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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  A  THESIS THE  ALDON  MCRAE  SUBMITTED IN PARTIAL FULFILMENT REQUIREMENTS  FOR  DOCTOR OF  THE DEGREE  OF  PHILOSOPHY  in THE  F A C U L T Y OF  GRADUATE  DEPARTMENT OF  We  accept to  THE  this  thesis  OF  August  ©  CHEMISTRY  the required  UNIVERSITY  Glenn  Aldon  STUDIES  as  conforming  standard  BRITISH  COLUMBIA  1984  McRae,  1984  In  presenting  requirements  this for  thesis  Columbia, I  freely  available  agree for  permission for  scholarly  purposes  Department  or  understood  that  financial  gain  partial  an a d v a n c e d d e g r e e a t  British  that  in  by  the  reference  extensive may his  copying  shall  that  not  be or  Date:  August  1984  Library and s t u d y .  I  granted  by  the.  allowed  Columbia  the  make  further  Head  of  this  without  of It  thesis my  of it  agree  thesis  representatives.  or p u b l i c a t i o n be  shall  this  her  of  The U n i v e r s i t y  of  CHEMISTRY  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  the  copying  permission.  DEPARTMENT OF  fulfilment  for my is for  written  ABSTRACT Microwave AsD ,  AsD H,  3  AsH D,  2  distortion  spectra  investigated  PD H and PH D  2  spectra  have been 2  (and P D ) .  2  the  frequency  m e a s u r e d and a n a l y z e d  for  GeH„  transitions, of  AsD H,  2  Harmonic  obtained  this  and v i b r a t i o n a l  force  fields  centrifugal  frequencies  of  correction  techniques  the  standard  shows t h e s e  techniques  corrections  of  structures  this  to  obtained  force  is  distinction different T  anharmonic i ty s  for  analysis various  equilibrium structures  also  presented  between  reductions;  frequencies  and so  average  special  of  light  z  Using a  have been top  reduction  t o make c l e a r  importance  is  linear  estimated.  l a b e l e d parameters  of  "r "  isotopic  symmetric  in order  similarly  constants  made.  field  the  data  studies. A  "harmonize" v i b r a t i o n a l  were not  for  constants  from o t h e r  "diatomic"  A c o m p a r i s o n of many of schemes  distortion  ammonia, p h o s p h i n e and a r s i n e .  extrapolation,  regions,  empirical  t o be u n s a t i s f a c t o r y  sort  harmonic  of  wave  as C o r i o l i s c o u p l i n g  have been o b t a i n e d  derivatives  rotational  have been p r o d u c e d  from v a r i o u s  work as w e l l  discussion  From t h e  been  2  the q u a r t i c in  Allowed  and m i l l i m e t e r  p h o s p h i n e and a r s i n e  including  3  8-26 GHz have  P D H and PH D have been m e a s u r e d and  2  analyzed.  and A s D .  the c e n t i m e t e r  AsH D,  2  ammonia,  in  range  GeH„,  Forbidden  3  in  for  a of  the  parameter  xxxz* To " f i t "  has  spectra  been s u g g e s t e d by p r e v i o u s  i i  easily authors  d i s t o r t e d molecules that  rational  it  fraction why  Pade a p p r o x i m a n t s might  these  approximants c o u l d prove  with a review general  be u s e f u l . A m o t i v a t i o n  of  fitting  earlier  attempts  schemes.  In  this  to  incorporate  review  we f i n d  them that  schemes and so a new m e t h o d , a method of  indeterminacies. general  study  converge.  is  proposed that  where  power  series  parameters  circumvents  T h i s method s h o u l d p r o v e  along into  these  separation,  various  given  relations  successive  between  is  indeterminacy earlier  exist  beneficial  useful  in N v a r i a b l e s  for  for are  in  these any slow  to  Table  of  Contents  A.  Introduction  1  B.  CHAPTER ONE: THEORY  7  1. The R i g i d R o t o r  7  2.  Centrifugal Distortion  3.  Hyperfine Perturbations  4. E f f e c t s 5.  C.  of  External  10 to  the Energy  Electric  Fields  34  6. D e t e r m i n a t i o n o f M o l e c u l a r S t r u c t u r e  38  7.  43  Summary and Comments  CHAPTER 2: EXPERIMENTAL METHODS  46  1.  Stark Modulation  47  2.  D i s t o r t i o n Spectrometer  48  3.  C e n t i m e t e r Wave S p e c t r o m e t e r  5.  Sundry  (UBC)  52  (JPL)  54  Items  55  CHAPTER 3: GERMANE G e H ,  56  1.  Germane Gas Sample  58  2.  Prediction  58  of  3. Measurement of  t h e Germane S p e c t r u m t h e Germane S p e c t r u m  4. O b s e r v e d Germane S p e c t r u m and A n a l y s i s  E.  18 22  Force F i e l d s  4. M i l l i m e t e r Wave S p e c t r o m e t e r  D.  Levels  5.  Isotope S p l i t t i n g  6.  Final  i n Germane  Comments on Germane  CHAPTER 4: ARSINE 1.  Preparation  2.  Prediction  3.  Initial  of of  60 ..'  64 71 83 86  t h e A r s i n e Gas Sample the AsD  3  Searches  Spectrum  89 90 94  i v  F.  G.  4.  Observed Arsine-D3 Spectrum  5.  Prediction Top A r s i n e  6.  O b s e r v e d A s H D and A s D H S p e c t r u m  7.  D i s c u s s i o n of Arsine  8.  D i s c u s s i o n of t h e S p i n - R o t a t i o n C o n s t a n t s of A r s i n e  and I n i t i a l Spectra 2  CHAPTER  5:  for  the  Asymmetric  117  2  Nuclear  Quadrupole C o u p l i n g  in  132  Coupling  134  PHOSPHINE of  113  136  1.  Preparation  2.  Prediction Spectra  3.  O b s e r v e d PH D and P D H S p e c t r a  141  4.  Analysis  149  of  P h o s p h i n e Gas  of  ....139  t h e A s y m m e t r i c Top P h o s p h i n e  2  C h a p t e r 6: MOMENTS 1.  140  2  the  Phosphine  FORCE F I E L D S ,  Spectra  STRUCTURES  AND DIPOLE  153  An E x a m i n a t i o n of S t a n d a r d A n h a r m o n i c i t y C o r r e c t i o n Techniques A p p l i e d to V i b r a t i o n a l Frequencies  156  2.  F u r t h e r C o n s i d e r a t i o n s When M a k i n g a Harmonic Force F i e l d C a l c u l a t i o n  172  3.  Harmonic F o r c e F i e l d P h o s p h i n e and A r s i n e  177  4.  Estimation  5.  Structure  of of  CHAPTER  7:  Refinements:  Ammonia,  t h e D i s t o r t i o n D i p o l e Moment Ammonia, P h o s p h i n e  6. D i s c u s s i o n of H.  Search  98  Structural  184  and A r s i n e  189  Parameters  195  INTERESTING EXTRAS  1.  Failures  of  2.  On t h e use of the F i t t i n g  197  the R e d u c t i o n s of  Rational  199  Fraction  Rotational  BIBLIOGRAPHY  Approximants  Spectra  in 206 227  v  L I S T OF TABLES Table  1.1 Definition i n e r t i a l axes  Table  representation  selection  rules  for  of  cartesian  asymmetric  top  transitions  and  31  66 rotational  transitions  3.2  Summary of conditions T a b l e 3.3  germane  line  parameters  and  experimental \  67  68  Tensor c e n t r i f u g a l Table  terms  3.1  O b s e r v e d germane Q - b r a n c h Table  in  1.2  Symmetry Table  of  distortion  constants  of  GeH«  3.4  C o m p a r i s o n of l i n e a r d i s t o r t i o n constants of r e f (r68)  c o m b i n a t i o n s of o b t a i n e d in the  germane c e n t r i f u g a l p r e s e n t work w i t h t h o s e  69 T a b l e 3.5 P r e d i c t i o n of germane IR-MW d o u b l e r e s o n a n c e t r a n s i t i o n s u s i n g c o n s t a n t s o b t a i n e d s o l e l y f r o m t h e microwave s p e c t r u m 70 T a b l e 3.6 C o m p a r i s o n of f o r c e c o n s t a n t s f o r m e t h a n e , s i l a n e and germane 75 T a b l e 3.7 C o m p a r i s o n of s p e c t r o s c o p i c p a r a m e t e r s : C H , S i H and GeH„ 85 F o r b i d d e n h y p e r f i n e t r a n s i t i o n s of A s D T a b l e 4.1 99 T a b l e 4.2 A r s i n e l i n e p a r a m e t e r s and e x p e r i m e n t a l c o n d i t i o n s 1 03 1 2  U  2 8  a  3  T a b l e 4.3 A n a l y s i s of  the  measured A s D  3  Spectrum  vi  (MHz)  11 1  L I S T OF TABLES T a b l e 4.4 O b s e r v e d s p e c t r u m of Table  s p e c t r u m of  2  1 23  1 28  AsD H 2  4.6  Empirical Table  AsH D  4.5  Observed Table  continued  131 spectroscopic  parameters  of  various  arsines 1 45  5.1  1 46  O b s e r v e d PD H s p e c t r u m i n MHz 2  Table  5.2  O b s e r v e d PH D s p e c t r u m i n MHz T a b l e 5.3 R e - a n a l y s i s of the measured P D s p e c t r u m i n MHz T a b l e 5.4 E a l s p e c t r o s c o p i c p a r a m e t e r s of v a r i o u s p h o s p h i n e s Tm a bpl ier i c6.1 2  147  A  3  Various frequency st i b i n e Table  phosphine,  arsine  152 and 174  force  field  study  of  ammonia  179  180 force  field  study  of  phosphine 181  6.4  Harmonic Table  ammonia,  6.3  Harmonic Table  of  6.2  Harmonic Table  ratios  force  field  study  of  arsine  6.5  Force  constants  Table  6.6  Force constants T a b l e 6.7 E s t i m a t i o n of i  182  182 (internal  symmetry  coordinates) 185  (valence) xxxz  vii  L I S T OF TABLES Table  6.8  | 3M/3Q-, |  continued  (amu A )  186  1  T a b l e 6.9 E s t i m a t i o n of the to the d i s t o r t i o n T a b l e 6.10 r structure z  r o t a t i o n a l and v i b r a t i o n a l d i p o l e moment  contributions 186  of  ammonia  structure  of  phosphine  1 92  T a b l e 6.12 r structure z  of  arsine  1 93  Table r  z  Table  191  6.13  structures  of  a m m o n i a , p h o s p h i n e and  arsine  7.1  1 94  204  Estimation Table  1  6.11  Equilibrium Table  ( "N)  of  the  contribution  of  £ to  h  3  7.2  Comparison of t r u n c a t e d T a y l o r s e r i e s f v e r s u s Pade fraction f i n f i t t i n g d a t a p r o d u c e d f r o m e q u a t i o n 7.12 T  p  viii  215  L I S T OF FIGURES Figure  1. The d i s t o r t i o n  and 26 G H z . I n t e r n a t i o n a l elements  spectrometer  u s e d between 8  symbols a r e used f o r the  various  (r57). 48  Figure resulting  2. The microwave  absorption  from the d i s t o r t i o n  o b t a i n e d by u s i n g t h e s i g n a l sample s c a n s and t h e n total  s c a n n i n g time  line  transition averager  subtract  of GeH„  18E <—18E . 2  1  It  was  t o a c c u m u l a t e 64  64 b a c k g r o u n d s c a n s . The  (sample p l u s background)  was 3.64 h o u r s . 63  Figure  3. Two of t h e K = ±1 <— *2 A s D  top reproduction  i s J=15. It  scans at a Stark  field  reproduction Stark  field  is  the r e s u l t  of 74 v o l t s / c m .  i s J=17 and i s t h e r e s u l t of  3  transitions.  The  o f 64 sample  The b o t t o m of  48 s c a n s a t a  200 v o l t s / c m . 1 02  Figure  4. The 2 •, -, —2 -, r o t a t i o n a l 2  T h i s was o b t a i n e d as a s i n g l e w i t h a time of  constant  of  0.1  transition  s c a n of 600 s e c o n d s  s e c . and a  of  AsD H. 2  duration  100 KHz S t a r k  field  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 carried Gerry.  out I  under  the  should l i k e  assistance  described  patient  this  and a b l e him f o r  his  and e n c o u r a g e m e n t ,  and f o r  the  Special enthusiatic  all  were of  Michael  invaluable large  degree  of  my w o r k .  thanks a l s o manner  thesis  direction  thank  independence allowed  to  in  goes t o  Irving Ozier  for  the  i n w h i c h he s h a r e d many i n s i g h t s  and  ideas. Many of in  the  presented  David  to acknowledge  Spencer  Alan Robiette  and h e l p w i t h ,  thesis.  I  patient  efforts  many of  should a l s o in  in  this  thesis  and Man Wong.  and t h a n k  am i n d e b t e d t o Ed C o h e n ,  Herb P i c k e t t , in,  ideas  company of  pleasure I  the  were honed  It  is  a  them.  Bob D a v i s ,  Robert  and Man Wong f o r  Kagann,  their  interest  t h e measurments p r e s e n t e d  wish  to  thank  Z o l Germann f o r  k e e p i n g the v a r i o u s  U.B.C.  in  this  his  spectrometers  alive. Finally, father,  I  should l i k e  mother and s i s t e r  encouragement  they  to dedicate  in  have g i v e n  this  r e c o g n i t i o n of  thesis the  me t h r o u g h o u t my  x  t o my  support studies.  and  1 A.  INTRODUCTION This  thesis  electromagnetic phase.  is  concerned w i t h the a b s o r p t i o n  radiation  The r a d i a t i o n  region,  by  small molecules  frequencies  between r a d i o and f a r  A b s o r p t i o n of  radiation  in  associated with molecular A classical  associated with  infrared,  this  is  to  or  radiate  the  angular  -  For  frequency region.  with  angular energy  affects  about  the  above argument  from an  radiation  the  the  the  we o f t e n  however,  absorption  features  lower p r e s s u r e s  (=*0.1mTorr  i n our  dipole  external radiated  by,  molecules.  any  heavy m o l e c u l a r  broad frequency  If  d i p o l e moment becomes  see  c o v e r i n g much of  pressure  —  rotational  we s h o u l d e x p e c t for  axis.  this  energy  can r o t a t e  pressures,  absorption p r o f i l e s the  rotation  d i p o l e moment t h i s  increasing  the microwaves  As we l o w e r  the  rotational  absorption,  and h i g h e r  J  microwave  be a b s o r b e d and g e n e r a l l y  intermediate  usually  21  or a b s o r b e n e r g y  d e c r e a s i n g or  by w h i c h  From t h e to  field.  momentum. F o r  a handle  is  bond has a r o t a t i o n a l  inertia  energy  absorbed energy  respectively,  microwave  it  t h e moment of  electromagnetic  gas  1-1000 GHz.  region  rotating  t h e m o l e c u l e p o s s e s s e s an e l e c t r i c moment can  the  roughly  frequency  molecule  • E ROT where I  in  the  rotation.  diatomic  momentum / p e r p e n d i c u l a r  used f a l l  in  of  gases rather  the  we n o t i c e  begin to  studies)  frequency at broad  microwave the  sharpen up.  gases  become  At  2 quite will  selective  to  a b s o r b and one  whose  widths  The l i n e part the  as  in  6  order  spectral  absorption  can u s u a l l y  This p r e c i s i o n  precision In  finds  which  associated to  lines,  of  measurement  w i t h microwave  we  these  invoke  is  they  lines, gigahertz.  better the  than basis  magic  of  constant  angular  momentum a v a i l a b l e  our m o l e c u l e s q u a n t i z e s  energy.  quantized  levels  instance,  for  It  that  27r, and t h e  is  molecule  rotational  energy  a b s o r p t i o n one h a s  are  R  equation moment o f  T  is  that  inertia  the  these For  momentum  constant  divided  2  w i t h the  angular  momentum.  where J c h a n g e s  by  one.  I frequency.  we c a n measure I.  the  J—>J+1 and  the m i c r o w a v e if  spectra.  Planck's  those  ^ROT *> Q  between  angular  J(J+,)M 21  ROT  Quantizing  becomes  a quantum number a s s o c i a t e d  where  the  where K e q u a l s  The o b s e r v e d t r a n s i t i o n s For  transitions  e x p l a i n microwave  „  J  then  a diatomic  becomes v/JTJ+TT by  is  'h'.  quantized  momentum and P l a n c k ' s  rotational  of  isolated  angular  to  one  spectroscopy.  seemingly  the  or  one h u n d r e d  be m e a s u r e d t o  rationalize today  frequencies  features,  can be one m e g a h e r t z a t  centers 10 .  how much of  For  a  ^RQT^ "^  diatomic  1  We see w  e  c  a  n  from the c  a  l  c  u  ^  a  t  above e  the  3  so knowing t h e bond l e n g t h . pendulum's In  effect  they  This  is  length  masses a l l o w s us t o c a l c u l a t e  not  unlike  from i t s  calculating  "resonant"  s h i n i n g microwaves  look It  various  on m o l e c u l e s  now.  lets  it.  in order  The c l a s s i c a l  is  responsible  transitions.  We w i l l  transitions.  We s h a l l  "forbidden" "weakly"  allowed  distortion is  has  the  the  what  also  these  observe  These of  sum of  two p a r t s ,  the  molecule  thus  the  "symmetrical" are  Studies centrifugal  rotational  so-called  distortion.  see,  this  a rotational  In  is  certain  to of  see  r i g i d molecule.  a  part  picture  distorting  cases  These  The  distortion  The c l a s s i c a l molecule  become  borrowing  e q u i l i b r i u m symmetry  very  allowed  called  these of  p r o d u c i n g s m a l l d i p o l e moments not  generally  spectrometers  forces.  rigid  a number o f  and v i b r a t i n g  can break  with a  these  transitions  borrowing p a r t .  rotating  tenuous  forbidden transitions  of  to  a number of  centrifugal  sorts  a  w i t h which  " n o r m a l " or a l l o w e d  be l o o k i n g a t  centrifugal  distortions  spectra  "see"  becomes a l i t t l e  d i p o l e moment. As we w i l l  b e c a u s e of  more  for  because  plus a vibrational here  argument  transitions.  d i p o l e moment f o r  moment  us  to absorb microwaves  " n o r m a l " d i p o l e moment we a s s o c i a t e  molecule  frequency.  like.  would seem t h e n ,  The  the  simple  absorption  m o l e c u l e must have a d i p o l e moment, a h a n d l e rotate  a  r,  the allowed  in  forbidden  weak and we r e q u i r e  specialized  them.  forbidden r o t a t i o n a l  distortion  date  from  spectra  due  1967 w i t h Hanson  to suggesting  4 the  p o s s i b i l i t y of  molecules  (rl).  attributed  s o - c a l l e d AK-±3 t r a n s i t i o n s  The d e v e l o p m e n t  t o Watson  (r2),  Fox  of (r3)  The f i r s t  experiment, a molecular  done t h a t  same y e a r  initial at  research  of  R e s e a r c h C o u n c i l of  British  d e c a d e has  institutions University Russian  seen a c t i v e  the  in  of  at  the  Gorky,  (r6).  The  r e s e a r c h at  such  Katholieke  Caltech,  Harvard, of  out  National  F y s i s c h Labs at  Institute  was  distortion  1974 by Oka  P r o p u l s i o n Lab a t  Sciences  Swiss F e d e r a l  Much of  the  is  i n 1971.  f i e l d was c a r r i e d  experimental  H o l l a n d , the J e t  (r4)  (r5).  C o l u m b i a and a t  as Duke U n i v e r s i t y ,  Academy of  recently  this  reviewed  v  beam r e s o n a n c e s t u d y ,  C a n a d a . The d e v e l o p m e n t  moment s p e c t r o s c o p y was last  in  C^  mechanisms  and A l i e v  on methane by O z i e r  experimental  the U n i v e r s i t y  transition  in  and  Technology  the  very in  Zurich. In  this  molecules. the  thesis  we s h a l l  The f i r s t  study  is  same symmetry as methane  moment e x i s t s  due t o t h e  f o r b i d d e n microwave continuation  of  look at  three  on germane, G e H „ . (Td)  for  tetrahedral  work  in  are  this  Germane  w h i c h no n o r m a l symmetry.  absorption lines  earlier  individual  In  this  reported  in a  labratory  has  dipole study  on g r o u p  IV  hydrides. The s e c o n d s t u d y and A s H D .  In  2  transitions a  large  is  the  first  on a r s i n e ,  specifically  case a s e r i e s  p r e s e n t e d . For  number of  frequency  is  allowed  the  partially  reported.  3  AsD H 2  of AK=±3 f o r b i d d e n  "quadrupole s p l i t "  r a n g e 8-303 GHZ i s  AsD ,  deuterated lines  in  forms the  5 The t h i r d to  the  arsine  study study  No new t r a n s i t i o n s re-analyze  in  on p h o s p h i n e . T h i s that  are  previous  standard data  is  r e p o r t e d here  work  reduction  From a n a l y s e s  of  vibration-rotation  they  reviewed  harmonic  we t r e a t  We t h e n c o n s i d e r  centrifugal  displacing  springs  These  force  to  the  in  by  distortion  more g e n e r a l be o f  the  force  displacement; later  positions specific  to  opposite  the c o n s t a n t s  a clear  mechanisms w o r k i n g  in  physically  call  by t h i s molecular  force  picture  symmetric  simple  as  studies in  quantities  of  the  of  to  there  thesis  For  estimate  are that  an a t t e m p t  the  use t h e  and  model. some should has  distortion  molecular  as many a u t h o r s  different  springs.  constants.  spectroscopists. First,  been made t o p a i n t  deemed n e c e s s a r y  tops.  This  same l a b e l  resulting  was for  in a c o n f u s i n g  literature. A l s o of attempt  to  interest  explain  spectroscopic  s h o u l d be t h e  the  parameter  as  from.some e q u i l i b r i u m p o s i t i o n .  discussions presented  interest  studies  and v i b r a t i o n  and p h o s p h i n e we have been a b l e  Besides  thesis.  "harmonic"  ammonia,  atomic  we  in a molecule  we s h a l l  molecular  2  arsine  proportionality arsine  PH D.  the  this  field  t h e atoms  s p r i n g s produce a r e s t o r i n g  proportional  with  ammonia, p h o s p h i n e and  spectra,  instead  3  techniques  similar  2  PD ,  consistent  is  P D H and  3  for  were c o n n e c t e d t o one a n o t h e r  these  PD ,  i n a way  have been p r o d u c e d . Here if  we c o n s i d e r  study  isotopic D  for  quasi-theoretical  variations the  of  tetrahedral  the spherical  6  tops.  T h i s d i s c u s s i o n might prove  similar  explanations  Third, force  in other  t h e p r o c e s s of  field  studies  The g e n e r a l  is  reviewed  s h o u l d be of  point  for  studies.  anharmonicity  conclusions that  unsatisfactory  a starting  for  these  corrections  ammonia-like corrections  interest  to other  in  molecules.  are  so  far  workers  in  the  field. Finally, than  the  recently  standard  accounting  for  "power"  centrifugal  Pade a p p r o x i m a n t s . Here this  subject  and p o i n t  gone wrong w h i l e a t discussion  for  it  the  the  has  been s u g g e s t e d t h a t  series  out where same t i m e  future.  e x p a n s i o n method o f  distortion  we s h a l l  rather  we s h o u l d c o n s i d e r  review  the  literature  on  these  early  attempts  have  giving a direction  to  the  7  B.  CHAPTER ONE;  1.  THEORY  THE RIGID ROTOR  A d i s c u s s i o n of usually  theory  of  rotational  begins with a c o n s i d e r a t i o n  molecule. which  the  in  From r o t a t i o n a l the case  of  spectra  a rigid  of  one  rotor  a rigidly infers  are  spectroscopy rotating  energy  consistent  levels  with  the  hamiltonian H  = B  r  where t h e c a r t e s i a n principal operator  inertial for  J + 2  xx  axis  B  J  + B  2  y y  t h e component of  the  axis  constants"  and a r e  proportional  moments of  inertia  about  By  convention  that  the  observes (e1 .1 ) ,  are  often  the  the  the  y  to  the  represents  the  momentum a b o u t called  the  "rotational  reciprocals  of  the  g axes. [1.2]  g  and J  axes are  labelled a,b,c  obey B >B,>B^. The a  renamed A , B , C  following  angular  to  principal  2  The J  7  constants  a b a s i s where J  refers  h . The B „ a r e  = KV2hI  three  rotational  constants In  the  g  of  axes.  g inertial  B  [1.1]  2  system x , y , z  (molecular)  in u n i t s  J  z z  z  D  so  rotational  C  s u c h t h a t A>B>C.  are  both d i a g o n a l  non-zero matrix  elements  of  one equation  [1.3]  <JK±2|H |JK>  = ,{(B -B  r  x  y  ){[J(J+1)-K(K±1)]  X[J(J+1)-(K±1)(K±2)]}  [1.4]  1 / 2  where J  and K a r e quantum numbers r e p r e s e n t i n g  angular  momentum and i t s m o l e c u l e  respectively,  J  = 0,1,2,3,....  The form o f t h e molecular 1.  types.  f i x e d component  , K = 0,±1,...±J.  matrix  now a l l o w s c l a s s i f i c a t i o n of  One h a s  1inear  B  = B  x 2.  spherical  tops  3.  prolate  4.  oblate  symmetric  tops  asymmetric  Linear  molecules  discussed  When B  x  y  x  here  special  attention  z  >B  B ,B^,B  require  further  = B x  tops  z  = B^ < B  x  z  =B y  B  ^  1  5.  B  tops  '  = B x  symmetric  = 0, B  y  B  ^  r  the t o t a l  z z  and w i l l  n o t be  (r7).  = B^ t h e o f f d i a g o n a l  terms  go t o z e r o a n d one  f inds Symmetric E(J,K)  x  = B J(J+1)  Spherical E(J)  Tops ( B = B , B ) y  z  + (B - B ) K  Tops  [1.5]  2  (B =B =B ) x  y  z  = BJ(J+1)  [1.6]  A.  For is  asymmetric  not d i a g o n a l  equation.  independent For Cross  (r9)  case  a change  (r8) b l o c k s  tri-diagonal  right  the general  form o f t h e h a m i l t o n i a n  and one i s f o r c e d t o s o l v e  In t h i s  Transformation  tops  the H matrix  known a s a Wang into  four  submatrices.  handed c o o r d i n a t e  identify  of b a s i s  the s e c u l a r  cartesian  systems  axes  x,y,z,  King, with  H a i n e r and principal  9 axes a , b , c  according to'three  systems.  Table Definition  of  1.1  representation  in  inertial I x  c  representation  basis.  In  the  case  of  is  often  rotational  constant  than  to the A c o n s t a n t .  it  and By  is  respectively  coefficient is  as  of  the  closer  i n the  terms  in  m i n i m i z e d when t h e  III  oblate  asymmetric  representations representation For  levels  are  Kc a r e  values  of  the  in H ,  is  top K i s most o f t e n  |K|  Whether Ka and Kc a r e  used  so c h o i c e  top (for  -  B^,  the  are an  All of alone.  not a good quantum number. labelled  (A=B>C)  by  limits  J  K  a  K  c  •  Here Ka and  "varying"  B to  respectively.  or odd can. be r e l a t e d  the asymmetric  x  shows t h a t  to A than C ) .  one would g e t  even  the  namely B  f  argument  same r e s u l t s  (A>B=C) and o b l a t e  s y m m e t r i e s of  b a s i s means t h a t  basis  closer  C constant  B and C w i t h B  an o b l a t e a s y m m e t r i c  or o b l a t e  B  b a s e d on c o m p u t a t i o n a l e f f i c i e n c y  The e n e r g y  prolate  of  an a s y m m e t r i c  the  Identifying  the  and  oblate  top the  in magnitude to  prolate  basis  t o as an  asymmetric  A similar  top B i s  give is  referred  o f f - d i a g o n a l terms  s m a l l as p o s s i b l e .  off-diagonal  c  often c a l l e d a prolate  a prolate  is  r  a  b  is  and  b  a  representation  III  III  a  z  the  r  II  e  cartesian  axes  b  y  The I  r  terms of  t o p wave f u n c t i o n s  to  the  (r9,rl0).  the  1 0 2.  CENTRIFUGAL  The r i g i d approximation that and  are it  at  will  effective  rotor for  low  hamiltonian  molecules  J).  distort  molecular  d e p e n d e n c e and as molecular  DISTORTION  that  an e x c e l l e n t  are  rotating  However  a molecule  because  of  constants  such w i l l  geometry  is  as  is  will  not  "slowly"  not  centrifugal  the m o l e c u l a r  (i.e.  a rigid  forces.  now e x h i b i t  have t h e  first  a  rotor  The  J  same r e l a t i o n  constants  of  the  to  rigid  rotor. From a d y n a m i c a l involved  in  reasonable  of  the  order  1  shown t h a t following  This the  H  is  method o f power  = Zh  are  molecular  0,1,2,....  For  a specific  1  Later  can  better  r  account  for  distortion  series  angular  expansion  momentum  (r12),  who  can be w r i t t e n  has in  by o m i t t i n g a l l and t i m e  the  terms  reversal,  r  q  p  z y x  molecular  7 we s h a l l  a rational  power  is  +J j j )  constants;  c o n s t r u c t e d by  in chapter  suggesting that  q  it  centrifugal  conjugation  p q r ( J x j y jz p  form"  forces  105ff)  Watson  series  "standard  to Hermitian  is  for  in a general  where h  hamiltonian  centrifugal  (r11,pg  to account  the  general  r  the  i n components of  so-called  invariant  of  molecule  terms  hamiltonian  operators .  not  a rotating to attempt  with higher  study  p,q,r type  retaining  challenge  fraction  centrifugal  L  of  are the  only  this  [1.7] • ' J  integers standard those  form  terms  a s s u m p t i o n by  general  distortion.  power  series  11 which  t r a n s f o r m as  molecular  point  the  fitting field  contains  rotational  free  of  the  in g e n e r a l  more terms  t h a n can be o b t a i n e d by  data  obtained For  t h e maximum number of data.  parameters  E a c h of  from  the  these  standard  (static  each m o l e c u l a r  can be r e d u c e d t o a  rotational  species  shown t h a t  measurements.  