where h i g h e r order c o r r e c t i o n s to Mo have been i g n o r e d . For s p h e r i c a l tops the r o t a t i o n a l mix ing i s ze ro because jue i s ze ro in equa t ion e1.34 and M d i s s o l e l y due to v i b r a t i o n a l " b o r r o w i n g " . 29 4 b . I n t e n s i t i e s and S e l e c t i o n Rules The parameter of i n t e r e s t f o r microwave a b s o r p t i o n i n t e n s i t i e s 2 i s the maximum or peak a b s o r p t i o n c o e f f i c i e n t which fo r a t r a n s i t i o n between an i n i t i a l l e v e l i w i th energy E i to a f i n a l l e v e l f wi th energy Ef i s ( r l 7 , r 3 l ) . „ r -E i/KT -Ef/KT-, r. /v 0 S . i f [ e -e J [1 .37] where v0 i s the t r a n s i t i o n f r equency , Q i s the product of the r o t a t i o n a l and v i b r a t i o n a l p a r t i t i o n f u n c t i o n s , K i s Bo l t zmann ' s cons tan t and c i s the speed of l i g h t . The t r a n s i t i o n f u l l - w i d t h at h a l f maximum Av (FWHM) d i v i d e d by the p r e s su re P i s a l s o known as the p r e s s u r e broaden ing parameter . The s e l e c t i o n r u l e s are c o n t a i n e d in the r o t a t i o n a l l i n e s t r eng ths S ^ . The l i n e s t r e n g t h s are d e r i v e d from the d i p o l e moment mat r ix e lements that coup le the e x t e r n a l o s c i l l a t i n g e l e c t r i c f i e l d to the r o t a t i n g m o l e c u l e , S i f = Z | <i | jug | f > | 2 [1 .38] where the summation i s over the th ree g d i r e c t i o n s and a l l components a of i and 0 of f . The M are the d i p o l e moment y o p e r a t o r s a long the g axes . 2 S t r i c t l y at low i n c i d e n t powers where l i n e s are on l y broadened by p ressu re e f f e c t s and e f f e c t s due to s a t u r a t i o n and power broaden ing can be i g n o r e d . max 8TT2P 3CQKTAIV 30 In o rder tha t the l i n e s t r e n g t h be non-zero one has the gene r a l t r a n s i t i o n s e l e c t i o n r u l e fo r the t o t a l angu la r momentum AJ = -1,0,1 [ 1 .39] g i v i n g s o - c a l l e d P, Q and R type t r a n s i t i o n s r e s p e c t i v e l y (when h y p e r f i n e s t r u c t u r e i s important one has AF = 0 , ± 1 ) . F u r t h e r s e l e c t i o n r u l e s depend on s p e c i f i c mo l e cu l a r t y p e . S p h e r i c a l Tops (Td) In o rder that the l i n e s t r e n g t h be non-zero one has the symmetry s e l e c t i o n r u l e s A ,—A 2 , E—E and F ,-F 2 a l ong wi th the angu la r momentum s e l e c t i o n r u l e AJ = 0, ± 1 . For a gene ra l Td .molecule w i th un i t d i p o l e Dorney and Watson (r17) have e va l ua t ed and t a b u l a t e d s o - c a l l e d reduced l i n e s t r e n g t h s . The a c t u a l l i n e s t r e n g t h s are e va l ua t ed by m u l t i p l y i n g these v a l ues by the nuc l ea r s p i n degeneracy f a c t o r s 5, 2 and 3 fo r A, E and F t r a n s i t i o n s r e s p e c t i v e l y . For s p e c i f i c examples one f u r t h e r m u l t i p l i e s these va lues by the square of the mo lecu l a r d i s t o r t i o n d i p o l e moment M D « Asymmetric Tops A long wi th the gene r a l s e l e c t i o n r u l e s AJ = 0,±1 C r o s s , Ha iner and K ing (r38) have shown tha t fo r non-zero components of the d i p o l e moment a long i n e r t i a l axes a , b, c the f o l l o w i n g symmetry r e s t r i c t i o n s app ly fo r non-zero l i n e s t r e n g t h s . 31 Tab le 1.2 Symmetry s e l e c t i o n r u l e s fo r asymmetric top t r a n s i t i o n s A l lowed Axes p a r a l l e l to T r a n s i t i o n s D i p o l e Moment KaKc<—>KaKc ee<—>eo a oo<—>oe a ee<—>oo b eo<—>oe b ee<—>oe c oo<—>eo c where o, e r e f e r to odd or even va lues of Ka and Kc . Asymmetric top l i n e s t r e n g t h s cannot in gene ra l be w r i t t e n in c l o s e d form. They are ob t a i ned by molecu le dependent l i n e a r combina t ion of the c l o s e d form symmetric top l i n e s t r e n g t h s ( r 38 ) . Symmetric Tops (Forb idden T r a n s i t i o n s ) The s e l e c t i o n r u l e s fo r C 3 V symmetric tops are AJ = 0,±1 and AK = 0,±3n (n=1,2,3 ). The AK s e l e c t i o n r u l e s can be seen p i c t o r i a l l y as due to the th ree e q u i v a l e n t atoms in a l l C^y mo lecu les ( r 6 ) . With a r i g i d r o t o r the AK = ± 3 , ± 6 , . . . t r a n s i t i o n s are s t r i c t l y f o r b i d d e n so tha t s p e c t r a w i th t r a n s i t i o n s of t h i s so r t are o f t e n c a l l e d f o rb idden or d i s t o r t i o n moment s p e c t r a . The l i n e s t r e n g t h s of the Q branch AK=±3 t r a n s i t i o n s are g i ven by Watson (r2) a s , S i f = T^D H J + K ) ( j + K - 1 ) ( J + K - 2 ) ( J ± K + 1 ) ( j ± K + 2 ) ( j ± K + 3 ) } 2J+1 4J(J+1) [1 .40] where the f a c t o r g R takes i n t o account the K degeneracy and 32 the nuc l ea r sp in s t a t i s t i c a l we igh t s . Equa t ion (e1.40) a l s o d e f i n e s the v a r i o u s numer i ca l f a c t o r s in the d e f i n i t i o n of , the d i s t o r t i o n d i p o l e moment ( e l . 3 6 ) . T h i s i s important because in e a r l i e r s t u d i e s (r39) the d i s t o r t i o n d i p o l e moment was d e f i n e d as one-ha l f the va lue g i ven by equa t ion ( e1 .36 ) . In t h i s case the l i n e s t r e n g t h was g i ven as four t imes the va lue g iven by equa t ion ( e1 .40 ) . 4 c . S t a r k E f f e c t In the presence of an a p p l i e d s t a t i c e x t e r n a l e l e c t r i c f i e l d a d e s c r i p t i o n of our system r e q u i r e s the i n t r o d u c t i o n of a new quantum number M, where M , the component of / a l ong the s p a c e - f i x e d a x i s as d e f i n e d by the e x t e r n a l f i e l d ,can take on i n t e g e r va lues rang ing from -J to J . The s e l e c t i o n r u l e s fo r t r a n s i t i o n are AM = 0, and ±1, c o r r e s p o n d i n g to the o s c i l l a t i n g e l e c t r i c f i e l d be ing r e s p e c t i v e l y p a r a l l e l t o , or p e r p e n d i c u l a r t o , the space f i x e d q u a n t i z a t i o n a x i s Z. In gene r a l f o r our expe r imen ta l s i t u a t i o n the s t a t i c and microwave e l e c t r i c f i e l d s have e s s e n t i a l l y the same d i r e c t i o n and hence the AM = 0 t r a n s i t i o n s dominate . For the sma l l S tark f i e l d s used the energy change i n the r o t a t i o n a l l e v e l s i s u s u a l l y expanded as a p e r t u r b a t i o n s e r i e s in powers of the e l e c t r i c f i e l d E. A f i r s t o rder S tark e f f e c t depends l i n e a r l y on the e l e c t r i c f i e l d and a second order e f f e c t depends on the square of the e l e c t r i c f i e l d et c e t e r a . Watson (r40) has g iven the symmetry requ i rements f o r f i r s t o rder S ta rk e f f e c t s in the case of n e g l i g i b l e 33 h y p e r f i n e e f f e c t s and no a c c i d e n t a l d e g e n e r a c i e s . L i n e a r S ta rk e f f e c t s are e s p e c i a l l y impor tant fo r s t u d y i n g mo l e cu l e s wi th " s m a l l " d i p o l e moments as one has a b e t t e r chance at see ing f requency s h i f t s due to S tark s h i f t e d l e v e l s w i th the e l e c t r i c f i e l d s a v a i l a b l e in most l a b o r a t o r i e s . Quad ra t i c S tark e f f e c t s a l though they depend on E 2 a re second order e f f e c t s and fo r the same f i e l d g e n e r a l l y produce much sma l l e r s h i f t s (the e x c e p t i o n i s when we have near degenerate energy l e v e l s , see ( r 3 l ) ) . For s p h e r i c a l tops on ly the doub ly degenera te E r o v i b r o n i c l e v e l s have the r e q u i r e d symmetry to show l i n e a r S ta rk e f f e c t s . For the E l e v e l s the Stark energy s h i f t s are g i v en by . 6 J m e ( t ' , ' t , ) = A J e ( t " ' t , ) M " D E [ 1 ' 4 1 ] w i t h 3 A., ( t " , f ) = 0.50344{C( J , e , f ) -C ( J , e , t " ) } [1 .42] where the C ( J , e , t ) a re the s i gned S ta rk c o e f f i c i e n t s as c a l c u l a t e d and t a b u l a t e d by Dorney and Watson ( r l 7 ) . For asymmetr ic tops where the r o t a t i o n a l l e v e l s are not a c c i d e n t a l l y c l o s e one sees s t r i c t l y second o rde r s h i f t s E <2> = (A "+B"M 2 ) E 2 [1 .43] 3 6 T ( t " , t ' ) i s in MHz fo r M n in Debye and E i n V o l t s / c m . 34 where A" and B" depend in an e l a b o r a t e way on J , M, Ka, Kc , d i p o l e mat r ix e lements and energy l e v e l d i f f e r e n c e s ( r 4 l ) . For symmetric tops the f i r s t o rder c o r r e c t i o n i s ( r 1 1 , r 31 ) E ( 1 > = - MKM r1 441 s J(J+1) fc 5. FORCE FIELDS One of the more a e s t h e t i c s i m p l i f i c a t i o n s in the unders tand ing of mo lecu l a r v i b r a t i o n - r o t a t i o n s p e c t r a has to be that of the f o r c e f i e l d . A l l the independent i s o t o p i c d e r i v a t i v e s of a molecu le can now be t r e a t e d as one prob lem. G e n e r a l l y i t i s q u i t e d i f f i c u l t to e s t a b l i s h q u a l i t a t i v e t r ends between the e m p i r i c a l f i t t i n g parameters of d i f f e r e n t i s o t o p i c d e r i v a t i v e s due to changes in symmetry, r o t a t i o n of p r i n c i p a l axes , and the somewhat a r b i t r a r y c h o i c e of r e d u c t i o n in the case of asymmetric t o p s . On the o ther hand a f o r c e f i e l d t reatment a l l ows a q u a n t i t a t i v e p i c t u r e to be drawn tha t accounts f u l l y f o r the v i b r a t i o n and r o t a t i o n s p e c t r a of a l l the i s o t o p i c s p e c i e s in terms of a much sma l l e r set of pa ramete rs . There i s an e x t e n s i v e l i t e r a t u r e on the s u b j e c t , much of which i s r e f e r e n c e d in the e x c e l l e n t rev iew of Duncan ( r 42 ) . Ammonia-l ike mo lecu les are e s p e c i a l l y we l l r ep resen ted in the l i t e r a t u r e o f t e n as textbook examples (see Woodward ( r 4 3 ) ) . For p r a c t i c a l problems one i s d i r e c t e d to the d e t a i l e d accounts in C y v i n ' s books ( r 4 4 , r 4 5 ) . 35 For our purposes i t s u f f i c e s to e s t a b l i s h the harmonic approx imat ion to the f o r c e f i e l d . In t h i s case the r e s t o r i n g f o r c e fo r i n t e r a tom i c motion i s l i n e a r in the d i sp lacement c o o r d i n a t e wi th a f o r c e cons tan t as the cons tan t of p r o p o r t i o n a l i t y . The v i b r a t i o n s of the mo lecu le in the q u a d r a t i c p o t e n t i a l f i e l d s are r ep resen ted as s imple l i n e a r s u p e r p o s i t i o n s of fundamental v i b r a t i o n s . For ammonia-l ike mo lecu les the re are on l y s i x d i s t i n c t q u a d r a t i c f i e l d f o r c e c o n s t a n t s . In a r e p r e s e n t a t i o n known as a va l ency f o r c e f i e l d these are f (the f o r c e cons tan t fo r the s t r e t c h i n g of a bond) , f^ (the f o r c e cons tan t fo r an i n c r ea se in a bond a n g l e ) , and the i n t e r a c t i o n f o r c e c o n s t a n t s . The i n t e r a c t i o n f o r c e cons t an t s r e l a t e how the changes of a bond or ang le a f f e c t changes of o ther bonds and a n g l e s , we have f t00, and f a where r i s pa r t of the 3 r r pp rp c a n g l e , and f r ^ ' where r i s o p p o s i t e the a n g l e . It i s o f t e n conven ien t to make use of the mo lecu l a r symmetry and d e f i n e i n t e r n a l symmetry c o o r d i n a t e s . In t h i s way v i b r a t i o n s tha t be long to d i f f e r e n t symmetry s p e c i e s can be t r e a t e d s e p a r a t e l y . The f i n a l f o r c e c o n s t a n t s are then s imple l i n e a r combina t ions of v a l ence f o r ce c o n s t a n t s (see (r43) p g . 193). In h i s review Duncan (r42) l i s t s the parameters tha t are we l l c h a r a c t e r i z e d by a q u a d r a t i c f o r c e f i e l d a p p r o x i m a t i o n . The expe r imen ta l data a v a i l a b l e a l l owed us to i n c l u d e from that l i s t q u a r t i c d i s t o r t i o n c o n s t a n t s ( c o e f f i c i e n t s of f o u r t h degree angu la r momentum o p e r a t o r s ) 36 fo r v a r i o u s i s o t o p i c spec i e s a long wi th C o r i o l i s c o u p l i n g c o n s t a n t s and v i b r a t i o n a l f r e q u e n c i e s fo r the symmetric t o p s . The v i b r a t i o n a l f r e q u e n c i e s are ob ta ined as the e i g e n v a l u e s of the mat r ix GF, where F i s the mat r ix of f o r c e c o n s t a n t s and G the inve rse k i n e t i c energy m a t r i x . The mat r ix of e i g e n v e c t o r s i s c a l l e d the L mat r ix or the normal c o o r d i n a t e t r a n s f o r m a t i o n mat r ix ( r 46 ) . The fo rma l i sm can c o n v e n i e n t l y be expressed in terms of the symmetr ized B matr ix ( i . e . in an i n t e r n a l symmetry c o o r d i n a t e r e p r e s e n t a t i o n ) tha t t r ans fo rms between C a r t e s i a n d i sp l acement c o o r d i n a t e s X and i n t e r n a l symmetry c o o r d i n a t e s S such tha t S = BX [1.45] The e lements of the B mat r ix are p u r e l y g e o m e t r i c a l q u a n t i t i e s determined as components of the W i l son S-vectors ( r 46 ) . The G mat r ix i s g i ven then in terms of atomic masses and mo lecu l a r geometry as G = Birr 1B T [1 .46] where irr 1 i s a d i agona l 3Nx3N (N=number of atoms) mat r ix of r e c i p r o c a l atomic masses ( r 44 ) . The C o r i o l i s c o u p l i n g c o n s t a n t s (the ze t a cons t an t s ) a re g i ven by 37 S a = ( L - 1 ) B m - 1 M a B T ( L " 1 ) T [1 .47] where the elements of M a are g i ven by ( M ( 1 ) j 5 7 = e a/3 7 (*'P>7 = x y or z) [1 .48] where e i s the L e v i - C i v i t a an t i - s ymmet r i ze r equa l ±1 fo r c y c l i c or a n t i c y c l i c permuta t ion of xyz and equa l to ze ro o therw ise ( r 4 7 , r 4 8 ) . The 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 a re l i n e a r comb ina t ions of T 0 ' s where apyo T a M = ("KV2hc ) ( I a I p I 7 I 6 )- ' rfa^6/XK [1 .49] and where X,, = 47r2c2o> 2 and uv i s the harmonic v i b r a t i o n f requency ( r 12 ) . The a j ^ ' s are e lements of the " s m a l l a " mat r ix f o r the Kfc^ v i b r a t i o n and are p a r t i a l d e r i v a t i v e s at e q u i l i b r i u m of the i n e r t i a l t ensor a,/3 component wi th r e spec t to the K t h normal c o o r d i n a t e Q R ( a ^ = ( 9 I a ^/9Q K ) 0 ) The e lements of the sma l l " a " mat r ix are g i ven by a/3 = L - , v a/3 [ K 5 0 ] where Y a / 3 = 2BM / 3 (M a ) T R e [1 .51] and where R e i s a 3Nx1 vec to r of e q u i l i b r i u m C a r t e s i a n c o o r d i n a t e s f o r the N atoms ( r 1 2 , r 4 4 ) . 38 6. DETERMINATION OF MOLECULAR STRUCTURE The r o t a t i o n a l cons t an t s ob t a i ned from microwave r o t a t i o n a l spec t roscopy are important sources of da ta in the d e t e r m i n a t i o n of mo lecu l a r s t r u c t u r e s . Two e x c e l l e n t rev iews on mo lecu la r s t r u c t u r e d e t e r m i n a t i o n are by K u c h i t s u and Cyv in (r44 Chapt 12) and Rob i e t t e ( r 49 ) . The two p h y s i c a l l y mean ingfu l s t r u c t u r e s c a l c u l a b l e from r o t a t i o n a l spec t roscopy are the r and r s t r u c t u r e s . In an r Z 6 Z s t r u c t u r e the atomic p o s i t i o n s are averaged over the zero p o i n t v i b r a t i o n in the ground v i b r a t i o n a l s t a t e . An s t r u c t u r e i s c o n s i s t e n t wi th the q u a d r a t i c f o r c e f i e l d a p p r o x i m a t i o n . In an e q u i l i b r i u m r g s t r u c t u r e the atomic p o s i t i o n s are those of a h y p o t h e t i c a l v i b r a t i o n l e s s m o l e c u l e . An r g s t r u c t u r e i s c o n s i s t e n t wi th h ighe r o rder p o t e n t i a l c o n s t a n t s . The r o t a t i o n a l cons t an t s determined from s p e c t r a are not g e n e r a l l y connected to a p h y s i c a l s t r u c t u r e . In forming the v a r i o u s r e d u c t i o n s , e m p i r i c a l l y de te rminab le r o t a t i o n a l cons t an t s are found to c o n t a i n r e d u c t i o n dependent l i n e a r combina t ions of 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 o r r e c t i n g fo r these d i s t o r t i o n c o n t r i b u t i o n s g i v e s us the e f f e c t i v e (KW) r o t a t i o n a l cons t an t s of K i v e l s o n and Wi l son ( r 50 ) . It i s these e f f e c t i v e cons t an t s tha t we s h a l l use in subsequent s t r u c t u r a l d e t e r m i n a t i o n s . " * In order to p r e se r ve a c l a s s i c a l model of a r i g i d r o t o r w i th c e n t r i f u g a l d i s t o r t i o n K i v e l s o n and Wi lson g i ve f u r t h e r 39 For symmetric tops in the ground v i b r a t i o n a l s t a t e e l i m i n a t i o n of the <K|H|K±3>, t h r e e - o f f - t h e d i a g o n a l , terms to g i ve our reduced h a m i l t o n i a n i n t r oduces c o r r e c t i o n s of the o rder of a s e x t i c d i s t o r t i o n cons tan t i n t o the r o t a t i o n a l c o n s t a n t s . For our purposes we can not d i s t i n g u i s h the e m p i r i c a l r o t a t i o n a l cons t an t s from the K.W. e f f e c t i v e cons t an t s of symmetric t o p s . For asymmetric t o p s , to a l l e v i a t e the problem of r e d u c t i o n dependent parameters Watson ( r 1 2 , r 2 l ) has d e f i n e d a set of " d e t e r m i n a b l e " pa ramete rs . The r o t a t i o n a l cons t an t s of t h i s s e t , c a l l e d Watson 's " S c r i p t " c o n s t a n t s , a re found to vary on l y s l i g h t l y when c a l c u l a t e d from e m p i r i c a l c ons t an t s d e r i v e d from v a r i o u s r e d u c t i o n s . For the S- reduc t ion one has , f o r example Bx = B x + 2 D J + D JK + 2 d < + 4 d * \ = B y + 2 D J + D JK " 2 d < + 4 d * B = + 2D + 6 d 2 [1 .52] z z J i S i m i l a r equa t i ons can be w r i t t e n f o r o ther r e d u c t i o n s . The S c r i p t c o n s t a n t s , a l t hough i n v a r i a n t to the c h o i c e of r e d u c t i o n , must s t i l l be c o r r e c t e d to g i ve the K.W. e f f e c t i v e pa ramete rs , in the ground v i b r a t i o n a l s t a t e , one h a s , ( r 1 2 ) B = B + (1/2)T' x x ' xxyy * ( c o n t ' d ) c o r r e c t i o n s to t h e i r e f f e c t i v e c o n s t a n t s . C o r r e c t i o n s of t h i s so r t were not made. 40 B = B + ( I / 2 ) T ' y y zzxx B = B + ( I / 2 ) T ' [1 .53 ] z z ' xxyy where r' = (T 0 / 3 + 2T „ „ ) a#/3. [1.54] The K.W. r o t a t i o n a l c o n s t a n t s determined from s p e c t r a are v i b r a t i o n a l s t a t e dependent and are r e l a t e d on l y rough ly to the e q u i l i b r i u m moments of i n e r t i a . A s t r u c t u r e determined s o l e l y from K.W. parameters of one v i b r a t i o n a l s t a t e , v, i s c a l l e d an r j s t r u c t u r e . In p a r t i c u l a r f o r the ground v i b r a t i o n a l s t a t e we have a r 0 s t r u c t u r e . These r 0 s t r u c t u r e s are on l y rough approx imat ions to the p h y s i c a l l y d e f i n a b l e r and r s t r u c t u r e s d i s c u s s e d below, z e For a po l ya tomic mo lecu le in the v v i b r a t i o n a l s t a t e the e q u i l i b r i u m r o t a t i o n a l cons tan t B i s r e l a t e d to B , the K.W. parameters of the v v i b r a t i o n a l s t a t e , by B E = Bp + E a s ( ^ s + d s / 2 ) + . . . [1 .55] where h ighe r order terms are neg l e c t ed and s and d g r ep resen t the number and degeneracy of the normal v i b r a t i o n modes. The formulas f o r the a lphas are g i ven by M i l l s ( r 5 l ) . For a s p e c i f i c v i b r a t i o n u>r the a lpha c o r r e c t i o n to the B r o t a t i o n a l cons tan t a long the b i n e r t i a l a x i s i s b 2 B 2 r 3 ( a ^ g ) 2 u {Zul+uD -a = r co c b I T S ' §T! +§{Srs)2 Cu) s [ 1 .56] where g runs over the C a r t e s i a n components x , y , z and <t>rss i s 41 a c u b i c p o t e n t i a l c o n s t a n t . The f i r s t two terms in the b racke t are the harmonic pa r t of the a l p h a s , the l a s t term i s the anharmonic p a r t . A l l th ree terms are of order B 2 / C J v i D r a t i o n * p h y s i - c a l i n t e r p r e t a t i o n of the t e rms .a re g i ven by M i l l s . For sma l l mo lecu les the anharmonic c o n t r i b u t i o n i s o f t e n dominant and of o p p o s i t e s i gn to the harmonic c o n t r i b u t i o n ( r 5 l ) . I d e n t i f y i n g a g as a sum of harmonic and anharmonic terms a l l ows us to w r i t e equa t ion (e1.55) fo r a molecu le in the ground v i b r a t i o n a l s t a t e as B e = B 0 + (1/2)Zd s a s (HARMONIC) + (1/2)Zd ga s (ANHARMONIC) [1.57] where B 0 r ep re sen t s the ground s t a t e K.W. 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 s . The harmonic pa r t of the a lphas can be c a l c u l a t e d from a harmonic f o r c e f i e l d f o r m a l i s m . T h i s c o r r e c t i o n to the K.W. e f f e c t i v e parameters d e f i n e s the r o t a t i o n a l cons t an t s c o n s i s t e n t w i th an r s t r u c t u r e , B , as z ' z B = B + (1/2)Id a (HARMONIC) [1.58] z o s s The s t r u c t u r e c a l c u l a t e d from the B 2 c ons t an t s (see equa t ion (e1 .2 ) ) r ep re sen t s to a good approx imat ion the mean p o s i t i o n of atoms in a ground v i b r a t i o n a l s t a t e wi th ze ro po in t v i b r a t i o n . ( r 4 9 ) The r z s t r u c t u r e i s we l l d e f i n e d fo r a g i ven mo l e cu l e , but s i n c e zero p o i n t v i b r a t i o n e f f e c t s are mass dependent 42 and the p o t e n t i a l f i e l d s are not s imp ly q u a d r a t i c r z s t r u c t u r e s are not i n v a r i a n t to i s o t o p i c s u b s t i t u t i o n . On the o ther hand the e q u i l i b r i u m r g s t r u c t u r e c o n s i d e r s the atoms f r ozen at the minima of an i s o t o p i c a l l y i n v a r i a n t p o t e n t i a l w e l l . Because of t h i s the r g s t r u c t u r e i s to a h igh degree of approx imat ion i s o t o p i c a l l y i n v a r i a n t . Except fo r very s imple m o l e c u l e s , i t i s d i f f i c u l t to make e i t h e r expe r imen ta l d e t e r m i n a t i o n or t h e o r e t i c a l e v a l u a t i o n of the anharmonic pa r t of the a lphas ( r 4 4 , r 5 2 ) . In t h i s case the e q u i l i b r i u m s t r u c t u r e i s e s t ima ted from changes in the s t r u c t u r e s under i s o t o p i c s u b s t i t u t i o n and v i b r a t i o n a l e x c i t a t i o n ( r 53 ) . I t has been suggested that anha rmon i c i t y i nhe ren t in a bond s t r e t c h , or ang le bend, i s f o r the most pa r t independent of the r e s t of the m o l e c u l e . T h i s a l l ows us to make a " d i a t o m i c " approx imat ion ( r 4 9 , r 5 2 , r 5 3 , r 5 4 ) f o r a bond s t r e t c h (or angle bend) in a po lya tomic m o l e c u l e . In the case of hydrogen bonded to some other atom, of mass M, wi th bond l e n g t h r „ , s u b s t i t u t e d Z r i with deuter ium to g i ve a bond l e n g t h r z D the e q u i l i b r i u m s t r u c t u r e r g in the d i a tomic approx ima t ion i s , r e = ^ 4 ^ [1.593 / M H V M D m. M where M- = TT7 i s the reduced mass. R e p l a c i n g the r ' s by l m. +M i r a l ang les in equa t i on (e1.59) g i v e s the c o r r e s p o n d i n g equa t i on f o r the e q u i l i b r i u m a n g l e . 43 Even though the concept of an e q u i l i b r i u m s t r u c t u r e i s h y p o t h e t i c a l the e q u i l i b r i u m s t r u c t u r e se rves as a s t andard fo r compar isons of d i f f e r e n t m o l e c u l e s . Be ing devo id of mass dependent e f f e c t s an r g s t r u c t u r e r e f l e c t s atomic p o s i t i o n s due s o l e l y to e l e c t r o n i c i n t e r a c t i o n s . Fu r the r i t i s t h i s e q u i l i b r i u m s t r u c t u r e where the atoms of a molecu le r e s i d e at the bottom of the e l e c t r o n i c energy p o t e n t i a l w e l l s , which i s p r e d i c t e d by "ab i n i t i o " c a l c u l a t i o n s ( r 4 9 , r 5 5 ) . 7. SUMMARY AND COMMENTS Many of the q u a l i t a t i v e a spec t s of v i b r a t i o n - r o t a t i o n s p e c t r a can be e x p l a i n e d in terms of the harmonic o s c i l l a t o r and r i g i d r o t o r a p p r o x i m a t i o n s . These approx imat ions break down f o r the three not n e c e s s a r i l y independent r e a sons : anharmonic v i b r a t i o n s , c e n t r i f u g a l d i s t o r t i o n and the i n t e r a c t i o n of v i b r a t i o n s and r o t a t i o n s . The expe r imen ta l focus of t h i s t h e s i s i s on microwave t r a n s i t i o n s between r o t a t i o n a l energy l e v e l s . The r o t a t i o n a l energy l e v e l s are g r e a t l y a f f e c t e d by c e n t r i f u g a l d i s t o r t i o n . Mo t i va ted by c l a s s i c a l arguments of d i s t o r t i n g r o t o r s the r o t a t i o n a l h a m i l t o n i a n i s expanded as a power s e r i e s in components of angu la r momentum o p e r a t o r s . Group t h e o r e t i c a l arguments are then used to p r o j e c t out from the gene ra l h a m i l t o n i a n on l y those terms tha t are a l l owed by the symmetry of the p a r t i c u l a r mo lecu l a r p rob lem. Fu r the r c o n s t r a i n t s on the form of our h a m i l t o n i a n come from the i n v a r i a n c e of energy e i g e n v a l u e s to a u n i t a r y 44 t r a n s f o r m a t i o n . Hence any parameter that can be e l i m i n a t e d through a u n i t a r y t r a n s f o r m a t i o n cannot be determined e m p i r i c a l l y . T h i s l eads to r e d u c t i o n s in the number of parameters of our symmetry a l l owed h a m i l t o n i a n . Of s p e c i a l i n t e r e s t are the s o - c a l l e d f o r b i d d e n t r a n s i t i o n s . These are f o rb idden on l y in a r i g i d ro to r-harmon ic o s c i l l a t o r f o rma l i sm . V i b r a t i o n a l - r o t a t i o n a l , or v i b r o n i c , energy l e v e l e i g e n f u n c t i o n s are w r i t t e n as l i n e a r comb ina t ions of r i g i d r o t o r and harmonic o s c i l l a t o r b a s i s f u n c t i o n s . Con fus ion a r i s e s because the r i g i d ro to r-harmon ic o s c i l l a t o r d e s c r i p t i o n s are such e x c e l l e n t app rox ima t ions tha t these d e s c r i p t i o n s p r e v a i l . Because of t h i s we t a l k in terms of pseudo s e l e c t i o n r u l e s , AK = ±3 t r a n s i t i o n s f o r C 3 V symmetric t o p s , and v i b r a t i o n a l a l lowed t r a n s i t i o n s between r o t a t i o n a l energy l e v e l s in the ground v i b r a t i o n a l s t a t e . From the parameters ob ta ined e m p i r i c a l l y by f i t t i n g these s p e c t r a , f o r c e cons t an t s can be c a l c u l a t e d . The fo r ce c o n s t a n t s r ep resen t " s p r i n g " f o r c e cons t an t s in a p i c t u r e of the mo lecu le where atoms are connected by s p r i n g s . A l s o from the e m p i r i c a l parameters a r e d u c t i o n independent set of r o t a t i o n c o n s t a n t s can be d e f i n e d and a s s o c i a t e d wi th a mo l e cu l a r s t r u c t u r e . I n t e n s i t i e s of t r a n s i t i o n s o f t e n determine the types of t r a n s i t i o n s tha t can be measured. T y p i c a l 7 v a lues fo r J r max normal t r a n s i t i o n s s t a r t around 1 0 " " c m " 1 . Fo rb idden t r a n s i t i o n s have 7 va lues of 1 0 " 9 c m " 1 or lower . The 45 extreme weakness of f o r b i d d e n t r a n s i t i o n s r e q u i r e s development of h i g h l y s e n s i t i v e spec t rome te r s . One such des ign i s d i s c u s s e d in the next s e c t i o n . The d i f f i c u l t y in o b t a i n i n g f o r b i d d e n s p e c t r a i s o f f s e t by new i n f o r m a t i o n . For symmetric tops f o r b i d d e n t r a n s i t i o n s a l low de t e rm ina t i on of the p e r p e n d i c u l a r r o t a t i o n cons tan t and fo r s p h e r i c a l t o p s , f o r b i d d e n t r a n s i t i o n s are the on ly ones a v a i l a b l e . 46 C. CHAPTER 2: EXPERIMENTAL METHODS The microwave t r a n s i t i o n s r epo r t ed in t h i s t h e s i s span a wide range of i n t e n s i t y , 10 " " - 1 0 " 1 1 c m " 1 , and f r equency , 8-303 GHz. In order to encompass t h i s spectrum of s p e c t r a , th ree spec t romete rs were used . A h i g h l y s e n s i t i v e d i s t o r t i o n moment spec t rometer l o c a t e d at the U n i v e r s i t y of B r i t i s h Columbia was used to measure the 8-26 GHz f o r b i d d e n t r a n s i t i o n s . The a l l owed asymmetr ic top s p e c t r a were taken on a cen t ime te r wave spec t rometer at the U n i v e r s i t y of B r i t i s h Columbia and on a m i l l i m e t e r wave spec t rometer at the C a l i f o r n i a I n s t i t u t e of T e c h n o l o g y ' s Je t P r o p u l s i o n Labo ra to r y ( JPL) . Common f e a t u r e s of a l l the spec t romete rs used i n c l u d e a f requency source and f requency measur ing d e v i c e , an a b s o r p t i o n c e l l , microwave d e t e c t o r , a m p l i f i e r and s p e c t r a l r e c o r d i n g a p p a r a t u s . S ince microwave a b s o r p t i o n l i n e s are weak compared to a b s o r p t i o n l i n e s in other f requency r e g i o n s , microwave spec t romete rs u s u a l l y modulate t r a n s i t i o n s . The t r a n s i t i o n s are moved on and o f f resonance so tha t microwave power i s s u c c e s s i v e l y absorbed and not absorbed p roduc ing an amp l i tude modulated microwave s i g n a l at the modu la t ion f r equency . The s i g n a l i s decoded by a phase s e n s i t i v e l o c k - i n a m p l i f i e r that s u b t r a c t s the " o f f " s i g n a l from the " o n " s i g n a l and averages t h i s r e s u l t over the t ime cons tan t of the a m p l i f i e r . Modu la t i on of t h i s s o r t i n c r e a s e s s i g n a l and dec reases no i se ( r 1 1 ) . The UBC spec t romete rs both used S ta rk modu l a t i on . The JPL spec t rometer had the advantage of e i t h e r S tark or "Tone 47 B u r s t " modu la t ions (see d i s c u s s i o n on JPL s p e c t r o m e t e r ) . 1. STARK MODULATION One of the most popu la r modu la t ion t echn iques i s S tark m o d u l a t i o n , which r e l i e s on the s h i f t i n g and s p l i t t i n g of t r a n s i t i o n f r e q u e n c i e s in the presence of an e x t e r n a l s t a t i c e l e c t r i c f i e l d . In the exper iments c a r r i e d out here a metal p l a t e was i n s e r t e d in the midd le of a r e c t a n g u l a r meta l waveguide a b s o r p t i o n c e l l . The c e l l s used were of X band (0 .90 x 0.40 i n . id) and K band (0.42 x 0.17 i n . id ) waveguide and were 1-13 meters in l e n g t h . The p l a t e ran the l eng th of the c e l l and was i n s u l a t e d from i t by T e f l o n s t r i p s . The metal p l a t e was p l a c e d p a r a l l e l to the broad face of the c e l l so that f o r the dominant p ropaga t i on mode the s t a t i c and o s c i l l a t i n g f i e l d s were p a r a l l e l and AM=0 t r a n s i t i o n s o c cu r r ed between S tark components. A 1-100 KHz ze ro based square wave was a p p l i e d to the p l a t e . Dur ing the h a l f c y c l e wi th non-zero v o l t a g e the t r a n s i t i o n s are s h i f t e d and s p l i t by the S tark f i e l d . When the source f requency matches tha t of an a b s o r p t i o n l i n e or one of i t s S tark components an ampl i tude modulated s i g n a l i s produced at the d e t e c t o r . An o s c i l l o s c o p e t r a c e of the phase d e t e c t o r output shows both the a b s o r p t i o n l i n e and the S tark components wi th the l a t t e r i n v e r t e d . The main d i f f i c u l t y w i th S tark modu la t ion i s fo r mo lecu les where Stark s h i f t s on the o rder of l i n e widths r e q u i r e h igh f i e l d s , say over 2000 v/cm, as the i n v e r t e d S tark p a t t e r n can i n t e r f e r e wi th the a b s o r p t i o n 48 l i n e . 2. DISTORTION SPECTROMETER The spec t rometer used to measure the d i s t o r t i o n s p e c t r a between 8-26 GHz i s shown s c h e m a t i c a l l y in F i g . 1 ( reproduced wi th p e r m i s s i o n from the au tho rs ( r 5 6 ) ) . T h i s spec t rometer has been d e s c r i b e d in d e t a i l ( r 2 0 , r 5 6 , r 5 8 ) . F i g u r e 1 RAMP 1kHz Reference Oscillator SIGNAL AVERAGER NICOLET 1072 OSCILLOSCOPE X-Y RECORDER F i g u r e 1. The d i s t o r t i o n spec t rometer used between 8 and 26 GHz. I n t e r n a t i o n a l symbols are used f o r the v a r i o u s e lements ( r 5 7 ) . 49 Source no i se was p a r t i a l l y c a n c e l l e d w i th a d e t e c t o r b r i d g e . No ise common to the r e f e rence and sample d e t e c t o r s i s r e j e c t e d on the pr imary of the t r a n s f o r m e r . The AM s i g n a l (1 KHz) i s de-modulated by a PAR 128 l o c k - i n a m p l i f i e r , whose output i s s t o r e d as a f u n c t i o n of f requency in a N i c o l e t 1072 S i g n a l Ave rage r . The s i g n a l averager produces a vo l t age ramp that d r i v e s a r e f e rence c r y s t a l in a Microwave Systems MOS-5 which when m u l t i p l i e d up and phase l o cked to a k l y s t r o n source causes the k l y s t r o n f requency to be swept. Because the t r a n s i t i o n moments were so sma l l h igh microwave powers c o u l d be used without s a t u r a t i o n . T y p i c a l power l e v e l s were 60-100 mWatts. Power l e v e l s at the end of the 13 m. c e l l were t y p i c a l l y 1OdB down. The k l y s t r o n s used were V a r i a n X-13 and X-12, and OKI 20V10 and 24V10A. Frequency windows ave rag ing 4 MHz were swept forward and back on the order of two minutes wi th a t ime cons tan t of 1 second . A measurement c o n s i s t e d of a ve rag ing a l a rge number of scans (order of hundreds) w i th the gas in the c e l l and aga in wi thout the gas in the c e l l . The d i f f e r e n c e of the gas and no gas scans gave the spectrum c o r r e c t e d fo r background. T y p i c a l scan t imes ranged from 1-31 hours per l i n e . For the e a r l y germane work f requency measurements were made on the MOS-5 c r y s t a l (=* 15 MHz). The microwave f requency c o u l d be ob ta ined from knowledge of the m u l t i p l i c a t i o n harmonic and the 40 MHz IF ( r 5 9 ) . D i r e c t f requency measurements were a l s o made wi th a Hewlet t-Packard 50 5256 A heterodyne c o n v e r t e r . L a t e r , fo r the a r s i n e work a Sys t ron Donner microwave f requency counter p e r m i t t e d d i r e c t f requency measurement over the e n t i r e f requency range of the spec t rome te r . A l l the measurements were done with the a b s o r p t i o n c e l l at room tempera tu re . Sometimes when we look at low J t r a n s i t i o n s the a b s o r p t i o n c e l l i s coo l ed to enhance the p o p u l a t i o n s of the lower J , s t a t e s , but in the case of the d i s t o r t i o n t r a n s i t i o n s , g i ven the range of J , room temperature measurements were most a p p r o p r i a t e . Be fore any measurement c o u l d be made the system had to be " t u n e d " . The f i r s t s tep in t u n i n g the spec t romete r was to set the cen te r of a f requency sweep at the " t o p " of a k l y s t r o n mode. For d e s c r i p t i o n s of k l y s t r o n s see r e f e r e n c e ( r 60 ) . On a p l o t of k l y s t r o n output power ve r sus f requency a k l y s t r o n mode i s d e s c r i b e d by an ups ide down l e t t e r U s i t t i n g on the f requency a x i s . Ou t s ide the arms of the U the power i s ze ro and the k l y s t r o n does not o s c i l l a t e i n these f requency r e g i o n s . At the top of a mode the k l y s t r o n power l e v e l w i l l change very l i t t l e . Wi th the k l y s t r o n s a v a i l a b l e fo r t h i s exper iment , mode widths were found to be from ten to a few hundred megahertz w ide . The k l y s t r o n f r e q u e n c i e s depended on tempera ture , k l y s t r o n c a v i t y s i z e and r e f l e c t o r v o l t a g e . The k l y s t r o n s used in t h i s exper iment had three modes. For a g i ven c a v i t y s i z e the modes at h i ghe r r e f l e c t o r v o l t a g e would g i ve more microwave power. For ve ry weak f o r b i d d e n t r a n s i t i o n s power l e v e l s of a few hundred 51 m i l l i w a t t s are d e s i r a b l e and so whenever p o s s i b l e the h ighe r power modes of our k l y s t r o n s would be used . A l s o k l y s t r o n s tend to become very hot and temperature d r i f t s and i n s t a b i l i t i e s o c c u r . The temperature problem was s o l v e d by p l a c i n g the k l y s t r o n s in o i l baths c o o l e d by c o l d water c o o l i n g c o i l s . The s e t t i n g of the cen te r of a f requency sweep to the top of a k l y s t r o n mode was accompl i shed by phase l o c k i n g the k l y s t r o n in a c e r t a i n mode to the c en t e r f requency and then v a r y i n g the c a v i t y s i z e ( m e c h a n i c a l l y ) , and the r e f l e c t o r vo l t age fo r the maximum k l y s t r o n output power. A f t e r the k l y s t r o n was put at the top of the mode the c e l l would be tuned to op t im ize the microwave power at the d e t e c t o r s . At the same time the b r i dge c i r c u i t c o u l d be ba lanced in order to c a n c e l some of the source n o i s e . A l l t h i s was accompl i shed w i th p lunge rs ( to vary the l e n g t h of the c e l l on the order of the microwave wavelength) E and H tuners and s l i d e screw tuners a l l of which are s tandard microwave waveguide tun ing and matching dev i c e s ( r 1 1 , r 6 0 ) . A f t e r the spect rometer was " t u n e d " the gas c o u l d be i n t r o d u c e d , the square wave S tark f i e l d swi tched on and the measurement begun. There were three main d i f f i c u l t i e s wi th t h i s spec t rome te r : o u t g a s s i n g , k l y s t r o n d r i f t and " g l i t c h e s " , and n o i s e . Over the p ro longed s i g n a l accumula t ion t imes sample gas p r e s s u r e s were found to i n c r ease s i g n i f i c a n t l y due to l e aks and o u t g a s s i n g . It was g e n e r a l l y found tha t a gas 52 sample was on l y good in the c e l l f o r about an hour of measurement. T h i s , p l u s the s u b j e c t i v e c o n c l u s i o n tha t the system was " s t a b l e " over two hours but g e n e r a l l y not as s t a b l e o v e r , say , f i v e hou r s , r e s u l t e d in the expe r imen ta l techn ique of one hour s i g n a l measurement w i th the gas in the c e l l f o l l owed by one hour background measurement wi th the c e l l e vacua ted . The problem of k l y s t r o n d r i f t was s o l v e d by c o n t i n u a l r e- tun ing of the system. The problem of " g l i t c h e s " , d i s c o n t i n u o u s power changes as a f u n c t i o n of f r equency , though not a f requent problem was o f t e n p a t h o l o g i c a l in nature - g l i t c h e s would come and go seemingly f o r no r e a son . The most d i f f i c u l t expe r imen ta l problem had to be the no i se p rob lem. In t h i s case t u r n i n g o f f the l i g h t s in the l a b , p l u g g i n g i n a fan or any e l e c t r i c a l t r a n s i e n t s were enough to d e s t r o y a measurement. The problem of d e s t r o y i n g a many hour measurement in the l a s t minute was so l v ed by accumu la t ing fo r a h a l f hour and s t o r i n g the r e s u l t in the s i g n a l averager memory. The u l t i m a t e spect rometer s e n s i t i v i t y was p robab l y 1 x I 0~ 1 1 cm~ 1 which i s the i n t e n s i t y of the weakest S tark modulated microwave a b s o r p t i o n l i n e measured to date and i s r e p o r t e d here as the GeH a E type t r a n s i t i o n J=20 3—>4. 