shown how  symmetric  group.  Watson has a l s o hamiltonian  totally  electric)  type  he  "reduced form",  has  containing  w h i c h c a n be o b t a i n e d  is  form  d i s c u s s e d in the  from  following  sections. 2a.Centrifugal Under standard are  Td symmetry  form s e r i e s  allowed  types.  terms  momentum) reduced.  (r14).  Watson  (r13) not  terms;  set;  has  (Td Symmetry) of  J  any  of  shown t h a t  the  the the  hamiltonian is  in  rotational  i n c l u d i n g degree  usually  so-called scalar  To an a p p r o x i m a t i o n up t o  momentum one has  for  hamiltonian  as  Tops  powers  the  form t h a n  (up t o but  The r o t a t i o n a l two  retained  e x p a n s i o n of  form a m i n i m a l  sum of  in S p h e r i c a l  the  much more complex i n  molecular  the  Distortion  our  hamiltonian other symmetry  12 i n is  completely  written  and t e n s o r  8th degree  in  angular  as  parts  angular  (rl5)  H = H r scalar  + H tensor  [1.8]  where H  scalar  =B J -D 0  2  s  J"+  H J +L J s s 6  [1.9]  12 and  H  tensor  =  [ D  t  +L The O  +  8 t  n  J  x  n that  2  +  L  4  J  t  < ^  +  [ H  6t  +  L  6t  j  2  ]  "« [1.10]  8  and J  z  transform  the Td p o i n t  momentum o p e r a t o r s , J ± ,  o f even o r d e r  in a t o t a l l y  up t o a n d i n c l u d i n g  symmetric  g r o u p . F o r example  = -1/2(35J«  0,  j  a r e combinations of angular  n  (= J ± i J y ) ,  of  4t  H  +25J )  + (15J  2  operations  (r16,rl7)  + 3)J  2  way under  2  -  3/2(J ) 2  2  - 5/4(J +J ) 2  Formulae  f o r J2  S  order  [1.11]  2  and fi  8  a r e t o be f o u n d i n  references  (rl3,r15,rl8). The m a t r i x a basis  elements  of our h a m i l t o n i a n a r e e v a l u a t e d  s e t o f Wang f u n c t i o n s  s y m m e t r i z e d under  g r o u p u s i n g t h e method o f Fox and O z i e r e i g e n f u n c t i o n s a r e l a b e l e d by J C angular group  2  different  is  2  and t  eigenfunctions  values  positive.  to other  and by Papousek and A l i e v The s c a l a r the  symmetry  part  species  i n the Td p o i n t  of t h e same J a n d C , t a k i n g on of i n c r e a s i n g energy  i s the notation  and h a s been r e l a t e d  species  total  i s a number t o d i s t i n g u i s h  1,2,3...in order  This  the Td p o i n t  ( r l 9 ) . The  where J i s t h e  momentum, C i s t h e symmetry  (A,,A ,E,F!,F )  integer  F C  in  fc  o f Dorney a n d Watson ( r l 7 )  schemes by H o l t  et a l (r20)  (r14).  of the h a m i l t o n i a n of our r o t a t i o n a l  components o f a g i v e n v a l u e  if D  i s independent of s t a t e and a f f e c t s  of J e q u a l l y .  The t e n s o r  all  1 3 hamiltonian rotational  gives level  rise  to a c e n t r i f u g a l  with given  J  into  e a c h c h a r a c t e r i z e d by symmetry It  is transitions  were o b s e r v e d  between  in t h i s  2b.Centrifugal  Distortion  standard  form power  series  symmetric  xyz  (e1.12)  (J  only  form  The e l e m e n t s all  J  2  J  2  x y z  2  J  symmetry  (e1.12)  thereby  the  2  J  2  J a aaa a 6  [1.12]  )  and T aaPP NO  = T„ /3/3aa 0  in c y l i n d r i c a l  tensor  tops are expressed in  but t h i s  of e m p i r i c a l  (el.12)  hamiltonian  data.  we c a n e l i m i n a t e  i s not  Through a  terms  from  r e d u c i n g t h e number o f e m p i r i c a l  i s no o b v i o u s u n i q u e  equation  parameters.  r e d u c t i o n and so t h e c h o i c e  is  Two o f t h e p r e f e r r e d c h o i c e s a r e known a s  A and S r e d u c t i o n s  symmetric t o p .  t r a n s f o r m as  (r12).  f o r the f i t t i n g  arbitrary.  i n our  z y x  c a n be w r i t t e n  transformation,  rather  that  /3 a  + J  are real  unitary  There  that  r  2  2  terms  of t h e o r t h o r h o m b i c h a m i l t o n i a n  have t h e r e q u i r e d  suitable  those  p  f o r m i n much t h e same way a s s p h e r i c a l tensor  J's  t.  i n D,,^ ( r l 2 )  aa/3 a /3  where t h e c o e f f i c i e n t s  spherical  label  Tops  1 L„T „J Jl + 4 ap a a p p a p  2  + *  sublevels  of g i v e n  expansion of H  species  a*/3  Equation  we r e t a i n  = LB J + a a a  r  sublevels  i n Asymmetric  tops,  H  its tetrahedral  work.  asymmetric  totally  of t h e  s p e c i e s C and l e v e l  these  For  the  splitting  - A f o r asymmetric  top, S for  14 The A r e d u c t i o n , (r2l),  first  i s t h e most p o p u l a r  The r e a s o n  and p e r h a p s t h e e a s i e s t  i s the r e l a t i v e  formalism into  asymmetric  ease  of i n c o r p o r a t i n g  e x i s t i n g asymmetric  diagonalization hamiltonian  p r o p o s e d by Watson i n 1968  routines.  matrix  In t h i s  top r i g i d case  i s t h e same a s t h a t  + <i> J ( j + 1 ) 3  - A  2  J  (J+1)K 2  + $ K  KJ  <JK±2|H^|JK> =  J R  + <i> J  3  + <I> J'(J+1 )K"  of the  K  2  2  2  + 1/2<£  K  2  [1.13] -l/28„[(K±2) +K ] + j\ 2  2  -K(K±1)][J(J+1)-(K±1)(K±2)]  energy  tri-diagonal  (r2l)  it  rotor  blocks  submatrices.  is  For  in Chapter  molecules  preferred choice non-zero energy four  /  [1.14]  2  into  four  the A  independent of t h e A r e d u c t i o n  i n the symmetric on " F a i l u r e  top l i m i t  of t h e  7).  that  are only  slightly  i s the S r e d u c t i o n matrix  1  A disadvantage  ill-conditioned  2  i n a Wang b a s i s  (see the d i s c u s s i o n l a t e r  Reductions"  two,  matrix  2  [J(J+1)  1  of a r i g i d  4  J(J+1 ) [ (K±2) + K ] +  JK  l^^n^^+K' ]  in the case  2  6  7  J  that  of t h e  rigid  +B*]}K y - A K  2  (J+1 ) K  1/4(B*-B*)-5 J(J+1) x y u  0 J (J+1)  is  rotor  rotor  - AjJMJ+D  reduction  its  the s t r u c t u r e  <JK|H*|JK> = 1 / 2 [ B * + B * ] J ( J + 1 ) + { B * - l / 2 [ B * L JL z x  As  to use.  elements  (r22).  asymmetric In  this  are found a l o n g ,  the  case  as w e l l as  and s i x o f f , t h e d i a g o n a l . To s i x t h d e g r e e t h e  non-zero matrix  elements  are  15  [1.15]  <JK±2|H^|JK> =  1/4[B^-By]+d,J(J+1)+h,  J (J+1) 2  2  •F(J,K)F(J,K±1 )  <JK±4|H^|JK> =  d +h j(j+l) 2  2  [1.16]  F(J,K)F(J,K±1)  • F(J,K±2)F(J,K±3)  [1.17]  <JK±6|H^|JK> = h F ( J , K ) F ( J , K ± 1 ) F J , K ± 2 ) F ( J , K ± 3 ) 3  • F(J,K±4)F(J,K±5) where  F(J,K±n)  one g e t s  four  T h e s e two set. of  degree the  = {J(J+1)-(K±n)(K±n±1)} independent h e p t a d i a g o n a l reductions  Characteristic  terms  of  two,  [1.18]  degree five  of  all  degree  just  useful  in angular  of  r e d u c t i o n s are  are  two out of reductions  and seven  but  it  is  of  not  an is  the  cases  observed t r a n s i t i o n s .  other  different  through l i n e a r  reductions  are  transformations  six.  surprising  limiting  of  of  degree  than  parameters  number  terms  more " a p p r o p r i a t e " of  basis  infinite  some a r e  and t y p e s  others  a Wang  submatrices.  momentum, t h r e e  four  equivalent  . In  All  that  d e p e n d i n g on  all  (r12).  related  The to  each  1 6 2c.Centrifugal Distortion For  symmetric  following H  terms  = B'J +(B' x H  +  z  J<  J 2  x  - D;  2  z  H^(J )J  +  s t a n d a r d form l e a v e s  ( J ) - D' J J 2  2  2  H^J'jJ  +  +  U  K z  H^JJ  +  "split"  [  (el.19)  specific T  ^  e  by Watson rotors,  s  is quite  point t t  for  C  3  general  n  term f o r  U s i n g the molecules  V  and  R  (order  <lpii  t  diagonalization contact  a sextic  schemes,  and Watson  H  (r12),  that  unduly c o m p l i c a t e  (el.19) removed.  in  entitled  Primes  "Failures  the c o n t a c t  form,  form  gp]_^  r  e  small  of  (r23)  treated  approximate theory  Unfortunately, of  terms  removed from t h e  the  Reductions")  identical in  new  (See  t r a n s f o r m a t i o n method of  but w i t h t h e  and  (examples:  literature.  the  in  and a r e  l e d to a host  the  reduced hamiltonian  are  a  t  respectively.  reductions  a final  in  degree  have been u s e d by O l s o n  O l s o n have g e n e r a l l y  one g e t s  the  ]  tops,  rigid  standard perturbation  of  Following  9  introduced for  equivalent  the. r e s u l t s  section  1  given  contribution)  Two e n t i r e l y  transformation,  r24,r25,r26)  of  '  groups i s  through s i x t h  elements  1  <J,K|H|J,K±6>.  of  as p e r t u r b a t i o n s .  symmetric  point  basis  The o f f - d i a g o n a l terms due t o magnitude  all  various  |J,K> H  non-zero matrix  <J,K|H |J,K±3>  for  group dependence b e i n g c o n t a i n e d  P-'-^ i 9  (r12).  introduces  later  - D;J  2  J K z  2  the  degree  J 2  3  Tops  - "split  Equation the  >  the  through s i x t h  -B')J  2  r  tops,  i n Symmetric  with  Watson equation  <J,K||J,K±3>  parameters  i n the  final  17 reduced h a m i l t o n i a n .  The  "split To a v e r y K=3 l e v e l s , given H  r  levels  e  ^  e  v  e  ^  s  t  to the  f o r the h  AE(K=±3)  are  6  [1-20]  >  o f an u n p r i m e d e q u a t i o n  3  it  does t o t h e  The j K j = 3 l e v e l s can be t r e a t e d  contribution  J K|= 3  to those  by t h e  (r27)  be t h e c a s e  equation.  higher  perturbation  [1.21]  3  as w i l l  are  e1.19.  are s p l i t  by a  of the  levels  |K|=3n where n = 1 , 2 , 3 . . . w h e n  of the s e c u l a r  identical  J  i s now g i v e n by  =2h J(J+1)[J(J+1)-2][J(J+1)-6]  o r more g e n e r a l l y , solution  +  term  t  name f r o m . w h a t  s  The K=3 l e v e l s  expression  3 ^  part  are considered).  term.  p]_^  c a n be i g n o r e d a n d t h e e n e r g y  g  (actually  degrees 3  c  h  S  good a p p r o x i m a t i o n , w i t h t h e e x c e p t i o n  by t h e d i a g o n a l  split  h  H  =  H  in this  The e n e r g y  of the asymmetric  study,  matrix  top S  by  elements  reduction  with d!=d =h,=h =0. 2  2  The p r o b l e m o f t h e f a i l u r e reduction in-depth  i n the s p h e r i c a l look  found l a t e r Reduct i o n s " .  at  top l i m i t  the mechanics  in a section  of the symmetric t o p a l o n g w i t h a more  of the r e d u c t i o n  entitled  "Failures  i s t o be  of the  18 3.  HYPERFINE PERTURBATIONS TO THE ENERGY LEVELS  In  this  rotational  section  energy  perturbations  spin the  rotation low  it  are  will  turn  of  s p i n s of  3a.Nuclear  of  (or  of  the  symmetry  spin  greater  is  field  the the  negative  "measurable"  consider  asymmetry will  and in  specific (AsH )  where  3  the  nuclear ignore  the  try  and t h e  of  equal  related  that  charge  d i s t r i b u t e d by t h e  an  with  an  axially a measure  of  from  q u a d r u p o l e moment.  axis  this  is  greater  moment i s  respectively.  If  the  or  positive nuclear  t o one t h e q u a d r u p o l e moment  to  the  reflects necessary  symmetry  For  density  experimentally  function  gradient  I  or  to a l i g n  charge .density  symmetry  than  p o s s e s s e s an  charge d i s t r i b u t i o n  intrinsic net  simple  field  The  be t o a r s i n e  gradient.  (oblate)  q u a d r u p o l e moment Q i s  the  studies  deuteriums).  nuclear  nuclear  or  is  the  be c o n s i d e r e d  only.  nucleus with uniform charge  (prolate) is  in  n u c l e u s and c a n  intrinsic  electric  a l o n g the  I  will  the  The  quadrupole e f f e c t s  limit  the a r s e n i c  q u a d r u p o l e moment t h a t  D e p e n d i n g on whether less  field  to  Q u a d r u p o l e Moments  the d e v i a t i o n spherical  important  we need o n l y  hydrogens  intramolecular symmetric  be  perturbations section.  These e f f e c t s  A n u c l e u s w i t h an electric  to  last  t h e s e methods w i l l  effects the  the  out  electric  be f o u n d t h a t  hyperfine  of  look at  due t o n u c l e a r  effects.  incident  application  levels  that  presented here  we s h a l l  axis.  determinable intrinsic  the a v e r a g i n g m o t i o n of  I  moment by a of  the  (r28),  about  19 If  our nucleus  external  field,  i s in a rotating  molecule,  i n a weak  we c a n g e t a c o u p l i n g o f / a n d / t o form a  resultant F [1.22]  F=I+J 1 /2 with eigenvalues values  [F(F+1)]  gradient  product Since  energy  to the  of our q u a d r u p o l e n u c l e i  of t h e m o l e c u l e  i s given  of the q u a d r u p o l e and f i e l d  the primary  gradient the  where F i s r e s t r i c t e d  J + I , J + I - 1 | J - I | .  The i n t e r a c t i o n field  '  will  contribution  splittings  ultimately  as the s c a l a r  gradient  tensors  to the e l e c t r i c  be due t o t h e e l e c t r o n s  i n the  nearest  (r29).  field the nucleus,  p r o d u c e d by o u r i n t e r a c t i o n  us some i d e a a s t o how t h e e l e c t r o n s  are.distributed  give in  bonding. A u s e f u l model t o have  i n mind d u r i n g d i s c u s s i o n s o f  quadrupole coupling  i s the s e m i - c l a s s i c a l  "vector"  this  the e l e c t r o n  are fixed  picture,  since  molecule,  rotation  averaging  of the f i e l d  Rapid  rotation  orbitals  of the molecule gradient  averages  will  about  result  axis  rotation  a symmetry  f o r the f i e l d  axis  e x p e r i e n c e d by t h e q u a d r u p o l a r dependent rotated inertial  field  gradients  from the n u c l e a r axes  i n the  axis.  t h e components o f t h e f i e l d  to the r o t a t i o n  the f i e l d fixed  in the molecule  frame fixed  gradient  t o z e r o a n d makes t h e  nuclei.  gradient  To a v o i d  gradient  time  tensor  is  to the p r i n c i p a l frame.  In  i n an  the r o t a t i o n  perpendicular axis  model.  The p r i n c i p a l  20 inertial  axes are  vector  / is  vector  /,  axis  the  axes a l o n g which  the a n g u l a r  d e c o m p o s e d . The r o t a t i o n a l  then,  p r e c e s s e s about  the  angular  total  momentum  momentum  angular  momentum  d e f i n e d by F. In  a first  order  assumed slow and t h e axis.  It  is  degeneracies  W  in  energy  the  levels  charges,  first  of  the  and 2 i s  nuclear  for  linear  averaged for  about  correction  potential the  field  nuclear  f i x e d frame  to  the  of  to  symmetry  the m o l e c u l e  the  on  axis. fixed  Rotating frame  2  =  \J¥ /J 7  tops  J  (nucleus ,K  2J + 3  on  axis)  XBi / 7  1  [ 1  2  v  R  7  <J  2  9  > are  ,  2  JTJ+TT  2  where t h e  yields  u.^4j  and f o r a s y m m e t r i c t o p s /9 0\ _ 2J /9 A <J > \J Z /J ~ (2J+3) (2J+1 ) g Xag / g I  from  r  Z  2  the  molecules,  2  /a <A  (r30).  extra-nuclear  /3 «A _ _ J /d <j>\ \WT /J ~ 2J+3 YaFV symmetric  /  H.23J  charge  the  the  near  o b t a i n e d by C a s i m i r  e is  is  moderate  no r o t a t i o n a l  order  first  J about F  (3/4)C(C+1 )-I (1 + 1 )J(J+1 )" \VF/J 2J12J-1)I(21-1)  <j> i s  the  is  approximation  where C = F ( F + 1 ) - I ( 1 + 1 ) - J ( J + 1 ) , electron,  p r e c e s s i o n of  gradient  the case  t o use t h e  = eQ  g  field  an e x c e l l e n t  q u a d r u p o l e moments  rotational  theory  calculated  u  as  2  i n G o r d y and Cook  -  2 5 ]  [1.26] (r3l).  21 When h i g h e r  order  effects  necessary  t o employ an e x a c t  this  the  case  precession gradient matrix  vector  of  is  are  solution  averaged  The s e l e c t i o n  about given  rules  The q u a d r u p o l e  in  it  is  is  The  that  the field  appropriate  are  AI=0. are  In  (r32).  transitions  where AJ=AF  often  the p r o b l e m .  i g n o r e d and t h e  reference  for  splitting  be  the F a x i s .  AF = 0 , ± 1 , The t r a n s i t i o n s  to  model i n t e r p r e t a t i o n  / about F cannot  elements are  important,  [1 . 2 7 ] f o u n d t o be t h e  c o n f i r m s the  strongest.  rotational  assignments. The e x p e r i m e n t a l l y coupling  determinable  i  trace  of  are  the  constants  y  i where  parameters  and j  refer  to  the  principal  the c o u p l i n g t e n s o r  Laplace's  constrained  g,  axis.  The  t o z e r o by  equation.  3b.Spin-rotation  Coupling  The n u c l e a r  magnetic  can  interact  the  nucleus  w i t h the  to  the  moment of  a nucleus with spin  magnetic  rotation  splitting  D e f i n i n g magnetic of  (/•/)  effective  by m o l e c u l a r  contributions  convention  is  inertial,  of  to give  (r31)  produced  at  further  rotational  coupling constants  Gordy and Cook  field  I  levels.  C^g w i t h t h e  one c a n w r i t e  the  sign first  22 order  correction  asymmetric  tops  .  to  are C  2J(J+1)  =C =C y i e l d s xx yy N  axial  the  J  U  EFFECTS  In  the  VT  of  g  coupling constants Choosing C  above  =C„ and ZZ  K  3  2  [F(F+1)-I(1+1)-j(j+1)]  ;  OF EXTERNAL  presence  the  gg  symmetry.  2 C'N + ( CK- CN )J-( J + 1 )  4.  levels  following K  m  g  t o p two of  b e c a u s e of XT  energy  F(F+1)-I(I+1)-J(J+1)  a symmetric  equal  rotational  as  m In  the  of  [1.30]  ELECTRIC FIELDS  an a p p l i e d e x t e r n a l  new t e r m must be added t o our  rotational  electric  energy  field  a  level  hamiltonian H = E  where M i s electric matrix  the  mixing  of is  For  tens  typical shifts  the  are  level.  this  laboratory of  so s l i g h t  retained  that  energy  rotor  the  and one t a l k s  final  energy  known as  fields  of  differences  For  these  zero f i e l d of  external  wavefunctions. is  electric level  the  linear  effect  megahertz. that  this  are  original  fields  t o h u n d r e d s of  generally  a certain  of  electric  one s e e s  numbers J , K , of  (mixings)  and E i s  D i a g o n a l i z a t i o n of  in e i g e n f u n c t i o n s  static  shift.  volts/cm. order  vector.  results  For Stark  d i p o l e moment o p e r a t o r  field  combinations  [1.31]  -M-E  the  the 2000  on  the  fields  the  quantum  Stark  effect  23 For coupled  oscillating  external  fields  rotational  i n a t i m e d e p e n d e n t way t o g i v e  levels  a transition  are  state  (r33). 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  Contributions)  Rotat ional The e l e c t r i c  field  i s c o u p l e d t o our r o t a t i n g  t h r o u g h t h e d i p o l e moment Classically linear that  a molecule  states  operator.  one p i c t u r e s  separation  a d i p o l e moment as due t o a  of o p p o s i t e c h a r g e s .  in d i f f e r e n t  s h o u l d have d i f f e r e n t  averaging  molecule  One s h o u l d e x p e c t  vibrational  and r o t a t i o n a l  d i p o l e moments r e f l e c t i n g t h e  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  states. The example o f a C ^ illustrate sort  the p o i n t .  of matrix  allowed  they  (AJ=1,AK=0)  d i p o l e moment p a r a l l e l is  symmetric  top w i l l  perhaps  The d i p o l e moments a r e l a b e l e d by t h e  elements  R branch  v  are associated with. transition  the  t o t h e symmetry a x i s  F o r an  "effective" of the m o l e c u l e ,  g i v e n by ( r 3 4 ) u (J,K) r  = Mo  +  MJ(J+1)  + M K ...  2  [1.32]  2  R  where MO i s t h e e q u i l i b r i u m d i p o l e moment a n d M j a n d M corrections  due t o c e n t r i f u g a l  distortion.  corrections  are t y p i c a l l y  to s i x orders  smaller  than  Mo ( r 3 5 ) .  allowed Q branches  four  A similar  (r35).  k  the  The d i s t o r t i o n of magnitude  expression exists for  24 Recall rotational  that  for  energy  Elimination  of  C^  v  levels  the  <JK|H  symmetric  tops  by J and K i s | J K ± 3 > terms  reduced h a m i l t o n i a n  meant  d e s c r i b e d as  c o m b i n a t i o n s of  mixing,  although  transitions dipole  linear  is  responsible  constant  of  drives  the  (r2,r12,r14) giving small  the  and t h e  admixture matrix  of  is  above  by  be  This  allowed distortion transitions.  element,  distortion r  , xxxz'  mechanics  of  < J K ± 3 | and e a c h  | J , K > we g e t  the  unitary  transformation  Since  each  state  state  |J,K±3> has a  a non-zero value  for  < J , K | has a  the  small Q branch  element  where t h e  above  retaining  the  order  implicit  in  dipole angular  a change require reason  had t o  the q u a r t i c  matrix  V  2  Y  V  7  °  M  in  equation  z  corresponds to  J K labeling,  perturbation). equation  because  (el.33)  momentum a l o n g  the  above,  the  speed about  transitions  distortion is  often  convention  mixing  is  t o change  the  symmetry the  the  a perpendicular  (i.e.  states  driven  by of  Classically  axis axis.  d i p o l e moment r e s p o n s i b l e called  of  component  axis.  symmetry to  small  m i x i n g of  transitions  a d i p o l e moment p e r p e n d i c u l a r the  our  O v e r l o o k i n g the  moment would a p p e a r  rotation  [1.33]  ( B -B ) X  the  the  ..  <JK|M1JK±3> =  this  at  |JK> states. above  of  correct.  states  so-called  set  T  first  quite  f o r b i d d e n AK=±3  reduced h a m i l t o n i a n .  admixture  labeling  to a r r i v e  the  the  the  i n t r o d u c e d by t h e 1  in  for  mixing  not  rotational  insignificant  moment t h a t  The m a g n i t u d e  that  the  for  moment.  would For  this  AK=±3  25  Labeling J  and K i s  our  then  symmetric  just  of  this  as  s u p e r p o s i t i o n s of  top r o t a t i o n a l  an a p p r o x i m a t i o n . I t  energy is  the  "forbidden"  states  that  explains  by  breakdown  a p p r o x i m a t i o n and s u b s e q u e n t d e s c r i p t i o n |JK>  levels  of  states  our  transitions.  Vibrat ional In rotor  the  J,K  last  section  we saw  d e s c r i p t i o n of  the  breakdown of  rotational  states.  d i s t o r t i o n coupled states  J,K  states  oscillating  where t h e  external  coupled to a r o t a t i o n a l were was  with states  state via  introduced. A perpendicular identified In  this  centrifugal  g i v i n g AK=±3  section  on t h e  terms  vibration-rotation  the  harmonic  a more a c c u r a t e Linear the of  "higher"  higher this  vibrational  states  allow J,K  states.  of  hamiltonian states  states  allows  that  (r14).  In  During t r a n s i t i o n  expanded i n a T a y l o r  series  the  state  is  the  as a means  admixtures  attributed  the  result  of  is  to  between  d i p o l e moment o p e r a t o r  about  case  This  small  transitions  couple  this  state  usually  new  can  states.  should include  ground r o t a t i o n a l Here  operator  rotational  c o u p l i n g in ammonia-like molecules  and J , K ± 3 .  was  d i p o l e moment  distortion  oscillator  harmonic o s c i l l a t o r  vibrational  again  state  effect  a vibrational  harmonic  ground v i b r a t i o n a l  the  vibration  d e s c r i p t i o n of  c o m b i n a t i o n of  t h e d i p o l e moment  centrifugal  oscillator  field  vibration-rotation  Including  the  electric  Transition  transitions.  interaction. in  rigid  Centrifugal  J,K±3.  distortion  we c o n s i d e r  distortion  the  equilibrium  is  to  26 configuration  as a f u n c t i o n  M = M  The  first  term  and e s p e c i a l l y is  evaluated  + I  e  is  6  the  basis  distortion  The s e c o n d t e r m  is  will  of  the  of  allowed  x  x  z  >  rigid  three  off  oscillator  this  case  e  diagonal  last  section.  operator dipole  contribution of  when ju  basis  the  the  The o v e r a l l  harmonic In  rotor  transition  term because  transitions  transitions  d i s c u s s e d in  states.  state.  seen then causes  oscillator  and t h a t  a  final  that  from  the  states  the  matrix the  slight  AK=±3  interaction  centrifugal  a breakdown  harmonic  linear  rotational  into  the  transitions p r o d u c e d by  distortion.  We have rotation  x  by a v i b r a t i o n - r o t a t i o n  centrifugal  by  T  standard  this  higher  ground v i b r a t i o n a l are  the  "mixed"  t h e n have a s m a l l  non-zero elements "admixture"  the  £K ,  [1.34]  i  for  of  terms  connecting v i b r a t i o n a l element  i  normal c o o r d i n a t e s  forbidden rotational  s t a t e s — a m i x i n g due t o centrifugal  the  (3M /3Q )Q  i  responsible  the  in  of  of  description  description  c o m b i n a t i o n of  rigid  both a r i g i d  of  of  distortion  rotation  rotor-harmonic  rotor  and  "ro-vibronic"  induced and  vibration  states  is  oscillator  as a  basis  states. The h a r m o n i c  oscillator  d e s c r i p t i o n of  breaks  down when a n h a r m o n i c i t y  effect  results  of for  harmonic  in v i b r a t i o n a l  oscillator  states  forbidden t r a n s i t i o n s .  is  taken  states  as  and a g a i n  These  into  vibration  also  account.  This  linear is  combinations  also  transitions  are  responsible often  27 called  M.V.  transitions  Venkateswarlu  (r36)  anharmonicity  is  transitions For  in  the  after  who f i r s t  to allow  degenerate  p u r p o s e s of  other  For  this  For  vibrations  angle  perpendicular  doubly  pictorial  vibrational  small  the  of  amounts of  allowed  degenerate  see  will  not  mixing  e q u i l i b r i u m symmetry  leading  to  book by  of  are  of  the  parallel  bond s t r e t c h cu,  antisymmetric  vibrations  corresponding  o> and a n g l e 3  change the  two  bend CJ^  into  the  our  ground  molecular  symmetry.  will  be t o a l t e r for  the the  amounts o f  an  perpendicular  vibration,  however,  means t h e of  state  small  will  in  (for  ground v i b r a t i o n a l  vibrations  we mix  to  plOO)).  d i p o l e moment r e s p o n s i b l e  If  description is  excellent  symmetric  (r37  into  parallel  'permanent'  This  be  centrifugal  symmetric  vibrations  e q u i l i b r i u m symmetry  adequate.  p214).  only  mechanisms  states  plus  bond s t r e t c h  transitions.  oscillator  the  2  e x c i t e d doubly degenerate the  the  totally  bend co r  parallel  state  effect  along axis  the  representation  Mixing  The n e t  see  corresponding to  antisymmetric  other  rotational  (r14  we s h a l l  m i x i n g c a u s e d by  vibrational  There are  and s y m m e t r i c  the  states  of  (r14).  ammonia t h e  symmetries.  forbidden  thesis  a d i s c u s s i o n of  and A l i e v  M i z u s h i m a and  p r o p o s e d t h e m . The e f f e c t  vibrational  forbidden t r a n s i t i o n s  Papousek  authors  previously  concerned with v i b r a t i o n a l distortion.  the  be d e s t r o y e d . rigid  rotor,  our m o l e c u l e  c o u r s e what we a r e  is  no  Destroying  the  harmonic longer  saying  when we  write  28 the  ro-vibronic  rigid  rotor In  to  the  state,  energy  harmonic  due 0  where t h e  small  inertia  = 9I -'/9Q„  of  rigid  transitions; molecule in the and  the  the  9M/9Q'S  at  field  to normal  are  is  of  the  mixing, [1.35]  the  coordinates  vibration in  c o u p l e d t o our  9M/9Q.  analogues  rotational vibrating  H e r e we have t h e  d i p o l e moment g e n e r a t e d  normal c o o r d i n a t e s  vibrational  e q u i l i b r i u m of  moment o p e r a t o r  operator  perpendicular  ground  du 1 x 9Q«  with respect  electric  through the  perpendicular  change  by a change vibrations CJ  in 3  cji,.  The n e t top  derivatives  dipole  contribution  induced v i b r a t i o n a l  are  the  states. following  3  . The  c o m b i n a t i o n s of  the  u'j  1  rotor  in  xx x _^ a ii 9Q w«  I -'  1  basis  9M  33  a's  Q ,;a*-' the  • xx 2  linear  gave t h e  d i p o l e moment  = 2B  x  moments o f T  (r2,r6)  to c e n t r i f u g a l l y  X X  as  oscillator  1969 Watson distortion  levels  is  the  distortion  sum of  the  d i p o l e moment  vibrational  and  for  a C^  symmetric  v  rotational  contributions,  D  where h i g h e r For because  =  order  spherical ju  e  vibrational  is  zero  0  X  X  x  +  T  xxxz 2(B -B ) x z  corrections tops in  the  M o have been  rotational  equation  "borrowing".  to  [1.36]  Mo  mixing  e 1 . 3 4 and M  d  is  ignored. is  zero  solely  due  to  29 4b.Intensities  and S e l e c t i o n  The p a r a m e t e r intensities which  for  energy  is  2  of  interest  to a f i n a l  „ /v S.  2  v  where the  0  is  the  rotational  0  the  P is  at  also  rotational  line  matrix  strengths  field  that to S  where t h e  2  Strictly  i  low  Ef  is  i  with  (rl7,r3l).  rules  the are  the  r. [1.37]  product  functions,  speed of  maximum Av  half  light.  K  of is  The  (FWHM) d i v i d e d by  pressure contained  broadening in  the  S^.  are  derived  from the  rotating  molecule,  g  over  and 0 of  the f.  d i p o l e moment  oscillating  [1.38]  2  three  g directions  The M a r e y  the  and  all  d i p o l e moment  g axes.  incident  b r o a d e n e d by p r e s s u r e and power  the  = Z | <i | ju | f > |  i f  a l o n g the  at  partition  external  the  coefficient  level  Q is  c o u p l e the  summation i s  c o m p o n e n t s a of operators  is  strengths  elements  electric  absorption  -Ei/KT -Ef/KT-, [ e -e J  known as  The s e l e c t i o n  initial  frequency,  and c  parameter.  The l i n e  microwave  with energy r  i f  an  and v i b r a t i o n a l  full-width  pressure  f  transition  Boltzmann's constant transition  between  level  8TT P 3CQKTAIV  max  for  t h e maximum o r peak a b s o r p t i o n  a transition  Ei  Rules  powers where l i n e s  effects  and e f f e c t s  b r o a d e n i n g can be i g n o r e d .  are  due t o  only saturation  30 In  order  general  that  the  transition  line  strength  selection  rule  be n o n - z e r o one has  for  the  total  the  angular  momentum AJ giving  s o - c a l l e d P,  (when h y p e r f i n e Further  In  order  symmetry angular  Tops  rules  that  the  A, one  and t a b u l a t e d line  by t h e  further  molecular Asymmetric  strength  A,—A , 2  molecular  type.  rule  be n o n - z e r o one has  E—E and F , - F AJ  so-called  strengths  nuclear  = 0,  distortion  are  evaluated  respectively. these  along with  2  ± 1 . For  values  For  Td  strengths.  5,  these  2 and 3 f o r  specific  by t h e  the  have  by m u l t i p l y i n g  factors  the  a general  (r17)  reduced l i n e  s p i n degeneracy  multiplies  examples  s q u a r e of  the  d i p o l e moment M « D  Tops  Along w i t h the and K i n g  general  (r38)  components of  the  the  symmetry  following  strengths.  respectively  one has AF = 0 , ± 1 ) .  d i p o l e D o r n e y and Watson  E and F t r a n s i t i o n s  Hainer  line  momentum s e l e c t i o n  The a c t u a l  important  d e p e n d on s p e c i f i c  rules  .molecule with u n i t  values  is  transitions  (Td)  selection  evaluated  [ 1 .39]  Q and R t y p e  structure  selection  Spherical  = -1,0,1  selection  have shown t h a t  d i p o l e moment a l o n g restrictions  rules for  AJ  Cross,  non-zero  inertial  apply  = 0,±1  for  axes a, non-zero  b,  c  line  31 Table Symmetry  selection  rules  1.2  for  asymmetric  Allowed  top  Axes p a r a l l e l  Transitions  transitions to  D i p o l e Moment  KaKc<—>KaKc  where o ,  ee<—>eo  a  oo<—>oe  a  ee<—>oo  b  eo<—>oe  b  ee<—>oe  c  oo<—>eo  c  e refer  Asymmetric written  t o odd or even v a l u e s top l i n e  in c l o s e d form.  dependent top l i n e Symmetric  linear  (Forbidden  C^y m o l e c u l e s  are  this  b r a n c h AK=±3 t r a n s i t i o n s if  =  T^D  the  C  3  ).  symmetric  the  three  With a r i g i d  sort  are  given  AJ = rules  equivalent  rotor  often  called  strengths by Watson  atoms  t h e AK =  f o r b i d d e n so t h a t  The l i n e  are  tops are  The AK s e l e c t i o n  spectra  forbidden of  the Q  (r2)  as,  or  HJ+K)(j+K-1)(J+K-2)(J±K+1)(j±K+2)(j±K+3)} 2J+1 4J(J+1)  where t h e  be  molecule  c l o s e d form  symmetric  V  strictly  d i s t o r t i o n moment s p e c t r a .  S  in general  o b t a i n e d by  as due t o  (r6).  transitions of  for  (n=1,2,3  c a n be seen p i c t o r i a l l y  with t r a n s i t i o n s  cannot  Transitions)  rules  0,±1 and AK = 0,±3n  ±3,±6,...  Ka and K c .  (r38).  The s e l e c t i o n  in a l l  They a r e  c o m b i n a t i o n of  strengths Tops  strengths  of  factor  [1.40] g  R  takes  into  account  the K degeneracy  and  32 the  nuclear  defines ,  the  spin  statistical  various  weights.  numerical  factors  t h e d i s t o r t i o n d i p o l e moment  because  in e a r l i e r  studies  Equation in  the  This  is  distortion  dipole  the  (e1.36).  s t r e n g t h was g i v e n as  times  the  4c.Stark In  this  value  case  the  given  by e q u a t i o n  the  the  external  our s y s t e m r e q u i r e s  a new quantum number M, where M , t h e  for  values  transition  r a n g i n g from -J  are  AM = 0,  the  oscillating electric  field  to,  or p e r p e n d i c u l a r t o ,  the  In  general  for  introduction  space  rotational  series Stark  i n powers effect  second order field  first  levels  et  of  is  situation  have e s s e n t i a l l y  fields usually  the e l e c t r i c  depends l i n e a r l y effect  The  ,can  selection  parallel  fixed quantization  and hence t h e AM = 0 t r a n s i t i o n s small Stark  field  being r e s p e c t i v e l y  direction  the  / along  and ± 1 , c o r r e s p o n d i n g t o  our e x p e r i m e n t a l fields  to J .  electric  For  four  electric  component of  microwave  the  the  s p a c e - f i x e d a x i s as d e f i n e d by t h e e x t e r n a l  rules  equation  (e1.40).  p r e s e n c e of an a p p l i e d s t a t i c  t a k e on i n t e g e r  Z.  by  Effect  f i e l d a d e s c r i p t i o n of of  line  value given  of  important  moment was d e f i n e d as o n e - h a l f In  also  the d e f i n i t i o n  (el.36).  (r39)  (e1.40)  the  the  axis  static  and  same  dominate.  u s e d t h e e n e r g y change e x p a n d e d as a f i e l d E.  on t h e  depends on t h e  perturbation  A first  electric  s q u a r e of  in  order  f i e l d and a the  electric  cetera.  Watson  (r40)  has g i v e n  order  Stark  effects  in  the  symmetry  the case  of  requirements negligible  for  33 hyperfine Stark  effects  effects  molecules  a n d no a c c i d e n t a l  are e s p e c i a l l y  with  at seeing  levels  with the e l e c t r i c  on E  2  we have For  Quadratic  shifts  fields Stark  near  for studying  effects  degenerate  spherical levels  effects.  available  i n most  although they  shifts  field  see  symmetry  the Stark  i s when  (r3l)).  the doubly degenerate  For the E l e v e l s  depend  (the e x c e p t i o n  levels,  the r e q u i r e d  better  shifted  a n d f o r t h e same  energy  tops only have  due t o S t a r k  effects  p r o d u c e much s m a l l e r  rovibronic Stark  frequency  a r e second order  generally  important  Linear  " s m a l l " d i p o l e moments a s one h a s a  chance  laboratories.  degeneracies.  E  t o show  energy  linear  shifts  are  g i v e n by  . Jme 6  with  ( t  where  (t",f)  t , )  =  A  Je  (  t  "'  t  ,  )  M  "D  E  [  = 0.50344{C(J, e , f ) - C ( J , e , t " ) }  the C ( J , e , t )  calculated For  asymmetric close E  T  are the signed Stark  and t a b u l a t e d  accidentally  6  ,  1  '  4  1  ]  3  A.,  3  ' '  (t",t')  coefficients  by D o r n e y a n d Watson  t o p s where one s e e s  the r o t a t i o n a l  strictly  <> = (A"+B"M )E 2  i s i n MHz f o r M  2  n  2  [1.42]  as  (rl7). levels  second o r d e r  a r e not  shifts [1.43]  i n Debye and E i n V o l t s / c m .  34 where A"  and B"  dipole matrix For  depend i n an e l a b o r a t e  elements  symmetric  and e n e r g y  tops  the  first  way  on J ,  M, K a ,  level  differences  order  correction  Kc,  (r4l). is  ( r 1 1 , r 31 ) E  5.  be t h a t  of  the  derivatives Generally trends  of  it  is  in  force  spectra  of  smaller  set  There which  is  All  the  treatment  all  the  of  fitting  somewhat  of  allows for  isotopic  the  species  in  rotation  On t h e  of  of other  picture  and  terms  different  hand to  be  rotation  of  a much  parameters. an e x t e n s i v e  referenced  in  Woodward  of  choice  vibration  problem.  qualitative  symmetry,  tops.  to  isotopic  as one  a quantitative  in  literature the  the  literature  (r43)).  detailed  For  accounts  on t h e  excellent  Ammonia-like molecules are  represented  in  the  s p e c t r a has  parameters  arbitrary  asymmetric  fully  in  independent  to e s t a b l i s h  due t o c h a n g e s  case  accounts  is  simplifications  difficult  and t h e  the  field  that  the  fc  vibration-rotation  empirical  derivatives  reduction  to  r1 441  MKM J(J+1)  field.  quite the  axes,  (see  -  =  a m o l e c u l e c a n now be t r e a t e d  principal  (r42).  >  molecular  force  between  isotopic  of  1  t h e more a e s t h e t i c  u n d e r s t a n d i n g of  drawn  (  FORCE F I E L D S  One of  a  s  review  especially  often  practical in C y v i n ' s  as  subject, of  Duncan  well  textbook  p r o b l e m s one books  much  examples is  (r44,r45).  directed  35 For  our p u r p o s e s  approximation to force  for  the  quadratic  of  field  as a v a l e n c y the  constant  fields  fundamental  force  force  of  The  c h a n g e s of  a bond or a n g l e  3  angle,  we have  f  and f ^ '  where r  r  It  is  often  f  f^  (the  constants  simple (r43)  linear pg.  In are  rp a  distinct  where r  how  other  is  force  valence  for  an  force  part  the  bonds and of  c  the  angle. the  molecular  symmetry c o o r d i n a t e s .  The f i n a l  known  constant  relate  t o make use of  c o m b i n a t i o n s of  symmetry  this  species  constants force  In  are  can  then  constants  (see  193).  his  r e v i e w Duncan  well characterized  approximation. include  linear  constant  changes of  belong to d i f f e r e n t  separately.  the  six  interaction  way  be t r e a t e d  in  force  force  internal  that  of  simple  only  (the  symmetry and d e f i n e vibrations  displacement  a representation  o p p o s i t e the  convenient  the  the m o l e c u l e  are  and f  is  in  case  the c o n s t a n t  In  affect  00  restoring  this  there are  force  t , pp  rr  the  of  and t h e  interaction  In  vibrations.  a bond),  constants.  harmonic  r e p r e s e n t e d as  these  i n a bond a n g l e ) ,  the  as  constants.  field  stretching  increase  angles,  are  to e s t a b l i s h  linear  ammonia-like molecules  quadratic  for  is  The v i b r a t i o n s  potential  superpositions  field.  motion  with a force  proportionality.  suffices  force  interatomic  coordinate  For  it  of  list  lists  the  by a q u a d r a t i c  The e x p e r i m e n t a l  from t h a t  (coefficients  (r42)  quartic  data  parameters  force  field  available  distortion  f o u r t h degree angular  that  allowed  us  constants  momentum  operators)  to  36 for  various  constants  isotopic  species along with C o r i o l i s  and v i b r a t i o n a l  frequencies  for  the  coupling  symmetric  tops. The v i b r a t i o n a l eigenvalues  of  frequencies  the m a t r i x GF,  constants  and G the  matrix  eigenvectors  of  coordinate  inverse is  are  where F i s  kinetic  called  transformation matrix  symmetrized B matrix  coordinate  representation)  displacement S such  (i.e.  the m a t r i x  the L matrix  or  of  force  The  the  normal  (r46). be e x p r e s s e d i n terms  i n an  that  the  energy m a t r i x .  The f o r m a l i s m can c o n v e n i e n t l y the  o b t a i n e d as  internal  transforms  c o o r d i n a t e s X and i n t e r n a l  of  symmetry  between  symmetry  Cartesian  coordinates  that S = BX  The e l e m e n t s quantities  of  [1.45]  the B m a t r i x  are  purely  d e t e r m i n e d as components of  geometrical the W i l s o n  S-vectors  (r46). The G m a t r i x and m o l e c u l a r  is  given  geometry  then  1  1  reciprocal  is  terms of  given  a diagonal  a t o m i c masses  by  masses  [1 . 4 6 ]  T  3Nx3N (N=number  of  atoms)  matrix  (r44).  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 are  atomic  as  G = Birr B where irr  in  (the  zeta  constants)  of  37  S where  = (L- )Bm- M B (L" )  a  1  the elements  ( M ( 1 )  where  j57  =  e  1  of M  a/3  1  [1.47]  T  (*'P>7 = x y o r z )  7  or a n t i c y c l i c  otherwise  T  a r e g i v e n by  a  e i s the L e v i - C i v i t a  cyclic  a  anti-symmetrizer  permutation  [1.48] equal  of xyz and equal  ±1 f o r to zero  (r47,r48).  The q u a r t i c  centrifugal  combinations of T  d i s t o r t i o n constants are linear  ' s where  0  apyo T  = ("KV2hc)(I I  aM  a  and where  X,, = 47r c o>  frequency  (r12).  matrix  2  2  I I )-'  p  7  and u  2  f o r the K ^ v i b r a t i o n  t h  The e l e m e n t s  tensor  a/3  =  L  vibration  of the " s m a l l a " derivatives  a,/3 component  normal c o o r d i n a t e Q of the small  [1.49]  K  i s the harmonic  v  and a r e p a r t i a l  fc  to the K  6  The a j ^ ' s a r e e l e m e n t s  e q u i l i b r i u m of the i n e r t i a l respect  rfa^ /X  6  R  at  with  (a^= ( 9I ^/9Q ) )  " a " matrix  a  K  0  a r e g i v e n by  -, a/3 v  [  K  5  0  ]  where Y and where  R  coordinates  e  a / 3  = 2BM (M ) R / 3  i s a 3Nx1 v e c t o r f o r t h e N atoms  a  T  e  of e q u i l i b r i u m (r12,r44).  [1.51] Cartesian  38 6.  DETERMINATION OF MOLECULAR  The r o t a t i o n a l rotational  spectroscopy are  determination on m o l e c u l a r Cyvin  constants  (r44  of m o l e c u l a r structure  Chapt  meaningful  12)  are  the  important  determination  calculable r  and r  point  the  atomic  vibration  structure  is  in  consistent  approximation.  In  are  those  molecule.  An r  potential  constants.  g  of  The r o t a t i o n a l not  generally  the  various  constants  are  for  these  (KW) is  effective  of  constants  structural  determinations."  * In  to preserve  order  with c e n t r i f u g a l  an  r  state. force  field  the  atomic  structure.  In  that  constants.  gives  us t h e  model of  Kivelson  use  forming  linear Correcting  effective  and W i l s o n  we s h a l l  are  rotational  r e d u c t i o n dependent  distortion  order  from s p e c t r a  determinable  Kivelson  zero  An  with higher  determined  a classical  distortion  the  vibrationless  consistent  contributions  constants  and  physically  over  structure  g  empirically  centrifugal  distortion  rotational  these  averaged  a hypothetical  found to c o n t a i n  reviews  Z  connected to a p h y s i c a l  c o m b i n a t i o n s of  In  with the q u a d r a t i c  is  the  rotational  structures.  constants  reductions,  The two  ground v i b r a t i o n a l  structure  in  by K u c h i t s u  6  an e q u i l i b r i u m r  positions  are  from  data  Two e x c e l l e n t  (r49).  p o s i t i o n s are  the  microwave  sources of  structures.  Z  structure  o b t a i n e d from  and R o b i e t t e  structures  spectroscopy  STRUCTURE  in  (r50).  It  subsequent  a rigid  and W i l s o n g i v e  rotor further  39 For  symmetric  tops  the  ground v i b r a t i o n a l  elimination  of  to give  reduced h a m i l t o n i a n  the  our  order  of  rotational  of  the  of  called  to vary  only  slightly  derived  S-reduction  B  x  =  \ B  B  =  B  2D  J  +  D  JK  y  +  2D  J  +  D  JK  equations  The S c r i p t of  + 2D  reduction,  effective  +  constants, still  parameters,  in  the  from t h e  has  constants, from  x  = B  x  *(cont'd)corrections Corrections  of  this  +  sort  constants found  empirical For  the  2 d  < <  +  +  4  d  4  d  * * [1.52]  2 i  for  although  other  invariant  be c o r r e c t e d the  reductions.  to give  to  the  the  ground v i b r a t i o n a l  their  effective  were n o t  made.  choice  K.W.  state,  (1/2)T' ' xxyy  to  defined  are  has,(r12) B  K.W.  p r o b l e m of  (r12,r2l)  reductions.  c a n be w r i t t e n  must  not  The r o t a t i o n a l  "Script"  2 d  "  + 6d  J  the  example  +  z  Watson  when c a l c u l a t e d  for  into  of  tops.  to a l l e v i a t e  from v a r i o u s  terms  corrections  constants  parameters.  x  =  z  rotational  Watson's  one h a s ,  constant  state  diagonal,  p u r p o s e s we c a n  parameters  set,  introduces  symmetric  "determinable"  constants  our  tops,  this  Similar  For  of  dependent  three-off-the  distortion  empirical  asymmetric  reduction set  a sextic  constants  For  a  <K|H|K±3>,  constants.  distinguish effective  the  in  constants.  one  40  where  B  y  = B  y  +  (I/2)T' zzxx  B  z  = B  z  +  (I/2)T' ' xxyy  r'  = (T  The K.W. are to  state  solely is  v,  structures definable For  K.W.  called  are r  and r  z  an r j  e  parameters  represent  = B  E  where h i g h e r  structure. we have a r  -a  r  =  B  2  co  r  vibrational  particular  below,  discussed  molecule  in  the  constant  s  terms  for  the u>  along 3(a^ )  c §T!  g  the  the 2  of are  state  to  ,  B  the  [1.55]  the  normal  given  b inertial  d  g  vibration  by M i l l s  axis  to  (r5l).  the  B  is  {Zul+uD  b § rs {S  0  by  alpha c o r r e c t i o n  u  +  related  state,  r  physically  n e g l e c t e d and s and  alphas r  is  the  These  + ...  s  are  the  for  v vibrational  B  v vibrational  vibration  2  one  structure.  0  roughly  structure  of  In  spectra  only  structures  s  constant  related A  parameters  number and d e g e n e r a c y  a specific  b  inertia.  + Ea (^ +d /2)  p  formulas  rotational  and a r e  from  to  the  order  the  The  of  determined  rough a p p r o x i m a t i o n s  a polyatomic  B  For  dependent  state  only  [1.54]  constants  equilibrium rotational  modes.  a#/3.  from K.W.  ground v i b r a t i o n a l  the  „)  e q u i l i b r i u m moments of  determined state,  „  rotational  vibrational the  2T  + 0 / 3  [1.53]  I T S '  )2  [ 1 .56] Cu)  where g r u n s o v e r  the  Cartesian  s components x , y , z  and <t>  rss  is  41 a cubic  potential  bracket  are  is B 2  /  the C J  v  given  i  the  D r a  tion*  is  s  For  allows  = B  0  terms a r e of  the  dominant and of  in  the of  the  the  last  term  order terms.are  anharmonic  o p p o s i t e s i g n to  +  harmonic  equation  state  and a n h a r m o n i c  (e1.55)  for  a molecule  (1/2)Zd a (HARMONIC) s  g  B  represents  0  s  the  ground s t a t e  constants.  The h a r m o n i c  calculated  from a h a r m o n i c  correction  to  rotational  constants  t h e K.W.  [1.57]  s  rotational  force  part  field  effective  consistent  K.W.  effective  of  the a l p h a s  formalism.  parameters w i t h an r  = B  z  The s t r u c t u r e (e1.2)) of  o  defines  represents  atoms  from t h e  B  2  be  the  structure,  B  '  + ( 1 / 2 ) I d a (HARMONIC) s s  calculated  can  This  z B  in  as  + (1/2)Zd a (ANHARMONIC)  where  the  (r5l).  us t o w r i t e  e  the a l p h a s ,  three  a s a sum of  g  two terms  interpretation  ground v i b r a t i o n a l B  of  small molecules  contribution a  All  l  c a  often  Identifying  the  part.  y i-  p h  by M i l l s .  harmonic  The f i r s t  harmonic p a r t  anharmonic  contribution  terms  constant.  ,  as  z  [1.58]  constants  (see  equation  t o a good a p p r o x i m a t i o n t h e mean p o s i t i o n  in a ground v i b r a t i o n a l  state  with zero  point  vibration.(r49) The r but  since  z  structure  zero point  is  well defined  vibration  effects  for are  a given  molecule,  mass d e p e n d e n t  42 and t h e  potential  structures the  other  atoms  are  Except  evaluation this  case  changes  in  very  the the  other  isotopically  part  of  structures  under  (r53).  inherent  It  of  the  us t o make a " d i a t o m i c " for  molecule.  atom,  of  is  or  theoretical  a bond s t r e t c h In  the case  of  difficult  alphas  is  from  substitution  been s u g g e s t e d or a n g l e  rest  to  (r44,r52).  estimated  i n a bond s t r e t c h ,  independent  to a  invariant.  isotopic  has  is  it  the  the  invariant  structure  g  equilibrium structure  (r49,r52,r53,r54) polyatomic  r  z  considers  isotopically the  r  s u b s t i t u t i o n . On  structure  g  determination  anharmonic  t h e most p a r t  This allows  isotopic  simple molecules,  excitation  anharmonicity  simply quadratic  an  this  approximation  the  vibrational  for  of  experimental of  to  equilibrium r  Because  for  make e i t h e r  not  t h e minima of  well. of  are  invariant  hand t h e  h i g h degree  In  not  f r o z e n at  potential  fields  of  the  and  that bend,  is  molecule.  approximation (or  angle  bend)  in a  h y d r o g e n bonded t o  mass M, w i t h bond l e n g t h  r „,  some  substituted  Zri  with deuterium to give structure  r  g  in  the r  e  a bond l e n g t h  diatomic  where M- = l  angles for  the  TT7 i s  m. +M  the  z D  approximation  equilibrium  is,  = ^4^  [1.593 / M  m. M  r  the  H  V M  D  reduced mass. R e p l a c i n g ir  the  r's  by  a  l  in equation equilibrium  (e1.59) angle.  gives  the  corresponding equation  43 Even t h o u g h t h e hypothetical for  the  effects  due s o l e l y  to  different an r  the  which  b o t t o m of is  7.  the  serves  molecules.  reflects  "ab  energy  initio"  atomic  Further  atoms of  electronic  as a  is  a molecule  calculations  mass  positions  it  potential  is  standard  B e i n g d e v o i d of  interactions.  where t h e  p r e d i c t e d by  this reside  wells, (r49,r55).  SUMMARY AND COMMENTS  Many of spectra  the q u a l i t a t i v e  c a n be e x p l a i n e d  and r i g i d down f o r  rotor the  anharmonic interaction  transitions  centrifugal  distortion  of  vibrations  invariance  are  greatly  Motivated rotational  of  this  by c l a s s i c a l hamiltonian angular  only  then those  thesis  energy  affected  arguments are  by  is  is  levels.  the  on  microwave  The  rotational  centrifugal distorting  e x p a n d e d as a power  used to p r o j e c t terms  on t h e  our  energy  and  momentum o p e r a t o r s .  molecular  form of  break  reasons:  a r g u m e n t s of  the p a r t i c u l a r  of  oscillator  rotations.  between r o t a t i o n a l  hamiltonian  constraints  and  focus  i n components of  of  harmonic  vibrations,  series  symmetry  the  independent  the  general  of  vibration-rotation  necessarily  rotors  theoretical  terms  of  t h r e e not  levels  distortion.  in  aspects  a p p r o x i m a t i o n s . These a p p r o x i m a t i o n s  The e x p e r i m e n t a l  energy  an e q u i l i b r i u m s t r u c t u r e  structure  g  electronic  equilibrium structure at  of  equilibrium structure  c o m p a r i s o n s of  dependent  concept  that  to a  from  are  allowed  problem.  Further  hamiltonian  eigenvalues  out  Group  come from  unitary  by  the  the the  44 transformation.  Hence any p a r a m e t e r  through a unitary  transformation  empirically.  leads  parameters Of  This  of  our  special  transitions.  or  vibronic,  linear basis  symmetry  interest  These are  rotor-harmonic  to  level  c o m b i n a t i o n s of functions.  cannot  reductions allowed  are  the  c a n be be  in  determined  the  so-called  formalism.  rotor  Confusion arises  number of  forbidden  in a  rigid  Vibrational-rotational,  eigenfunctions  rigid  are  written  and h a r m o n i c because  the  oscillator  d e s c r i p t i o n s are  approximations  that  descriptions prevail.  this  we t a l k  in  terms  transitions  for  transitions  between  vibrational  state.  From t h e these  force  molecular  "spring"  oscillator  such  excellent Because  rules,  energy  levels  in  the  by  constants  allowed ground  fitting  c a n be c a l c u l a t e d .  force  The  force  in a p i c t u r e  c o n n e c t e d by s p r i n g s . A l s o  a reduction  of  AK = ±3  and v i b r a t i o n a l  obtained empirically  constants  parameters  constants  tops,  rotational  where atoms a r e  empirical  rotation  pseudo s e l e c t i o n  symmetric  V  represent  the molecule the  3  parameters  spectra,  constants  C  of  as  rigid  rotor-harmonic  these  eliminated  hamiltonian.  forbidden only  oscillator  energy  that  independent  set  of  from  of  c a n be d e f i n e d and a s s o c i a t e d w i t h a  structure.  Intensities transitions  that  of  transitions  often determine  c a n be m e a s u r e d . T y p i c a l J  normal t r a n s i t i o n s transitions  have 7  start  around  values  of  types  values  7  for  max  r  1 0 " " cm" . 1  10"  the  9  cm"  1  Forbidden or  lower.  The  of  45 extreme  weakness  development design  is  obtaining For of  of  tops,  forbidden transitions  highly  sensitive  d i s c u s s e d in  the  forbidden spectra  symmetric the  of  tops  next is  spectrometers. section. offset  rotation  forbidden transitions  constant  are  the  such  allow  ones  in  information.  and f o r  only  One  The d i f f i c u l t y  by new  forbidden t r a n s i t i o n s  perpendicular  requires  determination spherical available.  46 C.  CHAPTER  2:  EXPERIMENTAL METHODS  The microwave a wide range  of  8-303 GHz. In three  transitions  intensity,  order  to  reported  10"" - 1 0 "  1 1  in  this  c m " , and  s p e c t r u m of  spectrometers  were u s e d .  A highly  moment s p e c t r o m e t e r  l o c a t e d at  the U n i v e r s i t y  C o l u m b i a was  u s e d t o measure  transitions.  The a l l o w e d  on a c e n t i m e t e r  the  asymmetric  wave s p e c t r o m e t e r  California  Laboratory  Institute  (JPL).  and s p e c t r a l lines  are  frequency  all  r e g i o n s , microwave  microwave  power  the  is  are  of  lines  measuring amplifier  in  absorption  other  usually  moved on and o f f  successively  modulate resonance  a b s o r b e d and not  a b s o r b e d p r o d u c i n g an a m p l i t u d e m o d u l a t e d m i c r o w a v e at  the m o d u l a t i o n  phase signal the  sensitive from t h e  time  increases  spectrometers spectrometer  The s i g n a l  lock-in amplifier "on" signal  constant signal  frequency.  of  the  is  amplifier.  both used S t a r k advantage  noise  this  either  "off" over  this  The UBC  m o d u l a t i o n . The of  the  result  M o d u l a t i o n of (r11).  signal  d e c o d e d by a  subtracts  and a v e r a g e s  and d e c r e a s e s  had t h e  that  at  spectrometers  S i n c e microwave  spectrometers  The t r a n s i t i o n s  taken  Propulsion  detector,  weak compared t o a b s o r p t i o n  transitions. so t h a t  of  microwave  recording apparatus.  were  wave s p e c t r o m e t e r  s o u r c e and f r e q u e n c y  an a b s o r p t i o n c e l l ,  British  forbidden  Technology's Jet  Common f e a t u r e s  used i n c l u d e a frequency device,  of  distortion  the U n i v e r s i t y  B r i t i s h C o l u m b i a and on a m i l l i m e t e r the  of  top spectra  at  spectra,  sensitive  8-26 GHz  span  frequency,  1  encompass t h i s  thesis  Stark  JPL or  "Tone  sort  47 Burst"  modulations  1.  