3. CENTIMETER WAVE SPECTROMETER (UBC) . The UBC microwave spec t rometer covered the f requency range 8-120 GHz. The S tark c e l l s were t y p i c a l l y 3 meters in 53 l eng th and the modu la t ion f requency was 100 KHz. Three d i f f e r e n t microwave sources were used . The low f requency was genera ted by a Hewle t t-Packard (HP) 8400 B p h a s e - s t a b i l i z e d microwave spec t roscopy sou r ce . Much of the 8-40 GHz f requency range c o u l d be covered wi th the a p p r o p r i a t e backward wave o s c i l l a t o r (BWO) p l u g - i n u n i t s : X-band (8-12.4 GHz) HP, H81-8694B, P-band (12.4-18 GHz) HP, H81-8695 A and R-band (26.540 GHz) HP 8697, A p l u g - i n . The middle f r e q u e n c i e s , 53-80 GHz, were ob t a i ned from a SPACEKOM OV-1 microwave f requency doub le r u s i n g as a pr imary source the R-band BWO. The h i g h f requency 75-120 GHz range was genera ted by a Siemens RW<i>110B BWO. A Rohde and Schwarz f requency s y n t h e s i z e r SMDW was m u l t i p l i e d and mixed to phase lock a P-band BWO which in tu rn was m u l t i p l i e d ' a n d mixed to l ock the h igh f requency BWO. A l l f requency measurements were made on the r e f e r ence o s c i l l a t o r . No d e t e c t o r b r i d g e was used fo r the a l lowed t r a n s i t i o n s . The d e t e c t o r s i g n a l was fed through a p r e a m p l i f i e r i n t o a PAR 120 l o c k - i n a m p l i f i e r . The output of the l o c k - i n was p r e sen ted as a f u n c t i o n of f requency on an o s c i l l o s c o p e . The d e t e c t o r s used depended on f requency r e g i o n s : X-band HP 406-X422A back d i o d e , P-band HP H06-P422 A back d i o d e , R-band HP 11586 A p o i n t con tac t d i o d e , 53-120 GHz R-band d e t e c t o r or a HUGHES 47314 H-1100. 54 4. MILLIMETER WAVE SPECTROMETER (JPL) The JPL m i l l i m e t e r spec t rometer covered the f requency range 60-800 GHz. The low f requency range , 60-123 GHz, was a v a i l a b l e , w i th a few e x c e p t i o n s , from a host of k l y s t r o n s . H igher f r e q u e n c i e s were genera ted as harmonics of the low f requency k l y s t r o n s by harmonic m u l t i p l i c a t i o n (Schot tky b a r r i e r d i o d e s ) . The k l y s t r o n s were phase l o cked to a m u l t i p l i e d R-band BWO s i g n a l genera ted by a commerc ia l HP 8460 A MRR Spectrometer ( r 6 l ) , a l a t e r v e r s i o n of the UBC cen t ime te r spec t rome te r . For f r e q u e n c i e s up to about 200 GHz R-band d e t e c t o r s (see 2.3 UBC spec t rometer s e c t i o n ) c o u l d be used fo r s t rong l i n e s however, he l ium c o o l e d InSb bo lometers were u s u a l l y used above 100 GHz. The a b s o r p t i o n c e l l s were s tandard 3 meter X-band Stark c e l l s . The square wave modu la t ion f requency was 33 KHz. The JPL spect rometer had the added advantage of " tone b u r s t " modu la t ion ( r 62 ) . In tone bu rs t modu la t ion a s i g n a l , on the o rder of s e v e r a l megahertz i s tu rned on and o f f at a ra te o f , fo r example, 30 KHz. T h i s s i g n a l i s added to the r e f l e c t o r v o l t a g e of the k l y s t r o n p roduc ing s idebands at i n t e r v a l s of the tone f requency at the modu la t ion f requency of 30 KHz. If the k l y s t r o n phase l o ck l oop cannot respond to the 30 KHz FM s i g n a l then we get a f requency modulated source w i th the advantage of the lower no i se and s t a b i l i t y of a phase l o cked sys tem. Tone bu r s t modu la t ion i s e s p e c i a l l y h e l p f u l when slow S tark e f f e c t s produce i n v e r t e d 55 S ta rk " l o b e s " that can i n t e r f e r e w i th the ze ro f i e l d t r a n s i t i o n f r e q u e n c i e s . C l o s e l y spaced h y p e r f i n e s t r u c t u r e can a l s o be masked by i n t e r f e r i n g S ta rk l obes so tone bu r s t modu la t ion a c t s as an a l t e r n a t i v e to ze ro f i e l d f requency e s t ima tes based on S tark m o d u l a t i o n . 5. SUNDRY ITEMS The gas samples were s t o r e d in g l a s s v e s s e l s f i t t e d w i th greased s top c o c k s . The gas h a n d l i n g systems were meta l or g l a s s vacuum l i n e s f i t t e d w i th o i l d i f f u s i o n pumps. The a b s o r p t i o n c e l l s were i s o l a t e d from the atmosphere by mica windows. The gas i s i n t r o d u c e d to the c e l l through a g l a s s to meta l or metal to meta l KOVAR type j o i n . Gas p r e s su r e s are t y p i c a l l y 10-150 mTORR (microns) which gave l i n e widths on the o rder of 500 KHz. T r a n s i t i o n f requency measurement e r r o r s were between 10-100 KHz depending on f requency r e g i o n , l i n e s t r e n g t h and o ther pa ramete rs , and w i l l be p resen ted wi th the measurements. 56 D. CHAPTER 3: GERMANE GeH a The most germane p r e c u r s o r f o r t h i s f o r b i d d e n t r a n s i t i o n study was done by Ka t tenberg et a l ( r 63 ) . In 1972 from an a n a l y s i s of the i n f r a r e d and Raman spectrum of GeH a they were ab le to r epor t amongst o ther parameters B 0 and D g . R e c a l l , to f o u r t h degree in angu la r momentum ope r a to r s that f o r a l e v e l ' J C 1 " , E = B 0 J 2 - D s J f t + D t<ft«> [3.1] For s p h e r i c a l tops (r64) the c o n t r i b u t i o n to the f requency from B 0 i s much g r ea t e r than the q u a r t i c c o n t r i b u t i o n , and so the i n f r a r e d R branch (AJ=+1) " f o r b i d d e n " spectrum of GeH, was then we l l p r e d i c t e d and in 1973 O z i e r and Rosenberg (r65) r epo r t ed pa r t of i t . In t h e i r work they were ab le to c o n f i r m the e a r l i e r r e s u l t s f o r B 0 and D g and es t imate the d i s t o r t i o n d i p o l e moment from es t ima tes of a b s o l u t e l i n e i n t e n s i t i e s . They c o u l d not r epo r t a va lue of D f c but showed t h e i r da ta were c o n s i s t e n t w i th a va lue of D t c a l c u l a t e d by Fox (r66) based on t h e o r e t i c a l e x p r e s s i o n s g i ven by Hecht (r67) (D f c ~ 2 . 2 x 1 0 " 6 c m - 1 ) . From a m i c rowave- in f r a red double resonance s tudy , K r e i n e r et a l ( r68 , r69 ) r epo r t ed s e v e r a l Q branch (AJ=0) " f o r b i d d e n " t r a n s i t i o n s and a " b e t t e r " va lue f o r the. d i s t o r t i o n d i p o l e moment. They were ab le to a s s i g n f i v e r o t a t i o n a l t r a n s i t i o n s in the ground v i b r a t i o n a l s t a t e . They t e n t a t i v e l y a s s i g n e d two f u r t h e r resonances but were unable 57 to account fo r s e v e r a l o t h e r s . From the f i v e a s s i g n e d t r a n s i t i o n s they c a l c u l a t e d s e v e r a l l i n e a r combina t ions of the t enso r d i s t o r t i o n c o n s t a n t s and from these r e s u l t s were ab l e to p r e d i c t two E type t r a n s i t i o n s at 9.6 and 22.6 GHz. R e c a l l tha t E type t r a n s i t i o n s have the a l l - i m p o r t a n t f i r s t o rde r S ta rk e f f e c t of consequence in the p resen t work. As w i l l be gathered from the d i s c u s s i o n of the p r e d i c t i o n and measurement of the spectrum these p r e d i c t e d E type t r a n s i t i o n s were very important in the comp le t i on of t h i s s t udy . A l s o in t h e i r work K r e i n e r et a l (r68) observed microwave t r a n s i t i o n s at 2356.68 ±.1 and 2356.60 ±.1 MHz w i th two d i f f e r e n t l a s e r c o i n c i d e n c e s 0.10934 c m - 1 a p a r t . The r o t a t i o n a l ass ignments made were the same, J=T1, E ( 2 ) - E ( 1 ) , and the two t r a n s i t i o n s were a t t r i b u t e d to d i f f e r e n t Ge s p e c i e s . T h i s i s ana logous i so tope s p l i t t i n g to tha t observed for 1 3 C H 4 and 1 2 C H a where s e p a r a t i o n s up to 10 MHz were observed ( r 7 0 ) . I f K r e i n e r et a l were c o r r e c t in t h e i r e x p l a n a t i o n then in the p resen t s tudy , where l i n e w idths were on the o rder of hundreds of k i l o h e r t z , i s o t o p e s p l i t t i n g s were expected to be sma l l i f n o t i c e a b l e a t a l l . The MW-IR techn ique of K r e i n e r et a l i s e s p e c i a l l y u s e f u l in the i n i t i a l phase of s t u d i e s of t h i s s o r t because of the h i g h s e n s i t i v i t y a v a i l a b l e (1 .0 ' 1 3 cm" 1 ) ( r 6 ) , bu t , s i n c e i t r e l i e s on l a s e r c o i n c i d e n c e s sys temat i c o b s e r v a t i o n s of t r a n s i t i o n f r e q u e n c i e s a re not p o s s i b l e . F u r t h e r s t u d i e s are perhaps best done on h i g h l y s e n s i t i v e 58 S tark spec t romete rs capab le of f requency sweeping such as the d i s t o r t i o n spect rometer d e s c r i b e d in Chapter 2. T h i s then was the m o t i v a t i o n fo r the germane study p resen ted h e r e . Us ing the c e n t r i f u g a l d i s t o r t i o n moment spect rometer we hoped to s y s t e m a t i c a l l y measure many more f o r b i d d e n r o t a t i o n a l t r a n s i t i o n s and from t h e i r a n a l y s i s e va l ua t e separa te r o t a t i o n a l parameters r a the r than l i n e a r c o m b i n a t i o n s . A l s o from the i n c r e a s e d data set i t was hoped ass ignments c o u l d be made fo r the un-ass igned resonances of R r e i ne r et a l ( r 68 ) . F i n a l l y i t was hoped tha t something more c o u l d be s a i d about the Ge i so tope s p l i t t i n g . 1. GERMANE GAS SAMPLE The sample used f o r t h i s s tudy was ob t a i ned from the Matheson Company as 99.9% GeH„ and was used wi thout f u r t h e r p u r i f i c a t i o n , the main i m p u r i t i e s be ing H 2 0 and H 2 ( r 65 ) . 2. PREDICTION OF THE GERMANE SPECTRUM I n i t i a l l y the germane spectrum was p r e d i c t e d u s i n g the p e r t u r b a t i o n methods of M o r e t - B a i l l y and h i s c o l l a b o r a t o r s ( r71 , r72 ) and Watson and h i s c o l l a b o r a t o r s ( r 1 7 , r l 8 ) . Watson 's t reatment has been extended by O z i e r to i n c l u d e o c t i c terms (r15) and we have fo r the s p l i t t i n g of a J l e v e l due to c e n t r i f u g a l d i s t o r t i o n , the energy of a l e v e l JC f c 59 E = [D t +H 4 t J ( j+1 )+L 4 f c J 2 ( j+1 ) 2 ] f + [ H 6 t + L 6 t J ( J + 1 ) ] g + L g t h + ( H 2 6 t / D t ) g . [3 .2 ] Va lues of the numer i ca l f a c t o r s f and g are t a b u l a t e d by K i r s c h n e r and Watson ( r l 8 ) , and those of h and g are t a b u l a t e d by O z i e r ( r 15 ) , up to J=20. The f requency of a t r a n s i t i o n i s , of c o u r s e , j u s t the energy d i f f e r e n c e between l e v e l s . The r o t a t i o n a l parameters were es t ima ted from the v a r i o u s l i n e a r combina t ions p r e sen ted by K r e i n e r et a l ( r 68 ) , and s c a l i n g arguments based on r e l a t i v e magnitudes of the r o t a t i o n a l parameters in the s i m i l a r molecu le s i l a n e , S i H 4 ( r 56 ) . As an example of a s c a l i n g argument L g t ( G e H 4 ) was e s t ima ted from the r e l a t i o n r L 8 t (GeH 1 ) )- . L 6 f c ( G e H 4 ) = L L B t ( S i H . ) J L g ( S i H „ ) [3 .3] where L g t ( G e H 4 ) was determined s e p a r a t e l y by K r e i n e r et a l . From the l i n e a r combina t ions of Tab l e III of K r e i n e r et a l (r68) and from the es t ima te of L 6 t ( G e H t t ) we c o u l d es t imate Hg f c . A s i m i l a r r a t i o s c a l i n g argument a l l owed the e s t i m a t i o n of H 4 t and L 4 f c f o r G e H 4 . F i n a l l y , the re were a v a i l a b l e two equa t ions r e l a t i n g D f c to v a r i o u s h ighe r order pa ramete rs . I n s e r t i n g i n t o these equa t ions the e s t ima tes of the h ighe r order parameters gave two va lues of D f c which were averaged to g i ve the i n i t i a l D f c e s t i m a t e . I t was from these e s t ima tes and equa t i on (e3.2) tha t the i n i t i a l Q branch d i s t o r t i o n spectrum was p r e d i c t e d . 60 As we s h a l l see in the next s e c t i o n these p r e d i c t i o n s were very good . As p r e d i c t e d t r a n s i t i o n s were found new l i n e a r r e l a t i o n s amongst the d i s t o r t i o n parameters c o u l d be produced u s i n g equa t ion ( e 3 . 2 ) . These l i n e a r r e l a t i o n s were then s o l v e d f o r b e t t e r e s t ima te s of the d i s t o r t i o n pa ramete rs . T h i s procedure was, of c o u r s e , i t e r a t e d . When the number of t r a n s i t i o n s , or independent l i n e a r e q u a t i o n s , found became g r ea t e r than the number of r o t a t i o n a l parameters the system of l i n e a r equa t ions became over-dete rmined and the r o t a t i o n a l parameters were then determined by l i n e a r l e a s t squa r e s . Then, i n s t e a d of us ing the p e r t u r b a t i o n approach symbo l i zed by equa t ion (e3.2) the complete d i a g o n a l i z a t i o n method of Fox and O z i e r ( r l 9 ) was used . 3. MEASUREMENT OF THE GERMANE SPECTRUM In t h i s s tudy on ly E type f o r b i d d e n Q branch t r a n s i t i o n s were conven ien t to measure. R e c a l l tha t on ly E type t r a n s i t i o n s have f i r s t o rder S tark e f f e c t s tha t we can use to S ta rk modulate t r a n s i t i o n s w i th the s t a t i c e l e c t r i c f i e l d s a v a i l a b l e e x p e r i m e n t a l l y (see Chapter 2 ) . The f i r s t l i n e measured in t h i s study was J=18, E ( 3 ) - E ( 2 ) , a t 9615.310(75) MHz. T h i s l i n e had been p r e d i c t e d by K r e i n e r et a l (r68) to be at 9615.29 MHz. T h i s p r e d i c t i o n was expec ted to be good so no r e a l new i n f o rma t i on was ga ined on the r o t a t i o n a l pa ramete rs . However, t h i s t r a n s i t i o n was p a r t i c u l a r l y h e l p f u l in o p t i m i z i n g the 6 1 expe r imen ta l c o n d i t i o n s , most no tab l y p r e s s u r e and accumula t ion t imes . Us ing the " i n i t i a l " p r e d i c t i o n scheme o u t l i n e d in the p r e v i o u s s e c t i o n , p r e d i c t i o n s were made of o ther E type Q branch t r a n s i t i o n s a c c e s s i b l e to our spec t rome te r . F o l l o w i n g t h i s the J = 1 4 , E ( 2 ) - E ( 1 ) l i n e was searched f o r w i th no r e s u l t : the i n i t i a l p r e d i c t i o n turned out to be 1 4 MHz h i g h . In t h i s case i t was found tha t the Stark f i e l d was c o n t r i b u t i n g to a l a r g e d i s t o r t e d background upon which the s i g n a l was d i f f i c u l t to see . T h i s c l e a r l y made the f requency r eg i on of t h i s l i n e un favou rab l e fo r s e a r c h i n g . I t was then dec i ded to search fo r J = 1 6 , E ( 3 ) - E ( 2 ) . The p r e d i c t i o n tu rned out to be on ly 7 MHz h igh and the lower f i e l d s necessa ry to modulate t h i s l i n e (see Tab l e 3 . 2 A ( t " , t ' ) ) made background e f f e c t s l e s s and t h e r e f o r e s e a r c h i n g e a s i e r . Once t h i s l i n e was found the i n f o r m a t i o n on the d i s t o r t i o n cons t an t s ga thered from i t was used to produce new p r e d i c t i o n s tha t turned out to be w i t h i n 1 MHz of subsequent measured f r e q u e n c i e s . Given such e x c e l l e n t p r e d i c t i o n s the remain ing expe r imen ta l d i f f i c u l t y was in o p t i m i z i n g the spect rometer fo r extreme s e n s i t i v i t y in the v a r i o u s r e g i o n s of the spec t rum. A testament to the success we had s o l v i n g t h i s problem was the measurement of the weakest r e p o r t e d S tark modulated microwave a b s o r p t i o n l i n e , J = 2 0 , E ( A > - E < 3 ) at a f requency of 1 0 8 9 7 . 1 8 0 ( 1 0 0 ) MHz wi th a 7 M A X of 1.0 x 1 0 " 1 1 cm" 1 . 62 Dur ing the course of t h i s study there was concern whether a l l these measured t r a n s i t i o n s were due to the same i s o t o p i c Ge s p e c i e s . In methane ( r 2 0 , r 6 4 , r 7 0 ) two spec t r a sepa ra ted by 2 to 10 MHz were a t t r i b u t e d to 1 3 C H „ and 1 2 C H „ . T h i s r e s u l t , which w i l l be examined l a t e r , can be accounted f o r in terms of zero po in t v i b r a t i o n e f f e c t s in D f c and the i s o t o p i c dependence of H^ t and H g t » S ince Ge has v a r i o u s i s o t o p e s of s i g n i f i c a n t n a t u r a l abundance (mass number/ % n a t u r a l abundance, 70/20.52 , 72/27 .43 , 73/7 .76 , 74/36.54 , 76/7.76 ( r73) ) i t was suggested tha t s i m i l a r " s i s t e r " s p e c t r a might e x i s t . The s e p a r a t i o n between germane s i s t e r s p e c t r a was expected to be l e s s than in methane as the change in reduced mass upon i s o t o p i c Ge s u b s t i t u t i o n was much sma l l e r than that of 1 3 C to 1 2 C . In t h i s case one shou ld c e r t a i n l y be ab le to take the maximum " s i s t e r " s e p a r a t i o n in methane as an upper l i m i t in a sea rch fo r s i s t e r s p e c t r a ne i ghbo r i ng s i m i l a r germane t r a n s i t i o n s . For germane, t r a n s i t i o n s where the i s o t o p i c s p l i t t i n g was expected to be l a r g e i n c l u d e d t r a n s i t i o n s where the c o n t r i b u t i o n of D f c to the f requency was l a r g e which in our case meant the E type t r a n s i t i o n s 18(1—>2), 19(1—>2) and 20(2—>3) in descend ing o r d e r ; a t r a c i n g of the J=18 (1—>2) t r a n s i t i o n i s shown in F i g u r e 2. P a r t i c u l a r l y c a r e f u l s t u d i e s were made of these l i n e s by s e a r c h i n g up and down f r equency , over the maximum methane s p l i t t i n g of 10 MHz. In these s t u d i e s no f u r t h e r t r a n s i t i o n s were found , suppo r t i ng the c o n t e n t i o n of K r e i n e r et a l tha t the i so tope s p l i t t i n g 6 3 i s w i t h i n our l i n e w id ths . The problem of i so tope s p l i t t i n g i s d i s c u s s e d more q u a n t i t a t i v e l y l a t e r . F i g u r e 2 1 1 1 1 1 1 1 1 225548 22555.7 225566 Frequency (MHz) F i g u r e 2. The microwave a b s o r p t i o n l i n e of GeH« r e s u l t i n g from the d i s t o r t i o n t r a n s i t i o n 18E2<—18E1. I t was o b t a i n e d by u s i n g the s i g n a l ave rager to accumulate 64 sample scans and then s u b t r a c t 64 background s c a n s . The t o t a l s cann ing time (sample p l u s background) was 3.64 h o u r s . 64 4. OBSERVED GERMANE SPECTRUM AND ANALYSIS The a s s i gned germane spectrum observed in t h i s work c o n s i s t s of ten Q branch (AJ=0) t r a n s i t i o n s "where J ranges from 14 to 20. The measured t r a n s i t i o n f r e q u e n c i e s are l i s t e d in Tab le 3.1 a long w i th t h e i r ass ignments and expe r imen ta l u n c e r t a i n t i e s o . The expe r imen ta l u n c e r t a i n t i e s are es t ima ted from observed s i g n a l to no i se r a t i o s , l i n e widths and r e p r o d u c i b i l i t i e s of the f r e q u e n c i e s o b t a i n e d from d i f f e r e n t r u n s . The l i n e w id ths , f u l l w idth at h a l f maximum, were t y p i c a l l y 420 KHz which compare w e l l w i th the t y p i c a l v a l u e of 430 KHz f o r 2 8 S i H f l ( r 5 6 ) . A summary of l i n e parameters and expe r imen ta l c o n d i t i o n s i s g iven in Tab le 3 .2 . For y the b roaden ing 3 'max 3 parameter was taken as 10.0 MHz/Torr and the d i p o l e moment (r69) as 3.33 x 1 0 " 5 D . The s i x tensor d i s t o r t i o n cons t an t s of Tab le 3.3 were o b t a i n e d by f i t t i n g the t r a n s i t i o n f r e q u e n c i e s , weighted a c c o r d i n g to V ° " m 2 , w i th an i t e r a t i v e l e a s t squares method ( r 1 5 ) . Examinat ion of Tab le 3.3 r e v e a l s that sepa ra te v a l ues of the s i x tensor d i s t o r t i o n c o n s t a n t s have been o b t a i n e d from the p resen t work a l o n e . These separa te v a l ues a l l ow the c a l c u l a t i o n of the v a r i o u s l i n e a r combina t ions de t e rm inab l e from the IR-MW data (Table 3 .4 ) . The e r r o r s g i ven i n the p resen t work are s tandard d e v i a t i o n s . The e r r o r s in the MW-IR work, not g i ven in the o r i g i n a l paper , have been e s t ima t ed from t h e i r s t a t e d f requency u n c e r t a i n t y . In Tab l e 65 3.5 the observed MW-IR t r a n s i t i o n s are p r e d i c t e d from the c o n s t a n t s ob ta ined from the ten Q branch t r a n s i t i o n s r e p o r t e d h e r e . The p r e d i c t i o n of the l i n e 7 F ( , 2 ) - 7 F 2 2 ) at 216.17 MHz con f i rms the p r e v i ous t e n t a t i v e ass ignment of K r e i n e r et a l . We can a l s o r e - a s s i g n the t r a n s i t i o n at 543.69(30) MHz to 20F 2 2 »-20F ( , 5 > . The v a r i o u s o ther unass igned MW-IR l i n e s c o u l d not be a s s i gned to any t r a n s i t i o n w i th J l e s s than 30. For the ground s t a t e , Boltzmann f a c t o r s make the i n t e n s i t y of the a b s o r p t i o n l i n e s too weak fo r J g r ea t e r than 30. Tab le 3.3 a l s o i n c l u d e s an a n a l y s i s of the microwave data combined wi th the MW-IR d a t a . The net r e s u l t i s to improve the s t a t i s t i c s . In a s i m i l a r study on 2 8 S i H „ ( r 56 ) , the h ighe r order d e c t i c c o n t r i b u t i o n s to the t r a n s i t i o n f r e q u e n c i e s were es t ima ted and found to be -3% of the c o n t r i b u t i o n s due to the o c t i c te rms . The a b s o l u t e parameter e r r o r a a t tempts t o i n c l u d e these h ighe r o rder e f f e c t s . 66 Tab le 3.1 Observed germane Q-branch r o t a t i o n a l t r a n s i t i o n s T r a n s i t i o n J Symmetry Frequency(MHz) t D e v i a t i o n ' 1 4 E< 2 ) _ E ( 1 ) 9 950.525(50) 0.010 16 E ( 2 ) _ E ( 1 ) 14 691.317(50) -0.013 1 6 E< 3 > _ E ( 2 ) 9 510.590(90) -0.010 1 7 E ( 2 ) _ E ( 1 ) 1 3 709.472(90) -0.005 18 E< 2 ) _ E ( 1 ) 22 555.780(80) 0.003 18 E ( 3 ) _ E < z ) 9 615.310(75) 0.008 19 E< 2 >_E< 1 ) 18 536.036(80) 0.005 19 E ( 3 ) _ £ ( 2 ) 1 5 293.350(100) 0.019 20 E< 3 ) _ E ( 2 ) 18 394.252(60) 0.029 20 E ( t ) _ E ( 3 > 1 0 897.180(100) -0.016 fObserved f requency minus f requency c a l c u l a t e d from d i s t o r t i o n c o n s t a n t s ob ta ined s o l e l y from microwave d a t a . Numbers in pa ren theses are e s t ima ted measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 6 7 Tab le 3 . 2 Summary of germane l i n e parameters and expe r imen ta l c o n d i t i o n s (see below fo r u n i t s ) T r a n s i t i o n J Symmetry 7 x 1 0 1 1 'max A ( t " , t ' ) S ta rk # of Mean cm" 1 F i e l d i ndep . runs time /run 1 4 E ( 2 : 1 — E ( 1 ' 3 . 7 - 2 . 7 9 1 5 0 - 2 0 0 3 1 .5 1 6 E ( 2 : ) _ E ( 1 ) 6 . 5 1 . 2 6 4 5 0 3 1 .6 1 6 E< 3 : >-E < 2 > 2 . 7 ' - 8 . 8 1 9 0 7 2 . 2 1 7 E< 2 » - E ' 1 1 7 . 8 - 4 . 4 4 1 2 0 3 2 . 4 1 8 E ( 2 >_ E< 1) 8 . 5 -1 . 0 7 1 0 0 0 4 2 . 5 1 8 E ( 3 > _ E <• 2 > 6.4 5 . 1 1 5 0 - 1 5 0 3 0 . 5 1 9 E L 2 ) _ E ( 1 > 1 0 . 6 2 . 0 4 4 0 0 3 6 . 5 1 9 E< 3 ) _ E 1 2 > 2 . 5 - 1 1 . 3 1 8 0 - 1 5 0 3 1 5 . 2 2 0 E ( 3 ) _ E < 2 ' 7.4 - 6 . 3 7 4 0 0 3 5 . 3 2 0 E ( 4 ) _ E ( 3 ) 1 .0 1 5 . 6 7 7 0 5 1 2 . 3 f P r e s s u r e ^ l 0 - 5 0 mTorr . U N I T S : A ( t " , t ' ) in (MHz D " ' ) (V/cm)- 1 S ta rk f i e l d in V/cm. Mean time per run in hours where an equa l amount of t ime was spent de te rm in ing the background. 6 8 Tab le 3.3 Tensor 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 of GeH a ( i n Her tz ) Combined microwave and Parameter H H 4T 6T L 4 T X 1 0 « L 6 T X 1 ° 4 L 8 T X 1 0 « Microwave + data o n l y 1 67 775.44(91) •5.3820(59) 2.9694(20) 3.985(94) •4. 127(51 ) •8.01(15) in f ra red-mic rowave data Va lue 67 775.54(86) -5.3827(55) 2.9693(19) 3.996(88) -4.122(50) -8.01(14) a 3.3 0.013 0.006 0.25 0.21 0.46 f O b t a i n e d from an a n a l y s i s of the t r a n s i t i o n s measured . in the p resen t work o n l y . * A n a l y s i s of the p resen t da ta toge the r w i th the seven a s s i g n e d IR-MW double resonance measurements l i s t e d in Tab le 3 .5 . v a i s the e s t ima ted a b s o l u t e e r r o r . See t e x t , a Numbers in pa ren theses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 69 Tab l e 3.4 Comparison of l i n e a r combina t ions of germane c e n t r i f u g a l d i s t o r t i o n cons t an t s ob ta ined in the p resen t work wi th those of r e f (r68) ( in Her tz ) Combinat ions of c o n s t a n t s t J=1 8 D T +342H 4 T +116 9 6 4 L 4 T H 6 T + 3 4 2 L 6 T L 8 T X 1 0 « J=18 and J=11 D -45 144L f l P n+1 320L 4T H 4 T + 4 7 4 L 4 T + 3 8 . 6 L 6 H 6 T + 3 4 2 L 6 T 6T Microwave 65 981(3) 2.828(3) -8.01(15) 67 762.9(10) -5.209(74) 2.8283 Inf rared-microwave * double resonance 65 980(3) 2.830(3) -6.0(40) 67 765.5 -5.220 2.830 f O b t a i n e d from an a n a l y s i s of the t r a n s i t i o n s measured in the p resen t work o n l y . Numbers in pa ren theses are s tandard d e v i a t i o n s in u n i t s of the l e a s t s i g n i f i c a n t f i g u r e s . * Obta ined from re f ( r 6 8 ) . Numbers in pa ren theses are es t ima ted o u t s i d e e r r o r l i m i t s from s t a t e d u n c e r t a i n t y in the measured f r e q u e n c i e s . 70 Tab le 3.5 P r e d i c t i o n of germane IR-MW doub le resonance t r a n s i t i o n s u s i n g cons t an t s ob t a i ned s o l e l y from the microwave spectrum T r a n s i t i o n J Symmetry 18 F (, * > — F 2 9 ' 18 F ^ - F ' , 2 ' 18 F ^ ' - F ' , 3 * 18 A ( 1 2 ) - A 2 2 ) 11 E < 2 )_g( 1 ) 7 a F\2)-F22> 1 3 b F ' / ' - F ^ 3 ' 2 0 b F 2 5 > - F ( , 5 > f O b t a i n e d from an a n a l y s i s of the t r a n s i t i o n s measured in the p resen t work o n l y . Numbers in pa ren theses are s t anda rd d e v i a t i o n s in the p r e d i c t i o n s . * Measured f requency from r e f . ( r 6 8 ) . Numbers in pa ren theses are e s t ima ted measurement u n c e r t a i n t i e s . ' a ' T e n t a t i v e assignment made in r e f . ( r 6 8 ) i s conf i rmed . ' b ' There are two p o s s i b l e ass ignments to the measured l i n e . The t e n t a t i v e assignment in r e f . ( r 6 8 ) to J=13 i s i n c o r r e c t ; the t r a n s i t i o n i s r e a s s i g n e d to J=20 in the p resen t work. (MHz) t P r e d i c t ion 1 6 633.147(17) 10 575.354(14) 2 973.111(10) ,895.768(6) 2 356 . 627(11) 216. 173(2) 541.707(1) 543.832(5) Measurement 6 633.15(10) 10 575.33(10) 2 973.15(10) 895.79(10) 2 356.68(10) 2 356.60(10) 216.17(20) 543.69(20) 71 5. ISOTOPE SPLITTING IN GERMANE The purpose of t h i s s u b - s e c t i o n i s to i n v e s t i g a t e the c l a i m tha t the i so tope s p l i t t i n g of the s p e c t r a l f e a t u r e s of germane i s not s i g n i f i c a n t in the measurements r e p o r t e d h e r e . The ev idence fo r a n e g l i g i b l e i so tope e f f e c t can be d i v i d e d i n t o two k i n d s ; expe r imen ta l and t h e o r e t i c a l . E x p e r i m e n t a l l y the e a r l i e s t ev idence s u p p o r t i n g the n e g l i g i b l e i so tope e f f e c t c l a i m was the d i s c o v e r y (r68) tha t two d i f f e r e n t l a s e r c o i n c i d e n c e s in the MW-IR double resonance exper iment y i e l d e d the same microwave f r equency . R e c a l l tha t K re ine r et a l (r68) a t t r i b u t e d t h i s to s i m i l a r r o t a t i o n a l t r a n s i t i o n s in d i f f e r e n t Ge i s o t o p i c s p e c i e s . If t h i s were the case then the i s o t o p i c s p l i t t i n g in the E type t r a n s i t i o n J=11, 2<—1 must be l e s s than the expe r imen ta l e r r o r in the two f requency measurements of around 140 KHz. A d d i t i o n a l expe r imen ta l ev idence suppo r t i ng t h i s c l a i m can be found in the p resen t work. F i r s t l e t us c o n s i d e r how w e l l the s i n g l e i so tope model used in the da ta r e d u c t i o n accounts f o r the s p e c t r a . S ince the rms d e v i a t i o n i s 38 KHz, compared w i th a t y p i c a l expe r imen ta l u n c e r t a i n t y of 80 KHz, we can conc lude that the model accounts fo r the da ta very w e l l . We can a l s o conc lude that e i t h e r a l l the t r a n s i t i o n s we have measured are from one i s o t o p i c s p e c i e s on l y and we have s imp ly missed the nearby " s i s t e r " s p e c t r a , or that the t r a n s i t i o n s measured are due to many i s o t o p e s , tha t w i t h i n our r e s o l u t i o n average to appear as one i s o t o p i c s p e c i e s . 72 In order to p r e c l ude the p o s s i b i l i t y of " s i s t e r " s p e c t r a c a r e f u l searches were made 10 MHz above and below s e v e r a l key t r a n s i t i o n s . The 10 MHz range was e s t a b l i s h e d as e x p l a i n e d e a r l i e r from the 1 3 C - 1 2 C methane work ( r 2 0 , r 6 4 , r 7 0 ) . The key t r a n s i t i o n s were de te rmined by no t i ng which parameters in the methane study were most a f f e c t e d by i s o t o p i c s u b s t i t u t i o n and then f i n d i n g which germane t r a n s i t i o n s were most r e l i a n t on these parameters (see the t h e o r e t i c a l argument tha t f o l l o w s ) . In no case was any ev idence of " s i s t e r " s p e c t r a found . If we accept the above n u l l r e s u l t then i t would appear that no i so tope s p l i t t i n g i s g r ea t e r than the expe r imen ta l l i n e w id th . To see i f i s o tope s p l i t t i n g c o u l d c o n t r i b u t e to the l i n e width of the measured t r a n s i t i o n s i t i s perhaps i n s t r u c t i v e to c o n s i d e r the average germane l i n e widths (FWHM) of 420 KHz and compare them to the t y p i c a l 2 8 S i H f t va lue of 430 KHz ( r56 ) . In the s i l a n e example such i so tope e f f e c t s were not expected as the re was on l y one i so tope of s i g n i f i c a n c e , namely 2 8 S i . T h i s compar ison must, however be taken "con g ranu lus s a l i s " as c e r t a i n l y p r e s s u r e s and number of scans in the s i l a n e and germane s t u d i e s depend to a l a r g e degree on how we l l the spec t rometer was working on a p a r t i c u l a r day in a p a r t i c u l a r f requency r e g i o n . More i m p o r t a n t l y low Stark f i e l d s were o f t e n p r e f e r r e d e x p e r i m e n t a l l y in order to min imize "background" e f f e c t s so tha t in some cases l i n e s were a r t i f i c i a l l y narrowed by "unde r " modu l a t i on . S t i l l , the f a c t tha t no anomalous l i n e 73 widths were observed suggests any s p l i t t i n g i s p robab ly we l l w i t h i n a t y p i c a l l i n e w id th . Be fore t u r n i n g to the q u e s t i o n of a t h e o r e t i c a l unde r s t and ing of why the Ge i so tope e f f e c t on the tensor s p l i t t i n g s shou ld be n e g l i g i b l e , i t shou ld prove b e n e f i c i a l as a t e s t to t r y to e x p l a i n the methane s i t u a t i o n where the e f f e c t s are q u a n t i t a t i v e l y known. In methane one f i n d s two s p e c t r a separa ted by 2 to 10 MHz fo r 1 3 C H „ and 1 2 C H „ . The mo lecu l a r cons t an t s fo r methane are g i ven in Tab le 3 .7 . Watson has .examined the methane r e s u l t s and accounts fo r them by ze ro po in t v i b r a t i o n e f f e c t s in D f c and i s o t o p i c r e l a t i o n s between the i s o t o p i c H 4 t and Hg 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 . Watson has shown (r74) f o r XY„ mo lecu les (of Td symmetry) where AH = -(256/35) [ m Am /m (M+Am ) ] B ^ $ 2 3 $ i « ( C J i 2 - a ) a 2 ) 2 [3 .4 ] V A A A C and M i s the t o t a l mass of the o r i g i n a l mo l e cu l e , that A H 4 f c = (7/88)AH [3 .5 ] A H 6 t = -(15/22)AH. [3 .6 ] Watson has demonstrated that equa t i ons (e3.4-e3.6) d e s c r i b e we l l the methane i s o t o p i c H 4 t and H g t r e s u l t s . He has f u r t h e r shown that the v a r i a t i o n in D f c i s c o n s i s t e n t wi th a ze ro p o i n t v i b r a t i o n a l e f f e c t but does not examine t h i s v a r i a t i o n t h e o r e t i c a l l y . A more p r e c i s e es t imate of the i s o t o p i c v a r i a t i o n of D f c i s c l e a r l y needed as the D f c c o n t r i b u t i o n to the t r a n s i t i o n f r e q u e n c i e s i s dominant . For 74 i n s t a n c e , of the measured methane spectrum the Q branch t r a n s i t i o n w i th the g r e a t e s t i s o t o p e s h i f t i s the E type l i n e J=1"6,3<—2 where out of a s h i f t of 9.6 MHz approx imate l y 5.4 MHz can be a t t r i b u t e d to the 0.03% change in D f c . Hecht (r67) has g iven the f o l l o w i n g exp re s s i on fo r D f c , TJ 3 2 y 2 •H " -^4 t3 .7 ] where $ i 3 =(2/3) (1-$ 3 ) and $| 4 = ( 2 / 3 ) ( 1 - $ „ ) a re second order c o n s t a n t s expressed in terms of the f i r s t o rder C o r i o l i s i n t e r a c t i o n c o n s t a n t s . Equa t ion (e3.7) i s s t r i c t l y an e q u i l i b r i u m q u a r t i c cons tan t and i s r e l a t e d to the e m p i r i c a l e f f e c t i v e cons tan t in much the same way as the 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 a re r e l a t e d to the B 0 v a l u e s . A c c o r d i n g to Watson (r74) the e q u i l i b r i u m parameters B , D g and D f c a re a l l independent of the c e n t r a l mass so any i s o t o p i c v a r i a t i o n in D f c due to the c e n t r a l mass must be from the c o r r e c t i o n s to D f c e q u i l i b r i u m that r e s u l t in D f c e f f e c t i v e . In o rde r to a p p r e c i a t e how D f c might change w i th respec t to changes i n the c e n t r a l mass i t i s i n s t r u c t i v e to examine the methane v i b r a t i o n problem in some d e t a i l . T h i s w i l l a l s o h e l p a v o i d c o n f u s i o n in the l a t e r t reatment of ammonia-l ike mo lecu les where i d e n t i c a l symbols w i l l co r r espond to d i f f e r e n t t h i n g s . Of p a r t i c u l a r i n t e r e s t w i l l be how the v a r i o u s parameters in equa t ion (e3.7) behave when the c e n t r a l atom i s i s o t o p i c a l l y changed. 75 The symmetry c o o r d i n a t e s and normal v i b r a t i o n s of Td mo lecu les have the f o l l o w i n g p o i n t group d e s i g n a t i o n s : (A ,+E+F 2 ) . In a symmetry c o o r d i n a t e r e p r e s e n t a t i o n the f o r c e cons tan t F , , in the A, s p e c i e s and F 2 2 in the E s p e c i e s are immediate ly determined from the a p p r o p r i a t e va lues of the A, and E v i b r a t i o n a l f r e q u e n c i e s . In Tab le 3.6 we see tha t F , , can be w r i t t e n in terms of bond s t r e t c h va l ence f o r c e cons t an t s on l y and t h e r e f o r e we a t t r i b u t e the A, v i b r a t i o n 60, and the F , , f o r c e cons tan t to the t o t a l l y symmetric b r e a t h i n g v i b r a t i o n . Tab le 3.6 Comparison of f o r c e cons t an t s f o r methane, s i l a n e and germane : see r e f . ( r 7 6 ) f o r d e f ' n of p o t e n t i a l C H „ a S i H « b G e H , b F, , = f +3f . mdyn/A 5.435(8) 2 .84(6) 2 .65(5) mdyn A 0 .584 5 ( 1 ) 0 .41(1) 0 .40(1) F 3 3 = f -f , mdyn/A 5.378(8) 2 .74(6) 2 .63(5) F 3 « = / 2 < f r r - f r p > mdyn -0.221(3) -0 .03(3) -0 . 12(3)0) mdyn A 0.548(1) 0 -50 5 (1 ) 0 .46(1) ' a ' r e f ( r 7 5 ) •b' r e f ( r 7 6 ) Numbers in pa ren theses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 76 S i m i l a r l y , the E v i b r a t i o n co2 co r responds to a t o t a l l y ang le bending v i b r a t i o n . These v i b r a t i o n s (normal c o o r d i n a t e s ) are shown s c h e m a t i c a l l y in Herzberg (r37 p g . 1 0 0 ) . The important p o i n t here i s that the X atom of XY,, i s always at the system cen t e r of mass and so the va lues of co, and co2 do "Qt depend  on the mass of X. For the F 2 s p e c i e s there are th ree f o r c e c o n s t a n t s and on ly two v i b r a t i o n s . From t a b l e 3 .6 , F 3 3 co r responds to " p u r e " bond s t r e t c h i n g and F f t f t to " p u r e " ang le bend ing ; however the i n t e r a c t i o n cons tan t F 3 a coup les these s t a t e s to g i ve two v i b r a t i o n s that are l i n e a r combina t ions of both angle bends and bond s t r e t c h e s . These v i b r a t i o n s co r respond to l i n e a r combina t ions of the "v3" and "vn" v i b r a t i o n s p i c t u r e d by Herzberg ( r 37 ) . The important p o i n t here i s that the X atom i s now d i s p l a c e d from the system cen te r of mass and hence fo r the v i b r a t i o n s co3 and co,,  the mass of X w i l l be impor t an t . From a harmonic d i a tomic approx imat ion one would expect the change in the co3 and a>„ f r e q u e n c i e s would vary rough ly as 1/VM where M i s the reduced mass. As an example see the 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 S i F „ , S i=28,29,30 { t i l ) . The p r e ced ing d i s c u s s i o n i s summarized by the T e l l e r and R e d l i c h product r u l e (r37 pg .232 , r78 ) which fo r XYj, Td mo lecu les i s i-m i 1 /2 [3.8] and 77 m r m 1 x [3.9] We now c o n s i d e r the C o r i o l i s cons t an t s in equa t ion ( e 3 . 7 ) . The r o t a t i o n a l l e v e l s of the t r i p l y degenerate F v i b r a t i o n s are s p l i t i n t o th ree s u b l e v e l s by a C o r i o l i s i n t e r a c t i o n c h a r a c t e r i z e d by the C o r i o l i s c o u p l i n g cons t an t s $ 3 , $« ( r37 ,p447 ) . The c o u p l i n g cons t an t s r ep resen t r o t a t i o n a l angu la r momentum a s s o c i a t e d wi th the v i b r a t i o n . The q u a n t i t y i s not n e c e s s a r i l y an i n t ege r and proves to be a f u n c t i o n of the normal mode of v i b r a t i o n ( r 79 ) . E x p r e s s i o n s fo r these c o u p l i n g ze t a cons t an t s as f u n c t i o n s of p o t e n t i a l c o n s t a n t s , f r e q u e n c i e s and mass are g i ven in r e f . ( r 7 7 , r 8 0 ) . The problem of i s o t o p i c dependence of C o r i o l i s c o u p l i n g has been examined in d e t a i l by Cyv in ( r 4 5 , p 3 4 2 f f ) . In C y v i n ' s f i g . 16.3 on h i s page 359 where the " t o p " graph i s what we c a l l $„ (and the "bot tom" our $ 2 « ) the change in C o r i o l i s c o u p l i n g expected when m^ changes from 1 2 C to 1 3 C i s sma l l and we l l w i t h i n the usua l "ha rmon ic " e r r o r a s s o c i a t e d w i th C o r i o l i s c o u p l i n g cons t an t s of ±.01 to ±.02 ( r 4 2 , r 8 l ) . T h i s sma l l change i s suppor ted by the expe r imen ta l numbers t a b u l a t e d by Gray and R o b i e t t e (r75) in t h e i r Tab l e 6. No t i c e however that t h e i r sum of $ 3 + $ f l f o r both 1 2 C H „ and 1 3 C H „ i s rough ly .509 . From the ze t a sum r u l e (r37) we expect the ze t a cons t an t s of Gray and R o b i e t t e to sum to e x a c t l y 1/2, so perhaps we can take the d i s c r e p a n c y as a measure of the r e l i a b i l i t y of these ze t a c o n s t a n t s . F i n a l l y , w i thout go ing i n t o a l l the d e t a i l , 78 examina t ion of the above r e f e r e n c e s shou ld a l l ow us to conc lude t h a t , w i t h i n " e r r o r " , the ze ta c o n s t a n t s do not  vary w i th i s o t o p i c s u b s t i t u t i o n 1 3 C - 1 2 C . We are now in a p o s i t i o n to summarize some key p o i n t s in the v a r i a t i o n of D f c w i th c e n t r a l mass. R e f e r r i n g aga in to the e x p r e s s i o n fo r D f c , equ . (e3.7) we have perhaps g i ven some i n d i c a t i o n that the X - i s o t o p i c v a r i a t i o n of the terms in the square b racke t fo r methane shou ld be s m a l l . I t shou ld a l s o be apparent tha t in the germane c a s e , where the change in reduced mass i s rough ly an order of magnitude l e s s , the change in the b racke ted term would be even s m a l l e r . I t shou ld now be h e l p f u l to o u t l i n e the argument l e a d i n g to an es t imate of the germane i s o t o p i c change in D f c . F i r s t f o r methane we wi l . l show, under the assumpt ion tha t the " square b r a c k e t " term in equa t ion (e3.7) i s i n v a r i a n t to c e n t r a l atom i s o t o p i c s u b s t i t u t i o n , that the change in D f c can be e x p l a i n e d in terms of zero po in t v i b r a t i o n a l changes in bond l eng ths a f f e c t i n g the va lue of B 5 . Then s i n ce the i n v a r i a n c e of the b racke t term shou ld be even more 5 As a check on t h i s procedure we w i l l put the v a r i o u s e m p i r i c a l parameters ( a l b e i t r a the r i n a c c u r a t e va lues ) i n t o equa t i on (e3.7) and f i n d , tha t w i t h i n e r r o r , the square b racke t term does not vary wi th i s o t o p i c s u b s t i t u t i o n of the c e n t r a l atom. T h i s does not mean that the above assumpt ion i s n e c e s s a r i l y c o r r e c t r a the r tha t the a v a i l a b l e data does not p r e c l u d e i t . 