d i s c u s s i o n on JPL  t h e most p o p u l a r m o d u l a t i o n t e c h n i q u e s  modulation, transition electric  which  was  relies  frequencies  field.  waveguide  In  inserted  in.  id)  t h e m i d d l e of  and K band  1-13 m e t e r s  l e n g t h of  the c e l l  and was  strips.  The m e t a l  face  the  of  static  cell  plate  length.  insulated  for  and o s c i l l a t i n g  transitions  x 0.17  from i t  by to  metal  metal X band id) ran  Teflon the  broad  were p a r a l l e l  Stark  and AM=0  c o m p o n e n t s . A 1-100 KHz  z e r o b a s e d s q u a r e wave was a p p l i e d t o t h e p l a t e .  During  half  are  cycle  and s p l i t matches  with non-zero v o l t a g e by t h e  that  of  Stark  field.  an a b s o r p t i o n  the  transitions  When t h e line  source  or one of  components an a m p l i t u d e m o d u l a t e d s i g n a l detector.  An o s c i l l o s c o p e t r a c e  shows b o t h t h e a b s o r p t i o n l i n e the  latter  modulation of the  line  inverted. is  for  widths  inverted  Stark  of  and t h e  m o l e c u l e s where  can  its  say  Stark  interfere  the  output  components  with  with  Stark  shifts over  shifted  produced at  Stark  Stark  high f i e l d s ,  pattern  is  the  frequency  t h e phase d e t e c t o r  The main d i f f i c u l t y  require  the  t h e d o m i n a n t p r o p a g a t i o n mode  fields  o c c u r r e d between  in.  of static  a  The p l a t e  was p l a c e d p a r a l l e l  so t h a t  here  u s e d were of  (0.42 in  out  a rectangular  The c e l l s  and were  Stark  an e x t e r n a l  experiments c a r r i e d  waveguide  is  s h i f t i n g and s p l i t t i n g  i n t h e p r e s e n c e of  the in  on t h e  absorption c e l l .  ( 0 . 9 0 x 0.40  the  spectrometer).  STARK MODULATION  One of  plate  (see  on t h e  order  2000 v / c m ,  w i t h the  as  absorption  48 line.  2.  DISTORTION  SPECTROMETER  The s p e c t r o m e t e r between  8-26 GHz i s  u s e d t o measure  shown s c h e m a t i c a l l y  ( r e p r o d u c e d w i t h p e r m i s s i o n from t h e spectrometer  the  has been d e s c r i b e d  Figure  distortion  in F i g .  authors  in d e t a i l  spectra  1  (r56)).  This  (r20,r56,r58).  1  1kHz Reference Oscillator RAMP  Figure  1.  The d i s t o r t i o n  and 26 G H z . I n t e r n a t i o n a l elements  (r57).  spectrometer  symbols are  SIGNAL AVERAGER NICOLET 1072  OSCILLOSCOPE X-Y RECORDER  u s e d between 8  used f o r  the  various  49 Source  noise  was p a r t i a l l y  N o i s e common t o  the  rejected  primary  KHz)  is  output  on t h e  c a n c e l l e d with a detector  reference of  and sample d e t e c t o r s  the  d e - m o d u l a t e d by a PAR is  Averager.  drives  of  The s i g n a l  a reference  Because  source causes the  the  transition  averager  crystal  klystron  c o u l d be u s e d w i t h o u t  levels  were 60-100 mWatts.  Varian  were t y p i c a l l y X-13 and X - 1 2 ,  Frequency and back on t h e  and OKI  1 second.  A measurement  number of  scans  and a g a i n  without  gas and no gas background.  (order the  t o be high  Typical at  the  s c a n s gave  hundreds) in  microwave  end of  the  13  u s e d were  the  the  averaging w i t h the  cell.  a  scan times  early  germane work  ranged  gas  in  the  The d i f f e r e n c e  from  1-31  of  large  spectrum c o r r e c t e d  Typical  forward  w i t h a time c o n s t a n t  c o n s i s t e d of  gas  swept.  power  4 MHz were swept  two m i n u t e s  of  Systems  locked to a  frequency  levels  voltage  20V10 and 24V10A.  averaging of  Nicolet  in a Microwave  saturation.  Power  (1  whose  produces a  1OdB down. The k l y s t r o n s  windows order  in a  moments were so s m a l l  powers  m. c e l l  The AM s i g n a l  frequency  MOS-5 w h i c h when m u l t i p l i e d up a n d phase klystron  is  128 l o c k - i n a m p l i f i e r ,  s t o r e d as a f u n c t i o n  1072 S i g n a l ramp t h a t  transformer.  bridge.  cell  of  the  for hours  per  line. For  the  made on t h e MOS-5 c r y s t a l frequency  (=* 15 M H z ) .  The  c o u l d be o b t a i n e d f r o m knowledge  multiplication frequency  frequency  h a r m o n i c and t h e  measurements were a l s o  40 MHz IF  measurements  were  microwave of  the  (r59).  made w i t h a  Direct  Hewlett-Packard  50 5256 A h e t e r o d y n e Systron  converter.  Donner microwave  frequency  measurement  Later,  for  frequency  over  the  the  counter  entire  arsine  work a  permitted  frequency  direct  r a n g e of  the  spectrometer. All at  t h e measurements were done w i t h t h e  room t e m p e r a t u r e .  transitions  the  lower  transitions,  temperature  given  measurements  set  the  center  klystron (r60).  On a p l o t  sitting  mode i s on t h e  is  for  this  t o a few  modes. voltage  in  klystron  frequency  case  J,  room  range of  the  of  sweep a t of  s y s t e m had  spectrometer  the  "top"  klystrons  output  axis.  the  power  of  little.  mode w i d t h s  hundred megahertz wide.  reference  versus  frequency  The k l y s t r o n s a given  would g i v e  used i n  cavity  a mode t h e  size  power  cavity  this  power  levels  of  For  a few  ten  frequencies  experiment  power.  available  be f r o m  s i z e and  t h e modes a t  more m i c r o w a v e  forbidden transitions  klystron  The k l y s t r o n  a  the U the these  klystrons  to  U  in  were f o u n d t o  klystron  was  see  d o e s not o s c i l l a t e  W i t h the  to  a  O u t s i d e t h e arms of  t o p of  the  appropriate.  t u n i n g the  klystron  change v e r y  experiment,  For  the  cell  J  d e s c r i b e d by an u p s i d e down l e t t e r  d e p e n d e d on t e m p e r a t u r e , voltage.  in  low  enhance  c o u l d be made t h e  descriptions  r e g i o n s . At  will  but  the  a frequency  of  cooled to  were most  step  z e r o and t h e  frequency level  of  mode. F o r  klystron  power  The f i r s t  is  J,states,  B e f o r e any measurement be " t u n e d " .  when we l o o k a t  the a b s o r p t i o n c e l l  p o p u l a t i o n s of distortion  Sometimes  absorption  reflector  had  three  higher very  reflector weak  hundred  51  milliwatts power  are  desirable  modes of  our  klystrons  t e n d t o become v e r y instabilities placing  the  sweep t o phase  l o c k i n g the  and t h e  and t e m p e r a t u r e  in o i l  The s e t t i n g  t o p of  the  a klystron  voltage  drifts  center  and s o l v e d by  a  water  frequency  mode was a c c o m p l i s h e d by  the for  of  higher  klystrons  p r o b l e m was  in a c e r t a i n  and t h e n v a r y i n g  Also  b a t h s c o o l e d by c o l d  of  klystron  reflector  p o s s i b l e the  would be u s e d .  The t e m p e r a t u r e  klystrons  the  frequency  hot  occur.  cooling c o i l s .  and so whenever  cavity  mode t o  size  the  center  (mechanically),  t h e maximum k l y s t r o n  output  power. After cell  the  was p u t  at  the  t o p of  would be t u n e d t o o p t i m i z e t h e microwave  detectors.  this  power  the  in order  to c a n c e l  some of  the  source n o i s e .  was a c c o m p l i s h e d w i t h p l u n g e r s  (to  vary  on t h e  order  and s l i d e  microwave  screw  waveguide  After  the the  measurement  begun.  Over  tuners  of  was  the  which are  klystron  gas p r e s s u r e s were f o u n d t o It  (r11,r60).  on and  this  times  significantly  was g e n e r a l l y  the  and " g l i t c h e s " ,  accumulation  increase  E and H  gas c o u l d be  with  drift  of  standard  switched  main d i f f i c u l t i e s  prolonged signal  and o u t g a s s i n g .  field  All  length  wavelength)  " t u n e d " the  s q u a r e wave S t a r k  outgassing,  the  all  c o u l d be  t u n i n g and m a t c h i n g d e v i c e s  T h e r e were t h r e e  noise.  t h e microwave  spectrometer  introduced,  spectrometer:  of  bridge c i r c u i t  at  the  tuners  leaks  the  same t i m e  cell  At  t h e mode  the  balanced  the  klystron  found t h a t  sample due  a  and  gas  to  52 sample was  only  measurement. s y s t e m was stable  This,  plus  "stable"  over,  technique  good i n  of  say,  followed  cell  evacuated.  five  of  the  nature  subjective  two h o u r s  hours,  klystron  - glitches  resulted  drift  experimental case  turning  averager  a half  which  3.  as  is  the  last  gas  in  with  the  the  continual  "glitches", of  frequency,  pathological for  the  lights  be in  transients  The p r o b l e m of m i n u t e was the  in  no r e a s o n .  p r o b l e m had t o off  as  experimental  s o l v e d by  hour and s t o r i n g  sensitivity  intensity  absorption  the GeH  a  of  line  E type  spectrometer  8-120 G H z . The S t a r k  cells  the  the the  were  destroying a  solved  by  result  in  the  was p r o b a b l y  weakest  1 x  Stark  measured to date  transition  CENTIMETER WAVE SPECTROMETER  . The UBC microwave range  the  spectrometer  modulated microwave reported here  the  the  memory.  The u l t i m a t e 1  in  of  not  w i t h the  electrical  a measurement.  many hour measurement for  was  p r o b l e m was o f t e n  this  enough t o d e s t r o y  1 1  in  The p r o b l e m of  p l u g g i n g i n a f a n or any  I0~ cm~  but g e n e r a l l y  would come and go s e e m i n g l y  n o i s e p r o b l e m . In  signal  an hour  conclusion that  changes as a f u n c t i o n  a frequent  accumulating  about  s i g n a l measurement  The most d i f f i c u l t  lab,  the  system.  d i s c o n t i n u o u s power t h o u g h not  for  by one hour b a c k g r o u n d measurement  The p r o b l e m of re-tuning  cell  over  one hour  cell  the  and  is  J=20 3—>4.  (UBC) covered  the  were t y p i c a l l y  frequency 3 meters  in  53 length  and the m o d u l a t i o n  frequency  sources  was  different  microwave  generated  by a H e w l e t t - P a c k a r d  (HP)  microwave  spectroscopy  Much of  frequency  range  c o u l d be c o v e r e d  wave o s c i l l a t o r  GHz)  H81-8694B,  HP,  R-band ( 2 6 . 5 4 0 frequencies, microwave  GHz)  (BWO)  doubler  The h i g h  by a Siemens  frequency  synthesizer  a P-band BWO w h i c h  in  lock  the  BWO.  made on t h e  frequency  reference  No d e t e c t o r  bridge  preamplifier  into  regions: A back  a PAR  from a SPACEKOM  All  frequency  used f o r  signal  the  was  R-band HP or  Schwarz phase  mixed  measurements  to were  allowed  fed through a amplifier.  as a f u n c t i o n  The d e t e c t o r s  GHz R-band d e t e c t o r  the  was  A Rohde and  of  11586 A p o i n t  The o u t p u t  frequency  u s e d d e p e n d e d on  X-band HP 406-X422A back d i o d e ,  diode,  OV-1  source  was m u l t i p l i e d ' a n d  120 l o c k - i n  was p r e s e n t e d  oscilloscope.  middle  75-120 GHz r a n g e  turn  was  The d e t e c t o r  lock-in  H81-8695 A and  oscillator.  transitions.  the  (8-12.4  SMDW was m u l t i p l i e d and m i x e d t o  lock  high  HP,  X-band  u s i n g as a p r i m a r y  RW<i>110B BWO.  was  appropriate  A p l u g - i n . The  frequency  generated  frequency  8-40 GHz  plug-in units: GHz)  Three  phase-stabilized  the  w i t h the  P-band ( 1 2 . 4 - 1 8 HP 8 6 9 7 ,  8400 B  53-80 G H z , were o b t a i n e d  frequency  R-band BWO.  were u s e d . The low  source.  backward  100 KHz.  a HUGHES 47314 H-1100.  on an  frequency  P-band HP  contact  of  H06-P422  diode,  53-120  54 4. MILLIMETER WAVE SPECTROMETER The J P L range  millimeter  spectrometer  60-800 GHz. The low  available, Higher  w i t h a few  frequencies  frequency barrier  diodes).  frequency  exceptions,  The k l y s t r o n s  8460 A MRR S p e c t r o m e t e r  For (see  however,  JPL  of,  of  for  several example, voltage  intervals  of  the  30 K H z .  a later  had t h e  If  the  In  section)  of  the  UBC  detectors  c o u l d be u s e d f o r  added a d v a n t a g e  tone  burst is  30 K H z . T h i s of tone  the  the  strong  usually  signal  phase  t h e n we g e t  at  "tone  burst"  at  added t o  on  a  lock  rate  lower  Tone b u r s t Stark  respond  to  modulated  n o i s e and  modulation  effects  at  frequency  l o o p cannot  a frequency  the  the  the m o d u l a t i o n  of  when slow  the  is  Stark  33 K H z . The  producing sidebands  of  helpful  of  X-band  modulation a s i g n a l ,  klystron  klystron  was  t u r n e d on and o f f  frequency  locked system.  3 meter  frequency  source w i t h the advantage  especially  locked to a  version  were s t a n d a r d  megahertz  30 KHz FM s i g n a l  a phase  low  (Schottky  200 GHz R-band  wave m o d u l a t i o n  (r62).  reflector  of  the  100 GHz.  spectrometer  order  klystrons.  h e l i u m c o o l e d InSb b o l o m e t e r s were  The s q u a r e  modulation  of  was  g e n e r a t e d by a c o m m e r c i a l HP  up t o a b o u t  The a b s o r p t i o n c e l l s cells.  60-123 GHz,  spectrometer.  frequencies  u s e d above  frequency  as h a r m o n i c s of  were phase  (r6l),  2.3 UBC s p e c t r o m e t e r  lines  range,  the  by h a r m o n i c m u l t i p l i c a t i o n  m u l t i p l i e d R-band BWO s i g n a l  centimeter  covered  from a h o s t  were g e n e r a t e d  klystrons  (JPL)  stability is  produce  inverted  55 Stark  "lobes"  transition can a l s o  that  be masked by  estimates  interfere  frequencies.  modulation acts  5.  can  The gas  zero  field  spaced h y p e r f i n e  interfering  as an a l t e r n a t i v e  b a s e d on S t a r k  SUNDRY  Closely  w i t h the  Stark to  lobes  structure  so tone  zero f i e l d  frequency  modulation.  ITEMS glass  vessels  w i t h g r e a s e d s t o p c o c k s . The gas h a n d l i n g  systems  or  glass  s a m p l e s were s t o r e d  vacuum l i n e s  fitted  were i s o l a t e d  windows.  The gas  is  to metal  or metal  typically  10-150 mTORR  order  errors  were between  region,  line  of  were  metal  d i f f u s i o n pumps. The  the c e l l  KOVAR t y p e (microns)  500 K H z . T r a n s i t i o n  join. which  measurements.  through a Gas gave  frequency  10-100 KHz d e p e n d i n g on  s t r e n g t h and o t h e r  p r e s e n t e d w i t h the  fitted  from t h e a t m o s p h e r e by m i c a  i n t r o d u c e d to  to metal  on t h e  in  with o i l  absorption c e l l s  are  burst  parameters,  glass  pressures line  widths  measurement  frequency and w i l l  be  56 D.  CHAPTER  3:  GERMANE  GeH  a  The most germane p r e c u r s o r transition  study  were a b l e  Recall, for  a  to  is  0  was  tops  al  (r63).  In  and Raman s p e c t r u m of  amongst o t h e r in angular  - D J  parameters  B  1972 GeH  and  0  momentum o p e r a t o r s  + D <ft«>  f t  a  D . g  that  (r64)  the  contribution  than  the q u a r t i c  (AJ=+1)  earlier  data  fc  In  results  They c o u l d not  B  0  were c o n s i s t e n t  report  b a s e d on t h e o r e t i c a l ~  et  "forbidden"  work  they  and D  and  were a b l e  and e s t i m a t e  g  of  absolute  a v a l u e of D  t  D  to the  line  but  fc  showed  calculated  expressions given  by  by  Hecht  2.2x10" cm- ). 6  1  al  (r68,r69)  double resonance  reported  transitions  several  and a " b e t t e r "  d i p o l e moment. They were a b l e  rotational  transitions  in  a s s i g n e d two  the  for  (AJ=0) the.  to a s s i g n  ground v i b r a t i o n a l  further  study,  Q branch  value  distortion  tentatively  frequency  contribution,  w i t h a v a l u e of  From a m i c r o w a v e - i n f r a r e d Kreiner  their  for  the  1973 O z i e r and R o s e n b e r g  d i p o l e moment from e s t i m a t e s  intensities.  (r66)  it.  to  " f o r b i d d e n " s p e c t r u m of  t h e n w e l l p r e d i c t e d and i n of  [3.1]  t  s  R branch  reported part  (D  2  0  infrared  distortion  (r67)  = B J  much g r e a t e r  c o n f i r m the  Fox  report  et  1  so t h e  their  infrared  forbidden  level'JC ",  from B  (r65)  the  f o u r t h degree  spherical  GeH,  of  to  E  For  this  was done by K a t t e n b e r g  from an a n a l y s i s they  for  r e s o n a n c e s but  five  state. were  They  unable  57 to  account  for several  transitions the  tensor  able  they  Recall order will  calculated  distortion  to predict  others.  From t h e f i v e  several  constants  two E t y p e  c o m b i n a t i o n s of  and from t h e s e  transitions  that  E type  transitions  Stark  effect  of consequence  be g a t h e r e d  linear  assigned  at  results  were  9.6 and 2 2 . 6 G H z .  have t h e a l l - i m p o r t a n t i n the present  first  work.  As  from t h e d i s c u s s i o n o f t h e p r e d i c t i o n and  measurement  of the spectrum these  transitions  were v e r y  important  predicted E  type  i n t h e c o m p l e t i o n of  this  study. Also  in their  microwave  work  transitions  w i t h two d i f f e r e n t The r o t a t i o n a l E  (  2  )  -E  (  1  )  ,  different that MHz  laser  Ge s p e c i e s . for  explanation  1 3  CH  of  (r70). then  in the i n i t i a l  it  Further  If  i s analogous 1 2  CH  a  - 1  Kreiner  apart.  available  study,  if  where  are perhaps  is  1 3  1  to  up t o 10 in  line isotope at a l l .  especially  of t h i s  (1.0' cm"  frequencies best  noticeable  et a l  coincidences  of t r a n s i t i o n  studies  splitting  of hundreds of k i l o h e r t z ,  of K r e i n e r  to  e t a l were c o r r e c t  phase of s t u d i e s  on l a s e r  isotope  where s e p a r a t i o n s  in the present  sensitivity  relies  observations  0.10934 c m  were a t t r i b u t e d  were e x p e c t e d t o be s m a l l  the high  since  coincidences  and  4  The MW-IR t e c h n i q u e useful  observed  2 3 5 6 . 6 8 ±.1 and 2 3 5 6 . 6 0 ±.1 MHz  This  were on t h e o r d e r  splittings  (r68)  a s s i g n m e n t s made were t h e same, J = T 1 ,  were o b s e r v e d  widths  at  et a l  and t h e two t r a n s i t i o n s  observed  their  Kreiner  )  sort  because  (r6), but,  systematic a r e not p o s s i b l e .  done on h i g h l y  sensitive  58 Stark the  spectrometers  distortion  t h e n was here.  capable  spectrometer  the m o t i v a t i o n  U s i n g the  we hoped t o rotational separate  the  centrifugal  transitions  rotational  et  and from t h e i r  parameters  (r68).  GERMANE GAS  M a t h e s o n Company as  2.  it  (r71,r72) Watson's  the  (r15)  due t o c e n t r i f u g a l  evaluate  linear set  it  was  hoped  un-assigned resonances  t h e Ge i s o t o p e  this  of  something  splitting.  s t u d y was o b t a i n e d  from  and was u s e d w i t h o u t  impurities  b e i n g H 0 and H 2  the  further (r65).  2  SPECTRUM  Moret-Bailly  and Watson and h i s  terms  forbidden  germane s p e c t r u m was p r e d i c t e d u s i n g  methods of  treatment  This  spectrometer  was hoped t h a t  PREDICTION OF THE GERMANE  perturbation  octic  the  9 9 . 9 % GeH„  t h e main  Initially  than  2.  as  SAMPLE  The sample u s e d f o r  purification,  rather  such  presented  analysis  increased data  Finally  more c o u l d be s a i d a b o u t  1.  germane s t u d y  measure many more  A l s o from t h e  al  sweeping  d i s t o r t i o n moment  a s s i g n m e n t s c o u l d be made f o r Rreiner  frequency  d e s c r i b e d in Chapter  for  systematically  combinations.  of  and h i s  collaborators  collaborators (r17,rl8).  has been e x t e n d e d by O z i e r and we have distortion,  for the  the  to  splitting  e n e r g y of  the  include of  a J  a level  JC  level fc  59 E = [D +H J(j+1)+L t  4 t  + [H Values  of  6 t  the  +L  6 t  numerical  and Watson  tabulated  by O z i e r is,  of  J (j+1  ) ]f  2  J(J+1)]g  Kirschner  transition  4 f c  2  + L  g t  factors  (rl8),  (r15), course,  h + (H  2  6 t  /D )g.  f and g a r e  and t h o s e  of  up t o J = 2 0 . just  the  [3.2]  t  tabulated  h and g  are  The f r e q u e n c y  energy  by  of  difference  a  between  levels. The r o t a t i o n a l various (r68), the SiH  linear  and s c a l i n g a r g u m e n t s  (r56).  L  where  4  linear  A similar  f c  and L  4 t  4 f c  similar  magnitudes  molecule  a s c a l i n g argument  of  silane,  Lg (GeH ) t  4  relation L (GeH )-. 8 t  LL  1 )  B  t  L  (SiH.)J  (SiH„)  g  [3.3]  al.  c o m b i n a t i o n s of  of  et  al  estimate  ratio for  into  GeH . 4  D  L  6 t  III  (GeH ) t t  Finally,  to various  fc  equations  gave  initial  of  Table  s c a l i n g argument  these  parameters  al  et  Inserting  the  the  et  by K r e i n e r  relating  to give  in  the  was d e t e r m i n e d s e p a r a t e l y  equations  order  from  b a s e d on r e l a t i v e  4  and from t h e  Hg .  r  (GeH ) =  t  (r68)  H  from t h e  Lg (GeH )  From t h e  of  6 f c  parameters  As an example of  was e s t i m a t e d  were e s t i m a t e d  c o m b i n a t i o n s p r e s e n t e d by K r e i n e r  rotational 4  parameters  D  higher  the of  D  estimate.  It  was  and e q u a t i o n  (e3.2)  that  s p e c t r u m was  predicted.  the  initial  fc  the  estimate estimation  were a v a i l a b l e order  estimates  two v a l u e s fc  we c o u l d  allowed  there  Kreiner  parameters. of  the  w h i c h were from t h e s e  Q branch  two  higher  averaged estimates  distortion  60 As we s h a l l were v e r y linear  good.  see  in  the  As p r e d i c t e d  relations  solved  for  parameters. the  better  f o u n d became g r e a t e r parameters  (e3.2).  the  than  s y s t e m of  o v e r - d e t e r m i n e d and t h e d e t e r m i n e d by the  linear  perturbation  These  estimates  transitions,  or the  predictions  were f o u n d new  of of  linear  the  course,  number of  relations  iterated. linear  equations,  Then,  were  then  i n s t e a d of  a p p r o a c h s y m b o l i z e d by e q u a t i o n  c o m p l e t e d i a g o n a l i z a t i o n method of  When  became  parameters  squares.  were  rotational  equations  rotational  c o u l d be  distortion  independent  linear  least  these  d i s t o r t i o n parameters  T h i s p r o c e d u r e was,  number o f  section  transitions  amongst t h e  produced u s i n g equation then  next  using  (e3.2)  Fox and O z i e r  (rl9)  the was  used.  3. MEASUREMENT In  this  transitions type  study  OF THE GERMANE only  were c o n v e n i e n t  transitions  have f i r s t  use t o S t a r k m o d u l a t e fields  available  The f i r s t E  9615.310(75)  by K r e i n e r  et  al  )  (  2  )  Recall  order  effects  measured i n  at  3  to measure.  experimentally  line  (r68)  forbidden Q branch  transitions  ,  (  -E  E type  to  SPECTRUM  Stark  w i t h the  Chapter  this  s t u d y was  transition  rotational  new  parameters.  was p a r t i c u l a r l y  helpful  we  can  electric  J=18,  had been  information  However,  E  2).  9 6 1 5 . 2 9 MHz. T h i s  was e x p e c t e d t o be good so no r e a l g a i n e d on t h e  line  only  that  static  (see  MHz. T h i s be a t  that  predicted prediction was  this  in o p t i m i z i n g  the  61  experimental  c o n d i t i o n s , most n o t a b l y  accumulation  times.  U s i n g the previous  "initial"  section,  predictions  branch t r a n s i t i o n s this  the J = 1 4 , E  result:  the  In  case  this  it  effects  this less  the  to  sensitivity  1  .  Table  3  )  -E  Following  w i t h no  be  field  1 4 MHz h i g h . was  (  2  made t h e  searching. )  .  lower  fields  It  was  to  in  in the  distortion  the  necessary  Once t h i s  absorption  10897.180(100)  predictions  the  o p t i m i z i n g the  the  reported  line,  E  M  A  X  that  spectrometer  t h e weakest  MHz w i t h a 7  line  remaining  r e g i o n s of  J=20,  to  measured  s u c c e s s we had s o l v i n g of  turned  constants  subsequent  various  then  made b a c k g r o u n d  searching easier.  was  the  frequency  The p r e d i c t i o n  3.2 A(t",t'))  1 MHz o f  t h e measurement  m o d u l a t e d microwave  cm"  (  for  such e x c e l l e n t  A testament  of  E type Q  was u s e d t o p r o d u c e new p r e d i c t i o n s  difficulty  frequency  to  This clearly  i n f o r m a t i o n on t h e  experimental  p r o b l e m was  see.  J=16, E  (see  Stark  for  the  d i s t o r t e d b a c k g r o u n d upon w h i c h  unfavourable  for  line  searched  found t h a t  Given  spectrum.  was  t u r n e d out  t o be w i t h i n  extreme  line  other  in  spectrometer.  prediction  frequencies.  for  )  t o our  7 MHz h i g h and t h e  from i t  t u r n e d out  1  and t h e r e f o r e  found the  gathered  line  search  be o n l y  modulate  was  was  (  to a l a r g e  this  decided to to  - E  )  was d i f f i c u l t  r e g i o n of  out  2  and  scheme o u t l i n e d  were made of  accessible  initial  contributing signal  (  prediction  pressure  (  A  of  >  - E  <  1.0  3  this Stark )  at  a  x 10"  1  1  62 During the whether  all  isotopic  for  these  by 2 t o  result, in  In  of  study  zero point H^  significant  and H g »  t  natural  76/7.76  (r73))  spectra  might e x i s t .  spectra  was e x p e c t e d t o be l e s s in  than  should c e r t a i n l y separation  meant  be a b l e  the  1 3  C  to  to  1 2  and  73/7.76,  take the  and  fc  where t h e  of  the  D  various  74/36.54,  germane  is  E type  shown  over  studies  contention  sister the  the  isotopic  frequency  transitions  in Figure these  no f u r t h e r Kreiner  lines  al  one  "sister"  in a search  splitting where  large  18(1—>2),  for  of  the  For  was  the  which  in  19(1—>2)  Particularly  J=18  our  and (1—>2)  careful  by s e a r c h i n g up and down  transitions et  was  a tracing 2.  case  was  germane t r a n s i t i o n s .  t h e maximum methane  of  this  included transitions  to  fc  the  "sister"  maximum  limit  CH„.  accounted  i n methane as  In  1 2  (mass number/ %  between  C.  same  spectra  CH„  in D  similar  than  neighboring similar  were made of  frequency, these  of  in descending order;  transition studies  suggested that  that  transitions  contribution  20(2—>3)  1 3  effects  72/27.43,  The s e p a r a t i o n  e x p e c t e d t o be l a r g e  case  two  c a n be  abundance  i n methane as an upper  spectra  germane,  was  to  the  r e d u c e d mass upon i s o t o p i c Ge s u b s t i t u t i o n  much s m a l l e r  sister  were due t o  S i n c e Ge has  t  70/20.52,  concern  (r20,r64,r70)  vibration  abundance, it  was  be examined l a t e r ,  natural  change  there  10 MHz were a t t r i b u t e d  d e p e n d e n c e of  i s o t o p e s of  this  methane  which w i l l  terms  isotopic  of  measured t r a n s i t i o n s  Ge s p e c i e s .  separated This  course  that  splitting  of  10 MHz.  In  were f o u n d ,  supporting  the  splitting  isotope  63  is  within  our  line  widths.  The p r o b l e m of  is  d i s c u s s e d more q u a n t i t a t i v e l y  1  225548  Figure resulting  2.  1  1  22555.7  absorption  from the d i s t o r t i o n  transition  signal  sample  subtract  s c a n s and t h e n s c a n n i n g time  1  1  Frequency (MHz)  o b t a i n e d by u s i n g t h e  total  2  1  The m i c r o w a v e  (sample  splitting  later.  