79 pronounced fo r germane the ze ro p o i n t e f f e c t in B w i l l be used to es t imate AD f c in germane. F i r s t l e t us c o n s i d e r the change in D t i f the square b racke ted term i s c o n s t a n t . The B r o t a t i o n a l cons tan t does not e x p l i c i t l y depend on the c e n t r a l mass but there i s an i m p l i c i t dependence in the i s o t o p i c a l l y dependent zero p o i n t averaged bond l eng ths (the g r ea t e r the reduced mass the s l i g h t l y sho r t e r the v i b r a t i o n a l l y averaged bond l e n g t h ) . For methane- l ike mo lecu les i t i s s t r a i g h t f o r w a r d to show that 6D AD f c = - Ar [3 .10] where r i s the bond l e n g t h and Ar the change in r on i s o t o p i c s u b s t i t u t i o n . The problem now i s to es t imate A r . One way i s to use the d i a tomic approx imat ion ( r54 , r82 ) 4/FTL VMi / M ; [3 .11] where F 2 and F 3 are the d i a tomic p o t e n t i a l c ons t an t s d e f i n e d by the p o t e n t i a l U ( r ) 2U(r) = F 2 ( r - r e ) 2 + F 3 ( r - r e ) 3 + [3.12] The c o n s t a n t s F 2 and F 3 are s imply r e l a t e d to the va l ence s t r e t c h i n g p o t e n t i a l c ons t an t s f (or f ) and f which ^ ^ r r r r r r come from a q u a d r a t i c and cub i c (anharmonic) a n a l y s i s r e s p e c t i v e l y . The methane Ar w i l l be es t ima ted from the p u b l i s h e d anharmonic f o r c e f i e l d of Gray and Rob i e t t e (r75) 80 and a l s o from the e m p i r i c a l parameters of Herschbach and L a u r i e ( r 52 ) . The l a t t e r g ive c redence to the Herschbach and L a u r i e e s t i m a t i o n s which , because of the l ack of a germane anharmonic f o r c e f i e l d , w i l l be the so l e e s t i m a t i o n of Ar in germane. Some care must be taken in e s t a b l i s h i n g the va l ues of F 2 and F 3 from the anharmonic f o r c e f i e l d study ( r 7 5 ) . T h i s i s because the p o t e n t i a l s are d e f i n e d d i f f e r e n t l y than in equa t i on ( e3 .12 ) . The p o t e n t i a l in (r75) i s d e f i n e d fo r the bond s t r e t c h as in r e f e r e n c e (r83) and compar ison w i th equa t i on (e3.12) shows a f a c t o r of 3 d i f f e r e n t in d e f i n i t i o n s . A l s o (r83) assumed a Morse p o t e n t i a l f o r the C-H s t r e t c h and r e f e r e n c e to (r49) equa t i on (3) and (r53) equa t ion (5) shou ld show f F 3 = " - f ^ [3.13] where f r r r i s g i ven in Tab le 4 of ( r 7 5 ) . The va lue of F 2 i s j u s t f which can be determined from F , , and F 3 3 i n (r75) and the equa t i ons i n . T a b l e 3 .6 . In t h i s way we get F 2 = 5.403 mdyn/A and F 3 = 10.49 mdyn/A2 and then from equ . ( e3 .11 ) A r , 2 . , 3 = 5.0 x 10~ 5 A. Us ing the e m p i r i c a l r e l a t i o n s of Herschbach and L a u r i e (r52) one f i n d s F 2 = 4.98 mdyn/A and F 3 = 7.96 mdyn/A2 and then A r , 2 . , 3 = 4.3 x 10~ 5 A e s t a b l i s h i n g some c o n f i d e n c e in the e m p i r i c a l method of Herschbach and L a u r i e . Us ing the va lue of 5.0 x 10~ 5 A fo r A r , 2 . 1 3 one c a l c u l a t e s a A D f c , 2 . 1 3 from equa t ion ( e3 . l 0 ) of -37 Hz . (r = 1.0858 A(r75)) which compares very we l l w i th 81 the e x p e r i m e n t a l l y de termined va lue of - 3 8 . 7 ( ± 3 . 0 ) Hz . In l i g h t of the approx imat ions l e a d i n g up to the AD t e s t ima te i t i s l i k e l y that t h i s agreement wi th exper iment i s somewhat f o r t u i t o u s . Now we shou ld c o n s i d e r the e f f e c t on AD f c when the square b racke ted term i s a l l owed to v a r y . The d i f f e r e n c e i s not s i g n i f i c a n t , the e r r o r which i s hard to es t imate be ing on the o rde r of the d i f f e r e n c e , but s t i l l one o b t a i n s a va lue of -75 Hz us ing the exper imenta l numbers of Gray and R o b i e t t e ( r 75 ) . T h i s i s very s i m i l a r to the v a r i a t i o n found i f the square b racke ted term i s h e l d c o n s t a n t . However the e r r o r i s very l a r g e and so the a v a i l a b l e expe r imen ta l da ta do not p r e c l u d e the assumpt ion of i n v a r i a n c e of the b racke t term in equa t ion ( e 3 . 7 ) , an assumpt ion expected to be even b e t t e r f o r germane. J us t be fo re t u r n i n g our a t t e n t i o n to the germane problem one f u r t h e r p o i n t shou ld be d i s c u s s e d . If the square b racke t in equa t ion (e3.7) i s cons tan t then one c o u l d a l s o w r i t e AD f c = (3D^./B) AB. For methane AB 0 i s known (r75) and one ge ts a A D f c , 2 . i 3 = -11.4 Hz . T h i s r e s u l t i s a l s o s i m i l a r to the -37 Hz p r e d i c t e d from zero p o i n t v i b r a t i o n . T h i s shows tha t our zero p o i n t d i f f e r e n c e in bond l eng th es t ima te of AB i s reasonab le f o r methane. F i n a l l y we are ready to es t imate AD .^ f o r germane w i t h i n the q u a l i f i c a t i o n s s t a t e d fo r methane. The parameter F 2 i s easy to c a l c u l a t e . From Herschbach and L a u r i e (r52) one f i n d s F 2 = 2.59 mdyn/A. T h i s compares very we l l w i th the 82 va lue of f r ( = f r r ) = F 2 = 2.63 mdyn/A c a l c u l a t e d from the va lues of F , , and F 3 3 of Duncan and M i l l s (r76) p l u s r e l a t i o n s in Tab le 3 .6 . T h i s l a t t e r va lue we s h a l l use in subsequent c a l c u l a t i o n s . To es t ima te F 3 from Herschbach and L a u r i e one needs a va lue of a 1 f one of the a lpha cons t an t s d e f i n e d in s e c t i o n 1.6 where 1 r e f e r s to the b r e a t h i n g mode C J , . However on l y the v a l ues a3 and a f l (r63) are known. S ince F 3 4 i s sma l l fo r germane, see Tab l e 3 .6 , the mode co3 i s a lmost pure bond s t r e t c h and hence a , ^ a 3 . T h i s i s a l s o seen to be the case in methane where F 3 „ i s much l a r g e r but s t i l l a 1 = a 3 ( r 7 5 ) . For our purposes then a , was taken to be equa l to a 3 (0.015 c m - 1 r e f . ( r 6 3 ) ) . For a bond l e n g t h of 1.527 A ( r 76 ) , F 3 = 2.97 mdyn/ A 2 and A r 7 0 . 7 , = 5.2 x 10~ 6 A . T h i s change in r co r responds to a change, AD f c of 1.4 Hz, or a .002% change in D f c . The s t anda rd d e v i a t i o n e r r o r in the e m p i r i c a l l y determined va lue of D f c i s 0.86 Hz so the change p r e d i c t e d in D f c due to ze ro p o i n t v i b r a t i o n a l e f f e c t s due to a c e n t r a l atom i s o t o p i c s u b s t i t u t i o n i s w i t h i n two s tandard d e v i a t i o n s . From the parameters of Ka t t enbe rg et a l (r63) and equa t i ons (e3.4-e3.6) the changes A H ^ t and AH^ t a re e s t i m a t e d . Assuming a .002% change in D f c a l l ows the c o n t r i b u t i o n s to an i s o t o p i c f requency s h i f t , and f i n a l l y a t o t a l expec ted s p l i t t i n g to be c a l c u l a t e d . In a l l cases the i so tope s p l i t t i n g i s s i m i l a r to or l e s s than the l i n e widths (~ 420 KHz) . The t o t a l s h i f t f o r the J=11 E ( 2 ) < - E < 1 > i s 44 KHz. R e c a l l t h i s i s the t r a n s i t i o n tha t K r e i n e r et a l (r68) 83 observed w i th two d i f f e r e n t l a s e r c o i n c i d e n c e s . The p r e d i c t e d s p l i t t i n g f o r t h i s t r a n s i t i o n i s then we l l w i t h i n the 100 KHz e r r o r s of K r e i n e r et a l . In c o n c l u d i n g t h i s d i s c u s s i o n on the i so tope s p l i t t i n g i n germane i t shou ld seem apparent that the ev idence p resen ted here suppor t s the e a r l y c l a i m of K r e i n e r et a l tha t the s p l i t t i n g i s , f o r our pu rposes , indeed n e g l i g i b l e . The n u l l r e s u l t s of sea rches fo r " s i s t e r " s p e c t r a and the s i m i l a r i t y of GeH„ l i n e widths to those of 2 8 S i H „ agree w i th a q u a s i - t h e o r e t i c a l study sugges t i ng tha t the spectrum r epo r t ed here i s an average over a l l Ge i s o t o p i c s p e c i e s . 6. FINAL COMMENTS ON GERMANE We shou ld now summarize, the th ree gene ra l r e s u l t s of the germane s tudy . F i r s t , a s u f f i c i e n t number of f o r b i d d e n d i s t o r t i o n moment Q branch t r a n s i t i o n s have been observed in the microwave spectrum to a l low the d e t e r m i n a t i o n of a l l s i x independent tensor d i s t o r t i o n c o n s t a n t s in equa t ion ( e l . 1 0 ) . These cons t an t s prove to be in e x c e l l e n t agreement w i th the MW-IR double resonance study of K r e i n e r et a l (r68) c o n f i r m i n g a l l t h e i r d e f i n i t e ass ignments p l u s one t e n t a t i v e l y a s s i g n e d l i n e as we l l as e n a b l i n g reass ignment of one other t e n t a t i v e l y a s s i gned t r a n s i t i o n . The f i n a l a n a l y s i s i s made from the combined MW-IR data and the data of t h i s s tudy . Second l y , the p resen t study adds new ev idence s u p p o r t i n g the c l a i m tha t the i so tope s p l i t t i n g in germane 84 due to Ge i so topes i s n e g l i g i b l e . The ev idence p r e sen ted here was d i v i d e d i n t o two t y p e s , expe r imenta l and theo re t i c a l . T h i r d l y the e m p i r i c a l parameters ob ta ined here can be used to p r e d i c t a ccu ra te tensor s p l i t t i n g s f o r J up to and i n c l u d i n g 20. These p r e d i c t e d s p l i t t i n g s have a l r e a d y been p u b l i s h e d ( r 84 ) . These p r e d i c t e d t ensor s p l i t t i n g s would be u s e f u l i n f u r t h e r MW-IR s t u d i e s and in i n f r a r e d and Raman AJ=1 R branch spec t r a of h igh r e s o l u t i o n . A l s o o r tho-pa ra t r a n s i t i o n s of the type found f o r methane ( r 85 , r5 ) shou ld be e a s i e r to f i n d as a r e s u l t of the tensor c o n s t a n t s p resen ted here — see re f ( r86 ) . (Here o r tho and para r e f e r to the t o t a l nuc l ea r sp in I produced by the p r o t o n s ; pa ra 1=0, o r tho 1=1 and meta 1=2.) F i n a l l y i t i s i n t e r e s t i n g to note the s i m i l a r i t y of the GeH a and S i H „ parameters in Tab l e 3 .7 , l i s t e d on the next page, and how d i f f e r e n t these va lues are when compared to C H „ . R e c a l l from Tab le 3.6 tha t the f o r ce c o n s t a n t s show a s i m i l a r t r e n d . La te r we w i l l f i n d tha t a s i m i l a r t r e n d e x i s t s in the s e r i e s N H 3 , PH 3 and A s H 3 . 85 Tab le 3.7 Comparison of s p e c t r o s c o p i c p a r a m e t e r s : 1 2 C H „ , 2 8 S i H „ and GeH, Parameter 1 2 CH, 2 8 S iH, GeH, B 0 ( cm" 1 ) 5 . 24 l0356 (96 ) a 2 . 8 5 9 0 6 5 d O ) b 2. 6969 (9 ) C D T (Hz) 132 9 4 3 . 4 ( 3 7 ) d 74 751 .4(36) 67 775. 5 4 ( 8 6 ) e H 4 T (Hz) -16.984(23) -6.044(15) - 5 . 3827(55) H g T (Hz) 11,034(19) 2.598(5) 2. 9693(19) L 4 T ( H z ) x l O a 20.3(9) 4.65(29) 3. 996(88) L 6 T (Hz ) x lO a -26.8(12) -3.79(19) -4 . 122(50) L g T ( H Z ) X 1 0 " -30.0(27) -7.66(44) -8 . 01(14) 0 X y ( D ) X 1 0 5 z 2 . 4 1 ( 5 ) h 3 . 7 3 ( 4 ) f 3. 3 3 ( 5 ) 9 ' a ' r e f e r e n c e (r85) ' b ' r e f e r ence (r56) ' c ' r e f e r ence (r63) ' d ' r e f e r ence (r20) ' e ' p resen t work. ' f C o r r e c t e d from va lue p resen ted in r e f . ( r 8 7 ) . T h i s d i p o l e moment was o r i g i n a l l y r epo r t ed as equa l to 3 . 3 4 ( 4 ) x 1 0 " 5D. T h i s va lue has been r e v i s e d in t h i s work to 3 . 7 3 ( 4 ) x 1 0 " 5D. The change i s a lmost e n t i r e l y due to a sma l l c o r r e c t i o n in the c e l l c a l i b r a t i o n but i n c l u d e s a sma l l c o n t r i b u t i o n from a h ighe r order c a l c u l a t i o n of the l i n e a r S tark c o e f f i c i e n t . ' g ' r e f e r e n c e (r69) ' h ' r e f e r ence (r5) — Numbers in pa ren theses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 86 E. CHAPTER 4: ARSINE A r s i n e i s a c o l o r l e s s , po i sonous gas pu rpo r t ed to have a m i l d g a r l i c odo r . I t s t o x i c i t y i s wel l-documented by the dub ious v i r t u e of i t be ing a m i l i t a r y po i son gas . Today i t i s used more as a dop ing agent ( r 8 8 ) . A h i s t o r y of a r s e n i c i s g i ven by J .W. M e l l o r (r89) who quotes T . Be rgmar i s ' "De a r s e n i c o " (Upsala 1777) sugges t i ng tha t a r s e n i c was d i s c o v e r e d by sme l t e r s of ore as i t would g i ve o f f a whi te smoke, g a r l i c s m e l l , deprave meta l s and des t roy l i f e . Our word " a r s e n i c " p robab l y d e r i v e s from the P e r s i an "ZARNICH" which became the Greek word fo r " v a l i a n t " or " b o l d " , a r e f e r e n c e perhaps to the way a r s e n i c a c t s on o ther m e t a l s . A r s i n e i t s e l f da tes from about 1775. A r s i n e was hard on many e a r l y r e s e a r c h e r s , such as A . F . Gehlen who, in o rder to d e t e c t l e a k s , s n i f f e d s t r o n g l y at the j o i n t s of h i s a p p a r a t u s . . . . The f i r s t r e p o r t e d microwave measurements of symmetric top a r s i n e s , measured in 1955, were the J=0—>1 t r a n s i t i o n s of both normal and f u l l y deu t e r a t ed forms ( r90 ) . Because of the sma l l d i p o l e moment (0.22 Debye ) , and hence sma l l t r a n s i t i o n moment, and the f u r t h e r i n t e n s i t y deg rada t i on due to the m u l t i p l e nuc l ea r quadrupo le s p l i t t i n g of l i n e s , f u r t h e r t r a n s i t i o n s c o u l d not be measured at tha t t ime . Fu r the r s tudy was c a r r i e d out i n 1971 by Helmi-nger et a l ( r 9 l ) who measured th ree a l l owed t r a n s i t i o n s fo r a r s i n e and s i x a l l owed t r a n s i t i o n s fo r a r s ine-D3 in the m i l l i m e t e r wave range . From t h e i r data they were a b l e to c a l c u l a t e the B 0 , 87 Dj and D j K r o t a t i o n a l parameters p l u s the h y p e r f i n e parameters eqQ (where q=<b2 4>/dz2>) , C M and C „ . In 1974 Chu and Oka (r82) r epo r t ed four l i n e s in the f o rb idden AK=±3 r o t a t i o n a l spectrum of AsH 3 and were ab le to c a l c u l a t e v a r i o u s l i n e a r combina t ions of r o t a t i o n a l pa ramete rs . O l s o n , Maki and Sams ( r92 ) , one year l a t e r , were ab l e to r epor t ground s t a t e r o t a t i o n a l parameters B 0 , C 0 , D T , D T T . , D.. p l u s h igher o rder terms, from an a n a l y s i s of t h e i r h igh r e s o l u t i o n i n f r a r e d measurements. In 1978 Helms and Gordy (r26) r e p o r t e d , but ana l yzed us ing an obscure r e d u c t i o n , e l even new fo rb idden AsH 3 t r a n s i t i o n s . F i n a l l y du r i ng the course of t h i s work an e x t e n s i v e study of both a l lowed and f o rb idden AsH 3 t r a n s i t i o n s was p resen ted by Buren in et a l ( r 93 ) . T h i s l a s t study i s p a r t i c u l a r l y impress i ve as s e x t i c as we l l as o c t i c r o t a t i o n a l parameters c o u l d be r epo r t ed because of the very h i g h f requency measurements (up to 900 GHz) made p o s s i b l e by t h e i r very h igh f requency sou r ce s . The i n f r a r e d spectrum of AsH 3 i s we l l c h a r a c t e r i z e d (r92 , r94 , r95 , r96 ) and, where a p p r o p r i a t e , the ground s t a t e mo lecu l a r cons t an t s c a l c u l a t e d from the i n f r a r e d spec t r a are in e x c e l l e n t agreement w i th those of Buren in et a l ( r 93 ) . The AsD 3 (r97) and A s T 3 (r98) i n f r a r e d s p e c t r a are l e s s w e l l e s t a b l i s h e d and on ly fundamental v i b r a t i o n f r e q u e n c i e s have been r e p o r t e d . The p r e v i o u s p a r t i a l l y deu te r a t ed asymmetric top a r s i n e data c o n s i s t of two l i n e s fo r AsH 2 D repo r t ed by Loomis and S t randberg (r99) ( 1 0 i _ 1 i i at 35.5 GHz and 3 0 3 - 3 1 3 at 29.5 88 GHz) . In t h i s case the observed t r a n s i t i o n s were a by-product of a study on s t i b i n e , S b H 3 . The most p robab le reason fo r t h i s l ack of i n f o r m a t i o n on asymmetric top a r s i n e s (and phosph ines , as i s d i s c u s s e d l a t e r ) stems from the d i f f i c u l t y in o b t a i n i n g the R ( A J = 1 ) branch t r a n s i t i o n s . It tu rns out that these R branches are to be found at f r e q u e n c i e s above 1 0 0 GHz, which i s above the f requency c a p a b i l i t y of most spec t rome te r s . R e c a l l tha t both R and Q ( A J = 0 ) branches are necessa ry in o rder to e va lua t e the r o t a t i o n a l parameters s e p a r a t e l y so that any s tudy r e p o r t i n g on ly low f requency Q branches c o u l d at best r e p o r t on ly l i n e a r comb ina t ions of pa ramete rs , a f ea tu re which i s not very u s e f u l f o r s t r u c t u r a l d e t e r m i n a t i o n . In t h i s c h a p t e r , a f t e r a b r i e f d i s c u s s i o n of how the a r s i n e gas samples used fo r t h i s s tudy were p r e p a r e d , are g i ven d e s c r i p t i o n s of f i r s t the AsD 3 spectrum and second the asymmetric top AsH 2 D and AsD 2 H s p e c t r a a long w i th important p o i n t s of p r e d i c t i o n , o b s e r v a t i o n , and a n a l y s i s . Because of the s i m i l a r i t y of the a r s i n e and phosphine s t u d i e s f u r t h e r a n a l y s i s of a r s i n e in terms of harmonic f o r ce f i e l d s , s t r u c t u r e s and d i s t o r t i o n d i p o l e moments w i l l be p resen ted l a t e r , so tha t a r s i n e and phosphine can be s t u d i e d toge ther (see Chapter 6 ) . 89 1. PREPARATION OF THE ARSINE GAS SAMPLE There are many p r e p a r a t i v e methods fo r a r s i n e , many of which can be found in the review by M e l l o r ( r 89 ) . For the p resen t s t u d i e s a m o d i f i e d v e r s i o n of the method of Drake and R i d d l e (rlOO) was u sed . The procedure was to drop water on a sodium a r sen ide a l l o y (Na 3 As) p repared " i n s i t u " under an i n e r t atmosphere of dry n i t r o g e n or argon at p r e s su r e s around 100 t o r r . The a l l o y was made by hea t i ng 3.0 g . As and 1.5 g . Na meta l very g e n t l y w i th a semi luminous Bunsen burner f lame u n t i l the sodium metal began to me l t . A r a p i d exothermic r e a c t i o n f o l l o w e d in which the r e a c t i o n mix ture glowed red-hot and a b l a ck smoke was g i ven o f f . Drake and R i d d l e suggest do ing the a l l o y p r e p a r a t i o n in a n i c k e l c r u c i b l e , i n s u l a t e d from the g l a s s r e a c t i o n v e s s e l by g l a s s wool in o rder to p r o t e c t the r e a c t i o n v e s s e l from e x c e s s i v e hea t . We found t h i s method u n s a t i s f a c t o r y . I t was d i f f i c u l t to see what was happening in the r e a c t i o n ; a l s o , heat t r a n s f e r was s low. I n s t e a d , we opted fo r a qua r t z r e a c t i o n v e s s e l which cou ld w i t h s t and the h igh " r e d - h o t " temperature of the r e a c t i o n forming the a l l o y and d i spensed wi th the n i c k e l c r u c i b l e . A f t e r the a l l o y had been formed the system was a l l owed to coo l and was then tho rough l y evacua ted . Con t r a r y to the method of Drake and R i d d l e the 100 mm. i n e r t atmosphere of n i t r o g e n or argon was r e - e s t a b l i s h e d (see be low) . To produce a r s i n e approx imate l y 15 m i s . of water , heavy water or mixtures of the two, as a p p r o p r i a t e , were a l l owed to d r i p s low ly onto the a l l o y . Gas bubbles evo l ved 90 in the r e a c t i o n m i x t u r e . The p r o g r e s s of the r e a c t i o n was f o l l owed by m o n i t o r i n g the p r e s s u r e . The 100 mm. argon or n i t r o g e n atmosphere was found necessa ry fo r the water to " d r o p " ; under vacuum the water drops f r o z e to the end of the d ropp ing f u n n e l . The r e a c t i o n p roduc t s were passed through a t r a p fo r water (C0 2 /pe t ro l eum e the r T ^-80 °C ) and f i n a l l y t rapped under l i q u i d n i t r o g e n ( -196 °C ) where any non-condensable gases were pumped o f f . Water was f u r t h e r removed v i a t r a p - t o - t r a p d i s t i l l a t i o n through a pe t ro leum e t h e r / l i q u i d n i t r o g e n s l u s h . The presence of AsD 3 produced by t h i s method was con f i rmed by a Pe rk in Elmer 457. low r e s o l u t i o n i n f r a r e d spectrum reco rded at a p r e s su re of 5 To r r f o r a path l e n g t h of 10 cm. showing the c h a r a c t e r i s t i c fundamental v i b r a t i o n f r e q u e n c i e s (see chapte r 6 ) . The presence of the p a r t i a l l y deu t e r a t ed a r s i n e s was e s t a b l i s h e d from t h e i r vapour p r e s s u r e of 35 mm. at - 1 1 1 ° C , (1-bromobutane, or carbon d i s u l f i d e s l u s h r101) or from the o b s e r v a t i o n of p r e v i o u s l y measured microwave r o t a t i o n a l t r a n s i t i o n s ( r 99 ) . The a r s i n e s were s t o r e d in g l a s s bu lbs f i t t e d w i th greased s t o p c o c k s . 2. PREDICTION OF THE ASD 3 SPECTRUM In o rder to a p p r e c i a t e the d i f f i c u l t i e s in p r e d i c t i n g f o r b i d d e n AK=±3 s p e c t r a i t i s i n s t r u c t i v e to c o n s i d e r the parameters i n v o l v e d in a l lowed and f o r b i d d e n t r a n s i t i o n s of o b l a t e r o t o r s l i k e a r s i n e (and phosphine in the next c h a p t e r ) . To degree 4 in components of angu la r momentum 91 o p e r a t o r s , v N ( J ,K ) = 2 ( J+1)B 0 -2( J+1)K 2 D J K -4( J+1 ) * D j [4 .1 ] P d ( J , K ) = (6K+9) ( B 0 -C 0 )+ (2K 2 +6K+9)D R +j ( J+1 )D J R [4 .2] where ? N ( J ,K ) i s the f requency of the a l lowed (J,K)—> (J+1,K) t r a n s i t i o n , and f D ( J , K ) i s the f requency of a fo rb idden Q branch d i s t o r t i o n ( J , ± K ) from ( J , ± K * 3 ) t r a n s i t i o n (here K i s the more nega t i ve va lue of K i n the upper s t a t e ) From the work of Helminger et a l ( r 9 l ) on the normal AsD 3 spectrum we i n i t i a l l y had va lues of B 0 , D_„ and Bj. S ince does not appear in equa t ion ( e 4 . 2 ) , on l y B 0 and D J K were u s e f u l in p r e d i c t i n g the AsD 3 d i s t o r t i o n moment spect rum. In order to e s t a b l i s h the s e a r c h ' r e g i o n f o r the AsD 3 study " g u e s s t i m a t e s " of the r o t a t i o n a l parameters C 0 and D„ were made. A very r easonab le es t imate of D„. can be r D K (PD 3 ) D„ (PH 3 ) D R ( A s H 3 ) . The made from the " r a t i o " r e l a t i o n va lues f o r P D 3 , PH 3 and AsH 3 were taken from r e f e r e n c e s ( r 2 5 , r l 0 2 , r 9 3 ) , g i v i n g an es t ima ted D R f o r AsD 3 of 0.856 MHz, which compares we l l w i th the c a l c u l a t e d f o r c e f i e l d va lue ob t a i ned l a t e r in t h i s work of 0.833 MHz. (See Chapter 6 Tab le 6 . 4 ) . However even a rough es t imate of D R was s u f f i c i e n t , s i n ce fo r the k=*2 to ±1 comb we were l o o k i n g f o r , the t o t a l c o n t r i b u t i o n of D„ i s on the order of 15 MHz and a ±5 MHz e r r o r here would not s u b s t a n t i a l l y a f f e c t the s e a r c h . 92 The important and d i f f i c u l t parameter t o es t ima te we l l was C 0 . R e c a l l that C 0 i s p r o p o r t i o n a l to the r e c i p r o c a l of the moment of i n e r t i a about the symmetry a x i s ( e 1 . 2 ) . S ince the a r s e n i c atom l i e s on t h i s a x i s i t makes no c o n t r i b u t i o n to C 0 . I f we make the assumpt ion tha t the As-H d i s t a n c e in AsH 3 equa l s the As-D d i s t a n c e in AsD 3 then s i n c e the moment of i n e r t i a i s l i n e a r in mass, and Mass(D)/Mass(H) =* 2, the va lue of C 0 f o r AsD 3 shou ld be approx imate l y one-ha l f the c o r r e s p o n d i n g va lue in A s H 3 . T h i s at l e a s t seemed a reasonab le s t a r t i n g p o i n t and "pos t f a c t o " we note tha t C 0 ( A s H 3 ) / 2 = 52442 MHz (r93) compares reasonab l y w i th the f i n a l measured va lue C 0 ( A s D 3 ) of 52642 MHz. The weakness of t h i s method i s tha t i t r e l i e s on the i n v a r i a n c e of r 0 s t r u c t u r e s to i s o t o p i c s u b s t i t u t i o n and in l i g h t of the d i s c u s s i o n in 1.6 t h i s i s t rue on l y f o r an e q u i l i b r i u m s t r u c t u r e . In order to improve our es t imate we can take i n t o account the s l i g h t l y s h o r t e r As-D bond l eng th due to ze ro p o i n t v i b r a t i o n a l a v e r a g i n g 6 . In t h i s case i t i s easy to show AC = - ^ A r [4 .3 ] For ND 3 and NH 3 the N-D bond was approx imate l y .0034 A s h o r t e r than the N-H ze ro p o i n t average d i s t a n c e ( r l 0 3 ) . In 6 We are assuming the anharmonic c o n t r i b u t i o n to the AsH 3 and AsD 3 e m p i r i c a l r o t a t i o n a l cons t an t s i s rough l y the same in the two m o l e c u l e s . 93 phosphine the zero p o i n t average bond l eng th d i f f e r e n c e was A r p = r D - r H = -.00444 A ( r 82 ) . C o n t i n u i n g t h i s t r end we expec ted | A r ^ s | > | A r p | . However, s i n ce a r s i n e i s more s i m i l a r to phosphine than phosphine i s to ammonia ( fo r i n s t ance i o n i z a t i o n p o t e n t i a l s and f o r c e c o n s t a n t s : see Chapter 6 ) i t was expected tha t A r A g would be c l o s e r to A r p than A r p was to A r N > In t h i s way an upper l i m i t fo r A r f t s was taken as -.005 A. For a bond l e n g t h of 1.52 A us ing equa t ion (e4.3) the c o r r e c t i o n in o u r . e s t i m a t e of C 0 ( A s D 3 ) due to the s h o r t e r As-D bond was 345 MHz. The r o t a t i o n a l cons tan t C 0 ( A s D 3 ) was then e s t a b l i s h e d to be between 52442 and 52787 MHz. Even though the v a r i a t i o n s in the e s t ima tes of C 0 ( A s D 3 ) are a l l w i t h i n one p e r c e n t , the e r r o r in the r e s u l t i n g d i s t o r t i o n moment s p e c t r a p r e d i c t i o n i s g r e a t l y a m p l i f i e d . T h i s i s because we are i n t e r e s t e d in B 0 - C o . In t h i s case B 0 - C 0 ranged from 4691 to 5036 MHz. I t i s i n s t r u c t i v e now to c o n s i d e r the c h a r a c t e r i s t i c form of the AsD 3 spec t rum. For AK = ± 1 <—+^2 , K=-1 and equa t ion (e4.2) becomes vD(J) = 3 ( B Q -C o ) + 15D R + 3 J ( J + 1 ) D J K [4 .4 ] T h i s e x p r e s s i o n g i v e s the t r a n s i t i o n s in the lowest f requency "comb" of the f o rb idden Q branch spect rum. S u c c e s s i v e "combs" are to be formed to h ighe r f requency at i n t e r v a l s of app rox ima te l y 6 ( B 0 _ C 0 ) . The t e e t h of a comb are genera ted by the D J R term and in gene ra l are separa ted by an 94 amount dv{J,K) = 6 (2K+3)D J K ( J+1) [4 .5] We now see tha t a 345 MHz range in the p r e d i c t i o n of C 0 t r a n s l a t e s i n t o a g i g a h e r t z range in the p r e d i c t i o n of the " comb" . It shou ld a l s o be apparent tha t e r r o r s in the e s t i m a t i o n of D R are not s i g n i f i c a n t . In l i g h t of the extreme weakness of these t r a n s i t i o n s and the very narrow expe r imen ta l f requency window a v a i l a b l e a g i g a h e r t z sea rch r eg i on would g e n e r a l l y be un t enab l e . However, from equa t ion (e4.5) we need on ly sea rch the order of the spac ing of t e e t h in the comb and then i f the sea rch f a i l s s h i f t the sea rch r eg ion the width of the comb. In t h i s case t h i s meant sea rch r eg ions on the order of 100 MHz fo r the s t r o n g e s t p r e d i c t e d l i n e J=16. N e v e r t h e l e s s , a search over 100 MHz i s a monumental e f f o r t e x p e r i m e n t a l l y . Very o f t e n , f o r example, a peak i n i t i a l l y thought to be a l i n e tu rned out to be n o i s e , so tha t in t h i s case the mean accumu la t ion time of 2-5 hours per 4 MHz sea rch r eg ion was m u l t i p l i e d to 5-15 h o u r s . Perhaps more s i g n i f i c a n t l y , r eg i ons i n i t i a l l y swept w i th no r e s u l t were o f t e n l a t e r found to c o n t a i n t r a n s i t i o n s ! 3. INITIAL SEARCHES The i n i t i a l sea rch was made fo r the J=14 l i n e based on a p r e d i c t i o n us i ng a va lue of C 0 ( A s D 3 ) equal to C 0 ( A s H 3 ) / 2 . The J=14 l i n e was not the s t r o n g e s t p r e d i c t e d t r a n s i t i o n but 95 r e f e r ence to Tab le 4.1 shou ld show that the d i f f e r e n c e in p r e d i c t e d i n t e n s i t y between the J=14 and the s t r o n g e s t t r a n s i t i o n J=16 i s not s i g n i f i c a n t . The r a t i o n a l e f o r choos ing the J=14 l i n e f o r the i n i t i a l search was a c c o r d i n g to equa t ion (e4.5) the r eg ion between l i n e s in the comb at J=14 was expected to be rough ly 80 MHz as compared to 100 MHz fo r the J=16 l i n e . T h i s meant tha t a narrower sea rch r eg ion c o u l d be e s t a b l i s h e d . The r eg i on 14.50 to 14.58 GHz was s ea r ched , ave rag ing a t ime of one hour per four megahertz window, wi th no r e s u l t . In l i g h t of the above n u l l r e s u l t i t was d e c i d e d that a more c a r e f u l e s t i m a t i o n of C 0 ( A s D 3 ) was in o r d e r . In t h i s case the e s t i m a t i o n scheme d i s c u s s e d in the l a s t s e c t i o n , t a k i n g i n t o account the s h o r t e r As-D r e l a t i v e to As-H bond l e n g t h , was used . A rough midd le va lue in the range of the C 0 ( A s D 3 ) e s t ima te was taken and i t was dec ided to s t a r t the sea rch aga in at 13.89 GHz and sea r ch to lower f r equency . T h i s sea rch i n c l u d e d the f requency where we would l a t e r f i n d the J=15 l i n e at 13.87 GHz. The p robab le causes f o r i n i t i a l l y not see ing t h i s l i n e are many but c e r t a i n l y the e a s i e s t excuse would be s imp ly un favourab l e no i s e d u r i n g those p a r t i c u l a r r uns . The f i r s t i n d i c a t i o n of a r e p r o d u c i b l e a b s o r p t i o n s i g n a l was found f i n a l l y a t 13.79 GHz. C a r e f u l s t u d i e s were then made of t h i s t r a n s i t i o n and from 4-5 hours of s i g n a l a ccumu la t ion time there were f a i n t i n d i c a t i o n s tha t the t r a n s i t i o n c o u l d be r e s o l v e d i n t o a d o u b l e t . T h i s was 96 p a r t i c u l a r l y g r a t i f y i n g in l i g h t of the p r e d i c t e d quadrupole s p l i t t i n g . R e c a l l tha t the a r s e n i c nuc leus has a sp in 1=3/2 and t h e r e f o r e we shou ld expect to see fo r AsD 3 a s p l i t t i n g of the r o t a t i o n a l t r a n s i t i o n i n t o 21+1 or 4 s t rong h y p e r f i n e components. From the c o u p l i n g cons tan t r epo r t ed in r e f e r ence ( r 9 l , X z z =-164 .75 (3 ) MHz) and equa t ion ( e1 .25 ) , fo r J up to approx ima te l y 25, the h y p e r f i n e s p l i t t i n g was p r e d i c t e d to appear on l y as a s i n g l e doub le t w i th our a v a i l a b l e r e s o l u t i o n . Two u n r e s o l v a b l e components are pushed up f requency the other two u n r e s o l v a b l e components down and the p r e d i c t e d s e p a r a t i o n of the measurable doub le t was taken as the i n t e n s i t y weighted average f requency of the upper p a i r minus the lower p a i r . From an o p t i m i s t i c e m p i r i c a l es t imate of the supposed h y p e r f i n e s p l i t t i n g t h i s t r a n s i t i o n was t e n t a t i v e l y a s s i gned as J=16±2; the assignment would l a t e r be con f i rmed as J=16. Assuming the above t r a n s i t i o n to be indeed J=16 p e r m i t t e d the p r e d i c t i o n of J=15 at 13.87 GHz in a r eg ion a l r e ady s ea r ched . A c a r e f u l second study of tha t f requency r eg ion r e v e a l e d a broad a b s o r p t i o n f ea tu r e tha t aga in appeared to be a d o u b l e t . From these two t r a n s i t i o n s J=14 was p r e d i c t e d and a s i m i l a r a b s o r p t i o n was found at the p r e d i c t e d f r equency . It would t u rn out l a t e r tha t the J=14 t r a n s i t i o n would be p a r t i c u l a r l y hard to a s s i g n because of another l i n e of s i m i l a r i n t e n s i t y j u s t 2.8 MHz away from one of the members of the quadrupo le d o u b l e t . 97 It seemed reasonab le now to assume we were d e a l i n g w i th AsD 3 d i s t o r t i o n t r a n s i t i o n s . The p l an then was to search to lower J ( to h ighe r f r e q u e n c y ) . U n f o r t u n a t e l y the J=13 l i n e c o u l d not be measured fo r in i t s p l a ce we found a very s t rong a b s o r p t i o n wi th a second order S tark e f f e c t . T h i s l i n e was f a r too s t rong to be long to the same s e r i e s as the o ther l i n e s . In a d d i t i o n the d i s t o r t i o n s p e c t r a here a l l were expected to have f i r s t o rder S tark e f f e c t s . T h i s l i n e was ignored in subsequent d i s t o r t i o n d i s c u s s i o n . The next l i n e in our s e r i e s , the J=12, was d i f f i c u l t to observe and the 600 KHz e r r o r a t t r i b u t e d to the f i n a l measurement in Tab le 4.1 shou ld r e f l e c t t h i s . The spec t rometer in t h i s r eg i on was ext remely no i s y and the measurement was not s a t i s f a c t o r i l y r e p r o d u c i b l e . The t r a n s i t i o n i s i n c l u d e d in the f i n a l f i t as i t was b e l i e v e d that i t c o u l d be found w i t h i n the s t a t e d e r r o r s . Rather than con t i nue wi th the sea rch to lower J , s i n c e the t r a n s i t i o n s were f a s t becoming too weak fo r our spec t rometer i t was dec ided to sea rch f o r the t r a n s i t i o n s wi th J g r e a t e r than 16. Ave rag ing rough ly 3 hours s i g n a l p l u s 3 hours background, absorpt ions* were found c o r r e s p o n d i n g to J=17 to 21 (assuming the i n i t i a l t e n t a t i v e a s s i gnmen t ) . These i n i t i a l measurements were at best p o o r . The s i g n a l to no i se r a t i o s were o f t e n on the o rder of 3 to 1. The O p t i m i s t s were conv inced of the ass ignments and mechanisms r e l a t i n g these a b s o r p t i o n s but the P ragmat i s t s wanted more a c cu ra t e measurements and the S k e p t i c s wanted to 98 see the h y p e r f i n e s t r u c t u r e . In t h i s case a f u r t h e r study was needed where c a r e f u l c o n s i d e r a t i o n was g i ven to tun ing and no i s e m i n i m i z a t i o n coup led wi th i n c r e a s e d data accumula t ion t imes . T h i s second phase of the i n v e s t i g a t i o n r e s u l t e d in the observed ars ine-D3 spectrum r e p o r t e d in the next s e c t i o n . 4. OBSERVED ARSINE-D3 SPECTRUM Four teen Q branch t r a n s i t i o n s in the f requency range 13.0 to 14.3 GHz wi th K=±]<-*2 and J in the range 8 to 24 have been measured and a s s i g n e d . The ass ignment was based on the e x p e r i m e n t a l l y observed h y p e r f i n e s t r u c t u r e and on the "goodness " of a g l o b a l f i t tha t i n c l u d e d both the d i s t o r t i o n spectrum r epo r t ed here and the a l l owed t r a n s i t i o n s of r e f e r e n c e ( r 9 l ) . In p r e s e n t i n g the d i s t o r t i o n spectrum of AsD 3 both of these c r i t e r i a w i l l be d i s c u s s e d . The measured h y p e r f i n e s p l i t l i n e s are g iven in Tab le 4.1 a long w i th t h e i r a ss ignments , expe r imen ta l e r r o r om and maximum a b s o r p t i o n c o e f f i c i e n t s 7 (see e 1 . 3 7 ) . The r max expe r imen ta l u n c e r t a i n t i e s were e s t ima ted from observed s i g n a l to no i s e r a t i o s , l i n e widths and f l u c t u a t i o n s in the f requency measurement ob ta ined from d i f f e r e n t r u n s . The 7 ^ J 'max va lues were c a l c u l a t e d fo r a temperature of 298 K wi th a broaden ing parameter of 20 MHz/Torr (r82) f o r a d i s t o r t i o n d i p o l e moment of 1 x 1 0 " 5 Debye. The va lue fo r the d i p o l e moment was chosen p r i n c i p a l l y fo r s c a l i n g purposes as the d i s t o r t i o n d i p o l e moment was not measured in t h i s s t udy . 99 C o n s i d e r a t i o n of the es t imate of t h i s d i p o l e moment produced in Chapter 6 suggests the va lue quoted above i s r e a s o n a b l e . The 7 v a l ues are fo r the r o t a t i o n a l t r a n s i t i o n s . S ince max fo r our purpose the quadrupo le i n t e r a c t i o n s p l i t s each r o t a t i o n a l l i n e i n t o a doub le t the 7 f o r each member of a 'max doub le t i s s imply the 7 of the r o t a t i o n a l t r a n s i t i o n c 1 max d i v i d e d by two. Tab le 4.1 Fo rb idden h y p e r f i n e t r a n s i t i o n s of AsD 3 (MHz) AK = ±1<-+2 J f , f 2 A expt A ( c a l c X1 0 1 max 1 cm 8 1 4 321 .22(15) 5 .07 1 . 1 4 1 1- 1 4 1 58 .25(20) 1 4 1 55 .10(20) 3.15(40) 2 .76 2 .93 1 2 14 093 .20(60) 2 .33 3 .53 1 4 1 3 951 .70(20) • 13 949 .87(10) 1.83(30) 1 .74 4 .41 1 5 13 873 .62(10) 1 3 872 .09(10) 1.53(20) 1 .52 4 .60 16 1 3 791 .31(10) 1 3 790 .07(5) 1.08(25) 1 .33 4 .62 17 1 3 704 .92(15) 1 3 703 .84(10) 1.08(25) 1 .19 4 .47 18 1 3 614 .54(10) 1 3 613 .46(15) 1.09(25) 1 .06 4 . 1 6 19 1 3 520 .29(10) 1 3 519 .39(10) 0.90(20) 0 .96 3 .75 20 1 3 422 .20(5) 1 3 421 .37(5) 0.83(10) 0 .86 3 .27 21 1 3 320 .27(10) 1 3 319 .45(10) 0.77(20) 0 .78 2 .77 22 13 214 .29(10) 0 .71 2 .27 23 13 104 .61(15) 0 .65 1 .81 24 1 2 991 .00(8) 1 2 990 .44(10) 0.56(18) 0 .60 1 .40 Numbers in pa ren theses are es t ima ted measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 00 A l s o g i ven in Tab le 4.1 are the p r e d i c t e d and e x p e r i m e n t a l l y determined h y p e r f i n e s p l i t t i n g s : ^ c a ] _ c and ^expt* T ^ e P r e d i c t e d s p l i t t i n g s were eva lua ted from the a r s e n i c quadrupo le c o u p l i n g cons tan t ( r9 l ) and equa t ion (e1.25) as was d i s c u s s e d in the p r e ced ing s e c t i o n . In t h i s c a l c u l a t i o n we d i d not need to i n c l u d e the o f f - a x i s deute r ium quadrupo le moments because the quadrupo le c o u p l i n g cons t an t of deu te r i um, at most a few hundred k i l o h e r t z , i s sma l l when compared to the a r s e n i c c o u p l i n g cons tan t of a few hundred megaher tz . The magnetic h y p e r f i n e c o n t r i b u t i o n i s at most a few k i l o h e r t z and, to the l e v e l of accuracy r e q u i r e d h e r e , can be i g n o r e d . N o t i c e a l s o tha t the e r r o r in the observed s p l i t t i n g s o f t e n accommodates at l e a s t two d i f f e r e n t J a ss ignments . Invoking the c o n s t r a i n t tha t " t e e t h in a comb" must be in sequence, tha t i s , n e i g h b o r i n g l i n e s are J+1 and J-1 fo r lower and h ighe r f requency ne ighbors r e s p e c t i v e l y , we f i n d there i s on l y one p o s s i b l e J ass ignment c o n s i s t e n t wi th a l l the d a t a . T h i s argument i s the r a t i o n a l e behind the statement tha t the h y p e r f i n e s t r u c t u r e suppor t s the ass ignments independent of the f requency f i t . T h i s i s of course not s t r i c t l y t rue as the d e f i n i t i o n of a ne ighbor i s assumed i m p l i c i t l y from the s t r u c t u r e of the spectrum g iven by equa t ion e 4 . 2 . A summary of l i n e parameters and exper imenta l c o n d i t i o n s i s g iven in Tab le 4 . 2 . Here the S tark s h i f t c o e f f i c i e n t i s the expected f requency s h i f t of the M=1 component fo r an e l e c t r i c f i e l d of 1 v o l t / c m . For t h i s 101 c a l c u l a t i o n the va lue used fo r the normal d i p o l e moment was 0.22 Debye ( r 99 ) . As an example of the r e s o l u t i o n and the s i g n a l to n o i s e a v a i l a b l e from t h i s exper iment the J=15 and J=17 l i n e s have been reproduced in F i g u r e 3. In these cases the doub le t c h a r a c t e r of the l i n e s i s e a s i l y r e s o l v a b l e . An important parameter in a l l the measurements was the S tark f i e l d . Tab le 4.2 shows t h a t , e s p e c i a l l y f o r low J l i n e s , measurements were o f t e n taken over a v a r i e t y of S tark f i e l d s . In o rder to a p p r e c i a t e why t h i s was done i t i s i n s t r u c t i v e to review the f i r s t o rder S tark p a t t e r n fo r a Q branch t r a n s i t i o n . From equa t ion (e1.44) the S tark r o t a t i o n a l energy l e v e l s h i f t s depend l i n e a r l y on M. With the e l e c t r i c f i e l d on we see AM=0 t r a n s i t i o n s between r o t a t i o n a l l e v e l s of the same J but d i f f e r e n t K ( i . e . each J l e v e l s p l i t i n t o M components ) . The r e l a t i v e s t r e n g t h of these AM=0 t r a n s i t i o n s vary as M 2 . The S tark s h i f t c o e f f i c i e n t s of Tab le 4.2 sum up the dependence on M of the S tark t r a n s i t i o n s ; the S tark f requency s h i f t in KHz i s j u s t M m u l t i p l i e d by the e l e c t r i c f i e l d s t r e n g t h m u l t i p l i e d by the S tark s h i f t c o e f f i c i e n t . 102 F i g u r e 3 13874 13873 13872 Frequency (MHz) 13705 13704 13703 Frequency (MHz) F i g u r e 3. Two of the K = ±1 '<- *2 AsD 3 t r a n s i t i o n s . The top r e p r o d u c t i o n i s J=15. I t i s the r e s u l t of 64 sample scans at a Stark f i e l d of 74 v o l t s / c m . The bottom r e p r o d u c t i o n i s J=17 and i s the r e s u l t of 48 scans at a Stark f i e l d of 200 vol t s / c m . 1 03 Tab le 4.2 A r s i n e l i n e parameters and expe r imen ta l c o n d i t i o n s J S ta rk S h i f t Frequency S tark # of # of T o t a l Coef f i c i en t Component F i e l d Indep. 2. Min Time KHz/V/cm V/cm Runs Sweeps (hr) 8 4.61 f 1 30 1 1 38 4.6 1 50 1 1 6 0.5 200 1 16 0.5 1 1 2.52 f 1 60 1 64 2.1 500 1 1 44 4.8 f 2 60 1 40 1 .3 400 1 80 2.7 1 2 2.13 f , , f 2 300 2 64,128 6.4 • - 400 2 32,128 5.3 1 4 1 .58 f 1 62 3 64,88,112 8.8 200 1 96 3.2 400 3 16,64,112 6.4 600 1 32 1 .0 f 2 62 1 32 1 .0 200 1 96 3.2 400 2 64,112 5.9 600 1 64 2. 1 800 1 32 1.0 15 1 .38 f 1 , f 2 74 1 64 2. 1 200 3 32 ,64 ,64 5.3 1 6 1 .22 f 1 r f 2 80 1 64 2.1 200 3 48,64,112 7^5 104 Tab le 4.2 con t i nued J S tark S h i f t Frequency S tark # of C o e f f i c i e n t Component F i e l d Indep. 1 7 18 19 20 21 22 23 KHz/V/cm 1 .09 0.97 0.87 0.79 0.72 0.66 0.60 24 0.55 Spur ious l i n e at 13954.50(20) f , , f : f , , f : f 1,1; f , , f : f , , f ; =f 1 r f : f 1 , f V/cm 300 60 200 200 240 400 100 240 280 100 300 310 32 68 200 400 300 320 300 310 320 62 400 Runs 2 1 # of 2. Min Sweeps 1 1 2 64 80 80 109 32,42 96 64 64 96 96 32 ,40 ,88 80 96 16,23 1 6 80 32 64 64 80 64,88 80 T o t a l Time (hr) 3.7 2.1 • 2.7 2.7 3.6 2.5 3.2 2.1 4.3 3.2 3.2 5.3 2.7 3.2 1 .3 0.5 2.7 0.5 2.1 2.1 2.7 5.1 2.7 Sample p r e s s u r e s were a l l 25 to 65 mTorr 1 05 Now r e c a l l that one r e s u l t of phase s e n s i t i v e d e t e c t i o n i s to i n v e r t the S tark p a t t e r n and add i t to the zero f i e l d a b s o r p t i o n p r o f i l e of the t r a n s i t i o n . In the a r s i n e study i t was important to s h i f t the S tark lobes f a r enough away, that i s f a r t h e r than the a b s o r p t i o n h a l f l i n e w id th , to ensure tha t the " n e g a t i v e " i n t e n s i t y of the S tark lobe would not d e t r a c t from the a b s o r p t i o n l i n e . In the same way i t was important not to s h i f t the i n t ense S tark l obes of one member of the doub le t the doub le t f requency s e p a r a t i o n as the nega t i ve l obes of one member of a doub le t would s u b t r a c t from the a b s o r p t i o n p r o f i . l e of the other (and v i c e v e r s a , at the same t i m e ) . The reason fo r the d i f f e r e n t S tark f i e l d s can now best be demonstra ted w i th the h e l p of an example. For the J=11 l i n e e s s e n t i a l l y two Stark f i e l d s were used , 60 and 450 v o l t s / c m . At 60 vo l t s / cm the s t r o n g e s t M=±J i n v e r t e d S tark lobe fo r each member of the doub le t i s s h i f t e d ±1.7 MHz. From Tab l e 4.1 the doub le t s e p a r a t i o n i s 2.8 MHz. The r e s u l t of t h i s S ta rk f i e l d i s to accen tua te the s p l i t t i n g by " s u b t r a c t i n g " away b a s e l i n e in between the d o u b l e t s . T h i s , of c o u r s e , i s e x a c t l y what we would want in o rder to c o n f i r m the doub le t nature of our t r a n s i t i o n s . A l s o the S tark lobes w i l l not i n t e r f e r e wi th the ze ro f i e l d a b s o r p t i o n l i n e s . At the h ighe r f i e l d the important S ta rk l obes have a l l been s h i f t e d f a r t h e r than the doub le t s e p a r a t i o n . Aga in the nega t i ve i n t e n s i t y of the S tark p a t t e r n i s removed and the h ighe r f i e l d r e s u l t can be used as a check of the low f i e l d 1 06 r e s u l t . The problem wi th h ighe r f i e l d s i s background e f f e c t s become more p r e va l en t so o f t e n d i f f e r e n t h ighe r f i e l d s would be t r i e d in hope of m i n i m i z i n g background (see Chapter 2 ) . At h i ghe r J w i th our 800 KHz l i n e widths i t i s not p r a c t i c a l to t r y to move the Stark l obes w i t h i n the much sma l l e r doub le t s e p a r a t i o n and so on l y h i ghe r f i e l d measurements were made. In o rde r to determine the r o t a t i o n a l parameters the t h e o r e t i c a l u n s p l i t " c e n t e r " , v^, f r e q u e n c i e s were e v a l u a t e d . These are l i s t e d i n Tab le 4 . 3 . For low J ' s (J<15) these c en t e r f r e q u e n c i e s were c a l c u l a t e d from the p r e d i c t e d s p l i t t i n g s . For J g rea te r than 14 the asymmetry in the p r e d i c t e d s p l i t t i n g about the cen t e r f requency i s we l l w i t h i n expe r imen ta l e r r o r so where a p p r o p r i a t e , s imple averages of the s p l i t f r e q u e n c i e s determined the cen te r f r equency . A l s o in Tab le 4.3 are the low J h y p e r f i n e s p l i t AsD 3 a l l owed t r a n s i t i o n s r e p o r t e d by Helminger et a l ( r 9 l ) . The normal t r a n s i t i o n s (J,K)->(J+1,K) were f i t to the equa t ion (the h y p e r f i n e e n e r g i e s of equa t ions ( e1 .25 ,e1 .30 ) were a l s o i n c luded ) , N ( J , K ) = 2 ( J + 1 ) [ B 0 - D J K K 2 + H K J K " ] - 4 ( J + 1 ) 3 [ D J - H J R K 2 + H J ] + Uj[(J+1)3-J3] [4 .6] and the d i s t o r t i o n Q branch J , ± 1 < — J , ^ 2 ) were f i t to v D ( J ,K ) = a R + b R J ( J+ l ) + c K J 2 ( J + D 2 + e R J 3 ( J + D 3 [4 .7] 1 07 where a b K K c K -3H JK e K -3L J J JK and L JJJK i s the c o e f f i c i e n t of the o c t i c term J 3 ( J + 1 ) 3 K 2 and was i n c l u d e d p r i n c i p a l l y as a f i t t i n g parameter . The f i t s were made w i th the data weighted at Wo  2 3 m a c c o r d i n g to the i t e r a t i v e method d e s c r i b e d in ( r l 2 ) . The e r r o r s f o r the a l lowed t r a n s i t i o n s are set at the va lue quoted by Helminger et a l ( r 9 l ) of 0.05 MHz fo r t h e i r measurement of the J=0—>1 t r a n s i t i o n at 115 GHz but s i n c e they do not g i ve e r r o r s f o r t h e i r o ther measurements we assume e r r o r s of 0.10 and 0.15 MHz fo r t h e i r r epo r t ed t r a n s i t i o n s at 230 and 345 GHz r e s p e c t i v e l y because i t i s g e n e r a l l y more d i f f i c u l t to make f requency measurements a t h ighe r f r e q u e n c i e s . The e r r o r s in the cen te r f r e q u e n c i e s of the d i s t o r t i o n t r a n s i t i o n s were determined from the e r r o r s , p r e v i o u s l y d i s c u s s e d , in the s p l i t f r e q u e n c i e s used to "determine the cen te r f r e q u e n c i e s p l u s a s u b j e c t i v e es t ima te of the s t a t i s t i c a l spread one might expect to see in measurements of such weak t r a n s i t i o n s . T h i s l a t t e r es t imate was deemed necessary as a few l i n e s reproduced the cen te r f r e q u e n c i e s q u i t e w e l l but s i n ce our s t a t i s t i c a l sampl ing s i z e was very sma l l i t d i d not seem reasonab le to weight these t r a n s i t i o n s so much more than t r a n s i t i o n s wi th 108 comparable s i g n a l to n o i s e , but poorer r e p r o d u c i b i l i t y of f r equency . The f i r s t concern was to r e _ f i t j u s t the a l l owed t r a n s i t i o n s and compare our r e s u l t s w i th those p u b l i s h e d by Helminger et a l ( r 9 l ) . The r e s u l t s of t h i s are reproduced he re . CALCULATED CONSTANTS FOR AsD 3 BASED ON ALLOWED TRANSITIONS ONLY (MHz) B 0 57477.600 (3) Dj 0.7413(2) D J R -0.928(1) eqQ -164.74(3) C X T 0.051(3) C R 0.079(15) These numbers are the same as those r epo r t ed by Helminger et a l ( r 9 l ) except our C R i s 0.079 whereas they r epo r t 0.069 MHz. An important number above i s D T „ as we s h a l l f i n d i t s va lue w i l l change s i g n i f i c a n t l y when we i n c l u d e the f o r b i d d e n t r a n s i t i o n s in the f i t . The f o rb idden t r a n s i t i o n s were next f i t to the power s e r i e s expans ion in J(J+1) d e s c r i b e d by equa t ion e 4 . 7 . The r e s u l t s of t h i s f i t a re not i n c l u d e d here except to say that they were t o t a l l y c o n s i s t e n t w i th the r e s u l t s of the g l o b a l f i t d i s c u s s e d nex t . C o n s i d e r a t i o n of equa t i ons e4 .6 and e4.7 r e v e a l tha t the f o r b i d d e n and a l l owed t r a n s i t i o n s both c o n t a i n i n f o r m a t i o n on the parameters D T „ , H T t , and H„ T and so i t would seem prudent to do a s imu l taneous g l o b a l f i t r a the r than two separa te f i t s and then compare the pa ramete rs . The 109 r e s u l t s of t h i s g l o b a l f i t are g i ven in Tab le 4 . 6 . Comparison w i th the " a l l o w e d " r e s u l t s above shows two important d i f f e r e n c e s . F i r s t D T U . has changed s i g n i f i c a n t l y . T h i s can perhaps be seen as a r e s u l t of the two d i f f e r e n t models at work here and the d i f f e r e n t ways the h ighe r order s e x t i c cons t an t s w i l l tend to " c o r r e c t " the va lue of D T t , in each model . The second d i f f e r e n c e between the " a l l o w e d " and the " g l o b a l " f i t s i s tha t in the l a t t e r the magnetic h y p e r f i n e c o u p l i n g c o n s t a n t s C k T and C„ were not d e t e r m i n a b l e . T h i s r e s u l t came as a s u r p r i s e as the h y p e r f i n e s p l i t t i n g of the a l l owed t r a n s i t i o n s was expected to be independent of the f o rb idden t r a n s i t i o n " c en t e r f r e q u e n c i e s " . One can r a t i o n a l i z e t h i s r e s u l t by r e c a l l i n g the a l l owed t r a n s i t i o n s w i l l be coup led to the f o rb idden t r a n s i t i o n s through D T „ (and l e s s through H J K and H K J ) - I t was noted in A s H 3 , AsD 2 H and AsH 2 D that the magnetic h y p e r f i n e cons t an t s v a r i e d rough ly as the r o t a t i o n a l c o n s t a n t s , so r a the r than set C N and C R to zero and mix the magnet ic e f f e c t i n t o the quadrupo le h y p e r f i n e c o n s t a n t s , the va lues of C V T and C„ were f i x e d at the va lues in Tab le 4.6 in a cco rd w i th the d i s c u s s i o n at the end of t h i s chapte r ( s e c t i o n 4 . 8 ) . A l s o in the g l o b a l f i t i t was necessa ry to c o n s t r a i n H R j . I t was e s t ima ted us i ng r a t i o s of H R j f o r PH 3 ( r l 0 2 ) , PD 3 (see phosphine s e c t i o n ) and AsH 3 ( r 93 ) . The cons tan t H K J w i l l a l t e r the d e t e r m i n a t i o n of D j K » T h i s cannot e x p l a i n the change noted e a r l i e r between the " a l l o w e d " and " g l o b a l " D J K 1 10 va lues because when H „ _ was f i x e d at the above es t imate D T O on ly changed we l l w i t h i n a s t andard d e v i a t i o n . T h i s c o u l d mean tha t our es t imate of H R J was poo r , which i s reasonab le as the H R J va lue ob t a i ned f o r PD 3 i s not r e l i a b l e , or that H J K has the more important e f f e c t in the d e t e r m i n a t i o n of D T „ from the normal spec t rum. Two spu r i ous l i n e s that c o u l d not be accounted fo r by the d i s t o r t i o n model were a l s o found . These i n c l u d e d an unknown " a l l o w e d " t r a n s i t i o n at 14026 MHz tha t comp le t e l y masked the J=13 f o rb idden t r a n s i t i o n . The other unknown l i n e was at 13954.50(20) MHz (see Tab le 4 . 2 ) . T h i s l i n e was p a r t i c u l a r l y p a t h o l o g i c a l as i t had the same r e l a t i v e i n t e n s i t y as our d i s t o r t i o n l i n e s . It would have been a good c and ida t e f o r one member of a h y p e r f i n e s p l i t p a i r except the r e s u l t i n g s p l i t t i n g was not c o n s i s t e n t w i th the o ther d a t a . 111 Tab le 4.3 A n a l y s i s of the measured AsD 3 Spectrum (MHz) T r a n s i t i o n 4-Frequency(Unc) Obs-Calc Normal Rotat i o n a l T r a n s i t i o n s r e f ( r 9 l ) J->J+1, K->K f , - > f 2 J=0 K=0 1 .5->2.5 1 14 960.55 (0 .05 ) -0.01 1.5->1.5 1 1 4 919.24 (0 .05 ) 0.01 1.5->0.5 1 1 4 993.30 (0 .05 ) 0.01 J=1 K=0 2.5->3.5 229 890.31 (0 .10 ) 0.03 2.5->2.5 229 848 .92 (0 .10 ) 0.03 1.5->1.5 229 919.53 (0 .10 ) 0.02 0.5->1.5 229 845 .47 (0 . 10) 0.02 0.5->0.5 229 886 .56 (0 .10 ) 0.01 J=1 K=1 2.5->3.5 229 900.45 (0 .10 ) -0.05 2.5->2.5 229 879 .68 (0 .10 ) -0.04 1.5->2.5 229 859 .26 (0 .10 ) 0.00 1 .5->1.5 229 873.77 (0 .10 ) -0.06 0.5->1 .5 229 911.01 (0 .10 ) 0.03 0.5->0.5 229 931 . 47 (0 . 10) -0.02 J=2 K=0 3.5->4.5 344 787.60 (0 .15 ) 0.09 2.5->2.5 344 806.52 (0 .15 ) 0.04 1 .5->2.5 344 777.28 (0 .15 ) 0.08 1 .5->1.5 344 818 .32 (0 .15 ) 0.07 J=2 K=1 3.5->4.5 344 795.53 (0 .15 ) -0.05 3.5—>3.5 344 764.52 (0 .15 ) 0.08 2.5->2.5 344 799.51 (0 .15 ) 0.06 1 .5->1.5 344 815.70 (0 .15 ) 0.07 1 1 2 Tab le 4.3 c o n t i n u e d T r a n s i t i o n Frequency(Unc) J=2 K=2 3.5->4.5 344 819.66 (0 .15 ) 2.5->3.5 344 778.44 (0 .15 ) 1.5->2.5 344 807 .80 (0 .15 ) 0.5—>1.5 344 848.96 (0 .15 ) Fo rb idden R o t a t i o n a l T r a n s i t i o n s J<-J, K=:H<-±2 t h i s work 8 14318.73 5 (0.21) -0.13 1 1 14156 .67 5 ( 0 .28 ) 0.29 1 2 14093.20(0.60) 0.78 1 4 13950.79(0.21) 0.29 1 5 13872.85(0.15) 0.04 1 6 13790.70(0.15) -0.09 1 7 13704.40(0.18) -0.17 18 13614.07(0.18) -0.19 19 13519.84(0.15) -0.12 20 13421.73(0.15) -0.04 21 13319.85(0.15) 0.08 22 13214.29(0.15) 0.28 23 13104.61(0.15) 0.08 24 12990.72(0.20) -0.64 t Numbers in pa ren theses are e s t ima ted measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . Obs-Calc -0.13 -0.10 -0.11 -0.08 1 13 5. PREDICTION AND INITIAL SEARCH FOR THE ASYMMETRIC TOP  ARSINE SPECTRA For the asymmetric top a r s i n e s accuracy in the p r e d i c t i o n s of t h e i r s p e c t r a was not nea r l y as c r u c i a l as i t was f o r A s D 3 . T h i s was because the l i n e s t r e n g t h s were so l a r g e as to make the t r a n s i t i o n s e a s i l y a c c e s s i b l e to s t anda rd microwave spec t romete rs capab le of r e l a t i v e l y r a p i d scann ing over the e n t i r e microwave r e g i o n . Us ing a symmetric top a r s i n e as a r e f e r e n c e an a r b i t r a r y C a r t e s i a n a x i s system was set up w i th the z a x i s a l ong the C^y a x i s . The x a x i s was chosen such tha t the zx p lane would cut through the middle of one of the hydrogen i s o t o p e s . For the asymmetric tops the unique hydrogen i s o t o p e ( i . e . H in AsHD 2 ) would always be s i t u a t e d in the zx p l a n e . In t h i s s i t u a t i o n the p r i n c i p a l axes of the asymmetr ic tops c o u l d always be found by a r o t a t i o n about the y a x i s . The moment of i n e r t i a t ensor of an asymmetric top was e va l ua t ed in the above c a r t e s i a n a x i s system which i s o r i g i n s h i f t e d to the new cen te r of mass but keeping the new axes p a r a l l e l to the o l d . The bond l eng ths and ang les were chosen to be the r z parameters of Chu and Oka ( r 82 ) . R o t a t i o n s through a de te rminab le ang le about the y a x i s were then made tha t d i a g o n a l i z e d the moment of i n e r t i a t e n s o r . From the d i a g o n a l p r i n c i p a l i n e r t i a l e lements and equa t ion (e1.2) we then had es t ima tes of the zero p o i n t average r o t a t i o n a l cons t an t s A , B and C . A p r e l i m i n a r y f o r ce f i e l d d e r i v e d from known symmetric top data was then used to 1 1 4 es t ima te the harmonic p a r t s of the a lphas which when added to the r r o t a t i o n a l cons t an t s gave " g u e s s t i m a t e s " of the e m p i r i c a l r o t a t i o n a l cons t an t s u s u a l l y to much b e t t e r than one per cent ( e l . 5 7 ) . For these s t u d i e s e r r o r s of up to a per cent in the r o t a t i o n a l cons tan t p r e d i c t i o n s d i d not s i g n i f i c a n t l y impede the d e t e r m i n a t i o n of the s p e c t r a . One of the grea t s i m p l i f y i n g f e a t u r e s of the a r s i n e asymmetr ic top s p e c t r a was the quadrupole h y p e r f i n e s t r u c t u r e super-imposed on each r o t a t i o n a l t r a n s i t i o n . Broad f requency sweeps c o u l d be made f o r these s t rong l i n e s , and any t r a n s i t i o n s found c o u l d immediate ly be a s s i gned by t h e i r ve ry d i s t i n c t i v e t r a n s i t i o n dependent quadrupo le h y p e r f i n e " f i n g e r p r i n t " . The asymmetr ic top quadrupole h y p e r f i n e c o n s t a n t s were e s t ima ted .by r o t a t i n g the AsH 3 ( r 9 3 ) (or AsD 3 ( r 9 l ) ) c o u p l i n g cons tan t t ensor through the same ang le needed to d i a g o n a l i z e the symmetric top moment of i n e r t i a t ensor upon i s o t o p i c s u b s t i t u t i o n . A l though the asymmetric top h y p e r f i n e e s t ima tes depended on whether one s t a r t e d w i th the AsH 3 or AsD 3 c o u p l i n g c o n s t a n t s , the p r e d i c t e d h y p e r f i n e p a t t e r n s were d i s t i n c t i v e enough fo r easy ass ignment . T h i s problem of the v a r i a b i l i t y of the c o u p l i n g cons t an t s in AsH 3 and AsD 3 i s d i s c u s s e d f u r t h e r in s e c t i o n 4 . 8 . A n t i c i p a t i o n of t h i s problem in AsH 2 D and AsD 2 H l e d to the exact t reatment of the quadrupo le e f f e c t r a the r than the usua l f i r s t or second o rder p e r t u r b a t i o n approach . As was d i s c u s s e d e a r l i e r w i th the c a l c u l a t i o n s of the quadrupole p a t t e r n s of A s D 3 , the o f f - a x i s deuter iums do not c o n t r i b u t e 1 1 5 to the s p l i t t i n g s we measure h e r e . F i n a l l y the magnetic h y p e r f i n e c o n t r i b u t i o n to the s p l i t t i n g s was on the order of s e v e r a l hundred k i l o h e r t z and so i n i t i a l l y t h i s e f f e c t was i g n o r e d . A f t e r a few t r a n s i t i o n s were measured rough va lues of these magnet ic c o n s t a n t s were e s t a b l i s h e d and put i n t o subsequent p r e d i c t i o n s . The i n i t i a l study of the asymmetric a r s i n e s was made in the f requency r eg ion below 100 GHz on the U .B .C . spec t rome te r . T h i s meant tha t i n i t i a l l y on ly Q branches were found . In gene r a l we c o u l d not f i t to a l l th ree r o t a t i o n a l cons t an t s as no R branches were then a v a i l a b l e so f i t s were made a l l o w i n g two r o t a t i o n a l cons t an t s to va ry and f i x i n g the t h i r d at the p r e d i c t e d v a l u e . The two " v a r i a b l e " r o t a t i o n a l cons t an t s were then j u s t f i t t i n g parameters w i th no obv ious p h y s i c a l meaning. As l i n e s were found and a s s i g n e d , i n t e r i m f i t s a l l owed the p r e d i c t i o n of new t r a n s i t i o n s wi th g r e a t e r and g rea t e r p r e c i s i o n . T h i s r e s u l t e d in an i t e r a t i v e method o f t e n c a l l e d " b o o t s t r a p p i n g " , where p r e d i c t i o n s are made f o r t r a n s i t i o n s that are sea rched fo r and h o p e f u l l y found, and then put i n t o a f i t tha t g i v e s new p r e d i c t i o n s . The method i s stopped when we reach the d e s i r e d l e v e l of d e s c r i p t i o n of the s p e c t r a . In our case i t was important to o b t a i n good q u a r t i c cons t an t s that c o u l d be used in a f o r c e f i e l d a n a l y s i s . T h i s meant we needed enough t r a n s i t i o n s to determine at l e a s t a p a r t i a l set of s e x t i c cons t an t s so as to a vo id h ighe r c o n t r i b u t i o n s be ing mixed i n t o the q u a r t i c s . With t h i s in mind the s e x t i c s 1 16 f i n a l l y r epo r t ed shou ld be c o n s i d e r e d on ly as f i t t i n g pa ramete rs . As i t tu rned out we c o u l d not determine a f u l l set of s e x t i c s so tha t the s e x t i c s r epo r t ed w i l l have an even more nebulous p h y s i c a l meaning than i s u s u a l l y a s s o c i a t e d wi th s e x t i c c o n s t a n t s . The f i n a l phase of the asymmetr ic top a r s i n e study was completed at the Jet P r o p u l s i o n Labo ra to r y of C a l t e c h . I t was the re us ing t h e i r h i gh f requency spec t rometer tha t the R branch t r a n s i t i o n s were measured. As a by-product of the sea rch fo r the R branch t r a n s i t i o n s v a r i o u s o ther h igh f requency Q branches were a l s o measured. F i n a l l y a word about the t ypes of t r a n s i t i o n s tha t e x i s t f o r these a r s i n e asymmetric t o p s . No t i c e tha t a l l these asymmetric tops have a p l ane of symmetry (the zx p lane r e f e r r e d to at the beg inn ing of t h i s s e c t i o n ) . C o n s i d e r a t i o n of the r e l a t i o n s fo r the e lements of the moment of i n e r t i a t ensor (see any of the c l a s s i c t e x t s (r 1 1 , r 31) ) shou ld make i t e v iden t that one of the p r i n c i p a l i n e r t i a l axes i s p e r p e n d i c u l a r to t h i s symmetry p l a n e . By the very nature of a symmetry p lane i t shou ld a l s o be apparent that no d i p o l e moment can e x i s t a long t h i s p a r t i c u l a r p r i n c i p a l a x i s . For the p r o l a t e asymmetric r o to r AsH 2 D t h i s means we have no d i p o l e moment a long the " b " a x i s and we see " a " and " c " type t r a n s i t i o n s on ly (see s e c t i o n 1.4b) . For A s D 2 H , an o b l a t e asymmetr ic r o t o r , the re i s no d i p o l e moment a long the " a " p r i n c i p a l i n e r t i a l a x i s and we see on l y " b " and " c " type r o t a t i o n a l t r a n s i t i o n s . P r e c i s e l y the same s i t u a t i o n e x i s t s 1 1 7 for PH 2D and PD 2H which w i l l be s t u d i e d l a t e r in Chapter 5. 6 . OBSERVED ASH3D AND ASD 2H SPECTRUM One hundred and f o r t y - e i g h t h y p e r f i n e l i n e s r e p r e s e n t i n g twenty-nine r o t a t i o n a l t r a n s i t i o n s w i th J up to 1 6 were measured fo r AsH 2 D (Table 4 . 4 ) . For AsD 2 H seven ty- th ree h y p e r f i n e l i n e s w i t h i n seventeen r o t a t i o n a l t r a n s i t i o n s wi th J up to 11, were recorded (Table 4 . 5 ) . The f r e q u e n c i e s of a l l these t r a n s i t i o n s ranged from 9.3 to 302 GHz. The ass ignments are suppor ted by the f requency f i t s and the a r s e n i c h y p e r f i n e s t r u c t u r e , both of which are d i s c u s s e d f u r t h e r be low. As an example of the d i s t i n c t i v e h y p e r f i n e s p l i t t i n g of the r o t a t i o n a l t r a n s i t i o n s , in F i g u r e 4 we f i n d a r e p r o d u c t i o n of a broad f requency scan showing the c type 2 1 1 - 2 2 1 r o t a t i o n a l t r a n s i t i o n of AsD 2 H c l e a r l y r e s o l v e d i n t o the 10 h y p e r f i n e components we were ab l e to r epo r t in Tab le 4 . 5 . The S ta rk lobes are above the base l i n e and j u s t to the r i g h t of the ze ro f i e l d a b s o r p t i o n s below the b a s e l i n e . R e c a l l tha t the Stark lobes "move" depending on the e l e c t r i c f i e l d s t r e n g t h so from F i g u r e 4 i t shou ld not be hard to imagine S ta rk lobes of one component i n t e r f e r i n g w i th the zero f i e l d a b s o r p t i o n s of another component. 118 F i g u r e 4 I I L_ I I I I _ l 1 6 7 9 0 1 6 8 1 0 1 6 8 3 0 Frequency (MHz) F i g u r e 4. The 2,,—2 2 i r o t a t i o n a l t r a n s i t i o n of A s D 2 H . T h i s was o b t a i n e d as a s i n g l e scan of 600 seconds d u r a t i o n w i th a t ime cons tan t of 0.1 s e c . and a 100 KHz S ta rk f i e l d of 2800 v o l t s / c m . The problem then i s to choose a f i e l d l a r g e enough to s h i f t the S ta rk lobe of the component you are t r y i n g to measure o u t s i d e the ze ro f i e l d a b s o r p t i o n l i n e w i d t h and a t the same t ime a v o i d s h i f t i n g any S tark l obes from n e i g h b o r i n g components i n . I t t u rned out tha t f o r the a r s i n e asymmetric 1 1 9 tops the main d i f f i c u l t y was in s h i f t i n g the a b s o r p t i o n l i n e ' s S ta rk lobes more than a l i n e w i d t h , o r , " m o d u l a t i n g " the l i n e . The problem of i n t e r f e r i n g Stark lobes from other t r a n s i t i o n s was not as important because the h y p e r f i n e s p l i t t i n g was g e n e r a l l y l a r g e r than the S tark s h i f t s we c o u l d produce wi th the a v a i l a b l e expe r imenta l f i e l d s t r e n g t h . T h i s " s l ow" S tark e f f e c t i s a consequence of the sma l l a r s i n e d i p o l e moment and the second order nature of the s h i f t (see s e c t i o n 1.4c. Note : a l though we can have a " q u a s i " f i r s t o rder S tark s h i f t in the case of near degenerate energy l e v e l s we d i d not expe r i ence t h i s i n . t h i s s tudy . ) The ze ro f i e l d a b s o r p t i o n l i n e s were u s u a l l y measured at four f i e l d s t r e n g t h s , 1000., 2400, 3000 and 4000 vo l t s / cm (the l a t t e r be ing the maximum f i e l d a v a i l a b l e ) . At the low modu la t ion f i e l d s the a b s o r p t i o n p r o f i l e s were not symmetric r e v e a l i n g the i n t e r f e r e n c e of the S tark l o b e . At h ighe r f i e l d s the t r a n s i t i o n s were very nea r l y f u l l y modulated and the l i n e p r o f i l e s n e a r l y symmetr ic . In cases where the l i n e s were s t i l l asymmetric even at 4000 vo l t s / cm e x t r a p o l a t i o n s were made based on how the f requency measurements v a r i e d w i th S ta rk f i e l d . In these cases the exper imenta l e r r o r was i n c r e a s e d a c c o r d i n g l y . The h igh f requency measurements made on the JPL m i l l i m e t e r wave spec t rometer were done u s i n g Tone Burs t modu la t ion so the problems of a slow Stark e f f e c t were a v o i d e d . The e r r o r s in these h igh f requency measurements are g e n e r a l l y lower than t h e i r lower f requency c o u n t e r p a r t s as 1 20 we no longer need to take i n t o account the e f f e c t s of i n s u f f i c i e n t S tark f i e l d in our zero f i e l d f requency measurement. The gene ra l expe r imen ta l procedure was to measure each l i n e roughly 10 to 20 t imes . The quoted expe r imen ta l u n c e r t a i n t i e s are an i n d i c a t i o n of the spread in these measurements. The s p e c t r a were f i t to an A r e d u c t i o n h a m i l t o n i a n (equat ions e1 .13 , e1 .14 ) us ing an i t e r a t i v e method ( r 1 2 ) . The quadrupo le c o u p l i n g cons t an t s were ob ta ined by i n c l u d i n g the mat r ix e lements of r e f . (r32) i n the f i t t i n g h a m i l t o n i a n . The magnet ic h y p e r f i n e c o u p l i n g cons t an t s were de te rmined s e p a r a t e l y (see below) to f i r s t order us ing equa t ion ( e1 .29 ) . The AsDH 2 and AsHD 2 f i t s were done in the 1 1 1 r and r . . II r e p r e s e n t a t i o n s r e s p e c t i v e l y to a v o i d the i n t r o d u c t i o n of imaginary quadrupole e lements i n t o our f i t t i n g h a m i l t o n i a n ( r 32 ) . The r e s u l t s of these f i t s are in Tab l e 4 . 6 . For each i s o t o p i c d e r i v a t i v e the data se t s i n c l u d e R branches from two d i f f e r e n t i n i t i a l J v a l u e s . C o e f f i c i e n t s of d i a g o n a l powers of J 2 terms ( r o t a t i o n a l c o n s t a n t , D J , $ J , . . . ) are determined from these R b ranches . From equa t i on (e1 . 13) w i th j u s t two of these R branches we see we cannot determine the " t h i r d " power of J 2 parameter , namely $ j , from our data s e t s . Because of t h i s the parameter $ j was c o n s t r a i n e d to zero in the f i t s . T h i s means the e m p i r i c a l va lue of D T w i l l i n c l u d e the e f f e c t of <i> , which from the s u b m i l l i m e t e r wave study of A s H 3 (r93) i s expected to be on 121 the o rder of 0.5 KHz or about .05% of D T . Look ing at Tab le 4.6 one n o t i c e s tha t the cho i c e of s e x t i c (<£>'s and c/>'s) cons t an t s r epo r t ed d i f f e r s in the two asymmetric t o p s . These s e x t i c s are i n c l u d e d here on l y as f i t t i n g parameters and no p h y s i c a l s i g n i f i c a n c e shou ld be a t t r i b u t e d to them. The cho i c e of which s e x t i c c o n s t a n t s to i n c l u d e in the f i t s was e s t a b l i s h e d by how w e l l the c o n s t a n t s c o u l d be determined and what c h o i c e of combina t ions of cons t an t s was the l e a s t i n t e r n a l l y c o r r e l a t e d . In the i n i t i a l data r e d u c t i o n the quadrupo le c o u p l i n g and the magnet ic h y p e r f i n e c o u p l i n g were t r e a t e d to f i r s t o rder u s i ng equa t ion ( e 1 . 2 6 ) . T h i s f i r s t o rde r t reatment seemed adequate but c e r t a i n anomal ies appeared in the v a l ues ob ta ined fo r the quadrupole c o u p l i n g c o n s t a n t s . I t was not p robab le but i t had to be a s c e r t a i n e d whether or not these anomalous r e s u l t s c o u l d be e x p l a i n e d by h ighe r o rder quadrupo le e f f e c t s . I t tu rned out they c o u l d not and the d i a g o n a l " e x a c t " c o u p l i n g cons t an t s r e p o r t e d here are not s i g n i f i c a n t l y d i f f e r e n t than those ob t a i ned from a f i r s t o rder f i t . A more complete d i s c u s s i o n of t h i s problem i s to be found under the t i t l e "Nuc lea r Hype r f i ne E f f e c t s in A r s i n e " ( s e c t i o n 4 . 8 ) . A consequence of t h i s exact s o l u t i o n of the quadrupo le e f f e c t was the e x t r a o r d i n a r y t reatment of the sp in r o t a t i o n magnetic h y p e r f i n e e f f e c t . The magnetic c o n s t a n t s are those ob t a i ned from the f i r s t o rder quad rupo l e-sp in r o t a t i o n f i t , a f t e r which the s p i n r o t a t i o n 1 22 c o n t r i b u t i o n to the s p l i t t i n g was s u b t r a c t e d o f f the expe r imen ta l f r e q u e n c i e s and these " f i x e d " f r equenc i e s were then f i t to the " e x a c t " quadrupo le A r e d u c t i o n h a m i l t o n i a n . S ince the p r i n c i p a l quadrupo le c o u p l i n g cons t an t s d i d not change s i g n i f i c a n t l y no r e a l d i f f i c u l t y i s expected from t h i s method. Problems may a r i s e in the s tandard d e v i a t i o n a t t r i b u t e d to the v a r i o u s o ther parameters as we have e l i m i n a t e d any c o r r e l a t i o n s between the sp in r o t a t i o n cons t an t s and the o ther pa ramete rs . However, the c o r r e l a t i o n s between the sp in r o t a t i o n cons t an t s and the quadrupo le c o n s t a n t s ob ta ined in the f i r s t o rder f i t were very sma l l and the c o r r e l a t i o n c o e f f i c i e n t s between the exact quadrupo le c o n s t a n t s and the r o t a t i o n a l parameters of the f i n a l f i t were a l s o sma l l so one would expect no s i g n i f i c a n t change in any parameter e r r o r as a r e s u l t of t r e a t i n g the sp in r o t a t i o n c o u p l i n g " s e p a r a t e l y " . The parameter most s u s c e p t i b l e to f a i l u r e s of t h i s method w i l l be X because i t , l i k e the s p i n r o t a t i o n paramete rs , i s s e n s i t i v e to sma l l changes in the h y p e r f i n e s p l i t t i n g . S t i l l , i t i s not expec ted tha t a new minimum found in a g l o b a l f i t , i n c l u d i n g the s p i n r o t a t i o n , would produce " b e t t e r " v a l ues of e i t h e r X or the sp in r o t a t i o n c o n s t a n t s . Tab le 4.4 Observed spectrum of AsH 2 D 1 23 R o t a t i o n a l T r a n s i t i o n F (upper )-F ( lower ) Frequency (MHz)^ QBS-CALC 1 1 o 0 o o 1.5 -- 1.5 2.5 — 1.5 0.5 -- 1.5 185689.428(0.015) -0.008 185715.534(0.015) -0.023 185736.006(0.015) 0.030 1 1 1 - 1 1 . 1 . 1 . 2. 2. 0. 0. o 1 5 5 5 5 5 5 5 0, 2, 1 , 2 1 . 0 1 35461 35456 35451 35436 35430 35423 3541 4 .030(0, .480(0, .290(0 .090(0 .930(0 .920(0 .140(0 030) 030) 030) 030) 030) 030) 030) •0.056 0.009 -0.026 •0.061 -0.048 -0.028 -0.039 2 O 2 1 10 0.5 • 5 5 5 5 5 0 0 2 2 1 1 257141 . 132(0.060) 257148.811(0.150) 257167.070(0.010) 257173.819(0.015) 257194.733(0.020) 257199.919(0.015) -0 0.033 0.577 0.003 0.016 0.041 005 2 1 2 2 o 0. 0, 3, 3, 1 , 1 , 2, 2, 2 .5 .5 5 .5 .5 5 ,5 .5 .5 32943. 32950. 32957. 32964. 32969. 32976. 32983. 32988. 32989. 570(0, 710(0, 520(0, 270(0, 640(0, 780(0, 170(0, 340(0, 910(0, 030) 030) 030) 030) 030) 030) 030) 030) 030) •0.005 •0.000 0.004 0.018 0.024 0.029 0.000 0.020 0.005 1 2 • 3 1 3 3.5 • 5 5 5 .5 5 .5 ,5 .5 1 . 5 -3, 3, 2 3. 2 4 2 4 1 1 1 5502 15511 15516 1 5521 1 5524 1 5530 15543 15549 1 5552 1 5570 .860(0 .350(0 .280(0 .930(0 .510(0 .270(0 .010(0 .260(0 .470(0 .950(0 .030) .030) .300) .060) .030) .060) .030) .030) .040) .030) 0.024 0.041 0.234 0.083 0.009 0.004 0.003 •0.017 0.011 •0.003 Tab le 4.4 con t i nued Observed spectrum of AsH 2 D R o t a t i o n a l Frequency (MHz) OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 3,3- 3 0 3 1 . 5 — 2.5 29499.490(0 .030) -0.006 4.5 — 3.5 29508.600(0.030) 0.019 1.5 -- 1.5 29509.100(0.030) 0.017 4.5 -- 4.5 29517.700(0.030) 0.022 2.5 -- 3.5 29522.800(0 .030) -0.001 2.5 — 2.5 29527.450(0.030) 0.013 3.5 — 3.5 29536.030(0.030) 0.019 2.5 — 1.5 29537.050(0.100) 0.027 3.5 -- 2.5 29540.660(0.030) 0.014 3.5 — 4.5 29545.120(0.030) 0.012 4 2 3 - 4 1 3 5.5 — 5.5 97368.932(0.100) -0.163 4 , 3 " 4 1 f l 4.5 -- 4.5 25826.540(0.030) 0.019 3.5 -- 3.5 25842.760(0.050) -0.006 5.5 -- 5.5 25872.820(0.050) -0.012 2.5 — 2.5 25889.100(0 .030) 0.014 4 « o " 4 3 2 4.5 -- 4.5 256951.355(0.015) 0.018 3.5 -- 3.5 256953.001(0.015) 0.035 5.5 — 5.5 256956.197(0.015) -0.011 2.5 — 2.5 256957.736(0.240) 0.007 4 , « " 4 0 u 2.5 --- 3 . 5 25378.520(0.030) -0.030 5.5 --- 4.5 25384.980(0.030) -0.006 2.5 --- 2.5 25391.320(0.030) -0.007 5.5 — - 5.5 25397.110(0.030) -0.021 3.5 — - 4.5 25403.070(0.050) -0.038 3.5 — - 3.5 25407.810(0.030) -0.005 4.5 --- 4.5 25413.590(0.030) -0.017 4.5 .— - 3.5 25418.320(0 .030) 0.006 3.5 — - 2.5 25420.600(0 .030) 0.008 4.5 --- 5.5 25425.770(0.030) 0.018 4.5 — 4.5 256922.816(0.015) 0.016 3.5 — 3.5 256924.363(0.015) -0.009 5.5 — 5.5 256927.523(0.015) 0.015 2.5 -- 2.5 256928.904(0.015) -0.065 125 Tab le 4.4 con t i nued Observed spectrum of AsH 2 D R o t a t i o n a l Frequency (MHz) OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 3 . 5 - - 3.5 91154.277(0.030) 0.153 4 . 5 - - 4.5 91174.250(0.030) 0.011 5 . 5 — 5.5 91179.960(0.030) 0.028 5 1 _ 5 1 5 5.5 -- 5.5 • 38662.235(0.030) 0.007 4.5 -- 4.5 38675.160(0.030) -0.062 6.5 -- 6.5 38708.250(0.030) -0.018 3.5 -- 3.5 38721.290(0.030) 0.024 5 2 3 _ 5 1 5 5.5 -- 5.5 134376.902(0.015) -0.012 4.5 -- 4.5 134387.266(0.015) 0.001 6.5 -- 6.5 134413.696(0.015) 0.014 3.5 -- 3.5 134424.004(0.015) -0.002 6 1 6 - 6 0 6 4.5 --7.5 --4.5 --7.5 --5.5 --5.5 --6.5 --6.5 --5.5 — 6.5 --633- 6 25 6.5 --5.5 --7.5 --4.5 — 7 2 6 - 7 , 6 5.5 --8.5 --6.5 --7.5 --71 7 _ 7 0 7 5.5 --8.5 --5.5 --8.5 --6.5 --5.5 16597.230(0.030) -0.086 6.5 16600.840(0.030) -0.057 4.5 16617.790(0.030) 0.009 7.5 16620.460(0.030) 0.010 6.5 16623.750(0.050) -0.054 5.5 16629.010(0.030) 0.003 6.5 16631.690(0.030) 0.024 5.5 16636.880(0.030) 0.010 4.5 16649.510(0.050) 0.039 7.5 16651.240(0.030) 0.021 6.5 185726.865(0.015) -0.004 5.5 185728.479(0.015) 0.011 7.5 185733.771(0.015) -0.001 4.5 185735.304(0.015) -0.015 5.5 75827.748(0.040) -0.003 8.5 75831.756(0.020) 0.042 6.5 75846.940(0.030) -0.050 7.5 75850.890(0.020) -0.033 6.5 12641.840(0.300) -0.095 7.5 12644.680(0.030) -0.060 5.5 12666.330(0.030) 0.017 8.5 12668.070(0.030) 0.006 6.5 12674.860(0.030) 0.007 Tab le 4.4 con t i nued Observed spectrum of AsH 2 D R o t a t i o n a l Frequency (MHz) OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 7.5 -- 7.5 12676.600(0.030) 0.004 7.5 — 6.5 12682.020(0.300) 0.025 6.5 -- 5.5 12698.610(0.500) -0.620 7.5 — 8.5 12699.600(0.500) -0.320 1 3 1 ~~ 1 2 6 7.5 -- 7.5 187808.628(0.015) 0.006 6.5 -- 6.5 187810.374(0.015) 0.041 8.5 -- 8.5 187817.121(0.015) -0.036 5.5 -- 5.5 187818.819(0.015) -0.005 g ^ _ g 6.5 -- 6.5 9318.710(0.030) 0.017 9.5 -- 9.5 9319.810(0.030) -0.000 7.5 — 7.5 9324.900(0.030) 0.022 8.5 -- 8.5 9325.990(0.030) 0.002 8 — 8 8.5 -- 8.5 24264.570(0.030) 0.028 7.5 -- 7.5 24268.500(0.030) 0.020 9.5 -- 9.5 24286.450(0.030) -0.053 6.5 — 6.5 24290.410(0.050) -0.041 8 — 8 6.5 -- 6.5 67080.220(0.050) -0.037 9.5 — 9.5 67083.480(0.050) -0.082 7.5 -- 7.5 67098.420(0.050) -0.074 8.5 -- 8.5 67101.660(0.050) -0.111 9 2 7 ~ 9 2 8 9.5 — 9.5 35848.010(0.050) 0.045 8.5 -- 8.5 35852.000(0.050) 0.053 10.5 -- 10.5 35872.810(0.050) -0.036 7.5 — 7.5 35876.810(0.050) -0.031 9 - 9 2 8 7 . 5 8 - - 7.5 57954.860(0.060) -0.060 10.5 -- 10.5 57957.600(0.060) -0.051 9?5 1 -- 1 9 . 5 39923.350(0.030) 0.014 12.5 -- 12.5 39925.140(0.030) 0.046 10.5 -- 10.5 39936.480(0.030) -0.004 11.5 -- 11.5 39938.280(0.030) 0.057 1 27 Tab le 4.4 con t i nued Observed spectrum o f . A s H 2 D R o t a t i o n a l Frequency (MHz) OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 12 2 , , -12 , 1 1 10.5 -- 10.5 31733.020(0.050) 0.026 13.5 -- 13.5 31734.370(0.050) 0.026 11.5 -- 11.5 31744.000(0.050) 0.029 12.5 — 12.5 31745.300(0.050) -0.005 13 2 ,2- 1 3 , 1 2 11.5 -- 11.5 24497.640(0.030) 0.020 14.5 -- 14.5 24498.650(0.030) 0.027 12.5 -- 12.5 24506.350(0.030) -0.067 13.5 — 13.5 24507.350(0.030) -0.057 1 5 2 , (i ~ 1 5 , 1 « 13.5 -- 13.5 13454.850(0.040) 0.036 16.5 -- 16.5 13455.280(0.040) -0.034 14.5 -- 14.5 13459.850(0.040) 0.020 15.5 -- 15.5 13460.290(0.040) -0.033 1 6 2 15~16 , 1 5 14.5 -- 14.5 9617.920(0.040) 0.017 17.5 — 17.5 9618.230(0.040) -0.011 15.5 -- 15.5 9621.560(0.050) 0.062 16.5 — 16.5 9621.810(0.040) -0.019 t Numbers in pa ren theses are es t imated measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 28 Tab le 4.5 Observed spectrum of AsD 2 H R o t a t i o n a l Frequency (MHz )^OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 1 , , - 0 o o ' 2 1 2 2 , " 1 2 2 3 3 , " 1 .5 -- 1 .5 2.5 -- 1 .5 0.5 -- 1 .5 1 , , 2.5 -- 1 .5 3.5 -- 2.5 "2, , 2.5 — 2.5 2.5 — • 1 .5 2.5 -- 3.5 1 .5 -- 2.5 3.5 -- 2.5 1 .5 -- 1 .5 3.5 -- 3.5 1 .5 -- 0.5 0.5 -- 1 .5 0.5 -- 0.5 3,3 3.5 -- 3.5 2.5 -- 2.5 4.5 -- 4.5 1 .5 -- 1 .5 3 2 , 3.5 -- 3.5 2.5 -- 2.5 4.5 -- 4.5 1 . 