Figure  1  isotope  averager  line  1  225566  of  GeH«  18E <—18E . 2  to accumulate  64 b a c k g r o u n d s c a n s .  plus  background)  It  1  was  was  64 The  3.64  hours.  64 4.  OBSERVED GERMANE SPECTRUM AND ANALYSIS  The a s s i g n e d germane s p e c t r u m o b s e r v e d i n consists from  of  14 t o  listed  ten Q branch  experimental  half the  along with t h e i r  uncertainties  uncertainties line  obtained  3.1  are  o . The  estimated  widths  v a l u e of  A summary of conditions parameter (r69)  as  runs.  in Table  was  taken  as  3  o b t a i n e d by  x  and  of  to  the  widths,  noise  frequencies  full  width  2 8  3.2.  SiH  at  with  (r56).  f l  and  experimental  For  y the 'max  10.0 M H z / T o r r and t h e  broadening 3  d i p o l e moment  10" D. 5  tensor  distortion  fitting  to  are  420 KHz w h i c h compare w e l l  parameters  given  The s i x  according  line  ranges  experimental  The l i n e  430 KHz f o r  is  3.33  assignments  from o b s e r v e d s i g n a l  maximum, were t y p i c a l l y typical  frequencies  and r e p r o d u c i b i l i t i e s  from d i f f e r e n t  work  t r a n s i t i o n s "where J  2 0 . The measured t r a n s i t i o n  in Table  ratios,  (AJ=0)  this  the  V°" , m  transition  w i t h an  2  constants  of  Table  frequencies,  iterative  least  3.3  were  weighted  squares  method  separate  values  (r15). Examination of  the  six  from the  tensor  present  calculation from the present  of  of  Table  distortion work  the  MW-IR work, estimated  are not  alone.  various  IR-MW d a t a work  3.3  (Table  reveals constants  from t h e i r  in  linear 3.4).  the  stated  have been  These separate  The e r r o r s  original frequency  allow  the  determinable  given  The e r r o r s paper,  obtained  values  combinations  standard d e v i a t i o n s . given  that  in in  have  uncertainty.  the the  been In  Table  65 3.5  t h e o b s e r v e d MW-IR t r a n s i t i o n s  constants  obtained  reported here.  from t h e t e n Q b r a n c h  The p r e d i c t i o n  2 1 6 . 1 7 MHz c o n f i r m s Kreiner  MHz t o 2 0 F  2  2  7F , (  2 )  -7F  2  the t r a n s i t i o n  » - 2 0 F , > . The v a r i o u s 5  c o u l d n o t be a s s i g n e d t o any  transition  than  Boltzmann  factors  t o o weak f o r J Table data  less  30. For the ground  make t h e i n t e n s i t y  greater  3.3 a l s o  the  contributions  a  lines  microwave  study  on  to the  2 8  i s to  SiH„  (r56),  transition  were e s t i m a t e d and f o u n d t o be - 3 % o f t h e  contributions error  dectic  of the  The n e t r e s u l t  In a s i m i l a r  frequencies  of the a b s o r p t i o n  i n c l u d e s an a n a l y s i s  improve the s t a t i s t i c s . order  state,  than 30.  c o m b i n e d w i t h t h e MW-IR d a t a .  higher  at  other  u n a s s i g n e d MW-IR l i n e s with J  at  2 )  t e n t a t i v e a s s i g n m e n t of  re-assign (  from t h e  transitions  of the l i n e  the p r e v i o u s  e t a l . We c a n a l s o  543.69(30)  are p r e d i c t e d  due t o t h e o c t i c  attempts  to include  terms.  these  The a b s o l u t e  higher  order  parameter  effects.  66 Table  3.1  O b s e r v e d germane Q - b r a n c h r o t a t i o n a l  transitions  Transition t  J  Symmetry  1 4  E< 2 ) _  E  ( 1 )  9 950.525(50)  Deviation' 0.010  16  E  E  ( 1 )  14 6 9 1 . 3 1 7 ( 5 0 )  -0.013  1 6  E<  3 > _  9 510.590(90)  -0.010  1 7  E  2 )_  18  E< 2 ) _  18  E  19  E< 2 >_ <  19  E  20  E<  20  E  (  (  (  2 )_  ( 2 )  E  E  ( 1 )  1 37 0 9 . 4 7 2 ( 9 0 )  -0.005  E  ( 1 )  22 5 5 5 . 7 8 0 ( 8 0 )  0.003  9 615.310(75)  0.008  18 5 3 6 . 0 3 6 ( 8 0 )  0.005  3 ) _ £ ( 2)  1 52 9 3 . 3 5 0 ( 1 0 0 )  0.019  3 ) _  18 3 9 4 . 2 5 2 ( 6 0 )  0.029  3  )_  E  < z ) 1 )  E  (  (  Frequency(MHz)  t)_ ( E  fObserved  E  ( 2) 3  >  frequency  d i s t o r t i o n constants Numbers uncertainties  1 08 9 7 . 1 8 0 ( 1 0 0 ) minus f r e q u e n c y  obtained solely  in parentheses in u n i t s  of  are  the  calculated  f r o m microwave  estimated  last  -0.016 from data.  measurement  significant  figures.  67  Table 3.2 Summary  of  germane  line  parameters  and  experimental  conditions (see  below  for  units)  Transition J  Symmetry  7  'max  x1 0 cm"  1 4  E  (  2 :— E  16  E  (  2 ) : _  16  : < > E< 3 >-E  2.7 '  1 7  E< 2 » - E '  18  E  (  2 >_ <  18  E  (  3 > _ E <• >  19  E  L  2 )_ (1>  19  E<  3 )_E  20  E  20  E  1  '  3.7  ( 1 )  6.5  ( 1  E  2  1 1  1)  E  2  A(t"  1 1  ,t')  1  -2.79  Stark  # of  Mean  Field  indep.  time  runs  /run  150-200  3  1 .5  450  3  1 .6  -8.81  90  7  2.2  7.8  -4.44  120  3  2.4  8.5  -1 . 0 7  1000  4  2.5  6.4  5.11  50-150  3  0.5  3  6.5  3  15.2  1.26  10.6  2.04  >  2.5  -11.31  (  3 )_E < '  7.4  -6.37  400  3  5.3  (  4 ) _  1 .0  15.67  70  5  12.3  E  1  2  2  E  ( 3 )  fPressure^l0-50 UNITS:A(t",t') Stark  of  80-150  mTorr. in  field  Mean t i m e  400  (MHz  D"')(V/cm)-  1  i n V/cm.  per  t i m e was  run  in  spent  h o u r s where  an e q u a l  amount  d e t e r m i n i n g the background.  68  Table Tensor  centrifugal  Parameter  distortion  3.3 constants  Microwave +  data 67 H H L  L  L  only  •5.3820(59)  6T  2.9694(20)  4  T  X10«  6T 8  T  X  1  °  •8.01(15)  f O b t a i n e d from an a n a l y s i s in  the p r e s e n t  work  * Analysis  of  Hertz)  Combined microwave  and  infrared-microwave  data  67  •4. 127(51 )  X10«  (in  a  Value  a  775.54(86)  3.3  -5.3827(55)  0.013  2.9693(19)  0.006  3.996(88)  0.25  -4.122(50)  0.21  -8.01(14)  0.46  3.985(94)  4  GeH  1  775.44(91)  4T  of  of  the  transitions  measured .  only. the  present  data  together  a s s i g n e d IR-MW d o u b l e r e s o n a n c e measurements  w i t h the listed  seven  in  Table  3.5. v  a  a  is  Numbers the  least  figures.  the  estimated absolute  in parentheses  squares  fits  are  in units  the of  error.  See  text,  standard deviations the  last  significant  of  69 Table C o m p a r i s o n of distortion  linear  c o m b i n a t i o n s of  constants  obtained  of Combinations  3.4  ref  in  (r68)  germane  the p r e s e n t (in  centrifugal work  with  those  Hertz)  of  Infrared-microwave  constants  Microwave  t  double  resonance  *  J=1 8 D +342H +116 T  H  L  4T  6T 8 T  + 3 4 2 L  and  +  6T  4 T +  320L  38.6L  67  6T  the  2.830(3) -6.0(40)  762.9(10)  67  -5.209(74)  6  present  work  standard deviations  only.  2.830  of  Numbers  in u n i t s  of  765.5  -5.220  2.8283  6T  f O b t a i n e d from an a n a l y s i s in  980(3)  J=11  474L + 3 4 2 L  65  -8.01(15)  flPn  4 T  981(3)  2.828(3)  6T  D -45 144L +1 4T  H  65  4 T  X10«  J=18  H  964L  the  transitions  in parentheses  the  least  measured  are  significant  figures. * O b t a i n e d from r e f estimated  outside  the measured  error  (r68). limits  frequencies.  Numbers  in parentheses  from s t a t e d  uncertainty  are in  70 Table Prediction  of  3.5  germane IR-MW d o u b l e r e s o n a n c e  using constants  obtained solely  transitions  from t h e m i c r o w a v e  spectrum  (MHz) Transition J  t  Symmetry  18  F ,*> —F (  '  9  2  18  F ^ - F ' ,  18  P r e d i c t ion  '  2  F^'-F', * 3  18  A  (  1  2 )  -A  2 )  2  Measurement  1  6  633.147(17)  10  575.354(14)  2  973.111(10)  6  633.15(10)  10  575.33(10)  2  973.15(10) 895.79(10)  ,895.768(6) 11  E < 2 )_g(  7  a  13  b  20  b  F\ -F >  216. 173(2)  F'/'-F^ '  541.707(1)  F  543.832(5)  2)  2  2  3  5 2  >-F , > (  fObtained in  the  present  5  from an a n a l y s i s work o n l y .  standard deviations  in  the  * Measured f r e q u e n c y parentheses 'a'  are  of  Numbers  356.68(10) 356.60(10)  216.17(20)  543.69(20)  the  transitions  in parentheses  measured  are  predictions. from r e f . ( r 6 8 ) .  e s t i m a t e d measurement  Tentative  2 2  2 356 . 627(11)  1 )  a s s i g n m e n t made i n  Numbers  in  uncertainties. ref.(r68)  is  conf irmed. 'b' line.  There are  The t e n t a t i v e  incorrect; present  the  work.  two p o s s i b l e a s s i g n m e n t s assignment  transition  is  in  ref.(r68)  to  the  t o J=13  r e a s s i g n e d t o J=20 i n  measured is the  71 5.  ISOTOPE SPLITTING IN  The p u r p o s e of claim  that  germane here.  is  isotope  not  significant  into  two  for  Recall  rotational this  Kreiner  transition in  et  case  J=11,  the  two  Additional  al  measurements  the  the  supporting  discovery  attributed  splitting  than  the  evidence  the  spectra.  compared w i t h a t y p i c a l we c a n c o n c l u d e t h a t well.  Since  from one nearby  measured a r e  r e s o l u t i o n average  let  for all  the  species  "sister"  spectra,  isotopes,  a s one  type  claim how  reduction  the  isotopic  to appear  If  140 K H z .  us c o n s i d e r  uncertainty  either  due t o many  the E  rms d e v i a t i o n  t h e model a c c o u n t s  have s i m p l y m i s s e d t h e transitions  the  in  supporting this  First  experimental  We c a n a l s o c o n c l u d e t h a t  we have m e a s u r e d a r e  our  work.  similar  species.  around  i s o t o p e model u s e d i n t h e d a t a  for  that  experimental  well  accounts  the  (r68)  to  Ge i s o t o p i c  the  single  be  frequency.  this  c a n be f o u n d i n the  can  theoretical.  measurements of  present  reported  same microwave  isotopic  experimental  of  i n t h e MW-IR d o u b l e  2<—1 must be l e s s frequency  and  the  features  isotope e f f e c t  in d i f f e r e n t  then  investigate  spectral  evidence  the  (r68)  to  the  c l a i m was  yielded  transitions  were t h e  error  earliest  coincidences  experiment  that  the  is  experimental  the  laser  in  of  a negligible  isotope effect  two d i f f e r e n t resonance  splitting  kinds;  Experimentally negligible  sub-section  the  The e v i d e n c e  divided  this  GERMANE  is  38 K H z ,  of  80 K H z ,  data  very  transitions only or that  isotopic  and we that  the  within  species.  72 In  order  to preclude  spectra  careful  several  key  searches  (r20,r64,r70).  which parameters isotopic  The  from t h e  The key in  and t h e n  theoretical  argument  that  evidence  "sister"  spectra  we a c c e p t  that  no i s o t o p e  line  width.  the  line  instructive (FWHM) of value  of  effects  of  430 KHz  in  importantly  "under"  study  day  2 8  no c a s e  silane  greater  then  than  splitting  the  there  spectrometer  fields  in order  some c a s e s  lines  the  it  any  was  is  perhaps widths 2 8  such  one  to  SiH  f t  isotope  isotope however  of be  p r e s s u r e s and number  studies  depend t o a  large  was w o r k i n g on a region.  More  preferred  "background" e f f e c t s  were a r t i f i c i a l l y fact  appear  experimental  typical  only  were o f t e n  the  would  line  example  frequency  to minimize  modulation. S t i l l ,  the  could contribute  as c e r t a i n l y  in a p a r t i c u l a r  low S t a r k  it  the  germane  silane  and germane  the  (see  was  T h i s comparison must,  salis"  by  germane  parameters  result  the average  Si.  noting  were most a f f e c t e d  In  as  work  f i n d i n g which  null  e x p e c t e d as  established  found.  is  In  below  were d e t e r m i n e d by  on t h e s e  isotope  (r56).  namely  the  experimentally in  methane  the measured t r a n s i t i o n s  on how w e l l  particular  that  if  "con granulus  degree  C  420 KHz and compare them t o  were not  scans  above  to c o n s i d e r  significance, taken  of  1 2  follows).  splitting  To see  width  reliant  the  "sister"  10 MHz above and  transitions  were most  If  C-  1 3  transitions  of  10 MHz range was  t h e methane  substitution  of  possibility  were made  transitions.  explained earlier  the  that  narrowed  so  by  no a n o m a l o u s  line  73 widths  were o b s e r v e d s u g g e s t s any  within  a typical  Before  line  splittings as  a test  effects  are  to  try  1 3  CH„  given  to  1 2  in Table  CH„.  effects  in D  and Hg  4 t  shown  fc  two  for  on t h e  methane  constants  between  (of  the  total AH AH  Watson well  A  mass of  4 f c  6 t  A  the  further  methane  zero point variation isotopic  to  the  C J  i  isotopic  Watson  has  where  2 - a )  a  )  2  [3.4]  2  C  original  equations 4 t  and H g  variation effect  theoretically.  contribution  methane  molecule,  that  [3.6]  the  of  methane  = -(15/22)AH.  vibrational  variation  10  [3.5]  isotopic H  shown t h a t  the  = (7/88)AH  has d e m o n s t r a t e d t h a t  the  for  the  Td symmetry)  AH = - ( 2 5 6 / 3 5 ) [ m Am /m (M+Am ) ] B ^ $ 2 3 $ i « ( and M i s  where  vibration  distortion constants.  A  beneficial  s e p a r a t e d by 2 t o  has.examined the  relations  XY„ molecules  V  tensor  situation  them by z e r o p o i n t  and i s o t o p i c  for  well  theoretical  should prove  spectra  Watson  centrifugal  (r74)  the  it  The m o l e c u l a r  3.7.  and a c c o u n t s  probably  known.  one f i n d s  and  a  isotope effect  explain  results  H  t h e Ge  quantitatively  methane  MHz f o r  t h e q u e s t i o n of  s h o u l d be n e g l i g i b l e ,  are  In  why  is  width.  t u r n i n g to  u n d e r s t a n d i n g of  splitting  D  fc  but  results.  t  in D  (e3.4-e3.6)  fc  d o e s not  A more p r e c i s e is  clearly  transition  is  He  has  consistent examine  estimate  needed as  frequencies  describe  is  the  with a  this  of D  the fc  dominant.  For  74 instance,  of  transition line 5.4  t h e measured methane  w i t h the  greatest  J=1"6,3<—2 where out of MHz c a n be a t t r i b u t e d Hecht  (r67) TJ  isotope a shift  to  has g i v e n  spectrum the Q branch  the  the  3  of  =(2/3)(1-$ )  order  c o n s t a n t s expressed in  3  Coriolis  and $ |  3  interaction  effective  equilibrium rotational values. B  , D  g  and D  isotopic  fc  are  variation  all  terms  from t h e c o r r e c t i o n s  of  to D  the  and i s  (r74)  D , fc  second order  (e3.7)  is to  strictly the  same way a s  related  the  the  first  to  the  equilibrium the c e n t r a l  central  e q u i l i b r i u m that  fc  fc  are  related  i n much t h e  due t o  fc  D .  t3.7]  Equation  independent of  in D  in  -^4  constants are  A c c o r d i n g t o Watson  9.6 MHz a p p r o x i m a t e l y  = (2/3)(1-$„)  constant  constant  type  y 2  constants.  an e q u i l i b r i u m q u a r t i c empirical  4  the E  following expression for  •H " $i  is  0 . 0 3 % change  2  where  shift  the B  0  parameters mass so any  mass must be result  in D  fc  effective. In  order  to changes  to appreciate  i n the  central  t h e methane v i b r a t i o n  different  the  identical  things.  various  parameters  central  atom i s  mass i t  problem in  help a v o i d c o n f u s i o n in m o l e c u l e s where  how D  fc  m i g h t change w i t h is  instructive  some d e t a i l .  later  treatment  symbols w i l l  Of p a r t i c u l a r in equation  i s o t o p i c a l l y changed.  examine  This w i l l  will  also  ammonia-like  correspond to  interest (e3.7)  of  to  respect  be how  behave when  the  the  75 The symmetry c o o r d i n a t e s m o l e c u l e s have (A,+E+F ).  In  2  constant  F,,  the  a symmetry in  immediately  following  the A,  and n o r m a l v i b r a t i o n s point  and E v i b r a t i o n a l  frequencies.  c a n be w r i t t e n  terms of  constants  in  breathing  F,,  force  Table  force  germane  constant  to  the  +3f  =f  constants  : see  .  -f  F «=/2 3  , <f  r r  -f p>  least  f igures.  of  we see  that  A, F,,  force  the A,  totally  the  vibration  symmetric  methane,  for  r  of  potential  SiH«  a  and  b  GeH,  b  5.435(8)  2 .84(6)  2 .65(5)  mdyn A  0.584 (1)  0 .41(1)  0 .40(1)  mdyn/A  5.378(8)  2 .74(6)  2 .63(5)  -0.221(3)  -0 . 0 3 ( 3 )  5  mdyn  •b'  0.548(1)  fits  in  0 -50 (1) 5  -0 . 1 2 ( 3 ) 0 ) 0 .46(1)  ref(r76)  in parentheses  squares  def'n  silane  mdyn/A  ref(r75)  Numbers the  values  are  3.6  for  ref.(r76)  mdyn A 'a'  force  E species  valence  we a t t r i b u t e  CH„  3 3  3.6  the  vibration.  C o m p a r i s o n of  F  the  appropriate In  Table  F, , = f  in  2 2  bond s t r e t c h  o n l y and t h e r e f o r e  60, and t h e  representation  and F  d e t e r m i n e d from t h e  Td  group d e s i g n a t i o n s :  coordinate  species  of  are  units  the of  standard deviations the  last  significant  of  76 Similarly,  the  E vibration  bending v i b r a t i o n .  here  center  of  is  that  constants  in Herzberg  X.  t h e X atom of  and o n l y  corresponds to  For  the F  b e n d i n g ; however  these  states  c o m b i n a t i o n s of vibrations  point  both angle  here  is  system c e n t e r  linear  SiF„,  Si=28,29,30  pg.232,r78)  which  for  3.6, to  f t f t  constant are  force F  3 3  "pure" F  couples  3 a  linear  {til).  (r37).  the  The  3  change 1/VM  the  in  from  the and co,,  co  3  diatomic  t h e co  3  where M i s  vibrational  and a>„ the  frequencies  The p r e c e d i n g d i s c u s s i o n  and R e d l i c h  product  XYj, Td m o l e c u l e s  and  important  From a h a r m o n i c  the  These "v "  the v i b r a t i o n s  i-m i 1 /2  and  and F  that  for  r o u g h l y as  summarized by t h e T e l l e r  system  three  now d i s p l a c e d  be i m p o r t a n t .  r e d u c e d m a s s . As an example see for  are  c o m b i n a t i o n s of  t h e X atom i s  would v a r y  the  bends and bond s t r e t c h e s .  a p p r o x i m a t i o n one would e x p e c t frequencies  there  interaction  of mass and hence  t h e mass of X w i l l  always at  From t a b l e  p i c t u r e d by H e r z b e r g  that  important  2  two v i b r a t i o n s  c o r r e s p o n d to  vibrations  "vn"  the  The  are  o f co, and co do "Qt depend  bond s t r e t c h i n g  angle  to give  is  species  2  coordinates)  pg.100).  XY,,  two v i b r a t i o n s .  "pure"  (normal  (r37  mass and so t h e v a l u e s  on t h e mass of  angle  2  These v i b r a t i o n s  shown s c h e m a t i c a l l y point  co c o r r e s p o n d s t o a t o t a l l y  rule  is  (r37  is [3.8]  77 m  r  m  [3.9]  1  x We now c o n s i d e r (e3.7).  The r o t a t i o n a l  vibrations  are  interaction  $3,  $«  split  angular  The q u a n t i t y  is  be a f u n c t i o n Expressions  of  the  C  In  to  1 3  C  ±.02  ±.01 t o  for  zeta  f l  their both  sum r u l e  Robiette  an  n o r m a l mode of  by a  frequencies  to  discrepancy constants.  fig.  F  Coriolis constants  represent  integer  vibration.  and p r o v e s  vibration  to  (r79).  coupling zeta constants  as  functions  and mass a r e  given  in  i s o t o p i c d e p e n d e n c e of  16.3 on h i s  $„  (and  the  by  page  Cyvin  359 where  " b o t t o m " our  s m a l l and w e l l w i t h i n  This  the  $ «) 2  usual  s m a l l change  numbers t a b u l a t e d  Table  6.  CH„  and  1 2  the  associated with C o r i o l i s coupling (r42,r8l).  experimental  $3+$  degenerate  Coriolis coupling  necessarily  Cyvin's  is  of  in  equation  i n C o r i o l i s c o u p l i n g e x p e c t e d when m^ c h a n g e s  error  (r75)  not  what we c a l l  "harmonic"  the  sublevels  The p r o b l e m of  graph i s  1 2  triply  c o u p l i n g has been examined i n d e t a i l  change  from  three  the  in  momentum a s s o c i a t e d w i t h t h e  constants,  (r45,p342ff). "top"  of  The c o u p l i n g c o n s t a n t s  these  (r77,r80).  Coriolis  levels  into  the  for  potential  ref.  C o r i o l i s constants  c h a r a c t e r i z e d by t h e  (r37,p447).  rotational  of  the  (r37)  sum t o  1 3  however  CH„  we e x p e c t exactly  as a measure Finally,  Notice  of  without  by G r a y  is the  1/2, the  and  that  roughly  is  constants  s u p p o r t e d by Robiette  their  sum of  . 5 0 9 . From  zeta constants  of  the Gray  so p e r h a p s we c a n t a k e reliability  going into  all  of the  these  the  zeta  detail,  and  78 examination  of  the  conclude that, vary  with  the  the  within  isotopic  We a r e in  variation  square  change  in  It leading First  is  to  C-  for in  (e3.7)  methane the  variation  germane  term  atom i s o t o p i c  bond l e n g t h s of  As a c h e c k  empirical  in  terms of  bracket  on t h i s  parameters (e3.7)  in equation  substitution,  affecting  the  under  value  of  (albeit  and f i n d ,  rather  that  within  central  a t o m . T h i s d o e s not mean t h a t  not  preclude  correct it.  (e3.7)  with  rather  terms  It  should  the  change  less,  change  is  5  the  .  in  fc  that  in D  to  fc  changes  Then s i n c e  the  more  the  error,  D .  invariant  the change  put  isotopic  that  given  the  vibrational B  to  argument  inaccurate  t e r m does not v a r y  necessarily  the  term s h o u l d be even  p r o c e d u r e we w i l l  again  the a s s u m p t i o n  zero point  the  points  smaller.  isotopic  that  bracket  is  where  of m a g n i t u d e  t o an e s t i m a t e  bracket"  of  s h o u l d be s m a l l .  to o u t l i n e  invariance  perhaps  b r a c k e t e d t e r m would be even  methane we w i l . l show,  do not  mass. R e f e r r i n g  germane c a s e ,  the  to  C.  we have  r o u g h l y an o r d e r  of  us  summarize some key  with c e n t r a l equ.  1 2  s h o u l d now be h e l p f u l  "square  equation  1 3  should allow zeta constants  the X - i s o t o p i c  that  c a n be e x p l a i n e d  5  fc  bracket  the  for  central  in  D  that  r e d u c e d mass  the  of  be a p p a r e n t  the  substitution  fc  also in  "error",  expression for D ,  the  references  now i n a p o s i t i o n  some i n d i c a t i o n in  above  various values)  the  square  substitution  t h e above  the a v a i l a b l e  into  of  the  assumption data  does  79 p r o n o u n c e d f o r germane t h e z e r o p o i n t used t o e s t i m a t e First  AD  t h e change  b r a c k e t e d term i s c o n s t a n t .  implicit  dependence  averaged  bond l e n g t h s  slightly  shorter  in D  (the g r e a t e r  constant  dependent  does  i s an  zero  point  t h e r e d u c e d mass t h e  the v i b r a t i o n a l l y it  the square  mass b u t t h e r e  i n the i s o t o p i c a l l y  molecules  if  t  The B r o t a t i o n a l  depend on t h e c e n t r a l  For methane-like  i n B w i l l be  i n germane.  fc  l e t us c o n s i d e r  not e x p l i c i t l y  effect  averaged  bond  length).  is straightforward  t o show  that AD where r  6D  = -  fc  Ar  [3.10]  i s t h e bond l e n g t h and Ar t h e change  isotopic  s u b s t i t u t i o n . The p r o b l e m now i s t o e s t i m a t e A r .  One way i s t o use t h e d i a t o m i c  approximation  2  and F  by t h e p o t e n t i a l 2U(r)  The c o n s t a n t s  2  2  e  and F  stretching potential ^ ^ come from a q u a d r a t i c respectively.  ;  potential  constants  defined  U(r)  = F (r-r )  F  /M  are the diatomic  3  (r54,r82) [3.11]  4/FTL VMi where F  i n r on  3  2  + F (r-r ) 3  e  are simply  constants  f  and c u b i c  r  force  field  +  [3.12]  related (or f  rr  to the  ) and f  (anharmonic)  The methane Ar w i l l  p u b l i s h e d anharmonic  3  rrr  which  analysis  be e s t i m a t e d of Gray  valence  from t h e  and R o b i e t t e  (r75)  80 and a l s o  from t h e e m p i r i c a l p a r a m e t e r s  Laurie  (r52).  Laurie  estimations  anharmonic  The l a t t e r  force  give  which,  field,  credence  of H e r s c h b a c h and t o t h e H e r s c h b a c h and  b e c a u s e of t h e l a c k  will  be t h e s o l e  o f a germane  estimation  of Ar i n  germane. Some c a r e F  2  is  and F  must be t a k e n  in e s t a b l i s h i n g  from t h e a n h a r m o n i c  3  because  equation  the p o t e n t i a l s  (e3.12).  bond s t r e t c h equation  definitions. C-H s t r e t c h equation  The p o t e n t i a l  (r75).  This  than  in  i n (r75) i s d e f i n e d f o r the  of 3 d i f f e r e n t  with  in  ( r 8 3 ) assumed a Morse p o t e n t i a l  and r e f e r e n c e  (5)  study  ( r 8 3 ) and c o m p a r i s o n  shows a f a c t o r  Also  field  are defined d i f f e r e n t l y  as i n r e f e r e n c e  (e3.12)  force  t h e v a l u e s of  to (r49) e q u a t i o n  f o r the  (3) a n d ( r 5 3 )  s h o u l d show f F  where f just  f  r  r  i s given  r  3  = " -f^  in Table  4 of  w h i c h c a n be d e t e r m i n e d  and t h e e q u a t i o n s 5.403 mdyn/A and F equ.(e3.11) relations  Ar, ., 2  in.Table 3  3  [3.13]  from F , ,  3 . 6 . In t h i s  = 1 0 . 4 9 mdyn/A  establishing  2  The v a l u e and F  and t h e n  =  5  ( r 5 2 ) one f i n d s F  = 7.96 mdyn/A and t h e n A r , . , 2  some c o n f i d e n c e  3  a A D , . 1 3 from e q u a t i o n  = 1.0858 A ( r 7 5 ) )  = 4.98 5  i n t h e e m p i r i c a l method o f  Ar, .13  one c a l c u l a t e s  2  = 4.3 x 10~ A  Using the value  -37 H z . ( r  2  from  H e r s c h b a c h and L a u r i e . 2  is  2  i n (r75)  3 3  way we g e t F  2  3  of F  = 5.0 x 1 0 ~ A. U s i n g t h e e m p i r i c a l  o f H e r s c h b a c h and L a u r i e  mdyn/A a n d F  (r75).  f c  o f 5.0 x 1 0 ~ A f o r  2  which c o m p a r e s v e r y  5  (e3.l0) well  of  with  81 the  experimentally  light  of  it  likely  is  the  determined value  of  -38.7(±3.0)  a p p r o x i m a t i o n s l e a d i n g up t o that  this  agreement  the  AD  Hz.  estimate  t  with experiment  In  is  somewhat  fortuitous. Now we s h o u l d c o n s i d e r square bracketed  term  not  the  significant,  on t h e value  order of  the  error  square is  very  This  for  Just  before  one g e t s to  the  of  AB  further  f c  (e3.7)  invariance  However  of  2  reasonable we a r e  is  constant AB  = -11.4 Hz. T h i s  for  to  the  the  the data  bracket even  0  difference  square  is  and  known  result  from z e r o p o i n t  the  also  is  (r75) also  vibration  .  i n bond l e n g t h  similar This estimate  methane.  ready  to estimate for  methane.  AD^. f o r  easy  From H e r s c h b a c h and L a u r i e  to c a l c u l a t e .  germane  t h e n one c o u l d  stated  2  and found  experimental  the q u a l i f i c a t i o n s  F  Gray  variation  s h o u l d be d i s c u s s e d . I f  F o r methane  our z e r o p o i n t  Finally  finds  a  an a s s u m p t i o n e x p e c t e d t o be  point  -37 Hz p r e d i c t e d  is  the  held constant.  t u r n i n g our a t t e n t i o n  a AD , .i3  shows t h a t  to  is  being  one o b t a i n s  numbers of  and so t h e a v a i l a b l e  = (3D^./B) AB.  fc  experimental  is  (e3.7),  The d i f f e r e n c e  still  term  in equation  w r i t e AD  but  a s s u m p t i o n of  the  hard to estimate  bracketed  the  when  fc  germane.  p r o b l e m one bracket  is  similar  in equation  better  which  very  large  on AD  to vary.  is  do not p r e c l u d e term  error  the d i f f e r e n c e ,  (r75).  effect  allowed  -75 Hz u s i n g t h e  Robiette if  of  is  the  germane  The p a r a m e t e r  = 2.59 mdyn/A. T h i s compares v e r y  (r52)  well with  within F  2  one the  is  82 value  of  values  f (=f r  of  F,,  relations  is  and F  in  only  pure  to  for  a =a (r75).  equal  to  1.527  A (r76),  a  This  a  of  (0.015  3  change  F  in  values  1  a  and a  3  see T a b l e and h e n c e  our  (r63)  in D .  