5 -- 1 .5 3 2 2 3.5 -- 3.5 2.5 -- 2.5 4.5 -- 4.5 1 .5 -- 1 .5 132628.022(0.015) -0.000 132641.702(0.015) 0.014 132652.180(0.030) -0.048 302880.654(0.015) 0.021 302887.548(0.015) -0.029 16783.520(0.030) -0.016 16793.370(0.030) 0.020 16796.820(0.040) 0.016 16797.930(0.040) 0.007 16804.140(0.030) -0.007 16807.760(0.030) 0.022 16817.410(0.030) -0.006 16821.330(0.030) -0.007 16827.980(0.030) -0.008 16841.620(0.030) 0.033 94764.338(0.015) -0.009 94768.923(0.015) -0.004 94774.226(0.015) -0.001 94778.787(0.015) 0.015 33393.890(0.030) -0.024 33409.490(0.030) 0.004 33427.290(0.030) 0.004 33442.850(0.030) -0.005 81006.936(0.015) -0.006 81015.038(0.015) 0.029 81024.294(0.015) 0.014 81032.298(0.015) -0.025 Tab le 4.5 con t inued Observed spectrum of AsD 2 H R o t a t i o n a l Frequency (MHz) OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 4 2 2 " " 4 1 3 2.5 -- 3.5 5.5 -- 4.5 2.5 — 2.5 5.5 -- 5.5 3.5 -- 3.5 4.5 -- 4.5 4 3 2 ' " 4 2 2 4.5 -- 4.5 3.5 -- 3.5 5.5 -- 5.5 2.5 -- 2.5 5 q 2 " 5 3 2 5.5 -- 5.5 4.5 -- 4.5 6.5 -- 6.5 3.5 -- 3.5 5 5 0 " 5 n i 5.5 -- 5.5 4.5 — 4.5 6.5 -- 6.5 3.5 -- 3.5 6« 3" " 6 3 3 6.5 -- 6.5 5.5 -- 5.5 7.5 -- 7.5 4.5 -- 4.5 ? 5 3 - ~l« t 7.5 — 7.5 6.5 -- 6.5 8.5 — 8.5 5.5 -- 5.5 83231.403(0.015) -0.014 83233.590(0.015) -0.007 83238.915(0.015) 0.013 83240.523(0.015) 0.017 83243.355(0.015) -0.006 83244.920(0.015) -0.017 15761.170(0.030) -0.014 15766.620(0.030) -0.023 15776.740(0.030) -0.010 15782.200(0.030) -0.013 31654.030(0.030) 0.005 31659.390(0.030) -0.000 31673.060(0.030) 0.017 31678.400(0.030) - -0.012 80981.343(0.015) -0.004 80989.149(0.015) -0.002 81009.034(0.015) 0.002 81016.836(0.015) 0.005 12408.800(0.030) 0.006 12410.870(0.040) 0.063 12417.290(0.030) -0.017 12419.330(0.040) 0.006 133555.028(0.030) 0.004 133555.976(0.030) 0.016 133559.702(0.015) 0.015 133560.586(0.015) -0.014 1 30 Tab le 4.5 con t i nued Observed spectrum of AsD 2 H R o t a t i o n a l Frequency (MHz) OBS-CALC T r a n s i t i o n F (upper )-F ( lower ) 8 6 2 8 5 3 8.5 -- 8.5 77233.644(0.015) 0.021 7.5 -- 7.5 77234.906(0.015) 0.009 9.5 -- 9.5 77240.756(0.015) -0.010 6.5 -- 6.5 77242.010(0.015) -0.023 9 6 3 9 5 fl 7.5 -- 7.5 94017.284(0.200) 0.116 10.5 -- 10.5 94017.284(0.200) -0.164 8.5 -- 8.5 94019.000(0.200) 0.075 9.5 — 9.5 94019.000(0.200) -0.206 10 7 3 _ 1 0 6 4 10.5 -- 10.5 93282.489(0.150) 0.253 9.5 -- 9.5 93282.489(0.150) -0.028 11.5 — 11.5 93284.246(0.150) 0.062 8.5 -- 8.5 93284.246(0.150) -0.219 1 1 7 5 - 1 1 6 5 11.5 -- 11.5 14348.590(0.040) 0.044 10.5 -- 10.5 14349.180(0.040) -0.000 12.5 -- 12.5 14353.340(0.040) -0.015 9.5 -- 9.5 14353.970(0.040) -0.024 1 1 1 0 1 - 1 1 9 2 11.5 -- 11.5 187226.127(0.015) -0.020 10.5 -- 10.5 187227.900(0.015) 0.013 12.5 -- 12.5 187239.396(0.015) -0.026 9.5 -- 9.5 187241.193(0.015) 0.039 t Numbers in pa ren theses a re es t imated measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 131 Tab le 4 . 6 E m p i r i c a l s p e c t r o s c o p i c parameters of v a r i o u s a r s i n e s AsH 2 D AsHD 2 AsD 3 X MHz 1 1 0 5 8 1 . 6 6 ( 1 ) 55511 . 8 9 ( 1 ) Y MHz 7 5 1 3 3 . 5 2 ( 1 ) 7 7 1 3 1 . 5 3 ( 1 ) 5 7 4 7 7 . 6 0 ( 1 ) Z MHz 7 2 5 4 8 . 7 0 ( 1 ) 7 1 5 2 5 . 6 4 ( 1 ) 5 2 6 4 2 . 0 8 ( 1 2 ) AJ KHz 1 2 3 5 . 5 ( 1 . 5 ) 1 1 3 1 . 3 ( 1 . 0 ) D J KHz 7 4 2 . 3 ( 7 ) A JK KHz 2 2 2 9 . 9 ( 8 ) - 1 8 7 5 . 8 ( 3 ) D JK KHz - 9 4 0 . 1 ( 1 3 ) A K KHz - 2 6 3 0 . 2 ( 3 ) 2 1 7 3 . 2 ( 3 ) D K KHz 8 3 3 . 4t 6J KHz 1 2 5 . 8 ( 2 ) - 4 0 7 . 0 4 ( 5 ) 6 K KHz 1 9 9 3 . 4 ( 2 ) 9 7 9 . 9 ( 2 ) *J Hz H J Hz *JK Hz - 3 4 6 ( 9 ) H JK Hz - 1 9 7 ( 4 ) <l> KJ Hz 6 5 7 ( 3 7 ) 66 ( 1 ) H K J Hz 1 2 3 . 8 * *K Hz - 4 4 3 ( 3 1 ) H K Hz *K Hz 6 3 ( 4 ) - 6 9 ( 1 ) *JK Hz 2 5 4 ( 7 ) *K Hz 9 8 ( 1 1 ) - 1 3 5 ( 6 ) L J J J K H Z 0 . 0 7 3 ( 4 ) X Z Z MHz - 1 0 3 . 6 9 ( 2 ) - 5 4 . 0 1 ( 3 ) - 1 6 4 . 7 6 ( 1 0 ) x x x ~ X Y y MHz - 6 0 . 7 8 ( 5 ) - 1 0 8 . 9 3 ( 5 ) x z x MHz 1 3 1 ( 2 6 ) - 4 9 ( 1 7 ) C AA KHz 1 0 4 . 7 ( 3 0 ) 6 9 . 9 ( 4 5 ) C N KHz * 5 5 . 5 C BB KHz 6 9 . 0 ( 1 6 ) 6 9 . 3 ( 3 5 ) C K KHz 5 0 . 8 * c c c KHz 7 0 . 7 ( 1 0 ) 5 6 . 8 ( 4 5 ) fHarmonic Force F i e l d va lue * F i x e d by a r a t i o argument, see t e x t . — Numbers in pa ren theses are the s t anda rd d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 32 7. DISCUSSION OF NUCLEAR QUADRUPOLE COUPLING IN ARSINE In the a r s i n e symmetric tops the mo lecu l a r quadrupo le c o u p l i n g t ensor i s d i a g o n a l in the symmetric top p r i n c i p a l a x i s sys tem. If we d e f i n e the z a x i s as the C 3 a x i s , the y a x i s as p e r p e n d i c u l a r to one of the symmetry p l anes tha t i n c l u d e s a hydrogen i so tope and b i s e c t s the ang le between the o ther two hydrogen i s o t o p e s , then by symmetry and L a p l a c e ' s equa t ion x x = X y = ~ ^X z -For the asymmetric tops we d e f i n e the y a x i s , as above, as p e r p e n d i c u l a r to the symmetry p lane tha t i n c l u d e s the unique hydrogen i s o t o p e . In t h i s case however, the p r i n c i p a l z a x i s of the asymmetric top c o u p l i n g tensor does not c o i n c i d e w i th what one might c a l l the symmetric top ( i f i t were one) C 3 a x i s . I t w i l l be very c l o s e w i th the d i f f e r e n c e r e f l e c t i n g the d i f f e r e n t " a ve rage " e l e c t r o n i c s t r u c t u r e s of the v a r i o u s s p e c i e s . G iven the e m p i r i c a l l y de te rmined l i n e a r comb ina t ions of the quadrupo le c o u p l i n g c o n s t a n t s found in Tab le 4.6 and L a p l a c e ' s equa t ion we can c o n s t r u c t the f o l l o w i n g t a b l e ( in MHz) . AsH 2 D (111 r ) AsD 2 H ( I i r ) X xx 21.46(9) -27.46(9) X xz 1 31(26) -49(17) X zz' -103.69(2) -54.01(3) 82.24(4) 81 .47(4) The l a b e l s x , y , z in the above t a b l e co r r e spond to the 1 33 p r i n c i p a l i n e r t i a l axes as d e s c r i b e d in 4 . 6 . The above i m p l i e d t ensor ( a l l the other e lements are set at zero ) i s a two by two and a one by one. The yy component above c o r r e s p o n d i n g to the " p e r p e n d i c u l a r " y a x i s d e f i n e d be fo re i s a lways then a p r i n c i p a l component of the mo lecu l a r quadrupo le tensor fo r a l l the i s o t o p i c s p e c i e s we are l o o k i n g a t . G iven that the e l e c t r o n i c env i ronments of our v a r i o u s s p e c i e s are very s i m i l a r we shou ld expect tha t t h i s p e r p e n d i c u l a r y component of the c o u p l i n g t enso r shou ld be ve ry s i m i l a r in a l l c a s e s . T h i s i s the expe r i ence of the above t a b l e . A l s o the y component of the p r i n c i p a l c o u p l i n g t enso r in the symmetric tops i s j u s t - -^-x • D i v i d i n g the lone symmetric top c o u p l i n g cons t an t s by minus two g i v e s x y va lues of 82.375(80) MHz r e f . ( r 9 l ) f o r AsD 3 and 81.3219(266) MHz(r93) f o r A s H 3 . The d i f f e r e n c e s are a l l in the order of a percent sugges t i ng the assumpt ion of an i s o t o p i c a l l y i n v a r i a n t e l e c t r o n i c p o t e n t i a l i s q u i t e r e a s o n a b l e . S i m i l a r d i s c r e p a n c i e s have been observed fo r ammonia ( r l 0 4 , r l 0 5 ) , s t i b i n e s ( r 9 l ) and CH 3C1 ( r l 0 6 ) . To d i a g o n a l i z e our asymmetric top c o u p l i n g tensor we r o t a t e about the y a x i s by an ang le 6' g i ven by 2x xz tan20 = X - X [4 .8 ] xx zz T h i s ang le shou ld approximate rough ly the ang le we must r o t a t e the symmetric top moment of i n e r t i a t ensor upon i s o t o p i c s u b s t i t u t i o n in order to aga in d i a g o n a l i z e i t . 1 34 Assuming .the -r s t r u c t u r e g iven in Chapter 6 the moment of i n e r t i a t enso r s r e q u i r e d r o t a t i o n s of - 4 2 . 2 ° and 2 9 . 3 ° fo r AsD 2 H and AsH 2 D r e s p e c t i v e l y . These ang les compare reasonab l y we l l w i th the va lues 9 of - 3 7 . 4 ° ± 3 ° and 3 2 . 3 ° ± 2 ° . 8. DISCUSSION OF THE SPIN-ROTATION COUPLING CONSTANTS OF  ARSINE The purpose of t h i s s e c t i o n i s to i n d i c a t e tha t the e m p i r i c a l l y de termined magnetic c o u p l i n g cons t an t s of A s H 3 , A sH 2 and AsD 2 H are a l l c o n s i s t e n t w i th one another by us ing them to es t imate the c o u p l i n g cons t an t s of A s D 3 . In the case of AsD 3 the magnetic s p i n - r o t a t i o n cons t an t s were not e m p i r i c a l l y de te rm inab le from our i n c r e a s e d data set (see s e c t i o n 4.4) so tha t in the f i n a l f i t s they were c o n s t r a i n e d to the va lues e s t ima ted below. The i n t e r a c t i o n we are d e a l i n g w i th i s between the sp in magnet ic moment of the a r s e n i c nuc leus and the mo lecu l a r magnet ic f i e l d genera ted by the mo lecu la r r o t a t i o n . For a f i r s t o rder d i s c u s s i o n we can assume tha t the mo lecu l a r magnet ic f i e l d genera ted by the r o t a t i o n i s p r o p o r t i o n a l to the angu la r momentum ( r31 , where we are assuming tha t the e l e c t r o n i c s t r u c t u r e i s i s o t o p i c a l l y i n v a r i a n t ) . That i s , the f a s t e r the molecu le r o t a t e s about an a x i s the g r ea t e r the magnet ic f i e l d and hence the g r ea t e r the i n t e r a c t i o n ( coup l i ng ) cons tan t about that a x i s . A l s o s i n ce the speed of r o t a t i o n i s d i r e c t l y r e l a t e d to the r o t a t i o n a l cons t an t s i t 135 shou ld not be s u r p r i s i n g tha t the s p i n - r o t a t i o n cons t an t s are a l s o r e l a t e d to the r o t a t i o n a l c o n s t a n t s . In f a c t we f i n d tha t the magnet ic c o u p l i n g cons tan t d i v i d e d by the " c o r r e s p o n d i n g " (same a x i s ) r o t a t i o n a l c o n s t a n t , f o r the examples p resen ted in t h i s work, i s a cons t an t r a t i o (w i th in 4%), (see fo r i n s t ance Garvey et a l ( r l 0 4 ) ) . Us ing t h i s r a t i o and the r o t a t i o n a l cons t an t s of AsD 3 a l l owed the d e t e r m i n a t i o n of the AsD 3 c o u p l i n g c o n s t a n t s p resen ted in Tab le 4 . 6 . 136 F. CHAPTER 5: PHOSPHINE Phosphine i s a c o l o r l e s s , po isonous gas pu rpo r t ed to have a p e c u l i a r " a c e t y l e n e l i k e " odo r . In f a c t , i t i s the o ther way a round , the odor we u s u a l l y a s s o c i a t e w i th a c e t y l e n e i s due to t r a c e amounts of PH 3 ( r l 0 7 ) . A h i s t o r y of phosphorus i s g i ven by J . Emsley ( r l 08 ) who n a r r a t e s that phosphorus s i n c e i t s d i s c o v e r y j u s t over 300 years ago has been used a s , amongst o ther t h i n g s , an a p h r o d i s i a c , a po i son and a magic " p i l l " to i n c r e a s e i n t e l l i g e n c e . Phosphorus i s perhaps best known fo r i t s r o l e in i n c e n d i a r y bombs. Phosphorus was e s p e c i a l l y popu la r in the c o u r t s of the 17th and 18th cen tu r y as i t would glow in the dark ( "phosus" meaning l i g h t - b e a r i n g ) , k i l l peop le and burs t i n t o f lame when exposed to a i r . It no doubt was d u r i n g t h i s t ime tha t phosphine was d i s c o v e r e d as the gas g iven o f f when white phosphorus was heated in an a l k a l i s o l u t i o n . When phosphine i s contaminated wi th P 2 H „ (produced in the same r e a c t i o n ) i t w i l l i g n i t e spontaneous l y at room temperature (r107) . The PH 3 spectrum has been we l l s t u d i e d in both the i n f r a r e d ( r l09-r113) and the microwave r eg ions ( r 82 , r114 , r102 ) where o b s e r v a t i o n of many normal as we l l as f o r b i d d e n t r a n s i t i o n s has l ed to a complete d e t e r m i n a t i o n of ground s t a t e r o t a t i o n a l cons t an t s as we l l as q u a r t i c and many h ighe r order d i s t o r t i o n c o n s t a n t s . The PD 3 spectrum has a l s o been s t u d i e d a l though not as e x t e n s i v e l y as tha t of P H 3 . A l lowed s u b m i l l i m e t e r wave 1 37 t r a n s i t i o n s in the ground v i b r a t i o n a l s t a t e were r epo r t ed by Helminger and Gordy (r1 14) in 1969. L a t e r in 1974 an i n i t i a l s tudy of f o r b i d d e n t r a n s i t i o n s in the ground v i b r a t i o n a l s t a t e was made by Chu and Oka ( r 82 ) . In 1977 Helms and Gordy (r25) i n c r e a s e d the number of measured f o r b i d d e n t r a n s i t i o n s and p re sen ted a s imul taneous a n a l y s i s of a l l the f o rb idden and a l l owed r o t a t i o n a l da ta of the ground v i b r a t i o n a l s t a t e . T h i s a n a l y s i s proved u n s a t i s f a c t o r y and we s h a l l p resen t a r e - a n a l y s i s based on the fo rma l i sm e s t a b l i s h e d in Chapter 1 (see a l s o the s e c t i o n e n t i t l e d " F a i l u r e s of the R e d u c t i o n s " ) . F i n a l l y , f o r b i d d e n and a l l owed v i b r a t i o n - r o t a t i o n t r a n s i t i o n s have been observed in the i n f r a r e d spectrum of the four fundamental v i b r a t i o n s of PD 3 by K i j i m a and Tanaka ( r115 ) . T h e i r a n a l y s i s was based on the approach of Helms and Gordy and would have a l s o been u n s a t i s f a c t o r y except tha t in p r a c t i c e they c o n s t r a i n e d to ze ro the o f f e n d i n g terms " f i t t i n g " on ly to the " s t a n d a r d " parameters e s t a b l i s h e d in Chapter 1 . The PT 3 spectrum i s not we l l s t u d i e d and on l y the fundamental v i b r a t i o n f r e q u e n c i e s have been r e p o r t e d in a c u r s o r y examinat ion of the i n f r a r e d spectrum ( r 98 ) . The asymmetric top phosph ines have been v i r t u a l l y i gno red i n the l i t e r a t u r e . For the p a r t i a l l y deu t e r a t ed phosph ines , l i k e t h e i r a r s i n e c o u n t e r p a r t s , the d i f f i c u l t y l i e s in o b t a i n i n g a whole c l a s s of t r a n s i t i o n s , namely R branches (AJ=1). These R branches are to be found at f r e q u e n c i e s above 100 GHz, and a l though s t r o n g , are beyond 1 38 the f requency c a p a b i l i t y of most spec t r ome te r s . In 1951 Loomis and S t r andberg (r99) r e p o r t e d one t r a n s i t i o n fo r PH 2 D. L a t e r , in 1953, S i r v e t z and Weston (r116) measured two other t r a n s i t i o n s fo r PH 2D and three new t r a n s i t i o n s fo r PD 2 H. They a l s o r epo r t ed s e v e r a l o ther t r a n s i t i o n s that they were unable to a s s i g n . Dur ing the course of the present work some t r a n s i t i o n s measured here fo r PH 2D were r epo r t ed by K u k o l i c h et a l ( r117 ) . The purpose of t h e i r exper iment was to i n v e s t i g a t e the deuter ium quadrupo le moment in PH 2D and consequen t l y they needed a h ighe r r e s o l u t i o n inst rument than was a v a i l a b l e fo r t h i s study (the deuter ium quadrupo le moment produces s p l i t t i n g s in the PH 2D spectrum on the o rder of 100 KHz) . In a few s p e c i a l cases our own measurements have been r e p l a c e d by these h ighe r r e s o l u t i o n measurements. S t i l l , K u k o l i c h et a l (r117) on l y i n c r e a s e d the number of PH 2D r o t a t i o n a l t r a n s i t i o n s to 9 and s i n ce a l l these t r a n s i t i o n s were Q branches (AJ=0) they c o u l d on l y determine l i n e a r combina t ions of the r o t a t i o n a l cons t an t s - to " s o l v e " t h i s molecu le R branches were s t i l l needed. F u r t h e r , from t h e i r f i t they r epo r t a va lue of one q u a r t i c c o n s t a n t , D T t , , J K d e f i n e d in a p r o l a t e b a s i s , a t 4.30 ± .04 MHz. Our subsequent data ana l y ses are a l l done in o b l a t e bases but i t i s s t r a i g h t f o r w a r d to c a l c u l a t e D J K in a p r o l a t e b a s i s . In t h i s case we get a very d i f f e r e n t va lue of 3.64 MHz. The problem w i th the a n a l y s i s of K u k o l i c h et a l i s tha t t h e i r data set was j u s t too s m a l l . T h i s f o r c e d them to c o n s t r a i n 1 39 to ze ro v a r i o u s parameters making the p h y s i c a l s i g n i f i c a n c e of the parameters they c o u l d determine u n c e r t a i n . In t h i s case the parameters they r epor t shou ld on l y have been c o n s i d e r e d as f i t t i n g parameters in p r e c i s e l y the same way as we s h a l l c o n s i d e r the s e x t i c c o n s t a n t s we r epo r t l a t e r f o r the phosphine asymmetric t o p s . 1. PREPARATION OF PHOSPHINE GAS The phosphine fo r t h i s i n v e s t i g a t i o n was p repared by the r e d u c t i o n of white phosphorus in a sodium hydrox ide s o l u t i o n . Sodium hydrox ide p e l l e t s , H 2 0 / D 2 0 and white phosphorus were de-gassed in a g l a s s vacuum l i n e . Under 150 mm. of n i t r o g e n ' t h e mix ture was heated w i th a semi-luminous flame u n t i l the phosphorus mel ted (m.p. 4 0 ° C ) . The phosphine bubbled o f f , was passed through a water t r a p and was condensed wi th l i q u i d n i t r o g e n . 1 4 0 2. PREDICTION OF THE ASYMMETRIC TOP PHOSPHINE SPECTRA Es t ima tes of the r o t a t i o n a l parameters fo r the asymmetric top phosphine s p e c t r a were made in the same way as in the a r s i n e asymmetric top s tudy . The zero p o i n t average r o t a t i o n a l cons t an t s A , B and C were c a l c u l a t e d z z z by d i a g o n a l i z i n g the moment of i n e r t i a t ensor where bond l eng ths and ang les were the r z parameters of Chu and Oka ( r 82 ) . From a p r e l i m i n a r y f o r c e f i e l d d e r i v e d from c o r r e s p o n d i n g symmetric top data the harmonic p a r t s of the a lphas were c a l c u l a t e d . Adding t h i s c o n t r i b u t i o n to the r z r o t a t i o n a l cons t an t s above gave us " g u e s s t i m a t e s " of the e m p i r i c a l r o t a t i o n a l cons t an t s u s u a l l y to much b e t t e r than one per c e n t . A u s e f u l by-product of the i n i t i a l f o r c e f i e l d s tudy was the p r e d i c t i o n of the asymmetr ic top q u a r t i c c o n s t a n t s . C e r t a i n l y du r i ng the i n i t i a l sea rches fo r Q and R b ranches , e r r o r s in the q u a r t i c c o n t r i b u t i o n to the t r a n s i t i o n f requency p r e d i c t i o n s r e l a t i v e to the e r r o r s in the r o t a t i o n a l cons tan t c o n t r i b u t i o n were not s i g n i f i c a n t . However, once the r o t a t i o n a l c o n s t a n t s A 0 , B 0 , and C 0 were e s t a b l i s h e d , the e x c e l l e n t q u a r t i c p r e d i c t i o n s shor tened s u b s t a n t i a l l y search t i m e s . I t was a l s o c o m f o r t i n g to watch the " e m p i r i c a l " q u a r t i c s " conve rge " to the f o r c e f i e l d va lues as the number of measured l i n e s i n c r e a s e d . In a r s i n e , ass ignments were e a s i l y made from the h y p e r f i n e s p l i t t i n g . T h i s was not the case in phosphine and c e r t a i n l y here ass ignments made e a r l y in the study were o f t en based on how 141 c l o s e subsequent q u a r t i c s came to the f o r ce f i e l d p r e d i c t i o n s . The phosphine study p resen ted here i s v i r t u a l l y i d e n t i c a l to the a r s i n e study of Chapter 4. The d i f f e r e n c e s are tha t in the phosphine case there i s no h y p e r f i n e s t r u c t u r e d i s c e r n i b l e w i t h i n our expe r imen ta l r e s o l u t i o n and tha t the PD 3 spectrum i s s imp ly r e-ana l yzed w i th no new measurements. As was the case f o r AsH 2 D, PH 2D i s a p r o l a t e asymmetr ic r o t o r wi th " a " and " c " type t r a n s i t i o n s . A l s o l i k e A s D 2 H , PD 2H i s an o b l a t e asymmetric top wi th " b " and " c " type t r a n s i t i o n s . 3. OBSERVED PH 2D AND PD 2H SPECTRA In the present study the number of measured r o t a t i o n a l t r a n s i t i o n f r e q u e n c i e s has been i n c r eased to 24 fo r PH 2D and 17 f o r PD 2H (Tab les 5.1 and 5 .2 ) . These new t r a n s i t i o n s i n c l u d e h i g h f requency R branches of the type J=0—>1. The new f r e q u e n c i e s range from 9-264 GHz and J v a r i e s from zero to 19 f o r PH 2D and 14 f o r PD 2 H. The t r a n s i t i o n ass ignments are suppor ted by the f requency f i t s , d i s c u s s e d in the next s e c t i o n , r e l a t i v e i n t e n s i t i e s and Stark e f f e c t s as d i s c u s s e d below. Not hav ing a t r a n s i t i o n dependent h y p e r f i n e s t r u c t u r e super imposed on the r o t a t i o n a l t r a n s i t i o n s made the ass ignment of the asymmetric top phosphine s p e c t r a much more d i f f i c u l t than the a r s i n e asymmetric top s p e c t r a ass ignment . 1 42 Whenever p o s s i b l e t h e n , ass ignments were made from a s u b j e c t i v e es t imate of r e l a t i v e i n t e n s i t i e s or more c o n v i n c i n g l y from an a n a l y s i s of S tark s h i f t s . For example, f o r a AM=0 Q branch t r a n s i t i o n the r e l a t i v e i n t e n s i t i e s of S tark s h i f t e d components vary as M 2 . A l so from equa t ion (e1.43) a p l o t of f requency s h i f t of a component ve rsus M 2 w i l l g i ve a s t r a i g h t l i n e fo r the c o r r e c t ass ignment . Of course when the c h o i c e s of ass ignment fo r a t r a n s i t i o n d i f f e r in J i t o f t e n s u f f i c e s to s imp ly count the S tark l o b e s . The i n i t i a l phase of the phosphine asymmetric top study was very s i m i l a r to the i n i t i a l phase of the a r s i n e asymmetric top s tudy . Low f requency Q branches were measured us ing the U .B .C . s p e c t r o m e t e r s . F i x i n g one r o t a t i o n a l cons tan t at the p r e d i c t e d va lue of the p r e ced ing s e c t i o n and a l l o w i n g the o ther two r o t a t i o n a l cons t an t s to v a r y , the 3 p r e v i o u s l y a s s i g n e d t r a n s i t i o n s f o r each i s o t o p i c spec i e s (r116) were " f i t " . The subsequent f i t t i n g parameters were then used to p r e d i c t , r o u g h l y , f u r t h e r t r a n s i t i o n s . These p r e d i c t i o n s tu rned out to on l y be reasonab le f o r t r a n s i t i o n s whose J v a l ues were on the o rder of the J v a l ues of the t r a n s i t i o n s used in the f i t s . T h i s i s perhaps i l l u s t r a t e d by the f a c t tha t t r a n s i t i o n s i n i t i a l l y a s s i gned h igh J quantum numbers, based on p r e d i c t i o n s made from f i t s i n c l u d i n g on ly low J l i n e s , sometimes had to be r e-as s i gned when t r a n s i t i o n s w i th i n t e rmed i a t e J were found . 143 The second phase of t h i s study was to measure some h i g h f requency R b ranches . These measurements were done at the Je t P r o p u l s i o n L a b o r a t o r y . We were ab l e to measure J=0—>1 R branch l i n e s . The AM=0, J=0—>1 R branches are easy to i d e n t i f y from t h e i r d i s t i n c t i v e S ta rk p a t t e r n s . In t h i s case one expec ts to see one S tark lobe i d e n t i c a l in s i z e and shape to the ze ro f i e l d l i n e . However, a n t i c i p a t i o n of the above p a t t e r n l e d to some d i f f i c u l t y in e s t a b l i s h i n g the 0 o o - > 1 i 1 t r a n s i t i o n of PD 2 H. For t h i s t r a n s i t i o n the on l y s u i t a b l e low J , R branch c and ida t e w i t h i n ±10 GHz had two Stark l o b e s . T h i s t r a n s i t i o n showed the c h a r a c t e r i s t i c R branch S ta rk p a t t e r n where lobe i n t e n s i t y goes down as M goes up, but u n f o r t u n a t e l y two lobes were one too many fo r a AM=0, J=0—>1 t r a n s i t i o n . R u l i n g out a s s i g n i n g t h i s t r a n s i t i o n as the J=0—>1 fo r the moment, another f e a tu r e was the 0—>1 t r a n s i t i o n we were l o o k i n g f o r had an expected i n t e n s i t y g r e a t e r than tha t of o the r Q branch t r a n s i t i o n s tha t were encounte red in t h i s ±10 GHz s e a r c h . It was i n c o n c e i v a b l e tha t the p r e d i c t i o n was f a r t h e r o f f than our sea rch r e g i o n and i t was improbable tha t we c o u l d c o n t i n u a l l y miss the l i n e on s u c c e s s i v e s e a r c h e s . The double l o b e d , low J , R branch was r e - i n v e s t i g a t e d , but i t c o u l d not be a t t r i b u t e d to any l i k e l y i m p u r i t y ; in a d d i t i o n the i n t e n s i t y was so grea t sugges t i ng tha t were i t an impur i t y l i n e i t would have an u n r e a l i s t i c a b s o r p t i o n c o e f f i c i e n t . F i n a l l y i t seemed i n e v i t a b l e tha t the double lobed mystery l i n e and the 0—>1 were one and the same. I t was then 1 4 4 sugges t ed , and l a t e r con f i rmed by measurement and c a l c u l a t i o n , tha t the second lobe was due to the supposed weak AM=±1 t r a n s i t i o n . T h i s r e s u l t f o r t h i s m i l l i m e t e r wave l i n e came as a s u r p r i s e fo r expe r i ence in the cen t ime te r wave r eg i on d u r i n g the i n i t i a l phase of t h i s study was AM=±1 t r a n s i t i o n s were so weak as to be n o n - e x i s t e n t . T h i s i s because the waveguide a b s o r p t i o n c e l l s used at the lower f r e q u e n c i e s had d imens ions on the same order as the microwave r a d i a t i o n wavelength and hence fo r our expe r imen ta l c o n f i g u r a t i o n (see Chapter 2) one mode of p ropaga t i on dominated p roduc ing on l y AM=0 t r a n s i t i o n s . However, we s t i l l used a low f requency waveguide a b s o r p t i o n c e l l (X band c e l l ) when we went to m i l l i m e t e r l eng th waves so the p ropaga t i on of o ther modes, p a r t i c u l a r l y the p e r p e n d i c u l a r modes r e s p o n s i b l e fo r AM=±1 t r a n s i t i o n s , became more p r e v a l e n t . La te r the ass ignment was con f i rmed by the p r e d i c t i o n and subsequent measurement of the second 0—>1 t r a n s i t i o n at 175.8 GHz. Not a l l t r a n s i t i o n s c o u l d be a s s i g n e d by i n t e n s i t y or S ta rk e f f e c t . In these cases the f requency f i t d i s c u s s e d in the next s e c t i o n e s t a b l i s h e d the ass ignment . 1 45 Tab le 5. 1 Observed PD 2H spectrum in MHz T r a n s i t i o n Frequency(Unc) Obs-Calc 1 1 1 - 1 0 1 1 2 0 0 5 . 0 3 0 ( 0 . 0 3 0 ) 0 .030 1 1 o _ 0 o o 1 7 5 8 2 2 . 2 8 4 ( 0 . 0 2 0 ) 0 .027 1 1 1 _ U 0 0 1 5 8 7 5 4 . 1 1 4 ( 0 . 0 2 0 ) - 0 . 0 2 7 1 1 o - 1 0 1 2 9 0 7 3 . 2 1 0 ( 0 . 0 5 0 ) * 0 .093 2 2 1 - 2 i 1 3 6 0 0 3 . 7 2 0 ( 0 . 0 3 0 ) - 0 . 0 5 7 3 3 0 - 3 2 2 1 1 1 5 4 4 . 0 8 4 ( 0 . 0 5 0 ) - 0 . 0 5 9 3 2 2 - 3 , 2 1 9 4 1 5 . 1 9 0 ( 0 . 0 5 0 0 .018 4« i_-43 i 1 1 2 6 1 6 . 0 6 5 ( 0 . 0 3 0 ) 0 .022 4 a o " 4 3 1 1 1 4 0 0 0 . 4 8 0 ( 0 . 1 0 0 ) - 0 . 0 8 7 5 3 3 ~ 5 2 3 2 4 0 7 9 . 4 8 0 ( 0 . 0 5 0 ) ^ 0.031 6 3 4 ^ 2 4 9 0 4 9 . 3 3 0 ( 0 . 0 3 0 ) 0 .020 9 5 5 - 9 4 5 2 8 7 5 9 . 3 5 0 ( 0 . 0 5 0 ) * 0 .045 9 8 2 ~ 9 7 2 2 6 3 7 5 7 . 2 7 6 ( 0 . 0 3 0 ) - 0 . 0 0 0 1 1 6 6 - 1 1 5 6 2 9 8 3 3 . 8 3 0 ( 0 . 0 3 0 ) - 0 . 0 2 5 1 3 7 7 - 1 3 6 7 3 0 5 3 1 . 3 3 0 ( 0 . 0 5 0 ) * 0 .023 1 4 7 7 - 1 4 6 8 2 5 4 5 8 8 . 0 4 0 ( 0 . 0 3 0 ) 0 .000 1 4 7 8 - 1 4 6 8 1 1 8 6 1 . 0 5 0 ( 0 . 0 5 0 ) 0.001 t r e f ( r 116 ) ' * ' unass igned t r a n s i t i o n s of r e f ( r 116 ) a s s i g n e d as pa r t of t h i s work ' * * ' as above except t h i s t r a n s i t i o n was re-measured. — Numbers in parentheses are es t imated measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 46 Tab le 5.2 Observed PH 2D spectrum in MHz Trans i t i-on Frequency (Unc) Obs-Calc I n " l o i 40561 .870 (0 .050 ) k - 0 . 124 1 10 " 111 6 0 2 4 . 6 4 5 ( 0 . 0 0 5 ) k -0.005 l o i " Ooo 172524.933(0.015) 0.0 2,2 - 2 02 35142 .100 (0 .030 ) k 0.022 2 , , " 2,2 18070.910(0.035) -0.038 3,2 " 3,3 36093.670(0.030) 0.015 3,3 " 3 0 3 28158 .500 (0 .030 ) k 0.002 3 2 2 " 3,2 112977.568(0.050) 0.056 4,« - 4 0 « 20815 .334 (0 .005 ) k 0.000 5 , 5 " 5 0 5 1 4 2 4 6 . 6 9 0 ( 0 . 0 l 0 ) k -0.005 5 2 3 " 5 2 tt 19611.360(0.100) 0.068 6,6 " 6 0 6 9 1 2 1 . 3 7 6 ( 0 . 0 0 5 ) k -0.003 6 2 n ~ 6 2 5 36331.040(0.040) 0.015 V , 7 " 7 0 7 5 5 3 6 . 3 8 2 ( 0 . 0 0 3 ) k 0.001 9 2 8 " 9,8 30957.770(0.030) -0.053 1 0 , 7 " 1 0 3 7 256864.972(0.015) -0.001 1 0 3 8 -1 0 2 8 112617.504(0.030) 0.006 1 0 2 9 -1 0 1 9 20754.570 (0 .050 )* 0.221 1 1 2 , 0 "" 1 1 \ 1 0 13259.560(0.030) -0.050 1 4 3 > , 2 - 1 4 2 f , 2 34520.190(0.030) 0.023 1 5 3 f , 3 - 1 5 2 f , 3 22821 .900 (0 .050 ) * -0.057 16 3 , 1 6 2 f , 4 14504.990(0.040) -0.009 1 7 3 f ,5- 1 7 2 f , 5 8916.190(0.060) 0.044 I 9 3 f , 6 34207.180(0.050) -0.003 > * * and as in Tab le 5.1 ' k ' r e f ( r 117 ) - Numbers in pa ren theses are e s t ima ted measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 47 Tab le 5.3 Re-ana l y s i s of the measured PD 3 spectrum in MHz + T r a n s i t i o n F requency (Unc ) 1 Obs-Calc Normal R o t a t i o n a l T r a n s i t i o n s J + K - J , K<-K re f (r1 1 4) J= 0 K=0 138938.17(0.20) 0.06 1 0 277851.71(0.20) -0.04 1 1 277856.93(0.20) -0.01 2 0 416716.39(0.20) 0.02 2 1 416724.18(0.20) 0.03 2 2 416747.81(0.20) -0.04 Forb idden R o t a t i o n a l T r a n s i t i o n s J<-J, K=*l<-±2 r e f ( r 8 2 ) J=13 30805.64(0.60) 0.30 14 30700.67(0.50) 0.10 15 30589.40(0.30) 0.24 16 30471.50(0.20) 0.21 17 30347.32(0.20) 0.21 18 30217.40(0.30) 0.59 19 30081.32(0.40) 0.76 20 29939.20(0.60) 0.67 J<-J, K=0<-3 r e f ( r 2 5 ) J=9 93498.65(0.02) 0.02 10 93265.70(0.02) 0.05 11 93009.29(0.02) -0.08 1 48 Tab le 5.3 con t i nued T r a n s i t i o n Frequency(Unc) Obs-Calc 1 2 92729.42(0.02) -0.04 13 92425.42(0.02) 0.04 -J, K=±1<-±4 r e f ( r 2 5 ) 9 155957.87(0.03) -0.03 10 155573.65(0.03) -0.02 1 1 155153.32(0.03) -0.02 1 2 154697.68(0.03) 0.13 13 154207.02(0.03) 0.01 1 4 153682.42(0.03) -0.05 1 5 153124.72(0.03) -0.03 1 6 152534.73(0.03) 0.04 1 7 151913.22(0.03) -0.01 18 151261.35(0.03) -0.00 f Numbers in pa ren theses are es t ima ted measurement u n c e r t a i n t i e s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 149 4. ANALYSIS OF THE PHOSPHINE SPECTRA The p a r t i a l l y deu t e r a t ed phosphine s p e c t r a were f i t to a S r e d u c t i o n h a m i l t o n i a n {e1.15-e1.18) us ing an i t e r a t i v e method ( r 1 2 ) . Even though PH 2D and PD 2H are o b l a t e and . p r o l a t e asymmetric r o t o r s r e s p e c t i v e l y a 1 1 1 r o b l a t e r e p r e s e n t a t i o n was chosen in order to be ab l e to make c u r s o r y compar isons of the e m p i r i c a l parameters of the asymmetric tops wi th those of the o b l a t e symmetric t o p s . The expe r imen ta l u n c e r t a i n t i e s (Tab les 5.1 and 5.2) were determined as in the a r s i n e s tudy . The t o t a l l y deu t e r a t ed phosph ine , P D 3 , spectrum r epo r t ed in r e f e r e n c e s ( r 25 , r82 , r114 ) and u n s a t i s f a c t o r i l y ana l yzed in r e f e r ence (r25) i s here r e-ana l yzed in p r e c i s e l y the same manner as in the AsD 3 study d e s c r i b e d e a r l i e r 7 . Here s i n c e the data set i n c l u d e s |K j = 3 l e v e l s the parameter h 3 must be i n c l u d e d in the g l o b a l f i t . Both complete d i a g o n a l i z a t i o n and a f i r s t order p e r t u r b a t i o n approach (e1 .21) l e d to e s s e n t i a l l y the same va lue of h 3 . The e m p i r i c a l parameters l i s t e d in Tab le 5.4 were ob ta ined by the complete d i a g o n a l i z a t i o n of the s e c u l a r e q u a t i o n . The expe r imen ta l f r e q u e n c i e s were weighted in the f i t s by 1/a 2 where a i s the expe r imen ta l u n c e r t a i n t y . For the 30 GHz comb r e p o r t e d by Chu and Oka (r82) we took t h e i r quoted 7 See the l a t e r s e c t i o n e n t i t l e d " F a i l u r e s of the R e d u c t i o n s " f o r an e x p l a n a t i o n of why t h i s p r e v i ous a n a l y s i s i s not s a t i s f a c t o r y . 1 50 exper imenta l e r r o r s . However, f o r the f o r b i d d e n t r a n s i t i o n s r epo r t ed by Helms and Gordy (r25) they c l a i m an accuracy of 1 in 10 8 which we c o n s i d e r too o p t i m i s t i c and we s u b s t i t u t e d , somewhat a r b i t r a r i l y , the e r r o r s g i ven in Tab le 5 .3 . The a l l owed PD 3 t r a n s i t i o n s r epo r t ed in r e f . ( r 1 1 4 ) were each g iven an expe r imen ta l u n c e r t a i n t y of 200 KHz. For the asymmetric top s p e c t r a we were ab le to o b t a i n on l y J=0—>1 R b ranches . From equa t ion (e1.15 ), and the s i m i l a r d i s c u s s i o n in the p r e ced ing chapte r conce rn i ng the a r s i n e asymmetric t o p s , i t shou ld be apparent tha t c o e f f i c i e n t s of d i a g o n a l ( J 2 ) n ( n = 1 , 2 , 3 . . . ) terms cannot be separa ted w i th our data s e t . In o rder to determine the i n d i v i d u a l r o t a t i o n a l c o n s t a n t s , in each c a s e , (S r educ t i on ) was f i x e d at the va lue c a l c u l a t e d from the harmonic f o r c e f i e l d (see Chapter 6 ) . E m p i r i c a l l y we determine l i n e a r combina t ions of r o t a t i o n a l cons t an t s w i th Dj and H^ te rms. G iven the f o r c e f i e l d v a lues of Dj a l l ows us to es t imate the Dj terms in these l i n e a r combina t ions and then determine the r o t a t i o n a l cons t an t s assuming Hj i s s m a l l . The u n c e r t a i n t y i n t r o d u c e d i n t o the r o t a t i o n a l cons t an t s we ob t a i n in t h i s way shou ld be on the o rder of the s e x t i c H-, c o n t r i b u t i o n and/or the e r r o r in the p r e d i c t i o n of D T — in our case we expect t h i s u n c e r t a i n t y in the r o t a t i o n a l c o n s t a n t s to be w i t h i n t h e i r s t andard e r r o r s quoted from the l e a s t squares f i t s . The cho i c e and i n t e r p r e t a t i o n of the s e x t i c (and h ighe r ) f i t t i n g parameters i s i d e n t i c a l to that of the 151 a r s i n e s tudy . These h igher o rde r parameters are to be thought of on l y as f i t t i n g c o n s t a n t s that enab le b e t t e r d e t e r m i n a t i o n of the q u a r t i c c o n s t a n t s . The r e s u l t s of the f i t s a re g i ven in Tab le 5.4. The O b s - C a l c ' s , tha t i s the best f i t f requency minus the expe r imen ta l f r equency , are l i s t e d in the t a b l e s of expe r imen ta l f r e q u e n c i e s . ( T a b l e s 5.1 and 5.2) F i n a l l y , in t h i s study we were ab le to d e f i n i t e l y a s s i g n a l l the unass igned l i n e s r epo r t ed by S i r v e t z and Weston (r116) . Some of these t r a n s i t i o n s have been re-measured as i n d i c a t e d in T a b l e s 5.1 and 5 .2 , but o therw ise the measurements of S i r v e t z and Weston, w i th t h e i r quoted expe r imen ta l u n c e r t a i n t y of 0.050 MHz, were used in the f i n a l f i t s . The t r a n s i t i o n s tha t S i r v e t z and Weston c o u l d not a s s i g n were " h i g h " J t r a n s i t i o n s . Because of the l a r g e c e n t r i f u g a l d i s t o r t i o n in phosphine e x t r a p o l a t i o n s from the lower J t r a n s i t i o n s a v a i l a b l e to S i r v e t z and Weston were not good enough to enable them to a s s i g n these h ighe r J t r a n s i t i o n s . In t h i s study the i d e n t i f i c a t i o n of many other h igh J t r a n s i t i o n s has a l l owed us to d e f i n i t e l y a s s i g n these p r e v i o u s l y unass igned t r a n s i t i o n s . 1 52 Tab le 5.4 E m p i r i c a l s p e c t r o s c o p i c parameters of v a r i o u s phosphines X Y Z D. L D JK K d 2 H, H JK H KJ H K h, h 3 L J J JK MHz MHz MHz KHz KHz KHz KHz KHz Hz Hz Hz Hz Hz Hz Hz Hz PH 2D 129843.14(4) 89279.34(2) 83250.52(2) 2726.04''* -3475.8(23) 1873.54(56) -226.7(1 .7) 449.1(6) •346( 18) 1 16(20) 67(20) -71(4) PHD 2 93917.79(2) 81910.92(2) 64841 .80(3) 1612.75 t -794.3(64) -352(19) -233.4(3) -326.20(17) 221(31 ) -1290(150) -52(3) -4.2(8) PD 3 69471.09(11) 58974.42(11) 1020.4(68) -1312.28(46) 1026.9(22) 171(19) 140(23) 4.63(3) 0.013(3) fHarmonic Force F i e l d va lue — Numbers in pa ren theses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 53 G. CHAPTER 6: FORCE FIELDS, STRUCTURES AND DIPOLE MOMENTS In t h i s chapte r we s h a l l look at the harmonic f o r c e f i e l d d e s c r i p t i o n s of the group V h y d r i d e s , namely ammonia, phosphine and a r s i n e . From the f o r c e f i e l d a n a l y s i s we immediate ly get three very important p i e c e s of i n f o r m a t i o n . F i r s t we get e x c e l l e n t va lues fo r the q u a r t i c d i s t o r t i o n cons t an t s we c o u l d not determine in chap te r s 4 and 5 (D^ fo r PH 2D and PD 2H and D R f o r A s D 3 ) . C o n s t r a i n i n g to these va lues in the f i t s to the s p e c t r a l data a l l ows t h e i r c o n t r i b u t i o n to the o ther e m p i r i c a l parameters to be removed. Second, from the f o r c e f i e l d we can c a l c u l a t e parameters necessa ry f o r the e s t i m a t i o n of the d i s t o r t i o n d i p o l e moments; and t h i r d , e v a l u a t i o n of the harmonic pa r t of the a lphas a l l ows the d e t e r m i n a t i o n of ze ro p o i n t averaged s t r u c t u r e s from which e q u i l i b r i u m s t r u c t u r e s can be e s t i m a t e d . The p o t e n t i a l cons t an t s themselves are a l s o u s e f u l in comparing bonding parameters of the v a r i o u s h y d r i d e s . Comparisons of t h i s so r t w i l l be made. There have been tens of harmonic f o r c e f i e l d s t u d i e s done on the group V h y d r i d e s . As e a r l y as the 1930's f o r c e cons t an t s were c a l c u l a t e d from v i b r a t i o n a l f r e q u e n c i e s ob ta ined from low r e s o l u t i o n s p e c t r a . Woodward has c o n s i d e r e d the problem of c a l c u l a t i n g these h yd r i de f o r c e c o n s t a n t s from XH 3 and XD 3 v i b r a t i o n a l f r e q u e n c i e s a lone ( r 43 ) . Subsequent s t u d i e s in the 1940's and 1950's r e p o r t e d " b e t t e r " f o r c e cons t an t s based on h ighe r r e s o l u t i o n s t u d i e s . S t i l l these f o r c e cons t an t s had a l l been ob t a i ned from j u s t 1 54 the v i b r a t i o n a l f r e q u e n c i e s of XH 3 and XD 3 s p e c i e s . In 1960 M i l l s (r118) p u b l i s h e d a method of i n c l u d i n g C o r i o l i s c o u p l i n g cons t an t s and 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 a long wi th v i b r a t i o n a l f r e q u e n c i e s i n t o an i t e r a t i v e l e a s t squares f i t t i n g procedure (see a l s o ( r 8 l , r 1 l 9 ) f o r a d e s c r i p t i o n of the method) . I t i s t h i s method tha t we have used fo r the harmonic f o r ce f i e l d a n a l y s i s p resen ted h e r e . I n c l u d i n g C o r i o l i s c o u p l i n g c o n s t a n t s and q u a r t i c cons t an t s a l ong wi th v i b r a t i o n a l f r e q u e n c i e s in a c a l c u l a t i o n of f o r c e cons t an t s enab led Duncan and M i l l s in 1964 (r76) to p resen t a d e f i n i t i v e study of the harmonic f o r c e f i e l d s of the XH 3 and XH„ m o l e c u l e s . For the XH 3 mo lecu les they were ab l e to i n c l u d e on l y data from XH 3 and XD 3 s p e c i e s . The q u a r t i c cons t an t s were incomple te (no D R v a l ues ) and the C o r i o l i s c o u p l i n g c o n s t a n t s i n a c c u r a t e . They a l s o d i d not i n c l u d e the v i b r a t i o n a l f r e q u e n c i e s fo r the XT 3 s p e c i e s r e p o r t e d by De A l t i et a l ( r 98 ) . S t i l l t h i s study was a marked improvement over e a r l i e r s t u d i e s that used on l y v i b r a t i o n a l f r e q u e n c i e s in c a l c u l a t i o n s of f o r c e c o n s t a n t s . For completeness we shou ld a l s o mention here another e x t e n s i v e study by Shimanouchi et a l (r77) in 1966. They a l s o d i d not i n c l u d e the XT 3 data and fur thermore d i d not i n c l u d e any q u a r t i c c o n s t a n t s in t h e i r re f inement of the f o r c e c o n s t a n t s . They d i d however attempt to account f o r " a n h a r m o n i c i t y " u s i ng what one might c a l l " s p e c t r o s c o p i c " masses. T h i s i s d i s c u s s e d f u r t h e r be low. 1 55 There have a l s o been s e v e r a l anharmonic f o r c e f i e l d s t u d i e s of group V h y d r i d e s . In t h i s case c u b i c and h igher terms as we l l as q u a d r a t i c terms are i n c l u d e d in the p o t e n t i a l . Of s p e c i a l i n t e r e s t to us here are the anharmonic f o r c e f i e l d s t u d i e s of phosphine by K u c h i t s u in 1961 (r120) and the s t u d i e s of ammonia, f i r s t by Mor ino et a l in 1967 ( r l 03 ) and l a t e r by Hoy et a l in 1972 (r121 ) . With such an e x t e n s i v e l i t e r a t u r e on the f o r c e f i e l d s of N H 3 , PH 3 and AsH 3 i t i s perhaps necessa ry to g i ve reasons why a f u r t h e r s tudy i s warranted h e r e . F i r s t , in the p e r i o d s i n c e the s tudy of Duncan and M i l l s the accu racy of the. parameters a v a i l a b l e to them has been g r e a t l y improved. More impo r t an t l y v a l ues of D R fo r the symmetric tops NH 3 and PD 3 ( the l a t t e r va lue p resen ted in. t h i s t h e s i s ) have j u s t r e c e n t l y become a v a i l a b l e . A l s o the q u a r t i c c o n s t a n t s of the p a r t i a l l y d e u t e r a t e d ammonias, phosph ines and a r s i n e s (the l a t t e r two p r e sen t ed as pa r t of t h i s t h e s i s ) i n c r e a s e by more than f i f t y per cent the data a v a i l a b l e from which to c a l c u l a t e the f o r c e c o n s t a n t s . The f i r s t reason then fo r a new f o r c e f i e l d s tudy i s thus tha t we now have a much more a c cu r a t e and g r e a t l y i n c r eased da ta s e t . A second reason fo r a new f o r c e f i e l d s tudy i s found in the f a c t tha t a lmost every study to d a t e , i n c l u d i n g the anharmonic s t u d i e s , has " a t t empted " to account f o r anharmonic e f f e c t s in v i b r a t i o n f r e q u e n c i e s u s i n g the e m p i r i c a l r e l a t i o n s of Dennison ( r 79 ) . In the next s e c t i o n we w i l l show tha t t h i s i s an unreasonab le t h i n g to do and 156 that f o r c e cons t an t s ob ta ined us ing t h i s method are u n s a t i s f a c t o r y . 