a, ^ a  are  3  predicted  in D  due t o  a central  atom i s o t o p i c  of  For  fc  zero point  is  is also  taken  7  0  . ,  = 5.2  7  AD  of  fc  of 6  1.4  error  Hz,  in  two  or  the change  effects  within  be  x 10~ A .  0.86 Hz so t h e  is  but  to  a bond l e n g t h  vibrational  substitution  is  Since  much l a r g e r  a , was  and A r  D  known.  . This  is  p u r p o s e s then  determined value fc  b r e a t h i n g mode  The s t a n d a r d d e v i a t i o n  fc  in  constants  3  3  2  use  t h e mode co  where F „  = 2.97 m d y n / A  alpha  the  3.6,  ref.(r63)).  - 1  to  f l  plus  we s h a l l  the  1 refers  the  from H e r s c h b a c h and  3  one of  f  i n methane  cm  value F  from  (r76)  r corresponds to a change,  .002% change  empirically  3  a  where  For  3  latter  To e s t i m a t e  germane,  case  mdyn/A c a l c u l a t e d  Duncan and M i l l s  bond s t r e t c h  be t h e 1  1.6  the  still  This  of  3 3  3.6.  section  small  almost seen  = 2.63  F  calculations.  C J , . However 3 4  = 2  one needs a v a l u e  defined  F  )  in Table  subsequent Laurie  r r  due  to  standard  deviations. From t h e equations  (e3.4-e3.6)  estimated.  splitting  420 K H z ) .  KHz.  Recall  is  is  changes A H ^  isotopic to  similar  The t o t a l this  Kattenberg  .002% c h a n g e  expected s p l i t t i n g  isotope (~  t o an  of  the  Assuming a  contributions total  parameters  the  shift  et t  for  or  less  the  transition  shift, In  than  J=11  that  t  allows  fc  be c a l c u l a t e d . to  (r63)  and A H ^  in D  frequency  al  E  ( 2 )  and  are the  and f i n a l l y all the  line  <-E  Kreiner  cases  < 1 >  et  al  a  the  widths is  44 (r68)  83 o b s e r v e d w i t h two d i f f e r e n t predicted the  splitting  100 KHz e r r o r s  for this  it  The n u l l  results  similarity  is,  o f GeH„  reported here  that  line  to those  over  germane s t u d y .  independent  First,  SiH„  agree  with  the spectrum  general  a sufficient  spectrum to allow  tensor  These c o n s t a n t s MW-IR d o u b l e  species.  prove  tentatively one o t h e r  analysis  study  definite  assigned line tentatively  of  forbidden  have been o b s e r v e d  the determination  t o be i n e x c e l l e n t  resonance  results  number of  d i s t o r t i o n constants  confirming a l l their  this  and the  FINAL COMMENTS ON GERMANE  t h e microwave  of  2 8  al  negligible.  a l l Ge i s o t o p i c  d i s t o r t i o n moment Q b r a n c h t r a n s i t i o n s  of  et  spectra  of  suggesting that  i s an a v e r a g e  evidence  indeed  for " s i s t e r "  widths  study  within  splitting  c l a i m of K r e i n e r  We s h o u l d now summarize, t h e t h r e e the  the  f o r our p u r p o s e s ,  of s e a r c h e s  a quasi-theoretical  6.  d i s c u s s i o n on t h e i s o t o p e  supports the e a r l y  the s p l i t t i n g  i s then w e l l  et a l .  s h o u l d seem a p p a r e n t  presented here that  c o i n c i d e n c e s . The  transition  of K r e i n e r  In c o n c l u d i n g t h i s i n germane  laser  of K r e i n e r  of a l l s i x  in equation agreement et a l  (el.10). with the  (r68)  a s s i g n m e n t s p l u s one  as w e l l a s e n a b l i n g assigned t r a n s i t i o n .  i s made from t h e c o m b i n e d MW-IR d a t a  reassignment The  final  and t h e d a t a  study.  Secondly,  the p r e s e n t  supporting the c l a i m  that  study  a d d s new e v i d e n c e  the i s o t o p e  splitting  in  i n germane  84 due t o Ge i s o t o p e s here  was d i v i d e d  is  negligible.  into  two t y p e s ,  The e v i d e n c e experimental  presented and  theoret i c a l . Thirdly  the  used t o p r e d i c t  empirical accurate  parameters  tensor  including  2 0 . These p r e d i c t e d  published  (r84).  useful  further  AJ=1  in  easier  to  of  the  ref  nuclear  ortho  1=1 and meta  GeH  a  CH„.  it  is  found f o r  similar exists  in  the  the  up t o  splittings infrared  methane tensor  p r o d u c e d by t h e  be and  been  would be  and Raman ortho-para  (r85,r5)  s h o u l d be  constants  presented  refer  to  protons; para  the 1=0,  1=2.) interesting  parameters  from T a b l e  trend.  of  J  can  have a l r e a d y  (Here o r t h o and p a r a  and how d i f f e r e n t Recall  and i n  for  high r e s o l u t i o n . Also  type  spin I  and S i H „  page,  of  (r86).  total  Finally  splittings  MW-IR s t u d i e s  f i n d as a r e s u l t  h e r e — see  splittings  These p r e d i c t e d t e n s o r  R branch spectra  transitions  obtained here  Later series  to  note  the  in Table  3.7,  listed  these 3.6  that  we w i l l NH , 3  values  PH  the  are force  f i n d that 3  and  similarity on t h e  constants  3  the  next  when c o m p a r e d  a similar  AsH .  of  to  show a  trend  85 Table  3.7  C o m p a r i s o n of  spectroscopic p a r a m e t e r s :  Parameter  12  B D  H H L L  T  4 T  5.24l0356(96)  (Hz)  132  4 T  74  d  and GeH,  GeH, 2. 6 9 6 9 ( 9 )  b  751 . 4 ( 3 6 )  67  775. 5 4 ( 8 6 )  C  e  (Hz)  11,034(19)  2.598(5)  2. 9693(19)  4.65(29)  3. 996(88)  -6.044(15)  - 5 . 3827(55)  (Hz)xlO  a  20.3(9)  (Hz)xlO  a  -26.8(12)  -3.79(19)  - 4 . 122(50)  (HZ)X10"  -30.0(27)  -7.66(44)  - 8 . 01(14)  3.73(4)  3. 3 3 ( 5 )  (D)X10  X y  z  2.41(5)  5  h  f  'a'  reference  (r85)  'b'  reference  (r56)  'c'  reference  (r63)  'd'  reference  (r20)  'e'  present  'f  C o r r e c t e d from v a l u e p r e s e n t e d  in  This value  has been r e v i s e d  3.73(4)x10" D.  The c h a n g e  is  5  correction  in  contribution Stark  the  cell  almost  calibration  from a h i g h e r  ref.(r87).  r e p o r t e d as e q u a l  3.34(4)x10" D. 5  9  work.  d i p o l e moment was o r i g i n a l l y  order  in  entirely but  to this  work  due t o a  includes a  calculation  This  of  to  small  small  the  linear  coefficient. 'g'  reference  (r69)  'h'  reference  — Numbers i n p a r e n t h e s e s a r e the  SiH„  -16.984(23)  6 T  T  943.4(37)  2 8  SiH,  2.859065dO)  a  CH„,  (Hz)  g T  Lg 0  (cm" ) 1  0  28  CH,  1 2  least  f igures.  squares  fits  in  units  of  the the  (r5)  standard deviations last  significant  of  86 E.  CHAPTER  4:  Arsine  ARSINE  is  a mild g a r l i c  a colorless, odor.  dubious v i r t u e is  of  it  quotes T.  of  arsenic  give  off  life.  Persian  "ZARNICH"  other  "De a r s e n i c o "  (Upsala  1777)  his  Arsine  to d e t e c t  itself  top a r s i n e s ,  to  of  ore as  deprave  Greek the  dates  (r89)  who  it  metals  sniffed  strongly  at  and the  "valiant"  way a r s e n i c  acts  1775.  s u c h as A . F .  would  from  word f o r  from a b o u t  r e p o r t e d microwave  measured i n  on  Arsine  Gehlen  the  who,  joints  s m a l l d i p o l e moment  transition  nuclear  further  transitions  Further  study  (r9l)  (0.22  further  of  was c a r r i e d  From t h e i r  data  and hence  intensity  out  be m e a s u r e d at in  symmetric  transitions  (r90).  quadrupole s p l i t t i n g  c o u l d not  transitions  J=0—>1  forms  Debye),  who m e a s u r e d t h r e e a l l o w e d  allowed  range.  deuterated  moment, and t h e  the m u l t i p l e  measurements of  1955, were t h e  b o t h n o r m a l and f u l l y  the  six  it  suggesting  probably derives  researchers,  leaks,  Today  apparatus.... The f i r s t  to  smell,  w h i c h became t h e  was h a r d on many e a r l y in order  smelters  "arsenic"  perhaps  the  (r88). Mellor  a reference  metals.  poison gas.  by J . W .  Our word  have  w e l l - d o c u m e n t e d by  given  smoke, g a r l i c  destroy  "bold",  is  was d i s c o v e r e d by  a white  is  being a m i l i t a r y  arsenic  Bergmaris'  that  of  toxicity  u s e d more as a d o p i n g a g e n t A history  or  Its  p o i s o n o u s gas p u r p o r t e d t o  B e c a u s e of small  degradation of  lines,  that  time.  1971 by Helmi-nger e t transitions  for  due  al  arsine  for  arsine-D3  i n the  millimeter  they  were a b l e  to c a l c u l a t e  the  and wave B , 0  87 Dj  and D j  rotational  K  parameters  and Oka  eqQ  (r82)  rotational various  parameters  (where q=<b  s p e c t r u m of  linear  rotational  higher  terms,  resolution (r26) eleven  new  course  of  forbidden AsH (r93).  This  3  last  the  The A s D  3  is  C ,  0  In  0  of  very  high  calculate  parameters.  D ,  very AsH  calculated  and A s T  e s t a b l i s h e d and o n l y  3  reduction, during  well  (r98)  of  as  al  sextic  (up t o 900 sources.  the  ground  infrared et  spectra  fundamental v i b r a t i o n  et  characterized  Burenin  infrared  and  reported  measurements  is  the  both allowed  c o u l d be  from t h e  with those  plus  high  impressive  a n d , where a p p r o p r i a t e ,  agreement  D..  T T  high frequency 3  Olson,  report  D .,  T  their  of  parameters  s p e c t r u m of  constants  to  Finally  study  frequency  Chu  1978 Helms and G o r d y  particularly  rotational  infrared  (r97)  B ,  1974  was p r e s e n t e d by B u r e n i n  made p o s s i b l e by t h e i r  in e x c e l l e n t  to  were a b l e  transitions.  3  an e x t e n s i v e  study  (r92 , r94 , r 9 5 , r 9 6 )  been  parameters  transitions  b e c a u s e of  molecular  later,  In  f o r b i d d e n AK=±3  rotational  from an a n a l y s i s  work  octic  The  the  C„.  but a n a l y z e d u s i n g an o b s c u r e  as w e l l as  GHz)  and were a b l e  3  in  hyperfine  and  M  AsH  forbidden AsH this  C  lines  i n f r a r e d measurements.  reported,  the  four  one y e a r  ground s t a t e order  2  c o m b i n a t i o n s of  Maki and Sams ( r 9 2 ) ,  ,  4>/dz >)  2  reported  plus  state  spectra al  are  are  (r93). less  frequencies  well have  reported. The p r e v i o u s  data c o n s i s t Strandberg  of  (r99)  partially  two  lines  (  1 i 1ii 0  _  deuterated  asymmetric  top  arsine  f o r A s H D r e p o r t e d by Loomis and 2  at  3 5 . 5 GHz and 3  0 3  -3  1 3  at  29.5  88 GHz).  In  this  case  by-product  of  reason  this  for  arsines the It  a study  turns  out  frequencies capability  low  very  given  most  this gas  of  later,  is  after  first  2  arsine  a brief this  so t h a t  the a r s i n e  Chapter  arsine  6).  b o t h R and Q  best  study report which  the reporting only is  not  how  the  were p r e p a r e d ,  are  d i s c u s s i o n of  study  the A s D  observation,  3  s p e c t r u m and s e c o n d along with  and a n a l y s i s .  harmonic  force  d i p o l e moments w i l l  the  important Because  and p h o s p h i n e s t u d i e s  terms of  and d i s t o r t i o n  any  a feature  2  in  frequency  to e v a l u a t e  so t h a t  t o p A s H D and A s D H s p e c t r a  of  that  at  determination.  samples used f o r  of  the  from  transitions.  t o be f o u n d  above  in order  top  stems  branch  Recall  parameters,  structural  prediction,  structures  R (AJ=1)  Q branches c o u l d at  chapter,  similarity  analysis  discussed later)  separately  probable  on a s y m m e t r i c  R branches are  necessary  d e s c r i p t i o n s of  points  is  were a  The most  3  spectrometers.  are  for  SbH .  information  1 0 0 GHz, w h i c h  frequency  asymmetric  (see  these  parameters  useful  arsine  of  c o m b i n a t i o n s of  In  the  above  branches  linear  on s t i b i n e ,  in o b t a i n i n g the  that  of  rotational only  lack  observed t r a n s i t i o n s  (and p h o s p h i n e s , as  difficulty  (AJ=0)  the  of  further  fields, be  presented  and p h o s p h i n e can be s t u d i e d  together  89 1.  PREPARATION OF THE ARSINE GAS  There are  many p r e p a r a t i v e  w h i c h c a n be f o u n d i n present  (rlOO)  inert  100 t o r r .  1.5  Na m e t a l flame  exothermic  reaction  to  We f o u n d t h i s  transfer  was  slow.  the  in  Instead,  of  reaction  nickel  crucible.  After  the  t o c o o l and was  Contrary  t h e method of  atmosphere below). heavy  nitrogen  or  To p r o d u c e a r s i n e  water  allowed  of  glass  or m i x t u r e s  to d r i p slowly  of  the  pressures 3.0  the alloy  then  Bunsen  high  A  mixture  nickel  vessel  by  was  difficult  also,  heat reaction  temperature  and d i s p e n s e d w i t h  a r g o n was  formed t h e  two,  the  system  100 mm.  15 m i s .  of  as a p p r o p r i a t e ,  onto the a l l o y .  the  evacuated.  re-established  approximately  glass  excessive  It  thoroughly  and  from  "red-hot"  had been  rapid  in a  a quartz  D r a k e and R i d d l e  the  g . As and  Drake  vessel  reaction;  under  off.  reaction  reaction  alloy  was a l l o w e d to  preparation  we o p t e d f o r  which c o u l d w i t h s t a n d forming the  situ"  reaction  method u n s a t i s f a c t o r y .  vessel the  the  the  Drake  began t o m e l t .  smoke was g i v e n  from t h e  see what was h a p p e n i n g  "in  or a r g o n a t  i n which  alloy  For  t o d r o p water  with a semiluminous  followed  many of  t h e method of  was made by h e a t i n g  sodium metal  to p r o t e c t  (r89).  prepared  nitrogen  and a b l a c k  insulated  in order  heat.  the  arsine,  The p r o c e d u r e was  gently  suggest doing the  crucible, wool  very  of  3  dry  for  by M e l l o r  (Na As)  The a l l o y  until  glowed r e d - h o t Riddle  alloy  a t m o s p h e r e of  around  burner  review  was u s e d .  on a s o d i u m a r s e n i d e  g.  methods  studies a modified version  and R i d d l e  an  the  SAMPLE  Gas b u b b l e s  inert  (see water, were evolved  90 in  the  reaction  followed  mixture.  by m o n i t o r i n g t h e  n i t r o g e n a t m o s p h e r e was "drop";  The p r o g r e s s of  funnel.  trap  water  for  t r a p p e d under  The  found n e c e s s a r y  under vacuum t h e  dropping  pressure.  water d r o p s  The r e a c t i o n 2  ether/  l i q u i d nitrogen  by t h i s  for  infrared  (-196°C)  where  a path  l e n g t h of  presence  of  from t h e i r  the  vapour  (1-bromobutane, of  transitions  (r99).  In  previously  with greased  parameters  of  rotors  chapter).  like  To d e g r e e  the  finally  further petroleum  AsD  3  produced  457. low  a pressure  arsines  of  5  characteristic 6).  was  The  established  -111°C,  s l u s h r101)  measured microwave were s t o r e d  or  from  the  rotational  in glass  bulbs  stopcocks.  to a p p r e c i a t e  involved  to  any  chapter  35 mm. a t  The a r s i n e s  f o r b i d d e n AK=±3 s p e c t r a  oblate  of  PREDICTION OF THE A S D order  (see  deuterated  pressure  or  end of  and  10 cm. s h o w i n g t h e  or c a r b o n d i s u l f i d e  observation  2.  the  through a  The p r e s e n c e  frequencies  partially  water  Water was  spectrum recorded at  fundamental v i b r a t i o n  fitted  the  T ^-80°C)  distillation  slush.  for  method was c o n f i r m e d by a P e r k i n E l m e r  resolution Torr  trap-to-trap  was  100 mm. a r g o n  f r o z e to  n o n - c o n d e n s a b l e g a s e s were pumped o f f . removed v i a  reaction  p r o d u c t s were p a s s e d t h r o u g h a  (C0 /petroleum ether l i q u i d nitrogen  the  it  3  the is  in allowed arsine  SPECTRUM difficulties  instructive  in  predicting  to c o n s i d e r  the  and f o r b i d d e n t r a n s i t i o n s  (and p h o s p h i n e  4 i n components of  in  the  angular  next momentum  of  91 operators, v (J,K)  = 2(J+1)B -2(J+1)K D -4(J+1)*Dj  N  P (J,K)  = (6K+9)  d  where  ? (J,K)  0  transition,  2  0  upper  D  3  From t h e work  Since  D  were u s e f u l  spectrum. AsD  3  "guesstimates"  made from t h e " r a t i o " f o r P D , PH 3  (r25,rl02,r93),  3  obtained later  6 Table  6 . 4 ) . However  sufficient, for,  since  the t o t a l  ( r 9 l ) on t h e  3  0  (e4.2),  K D„  r D  only  0  and  f o r the  parameters C  0  o f D„. c a n be  3  3  R  R  from  3  references  f o r AsD  3  o f 0.856  force  field  o f 0.833 MHz. (See C h a p t e r  even a r o u g h e s t i m a t e  o f D was R  f o r t h e k=*2 t o ±1 comb we were  here  B  (PD ) ( P H ) D ( A s H ) . The  were t a k e n  work  D_„ and  d i s t o r t i o n moment  3  with the c a l c u l a t e d  in t h i s  of B ,  c o n t r i b u t i o n o f D „ i s on t h e o r d e r  and a ±5 MHz e r r o r search.  et a l  estimate  g i v i n g an e s t i m a t e d D  MHz, w h i c h c o m p a r e s w e l l  of K i n the  the s e a r c h ' r e g i o n  reasonable  and A s H  of a  value  of the r o t a t i o n a l  relation  (J,K)—>  (J,±K*3)  had v a l u e s  the AsD  to e s t a b l i s h  and D „ were made. A v e r y  value  from  in equation  in predicting  In o r d e r  study  values  (J,±K)  of Helminger  does n o t a p p e a r  [4.2]  J R  i s the frequency  s p e c t r u m we i n i t i a l l y  Bj. J K  of the allowed  ( h e r e K i s t h e more n e g a t i v e  state)  normal A s D  R  and f ( J , K )  forbidden Q branch d i s t o r t i o n transition  [4.1]  J K  (B -C )+(2K +6K+9)D +j(J+1)D  i s the frequency  N  (J+1,K)  2  0  would n o t s u b s t a n t i a l l y  looking o f 15 MHz  affect  the  92 The was C .  important Recall  0  t h e moment of the to  arsenic C .  If  0  AsH of  of  C  C  is  0  inertia  the  on t h i s  linear  for  0  AsD  in A s H .  reasonable  starting  point  C (AsH )/2  = 52442 MHz ( r 9 3 )  3  final this  that  3  and " p o s t  the  it  of  3  relies  well  reciprocal  axis  (e1.2).  makes no  Since  t h e As-H d i s t a n c e then  least facto"  since  =* 2,  the  one-half  the  seemed a we n o t e  that  with  the  52642 MHz. The weakness  on t h e  invariance  of  r  isotopic  s u b s t i t u t i o n and i n  light  discussion  in  1.6  is  equilibrium  structure.  In  order  account point  the  to  slightly  vibrational  only  improve our  shorter  for  an  estimate  of  the  we c a n t a k e  As-D bond l e n g t h due t o  averaging . 6  In  this  case  it  is  of  0  to  true  in  t h e moment  structures  this  of  contribution  compares r e a s o n a b l y  C (AsD ) 0  estimate  and M a s s ( D ) / M a s s ( H )  T h i s at  3  measured v a l u e method i s  in AsD  to  s h o u l d be a p p r o x i m a t e l y  3  corresponding value  0  it  assumption that  i n mass,  to  symmetry  axis  t h e As-D d i s t a n c e is  parameter  proportional  about  we make t h e  inertia  value  that  atom l i e s  equals  3  and d i f f i c u l t  into  zero  easy  to  show AC = - ^ A r  For  ND  and N H  3  shorter  6  We a r e  and A s D in  than  the  3  3  the  t h e N-D bond was a p p r o x i m a t e l y N-H z e r o p o i n t  assuming the empirical  two  [4.3]  distance  anharmonic c o n t r i b u t i o n  rotational  molecules.  average  constants  .0034 A  is  to  (rl03).  the  roughly  In  AsH  the  3  same  93 phosphine Ar  the  = r -r  p  D  expected  zero point  average  = -.00444 A ( r 8 2 ) .  H  s  ionization was  was  to A r  potentials  In  N >  For  this  0  Ar  was  3  A g  arsine  t o ammonia  and f o r c e  limit  1.52  in o u r . e s t i m a t e  of  is  to A r  for  Ar  to  than  p  was  f t s  A using equation C (AsD ) 0  similar  instance  due t o  3  345 MHz. The r o t a t i o n a l  then e s t a b l i s h e d  more  see C h a p t e r  w o u l d be c l o s e r  be between  was  t r e n d we  (for  constants:  way an upper  As-D bond was  C (AsD )  is  a bond l e n g t h of  correction  shorter  since  p  expected that  -.005 A. the  Continuing this  | A r ^ | > | A r | . However,  to p h o s p h i n e than phosphine  it  bond l e n g t h d i f f e r e n c e  6 ) Ar  p  taken  as  (e4.3)  the  constant  52442 and 52787  MHz. Even though the are  all  within  distortion This B -C 0  is  one p e r c e n t ,  moment s p e c t r a  b e c a u s e we a r e  is  form of  instructive  the AsD  equation  (e4.2)  v (J)  3  Successive  error  prediction in  estimates in the  is 0  In  o  C (AsD ) 0  3  resulting  greatly  B -C .  of  amplified. this  case  5036 MHz.  spectrum. For  the  characteristic  AK = ± 1 <—+^2 , K=-1  and  becomes  Q  o  expression gives  frequency  the  now t o c o n s i d e r  = 3(B -C )  D  i n the  interested  r a n g e d from 4691 t o  0  It  This  variations  " c o m b " of  the  the  "combs" are  intervals  of  generated  by t h e D  + 15D  transitions  [4.4]  J K  in the  lowest  forbidden Q branch spectrum. to  approximately J R  + 3J(J+1)D  R  be formed t o h i g h e r 6(B  0  _  C ). 0  frequency  The t e e t h  t e r m and i n g e n e r a l  are  of  at  a comb a r e  s e p a r a t e d by  an  94 amount = 6(2K+3)D (J+1)  dv{J,K)  We now see  that  translates  into a gigahertz  "comb".  should also  It  estimation In  of D  light  and t h e  very  a gigahertz However, of  the  fails case the over  a 345 MHz range  R  of  this  for  t u r n e d out  r e g i o n would g e n e r a l l y (e4.5)  teeth  predicted  in the  time  INITIAL  the  width  J=16.  of  in  available  untenable.  search if  the  order  the  the comb.  of  thought  this  case  w i t h no r e s u l t  In  this  100 MHz f o r  to  a  search Very  be a  the  4 MHz s e a r c h more  order  search  experimentally.  initially  Perhaps  transitions  Nevertheless,  2-5 h o u r s p e r  swept  the  line  mean region  was  significantly, were o f t e n  later  transitions!  SEARCHES  The i n i t i a l  line  of  be  comb and t h e n  so t h a t  5-15 h o u r s .  found to c o n t a i n  a prediction  line  a peak  be n o i s e ,  0  the  window  we need o n l y  a monumental e f f o r t  example,  in  these  frequency  r e g i o n s on t h e  initially  The J=14  experimental  search  to  errors  of  meant  m u l t i p l i e d to  3.  e x t r e m e weakness  C  significant.  region  accumulation  regions  the  of  p r e d i c t i o n of  that  search  100 MHz i s  prediction  the  the  strongest  often,  in  be a p p a r e n t  from e q u a t i o n  shift  range  the  not  narrow  s p a c i n g of  in  are  search  [4.5]  J K  s e a r c h was made f o r  u s i n g a v a l u e of was  not  the  the  C (AsD ) 0  strongest  3  J=14  equal  line to  b a s e d on  C (AsH )/2. 0  3  predicted transition  but  95 reference  to Table  predicted  intensity  transition  J=16  c h o o s i n g the to  equation  J=14  4.1  is  J=14  between not  line  (e4.5)  s h o u l d show t h a t  the  the  significant. for  the  region  the  J=16  line.  was  searched,  averaging  m e g a h e r t z window, In  light  more c a r e f u l case  the  taking  account  was  C (AsD ) 0  estimation  used.  estimate  3  search again This  search  t h e J=15  of  result  C (AsD ) 0  was  3  shorter  14.50 t o  14.58 GHz  was t a k e n  and  it  The f i r s t s i g n a l was  four  was d e c i d e d t h a t  in o r d e r . the  last  it  frequency  was d e c i d e d t o to  are  lower  of  the  start  the  frequency.  where we would l a t e r  find  for  many but c e r t a i n l y noise  the  during  runs.  indication  found f i n a l l y  of at  a reproducible  absorption  13.79 GHz. C a r e f u l  studies  t h e n made of  this  transition  and f r o m 4-5 h o u r s of  accumulation  time  there  faint  transition  this  section,  range  13.87 G H z . The p r o b a b l e c a u s e s line  In  were  c o u l d be r e s o l v e d  into  indications a doublet.  a  t o As-H bond  in the  e x c u s e would be s i m p l y u n f a v o u r a b l e  those p a r t i c u l a r  100  search  As-D r e l a t i v e  A rough m i d d l e v a l u e  seeing this  comb a t  a narrower  one hour p e r  13.89 GHz and s e a r c h  at  not  null  of  i n c l u d e d the  line  initially easiest  at  that  scheme d i s c u s s e d i n the  in the  result.  t h e above  estimation  into  length,  of  w i t h no  according  80 MHz as compared t o  The r e g i o n  a time  for  s e a r c h was lines  in  strongest  The r a t i o n a l e  between  T h i s meant  r e g i o n c o u l d be e s t a b l i s h e d .  and the  initial  was e x p e c t e d t o be r o u g h l y  MHz f o r  J=14  the d i f f e r e n c e  signal  that  This  were  the  was  96 particularly splitting.  gratifying  Recall  and t h e r e f o r e of  the  only  25, the  Two u n r e s o l v a b l e  frequency  the  predicted  separation  other  intensity  the  of  weighted  lower  tentatively  pair.  average  in  (e1.25),  for  hyperfine reference J  up t o  splitting  was p r e d i c t e d  w i t h our  available  frequency  splitting  this  the  to  p u s h e d up  components down and  From an o p t i m i s t i c  of  was  the  taken  upper  empirical  as  pair  estimate  transition  assignment  the  was  would  later  J=16. transition  the p r e d i c t i o n  of  searched. A careful revealed  1=3/2  splitting  reported  the measurable d o u b l e t  A s s u m i n g the above  region  a  3  4 strong  components a r e  a s s i g n e d as J=16±2;  be c o n f i r m e d as  already  21+1 or  two u n r e s o l v a b l e  supposed h y p e r f i n e  permitted  for AsD  and e q u a t i o n  doublet  p r e d i c t e d quadrupole  n u c l e u s has a s p i n see  into  hyperfine  as a s i n g l e  resolution.  of  MHz)  z z  minus t h e  to  the  From t h e c o u p l i n g c o n s t a n t  approximately  the  of  the a r s e n i c  transition  X =-164.75(3)  appear  light  we s h o u l d e x p e c t  rotational  components. (r9l,  that  in  J=15  t o be i n d e e d  at  13.87 GHz i n a  second study  a broad a b s o r p t i o n  J=16  of  feature  that that  region  frequency again  a p p e a r e d t o be a d o u b l e t . From t h e s e similar  a b s o r p t i o n was  would t u r n  out  particularly similar of  two t r a n s i t i o n s  later  found at  that  the  J=14 the  J=14  was p r e d i c t e d and a predicted  transition  hard to a s s i g n because of  intensity  the q u a d r u p o l e  just  frequency. w o u l d be  another  2.8 MHz away from one of  doublet.  It  line the  of members  97 It AsD  seemed r e a s o n a b l e  distortion  3  lower  J  (to  c o u l d not  now t o assume we were d e a l i n g  transitions.  higher  The p l a n  frequency).  be measured f o r  in  t h e n was  Unfortunately  its  place  s t r o n g a b s o r p t i o n with a second order line  was  other  far  too s t r o n g to belong to  lines.  In  addition  were e x p e c t e d t o have was  ignored in The n e x t  line  measurement transition that  it  in  is  4.1  this  it  the  the  The s i g n a l  to  as  here  the  all  This  line  this.  fit  These  stated search  search  16. A v e r a g i n g  21  initial  noise  for  roughly  ratios  (assuming  wanted more a c c u r a t e  these  it  and  the  were o f t e n the  measurements  to  lower  the  J,  since  our  transitions  3 hours  signal  found initial  the  tentative  best  order  assignments  and t h e  believed  errors.  on t h e  a b s o r p t i o n s but  the  The  was  measurements were a t  The O p t i m i s t s were c o n v i n c e d of  mechanisms r e l a t i n g  The  noisy  as  to  final  b e c o m i n g t o o weak f o r  was d e c i d e d t o  to  the  reproducible.  final the  was d i f f i c u l t  to  extremely  w i t h the  were f a s t  than  J=12,  3 h o u r s b a c k g r o u n d , a b s o r p t i o n s * were  assignment).  line  This  same s e r i e s  effects.  to  very  effect.  spectra  Stark  satisfactorily  c o r r e s p o n d i n g t o J=17  1.  