1. AN EXAMINATION OF STANDARD ANHARMONICITY CORRECTION  TECHNIQUES APPLIED TO VIBRATIONAL FREQUENCIES A p r i n c i p a l concern in the c a l c u l a t i o n of harmonic f o r c e f i e l d s i s whether the e m p i r i c a l parameters used in the c a l c u l a t i o n i n c l u d e s i g n i f i c a n t c o n t r i b u t i o n s from e f f e c t s or i n t e r a c t i o n s o u t s i d e the harmonic o s c i l l a t o r a p p r o x i m a t i o n . The l a r g e s t c o n t r i b u t i o n s of t h i s so r t are g e n e r a l l y the " a c c i d e n t a l " f requency s h i f t s of v i b r a t i o n a l f r e q u e n c i e s due to Fermi resonance ( d i s cus sed in the next s e c t i o n ) and the c o n t r i b u t i o n s to the parameters from the anharmonic terms of the p o t e n t i a l - terms we of course ignore in the harmonic a p p r o x i m a t i o n . I d e a l l y when c a l c u l a t i n g harmonic f o r c e cons t an t s one would p r e f e r to " c o r r e c t " the e m p i r i c a l data fo r these ext ra-harmonic e f f e c t s so tha t the f i n a l f o r c e cons t an t s would be c l o s e r to a " t r u e " mo lecu l a r f o r c e f i e l d . In t h i s s e c t i o n we s h a l l look at how c o r r e c t i o n s due to anharmonic e f f e c t s have been made in the p a s t . The r e s u l t of t h i s shou ld be that the methods promoted by p r e v i ous au thors are found to be u n s a t i s f a c t o r y . The v i b r a t i o n a l f r e q u e n c i e s are the expe r imen ta l parameters g e n e r a l l y f e l t to have the l a r g e s t anharmonic c o n t r i b u t i o n and so i t i s on l y to these f r equenc i e s that anharmonic c o r r e c t i o n s have been a p p l i e d . The anharmonic 1 57 c o n t r i b u t i o n to the q u a r t i c and C o r i o l i s c o u p l i n g cons t an t s i s g e n e r a l l y assumed to be l e s s than the anharmonic c o n t r i b u t i o n to the v i b r a t i o n a l f r e q u e n c i e s and so these c o n s t a n t s are g e n e r a l l y used in f o r c e f i e l d c a l c u l a t i o n s w i thout c o r r e c t i o n . T h i s tu rns out to be an important p o i n t . As we s h a l l see l a t e r there e x i s t v a r i o u s " t r a c e " r e l a t i o n s , sum r u l e s or i n v a r i a n c e r e l a t i o n s amongst the "ha rmon ic " pa ramete rs . If we assume n e g l i g i b l e anharmonic c o n t r i b u t i o n to some parameters we can then use these parameters in i n v a r i a n c e r e l a t i o n s to es t imate the anharmonic c o n t r i b u t i o n s in other pa ramete r s . An o b j e c t i o n to these anharmonic c o r r e c t i o n s w i l l then be based on the c r i t e r i o n used to determine the r e l a t i v e magnitudes of the anharmonic c o n t r i b u t i o n s in v a r i o u s expe r imen ta l pa ramete rs . C o n s i d e r a t i o n of anha rmon i c i t y u s u a l l y s t a r t s wi th the example of a d i a tomic molecu le where the energy a s s o c i a t e d w i th a v i b r a t i o n of quantum number v i s ( r l22 pg 274) E(v) = C J (V+1/2) - x e w ( v + l / 2 ) 2 + [6 .1 ] where u> i s the "harmon ic " or " n o r m a l " f requency and x^co i s the c o r r e c t i o n due to a n h a r m o n i c i t y . The " fundamenta l " or e x p e r i m e n t a l l y measured f requency i s E (1 )-E (0 ) , or v = u ( l -2x ) [6 .2] Now r e c a l l f o r a py ramida l XY 3 mo lecu le there are four fundamental v i b r a t i o n s (2A, + 2E) which can be c h a r a c t e r i z e d by symmetric and an t i symmet r i c bond s t r e t c h e s and angle 1 58 bends. Furthermore the re are s i x d i s t i n c t f o r c e c o n s t a n t s , th ree fo r the A, s p e c i e s and th ree f o r the E s p e c i e s v i b r a t i o n s . For argument 's sake we s h a l l now c o n s i d e r on ly the E v i b r a t i o n s . For the E spec i e s the 3 f o r c e cons t an t s F 3 3 , F f l 4 and F 3 q account fo r the two v i b r a t i o n s co3 and co„. The cons t an t s F 3 3 and F M co r respond to bond s t r e t c h i n g and ang le bending r e s p e c t i v e l y . The F 3 „ cons t an t mixes the bond s t r e t c h i n g and ang le bending mot ions . I f the i n t e r a c t i o n f o r c e cons tan t F 3 f t i s sma l l then co3 i s e s s e n t i a l l y bond s t r e t c h and CJ , i s e s s e n t i a l l y angle bend in c h a r a c t e r . In t h i s case because of the s i m i l a r i t y of a l l the bonds and ang les in py ramida l XY 3 molecu les i t has been assumed in the past tha t anha rmon i c i t y in the v i b r a t i o n a l f r e q u e n c i e s may be accounted fo r u s i n g equa t ion ( e 6 . 2 ) : that i s a l l bond s t r e t c h e s and ang le bends are independent and each behave in a " d i a t o m i c - l i k e " way. I t might prove i n s t r u c t i v e f o r l a t e r d i s c u s s i o n to note in Tab le 6.5 that the " o f f - d i a g o n a l " i n t e r a c t i o n fo r ce cons t an t s u n f o r t u n a t e l y do not g e n e r a l l y appear to be sma l l w i th r e spec t to the d i a g o n a l c o n s t a n t s . T h i s shou ld suggest tha t the bond s t r e t c h e s and ang le bends cannot be t r e a t e d as independent . If we take fo r the p o t e n t i a l f u n c t i o n , in which our " d i a t o m i c " o s c i l l a t e s , a Morse P o t e n t i a l then x i s found to e be p r o p o r t i o n a l to co ( r l 22 pg 274,r123 pg 101) so tha t we can w r i t e equa t ion (e6.2) f o r the E v i b r a t i o n s 159 V 3 = C J 3 ( 1 +X 3 ( J 3 ) V „ = CJ<, ( 1 +X a C J a ) [6 .3 ] where the x^ rep resen t new cons t an t s d i f f e r e n t from x g . The parameters x^ are i so tope independent so tha t i s o t o p i c data shou ld be u s e f u l in s o l v i n g equa t ion ( e 6 . 3 ) . G iven the fundamental E f r e q u e n c i e s fo r XH 3 and XD 3 Dennison proposed the f o l l o w i n g method of s o l v i n g equa t ion (e6.3) ( r 79 ) . ( A c t u a l l y f o r t h i s d i s c u s s i o n we have adopted the " s l i g h t l y more l o g i c a l " approach of Denn i son , see f oo tno te on h i s page 197 r79 . ) For two i s o t o p i c s p e c i e s equa t ion (e6.3) produces 4 equa t ions in s i x unknowns. C l e a r l y two f u r t h e r r e l a t i o n s are needed. F i r s t , the w 's f o r the two i s o t o p i c s p e c i e s are not independent . The R e d l i c h - T e l l e r product r u l e fo r py ramida l XY 3 E type v i b r a t i o n s i s (r37 pg232,r78) where M i s the mass of the X atom. As a second r e l a t i o n Dennison produced a r e l a t i o n between the harmonic f r e q u e n c i e s and the C o r i o l i s (Zeta) c o u p l i n g cons t an t s ( r e f . (r79) page 197, see a l s o ( r l 2 4 ) ) . In t h i s way i f the C o r i o l i s c o u p l i n g cons t an t s c o u l d be determined e x p e r i m e n t a l l y then the anharmonic c o r r e c t i o n s of equa t ion (e6.3) c o u l d be es t ima ted and the proper harmonic f r e q u e n c i e s c o u l d be used in the f o r ce f i e l d d e t e r m i n a t i o n . For the p a r a l l e l symmetric A, v i b r a t i o n s where by symmetry there are no C o r i o l i s c o u p l i n g cons t an t s Dennison suggested tha t s i n c e v2-v 1 and both v i b r a t i o n s are r (M+3mu) (M+3mu) V H ] 1 / 2 TT7^ J [6 .4 ] 160 " e s s e n t i a l l y " bond s t r e t c h e s that the anha rmon i c i t y c o r r e c t i o n s in shou ld be p r o p o r t i o n a l l y equa l to those of v2. The va lue of co2 would then be c a l c u l a t e d from the c o r r e s p o n d i n g product r u l e fo r the A, v i b r a t i o n s . L a t e r McConaghie and N i e l s e n (r97) were ab le to produce a second r e l a t i o n -for the A, v i b r a t i o n s us i ng the t h e o r e t i c a l r e s u l t s of Sha f f e r (r125) w i th the f u r t h e r assumpt ion tha t $ 1 3 ( a c t u a l l y $ 1 2 in the n o t a t i o n of the t imes) was sma l l enough to be set to zero (see Hayne and N i e l s e n ( r l 26 ) fo r m i s p r i n t s ) 8 . The above method has been by f a r the most popu la r techn ique by which anharmonic c o r r e c t i o n s have been made to the v i b r a t i o n a l f r e q u e n c i e s . T h i s method as used up to now however, i s u n s a t i s f a c t o r y . F i r s t the second r e l a t i o n of Dennison (r79 h i s p g . 197) r e l a t i n g C o r i o l i s c o u p l i n g cons t an t s to masses and harmonic f r e q u e n c i e s fo r the E v i b r a t i o n s i s not c o n s i s t e n t wi th o ther r e l a t i o n s g i ven by Denn i son . In p a r t i c u l a r i t seems tha t the + s i gns on the r i g h t hand s i de of the l a s t equa t ion on D e n n i s o n ' s page 197 shou ld be - s i g n s . Fur thermore the r e l a t i o n s fo r the C o r i o l i s c o u p l i n g c o n s t a n t s on h i s page 199 are not 8 The r e l a t i o n of McConaghie and N i e l s e n needs to be c o r r e c t e d as t h e i r term M ' / 2 shou ld read ju'm/m'. A l s o the C tha t they d e f i n e can b e t t e r be c a l c u l a t e d from C=(2-b 2 ) / (3b 2 -2 ) where -b 2 =$ 3 +$ a =I / U l )-\ (the ze t a sum r u l e fo r ammonia-l ike m o l e c u l e s . ) 161 o b v i o u s l y e q u i v a l e n t to supposed ly i d e n t i c a l equa t ions produced by Sha f f e r ( r125 ) , h i s equa t i on 16. A l s o i f we use S h a f f e r ' s fo rma l i sm to d e r i v e D e n n i s o n ' s r e l a t i o n i t appears we need to i n c l u d e C o r i o l i s da ta from both XH 3 and XD 3 in c o n t r a s t to Denn i son ' s r e s u l t which on l y r e q u i r e s data from one s p e c i e s . T h i s l a s t p o i n t i s c o n s i s t e n t w i th the " t r a c e " i n v a r i a n c e r e l a t i o n of Meal and Po lo (r127 equ . III-6) m I. X . ( l - $ . ) = cons tan t [6 .5 ] 3 1 1 1 where m i s the mass of the o f f a x i s atoms and where X. = 9 1 47r 2c 2w?. U n l i k e the " s e cond " equa t i ons of Dennison and McConaghie and N i e l s e n the t r a c e r e l a t i o n of Meal and Po lo i s easy to v e r i f y and i t i s sugges ted that i f one i s i n t e r e s t e d in p r e s e r v i n g the above method of anha rmon i c i t y c o r r e c t i o n tha t equa t ion (e6.5) r e p l a c e D e n n i s o n ' s " s e c o n d " equa t ion f o r the c o r r e c t i o n of the E v i b r a t i o n s . A p r e l i m i n a r y study was made of the e f f e c t s of the above c o r r e c t i o n s by t a k i n g v a r i o u s comb ina t ions of c o r r e c t i o n s and then i n c l u d i n g the "ha rmon ized " f r e q u e n c i e s i n the harmonic f o r c e f i e l d f i t s d e s c r i b e d l a t e r . An important f e a tu r e of these f i t s was tha t they i n c l u d e d a l a r g e amount of " o t h e r " k inds of d a t a , namely C o r i o l i s and q u a r t i c d i s t o r t i o n c o n s t a n t s . In a l l cases the f i t s were q u i t e s i g n i f i c a n t l y worse than when we used u n c o r r e c t e d anharmonic f r e q u e n c i e s . C e r t a i n l y the above r e s u l t suggests tha t t h i s method of anha rmon i c i t y c o r r e c t i o n i s s u s p e c t . Fu r the rmore , one might 1 62 ask why t h i s problem has not been n o t i c e d be fo re ? The reason t h i s problem i s in t h i s case n o t i c e a b l e a r i s e s from our e x t e n s i v e and a c c u r a t e l y measured data s e t . In the work p r i o r t o , say , 1960 on l y v i b r a t i o n a l f r e q u e n c i e s were used in c a l c u l a t i n g f o r c e f i e l d s . In these cases the 8 v i b r a t i o n a l f r e q u e n c i e s of XH 3 and XD 3 a l l owed the unique d e t e r m i n a t i o n of the 6 f o r c e cons t an t s ( r e c a l l the re are 2 product r u l e s , one fo r each of the A and E v i b r a t i o n s and so the re are on l y 6 independent p i e c e s of d a t a , see Woodward ( r 4 3 ) ) . For c a l c u l a t i o n s of t h i s so r t w i th 6 equa t i ons and 6 unknowns the v a l i d i t y of any " h a r m o n i z i n g " scheme co.uld not be a s c e r t a i n e d . C e r t a i n l y l a t e r , in the mid 6 0 ' s , in the e x t e n s i v e study of Duncan and M i l l s (r76) in which they - i n c l u d e d some C o r i o l i s and q u a r t i c d a t a , we shou ld f i n d some i n d i c a t i o n tha t t h i s " h a r m o n i z i n g " procedure was u n s a t i s f a c t o r y , even more so perhaps because they used the " u n c o r r e c t e d " second equa t ion of Denn i son . I t i s then s u r p r i s i n g to note t h e i r r e s u l t s , tha t f i t s i n c l u d i n g anharmonic c o r r e c t i o n s to the v i b r a t i o n a l f r e q u e n c i e s were always b e t t e r than f i t s where no such c o r r e c t i o n s were made; these r e s u l t s d i r e c t l y c o n t r a d i c t what we see h e r e . The reason f o r t h i s c o n t r a d i c t i o n i s p robab ly to be found in the very sma l l and p o o r l y determined c o r r o b o r a t i n g data set of C o r i o l i s and q u a r t i c cons t an t s a v a i l a b l e to Duncan and M i l l s . The anharmonic c o r r e c t i o n s to the v i b r a t i o n a l f r e q u e n c i e s are s m a l l , on the o rder of a few p e r c e n t , but one very important aspec t of the harmon iz ing method i s tha t 1 63 the f r e q u e n c i e s are " f i d d l e d " w i th u n t i l they obey the product r u l e . Now the product r u l e i s an i n v a r i a n c e r e l a t i o n i m p l i c i t in the harmonic f o r c e f i e l d c a l c u l a t i o n s so what the ha rmon iz ing does i s to produce a f requency data set e x a c t l y c o n s i s t e n t w i th a p a r t i c u l a r harmonic f o r c e f i e l d . A l s o s i n c e the c o r r e c t i o n s to the f r e q u e n c i e s are sma l l t h i s p a r t i c u l a r f o r c e f i e l d shou ld produce reasonab le C o r i o l i s c o u p l i n g c o n s t a n t s and q u a r t i c c o n s t a n t s , even though the c o r r e c t i o n s may be i n c o r r e c t . F i n a l l y i f the expe r imen ta l C o r i o l i s and q u a r t i c data are p o o r l y determined and weighted a c c o r d i n g l y then i t i s not unreasonab le tha t the harmonized f r e q u e n c i e s produce a b e t t e r o v e r - a l l f i t - c e r t a i n l y the v i b r a t i o n a l f r e q u e n c i e s w i l l f i t b e t t e r ! T h i s d i s c u s s i o n shou ld i n t r o d u c e the importance of the " w e i g h t i n g " of expe r imen ta l data - a problem c o n s i d e r e d in the next sec t i o n . I f the harmon iz ing method does not work, why does i t not work? One can c e r t a i n l y argue tha t the i n i t i a l d i a tomic app rox ima t ion i s not a p p l i c a b l e and tha t argument w i l l be made but f i r s t there i s a fundamental f law in the method we are c o n s i d e r i n g , even i f one uses the " c o r r e c t e d " method w i th the t r a c e r e l a t i o n of Meal and P o l o . In order to es t imate the co's from the v ' s we use e x p e r i m e n t a l l y de termined and anharmonic $ c o n s t a n t s . T h i s method w i l l on l y beg in to produce reasonab le to's i f the anharmon ic i t y i nhe ren t in a C o r i o l i s £ cons tan t i s l e s s than in a fundamental f requency v. If the anha rmon i c i t y i n a $ were 1 64 g rea te r than in a v then t h i s method would be b e t t e r used to g i ve harmonized $ c o n s t a n t s . I t i s d i f f i c u l t to es t imate the r e l a t i v e anha rmon i c i t y in the $ or v parameters but e x p e r i m e n t a l l y i t i s found tha t f r e q u e n c i e s are u s u a l l y a few o rde r s of magnitude b e t t e r determined than C o r i o l i s cons t an t s and so i t shou ld seem unreasonab le to attempt to c o r r e c t f r e q u e n c i e s u s i ng C o r i o l i s c o n s t a n t s . T h i s completes the argument a g a i n s t the s t anda rd anharmonic c o r r e c t i o n method due to Denn i son . We shou ld now suggest tha t a l l p r e v i ous f o r c e f i e l d s e va l ua t ed from data harmonized us i ng the method of Dennison ( e s s e n t i a l l y every p r e v i ous study) shou ld be regarded as on ly approx ima te . If one i s s t i l l i n t e r e s t e d in making anharmonic c o r r e c t i o n s are there any a l t e r n a t e s o l u t i o n s ? One c o u l d use the method A rne t t and Crawford J r . (r128) proposed fo r e thy l ene in which they assumed a l l the anha rmon i c i t y was in the bond s t r e t c h f requency and v a r i e d t h i s f requency w i t h i n the d i a tomic approx imat ion so tha t the product r u l e was s a t i s f i e d . T h i s method they admit i s a r b i t r a r y and , in l i g h t of the d i s c u s s i o n on D e n n i s o n ' s method a l s o , u n s a t i s f a c t o r y . There i s a s o l u t i o n that does manage to a v o i d many of the o b j e c t i o n s of the Dennison method. T h i s i s s imply to i n c l u d e the XT 3 v i b r a t i o n a l f r equency d a t a . In t h i s case the " s e cond " equa t ion becomes another product r u l e . A f u r t h e r advantage of us ing the XT 3 da ta i s tha t i t a l l ows p a r a l l e l c a l c u l a t i o n s to be made fo r both the E and A, v i b r a t i o n s i n d e p e n d e n t l y . A l s o a c o n s i s t e n c y of t reatment i s ma in ta ined 1 65 in that we on ly use the assumpt ions of equa t i ons l i k e equa t ion (e6.3) and product r u l e s in o rder to harmonize our f r e q u e n c i e s . A p r e l i m i n a r y study was made on the anharmonic c o r r e c t i o n s to the fundamental f r e q u e n c i e s u s i n g the above i d e a s . The c a l c u l a t i o n of the co's i n v o l v e d s o l u t i o n of a q u a d r a t i c equa t ion which , in our examples, produced no r e a l r o o t s . There are s e v e r a l reasons r a t i o n a l i z i n g t h i s r e s u l t . F i r s t the XT 3 d a t a , see Tab l e s 6 .2 , 6.3 and 6 .4 , are not known to anywhere near the same accuracy as the XH 3 and XD 3 d a t a , and c e r t a i n l y " l a r g e " e r r o r s here would mask any anharmonic c o n t r i b u t i o n . A l s o sma l l p e r t u r b a t i o n s due to Fermi resonance , d i s c u s s e d in the next s e c t i o n , c o u l d s h i f t f r e q u e n c i e s and mask anharmonic e f f e c t s . F i n a l l y , always suspect i s our i n i t i a l " d i a t o m i c " a p p r o x i m a t i o n ; g i ven the magnitude of the o f f d i a g o n a l f o r c e cons t an t s (see e a r l i e r d i s c u s s i o n ) the d i a tomic approx imat ion may not be a p p l i c a b l e to ammonia-l ike m o l e c u l e s . We shou ld now i n v e s t i g a t e t h i s d i a tomic a p p r o x i m a t i o n . For t h i s d i s c u s s i o n we s h a l l r e-wr i t e equa t ion (e6.3) in the form vi = (1/2*) r K • -, 1 /2 1 ' [6 .6] where K. i s the f o r c e cons tan t and u' = u- (1 - 2 x .co. ) l * i i i [6 .7 ] and w i = (1/2TT) rK.ni/2 1 1 66 [ 6 . 8 ] as i s the d i a tomic reduced mass. The assumpt ion tha t the anharmonic c o r r e c t i o n s are " s m a l l " has a l s o been made. From equa t ions (e6 .6-e6.8 ) i t might seem reasonab le to c o n s i d e r i so tope 1 the r a t i o s • „ . ^ — ~ . If we make the assumpt ion tha t the i so tope 2 ^ f o r c e c o n s t a n t s are i s o t o p i c a l l y i n v a r i a n t then the above f requency r a t i o s depend on l y on r a t i o s of M ' from which the can be c a l c u l a t e d . A problem wi th c a l c u l a t i o n s of t h i s so r t i s tha t we i m p l i c i t l y i n t r oduce an approximate product r u l e . T h i s approximate product r u l e tu rns out to d e v i a t e from the exact product r u l e on the same order as the exper imenta l f r e q u e n c i e s . In o ther words the d i a tom i c  approx imat ion beg ins to f a i l at the l e v e l of the anharmonic  c o r r e c t i o n s . As an example we c o n s i d e r the c, and v2 v i b r a t i o n s f o r XH 3 and X D 3 . For these A, v i b r a t i o n s the exact p roduct r u l e of R e d l i c h and T e l l e r (r37 pg 232, r78) i s ( u 3 u , ) H m D r (M+3mu) -,1/2 (co 3a>„ ) D m H H' (M+3mD) [6 .9 ] From equa t ion (8) the " d i a t o m i c " approx imat ion product r u l e i s ( W 3 C J „ ) h m D (M+mH) ( u 3 « « ) D = ^ (M+mD) [ 6 ' 1 0 ] We can now c o n s t r u c t the f o l l o w i n g t a b l e . 167 NH 3 /ND 3 PH 3 /PD 3 ASH 3 /ASD 3 Exact Product Rule 1 .844 1 .917 1 .962 Expe r imen ta l 1 .851 1 .880 1 .908 D ia tomic Product Rule 1 .875 1 .939 1 .974 From the above numbers i t shou ld be apparent that the d i a tomic approx imat ion o f f e r s no r e a l improvement over s imply t a k i n g the fundamental f r equenc i e s as equa l to the harmonic f r e q u e n c i e s . An argument c o u l d perhaps be made fo r the v a l i d i t y of the d i a tomic approx imat ion in the case of l a r g e reduced mass, f o r i n s t ance a r s i n e seems more " d i a t o m i c " than ammonia, but in genera l we s h a l l conc lude tha t the d i a tom i c approx imat ion i s not a p p l i c a b l e to our problem of anharmonic c o r r e c t i o n s to the fundamental v i b r a t i o n a l f r e q u e n c i e s . The problem wi th ammonia w i l l l a t e r be seen as p a r t i a l l y due to i n v e r s i o n , where the n i t r o g e n atom passes through the p lane of the hydrogens , making the v2 v i b r a t i o n much more anharmonic than would o r d i n a r i l y be the c a s e . The above r a t i o method can r e s u l t in " s p e c t r o s c o p i c a l l y " d e f i n e d reduced masses (e6.7) which are used to d e f i n e " s p e c t r o s c o p i c " masses. That i s , one d e f i n e s masses so tha t the " e x a c t " p roduct r u l e f requency r a t i o agrees w i th the e x p e r i m e n t a l l y determined r a t i o . For i n s t ance m Q equa ls 2.126 i n s t e a d of 2.014 a .m .u . (see Sve rd lov et a l (r129) p g . 4 3 f f . ) . The hope i s that we can c a r r y over these "new" masses to d i f f e r e n t p rob lems . T h i s 1 68 procedure i s e q u i v a l e n t to the d i a tom i c method d i s c u s s e d above . Ins tead of d e f i n i n g s p e c t r o s c o p i c masses, g i ven equa t i on (e6.6) one c o u l d use " no rma l " masses and d e f i n e i s o t o p i c f o r c e c o n s t a n t s . In t h i s case an equa t ion e6.7 would be in terms of f o r c e c o n s t a n t s . T h i s i s the method of Shimanouchi et a l ( r 77 ) . In t h e i r case they d e f i n e d r a t i o s of f o r c e cons t an t s fo r the v a r i o u s i so topes such tha t the exact p roduct r u l e was s a t i s f i e d . T h i s i s the same as the method of s p e c t r o s c o p i c masses d i s c u s s e d above and t h e r e f o r e shou ld s u f f e r from the same shor tcomings ( i . e . the problems of the. d i a tomic a p p r o x i m a t i o n ) . Be fore c o n c l u d i n g t h i s s e c t i o n on anharmon ic i t y c o r r e c t i o n s we shou ld c o n s i d e r the work of C u r t i s ( r130, see a l s o Sve rd lov (r129) p g . 4 5 f f . ) . C u r t i s has taken a f i r s t s t ep towards s o l v i n g some of the problems i n t r o d u c e d by the above methods. To t h i s end he has i n t r oduced two methods: one c a l l e d ' c o n v e n t i o n a l a n h a r m o n i c ' , the other ' amp l i t ude a n h a r m o n i c ' . H i s ' c o n v e n t i o n a l anharmonic ' method assumes a d i a t o m i c - l i k e c o r r e c t i o n wi th both bending and s t r e t c h i n g c o r r e c t i o n s be ing made on each v i b r a t i o n f requency to accommodate the " m i x i n g " of the modes by the o f f d i a g o n a l f o r c e c o n s t a n t s . H i s 'anharmonic amp l i t ude ' method uses phenomeno log ica l h ighe r order anharmonic f requency c o r r e c t i o n s expressed in terms of root mean square amp l i t udes (see a l s o Nakata et a l (r53) fo r an i n t e r e s t i n g s i m i l a r a p p l i c a t i o n to r s t r u c t u r e s ) . T h i s approach of C u r t i s i s more s a t i s f y i n g than the methods we have d e s c r i b e d 169 p r e v i o u s l y as anharmonic c o n t r i b u t i o n s to the f r e q u e n c i e s from v a r i o u s h ighe r order terms in the p o t e n t i a l can be s t u d i e d s y s t e m a t i c a l l y . These c o n t r i b u t i o n s can be pa r ame te r i z ed and e a s i l y ob t a i ned e m p i r i c a l l y from a l e a s t squares i t e r a t i v e f i t . At some l e v e l of s o p h i s t i c a t i o n of t h i s method however, i t can be argued tha t one would do b e t t e r to s imply do a " comp le te " anharmonic f o r c e f i e l d s t udy . Even though the methods of C u r t i s r ep resen t a s tep in the r i g h t d i r e c t i o n t h i s s t ep does not appear l a r g e enough to warrant adop t ing h i s methods in the p resen t s tudy . It would a l s o appear tha t in order to account c o r r e c t l y fo r v a r i o u s anharmonic e f f e c t s would n e c e s s a r i l y demand too many new f i t t i n g parameters and the problem would then a r i s e whether we were de te rm in ing something " r e a l " or j u s t f i t t i n g the e r r o r . F i n a l l y there i s one l a r g e i n c o n s i s t e n c y w i t h i n the C u r t i s method, as he found tha t h i s anha rmon i c i t y c o r r e c t i o n s were d i f f e r e n t depending on the harmonic p o t e n t i a l f u n c t i o n used . T h i s suggests that f o r the cases he s t u d i e d , h i s method pa ramete r i zes the " e r r o r " more than the anharmonic c o r r e c t i o n s . Where then does t h i s d i s c u s s i o n leave us? It would seem tha t none of the p r e v i ous anha rmon i c i t y c o r r e c t i o n schemes, even when " c o r r e c t e d " , are a p p l i c a b l e to the ammonia-l ike mo l e cu l e s of t h i s s tudy . Fur thermore in o rder to do the anharmonic c o n t r i b u t i o n s c o r r e c t l y r e a l l y r e q u i r e s an anharmonic f o r c e f i e l d study to be done; which i s not tha t much more comp l i c a t ed than us ing an e l a b o r a t e approx imat ion 1 70 scheme, in the s p i r i t of C u r t i s ' method, that would s t i l l need to be deve l oped . I t would seem then tha t our answer i s , i f one wants to i n c l u d e anharmonic c o r r e c t i o n s one had b e t t e r do an anharmonic f o r ce f i e l d s tudy . T h i s r a the r obv ious s o l u t i o n addresses perhaps the most damning concern r e g a r d i n g the making of anharmonic c o r r e c t i o n s to v i b r a t i o n a l f r e q u e n c i e s , that b e i n g , the o ther r e l e v a n t d a t a , C o r i o l i s c o u p l i n g cons t an t s and q u a r t i c d i s t o r t i o n c o n s t a n t s fo r i n s t a n c e , shou ld a l l have anharmonic c o n t r i b u t i o n s t o o . I t seems r a the r a r b i t r a r y to c o r r e c t on ly the v i b r a t i o n a l f r e q u e n c i e s fo r anharmon ic i t y and l eave the o ther data u n c o r r e c t e d . Even i f arguments c o u l d be made that the anharmonic c o n t r i b u t i o n to the v i b r a t i o n a l f r e q u e n c i e s was l a r g e r than in o ther parameters i t would seem unreasonab le to make t h i s a gene ra l r u l e . It shou ld now be apparent tha t the e f f e c t s of anha rmon i c i t y are beyond the scope of the harmonic f o r c e f i e l d study p resen ted in t h i s t h e s i s . No anha rmon i c i t y c o r r e c t i o n s have been made and the f i n a l "ha rmon ic " f o r c e c o n s t a n t s can be expec ted to c o n t a i n smal l c o n t r i b u t i o n s due to a n h a r m o n i c i t y . F i n a l l y , a f t e r t h i s r a the r l engthy d i s c u s s i o n on what we cannot say about anharmonic f requency c o r r e c t i o n s , i s t he re any th ing we can say? Perhaps . Enough C o r i o l i s c o u p l i n g da ta and q u a r t i c d i s t o r t i o n cons tan t data e x i s t f o r reasonab le harmonic f o r c e f i e l d s t u d i e s (see s e c t i o n 6 . 3 ) t c be done without i n c l u d i n g the fundamental v i b r a t i o n 171 f r e q u e n c i e s in the f i t s . T h i s p rocedure a l l ows us to es t ima te the v i b r a t i o n a l f r e q u e n c i e s from C o r i o l i s and q u a r t i c data a l o n e . C e r t a i n l y t h i s method c o u l d not be used as a method of anharmonic c o r r e c t i o n but i t might prove u s e f u l in i n d i c a t i n g which f r e q u e n c i e s the C o r i o l i s and q u a r t i c data would expect to be the most anharmonic . In t h i s c a l c u l a t i o n i t i s hoped tha t the non-harmonic e f f e c t s in the C o r i o l i s and q u a r t i c data are sma l l and tha t these e f f e c t s j u d i c i o u s l y c a n c e l in the c a l c u l a t i o n s of the v i b r a t i o n a l f r e q u e n c i e s . If t h i s were the case then these c a l c u l a t e d f r e q u e n c i e s might b e t t e r r ep resen t harmonic f r e q u e n c i e s . T h e . r e s u l t of t h i s c a l c u l a t i o n f o r the bending modes suggests tha t the anha rmon i c i t y in the bending f r e q u e n c i e s i s s m a l l . The bending f r e q u e n c i e s were a l l p r e d i c t e d from the C o r i o l i s and q u a r t i c data a lone to be w i t h i n ±30 c m " 1 of the a c t u a l measurements. The e r r o r in the symmetric s t r e t c h p r e d i c t i o n of v-i was p r o p o r t i o n a l l y the same as fo r the bending v i b r a t i o n s w i th the f o l l o w i n g p robab l y not s i g n i f i c a n t p e c u l i a r i t y : the p r e d i c t e d f r e q u e n c i e s fo r ammonia were a l l h ighe r than the measured f r e q u e n c i e s whereas fo r phosphine and a r s i n e they were a l l lower . A s u r p r i s i n g r e s u l t was found in the va lues p r e d i c t e d from the C o r i o l i s and q u a r t i c da ta fo r the an t i s ymmet r i c s t r e t c h i n g v i b r a t i o n v3. In a l l cases the p r e d i c t e d f requency was much l a r g e r than the measured f requency and the d i f f e r e n c e s s c a l e d rough ly as the square root of the reduced mass fo r the s e r i e s X H 3 , XD 3 and X T 3 . The d i f f e r e n c e s fo r the s e r i e s fo r 1 72 X=N were 236, 145, 113 c m " 1 ; X=P, 489, 330, 279 c m " 1 ; X=As, 289, 191, 157 c m - 1 . The sugges t i on here might be tha t anharmonic c o r r e c t i o n s to f r e q u e n c i e s tha t are p r i m a r i l y bending in c h a r a c t e r can p r o b a b l y , to. a f i r s t a p p r o x i m a t i o n , be i g n o r e d . Furthermore these r e s u l t s might suggest anharmonic e f f e c t s are more important in bond s t r e t c h i n g modes and most important i f those modes are a n t i s y m m e t r i c . 2. FURTHER CONSIDERATIONS WHEN MAKING A HARMONIC FORCE  FIELD CALCULATION It i s alway.s d e s i r a b l e when making any harmonic f o r c e f i e l d c a l c u l a t i o n to be aware of ext ra-harmonic e f f e c t s that may " p e r t u r b " e m p i r i c a l parameters tha t can r e s u l t in u n - r e p r e s e n t a t i v e f o r c e c o n s t a n t s . C e r t a i n l y the g e n e r a l q u e s t i o n of anha rmon i c i t y as d i s c u s s e d in the p r e v i o u s s e c t i o n i s one example of ext ra-harmonic e f f e c t s . Two more s p e c i f i c examples that we s h a l l now c o n s i d e r are the f requency p e r t u r b a t i o n s due to Fermi resonance and the added anha rmon i c i t y i n t r o d u c e d by the i n v e r s i o n of the X atom through the p lane of the Y atoms in py ramida l XY 3 m o l e c u l e s . Fermi resonance may occur when two v i b r a t i o n a l l e v e l s , u s u a l l y one fundamental and one ove r t one , of the same symmetry s p e c i e s have n e a r l y the same energy , tha t i s a re " a c c i d e n t a l l y degenera te " (r37 pg 215 ,216 ) . In such cases the weaker over tone borrows i n t e n s i t y from the fundamenta l . More i m p o r t a n t l y f o r f o r c e f i e l d c a l c u l a t i o n s these v i b r a t i o n a l l e v e l s tha t have in the zero approx imat ion 1 73 n e a r l y the same energy " r e p e l " each other so tha t the l e v e l w i th the g rea te r energy moves to h igher f requency and the l e v e l w i th the lower energy moves to lower f r e q u e n c y . As ev idence sugges t i ng the e f f e c t s of Fermi resonances on our fundamental v i b r a t i o n s are indeed " s m a l l " , we present Tab le 6 . 1 . Except fo r the v2 v i b r a t i o n s e s p e c i a l l y in ammonia we n o t i c e that the v i b r a t i o n s fo r the mo lecu l a r s p e c i e s XY 3 a l l s c a l e by about the same f a c t o r . Fur thermore t h i s s c a l i n g f a c t o r i s c o n s i s t e n t , at l e a s t a p p r o x i m a t e l y , w i th a d i a tomic a p p r o x i m a t i o n . For i n s t ance v W v 2 f o r a g i ven XY 3 spec i e s a c c o r d i n g to equa t ion (e6.6) shou ld be 1 /2 g i ven rough ly by the r a t i o of (f^/f^) 7 where f^ and f^ are the va l ence bond s t r e t c h i n g and bending f o r c e c o n s t a n t s r e s p e c t i v e l y , see Tab le 6 . 6 . A l s o v - x / v \ > f ° r t w o s p e c i e s 1 /2 square root of the d i a tomic reduced masses ( n ' / u ) XY 3 and X Y 3 , s i m i l a r l y shou ld be rough ly the r a t i o of the . The f a c t tha t these f requency r a t i o s ( excep t ing v2 f o r the p resen t ) are a l l rough ly the same suggests we need not worry about l a r g e Fermi resonance i n t e r a c t i o n s p e r t u r b i n g i n d i v i d u a l fundamenta ls . A l s o the d e v i a t i o n s of these v a r i o u s r a t i o s suggest any Fermi resonance f requency s h i f t shou ld be l e s s than one pe rcen t of the v i b r a t i o n a l f r equency . 1 7 4 Tab le 6.1 V a r i o u s f requency r a t i o s of ammonia, phosph ine , a r s i n e and st i b i n e NH 3 ND 3 NT 3 PH 3 PD 3 PT 3 AsH 3 AsD 3 A s T 3 SbH 3 SbD 3 i i i 3.51 3.24 3.07 2.34 2.31 2.24 2. 33 2.31 2. 27 2.42 2.42 i i i . v n 2.12 2.15 2.19 2.08 2.11 2.10 2. 1 3 2.14 2. 13 2.28 2.30 rf i 1/2 r f 3.42 2.18 2.06 v2/vz v n/vu / b / a^1/2 (y /M ) NHf/NDa 1 .38 1 .27 1 . 34 1 .37 1 .37 N H 3 / N T 3 1 .66 1 .45 1 .58 1 .63 1 .63 N D 3 / N T 3 1 .20 1.14 1.17 1.19 1.19 P H 3 / P D 3 1 .38 1 .36 1 .37 1 .39 1 .39 P H 3 / P T 3 1 .66 1 .59 1 .66 1 .67 1 .68 P D 3 / P T 3 1 .20 1.17 1.21 1 .20 1.21 A s H 3 / A s D 3 1 .39 1 .37 1 .39 1 .40 1 .40 AsH 3 / A S T 3 1 .68 1 .64 1 .69 1 .69 1 .71 AsD 3 / A s T 3 1.21 1.19 1 .22 1.21 1 .22 R e c a l l : / 2 = 1.414 /3= 1.732 v/3/v/2= 1.22 5 NOTE: The va lues of the va lence f o r c e cons t an t s f f and f a re from Tab l e 6.6 1 75 A r a t h e r s t r i k i n g f e a t u r e of Tab le 6.1 i s that any f requency r a t i o i n v o l v i n g v2 i s s i g n i f i c a n t l y d i f f e r e n t from s i m i l a r r a t i o s i n v o l v i n g o the r f r e q u e n c i e s . A reason fo r t h i s can be found when one r e c a l l s that the co2 v i b r a t i o n i s e s s e n t i a l l y a symmetric b e n d i n g , or b r e a t h i n g , v i b r a t i o n and that the mot ion of the atoms d u r i n g t h i s v i b r a t i o n i s s i m i l a r to the i n v e r s i o n m o t i o n . Dennison (r79) has suggested tha t somehow the i n v e r s i o n makes the v2 v i b r a t i o n more anharmonic . A p o s s i b l e "somehow" i s tha t du r i ng the v i b r a t i o n a>2 the e f f e c t i v e i n v e r s i o n b a r r i e r i s lowered . T h i s can be v i s u a l i z e d by n o t i n g at c e r t a i n t imes d u r i n g the u>2 v i b r a t i o n the X atom w i l l be c l o s e r to the p lane of the Y atoms (XY 3 ) and i n v e r s i o n w i l l be e a s i e r . Now r e c a l l tha t the energy of the o 2=1 v i b r a t i o n fo r a l l our XY 3 s p e c i e s i s below the b a r r i e r (see Herzberg r37 page 2 2 2 f ) . Lower ing the b a r r i e r means tha t the u>2 v i b r a t i o n w i l l be " c l o s e r " to the "anharmonic " top of the b a r r i e r . The r e s u l t of the lower e f f e c t i v e i n v e r s i o n b a r r i e r i s tha t the p o t e n t i a l e x p e r i e n c e d f o r t h i s v i b r a t i o n i s much more anharmonic . T h i s means t ha t the measured f requency v2 shou ld be at an even lower f requency than one might expect from gene ra l anharmonic f requency c o n s i d e r a t i o n s . Fur thermore s i n ce the lower weight hydrogen i s o t o p e s s i t p r o p o r t i o n a l l y h ighe r i n the i s o t o p i c a l l y i n v a r i a n t p o t e n t i a l w e l l s , because of ze ro p o i n t e f f e c t s , one shou ld expec t the anharmonic c o r r e c t i o n s to dec rease in the order X H 3 , X D 3 , X T 3 . F i n a l l y the lower the i n v e r s i o n b a r r i e r to beg in w i th the more n o t i c e a b l e 1 76 these p e r t u r b a t i o n s w i l l be and hence we shou ld expect the b i gges t anharmonic c o r r e c t i o n s to be in ammonia f o l l owed by much sma l l e r c o r r e c t i o n s in phosphine and a r s i n e . T h i s i s the expe r i ence of Tab le 6 . 1 . I t i s not immediate ly obv ious how one c o u l d c o r r e c t f o r t h i s added anha rmon i c i t y due to the lower ing of the i n v e r s i o n b a r r i e r . C e r t a i n l y a quas i u>2 c o u l d be es t ima ted from the s c a l i n g r a t i o s of o ther f r e q u e n c i e s but one shou ld expect c o r r e c t i o n s to the q u a r t i c d i s t o r t i o n cons t an t s to a l s o be n e c e s s a r y . I t i s not c l e a r how c o r r e c t i o n s to these cons t an t s c o u l d be made, e s p e c i a l l y the q u a r t i c cons t an t s of the asymmetric t o p s . So, fo r the purposes of t h i s t h e s i s the main emphasis w i l l be on harmonic f o r ce f i e l d s c a l c u l a t e d from data where no e f f o r t has been made to " c o r r e c t " f o r the added anha rmon i c i t y in the v2=1 v i b r a t i o n . T h i s means tha t some ca re must be taken in comparing the f o r c e cons t an t s of the group V hyd r i des c a l c u l a t e d here wi th each other and w i th those c a l c u l a t e d by "ab i n i t i o " methods; t h i s w i l l c e r t a i n l y be the case fo r the f^ c o n s t a n t . Whereas i g n o r i n g these " i n v e r s i o n " e f f e c t s in ammonia might l e ad to u n r e l i a b l e r e s u l t s , i t i s hoped tha t fo r phosphine and a r s i n e , the pr imary concerns of these harmonic f o r ce f i e l d s t u d i e s , where the e f f e c t s are sma l l e r that the f i n a l f o r c e cons t an t s shou ld s t i l l be r e p r e s e n t a t i v e . Fu r the r comments on t h i s problem w i l l be made in the next s e c t i o n . 177 3. HARMONIC FORCE FIELD REFINEMENTS: AMMONIA, PHOSPHINE  AND ARSINE The h i ghes t symmetry common to a l l the v a r i o u s i s o t o p i c d e r i v a t i v e s ( excep t ing XHTD fo r which no data i s yet a v a i l a b l e ) i s Cs . In t h i s case the f o l l o w i n g set of symmetry c o o r d i n a t e s may be t a k e n 9 A' B l o ck : S 1 =^ (A r T+Ar 2 +Ar 3 ) 5 2 = 7l(A/3,+A/3 2+A^ 3) 5 3 = 7 6 " ( 2 A r i " A r 2 " ^ 3 ) 5 4 =-7l(2A/3,-A/32-A/33) A" B l o ck : S g = ^ ( A r 2 - A r 3 ) S 6 ^ I t A ^ - A ^ ) where the change in a bond l eng th Ar^ = Ar (X-Y^) , and the change in a bond ang le A/3. = A/3(Y .-X-Y, ) , w i th X the heavy 1 ] K atom and Y r e p r e s e n t i n g H, D or T . A ' and A" r ep resen t symmetry s p e c i e s in the Cs group . For the symmetric top case the symmetry d e s i g n a t i o n becomes 2E + 2A, and S^ above becomes S-, , S.- becomes S o u , S. becomes S. and Sr becomes 3a b 3D 4 4a 6 S 4 b * F o r ^ e s y m m e t r ^ c tops S 1 and S 2 are the symmetry c o o r d i n a t e s fo r the A, t o t a l l y symmetric s p e c i e s and the S-. , S^, and S. , S., are the symmetry c o o r d i n a t e s used to 9 The d e f i n i t i o n of symmetry c o o r d i n a t e s d e f i n e s a l s o the U mat r ix which when m u l t i p l i e d by the W i l son B mat r ix g i ves the " symmet r i zed " B mat r ix of equa t i on e1.45 (see Woodward ( r 4 3 ) ) . 1 78 rep resen t the doubly degenerate E s p e c i e s . Each input datum was a s s i g n e d an u n c e r t a i n t y and weighted as the i n ve r se square of the u n c e r t a i n t y . The ass ignments of u n c e r t a i n t i e s are an attempt to account fo r ext ra-harmonic e f f e c t s . The u n c e r t a i n t i e s were chosen f o l l o w i n g the sugges t i ons of A ldous and M i l l s ( r 8 l ) . Here we a s s i g n an e r r o r of 1% to the f r e q u e n c i e s . The ze t a cons t an t s are a s s i gned u n c e r t a i n t i e s of ±0.01 to ±0 .02 . The q u a r t i c d i s t o r t i o n cons t an t s have a l s o been a ss igned u n c e r t a i n t i e s of = 1%. The advantage of t h i s somewhat a r b i t r a r y we igh t ing scheme i s tha t no p a r t i c u l a r type of parameter , that i s v, or D, was found to " p o l a r i z e " the f i t s . A l l input data were e m p i r i c a l l y de t e rm ined ; no c o r r e c t i o n s fo r anha rmon i c i t y or o ther non-harmonic e f f e c t s were made. The data used in c o n s t r u c t i n g the v a r i o u s f o r c e f i e l d s are p resen ted in Tab l e s 6 .2 , 6.3 and 6.4 a long w i th the c a l c u l a t e d "bes t f i t " v a l u e s . The f i t s were done us i ng the i t e r a t i v e approach rev iewed by A ldous and M i l l s ( r 8 l ) and d i s c u s s e d in d e t a i l by Gans ( r119) . The d e r i v e d f o r c e f i e l d s in the i n t e r n a l symmetry c o o r d i n a t e b a s i s are g i ven i n Tab le 6 . 5 . The more i n t u i t i v e va lence f o r c e f i e l d s are g i ven in Tab le 6 .6 . (r43 pg 193) 1 79 Tab le 6.2 Harmonic f o r c e f i e l d study of ammonia N H 3 N D 3 N T 3 Obs O - C Ref Obs O -c Ref Obs o-c Ref V 1 3337 -74 131 2421 1 4 131 2014 6 1 32 v2 950 -7 748 20 657 25 1 33 "3 3444 -53 2564 1 0 " 2185 2 1 32 1 627 -7 1191 5 " 1000 1 1 98 D J 25. 21 0.18 134 5.91 0.01 135 2.598 -0.072 1 36 D JK -46. 61 -0.44 -10.49 0.25 " -4.472 0.15 D K 26. 93 0.02 -6.16*'" - 2 . 5 9 t $3 0.042 0.003 131 0.133 0.009 131 0.228^ S« -0.255 0.002 " -0.327 0.003 137 -0.383 -0.012 1 33 N H ; 2 D N D 2 H Obs o-c Obs o-c D J 1 5.82 0.11 10.03 -0.04 D JK -23.95 -0.43 -14.79 -0.40 D K 1 0.95 0.25 6.49 0.34 d , 4.181 0.77 3.38 -0.06 d 2 0.1324 0.0003 -1 .59 0.03 Asymmetric top data from r e f ( r ! 0 5 ) f Harmonic f o r c e f i e l d va lue U n i t s : j v , c m _ 1 , D 's MHz and $'s no u n i t s . Tab le 6.3 Harmonic f o r c e f i e l d study of phosphine PH 3 PD 3 PT 3 Obs o-c Ref Obs O-C Ref Obs O-C Ref 2321 -28 1 1 2 1 682 -0.3 115 1398 7 98 992 -4 1 13 728 2 623 13 II " 3 2327 -23 1 1 2 1 693 4 1401 -1 H "» 1118 -3 1 1 3 803 4 668 10 H DJ 3.94 -0.03 1 02 1 .020 -0.026 0.491^ D JK -5.18 -0.06 -1.312 0.008 -0 .611^ D K 4.13 -0.04 ?i 1 .027 -0.002 0.457*1* $ 3 0.017 -0.007 1 1 2 0.056 -0.009 115 0. 1031* $4 -0.442 -0.013 1 1 3 -0.463 -0.012 " 0.495*1* $24 0.526 0.004 ?i 0.512*1* 0.501 1* 5,3 0.004 t 0.062 0.03 0.062*^* PH 2D PD 2H Obs o-c Obs o-c DJ 2.72*1* 1 .62*1" D JK -3.476 0.06 -0.794 0.04 D K 1 .874 0.06 -0.353 -0.06 d , -0.227 0.006 -0.233 0.003 d 2 0.449 0.002 -0.326 0.004 P D 3 , PD 2H and PDH 2 q u a r t i c cons t an t s were ob ta ined as pa r t of t h i s work. t Harmonic f o r c e f i e l d va lue U n i t s : i>,cnr 1 , D 's MHz and $'s no u n i t s . 181 Tab le 6.4 Harmonic f o r c e f i e l d study of a r s i n e AsH 3 AsD 3 A s T 3 Obs o-c Ref Obs O -c Ref Obs o-c Ref V 1 2115 -24 92 1 523 2 97 1 256 6 98 v2 907 -6 95 660 6 i t 553 1 2 i t v3 21 26 -23 92 1 529 -1 IT 1 258 -1 n 999 -7 95 714 -1 II 590 4 i t D J 2.930 -0.01 93 0.742 -o . 01 1 0.344* D JK -3.724 -0.026 i t -0.940 0. 008 -0 .423* D K 3.357 -0.004 0.833* 0 .369* $3 -0.013 -0.002 92 -0.04 0. 008 96 0 .049* -0.446 -0.03 95 0.485* 0 .493* i> 2 « 0.502 -0.009 II 0.507* 0.504* AsH AsD 2 H Obs o-c Obs o--c AJ 1 .236 -0.02 1.131 -0 .02 A JK 2.230 0.002 -1 .876 0 .007 A K -2.630 -0.005 2. 173 -0 .02 6J 0. 126 0.004 -0.407 -o .003 5 K 1.994 0.03 0.980 0 .008 A s D 3 , AsD 2 H and AsDH 2 q u a r t i c cons t an t s were ob ta ined as pa r t of t h i s work. t Harmonic f o r c e f i e l d va lue U n i t s : C j C i n " 1 , D 's MHz and $'s no u n i t s . 182 Tab le 6.5 Fo r ce cons t an t s ( i n t e r n a l symmetry c o o r d i n a t e s ) NH 3 PH 3 AsH 3 1 mdyn A" 1 6 .732(106) 3. 194(20) 2 . 682(17) mdyn 0 .560(30) 0. 156(10) 0.1 14(8) F 2 2 mdyn A 0 .439(8) 0. 594(4) 0.575(3) F 3 3 mdyn A" 1 6 .671(104) 3. 165(18) 2.705(19) F 3 « mdyn -0 .161(26) - 0 . 036(10) -0.005(10) F n n mdyn A 0 .636(3) 0. 710(4) 0.699(3) Tab le 6.6 Force cons t an t s (va lence) NH 3 PH 3 AsH 3 f r mdyn A" 1 6 .691(250) 3 . 175(31 ) 2.697(31) f , r r mdyn A- 1 0 .020(70) 0 .010(13) -0.008(12) h mdyn A 0 .571(14) 0 .671(7) 0.638(5) f<3<r mdyn A -0 .066(3) -0 .039(3) -0.031(2) mdyn 0 .240(19) 0 .064(7) 0.040(6) ' r f mdyn 0 .080(70) 0 .028(17) 0.035(16) G iven : F , 1=f + 2f , r r r F - = fr/3 ,+2t r/3 F 2 2 =f F 3 3 = f r - f , r r F 3 «=f r/3 '~ f r/3 F (, « = f Q-fP0' For Tab l es 6.5 and 6.6 the numbers in parentheses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l e a s t s i g n i f i c a n t f i g u r e s . See the text f o r a d i s c u s s i o n of the we ight ing of the v a r i o u s input d a t a . 183 When comparing the va l ence f o r c e cons t an t s of the v a r i o u s h y d r i d e s of Tab le 6.6 we see t h a t , except fo r f^, the gene ra l t r end i s the p o t e n t i a l cons t an t s of ammonia are much h ighe r than those of phosphine which in tu rn are r e l a t i v e l y on ly s l i g h t l y h ighe r than those of a r s i n e . T h i s i s not a s u r p r i s i n g r e s u l t as o ther r e l a t e d parameters such as i o n i z a t i o n p o t e n t i a l s a l s o show t h i s t r end ( i . P . ' s fo r N, P, As are 14 .53 , 10.484, 9.81 v o l t s r e s p e c t i v e l y ( r 7 3 ) ) . R e c a l l a l s o a s i m i l a r t r end was found in v a r i o u s parameters of CH„ , S i H „ and GeH„ in Chapter 3. The excep t i on to the above g e n e r a l i z a t i o n i s f^ fo r ammonia. Based on the o the r f o r c e cons t an t s i t would appear tha t £p f o r ammonia i s u n u s u a l l y s m a l l . The f^ parameter , the harmonic " s p r i n g " f o r c e cons tan t fo r the open ing and c l o s i n g of the H-N-H a n g l e , i s d i r e c t l y r e l a t e d to the symmetric bending v i b r a t i o n v2 ~ the same v2 v i b r a t i o n that gave the anomalous r a t i o s of Tab l e 6 . 1 . In the l a s t s e c t i o n i t was suggested tha t these anomalous r a t i o s were due to the v2 v i b r a t i o n be ing more anharmonic than o therw ise expec ted . T h i s e x t r a anha rmon i c i t y was then t e n t a t i v e l y t r a c e d to the i n v e r s i o n motion of ammonia. We see now, from compar isons of group V h yd r i de f o r c e c o n s t a n t s , that we can e x p l a i n the anomalous ammonia r e s u l t s wi thout recourse to a n h a r m o n i c i t y . A . " f i r s t o r d e r " e x p l a n a t i o n now would be that i n v e r s i o n e f f e c t i v e l y lowers the f^ p o t e n t i a l c o n s t a n t . The term " f i r s t o r d e r " i s to connote tha t t h i s e x p l a n a t i o n i s too s i m p l i s t i c , because i f i t were c o r r e c t we shou ld be ab le to 184 f i n d a f^ that would produce reasonab le r a t i o s of the so r t found in Tab le 6 . 1 . As an e x e r c i s e , p r e d i c t i o n s of v i b r a t i o n a l f r e q u e n c i e s were made f i x i n g a l l the f o r ce c o n s t a n t s to the ammonia va lues of Tab le 6.5 but v a r y i n g F 2 2 . I t was found tha t i n c r e a s i n g F 2 2 to 0 .8-1 .0 mdyn A, produced f requency r a t i o s approach ing e x p e c t a t i o n s based on the phosph ine , a r s i n e and s t i b i n e r a t i o s . C l e a r l y more work on the ammonia f o r c e f i e l d and the i m p l i c a t i o n s of i n v e r s i o n i s r e q u i r e d ( r138 ) . T h i s shou ld c e r t a i n l y be of i n t e r e s t to peop le c a l c u l a t i n g f o r c e c o n s t a n t s u s i ng "ab i n i t i o " methods as these methods do not i n c l u d e the e f f e c t s of i n v e r s i o n . 4. ESTIMATION OF THE DISTORTION DIPOLE MOMENT In s e c t i o n 1.4 we d e f i n e d the symmetric top d i s t o r t i o n d i p o l e moment as the sum of two te rms, a s o - c a l l e d r o t a t i o n a l pa r t and v i b r a t i o n a l p a r t . The r o t a t i o n a l c o n t r i b u t i o n can be e s t ima ted from equa t i on e 1 . 3 3 . Here r i s c a l c u l a t e d u s i n g the sma l l " a " ^ xxxz ^ mat r ix (e1.50) and equa t i on 1.49. A n a l y t i c a l forms fo r r tha t bypass the n e c e s s i t y of a f o r c e f i e l d a n a l y s i s are g i ven by A l i e v and Watson ( r139 ) . E s t ima tes of r are A. A A 2 g i ven in Tab le 6.7 below a long w i th an upper bound c a l c u l a t e d by (r140) xxxz T I = 2B 3 (2-B /B ) x x x z 1 x x' z 1/2 ZP~. CET min 1 max CJI_ ..TIT [6 .12] It would seem that the upper bound es t imate approx imates the " t r u e " va lue q u i t e w e l l . 1 85 Table 6.7 (MHz) E s t i m a t i o n of r xxxz | T X X X Z | Upper Bound NH 3 11.26 11.37 ND 3 2.72 2.76 PH 3 2.92 3.11 PD 3 0.78 0.83 AsH 3 2.29 2.45 AsD 3 0.59 0.63 In o rder to c a l c u l a t e the v i b r a t i o n a l c o n t r i b u t i o n to the d i s t o r t i o n moment we need v a l ues of 9M /9Q-, , where 1=3,4. These v a l u e s , ob ta ined from i n f r a - r e d i n t e n s i t y measurements, have been measured fo r NH 3 and PH 3 (r141) . The r e l a t i v e s i gns of these d i p o l e moment d e r i v a t i v e s are de te rminab le from q u a l i t a t i v e examinat ion of i n f r a - r e d i n t e n s i t y p e r t u r b a t i o n s (r142) but r ecorded s p e c t r a have not been p u b l i s h e d . As a r e s u l t the 9M/9Q'S in Tab le 6.8 are g i ven as a b s o l u t e v a l u e s . The 9M/9Q'S fo r the f u l l y deu t e r a t ed compounds are c a l c u l a t e d from the i s o t o p i c i n t e n s i t y i n v a r i a n c e r u l e ( r l 2 9 , r 1 4 3 ) |9M/9Q| r |3M/9Q| -, H "1 [6 .13] The d i p o l e d e r i v a t i v e s of a r s i n e are es t imated by assuming, a r s i n e to be an i s o t o p i c d e r i v a t i v e of phosph ine . T h i s assumpt ion i s expected to be reasonab le (say to w i t h i n 25%) in l i g h t of the s i m i l a r i t y of phosphine and a r s i n e (example: 186 f o r c e c o n s t a n t s , s t r u c t u r a l p a r a m e t e r s . . . . ) . Tab le 6.8 I S M / B Q - L | (amu A ) 1=3 1=4 NH 3 0.1857 0.5402 ND 3 0.1367 0.3951 PH 3 0.8304 0.5209 PD 3 0.6043 0.3741 AsH 3 0.7589 0.4655 AsD 3 0.5457 0.3326 The v i b r a t i o n a l c o n t r i b u t i o n to the d i p o l e moment was es t ima ted us i ng equa t ion e 1 . 3 5 . Va l ues of © V I B a l ong w i th © . a re g i ven in Tab le 6 . 9 . The two va lues of © X T T „ ro t 3 VIB co r r e spond to whether the 3 M / 9 Q ' S have the same or o p p o s i t e s i gns r e s p e c t i v e l y . Tab le 6.9 (X10 5 Debye) E s t i m a t i o n of the r o t a t i o n a l and v i b r a t i o n a l c o n t r i b u t i o n s to the d i s t o r t i o n d i p o l e moment I 0 R O T I I 0 V I B I NH 3 7.48 5.35,4.71 ND 3 3.32 2 .90 ,2 .59 PH 3 5.02 3 .51 ,2 .02 PD 3 2.04 1.89,1.12 AsH 3 3.31 2 .94 ,1 .74 AsD 3 1.35 1.53,0.91 1 87 The v a r i o u s moments of i n e r t i a and r o t a t i o n a l cons t an t s used in these c a l c u l a t i o n s were c a l c u l a t e d us i ng the e q u i l i b r i u m s t r u c t u r e s of Tab le 6 .13 . The u n c e r t a i n t i e s of these i n e r t i a l parameters a long w i th the u n c e r t a i n t i e s in v3, and M , ( M „ NH 3=1.47 D ( r 1 0 5 , r 1 4 4 1 0 , r 1 4 5 ) , M , PH 3 =0.55 D ( r 9 9 ) , M z AsH 3 =0.22 D (r99) ) and the " a ' s " are expected to be n e g l i g i b l e when compared to the o ther q u a n t i t i e s . The e r r o r in the a ' s shou ld be l e s s than a few p e r c e n t . T h i s seems reasonab le•as q u a d r a t i c p roduc t s of the v a r i o u s " a " mat r ix e lements reproduce d i s t o r t i o n c o n s t a n t s u s u a l l y w i t h i n 1 or 2 p e r c e n t . The main u n c e r t a i n t i e s in our d i s t o r t i o n d i p o l e moment e s t ima tes shou ld come from the 3M/9Q'S and the assumpt ion that a r s i n e i s an " i s o t o p i c " d e r i v a t i v e of phosph ine . The e f f e c t of the l a t t e r i s hard to q u a n t i f y . The accuracy of the 3M/9Q'S i s quoted by the au thors (r141) to be about 10 p e r c e n t . T h i s shou ld mean that the u n c e r t a i n t i e s of the v a r i o u s 0 R Q T ' S and 0 V I B ' s i s o p t i m i s t i c a l l y on the order of 5 and 13 percent r e s p e c t i v e l y . The on l y ambigu i t y in the d i s t o r t i o n d i p o l e e s t ima tes i s in the r e l a t i v e s i gns of the two c o n t r i b u t i o n s . Oka et a l (r39) have i n t r oduced a model from which they i n f e r tha t in ammonia the two terms have the same s i g n and the e f f e c t s of 1 0 In re f r144 the au thors ignore the v i b r a t i o n a l c o n t r i b u t i o n to the d i s t o r t i o n d i p o l e moment and t h e i r subsequent e s t ima tes of r a re i n c o r r e c t . 188 r o t a t i o n a l and v i b r a t i o n a l i n t e n s i t y borrowing enhance each o t h e r . The c o n t e n t i o n tha t we can add the v i b r a t i o n a l and r o t a t i o n a l c o n t r i b u t i o n s to g i ve the f i n a l d i s t o r t i o n d i p o l e moment i s suppor ted by the expe r imen ta l e s t imate of the PH 3 d i s t o r t i o n d i p o l e moment of 7.2 x 10 " 5 D made by Chu and Oka ( r 82 ) . T h i s va lue can on l y be a t t r i b u t e d to a sum of | © R O T I and | 0 T 7 T T J in Tab le 4 . 9 . Fur thermore a s u b j e c t i v e es t ima te of the s t r e n g t h of the AsD 3 a b s o r p t i o n l i n e s compared w i th the known l i n e s t r eng ths of the germane spectrum a l s o suggest tha t in the AsD 3 case the two c o n t r i b u t i o n s enhance one a n o t h e r . In t h i s case a d i s t o r t i o n d i p o l e moment c a l c u l a t e d as the d i f f e r e n c e of the two e f f e c t s would be below the s e n s i t i v i t y of our spec t rome te r . 189 •5. STRUCTURE OF AMMONIA, PHOSPHINE AND ARSINE The r e s u l t s of a zero po in t s t r u c t u r a l de t e rm ina t i on are g i ven in Tab l e s 6 .10 , 6.11 and 6 .12 . There r H , r D and r T are the n i t r o g e n , phosphorus or a r s e n i c to hydrogen, deute r ium and t r i t i u m r bond l eng ths r e s p e c t i v e l y . The ang le /3 i s the <XMX where M = N, P, As and X = H, D and T . The r z s t r u c t u r e s were e s t ima ted in the f o l l o w i n g way. From the e m p i r i c a l r o t a t i o n a l c o n s t a n t s , B 0 c ons t an t s were c a l c u l a t e d . For symmetric tops the e m p i r i c a l c ons t an t s equa l the B 0 c o n s t a n t s and so fo r these cases the c a l c u l a t i o n was r a the r easy . For the asymmetric tops Watson 's S c r i p t Cons tan ts were f i r s t e va lua ted and then c o r r e c t e d f o r the sma l l " r " c o n t r i b u t i o n to g i ve the K i v e l son-W i l son e f f e c t i v e parameters ( e 1 . 5 2 , e 1 . 5 3 ) . The " r ' s " were c a l c u l a t e d u s i n g equa t i on e 1 . 4 9 . For a more complete d i s c u s s i o n see s e c t i o n 1.6. These ground s t a t e B 0 r o t a t i o n a l cons t an t s were then conve r t ed to the zero p o i n t average B z va lues us ing the c a l c u l a t e d harmonic p a r t s of the a c ons t an t s ( e1 .58 ) . The r z s t r u c t u r e s were determined by a d i r e c t i t e r a t i v e f i t of the v a r i o u s g e o m e t r i c a l parameters to the t a b u l a t e d B z v a l u e s . In the NT 3 case there were not enough r o t a t i o n a l c o n s t a n t s to determine both the bond ang le and bond l e n g t h . The bond ang le was e s t ima ted from a l i n e a r e x t r a p o l a t i o n of 0 H and /3D ( r 5 4 ) . T h i s bond ang le was then used wi th the lone NT 3 r o t a t i o n a l cons tan t in the c a l c u l a t i o n of r T . E q u i l i b r i u m s t r u c t u r e s were e s t ima ted us ing the d i a tom i c approx imat ion (see equa t i on e1.59 and Tab le 6 . 1 3 ) . 1 90 S ince the c o r r e c t i o n s to the r z s t r u c t u r e s to g i ve the e q u i l i b r i u m s t r u c t u r e a re much l a r g e r than the i s o t o p i c changes i n the r parameters the e q u i l i b r i u m s t r u c t u r e s shou ld be t r e a t e d wi th c a u t i o n . S t r i c t l y the e x t r a p o l a t i o n to g i ve the e q u i l i b r i u m bond ang le i s a l lowed on l y f o r the case where the anharmonic p o t e n t i a l fo r the bending motion c o n t a i n s ma in ly cub i c terms w i th sma l l e r h ighe r terms ( r 54 ) . A l though t h i s s i t u a t i o n has not been e s t a b l i s h e d s i n ce the B ' s w i th d i f f e r e n t Y are n e a r l y the same the c o r r e c t i o n to z J the l i n e a r e x t r a p o l a t i o n shou ld be s m a l l . The s tandard d e v i a t i o n e r r o r s in the bond l eng th parameters are on the order of 10" " A. The bond ang le s t andard e r r o r i s around 0 . 0 1 ° . When a l l sources of random and sys t emat i c e r r o r are taken i n t o account an accuracy of at best 0.001 A fo r bond l eng ths (r49) i s more r e a l i s t i c . The bond ang le i s perhaps good to about ±0.05 ° . As an i n d i c a t i o n of the r e l i a b i l i t y of the d i a tomic approx imat ion (e1.59) i n e s t i m a t i n g the e q u i l i b r i u m s t r u c t u r e s of phosphine and a r s i n e we can compare the e q u i l i b r i u m s t r u c t u r e s c a l c u l a t e d from the p o s s i b l e p a i r combina t ions of ammonia r ' s where we have e x t r a i n f o rma t i on from the z t r i t i a t e d ammonia d a t a . For the combina t ions r^r^, r H r T and r_.rm one has r v a lues of 1.013, 1.012 and 1.010 A D T e r e s p e c t i v e l y . The r g c a l c u l a t e d from the r D r T average s t r u c t u r e p a i r i s p robab l y f avoured as l a r g e r reduced mass systems v i b r a t e c l o s e r to the bottom of the p o t e n t i a l we l l s where the l i n e a r e x t r a p o l a t i o n of the d i a tomic approx imat ion 191 works b e t t e r . The r g bond l e n g t h s . s h o u l d be r e l i a b l e to ±0.002 A. Tab le 6.10 (MHz) r s t r u c t u r e of ammonia ( 1 U N) z r H=1.02304(7)A r D =1.02030(10)A r T =1.01876(22)A* 0 H=1O7.O39( 16)° /3D= 107. 085(19)° 0 = 1 07 . 1 0 ( 3 ) o t E m p i r i c a l S c r i p t B 0 B O-C ND 2H 223 217 . I 1 a 223 222. 80 223 219 .16 219 970. 8 -44 i u r s 160 208 .10 160 208. 10 1 60 206 .58 1 58 304. 5 -1 16 1 1 2 510 .58 112 521 . 1 3 1 1 2 514 .03 1 1 1 804. 8 300 NH 2D 290 1 25 .38 a 290 141. 97 290 1 38 .35 285 549. 7 198 i n r s 1 92 1 94 .18 192 194. 04 192 189 . 1 1 189 666. 0 -126 1 40 795 .26 140 827. 70 1 40 787 .32 1 39 750. 3 1 23 NH 3 298 1 1 7 . I 0 b 293 240. 3 -76 186 726 .36 185 091 . 7 -180 ND 3 1 54 1 73 . 3 8 ° 1 52 375. 6 6 NT 3 1 05 565 . 3 7 d 1 04 546. 5 0 a r e f ( r l 0 5 ) ,b r e f ( r134 ) , c r e f ( r 1 3 5 ) , d r e f ( r 1 3 6 ) ' f C a l c u l a t e d from a l i n e a r e x t r a p o l a t i o n of /3U and (see M JJ t ex t ) ' * ' c a l c u l a t e d assuming the ang le 0 T Standard d e v i a t i o n of the f i t i s 154 MHz. . 1 92 Tab le 6.11 (MHz) r z s t r u c t u r e of phosphine . r H =1.42774(9 )A r Q =1.42373(13)A 0 H =93. 2 8 6 ( 1 2 ) ° /3D=93.332{ 1 8 ) ° E m p i r i c a l S c r i p t B 0 B O-C PD 2H 93 9 1 7 . 7 9 a 93 918.43 93 916.19 92 979.7 54 I I I r S 81 910.92 81 912.50 81 910.17 81 171.8 111 64 841.80 64 843.07 64 843.56 64 425.3 -5 PH 2D 129 8 4 3 . 1 4 a 129 846.48 129 845.33 128 397.0 58 I I l r S 89 279.34 89 283.56 89 277.60 88 471.9 -190 83 250.52 83 258.66 83 250.93 82 592.7 -168 PH 3 133 4 8 0 . 1 3 b 131 966.5 -11 117 488.69 116 346.9 16 PD 3 69 4 7 1 . 0 9 a 68 905.0 43 58 974.42 58 577.2 83 a t h i s work, b r e f ( r l 0 2 ) S tandard d e v i a t i o n of the f i t i s 96 MHz. . 1 93 Tab le 6.12 (MHz) r s t r u c t u r e of a r s i n e r H = 1 .52765(8)A r D = 1.52337(11 )A 0H=91 . 9 4 1 5 ( 9 5 ) ° 91 .982 6 ( 1 4 ) ° E m p i r i c a l S c r i p t B 0 B z o-c AsD 2 H 77 131 .53 a 77 133.06 77 131.20 76 417 .8 24 I i r A 71 525.64 71 527.90 71 526.01 70 909 .0 60 55 511.88 55 511.12 55 511.73 55 171 .6 -38 AsH 2 D 1 1 0 5 8 1 . 6 7 a 110 582.13 1 10 581.32 109 425 .5 1 55 I I l r A 75 133.53 75 142.48 75 136.64 74 51 1 . 1 -58 72 548.71 72 551.19 72 545. 17 71980. 57 -56 AsH 3 1 1 2 4 7 0 . 6 1 b 1 1 1 282 .7 -54 •1 04 884.07 1 03 878 .0 -39 AsD 3 57 4 7 7 . 6 0 a 57 045 .9 1 6 52 641.98 52 287 .4 33 a t h i s work, b r e f ( r 9 3 ) S tandard d e v i a t i o n of the f i t i s 65 MHz. . 1 94 Tab le 6.13 E q u i l i b r i u m s t r u c t u r e s of ammonia, phosphine and a r s i n e Ammonia T h i s work Bened i c t and P l y l e r (1957) Helminger et a l (1974)* r ( A ) e 1.011 7 (2 ) 1.0116(8) 1.0135(4) 0 ( ° ) e 107.21(11) 106.99(2) 107.14(9) Phosphine T h i s work K i j i m a and Tanaka (1981)t Helms and Gordy (1977) f t Chu and Oka (1974) f f 1 . 4 1 3 5 ( 2 ) 1 .41175(50) 1.4115 9 (6 ) 1.41154(50) 93.45(9) 93.421(60) 93 .32 8 ( 2 ) 93.36(8) A r s i n e T h i s work C a r l o t t i et a l (1 983)f t t O l s o n , Maki (1975)t t t and Sams 1 . 51 2 8 ( 2 ) 1-511060(14) 1 .5108(4) 92.08(7) 92.0690(14) 92.083(43) * T h i s s t r u c t u r e i s an average of v a r i o u s s u b s t i t u t i o n s t r u c t u r e s , s o - c a l l e d " r " s t r u c t u r e s , and can on ly be c o n s i d e r e d as an approximate r g s t r u c t u r e . t These e q u i l i b r i u m s t r u c t u r e s are e s t ima ted by e m p i r i c a l l y de t e rm in ing the a c ons t an t s from a n a l y s i s of r o t a t i o n a l s t r u c t u r e of v i b r a t i o n a l bands. R e c a l l tha t in t h i s work we use a l i n e a r e x t r a p o l a t i o n of r s t r u c t u r e s to es t imate the e q u i l i b r i u m s t r u c t u r e . t t These two s t r u c t u r e s fo r t h i s molecu le are not independent as the l a t e r au thors use in t h e i r s t r u c t u r a l d e t e r m i n a t i o n r e s u l t s ob ta ined by the e a r l i e r a u t h o r s . — Numbers in pa ren theses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 1 94 Tab le 6.13 E q u i l i b r i u m s t r u c t u r e s of ammonia, phosphine and a r s i n e Ammonia T h i s work Bened i c t and P l y l e r (1957) Helminger et a l (1974)* r (A) e 1.011 7 (2 ) 1.0116(8) 1.0135(4) 0 ( ° ) e 107.21(11 ) 106.99(2) 107.14(9) Phosphine T h i s work K i j i m a and Tanaka (1981)t Helms and Gordy ( I977 ) f t Chu and Oka (1974) f f 1 .413 5 (2 ) 1.41175(50) 1 .4115 9 ( 6 ) 1.41154(50) 93.45(9) 93.421(60) 93 .32 8 ( 2 ) 93.36(8) A r s i n e T h i s work C a r l o t t i et a l ( I983) f t t O l s o n , Maki (1975 ) t t t and Sams 1.512 8 ( 2 ) 1.511060(14) 1.5108(4) 92.08(7) 92.0690(14) 92.083(43) * T h i s s t r u c t u r e i s an average of v a r i o u s s u b s t i t u t i o n s t r u c t u r e s , s o - c a l l e d " r " s t r u c t u r e s , and can on l y be c o n s i d e r e d as an approximate r g s t r u c t u r e . t These e q u i l i b r i u m s t r u c t u r e s are es t ima ted by e m p i r i c a l l y , d e t e r m i n i n g the a c ons t an t s from a n a l y s i s of r o t a t i o n a l s t r u c t u r e of v i b r a t i o n a l bands. R e c a l l that in t h i s work we use a l i n e a r e x t r a p o l a t i o n of r s t r u c t u r e s to e s t ima te the e q u i l i b r i u m s t r u c t u r e . t t These two s t r u c t u r e s f o r t h i s molecu le are not independent as the l a t e r au tho rs use in t h e i r s t r u c t u r a l d e t e r m i n a t i o n r e s u l t s ob t a i ned by the e a r l i e r a u t h o r s . — Numbers in paren theses are the s tandard d e v i a t i o n s of the l e a s t squares f i t s in . u n i t s of the l a s t s i g n i f i c a n t f i g u r e s . 195 6. DISCUSSION OF STRUCTURAL PARAMETERS From Tab le 6.13 we see tha t the e q u i l i b r i u m s t r u c t u r e s determined as pa r t of t h i s work compare w e l l w i th the r e s u l t s of p r e v i o u s s t u d i e s . I t would a l s o seem that the v a l ues p resen ted here are s i g n i f i c a n t l y l e s s a c cu ra t e than the p r e v i o u s r e s u l t s . When making these compar isons we shou ld be aware of what peop le .mean when they quote mo lecu l a r bond l eng ths to an accu racy of the o rder of atomic r a d i i . I t seems unders tood tha t these bond l eng ths are not meant to be i n t e r p r e t e d as a p h y s i c a l l eng th a c c u r a t e l y de te rm ined , but r a the r as a parameter tha t w i l l reproduce the e q u i l i b r i u m r o t a t i o n a l cons tan t w i t h i n i t s s t a t e d e r r o r . For the case where the number of s t r u c t u r a l parameters equa l s the number of r o t a t i o n a l cons t an t s then the s t r u c t u r a l parameters are o f t e n quoted to the h igh p r e c i s i o n of the r o t a t i o n a l c o n s t a n t s wi thout regard to p h y s i c a l s i g n i f i c a n c e . In t h i s study we have seen tha t the l i n e a r e x t r a p o l a t i o n t echn ique when a p p l i e d to r H , r D and r T fo r ammonia produced e q u i l i b r i u m s t r u c t u r e s c o n s i s t e n t to on l y 1 0 " 3 A, even though the p r e c i s i o n of these numbers was much b e t t e r ( 10 " " -10~ 5 A ) . In l i g h t of t h i s one shou ld suggest tha t the e a r l i e r s t u d i e s tha t used the l i n e a r e x t r a p o l a t i o n have quoted bond l eng ths and ang les to a p r e c i s i o n beyond a c c u r a c y . These comments are d i r e c t e d to the work of Helms and Gordy and Chu and Oka on phosph ine . The s t u d i e s denoted by the f in Tab le 6.13 have e m p i r i c a l l y de termined va lues f o r many of the a cons tan t s 196 and so no f o r c e f i e l d ( to g i ve the harmonic pa r t of the a ' s ) and no l i n e a r e x t r a p o l a t i o n to c o r r e c t for the anharmonic pa r t of the a lphas i s r e q u i r e d . The s t r u c t u r a l parameters p resen ted in these s t u d i e s are of very h igh p r e c i s i o n but they have been c a l c u l a t e d from on ly one i s o t o p i c s p e c i e s . It i s suggested tha t s i m i l a r c a l c u l a t i o n s on d i f f e r e n t i s o t o p i c s p e c i e s would produce s t r u c t u r a l parameters c o n s i s t e n t more on the order of the c o n s i s t e n c y we found fo r ammonia =10" 3 A. T h i s i s the c o n c l u s i o n one draws from the work of Bened ic t and P l y l e r ( r131 ) . In t h e i r work on ammonia they were ab le to make good es t ima tes of the a c o n s t a n t s f o r NH 3 and ND 3 from which they c a l c u l a t e d e q u i l i b r i u m s t r u c t u r a l parameters tha t d i s a g r e e d by .0016 A and . 0 4 ° . From the above d i s c u s s i o n we shou ld conc lude that the e q u i l i b r i u m s t r u c t u r e s quoted by p r e v i ous au tho r s are p robab ly r e l i a b l e on ly to the accu racy we quote fo r our r e s u l t s . Fur thermore the s t r u c t u r e s p resen ted as pa r t of t h i s work, w i th an emphasis on the r z s t r u c t u r e s , a re we l l e s t a b l i s h e d as they are c a l c u l a t e d from four or f i v e i s o t o p i c mo lecu les r a the r than from one or two as in e a r l i e r s t u d i e s . 1 97 H. CHAPTER 7: INTERESTING EXTRAS In t h i s chapte r we s h a l l review two methods of a n a l y s i s of r o t a t i o n a l s p e c t r a that are at v a r i a n c e wi th the method p re sen ted in t h i s t h e s i s . These methods are we l l r ep re sen t ed in the l i t e r a t u r e and s i n ce they have been used p r i m a r i l y in r a t i o n a l i z i n g group V hyd r ide symmetric top s p e c t r a i t shou ld be necessa ry to e s t a b l i s h why we d i d not use them. The f i r s t method i s tha t of O lson as used by Helms and Gordy fo r t h e i r s t u d i e s on phosphine and a r s i n e ( r 2 5 , r 2 6 ) . It was l a t e r used by Poynter and M a r g o l i s in t h e i r study of ammonia (r24) and by K i j i m a and Tanaka in t h e i r a n a l y s i s of the i n f r a - r e d spectrum of PD 3 ( r115 ) . Very r e c e n t l y Ta r rago (r146) has suggested a s i m i l a r scheme as perhaps u s e f u l in the a n a l y s i s of the r o t a t i o n a l s p e c t r a of o ther q u a s i - s p h e r i c a l symmetric tops such as O P F 3 . A l l these methods have in common a way of e v a l u a t i n g r e m p i r i c a l l y from ( f i e l d f r ee ) f requency data a l o n e . As we s h a l l see , to the l e v e l of approx imat ion necessa ry to r a t i o n a l i z e the a v a i l a b l e f requency d a t a , T X K y L Z i s not a de te rminab le parameter . T h i s " f i r s t method" i s d i s c u s s e d f u r t h e r in the s e c t i o n f o l l o w i n g , e n t i t l e d " F a i l u r e s of the R e d u c t i o n s . " The second method we wish to dea l w i th was d e v i s e d to " a c c e l e r a t e " the convergence of the power s e r i e s e x p a n s i o n , in terms of angu la r momentum o p e r a t o r s , of the r o t a t i o n a l energy l e v e l s . The need fo r t h i s " a c c e l e r a t i o n " i s p a r t i c u l a r l y apparent fo r the l i g h t e a s i l y d i s t o r t e d h y d r i d e s (H 2X where X=0, S, Se, Te or XH 3 where X=N, P, As 198 or XH„ where X=C, S i and Ge) where o f t en the number of phenomeno log ica l parameters of our power s e r i e s model necessa ry to " f i t " the spectrum i s on the same order as the number of measured f r e q u e n c i e s . In t h i s c a s e , when the d i s t o r t i o n cons t an t s are l a r g e and slow to converge wi th s u c e s s i v e l y h ighe r degree of angu la r momentum o p e r a t o r , we f i n d o u r s e l v e s in the somewhat embarrass ing p o s i t i o n of a lmost add ing a new f i t t i n g parameter fo r each new data p o i n t . In o rder to a vo id t h i s s i t u a t i o n convergence a c c e l e r a t i o n schemes have been i n c o r p o r a t e d i n t o the s tandard fo rma l i sm of Watson and o t h e r s . T h i s l a t t e r fo rma l i sm was p resen ted in Chapter 1 and has been the method of a n a l y s i s used through-out t h i s t h e s i s . In the a l t e r n a t i v e method, a t tempts are made to a c c e l e r a t e the convergence of the power s e r i e s e x p r e s s i o n f o r the energy l e v e l s by c o n s t r u c t i n g r a t i o n a l f r a c t i o n s , or Pade approx imants ( P . A . ' s ) . Us ing t h i s approach v a r i o u s groups have made s t u d i e s of PH 3 ( r147 ) , H 2 0 ( r148 ) , H 2 S (r149) and the i n v e r s i o n spectrum of ammonia ( r150 ) . The method i s reviewed somewhat e x t e n s i v e l y in a l a t e r s e c t i o n e n t i t l e d "On the use of R a t i o n a l F r a c t i o n Approximants in the F i t t i n g of R o t a t i o n a l S p e c t r a . " There i t w i l l be found tha t the method used in the above s t u d i e s i s u n s a t i s f a c t o r y . R e d u c t i o n s , in the s p i r i t of Watson and Chapter 1, w i l l be shown to be necessa r y in order to d e f i n e un ique l y the approx imants these e a r l i e r s t u d i e s use . As we s h a l l see the need f o r these r e d u c t i o n s w i l l e x p l a i n the " u n i t y " c o r r e l a t i o n s r epo r t ed 1 9 9 between v a r i o u s "Pade" parameters in the above s t u d i e s . 1. FAILURES OF THE REDUCTIONS Two of the r e d u c t i o n s p resen ted in Chapter 1 become r a the r unwie ldy in s p e c i f i c l i m i t s . The asymmetric top A r e d u c t i o n i s undef ined i n the symmetric top l i m i t and the symmetric top r educ t i on i s unde f ined i n the s p h e r i c a l top l i m i t . In both cases the u n i t a r y t r a n s f o r m a t i o n s r e s p o n s i b l e f o r the r e d u c t i o n s c o n t a i n terms 1/(V-W) where V and W are r o t a t i o n a l c o n s t a n t s . In the l i m i t i n g cases these r o t a t i o n a l cons t an t s are d e f i n e d as equa l and so the r e d u c t i o n becomes i n a p p r o p r i a t e . For mo lecu les tha t are c l o s e to these l i m i t i n g cases use of these r e d u c t i o n s r e s u l t s i n some parameters of a g iven degree be ing very much l a r g e r than o t h e r s ; t h i s r e f l e c t s the c o n t r i b u t i o n of the " r e d u c t i o n " to tha t p a r t i c u l a r parameter . Fur thermore t h i s can mean tha t o f f - d i a g o n a l mat r ix e lements can become very l a r g e . For asymmetric tops the problem wi th the A r e d u c t i o n in the symmetric top l i m i t was so l v ed wi th the S r e d u c t i o n (see Chapter 1 ) . Strow (r 151 ) has made a compar ison of the A and S r e d u c t i o n s in v a r i o u s r e p r e s e n t a t i o n s and g i v e s a q u a n t i t a t i v e method of de t e rm in i ng i n a p p r o p r i a t e r e d u c t i o n s and r e p r e s e n t a t i o n s . For the s t u d i e s p resen ted in t h i s t h e s i s a l l th ree r e p r e s e n t a t i o n s ( r i g h t handed systems) in both r e d u c t i o n s produced e q u a l l y good f i t s so that cho i c e of f i n a l r e p r e s e n t a t i o n and r e d u c t i o n c o u l d be made a r b i t r a r i l y . 200 We s h a l l now c o n s i d e r symmetric t o p s . R e c a l l from s e c t i o n 1.2c that to s i x t h degree the p r e - r e d u c t i o n C^y symmetric top h a m i l t o n i a n c o n t a i n e d d i a g o n a l mat r ix e l ements , tha t were power s e r i e s expans ions of the d i a g o n a l J and J o p e r a t o r s , p l u s e lements th ree o f f the d i a g o n a l ( in z K ) w i th c o e f f i c i e n t r /4 ( a c t u a l l y to s i x t h degree we X X X z shou ld a l s o i n c l u d e and e R of Watson 's equa t ion 25 (r12) but t h i s omiss ion does not a f f e c t the presen t a rgument ) , p l u s f i n a l l y terms s i x o f f the d i a g o n a l w i th c o e f f i c i e n t h'3. F u r t h e r , a u n i t a r y t r a n s f o r m a t i o n was d e s c r i b e d tha t a l l owed the h a m i l t o n i a n to be r e-wr i t t en in a s tandard reduced form. H r = B x J ' + ( B 2 - B x ) J z ' - D j ( J 2 ) 2 - B J K J 2 J z 2 - D K J z « + H J ( J 2 ) 3 + H J K ( J 2 ) V + H K J J 2 V + H K V + h 3 ( J *+ J 6 ) [7 .1] where (r12) B x = K~2^ ' B z = B z " 7 2 * ' D J = D j " 1 6 * ' D J K = D J K " 1 2 6 * > D R = D^-130£ , H a = H j - 2 | , H J R = HJ R + 1 8 £ , H K J = H K J " 3 0 ^ ' H K = H K + 1 4 * ' H 3 = H**~* [ 7 ' 2 ] with r 2 * - i 6 ( " " ) [ 7 - 3 ] z x If we count the number of parameters be fo re the u n i t a r y t r a n s f o r m a t i o n we see there are e l even : 10 "p r imed " c o n s t a n t s p l u s T (or £ ) . A f t e r the u n i t a r y XXXZ t r a n s f o r m a t i o n there are on l y 10 pa ramete rs , the unprimed c o n s t a n t s . S ince a u n i t a r y t r a n s f o r m a t i o n cannot a f f e c t the e igenva lues of a h a m i l t o n i a n , the energy l e v e l s and hence 201 t r a n s i t i o n f r e q u e n c i e s depend on on ly 10 independent pa ramete rs . T h i s a l s o means that we can determine on ly 10 independent parameters e m p i r i c a l l y . Which 10 l i n e a r comb ina t ions o f . t h e 11 symmetry a l lowed (p re- reduc t i on ) c o n s t a n t s we choose to determine i s comp le te l y a r b i t r a r y . From equa t ions (e7.2) we can see that the s tandard r e d u c t i o n makes £ (and hence r ) i nde te rmina te in order to reduce s xxxz the number of parameters to the r e q u i r e d 10. The advantage of t h i s r e d u c t i o n i s tha t i t agrees wi th the asymmetric top S r e d u c t i o n in the symmetric top l i m i t . Of course o ther r e d u c t i o n s are p o s s i b l e . Helms and Gordy ( r25 , r26 ) appear to d e f i n e H' as equa l to some i nde te rm inab l e c o n s t a n t . In a l a t e r work on ammonia we i n f e r tha t Poynter and M a r g o l i s (r24) " f o l l o w i n g " Helms and Gordy reduce t h e i r h a m i l t o n i a n by s e t t i n g h'3 to z e r o . Us ing the above r e d u c t i o n s these au thors and o thers who have adopted t h e i r approach e m p i r i c a l l y determine £ and then us ing equa t i on (e7.3) va lues they l a b e l T . The problem here i s X X X Z tha t the va lues of r c a l c u l a t e d in t h i s way are xxxz 1 dependent on the r e d u c t i o n used . T h i s i s i d e n t i c a l to the problem of the r e d u c t i o n dependent parameters of both the A and S r e d u c t i o n s f o r asymmetric tops (see s e c t i o n ( 1 . 