the  attributed  r e g i o n was  than c o n t i n u e  greater  Stark  should r e f l e c t  c o u l d be f o u n d w i t h i n  spectrometer  plus  series,  included in  transitions  with J  order  t h e J=13  we f o u n d a  distortion  600 KHz e r r o r  was not  Rather the  i n our  in Table  spectrometer  first  search  subsequent d i s t o r t i o n d i s c u s s i o n .  observe and the measurement  the  to  with  poor.  of  3 to  and  Pragmatists  Skeptics  wanted  to  98 see  the  hyperfine  structure.  was  needed where c a r e f u l  In  this  resulted next  in  times. the  a further  c o n s i d e r a t i o n was g i v e n  and n o i s e m i n i m i z a t i o n c o u p l e d w i t h accumulation  case  This  increased  s e c o n d phase  observed arsine-D3  of  the  study  to  tuning  data investigation  spectrum r e p o r t e d  in  the  section.  4. OBSERVED ARSINE-D3 SPECTRUM Fourteen 13.0 t o  Q branch t r a n s i t i o n s  14.3 GHz w i t h K=±]<-*2  in  and J  the in  frequency  the  range  range  8 to  have been measured and a s s i g n e d . The a s s i g n m e n t  was  the  and on  experimentally  "goodness"  of  observed hyperfine  a global  fit  spectrum r e p o r t e d here reference AsD  3  (r9l).  b o t h of  In  these  that  and t h e  along with their  assignments, 7  r  experimental signal  uncertainties  to noise  ratios,  f r e q u e n c y measurement ^ v a l u e s were c a l c u l a t e d broadening parameter d i p o l e moment of  of  for  5  distortion  are  max  (see  given  in  Table  error  e1.37).  o  Debye. for  and  m  The  from o b s e r v e d  from d i f f e r e n t  a temperature  moment was c h o s e n p r i n c i p a l l y  s p e c t r u m of  experimental  20 M H z / T o r r  1 x 10"  of  w i d t h s and f l u c t u a t i o n s  obtained  J  lines  were e s t i m a t e d  line  the  be d i s c u s s e d .  split  maximum a b s o r p t i o n c o e f f i c i e n t s  b a s e d on  distortion  transitions  the d i s t o r t i o n  will  The measured h y p e r f i n e 4.1  i n c l u d e d both the allowed  presenting criteria  structure  24  of  (r82)  The v a l u e  runs.  in  the  The 7 'max  298 K w i t h a  for  a  for  distortion the  dipole  s c a l i n g p u r p o s e s as  d i p o l e moment was not m e a s u r e d i n  this  the  study.  99 Consideration in Chapter The 7 for  max  of  the  estimate  6 suggests the  values  are  for  of  value  the  this  d i p o l e moment p r o d u c e d  q u o t e d above  rotational  line  doublet  s i m p l y the  is  d i v i d e d by  c  into a doublet 1  of  7 max  the the  reasonable.  transitions.  our purpose the q u a d r u p o l e i n t e r a c t i o n  rotational  is  splits  for  7  'max rotational  Since  each  e a c h member of a transition  two.  Table Forbidden hyperfine  4.1  transitions  of  AsD  (MHz)  3  AK = ±1<-+2 J  f , 8  1 4321 . 2 2 ( 1 5 )  1 1-  1 41 58 .25(20)  1 2  A  f2  A expt  1 41 55 .10(20)  3.15(40)  14 093 . 2 0 ( 6 0 )  (  calc  max X10  1  1  cm  5 .07  1. 1 4  2 .76  2 .93  2 .33  3 .53  1 4  1 3951 . 7 0 ( 2 0 )  • 13 949 . 8 7 ( 1 0 )  1.83(30)  1 .74  4 .41  1 5  13 873 . 6 2 ( 1 0 )  1 3872 . 0 9 ( 1 0 )  1.53(20)  1 .52  4 .60  16  1 3791 . 3 1 ( 1 0 )  1 3790 . 0 7 ( 5 )  1.08(25)  1 .33  4 .62  17  1 3704 . 9 2 ( 1 5 )  1 3703 . 8 4 ( 1 0 )  1.08(25)  1.19  4 .47  18  1 3614 . 5 4 ( 1 0 )  1 3613 . 4 6 ( 1 5 )  1.09(25)  1 .06  4. 1 6  19  1 3520 . 2 9 ( 1 0 )  1 3519 . 3 9 ( 1 0 )  0.90(20)  0 .96  3 .75  20  1 3422 . 2 0 ( 5 )  1 3421 . 3 7 ( 5 )  0.83(10)  0 .86  3 .27  21  1 3320 . 2 7 ( 1 0 )  1 3319 . 4 5 ( 1 0 )  0.77(20)  0 .78  2 .77  22  13 214 . 2 9 ( 1 0 )  0 .71  2 .27  23  13 104 . 6 1 ( 1 5 )  0 .65  1 .81  0 .60  1 .40  24  1 2991 . 0 0 ( 8 ) Numbers  uncertainties  1 2990 . 4 4 ( 1 0 )  in parentheses are in u n i t s  of  the  0.56(18)  e s t i m a t e d measurement  last  significant  figures.  1 00 Also given experimentally ^expt*  T  ^  P  e  r e  in Table  4.1  are  the  determined hyperfine  dicted splittings  quadrupole coupling constant  (e1.25)  as was d i s c u s s e d i n we d i d not  and  splittings:  ^  were e v a l u a t e d  arsenic  calculation  predicted  (r9l)  c a  ]_  from  and  the  equation  the p r e c e d i n g s e c t i o n .  need t o  include  the  and  c  In  this  off-axis  d e u t e r i u m q u a d r u p o l e moments b e c a u s e  the q u a d r u p o l e  constant  hundred k i l o h e r t z ,  of  deuterium,  at  s m a l l when compared t o few is  at  most a few  k i l o h e r t z and,  can be  J  J+1  and J-1  respectively, assignment  for  sequence,  that  is  with a l l  behind the  only the  statement  structure  s u p p o r t s the assignments  frequency  fit.  definition structure  of of  the  is  coefficient component  is  for  of  a neighbor  line  given the  also  of  that  is  in Table  and Here  expected frequency  an e l e c t r i c  strictly  field  of  that  "teeth lines  neighbors J  of  the  true  as  from  shift  is  e4.2.  Stark of  1 volt/cm.  shift  t h e M=1 For  the  the  experimental the  in  two  hyperfine  independent  by e q u a t i o n  4.2.  least  T h i s argument the  a  error  neighboring  assumed i m p l i c i t l y  parameters  the  one p o s s i b l e  that  of  accuracy  constraint  data.  is  contribution  frequency  c o u r s e not  spectrum given  A summary of conditions  is  level  is,  l o w e r and h i g h e r  consistent  This  the  Invoking the  we f i n d t h e r e  rationale  to  hyperfine  o f t e n accommodates a t  assignments.  i n a comb" must be i n  the  coupling constant  ignored. Notice  observed s p l i t t i n g s  different  are  the a r s e n i c  h u n d r e d m e g a h e r t z . The m a g n e t i c  required here, the  most a few  coupling  this  101 calculation 0.22 Debye signal J=17 the  the  value  (r99).  have  doublet  An i m p o r t a n t  lines,  In  4.2  rotational  to  review  energy  electric  rotational  in a l l  level  field  levels  of  the  AM=0 t r a n s i t i o n s  i n t o M components).  of  Table  transitions;  M m u l t i p l i e d by t h e Stark  same J  shift  the  vary 4.2  over  electric  coefficient.  for  but  was done  Stark  different  The S t a r k  it  Stark  is for  shift  a Q  Stark on M.  With  between K (i.e. strength  each of  shift  d e p e n d e n c e on M of  frequency  the  J  of  pattern  the  The r e l a t i v e  field  cases  low  a variety  depend l i n e a r l y  sum up t h e  Stark  these  and  resolvable.  especially  (e1.44)  2  the  t h e measurements was  order  as M .  was  t h e J=15  In  easily  why t h i s  first  shifts  is  3.  on we see AM=0 t r a n s i t i o n s  these  coefficients  r e s o l u t i o n and  experiment  taken  From e q u a t i o n  split  the  lines  shows t h a t ,  the  level  Stark  the  to a p p r e c i a t e  branch t r a n s i t i o n .  the  of  parameter  Table  order  instructive  the  from t h i s  measurements were o f t e n  fields.  n o r m a l d i p o l e moment  been r e p r o d u c e d i n F i g u r e  character  field.  the  As an example of  to noise a v a i l a b l e  lines  Stark  used f o r  i n KHz i s  strength multiplied  the just by  J  102 Figure  13874  3  13873  13872  Frequency (MHz)  13705  13704  13703  Frequency (MHz)  F i g u r e 3. Two o f t h e K = ±1 '<- *2 A s D t r a n s i t i o n s . t o p r e p r o d u c t i o n i s J=15. I t i s t h e r e s u l t o f 64 sample s c a n s a t a S t a r k f i e l d o f 74 v o l t s / c m . The b o t t o m r e p r o d u c t i o n i s J=17 and i s t h e r e s u l t o f 48 s c a n s a t a S t a r k f i e l d o f 200 v o l t s / c m . 3  The  1 03 Table Arsine J  Stark  line  parameters  4.2  and e x p e r i m e n t a l  conditions  Shift  Frequency  Stark  # of  # of  Total  Coef f i c i e n t  Component  Field  Indep.  2.  Min  Time  V/cm  Runs  Sweeps  (hr)  30  1  1 38  4.6  1 50  1  1 6  0.5  200  1  16  0.5  60  1  64  2.1  500  1  1 44  4.8  60  1  40  1 .3  400  1  80  2.7  300  2  64,128  6.4  400  2  32,128  5.3  62  3  64,88,112  8.8  200  1  400  3  600  1  32  1 .0  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  74  1  64  2. 1  200  3  80  1  200  3  KHz/V/cm 8  1 1  4.61  f1  2.52  f1  f2  1 2  2.13  f •  1 4  1 .58  ,,f  2  -  f1  f2  15  1 6  1 .38  1 .22  f 1,f  f 1  r  2  f2  96 16,64,112  32,64,64 64 48,64,112  3.2 6.4  5.3 2.1 7^5  104  J  Stark S h i f t Coefficient KHz/V/cm  T a b l e 4.2 c o n t i n u e d Frequency Stark # of Component Field Indep. V/cm Runs  11 2  300 1 7  18  19  20  1 .09  f  ,,f:  0.97  0.87  f  0.79  f  ,,f:  1,1;  22  23  24  0.72  0.66  f  0.60  ,,f:  ,,f;  =f 1 r f :  0.55  Spurious at  f  f 1 , f  line  13954.50(20) Sample  pressures  Total Time (hr) 3.7  60  64  2.1  200  80  • 2.7  200  80  2.7  240  109  3.6  400  32,42  2.5  100  96  3.2  240  64  2.1  280  64  4.3  100  96  3.2  300  96  3.2  310 21  # of 2. M i n Sweeps  32,40,88  5.3  32  80  2.7  68  96  3.2  200  16,23  1 .3  400  1 6  0.5  300  80  2.7  320  32  0.5  300  64  2.1  310  64  2.1  320  80  2.7  62  2  400  1  were a l l  25 t o  64,88 80 65 mTorr  5.1 2.7  1 05 Now r e c a l l is  to  invert  that  the  Stark pattern  absorption p r o f i l e was is  important farther  that  the  detract  the  of  the  from the not  the  Stark  intensity  the  from t h e  absorption profi.le  same  of  the  one member of of  for  essentially  volts/cm. for  At  this  two S t a r k  60 v o l t s / c m t h e  4.1  Stark  the  the d o u b l e t field  is  exactly  course,  is  the d o u b l e t  field  farther  negative higher  nature  interfere  higher  shifted  to  study  it  that  ensure  l o b e would  not  same way  it  lobes  one member  of  separation  other  as  would  was  the  subtract  (and v i c e  result  doublet  versa,  our  the  is  at  is the  zero f i e l d Stark  the  J=11  Stark  ±1.7 MHz.  2.8 MHz. The splitting  in order  A l s o the  result  by  the d o u b l e t s .  This,  to  confirm  Stark  lobes  absorption l i n e s .  l o b e s have  separation. pattern  best  60 and 450  shifted  transitions.  Stark  c a n now  M=±J i n v e r t e d  i n between  the d o u b l e t  of  were u s e d ,  strongest  important  than  fields  what we w o u l d want  w i t h the  the  intensity  field  of  Stark  to accentuate  of  the  Stark  separation  away b a s e l i n e  not  the  field  enough away,  an e x a m p l e . F o r  fields  "subtracting"  will  In  the  the d i f f e r e n t  e a c h member of  From T a b l e of  Stark  zero  arsine  width,  a doublet  be d e m o n s t r a t e d w i t h t h e h e l p of  lobe  far  the  detection  time).  The r e a s o n  line  to the  line  frequency  lobes  In  lobes  intense  the d o u b l e t  sensitive  and add i t  absorption half  shift  of  phase  transition.  negative  the  of  absorption l i n e .  to  doublet  the  shift  "negative"  important of  to  than  one r e s u l t  is  can be u s e d as a c h e c k  all  Again  been the  removed and of  the  At  low  the field  1 06 result.  The problem w i t h h i g h e r  become more p r e v a l e n t be t r i e d  so o f t e n  i s background  different  higher  i n hope o f m i n i m i z i n g b a c k g r o u n d  At  higher  to  t r y t o move t h e S t a r k  doublet  fields  J  w i t h o u r 800 KHz l i n e  separation  widths  effects  fields  would  (see Chapter 2 ) . it  i s not p r a c t i c a l  lobes  within  t h e much  a n d so o n l y  higher  field  smaller  measurements  were made. In  order  theoretical evaluated. these  to determine  unsplit  splittings.  within  frequencies For J  v^,  "center",  These a r e l i s t e d  center  predicted  the r o t a t i o n a l  frequencies  in Table  than  were  4 . 3 . F o r low J ' s  were c a l c u l a t e d  greater  parameters the  from t h e p r e d i c t e d  14 t h e asymmetry  splitting  about  the center  experimental  error  so where a p p r o p r i a t e ,  averages  of the s p l i t  frequencies  (J<15)  in the  frequency  determined  is  well  simple  the center  frequency. Also allowed  in Table  transitions  The n o r m a l equation were a l s o , (J,K) N  transitions  hyperfine  by H e l m i n g e r  et a l  (J,K)->(J+1,K)  energies  split  were  of equations  AsD  3  (r9l). f i t to the  (e1.25,e1.30)  included) = 2(J 1)[B -D +  0  = a  R  J K  K  2  +  4(J+1) [D -H 3  and t h e d i s t o r t i o n  D  reported  (the hyperfine  -  v (J,K)  4 . 3 a r e t h e low J  J R  K J  K"] + Uj[(J+1) -J ]  K +H ] 2  Q branch  + b J(J+l) R  J  H  3  J  J,±1<—J,^2)  + c J (J+D K  2  2  [4.6]  3  were  f i t to  + e J (J+D R  3  3  [4.7]  1 07 where aK b  K  cK  -3H  eK  -3L  and L J (J+1) K 3  3  JK JJJK  JJJK  is  the c o e f f i c i e n t  of the o c t i c  a n d was i n c l u d e d p r i n c i p a l l y  2  as a  term fitting  parameter. The f i t s  were made w i t h t h e d a t a w e i g h t e d  a t Wo  2  m  3  a c c o r d i n g t o t h e i t e r a t i v e method d e s c r i b e d i n ( r l 2 ) . The errors  f o r the allowed  q u o t e d by H e l m i n g e r measurement they  assume e r r o r s transitions  higher the  errors  transitions  the center  the s t a t i s t i c a l  measurements  were d e t e r m i n e d  in the s p l i t frequencies  was v e r y  well  small  transitions  but s i n c e  frequencies of  from t h e e r r o r s , used t o  plus a subjective  a s a few l i n e s  estimate  t o see i n  This latter  estimate  reproduced the center  our s t a t i s t i c a l  i t d i d n o t seem r e a s o n a b l e  so much more t h a n  it is  measurements a t  frequencies  o f s u c h weak t r a n s i t i o n s .  quite  reported because  i n the center  s p r e a d one m i g h t e x p e c t  was deemed n e c e s s a r y  these  measurements we  t o make f r e q u e n c y  The e r r o r s  discussed,  frequencies  other  a t 230 a n d 345 GHz r e s p e c t i v e l y  distortion  size  for their  a t 115 GHz b u t s i n c e  o f 0.10 a n d 0.15 MHz f o r t h e i r  frequencies.  "determine of  e t a l ( r 9 l ) o f 0.05 MHz f o r t h e i r  more d i f f i c u l t  previously  are set at the value  o f t h e J=0—>1 t r a n s i t i o n  do n o t g i v e  generally  transitions  transitions  sampling  t o weight with  108 comparable  signal  to  noise,  but  poorer  reproducibility  of  frequency. The f i r s t transitions Helminger  c o n c e r n was  to  and compare our  et  al  (r9l).  re fit  just  _  results  The r e s u l t s  the  allowed  w i t h t h o s e p u b l i s h e d by of  this  are  reproduced  here. CALCULATED CONSTANTS FOR A s D  BASED ON  3  ALLOWED TRANSITIONS ONLY B  57477.600  0  Dj D  C T h e s e numbers a r e al  (r9l)  except  the  our C  -0.928(1)  J R  -164.74(3) 0.051(3)  XT  0.079(15)  R  same as is  R  those  number above  value  significantly  forbidden  change  transitions  in  the  is  expansion i n J(J+1)  results they fit  of  this  fit  were t o t a l l y discussed  on t h e  would seem p r u d e n t than  two  as we s h a l l  when we i n c l u d e  0.069  find  its  the  were n e x t  fit  to  the  d e s c r i b e d by e q u a t i o n  not  i n c l u d e d here  consistent  of  w i t h the  equations  f o r b i d d e n and a l l o w e d  information  T  report  except  results  of  power  e4.7. to  The  say  the  that  global  next.  Consideration the  are  D „  they  et  fit.  The f o r b i d d e n t r a n s i t i o n s series  r e p o r t e d by H e l m i n g e r  0 . 0 7 9 whereas  MHz. An i m p o r t a n t will  (3)  0.7413(2)  eqQ C  (MHz)  separate  e 4 . 6 and e 4 . 7  transitions  parameters  D „, T  both  T  t o do a s i m u l t a n e o u s g l o b a l fits  that  contain  H , and H „ T t  reveal  and so fit  it  rather  and t h e n compare t h e p a r a m e t e r s .  The  109 results  of  this  global  Comparison w i t h the important  fit  are  "allowed"  differences.  First  given  in Table  results  sextic each  work  here  constants  fits  is  that  coupling constants result  C  in  transitions  forbidden  this  was  the  v a l u e of  be c o u p l e d t o  (and  less  as  J K  and  H  K J  r o u g h l y as  rotational  and C  R  to  the v a l u e s  d i s c u s s i o n at Also It (see  hyperfine  order D ,  in  T t  in  the the  )-  It  hyperfine  constants, in Table  end o f global  fit  the  4.6  this  determination  change n o t e d e a r l i e r  the  constants,  phosphine s e c t i o n ) the  the  hyperfine This  allowed  it  between  3  constants  values  of  of the  C  (section  was n e c e s s a r y  and A s H  3  Dj » K  AsD H 2  set  C  N  the 4.8).  PH  constrain 3  (rl02),  The c o n s t a n t  T h i s cannot  "allowed"  T  and C „ were  VT  to  for  (r93).  D „  the  in accord with  R  the  varied than  into  H j  the  transitions  in A s H ,  effect  of  of  through  so r a t h e r  chapter  of  One c a n  was n o t e d  was e s t i m a t e d u s i n g r a t i o s  alter  and  splitting  frequencies".  z e r o and mix t h e m a g n e t i c  quadrupole hyperfine f i x e d at  the  forbidden transitions  the magnetic  the  "allowed"  the magnetic  by r e c a l l i n g  and A s H D t h a t 2  latter  "center  the  through H  the  e x p e c t e d t o be i n d e p e n d e n t  result  will  will  different  and C „ were not d e t e r m i n a b l e .  kT  transition  rationalize  3  two  ways t h e h i g h e r  "correct"  between  the  came as a s u r p r i s e  allowed  PD  the  model.  "global"  R  different  tend to  The s e c o n d d i f f e r e n c e  H j.  of  two  significantly.  TU  and t h e  will  shows  D . has c h a n g e d  T h i s can p e r h a p s be seen as a r e s u l t models at  above  4.6.  explain  and " g l o b a l "  H  K J  the D  J K  1 10 values only  b e c a u s e when H „ _ was  changed w e l l w i t h i n  mean t h a t as H  J K  D „ T  the has  H  R J  our  estimate  t h e more  from t h e  of  H  important  normal  "allowed"  masked t h e was a t  PD  J=13  that  which  is  T O  could  reasonable  is  not  in  the d e t e r m i n a t i o n  c o u l d not  at  found.  reliable,  MHz (see as  intensity  as  our d i s t o r t i o n  candidate  for  one member of splitting  Table it  be a c c o u n t e d These  or  that of  had t h e  lines.  It  This same  unknown  line  line  was  relative  would have been a good  a hyperfine  was not  by  completely  The o t h e r  4.2).  for  i n c l u d e d an  14026 MHz t h a t  forbidden t r a n s i t i o n .  pathological  data.  This  D  spectrum.  transition  13954.50(20)  resulting  3  effect  particularly  the  was p o o r ,  R J  d i s t o r t i o n model were a l s o  unknown  t h e above e s t i m a t e  a standard d e v i a t i o n .  value obtained for  Two s p u r i o u s l i n e s the  f i x e d at  split  consistent  pair  w i t h the  except other  111 Table Analysis  of  4.3  t h e measured A s D  3  Spectrum  Transition  Frequency(Unc) Normal R o t a t i o n a l T r a n s i t i o n s  J->J+1, J=0  J=1  J=1  J=2  J=2  (MHz) 4-  Obs-Calc  ref(r9l)  K->K  f,->f  K=0  1.5->2.5  1 14 9 6 0 . 5 5 ( 0 . 0 5 )  -0.01  1.5->1.5  11 4 919.24(0.05)  0.01  1.5->0.5  1 1 49 9 3 . 3 0 ( 0 . 0 5 )  0.01  2.5->3.5  229 8 9 0 . 3 1 ( 0 . 1 0 )  0.03  2.5->2.5  229 8 4 8 . 9 2 ( 0 . 1 0 )  0.03  1.5->1.5  229 9 1 9 . 5 3 ( 0 . 1 0 )  0.02  0.5->1.5  229 8 4 5 . 4 7 ( 0 . 10)  0.02  0.5->0.5  229 8 8 6 . 5 6 ( 0 . 1 0 )  0.01  2.5->3.5  229 9 0 0 . 4 5 ( 0 . 1 0 )  -0.05  2.5->2.5  229 8 7 9 . 6 8 ( 0 . 1 0 )  -0.04  1.5->2.5  229 8 5 9 . 2 6 ( 0 . 1 0 )  0.00  1 .5->1.5  229 8 7 3 . 7 7 ( 0 . 1 0 )  -0.06  0.5->1.5  229 9 1 1 . 0 1 ( 0 . 1 0 )  0.03  0.5->0.5  229 931 . 4 7 ( 0 . 10)  -0.02  3.5->4.5  344 7 8 7 . 6 0 ( 0 . 1 5 )  0.09  2.5->2.5  344 8 0 6 . 5 2 ( 0 . 1 5 )  0.04  1.5->2.5  344 7 7 7 . 2 8 ( 0 . 1 5 )  0.08  1 .5->1.5  344 8 1 8 . 3 2 ( 0 . 1 5 )  0.07  3.5->4.5  344 7 9 5 . 5 3 ( 0 . 1 5 )  -0.05  3.5—>3.5  344 7 6 4 . 5 2 ( 0 . 1 5 )  0.08  2.5->2.5  344 7 9 9 . 5 1 ( 0 . 1 5 )  0.06  1 .5->1.5  344 8 1 5 . 7 0 ( 0 . 1 5 )  0.07  K=0  K=1  K=0  K=1  2  11 2 Table  4.3  continued  Transition J=2  Frequency(Unc)  K=2  3.5->4.5  344  819.66(0.15)  -0.13  2.5->3.5  344  778.44(0.15)  -0.10  1.5->2.5  344  807.80(0.15)  -0.11  0.5—>1.5  344  848.96(0.15)  -0.08  Forbidden Rotational J<-J,  K= H<-±2  this  :  Obs-Calc  Transitions  work  8  14318.73 ( 0 . 2 1 )  -0.13  1 1  14156.67 (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  17  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  5  5  t  Numbers  uncertainties  in parentheses in  units  of  the  are last  estimated  measurement  significant  figures.  1 13 5.  PREDICTION AND I N I T I A L SEARCH FOR THE ASYMMETRIC ARSINE  For  was  for  large  SPECTRA  the asymmetric  predictions  of  AsD . 3  as  their  T h i s was  s t a n d a r d microwave the  top arsines  spectra  t o make t h e  scanning over  was  because  axis  a l o n g the  C^y a x i s .  The x a x i s  isotopes. isotope plane.  For  (i.e. In  this  asymmetric the  of  the  The moment of  set  up w i t h t h e  the  principal  the  unique  hydrogen  a x e s of  tensor  of  the  above c a r t e s i a n  is  shifted  to  the  new c e n t e r  parallel  to  the  o l d . The bond l e n g t h s  were c h o s e n t o be t h e Rotations  (e1.2)  hydrogen  i n the  angle  constants  A , B  inertial of  the  system  which  keeping  the  angles  Chu and Oka  (r82).  about  the  y axis  inertia  elements  and  zero point  top data  was  were  tensor. equation  average  and C . A p r e l i m i n a r y  from known s y m m e t r i c  about  and  d i a g o n a l i z e d t h e moment of  we t h e n had e s t i m a t e s  derived  of  zx  asymmetric  of mass b u t  parameters  diagonal p r i n c i p a l  rotational field  z  through a determinable  t h e n made t h a t From t h e  r  axis  zx  the  an  in  new a x e s  the  one of  t o p was e v a l u a t e d origin  z axis  was c h o s e n s u c h t h a t  inertia  rapid  an  t o p s c o u l d a l w a y s be f o u n d by a r o t a t i o n  y axis.  it  to  relatively  as a r e f e r e n c e  tops  as  were so  would a l w a y s be s i t u a t e d  2  situation  crucial  region.  t h r o u g h t h e m i d d l e of  H in AsHD )  as  the  accessible  capable  s y s t e m was  the asymmetric  in  strengths  easily  top arsine  Cartesian  nearly  line  microwave  arbitrary  would c u t  the  spectrometers entire  accuracy  not  transitions  U s i n g a symmetric  plane  TOP  force  then  used  to  11 4  estimate to  the  the  r  empirical one p e r per  harmonic p a r t s  the a l p h a s  rotational  constants  gave  rotational  constants  usually  cent  cent  of  in  (el.57). the  significantly One of asymmetric  For  these  rotational  impede t h e  the  great  top s p e c t r a  "guesstimates"  constant  was  errors  of  the  features  the quadrupole  s u p e r - i m p o s e d on e a c h r o t a t i o n a l  frequency  sweeps c o u l d be made f o r  transitions  very  "fingerprint". constants  transition  needed t o d i a g o n a l i z e tensor  upon i s o t o p i c  top hyperfine the A s H  patterns  of  or  enough f o r of  discussed further 2  of  the in  the A s H  easy  discussed earlier  w i t h the  patterns  the  of A s D , 3  3  same  the  inertia  the  with  hyperfine This in  AsH  Anticipation exact  than  3  asymmetric  the  a p p r o a c h . As of  AsD  angle  one s t a r t e d  4.8.  rather  calculations  off-axis  (or  (r93)  assignment.  2  their  hyperfine  the p r e d i c t e d  section  perturbation  and  hyperfine  coupling constants  the q u a d r u p o l e e f f e c t  second o r d e r  lines,  t o p moment of  coupling constants,  Broad  be a s s i g n e d by  p r o b l e m i n A s H D and A s D H l e d t o  treatment first  strong  t h r o u g h the  symmetric  arsine  transition.  d e p e n d e d on whether  the v a r i a b i l i t y  is  3  this  3  the  s u b s t i t u t i o n . A l t h o u g h the  were d i s t i n c t i v e  p r o b l e m of and A s D  the  estimates  or AsD  3  tensor  not  spectra.  of  top quadrupole  were e s t i m a t e d .by r o t a t i n g  than  up t o a  did  dependent q u a d r u p o l e  The a s y m m e t r i c  coupling constant  (r9l))  these  found c o u l d immediately  distinctive  the  hyperfine  structure  any  of  predictions  determination  of  t o much b e t t e r  studies  simplifying  w h i c h when added  usual  was quadrupole  d e u t e r i u m s do not  contribute  3  11 5 to  the  splittings  hyperfine several  contribution  After  of  magnetic  these  subsequent The  initial  In  as  constants  two  precision.  of  This  gives  the  our c a s e  it  effect  of  was values  and p u t  fit  into  sextic  Q branches  three  in  so f i t s  to v a r y  were  rotational  and  were  fixing  "variable"  fitting  parameters  i n an  interim  fits  with greater  and  with  for  and h o p e f u l l y  new p r e d i c t i o n s . level  important  of to  found,  The method i s  obtain field  analysis. at  into  s t o p p e d when  the  spectra.  In  constants  T h i s meant  least  to a v o i d h i g h e r With t h i s  called  transitions  good q u a r t i c  to determine  the q u a r t i c s .  greater  and t h e n put  d e s c r i p t i o n of  so as  allowed  i t e r a t i v e method o f t e n  where p r e d i c t i o n s a r e made f o r  constants  b e i n g mixed i n t o  only  The two  just  was made  U.B.C.  to a l l  constants  new t r a n s i t i o n s  desired was  arsines  were t h e n a v a i l a b l e  c o u l d be u s e d i n a f o r c e  of  order  meaning.  needed enough t r a n s i t i o n s set  initially  were t h e n  resulted  searched  we r e a c h  that  on t h e  were m e a s u r e d r o u g h  were f o u n d and a s s i g n e d ,  "bootstrapping",  that  that  rotational  constants  prediction  fit  magnetic  this  100 GHz on t h e  the p r e d i c t e d v a l u e .  As l i n e s  a  was  asymmetric  we c o u l d not  no o b v i o u s p h y s i c a l  are  the  were e s t a b l i s h e d  the  no R b r a n c h e s  at  rotational  that  splittings  transitions  T h i s meant  made a l l o w i n g third  Finally  and so i n i t i a l l y  study of  general  constants  the  the  r e g i o n below  spectrometer.  the  a few  here.  predictions.  frequency  found.  to  hundred k i l o h e r t z  ignored.  the  we measure  a  we  partial  contributions  i n mind t h e  sextics  1 16 finally  reported  parameters. set  of  s h o u l d be c o n s i d e r e d o n l y as  As i t  sextics  t u r n e d out we c o u l d not d e t e r m i n e  so t h a t  the  sextics  even more n e b u l o u s p h y s i c a l a s s o c i a t e d with s e x t i c The f i n a l completed at was  there  Jet  branch t r a n s i t i o n s search  for  frequency  the  these  asymmetric  the  tensor it  these  to at  (see  evident  a symmetry  for  to  moment c a n e x i s t  the  the  this it  the  asymmetric principal rotational  inertial  of  this of  texts  should also  (see  of  principal  along t h i s  rotor,  study  there axis  transitions.  is  of  other  the R  the  high  all  (the  section).  zx  inertia  (r 1 1 , r 31) ) s h o u l d make inertial By  axes  the very that  principal  2  For  is nature  no  dipole  axis.  Precisely  the  For  " a " and " c "  AsD H, 2  an  type  oblate  no d i p o l e moment a l o n g t h e only  of  means we have no  and we see  and we see  plane  Consideration  t h e moment of  AsH D t h i s  1.4b).  