2 b ) ) . In gene ra l we w i l l f i n d that the e m p i r i c a l l y determined £ 's w i l l be equa l to the £ d e f i n e d by equat ion (e7.3) p l u s some o ther r e d u c t i o n dependent te rms. These r e d u c t i o n dependent terms w i l l be c a l c u l a b l e from i n fo rma t i on other than ( f i e l d f r ee ) f requency measurements. The above argument w i l l a l s o 202 be t rue fo r a l l the pr imed parameters so tha t in gene ra l the " t r u e " p r e - r e d u c t i o n parameters a re not de te rminab le from f requency data a l o n e . We are now in a p o s i t i o n to reason w h y ' i t was dec ided to r e - f i t the p u b l i s h e d PD 3 spectrum to a s tandard reduced h a m i l t o n i a n . The reason was to a v o i d the problem of r e l a t i n g parameters of d i f f e r e n t r e d u c t i o n s . By r e - f i t t i n g the PD 3 spectrum we would have the same d e f i n i t i o n fo r a l l the parameters fo r a l l the i s o t o p i c s p e c i e s . A l l the r e d u c t i o n s are a n a l y t i c a l l y e q u i v a l e n t , a l t hough some might " f i t " l i m i t e d data se t s b e t t e r than o t h e r s , so i t shou ld s t i l l remain d i f f i c u l t to choose one r e d u c t i o n over a n o t h e r . The advantage of the s tandard r e d u c t i o n (the un-primed cons t an t s of equa t ion (e7.2) ) i s tha t fo r most mo lecu les £ i s very sma l l ( that i s , B^-B^ i s l a r g e ) and we can make the approx ima t ion tha t £=0. For mo lecu les of t h i s type the " t r u e " p r e - r e d u c t i o n parameters a re then very c l o s e l y g iven by the s tandard r educ t i on f i t t i n g pa ramete rs . A problem wi th t h i s argument i s that in the s p h e r i c a l top l i m i t the denominator in equa t ion (e7.3) goes to ze ro making £ r a the r l a r g e . T h i s i s of p a r t i c u l a r conce rn here as we choose to do exper iments on mo lecu les near t h i s l i m i t in o rder to enhance the r o t a t i o n a l borrowing d i p o l e moment c o n t r i b u t i o n to the f o r b i d d e n r o t a t i o n a l t r a n s i t i o n i n t e n s i t i e s (see s e c t i o n 1.4a) . Ta r rago ( r l46 ) has suggested tha t fo r mo lecu les where B' ~ B' tha t we t r y to f i t our s p e c t r a to terms up to f o u r t h 203 degree : B' B' , D ' , D ' , D' and Ir I. Ta r rago i n c l u d e s A L U J l \ I\ X X X Z the t h r e e - o f f the d i a g o n a l r terms e x p l i c i t l y in h i s X X X Z h a m i l t o n i a n and then d i a g o n a l i z e s the whole m a t r i x . Of c o u r s e , i m p l i c i t in t h i s method i s the u n i t a r y t r a n s f o r m a t i o n r e s p o n s i b l e fo r the r e d u c t i o n g i ven by equa t ions ( e7 .1 )- (e7 .3 ) and t h e r e f o r e i t i s an i d e n t i c a l method to f i t to B' , B ' , D ' , D ' , D' and £ w i th a l l the pr imed s e x t i c cons t an t s f i x e d at z e r o . Us ing h i s method Tar rago was ab l e to produce a very r easonab le f i t to p u b l i s h e d OPF 3 data ( r152 ) . The advantage of h i s method i s tha t w i th on l y s i x parameters he c o u l d produce a f i t tha t compared we l l w i th a f i t u s ing a s t andard reduced h a m i l t o n i a n r e q u i r i n g 9 parameters - i . e . s e x t i c terms were i n c l u d e d . The reason T a r r a g o ' s method works so we l l shou ld be c l e a r . For near s p h e r i c a l tops where B'z ~ B^ (OPF 3 AB' = 2 1 4 MHz) the term £ can become very l a r g e and be the dominant c o n t r i b u t i o n to the unprimed s tandard r e d u c t i o n s e x t i c pa ramete rs . In these cases f i t t i n g to £ shou ld a l low us to account fo r much of the s e x t i c c o n t r i b u t i o n to the f r e q u e n c i e s . T a r r a g o ' s r e d u c t i o n i s e s s e n t i a l l y the same as the " o t h e r " r e d u c t i o n s mentioned p r e v i o u s l y . H i s a b i l i t y to f i t the OPF 3 spectrum to fewer parameters p robab l y r e f l e c t s the u s e f u l n e s s of some r e d u c t i o n s over o the r s f o r some data s e t s . S ince AsD 3 i s a l s o a near s p h e r i c a l top (AB' = 4800 MHz) a study was made us ing the above approach in hopes of 204 f i t t i n g t h i s spectrum to a lower number of pa ramete r s . No improvements in the f i t s were found and the same number of parameters was r e q u i r e d . A s i m i l a r study of PD 3 gave i d e n t i c a l r e s u l t s . It shou ld now prove i n s t r u c t i v e to c o n s i d e r the magnitude of the £ te rm. G iven equa t i on ( e 7 . 3 ) , Tab l e 6.7 and the r o t a t i o n a l cons t an t s of Tab l e s 6 .10 , 6.11 and 6.12 we can c o n s t r u c t the f o l l o w i n g t a b l e . Tab le 7.1 E s t i m a t i o n of the c o n t r i b u t i o n of £ to h 3 % c o n t . Ref . 39 111 78 t h i s work 80 94 * e s t ( H 2 ) (Hz) PH 3 PD 3 AsH 3 33.3 3.6 43. 1 86.3 4.63 53.81 We see tha t fo r the h e a v i e r , more s p h e r i c a l h y d r i d e s , the assumpt ion that h 3 i s sma l l becomes r e a s o n a b l e . I t shou ld a l s o be apparent that fo r ammonia, h 3 c o u l d make a s i g n i f i c a n t c o n t r i b u t i o n to h 3 . T h i s i s seen i n the study by Poynter and M a r g o l i s (r24) where f o l l o w i n g the method of O lson as o u t l i n e d by Helms and Gordy ( r25 , r26 ) they i m p l i c i t l y assume h 3 to be ze ro and c a l c u l a t e a va lue of r f o r NH 3 of 0.44 MHz in d isagreement w i th the va lue p resen ted in Tab le 6.7 of 11.26 MHz. . We suspect t h i s d isagreement to be a r e s u l t of u s i n g the same symbols to r ep resen t " s i m i l a r " parameters in d i f f e r e n t r e d u c t i o n s . For 205 a more d e t a i l e d f o r m u l a t i o n of h 3 see r e f . ( r l 3 9 ) where t h e i r A 3 i s equa l to h 3 / 2 . F i n a l l y , through-out t h i s t h e s i s , T has been XXXZ l a b e l e d as not de te rminab le from s t a t i c e l e c t r i c f i e l d f r ee f requency measurements. A f t e r t h i s d i s c u s s i o n on how not to determine T one might ask how one might . One way i s to XXXZ measure the d i s t o r t i o n d i p o l e moment. R e c a l l from equat ion (e1.33) tha t the r o t a t i o n a l c o n t r i b u t i o n to the d i s t o r t i o n d i p o l e moment depends on T . T h i s d i p o l e moment can be XXXZ determined from abso lu t e i n t e n s i t y measurements (e1.37) or from Stark s h i f t measurements (us ing s t a t i c e l e c t r i c f i e l d s ) ( e 1 . 4 4 ) . The d i f f i c u l t y w i th t h i s c a l c u l a t i o n i s the d i s t o r t i o n d i p o l e moment c o n t a i n s a v i b r a t i o n a l borrowing c o n t r i b u t i o n (e1.35) — a c o n t r i b u t i o n which i s d i f f i c u l t to determine and hard to separa te from the r o t a t i o n a l , c o n t r i b u t i o n ( e1 .36 ) . T h i s i n s i g h t i n t o the behav iour of the r e d u c t i o n s in the v a r i o u s l i m i t s might h e l p in j u s t i f y i n g the concern made e a r l i e r tha t fo r the type of mo lecu les we are i n t e r e s t e d i n , the h ighe r degree d i s t o r t i o n cons t an t s can be q u i t e l a rge r e s u l t i n g in a slow convergence of our power s e r i e s e x p a n s i o n . Ways of a c c e l e r a t i n g t h i s convergence are d i s c u s s e d in the next s e c t i o n . 206 2. ON THE USE OF RATIONAL FRACTION APPROXIMANTS IN THE  FITTING OF ROTATIONAL SPECTRA A gene ra l concern when f i t t i n g molecu la r s p e c t r a to power s e r i e s e x p r e s s i o n s i s where to t runca te the i n f i n i t e s e r i e s so as to accommodate the f i n i t e data s e t s . The usua l p rocedure i s to i n c l u d e e m p i r i c a l parameters to as h igh a degree as necessa ry to account fo r the s p e c t r a l f r e q u e n c i e s to w i t h i n t h e i r expe r imen ta l u n c e r t a i n t y . The problem fo r l i g h t mo lecu les (where h ighe r degree e f f e c t s can be q u i t e l a rge ) i s that we sometimes must go to very h igh degree to i n c l u d e a l l s i g n i f i c a n t d i s t o r t i o n c o n t r i b u t i o n s to the f r e q u e n c i e s . The d i f f i c u l t y i s tha t i n t r o d u c i n g terms of h ighe r degree i n c r e a s e s the number of e m p i r i c a l parameters we must de te rm ine . T h i s i n ' t u r n means we need b igge r data s e t s . The problem i s f u r t h e r aggrava ted when these . " e x t r a " data are u n a v a i l a b l e , or e s p e c i a l l y when the on l y new data a v a i l a b l e are fo r " h i g h e r " J t r a n s i t i o n s , where even h igher degree d i s t o r t i o n e f f e c t s may be s i g n i f i c a n t . T h i s problem i s bad enough fo r l i n e a r mo lecu les where the f i t t i n g equa t i ons are power s e r i e s in on l y one v a r i a b l e J , but fo r symmetric t o p s , where we f i t w i th two v a r i a b l e s e r i e s , go ing to h ighe r degree d r a m a t i c a l l y i n c r e a s e s the number of new parameters we must determine because of a l l the J , K c r o s s te rms . For symmetric t o p s , in o rder to "de te rm ine " some parameters o ther parameters are o f t e n c o n s t r a i n e d to z e r o . The c h o i c e of which parameters to f i x to zero i s i n e v i t a b l y made through t r i a l and e r r o r m i n i m i z i n g of parameter 207 c o r r e l a t i o n s and s tandard d e v i a t i o n s . These t r i a l and e r r o r p rocedures are necessary because at h igh degree the p h y s i c a l i n t e r p r e t a t i o n of the f i t t i n g parameters i s r a the r nebulous and there i s no a priori p h y s i c a l reason f o r l e a v i n g c e r t a i n c o n s t a n t s , over o t h e r s , out of a f i t . Of course the e f f e c t of c o n s t r a i n i n g parameters can be to modify how we i n t e r p r e t the o ther e m p i r i c a l c o n s t a n t s . How much of a problem a l l t h i s i s f i n a l l y depends on what we are t r y i n g to a c c o m p l i s h . If we wish s imply to f i t da ta in order to p r e d i c t new data then h ighe r order expans ions w i th c e r t a i n parameters c o n s t r a i n e d to zero shou ld seem r e a s o n a b l e . I f , however we wish to r e t a i n some semblance of the p h y s i c a l o r i g i n in these f i t t i n g parameters then c l e a r l y something new i s needed. The problem then , i s t w o f o l d . F i r s t we would l i k e a s o l u t i o n to the problem of s l ow l y conve rg ing s e r i e s f o r the s p e c t r a l f r e q u e n c i e s of l i g h t mo lecu les and second , we shou ld l i k e to ma in ta in some degree of p h y s i c a l i n t e r p r e t a t i o n fo r the parameters we e m p i r i c a l l y de t e rm ine . In numer i ca l a n a l y s i s one i s o f t en concerned wi th convergence . Of s p e c i a l impor tance , e s p e c i a l l y in computer s c i e n c e , a re methods by which we can a c c e l e r a t e the convergence of s low ly conve rg ing sequences . Shanks (rl53) d i s c u s s e s a f am i l y of n o n - l i n e a r sequence-to-sequence t r a n s f o r m a t i o n s which are found to e f f e c t such an a c c e l e r a t i o n . Of r e l e vance to s p e c t r o s c o p i s t s i s the set of t r a n s f o r m a t i o n s known c o l l e c t i v e l y as r a t i o n a l " f r a c t i o n s or Pade approx imants . As we s h a l l see these Pade approximants 2 0 8 c o u l d prove u s e f u l in a c c e l e r a t i n g the convergence of s low ly conve rg ing angu la r momentum power s e r i e s f requency e x p r e s s i o n s . The u s e f u l n e s s of r a t i o n a l f r a c t i o n approximants as a method of a c c e l e r a t i n g the convergence of s e r i e s i s we l l e s t a b l i s h e d in such f i e l d s as s c a t t e r i n g t heo r y , f i e l d t h e o r y , computer s c i e n c e , c r i t i c a l phenomena and the study of con t i nued f r a c t i o n s . They have been used in chemis t r y in t h e o r e t i c a l accounts of l i q u i d s ( r154 ) . However in r o t a t i o n a l spec t roscopy r a t i o n a l f r a c t i o n approximants are s t i l l q u i t e new. Pade approximants d e r i v e t h e i r name from Henr i Pade, who in 1892 (r155) p u b l i s h e d an a r t i c l e conce rn ing the approximate r e p r e s e n t a t i o n of a f u n c t i o n by r a t i o n a l f r a c t i o n s . Much l a t e r Shanks, in 1955 ( r153 ) , gave a h e u r i s t i c m o t i v a t i o n f o r , and showed many of the advantages o f , Pade ' s method. S ince then the re has been a growing i n t e r e s t in Pade approximants as an a l t e r n a t i v e to the s tandard r e p r e s e n t a t i o n of f u n c t i o n s by Tay l o r s e r i e s . For a more complete d i s c u s s i o n of how and where Pade approximants are be ing used one i s r e f e r r e d to the f o l l o w i n g tex tbooks and books of c o l l e c t e d papers ( r156-r160) . In o rder to get a f e e l f o r what a Pade approximant i s and why i t works i t i s i n s t r u c t i v e f i r s t to c o n s i d e r a numer i ca l techn ique fo r f i n d i n g the roo t s of equa t i ons by s u c c e s s i v e approx imat ion known as A i t k i n ' s S 2 p r o c e s s , or " e x t r a p o l a t i o n to the l i m i t " . Here we choose a gene r a l form 209 fo r an equa t ion to be so l v ed by i t e r a t i o n g(x)=0 It i s o f t e n more conven ien t to w r i t e t h i s as x=G(x) where G i s a genera tor tha t g i v e s the f u n c t i o n a l dependence of the i t e r a n c e s . T h i s i s u s u a l l y expressed in r e c u r s i o n form G ( x L ) = X L + 1 where x L r ep r e sen t s v a lues of x produced by some i t e r a t i v e i p r o c e s s . For the t rue root r, G(r )=r 0 A p p l y i n g the T a y l o r expans ion and t r u n c a t i n g a f t e r the l i n e a r term one has G ( x L ) = G(r ) + ( x L - r ) G ' ( r ) or r " XL+1 = G ' ( r ) ( r _ X L ) and in the same way we a l s o have r - x L = G ' ( r ) ( r - x L _ 1 ) D i v i d i n g these l a s t two equa t ions and s o l v i n g f o r r y i e l d s X L+1 X L-1 " X L 210 or where A x L = * L + 1 - x L > T h i s d e r i v a t i o n of equa t ion (e7.4) i s e q u i v a l e n t to drawing a s t r a i g h t l i n e through the p o i n t s ( x T , A x r ) and (x T , , A x T ,) on a (x ,Ax) a x i s system and Li i_i Li 1 Li I e x t r a p o l a t i n g to Ax = 0 ( r 1 6 1 ) . Equa t i on (e7.4) i s the f a m i l i a r 8 2 fo rmula of A i t ken ( r162 ) . In t h i s c a s e , g i ven some i t e r a t i v e p rocess that produces va lues of x each of which i s c l o s e r to some r e a l root than the l a s t , we can take any th ree s u c c e s s i v e i t e r a n t s and put them i n t o equa t ion (e7.4) and produce a " b e t t e r " approx imat ion to the r e a l root than i f we s imp ly took the l a s t of our s u c c e s s i v e i t e r a n t s . S e r i e s fo r which equa t ion (e7.4) i s a good approx imat ion are o f t en c a l l e d " g e o m e t r i c " s e r i e s . In d e f i n i n g Pade approximants we c o n s i d e r s u c c e s s i v e approx imat ions ( A L : L = 0 , 1 , 2 , . ..} genera ted by the p a r t i a l sums of a power s e r i e s expans ion of the f u n c t i o n g ( x ) . A L = i § 0 a i x l [ 7 - 5 ] where g(x) i s equa l to A L in the l i m i t as L goes to i n f i n i t y . For convergent s e r i e s , s u c c e s s i v e A L ' s w i l l be c l o s e r and c l o s e r to the r e a l va lue of the f u n c t i o n g ( x ) . Given then any th ree suc ces s i v e expans ion c o e f f i c i e n t s of equa t ion (e7.5) and A i t k e n ' s 5 2 p rocess (e7.4) we can " e x t r a p o l a t e " from i n fo rma t i on i nhe ren t in these th ree 21 1 c o e f f i c i e n t s to h igher order by forming the r a t i o B = A L - 1 A A L " V ^ L - I AAT AA [7.6] L-1 where AA r = A T . - A T . U s u a l l y i f we knew th ree s u c c e s s i v e T a y l o r expans ion c o e f f i c i e n t s they would be the f i r s t t h r e e : a 0 , a , and a 2 . In t h i s case in the above r a t i o L would be equa l to one. When we know more than th ree T a y l o r c o e f f i c i e n t s we have more A L ' s to e x t r a p o l a t e from and in t h i s gene r a l case we have a gene ra l form approximant ( r ! 5 3 , r l 5 8 ) , B. M,L L-M AA L-M AA L-M+1 AA L-1 AA L-M AA L-M+1 AA L-1 L-1 AA L-1 AAT AA L-1 AA. L AAI AA L+1 AA L+M- 1 AAT AA L+1 AA L+M-1 [7.7] where s p e c i a l d e f i n i t i o n s are invoked whenever the denominator v a n i s h e s . 212 S u b s t i t u t i n g the d e f i n i t i o n of A T (e7.5) i n t o equa t ion { e l .1) and expanding the de te rminants we f i n d B M L can be w r i t t e n as the r a t i o of two p o l y n o m i a l s ; tha t i s , as a r a t i o n a l f r a c t i o n . PT (x) B M , L ' [ L / M ] = Q^TT f 7 ' 8 ^ Where P L ( x ) i s a po l ynomia l of degree at most L, and Q M a po l ynomia l of degree at most M, in the v a r i a b l e x P L =a ?0 P a a n d Q M = 1 + ^ ^ [ 7 ' 9 ] where the s tandard n o r m a l i z a t i o n q 0=1 has been adopted ( r158 ) . The c o e f f i c i e n t s p and q „ can be c a l c u l a t e d from a p equa t i on (e7.7) or in an e n t i r e l y e q u i v a l e n t way from A<x> " TTT^T = 0 ( x L + M + 1 ) [7 .10] where O means " terms of o r d e r " . T h i s i s the s tandard d e f i n i t i o n of the [L/M] Pade approximant to the T a y l o r s e r i e s expans ion (e7.5) ( r158 ) . M u l t i p l y i n g both s i d e s of equa t i on ( e7 . l 0 ) by Q M and equa t ing powers of x we f i n d , a 0 = Po a i + a 0 q i = P i a 2 + a i q 1 + a 0 q 2 = p 2 a L + a L - l q i + + a ° q L = P L a L + 1 + a L q i + + a L - M + l S M = ° 213 a L+M ,+a L+M-1 q i + = 0 w i th a^ = 0" i f i<0 and q . = 0 i f j>M. [7 .11] C l e a r l y the number of Pade c o e f f i c i e n t s to be de te rm ined , L+M+1 (as q 0 = 1 ) , must be equa l to the number of known T a y l o r expans ion c o e f f i c i e n t s . T h i s i s an important po in t tha t the t r a n s f o r m a t i o n of A(x) (e7.5) to a r a t i o n a l f r a c t i o n P L ( x ) / Q M (x ) i s a one to one mapping. T h i s p o i n t w i l l concern us aga in when we i n v e s t i g a t e approximants in two v a r i a b l e s . As an example of how u s e f u l these convergence a c c e l e r a t i o n methods can be we c o n s i d e r the s e r i e s For x=1 t h i s s e r i e s sums to TT/4 ( r153 ) . In t h i s case we have the L e i b n i t z s e r i e s T h i s i s a ve ry poor a l g o r i t h m fo r rr/4 because i t i s very slow to conve rge . In o rder to get TT/4 c o r r e c t to e i gh t d i g i t s ( tha t i s i g n o r i n g terms l e s s than 10~ 7) r e q u i r e s 5 m i l l i o n te rms ! On the o ther hand, from jus t the f i r s t 10 terms of the s e r i e s and equa t ion (e7.6) we can c a l c u l a t e TT/4 to the same e i g h t d i g i t accuracy ( r153 ) . I t shou ld be c l e a r tha t the f i r s t few terms of t h i s s e r i e s possess much more i n f o r m a t i o n than the T a y l o r s e r i e s would have us b e l i e v e . 4 3 5 7 214 U n f o r t u n a t e l y , in spec t ro s copy we do not as yet have a c cu r a t e a n a l y t i c a l methods of p roduc ing the r o t a t i o n a l f requency T a y l o r s e r i e s expans ion c o e f f i c i e n t s . These c o e f f i c i e n t s are u s u a l l y de te rmined e m p i r i c a l l y by l e a s t squares f i t t i n g of observed f r e q u e n c i e s to t r unca t ed power s e r i e s . These e m p i r i c a l T a y l o r c o e f f i c i e n t s w i l l c o n t a i n e f f e c t s due to the convergence and t r u n c a t i o n problems of the T a y l o r s e r i e s so i t i s not c l e a r that forming Pade approximants w i th these T a y l o r c o e f f i c i e n t s w i l l produce any th ing b e n e f i c i a l . I t would seem then a b e t t e r idea to f i t to a Pade approx imant , r a the r than a t r unca t ed T a y l o r s e r i e s , and take advantage of these s p e c i a l p r o p e r t i e s of Pade approx imants . To see how Pade approx imants c o u l d be used in f i t t i n g p rocedures we s h a l l c o n s i d e r some nove l p h y s i c a l phenomenon tha t d u r i n g some measurement p roces s y i e l d s obse r vab l e s g i ven by the f u n c t i o n a l form f (x ) = (1 + 4 X ) 1 / 2 [7 .12] = 1 + 2x - 2x 2 + 4x 3 - [7.13] Now suppose, as i s o f t e n the c a s e , we are unaware that t h i s God-given f u n c t i o n a l form i s r e s p o n s i b l e fo r our o b s e r v a t i o n s . Given a da ta set as a f u n c t i o n of one parameter , x, we might t r y to f i t t h i s data to a power s e r i e s in x. For t h i s d i s c u s s i o n we w i l l concen t r a t e on de t e rm in ing the f i r s t th ree power s e r i e s parameters a 0 , a , and a 2 in 215 f (x) = a 0 + a,x + a 2 x : [7 .14] where f T s tands fo r " t r u n c a t e d T a y l o r s e r i e s " . The [1/1] Pade approximant to equa t ion (7.14) i s a 2 " f p U ) = a 1 - a. [7 .15] 1 -To produce a data set we use equa t ion (e7.12) fo r va lues of x=0.1 to 1.0 in s teps of 0 . 1 . We a s s i g n a r b i t r a r y u n c e r t a i n t i e s to a l l the data of 0.01 and weight each datum as the i n v e r s e square of t h i s u n c e r t a i n t y . The r e s u l t s of l e a s t squares f i t t i n g f T and f p to t h i s data set are shown below a long wi th the " t r u e " T a y l o r expans ion c o e f f i c i e n t s (el.13). Tab le 7.2 Comparison of t r unca t ed T a y l o r s e r i e s f T v e r sus Pade f r a c t i o n fp in f i t t i n g data produced from equat ion 7.12 a 0 a , a 2 v 2 T 1.037(7) 1.583(30) -0.389(25) 2.28 1.022(4) 1.715(23) -0.714(35) 0.63 T r u t h 1 .000 2.000 -2.000 0.000 where the numbers in pa ren theses are s t anda rd d e v i a t i o n s in the l e a s t s i g n i f i c a n t f i g u r e s and x 2 i s the weighted sum of the squares of the d e v i a t i o n s . 216 The r e s u l t s in Tab le 7.2 are e n c o u r a g i n g . I t would seem that in t h i s case we have found a method tha t addresses the two fo ld problem of the convergence of the s e r i e s expans ion and the p h y s i c a l i n t e r p r e t a t i o n of the f i t t i n g paramete rs . In the f i r s t p l ace x 2 i s rough ly 4 t imes s m a l l e r f o r the Pade f i t than fo r the t r u n c a t e d T a y l o r f i t . T h i s would be because the Pade approx imant , s i n ce i t converges f a s t e r , b e t t e r accounts fo r h ighe r order e f f e c t s making i t a b e t t e r model f o r the d a t a . The other important p o i n t of Tab le 7.1 i s tha t the e m p i r i c a l parameters of the Pade f i t are b e t t e r de termined and c l o s e r to the " t r u e " a n a l y t i c a l v a lues (see equa t ion e 7 . 1 3 ) . T h i s i s , of c o u r s e , j u s t what we would want fo r spec t roscopy as we l i k e to a t t a c h p h y s i c a l meaning to the v a r i o u s expans ion pa ramete rs . As a f u r t h e r i n s i g h t i n t o the advantages of Pade approximants over t r unca t ed T a y l o r s e r i e s we n o t i c e the [1/1] Pade approximant to equa t ion (e7.13) i s I f we expand t h i s r a t i o we f i n d [ 1 / I ] f ( x ) = 1 + 2x - 2x 2 + 2x 3 [7 .17] In order to form the r a t i o n a l f r a c t i o n we on l y needed to know the f i r s t three T a y l o r c o e f f i c i e n t s , T runca ted Tay l o r f (x ) = 1 + 2x - 2x 2 [7 .18] 217 Comparing equa t i ons ( e7 . l 7 ) and ( e7 . l 8 ) we n o t i c e that the Pade approximant d i f f e r s from the t rue expans ion ( e7 . l3 ) by on l y = 2 x 3 , whereas the t r unca t ed T a y l o r s e r i e s d i f f e r s by = 4 x 3 , rough ly twice the e r r o r of the Pade f r a c t i o n . We can see t h i s in a gene ra l way by p a r t i a l l y expanding equa t ion (e7.15) a l x3 a 1 f p = a 0 + a,x + a 2 x 2 + 1 a2 1 - — x a. The beauty of the Pade approximant i s that w i th on ly a f i n i t e number of parameters we can attempt to account fo r h ighe r degree e f f e c t s wi th the " h i g h e r " degree components i m p l i c i t in the r a t i o n a l f r a c t i o n . In gene r a l the most e f f i c i e n t , or the most p o w e r f u l , Pade approx imants are those where L=M, the s o - c a l l e d d i a g o n a l approximants ( r163 ) . A l s o fo r a s p e c i f i c type of s e r i e s , known as a " s e r i e s of S t i e l t j e s " , the [N/N] and [N-1,N] Pade approximants are upper and lower l i m i t s r e s p e c t i v e l y ( r 1 5 8 , r 1 6 4 ) . L i m i t s of t h i s so r t would c e r t a i n l y be u s e f u l f o r p r e d i c t i o n s . With a l l the advantages of Pade approx imants , why then i s t h e i r use not more common? There are p robab l y two answers f o r t h i s . F i r s t , even though Pade approximants have been around f o r a long time i t i s on l y in the. l a s t 10-15 years tha t we can see an ex t ens i v e i n t e r e s t in t h e i r a p p l i c a t i o n . Second, the re are c e r t a i n f u n c t i o n s fo r which great care must be taken when c o n s t r u c t i n g approx imants . For i n s t a n c e , 218 i f i n s t e a d of equat ion (e7.12) we had b u i l t a data set from the f u n c t i o n a l form e then fo r some va lues of x the t r u n c a t e d T a y l o r s e r i e s would have produced a b e t t e r f i t than the Pade f r a c t i o n . Cons ide r the exact [1/1] Pade approx imant to the T a y l o r expans ion of e r , i 2 + x [ 1 / I ] e x = T^-TL We see tha t fo r x=2 t h i s [1/1] Pade approximant has a p o l e . If we were e m p i r i c a l l y t r y i n g to determine the Pade c o e f f i c i e n t s in the v i c i n i t y of x=2 we would have some t r o u b l e . A l s o , fo r x g r ea t e r than 2 the [1/1] approximant c h a n g e s . s i g n whereas the f u n c t i o n does n o t . T h i s does not mean tha t a Pade approximant i s a poor a l g o r i t h m fo r e fo r a l l x ( f o r i ns tance see equa t i on 1.29 of r e f e r e n c e ( r158)) but r a t h e r tha t we must be wary of t h e i r a p p l i c a b i l i t y in v a r i o u s r e g i o n s , (see Gammel (r165) fo r t e chn iques fo r h a n d l i n g these p o l e s . ) Now we r e-cons ide r the s e r i e s fo r rr/4 and the s e r i e s g i ven by equa t ion ( e7 .13 ) . In these s e r i e s s u c c e s s i v e terms have o p p o s i t e s i g n s . These are examples of s e r i e s of S t i e l t j e s ( r l 5 8 ) ; s e r i e s fo r which Pade approximants are most a p p r o p r i a t e . For s e r i e s of t h i s type i t can be shown that a l l the po l e s of the Pade approximants are on the nega t i v e r e a l a x i s - in o ther words, f o r p o s i t i v e x there are no p o l e s ( r l 5 8 ) . A l s o , a long w i th the we l l d e f i n e d upper and lower l i m i t s ment ioned e a r l i e r , s e r i e s of t h i s so r t have w e l l e s t a b l i s h e d convergence p r o p e r t i e s ( r158 ) . What i s 219 p a r t i c u l a r l y encourag ing i s that there i s c o n s i d e r a b l e ev idence sugges t ing the d i s t o r t i o n c o r r e c t i o n s fo r d i a tomic r o t o r s a l s o change s ign w i th s u c c e s s i v e order ( r123 ) . If t h i s be the case then c e r t a i n l y f o r d i a tom i c s Pade f r a c t i o n s r a the r than t runca t ed T a y l o r s e r i e s would indeed be the b e t t e r cho i c e fo r e m p i r i c a l f i t t i n g f u n c t i o n s . In the d i s c u s s i o n so f a r we have been concerned w i th f i t t i n g schemes fo r f u n c t i o n s of one v a r i a b l e . We have seen tha t a Pade approximant can o f t e n b e t t e r r ep resen t a f u n c t i o n in one v a r i a b l e then can a t r unca t ed T a y l o r s e r i e s . The reason fo r t h i s , we have seen , i s wi th a Pade approximant we ana l yze how s u c c e s s i v e terms in a T a y l o r s e r i e s h e l p the Tay l o r s e r i e s to converge , and then e x t r a p o l a t e the s e r i e s to a va lue c l o s e r to the " t r u t h " . F i n a l l y i t has been suggested that Pade approximants may be very a p p r o p r i a t e fo r r o t a t i o n a l f r e q u e n c i e s as t h e i r power s e r i e s expans ions may emulate s e r i e s of S t i e l t j e s , s e r i e s fo r which Pade approximants have many advantages . S t i l l , the d i s c u s s i o n ' t i l l now i s on l y g e n e r a l l y a p p l i c a b l e to l i n e a r mo lecu les where the r o t a t i o n a l f r e q u e n c i e s can be w r i t t e n in terms of one v a r i a b l e . U l t i m a t e l y we shou ld l i k e a s i m i l a r f i t t i n g method in N v a r i a b l e s . As a s tep in tha t d i r e c t i o n we now c o n s i d e r the two v a r i a b l e symmetric top c a s e . The " s t a n d a r d " r e p r e s e n t a t i o n of a one v a r i a b l e f u n c t i o n by a one v a r i a b l e Pade approximant i s as the r a t i o 220 of one v a r i a b l e po l ynomia l s whose c o e f f i c i e n t s are ob ta ined by c o n s t r a i n i n g the expans ion a f t e r d i v i s i o n of the r a t i o n a l f r a c t i o n to agree w i th the T a y l o r expans ion of the f u n c t i o n to as h i g h a degree as p o s s i b l e (see equa t ion ( e7 .10 ) ) . It i s not s u r p r i s i n g t h e n , on go ing to two v a r i a b l e s , that we might t r y to form two v a r i a b l e approximants as the r a t i o s of two v a r i a b l e po l ynomia l s whose expans ion a f t e r d i v i s i o n shou ld agree to as h igh a degree as p o s s i b l e w i th the r e l e v a n t T a y l o r e x p a n s i o n . Fu r the rmore , s i n ce i n the one v a r i a b l e case the most e f f i c i e n t approximants are the " d i a g o n a l " L=M approx imants , we shou ld i n i t i a l l y c o n s i d e r two v a r i a b l e approximants where the o rde rs of the numerator and denominator are e q u a l . T h i s i s the method of Ch isho lm (r166) . The d i f f i c u l t y w i th the Ch i sho lm approximant i s tha t to have terms of s i m i l a r degree i n the numerator and denominator i n t r o d u c e s an inde te rminacy i n t o the s o l u t i o n fo r the approximant c o e f f i c i e n t s ( n 6 6 ) . T h i s means the mapping from T a y l o r c o e f f i c i e n t to Ch isho lm c o e f f i c i e n t i s no longer one to one. T h i s inde te rminacy w i l l come up l a t e r when we review v a r i o u s other s t u d i e s where t h i s so r t of approximant i s used to f i t r o t a t i o n a l f r e q u e n c i e s . Ch isho lm chose to so l v e t h i s problem in a way that was symmetr i ca l f o r the two v a r i a b l e s by c o n s t r a i n i n g the c o e f f i c i e n t s of v a r i o u s " c r o s s " terms to z e r o . For symmetric tops i t i s not immediate ly obv ious tha t we need t r e a t the parameters J and K i n a symmetr i ca l way and so we shou ld expect d i f f e r e n t 221 " r e d u c t i o n s " might prove b e t t e r ( r167 ) . Now we s h a l l r e - cons ide r the b r i e f h i s t o r y of r a t i o n a l f r a c t i o n approximants as a p p l i e d to r o t a t i o n a l s p e c t r o s c o p y . The i n t r o d u c t i o n of Pade approx imants to the s p e c t r o s c o p i c l i t e r a t u r e was p robab ly due to M.Mizushima ( r l 6 8 ) . Mizushima was i n t e r e s t e d in us ing Pade approximants to a c c e l e r a t e the convergence of s tandard p e r t u r b a t i o n expans ions (see a l s o r l 64 and r169 ) . The f i r s t attempt at us ing r a t i o n a l f r a c t i o n s in f i t t i n g r o t a t i o n a l s p e c t r a was by Young and Young in 1978 ( r150 ) . In t h e i r study they used a two v a r i a b l e r a t i o n a l f r a c t i o n w i th s i m i l a r terms in powers of J and K in numerator and denominator to account f o r the i n v e r s i o n spectrum of ammonia ( i . e . s i m i l a r to C h i s h o l m ) . At the time the s t anda rd power s e r i e s cou ld not account fo r the ammonia i n v e r s i o n spectrum to w i t h i n expe r imenta l e r r o r (see Sp i r ko (r138) f o r an updated s t u d y ) . T h i s same data set was found to f i t much b e t t e r (an improvement in the s tandard d e v i a t i o n of the f i t of two o rde r s of magnitude) when the l oga r i t hms of the f r e q u e n c i e s were f i t to the s tandard t r unca t ed power s e r i e s in J and K — the s o - c a l l e d " e x p o n e n t i a l " form of C o s t a i n ( r170 ) . Young and Young found that t h e i r r a t i o n a l f r a c t i o n f i t t i n g f u n c t i o n appeared to r ep resen t the da ta even b e t t e r than the e x p o n e n t i a l C o s t a i n f u n c t i o n . L a t e r , in 1981, Belov et a l (r147) i n t r oduced a Pade h a m i l t o n i a n . That i s , r a the r than w r i t i n g Pade approximants to the e i genva lue equa t ion they wrote Pade approximants to 222 the h a m i l t o n i a n . The r e s u l t i n g method i s the same as the method of Young and Young, a l t hough somewhat more complex. With t h e i r method phosphine was s t u d i e d , and l a t e r H 2 S (r149) and most r e c e n t l y H 2 0 ( r148 ) . The r e s u l t of these v a r i o u s r a t i o n a l f r a c t i o n s t u d i e s was an i n d i c a t i o n that r a t i o n a l f r a c t i o n s c o u l d b e t t e r account to r o t a t i o n a l s p e c t r o s c o p i c f r e q u e n c i e s . U n f o r t u n a t e l y , a l l these s t u d i e s s u f f e r e d from the same compla in t — they a l l found very h i g h , e s s e n t i a l l y u n i t y , c o r r e l a t i o n s amongst some parameters of the numerators wi th some of those in the denomina to rs . The reason f o r t h i s shou ld be c l e a r from C h i s h o l m ' s inde te rminacy argument fo r approximants where we f i n d s i m i l a r terms in both numerator and denominator . The problem wi th the p r e v i o u s s p e c t r o s c o p i c s t u d i e s tha t use these C h i s h o l m - l i k e two v a r i a b l e r a t i o n a l f r a c t i o n s i s that there i s no attempt to make a one to one mapping from T a y l o r c o e f f i c i e n t to Ch isho lm r a t i o n a l f r a c t i o n c o e f f i c i e n t . Other two v a r i a b l e approximant f o r m u l a t i o n s are a v a i l a b l e . For i n s t ance we have the f o r m u l a t i o n s of L u t t e r o d t (r171 ) and K a r l s s o n and W a l l i n ( r172 ) . In us ing both these methods, as in the method of C h i s h o l m , we beg in by w r i t i n g the two v a r i a b l e approximant as the r a t i o of two v a r i a b l e p o l y n o m i a l s . U l t i m a t e l y , we shou ld expect any two v a r i a b l e approximant to end up as the r a t i o of two v a r i a b l e p o l y n o m i a l s , but i t would be more s a t i s f y i n g to have a s t ronge r h e u r i s t i c m o t i v a t i o n beh ind these approx imants , 223 s i m i l a r to what we had e a r l i e r f o r the one v a r i a b l e c a s e . As an example of a more p h y s i c a l approach we c o n s i d e r the sugges t i on of Tuy l ( r173 ) . The s tandard two v a r i a b l e T a y l o r expans ion f < x ' ^ =a§O0io aaB ^ [7'19] T u y l r e-wr i t e s as f ( x , y ) * , ( x ) y*3 [7 .20] w i th , = i j a „ x a [7.21 ] YB(x) a=0 aB The idea i s to decompose the two v a r i a b l e s e r i e s i n t o two "one " v a r i a b l e s e r i e s so we can use the s t anda rd one v a r i a b l e Pade approx imants . T h i s approach i s q u i t e e f f e c t i v e . Fur thermore now that we are us ing one v a r i a b l e approx imants , we no longer have an inde te rminacy p rob lem. The d i sadvantage of t h i s techn ique i s tha t we d e a l w i th powers of x and powers of y and not t o t a l powers of x and y . For example, we d e f i n e the 1,1 .Tuyl approximant as the [1/1] approximant of equa t ion (e7.20) where the c o e f f i c i e n t s ^p(x) are g i ven by the [1/1] approximant of equa t ion ( e7 .21 ) . When we form t h i s approximant we f i n d the h i g h e s t power of x i s x 2 and of y i s y 2 so that we need to know the T a y l o r c o e f f i c i e n t of the f o u r t h degree x 2 y 2 te rm, a 2 2 . However, we do not take i n t o account the x 3 and y 3 t e rms ; terms of t h i r d 224 degree in x and y . In s p e c i f i c cases t h i s may be a r easonab le t h i n g to do , but g e n e r a l l y we would l i k e to account fo r t h i r d degree e f f e c t s be fo re we go on to f o u r t h deg ree . The above d i f f i c u l t y w i th the T u y l method i s an example of a more gene ra l problem wi th many of these two v a r i a b l e approximant schemes. That i s , these approximants were des igned fo r f u n c t i o n s where a n a l y t i c a l methods e x i s t f o r c a l c u l a t i n g the v a r i o u s T a y l o r c o e f f i c i e n t s . The r e s u l t has been tha t we have r i g i d l y d e f i n e d approximants tha t r e q u i r e knowledge of r i g i d combina t ions of T a y l o r c o e f f i c i e n t s . For s p e c t r o s c o p i c a p p l i c a t i o n t h i s i s not s a t i s f a c t o r y as o f t e n the a v a i l a b i l i t y of data d i c t a t e s which T a y l o r c o e f f i c i e n t s we can de te rm ine . As an a l t e r n a t i v e to these r i g i d a n a l y t i c a l two v a r i a b l e approx imants one i s tempted to suggest a more f l e x i b l e method, one that r e l i e s on " s u c c e s s i v e s e p a r a t i o n " . The m o t i v a t i o n fo r t h i s new t echn ique comes from r e - w r i t i n g the s tandard two v a r i a b l e s e r i e s , equa t ion ( e 7 . l 9 ) , as f ( x , y ) = L?Q ( x y ) L [ - a L L a i + L f L x 1 a ^ j + L y* ] [7 .22] For any a r b i t r a r y t r u n c a t i o n of f ( x , y ) the summations in equa t ion (el .22) become f i n i t e . The ardent Pade a p p r o x i m a n t i s t now knows p r e c i s e l y what to do : g i ven any one v a r i a b l e t r u n c a t e d s e r i e s , of th ree terms or more, s imply r e p l a c e i t w i th the a p p r o p r i a t e one v a r i a b l e Pade f r a c t i o n . For the cases where the t r unca t ed s e r i e s c o n t a i n s fewer than 225 th ree te rms, s i n ce a Pade approximant cannot be formed, we s h a l l s imp ly leave these terms a l o n e . Approximants b u i l t in t h i s way have many advantages . F i r s t , because we are u s i ng on l y one v a r i a b l e approximants we have no problem in de t e rm in i ng the v a r i o u s Pade c o e f f i c i e n t s ( u n l i k e wi th Ch isho lm app rox iman t s ) . A l s o , these approximants are e s p e c i a l l y we l l s u i t e d to computers . As an example, say we wanted to i n c l u d e a new parameter a^j in a f i t . From the v a l ues of i and j we determine in which s u b - s e r i e s of equa t i on (e7.22) t h i s term be longs and then s imp ly c a l l , from a Pade approximant l i b r a r y , the new approximant tha t r e p l a c e s t h i s s u b - s e r i e s . F i n a l l y , these approximants are q u i t e f l e x i b l e as the degrees of the v a r i o u s s u b - s e r i e s are a l l i ndependent . We have suggested tha t Pade approximants or r a t i o n a l f r a c t i o n s c o u l d be u s e f u l as f i t t i n g f u n c t i o n s fo r r a t i o n a l i z i n g s p e c t r o s c o p i c d a t a . 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