that  that  symmetry  be a p p a r e n t  "b" axis  section  It  that  transitions  particular  rotor  was  Caltech.  spectrometer  symmetry p l a n e .  asymmetric  only  of  tops. Notice  elements  classic  d i p o l e moment a l o n g t h e transitions  top a r s i n e  various  types  asymmetric  one of  plane  prolate  the  b e g i n n i n g of  any of that  usually  measured.  t o p s have a p l a n e  the  full  have an  were m e a s u r e d . As a b y - p r o d u c t  arsine  relations  perpendicular  the  high frequency  a word a b o u t  for  is  Laboratory  R branch t r a n s i t i o n s  exist  of  the asymmetric  Q b r a n c h e s were a l s o  Finally  referred  meaning t h a n  Propulsion  using their  reported w i l l  a  constants.  phase of  the  fitting  " b " and " c "  same s i t u a t i o n  "a"  type exists  1 1 7  for  PH D a n d P D H w h i c h w i l l 2  be s t u d i e d l a t e r  2  OBSERVED A S H 3 D AND A S D H  6.  One h u n d r e d and f o r t y - e i g h t twenty-nine  (Table  2  seventy-three  hyperfine  hyperfine  rotational  1 6 were m e a s u r e d f o r A s H D  lines  4.4).  within  For  up t o  11, were r e c o r d e d  frequencies  of  these  transitions  the  The a s s i g n m e n t s a r e arsenic  further  hyperfine  rotational  1 1  -2  the  10 h y p e r f i n e  4.5.  The S t a r k  right  of  Recall field  that  (Table  4.5).  frequency which  are  in Figure  transition  The  to  302  fits  and  discussed  of  above  Stark  lobes  scan showing the AsD H c l e a r l y 2  the  base  to  4 it  the  s h o u l d not  one component  a b s o r p t i o n s of  report  line  another  of  type  in  and j u s t  into  Table to  the  baseline. electric  be h a r d  interfering  component.  c  resolved  "move" d e p e n d i n g on t h e  so from F i g u r e l o b e s of  splitting  4 we f i n d a  components we were a b l e  lobes are  the  Stark  field  hyperfine  z e r o f i e l d a b s o r p t i o n s below  strength  imagine zero  the  distinctive  a broad frequency  rotational  2 1  the  transitions,  r e p r o d u c t i o n of 2  rotational  below.  As an example of the  b o t h of  up t o  2  r a n g e d f r o m 9.3  s u p p o r t e d by t h e  structure,  with J  AsD H  seventeen  with J  GHz.  lines  transitions  transitions  all  5.  SPECTRUM  2  representing  in Chapter  with  to the  118 Figure  I  I  L_  4  I  16790  I  I  16810  I  _l  16830  Frequency (MHz) Figure  4. The 2 , , — 2 i 2  rotational  T h i s was o b t a i n e d a s a s i n g l e w i t h a time c o n s t a n t of  of  0.1  s c a n of  sec.  transition  of  600 s e c o n d s  and a  AsD H. 2  duration  100 KHz S t a r k  field  2800 v o l t s / c m .  The  problem then  the  Stark  outside  l o b e of  the  time a v o i d  is  to choose a f i e l d  large  t h e component y o u a r e  trying  zero f i e l d absorption linewidth s h i f t i n g any  components i n .  It  Stark  t u r n e d out  lobes that  enough t o to  shift  measure  and a t  the  same  from n e i g h b o r i n g  for  the a r s i n e  asymmetric  11 9 tops  the  line's the  main d i f f i c u l t y  Stark  line.  was  splitting  shifting  was  not as  interfering  important  was g e n e r a l l y  larger  than  This  small arsine the  shift  "quasi"  "slow"  first  section  effect  order  energy  1.4c.  Stark  is  levels  lobes  the  from  other  hyperfine  Stark  shifts  we  field  a c o n s e q u e n c e of  second order  Note:  shift  "modulating"  experimental  d i p o l e moment and t h e  (see  degenerate  Stark  the  absorption  or,  Stark  because  c o u l d produce w i t h the a v a i l a b l e strength.  the  l o b e s more t h a n a l i n e w i d t h ,  The p r o b l e m of  transitions  in  nature  the of  a l t h o u g h we c a n have a  in  the  we d i d not  case  of  near  experience  this  in.this  study.) The z e r o f i e l d at  four  (the  field  latter  modulation revealing fields the  line  strengths,  fields  2400,  were u s u a l l y  transitions  the  were v e r y  nearly  asymmetric  of  available).  At  were not  volts/cm the  low  symmetric  Stark  lobe.  At  nearly  fully  m o d u l a t e d and  symmetric.  even a t  measured  3000 and 4000  the a b s o r p t i o n p r o f i l e s  interference  profiles  were s t i l l  1000.,  b e i n g t h e maximum f i e l d  the  the  absorption lines  In  cases  4000 v o l t s / c m  higher  where t h e  lines  extrapolations  were made b a s e d on how t h e  frequency  measurements  varied  with Stark  cases  experimental  error  field.  In  these  increased accordingly. on t h e J P L Burst  millimeter  generally  The e r r o r s  wave s p e c t r o m e t e r  lower  than  in  was  The h i g h f r e q u e n c y measurements made  m o d u l a t i o n so t h e  avoided.  the  p r o b l e m s of these  their  were done u s i n g Tone  a slow  Stark  effect  h i g h f r e q u e n c y measurements  lower  frequency  counterparts  were are as  1 20 we no l o n g e r  need t o t a k e  insufficient  Stark  measurement.  The g e n e r a l  measure  line  each  experimental in  these  field  into  account  i n our  10 t o  uncertainties  are  effects  zero f i e l d  experimental  roughly  the  frequency  p r o c e d u r e was  20 t i m e s . an  (equations  The q u o t e d  indication  of  were f i t  e1.13,e1.14)  t o an A r e d u c t i o n  u s i n g an  iterative  spread  method ( r 1 2 ) .  were o b t a i n e d by  matrix  in  elements  The m a g n e t i c separately  of  ref.  hyperfine  (see  (r32)  to  first  and A s H D  2  .  hamiltonian  fitting  2  order  fits  including  the  hamiltonian.  were  using  The  determined  equation  were done i n  the  111  r  and  .  representations imaginary  the  coupling constants  below)  The A s D H  r  of  the  hamiltonian  quadrupole coupling constants  II  to  measurements.  The s p e c t r a  (e1.29).  of  respectively  to avoid  quadrupole elements (r32).  The r e s u l t s  into  of  the  our  these  introduction  fitting  fits  are  in  Table  4.6. For branches of  each  $j,  powers are  DJ,$J,...)  cannot  (e1 . 13)  from o u r d a t a  constrained  to  value  will  D  T  submillimeter  of  J  sets.  zero in  include  wave s t u d y  of  fits.  the  3  constant,  J  this  of  (r93)  From  R b r a n c h e s we see we parameter,  2  the  <> i , which is  namely  parameter  T h i s means t h e  effect  of A s H  of  include R  Coefficients  R branches.  these  power  Because the  sets  (rotational  two of  "third"  data  J values.  from t h e s e  with just the  the  initial  terms  2  determined  determine  of  derivative  f r o m two d i f f e r e n t  diagonal  equation  isotopic  $j  was  empirical from  expected to  the be on  121 the  order  of  0.5  Looking at sextic  KHz or a b o u t Table  . 0 5 % of  fitting  tops.  These s e x t i c s  parameters  attributed  to  include  the  in  constants  them. The c h o i c e  that  the  are  of  only  significance which  sextic  was  the  least  two as  s h o u l d be constants  was e s t a b l i s h e d by how w e l l  constants  of  i n the  i n c l u d e d here  c o u l d be d e t e r m i n e d and what c h o i c e  c o m b i n a t i o n s of  choice  reported d i f f e r s  and no p h y s i c a l  fits  T  4 . 6 one n o t i c e s  (<£>'s and c/>'s) c o n s t a n t s  asymmetric  D .  to  the  of  internally  correlated. In  the  initial  and t h e m a g n e t i c order  obtained  for  p r o b a b l e but  it  had t o  "exact"  significantly  of the  order  whether  It  t u r n e d out  they  than  those  title  4.8).  rotation are  in  obtained  was  the  Hyperfine  hyperfine  fit,  after  not  and  the  are  exact  not  first  first  order  to  in solution  treatment  The  the  these  problem i s  effect.  which  not  order  Effects  this  values  was  from a  extraordinary  o b t a i n e d from t h e  rotation  It  or  this  first  the  r e p o r t e d here  A c o n s e q u e n c e of  magnetic  those  quadrupole-spin  "Nuclear  to  c o u l d not  A more c o m p l e t e d i s c u s s i o n of the  coupling  treatment  anomalies appeared  c o u l d be e x p l a i n e d by h i g h e r  the q u a d r u p o l e e f f e c t  constants  first  coupling constants  (section  spin  This  be a s c e r t a i n e d  different  be f o u n d under Arsine"  (e1.26).  the quadrupole c o u p l i n g c o n s t a n t s .  quadrupole e f f e c t s .  fit.  the q u a d r u p o l e  c o u p l i n g were t r e a t e d  but c e r t a i n  anomalous r e s u l t s  diagonal  reduction  hyperfine  using equation  seemed a d e q u a t e  order  data  of  magnetic  spin  rotation  1 22 contribution  to  experimental  frequencies  then  fit  Since  the  change this  to  the  the  splitting  "exact"  principal  method. Problems to  eliminated  any  constants  the  correlations  the  no r e a l  may a r i s e  other  between  difficulty  fit  the  significant treating  the  parameter be X  change spin  most  because  spin  small  like  to  s m a l l changes  Still,  it  is  not  global  fit,  constants.  to  in  of  either  spin X  standard  did  spin  first  rotation the  order  and  the  fit  were  between  rotational  the  rotation  this  of  The  method  parameters,  hyperfine  will is  splitting.  a new minimum f o u n d i n a rotation, or  the  spin  would p r o d u c e rotation  of  no  as a r e s u l t  "separately". of  the  parameters  so one would e x p e c t  failures  from  deviation  constants  error  not  as we have  coefficients  spin  expected that  i n c l u d i n g the  values  the  coupling  the  sensitive  "better"  in  were  hamiltonian.  expected  However,  i n any p a r a m e t e r  susceptible  is  the  rotation  and t h e  rotation  it,  the  frequencies  parameters  between  obtained  were a l s o  the  parameters.  quadrupole constants  final  in  other  s m a l l and t h e c o r r e l a t i o n  exact  "fixed"  quadrupole A reduction  correlations  quadrupole constants very  and t h e s e  various  and t h e  subtracted off  quadrupole coupling constants  significantly  attributed  was  1 23 Table Observed Rotational Transition F(upper)-F(lower)  1  o 0oo 1.5 -2.5 — 0.5 --  1  1 1 1- 1 o 1 .5 1 . 5 1 . 5 2. 5 2. 5 0. 5 0. 5  1  1 ,  1  1 10 0.5 • 5 5 5 5 5  21  2o 0. 0, 3, 3, 1 , 1 , 2, 2, 2  1 2•  0, 2,  2 1 . 0  2O2  2  1.5 1.5 1.5  3 3.5 • 5 5 5 .5 5 .5 ,5 .5 1.51 3  0 0 2 2 1 1 .5 .5 5 .5 .5 5 ,5 .5 .5 3, 3, 2 3. 2 4 2 4 1 1  4.4  spectrum  of  AsH D  Frequency  2  (MHz)^  QBS-CALC  185689.428(0.015) 185715.534(0.015) 185736.006(0.015)  -0.008 -0.023 0.030  35461 .030(0, 35456 .480(0, 35451 .290(0 35436 .090(0 35430 .930(0 35423 .920(0 3541 4 .140(0  030) 030) 030) 030) 030) 030) 030)  •0.056 0.009 -0.026 •0.061 -0.048 -0.028 -0.039  257141 . 132(0.060) 257148.811(0.150) 257167.070(0.010) 257173.819(0.015) 257194.733(0.020) 257199.919(0.015)  0.033 0.577 0.003 0.016 0.041 -0 005  32943. 570(0, 030) 32950. 710(0, 030) 32957. 520(0, 030) 32964. 270(0, 030) 32969. 640(0, 030) 32976. 780(0, 030) 32983. 170(0, 030) 32988. 340(0, 030) 32989. 910(0, 030)  •0.005 •0.000 0.004 0.018 0.024 0.029 0.000 0.020 0.005  1 5502.860(0 .030) 15511 .350(0 .030) 15516 .280(0 .300) 1 5521.930(0 .060) 1 5524.510(0 .030) 1 5530.270(0 .060) 15543 .010(0 .030) 15549 .260(0 .030) 1 5552.470(0 .040) 1 5570.950(0 .030)  0.024 0.041 0.234 0.083 0.009 0.004 0.003 •0.017 0.011 •0.003  T a b l e 4.4 c o n t i n u e d O b s e r v e d s p e c t r u m of A s H D 2  Rotational Transition F(upper)-F(lower) 3,3-  4  2 3  -  30 3 1.5— 4.5 — 1.5 -4.5 -2.5 -2.5 — 3.5 — 2.5 — 3.5 -3.5 — 4  4«o"  4,«"  (MHz)  OBS-CALC  2.5 3.5 1.5 4.5 3.5 2.5 3.5 1.5 2.5 4.5  29499.490(0.030) 29508.600(0.030) 29509.100(0.030) 29517.700(0.030) 29522.800(0.030) 29527.450(0.030) 29536.030(0.030) 29537.050(0.100) 29540.660(0.030) 29545.120(0.030)  -0.006 0.019 0.017 0.022 -0.001 0.013 0.019 0.027 0.014 0.012  —  5.5  97368.932(0.100)  -0.163  ---—  4.5 3.5 5.5 2.5  25826.540(0.030) 25842.760(0.050) 25872.820(0.050) 25889.100(0.030)  0.019 -0.006 -0.012 0.014  --— —  4.5 3.5 5.5 2.5  256951.355(0.015) 256953.001(0.015) 256956.197(0.015) 256957.736(0.240)  0.018 0.035 -0.011 0.007  25378.520(0.030) 25384.980(0.030) 25391.320(0.030) 25397.110(0.030) 25403.070(0.050) 25407.810(0.030) 25413.590(0.030) 25418.320(0.030) 25420.600(0.030) 25425.770(0.030)  -0.030 -0.006 -0.007 -0.021 -0.038 -0.005 -0.017 0.006 0.008 0.018  256922.816(0.015) 256924.363(0.015) 256927.523(0.015) 256928.904(0.015)  0.016 -0.009 0.015 -0.065  1 3  5.5 4,3"  Frequency  4 4.5 3.5 5.5 2.5 4 4.5 3.5 5.5 2.5  1 f l  3 2  4 u 0  2.5 5.5 2.5 5.5 3.5 3.5 4.5 4.5 3.5 4.5  ------———--.——---  4.5 3.5 5.5 2.5  — — — --  3 .5 4.5 2.5 5.5 4.5 3.5 4.5 3.5 2.5 5.5 4.5 3.5 5.5 2.5  125 T a b l e 4.4 c o n t i n u e d O b s e r v e d s p e c t r u m of A s H D 2  Rotational Transition F(upper)-F(lower) 3.5-4.5-5.5—  Frequency  (MHz)  OBS-CALC  3.5 4.5 5.5  91154.277(0.030) 91174.250(0.030) 91179.960(0.030)  0.153 0.011 0.028  -----  5.5 4.5 6.5 3.5  • 38662.235(0.030) 38675.160(0.030) 38708.250(0.030) 38721.290(0.030)  0.007 -0.062 -0.018 0.024  51 5 5.5 -4 . 5 -6 . 5 -3.5 --  5.5 4.5 6.5 3.5  134376.902(0.015) 134387.266(0.015) 134413.696(0.015) 134424.004(0.015)  -0.012 0.001 0.014 -0.002  --------— --  5.5 6.5 4.5 7.5 6.5 5.5 6.5 5.5 4.5 7.5  16597.230(0.030) 16600.840(0.030) 16617.790(0.030) 16620.460(0.030) 16623.750(0.050) 16629.010(0.030) 16631.690(0.030) 16636.880(0.030) 16649.510(0.050) 16651.240(0.030)  -0.086 -0.057 0.009 0.010 -0.054 0.003 0.024 0.010 0.039 0.021  633-  6 5 6 . 5 -5.5 -7.5 -4.5 —  6.5 5.5 7.5 4.5  185726.865(0.015) 185728.479(0.015) 185733.771(0.015) 185735.304(0.015)  -0.004 0.011 -0.001 -0.015  726  7, 5.5 8.5 6.5 7.5  -----  5.5 8.5 6.5 7.5  75827.748(0.040) 75831.756(0.020) 75846.940(0.030) 75850.890(0.020)  -0.003 0.042 -0.050 -0.033  707 5.5 -8.5 -5.5 -8.5 -6 . 5 --  6.5 7.5 5.5 8.5 6.5  12641.840(0.300) 12644.680(0.030) 12666.330(0.030) 12668.070(0.030) 12674.860(0.030)  -0.095 -0.060 0.017 0.006 0.007  5  1  _ 5  1 5  5.5 4.5 6.5 3.5 52 3  _  6 1 6  71 7_  -  -  6 6  4.5 7.5 4.5 7.5 5.5 5.5 6.5 6.5 5.5 6.5  0  2  6  T a b l e 4.4 c o n t i n u e d O b s e r v e d s p e c t r u m of A s H D 2  Rotational Transition F(upper)-F(lower) 7.5 7.5 6.5 7.5  Frequency  (MHz)  OBS-CALC  -— -—  7.5 6.5 5.5 8.5  12676.600(0.030) 12682.020(0.300) 12698.610(0.500) 12699.600(0.500)  0.004 0.025 -0.620 -0.320  -----  7.5 6.5 8.5 5.5  187808.628(0.015) 187810.374(0.015) 187817.121(0.015) 187818.819(0.015)  0.006 0.041 -0.036 -0.005  6.5 9.5 7.5 8.5  --— --  6.5 9.5 7.5 8.5  9318.710(0.030) 9319.810(0.030) 9324.900(0.030) 9325.990(0.030)  0.017 -0.000 0.022 0.002  8  —8 8.5 7.5 9.5 6.5  ---—  8.5 7.5 9.5 6.5  24264.570(0.030) 24268.500(0.030) 24286.450(0.030) 24290.410(0.050)  0.028 0.020 -0.053 -0.041  8  —8 6.5 9.5 7.5 8.5  -— ---  6.5 9.5 7.5 8.5  67080.220(0.050) 67083.480(0.050) 67098.420(0.050) 67101.660(0.050)  -0.037 -0.082 -0.074 -0.111  — --—  9.5 8.5 10.5 7.5  35848.010(0.050) 35852.000(0.050) 35872.810(0.050) 35876.810(0.050)  0.045 0.053 -0.036 -0.031  - 9 7.5 -10.5 --  7.5 10.5  57954.860(0.060) 57957.600(0.060)  -0.060 -0.051  9.5 12.5 10.5 11.5  39923.350(0.030) 39925.140(0.030) 39936.480(0.030) 39938.280(0.030)  0.014 0.046 -0.004 0.057  1  3 1 ~~  12 7.5 6.5 8.5 5.5  6  g^ _ g  92  92 9.5 8.5 10.5 7.5  7 ~  9 2 8  8  8  9?5 12.5 10.5 11.5  1  -----  1  1 27 T a b l e 4.4 c o n t i n u e d Observed spectrum o f . A s H D 2  Rotational Transition F(upper)-F(lower) 12  2  , ,-12, 10.5 -13.5 -11.5 -12.5 —  10.5 13.5 11.5 12.5  ,2-13, 11.5 -14.5 -12.5 -13.5 —  1 1  Frequency  (MHz)  OBS-CALC  31733.020(0.050) 31734.370(0.050) 31744.000(0.050) 31745.300(0.050)  0.026 0.026 0.029 -0.005  11.5 14.5 12.5 13.5  24497.640(0.030) 24498.650(0.030) 24506.350(0.030) 24507.350(0.030)  0.020 0.027 -0.067 -0.057  1 5 , (i ~ 1 15«, 13.5 -- 13.5 16.5 -- 16.5 14.5 -- 14.5 15.5 -- 15.5  13454.850(0.040) 13455.280(0.040) 13459.850(0.040) 13460.290(0.040)  0.036 -0.034 0.020 -0.033  1 6 1 5 ~ 1 6 ,1 5 14.5 -- 14.5 17.5 — 17.5 15.5 -- 15.5 16.5 — 16.5  9617.920(0.040) 9618.230(0.040) 9621.560(0.050) 9621.810(0.040)  0.017 -0.011 0.062 -0.019  13  2  1 2  2  2  t Numbers i n p a r e n t h e s e s u n c e r t a i n t i e s i n u n i t s of t h e  a r e e s t i m a t e d measurement last significant figures.  1 28 Table  4.5  O b s e r v e d s p e c t r u m of  Rotational Transition F(upper)-F(lower)  AsD H 2  Frequency  (MHz)^OBS-CALC  1,,-0 oo 1 .5 -2.5 -0.5 --  1 .5 1 .5 1 .5  132628.022(0.015) 132641.702(0.015) 132652.180(0.030)  -0.000 0.014 -0.048  1, , 2.5 -3.5 --  1 .5 2.5  302880.654(0.015) 302887.548(0.015)  0.021 -0.029  2.5 1.5 3.5 2.5 2.5 1 .5 3.5 0.5 1 .5 0.5  16783.520(0.030) 16793.370(0.030) 16796.820(0.040) 16797.930(0.040) 16804.140(0.030) 16807.760(0.030) 16817.410(0.030) 16821.330(0.030) 16827.980(0.030) 16841.620(0.030)  -0.016 0.020 0.016 0.007 -0.007 0.022 -0.006 -0.007 -0.008 0.033  3.5 -2.5 -4.5 -1 .5 --  3.5 2.5 4.5 1 .5  94764.338(0.015) 94768.923(0.015) 94774.226(0.015) 94778.787(0.015)  -0.009 -0.004 -0.001 0.015  3 , 3.5 2.5 4.5 1.5  -----  3.5 2.5 4.5 1 .5  33393.890(0.030) 33409.490(0.030) 33427.290(0.030) 33442.850(0.030)  -0.024 0.004 0.004 -0.005  322 3.5 -2.5 -4 . 5 -1 .5 --  3.5 2.5 4.5 1 .5  81006.936(0.015) 81015.038(0.015) 81024.294(0.015) 81032.298(0.015)  -0.006 0.029 0.014 -0.025  ' 2 1  2 , " "2, , 2.5 2.5 2.5 1 .5 3.5 1 .5 3.5 1 .5 0.5 0.5 2  1  22  3 ," 3  — — • ---------  3,3  2  T a b l e 4.5 c o n t i n u e d O b s e r v e d s p e c t r u m of A s D H 2  Rotational Transition F(upper)-F(lower) 4  2  2  ""  4 2.5  1 3  5.5 2.5 5.5 3.5 4.5 4  3  2  '"  4 4.5 3.5 5.5 2.5  2 2  5q " 5 2 2  0"  6«  3""  5n 5.5 4.5 6.5 3.5 6 6.5  3 3  5.5 7.5 4.5 ?53-  i  ~l« 7.5 6.5 8.5 5.5  (MHz)  OBS-CALC  -— ----  3.5 4.5 2.5 5.5 3.5 4.5  83231.403(0.015) 83233.590(0.015) 83238.915(0.015) 83240.523(0.015) 83243.355(0.015) 83244.920(0.015)  -0.014 -0.007 0.013 0.017 -0.006 -0.017  -----  4.5 3.5 5.5 2.5  15761.170(0.030) 15766.620(0.030) 15776.740(0.030) 15782.200(0.030)  -0.014 -0.023 -0.010 -0.013  -----  5.5 4.5 6.5 3.5  31654.030(0.030) 31659.390(0.030) 31673.060(0.030) 31678.400(0.030)  0.005 -0.000 0.017 - -0.012  -— ---  5.5 4.5 6.5 3.5  80981.343(0.015) 80989.149(0.015) 81009.034(0.015) 81016.836(0.015)  -0.004 -0.002 0.002 0.005  --  6.5 5.5 7.5 4.5  12408.800(0.030) 12410.870(0.040) 12417.290(0.030) 12419.330(0.040)  0.006 0.063 -0.017 0.006  7.5 6.5 8.5 5.5  133555.028(0.030) 133555.976(0.030) 133559.702(0.015) 133560.586(0.015)  0.004 0.016 0.015 -0.014  3  5.5 4.5 6.5 3.5  55  --  Frequency  ---t  — -— --  1 30 T a b l e 4.5 c o n t i n u e d O b s e r v e d s p e c t r u m of A s D H 2  Rotational Transi tion F(upper)-F(lower) 8  6  2  6  3  8 8.5 7.5 9.5 6.5  5  3  Frequency  (MHz)  OBS-CALC  -----  8.5 7.5 9.5 6.5  77233.644(0.015) 77234.906(0.015) 77240.756(0.015) 77242.010(0.015)  0.021 0.009 -0.010 -0.023  9 5 fl 7.5 -10.5 -8.5 -9.5 —  7.5 10.5 8.5 9.5  94017.284(0.200) 94017.284(0.200) 94019.000(0.200) 94019.000(0.200)  0.116 -0.164 0.075 -0.206  -- 10.5 -9.5 — 11.5 -8.5  93282.489(0.150) 93282.489(0.150) 93284.246(0.150) 93284.246(0.150)  0.253 -0.028 0.062 -0.219  11.5 10.5 12.5 9.5  14348.590(0.040) 14349.180(0.040) 14353.340(0.040) 14353.970(0.040)  0.044 -0.000 -0.015 -0.024  1 11 0 1 1 1 29 11.5 -- 11.5 10.5 -- 10.5 12.5 -- 12.5 9.5 -9.5  187226.127(0.015) 187227.900(0.015) 187239.396(0.015) 187241.193(0.015)  -0.020 0.013 -0.026 0.039  9  10 7  1 17  10 10.5 9.5 11.5 8.5 3  _  5  -  6  1 1  11.5 10.5 12.5 9.5  4  65  -----  -  t Numbers i n p a r e n t h e s e s u n c e r t a i n t i e s i n u n i t s of t h e  a r e e s t i m a t e d measurement last significant figures.  131 Table Empirical  4.6  s p e c t r o s c o p i c parameters  AsH D  AsHD  2  of  various  arsines AsD  2  3  X  MHz  110581.66(1)  55511 . 8 9 ( 1 )  Y  MHz  75133.52(1)  77131.53(1)  57477.60(1)  Z  MHz  72548.70(1)  71525.64(1)  52642.08(12)  A  J  A  JK  A  K  6  J  6  K  *J *JK <> l  KJ *K *K *JK *K X  x  x  ZZ xx~ zx  C  AA  C  BB  c  cc  KHz  1235.5(1.5)  KHz  2229.9(8)  -1875.8(3)  KHz  -2630.2(3)  2173.2(3)  KHz  125.8(2)  KHz  1993.4(2)  1131.3(1.0)  657(37)  Hz  -443(31)  Hz  63(4)  MHz X MHz Y y  66 ( 1 )  D  K  H  J  H  JK  H  KJ  H  K  L  JJJK  KHz  -940.1(13)  KHz  833.4  t  Hz Hz  -197(4)  Hz  123.8*  Hz  254(7) 98(11)  -135(6)  -103.69(2)  -54.01(3)  -60.78(5)  -108.93(5)  MHz  131(26)  KHz  104.7(30)  69.9(45)  KHz  69.0(16)  69.3(35)  KHz  70.7(10)  56.8(45)  * F i x e d by a r a t i o  0.073(4)  H Z  -164.76(10)  -49(17)  fHarmonic Force F i e l d  the  JK  742.3(7)  -69(1)  Hz Hz  D  KHz  979.9(2)  -346(9)  Hz  J  -407.04(5)  Hz Hz  D  C  N  C  K  *  KHz  55.5  KHz  50.8*  value  argument,  see  text.  — Numbers i n p a r e n t h e s e s a r e  the  standard deviations  least  f igures.  squares  fits  in u n i t s  of  the  last  significant  of  1 32 7.  DISCUSSION OF NUCLEAR QUADRUPOLE COUPLING  In  the  coupling  arsine  tensor  symmetric  is  If  diagonal  axis  system.  we d e f i n e  axis  as p e r p e n d i c u l a r  i n c l u d e s a hydrogen the  other  For  equation the  unique hydrogen z axis  of  the  the  the m o l e c u l a r  the  z axis the  x  x  isotopes,  C  reflecting the  = X  to  the  isotope.  asymmetric  axis.  the  various Given  3  y  the  It  C  symmetry the  principal  axis,  3  planes angle  that  z  symmetry  that  this  plane case  includes  however,  top c o u p l i n g tensor the  be v e r y  "average"  the  does  symmetric  close  as  principal not  top  with the  electronic  (if  the  empirically  equation  determined l i n e a r  difference  structures  of  c o m b i n a t i o n s of  found in T a b l e  we can c o n s t r u c t  2  (111 ) r  21.46(9)  xx  1 31(26)  xz  the  4.6  following  82.24(4) x,y,z  in  t h e above  AsD H 2  table  and  table  (Ii ) r  -27.46(9) -49(17)  -103.69(2)  X zz'  The l a b e l s  it  species.  AsH D  X  above, the  MHz) .  X  y  and  the y a x i s ,  In  the  between  t o p s we d e f i n e  will  different  top  = ~ ^X -  the q u a d r u p o l e c o u p l i n g c o n s t a n t s Laplace's  as  ARSINE  quadrupole  t h e n by symmetry  c o i n c i d e w i t h what one m i g h t c a l l were one)  symmetric  i s o t o p e and b i s e c t s  asymmetric  as p e r p e n d i c u l a r  in  t o one of  two h y d r o g e n  Laplace's  tops  IN  -54.01(3) 81 . 4 7 ( 4 ) correspond to  the  (in  1 33 principal  inertial  implied tensor two  (all  the  other  elements  by two and a one by o n e . The yy  c o r r e s p o n d i n g to is  a x e s as d e s c r i b e d i n  the  quadrupole  tensor  looking at. various  Given  similar  above tensor lone  in the  of  the  very  isotopic  electronic  similar  cases.  A l s o the  the  is  is  MHz r e f .  just  (r9l)  of  a percent  s u g g e s t i n g the  isotopically  invariant  electronic  ammonia  Similar  the  stibines  y axis tan20 =  This  angle  rotate  the  isotopic  of  our  that  of  the  coupling the  by minus two g i v e s  by an a n g l e  are  is  all  substitution  3  X  xx  xz - X  an  in order  for  (rl06).  top c o u p l i n g tensor  we  by [4.8]  zz  roughly  t o p moment of  in  quite  2x  should approximate symmetric  and C H C 1  6' g i v e n  x  and  3  a s s u m p t i o n of  potential  (r9l)  this  s h o u l d be  d i s c r e p a n c i e s have been o b s e r v e d  (rl04,rl05),  about  are  principal  The d i f f e r e n c e s  To d i a g o n a l i z e our a s y m m e t r i c rotate  we  experience  AsD  the angle  inertia  to again  a  molecular  environments  for  the  3  is  before  - -^-x • D i v i d i n g  MHz(r93)  reasonable.  AsH .  defined  the  the  81.3219(266) order  for  zero)  above  species  the  top coupling constants  82.375(80)  at  coupling tensor  This  tops  above  we s h o u l d e x p e c t  y component of  symmetric  symmetric  values  the  set  y axis  component of  y component of in a l l  table.  all  that  species are  perpendicular very  for  are  The  component  "perpendicular"  always then a p r i n c i p a l  4.6.  we must  tensor  upon  diagonalize  it.  y  1 34  A s s u m i n g .the -r  structure  inertia  required  tensors  given  rotations  A s D H and A s H D r e s p e c t i v e l y . 2  w e l l w i t h the  of  6 t h e moment of  -42.2°  These a n g l e s  2  reasonably  in Chapter  values  9 of  and 2 9 . 3 °  for  compare  -37.4°  ± 3°  and 3 2 . 3 ° ±  2°.  8.  DISCUSSION OF THE SPIN-ROTATION COUPLING  CONSTANTS OF  ARSINE The p u r p o s e of empirically AsH  2  them t o  estimate  of  the  AsD  3  to  the  all  the  magnetic  empirically section  4.4)  so t h a t  values  moment of  magnetic  field  order  magnetic the  angular  in  the  w i t h one a n o t h e r of  the magnetic  field  directly  were  by  using  3  the  case  not set  fits  constrained  they  were  is  rotation  where we a r e  isotopically rotates  about  (see  related  to  the  is  spin  For  a  molecular  proportional  assuming that  an a x i s  greater  axis.  rotation. the  the  molecular  invariant).  about  the  that  between  n u c l e u s and t h e  by t h e  and hence  constant  AsH ,  increased data  by t h e m o l e c u l a r  momentum ( r 3 1 ,  the m o l e c u l e  of  In  3  constants  dealing with  arsenic  generated  is  AsD .  the  below.  generated  the  is  final  we a r e  structure  rotation  from our the  that  coupling constants  spin-rotation  electronic  (coupling)  indicate  d i s c u s s i o n we can assume t h a t  field  faster  to  coupling constants  estimated  magnetic  is  consistent  determinable  The i n t e r a c t i o n  first  section  determined magnetic  and A s D H a r e  2  this  the  the  the is,  greater  interaction  Also since rotational  That  to  the  s p e e d of  constants  it  135 s h o u l d not are  also  be s u r p r i s i n g t h a t  related  f i n d that  to  the magnetic  "corresponding"  ratio  (see  for  and t h e  determination Table  4.6.  in  instance  constants.  coupling constant  this  rotational work,  Garvey  rotational of  spin-rotation  rotational  (same a x i s )  examples p r e s e n t e d 4%),  the  the  the AsD  et  3  In  constant,  a constant  al  (rl04)). of  fact  d i v i d e d by  is  constants  constants  AsD  3  ratio  allowed  coupling constants  the  for  Using  we  the (within  this the  presented  in  136 F.  CHAPTER  5:  PHOSPHINE  Phosphine  is  have a p e c u l i a r other  is  the  due t o  A history who n a r r a t e s  p o i s o n o u s gas p u r p o r t e d  "acetylenelike"  way a r o u n d ,  acetylene  a colorless,  of  that  odor.  odor we u s u a l l y  t r a c e amounts of  phosphorus i s  phosphorus s i n c e  Phosphorus  in  bombs.  incendiary  the  courts  the  dark  burst this  of  as  It  is  contaminated with P H „  reaction)  it  will  2  over an  increase  it  its  in  w o u l d glow  kill  people  no d o u b t was the  (produced  s p o n t a n e o u s l y at  role  popular  i n an a l k a l i  phosphine  just  things,  especially  p h o s p h i n e was d i s c o v e r e d as  when w h i t e p h o s p h o r u s was h e a t e d  the  (rl08)  known f o r  meaning l i g h t - b e a r i n g ) ,  ignite  Emsley  to  perhaps best  18th c e n t u r y  is  with  discovery  "pill"  f l a m e when e x p o s e d t o a i r . that  it  (rl07).  3  by J .  its  P h o s p h o r u s was  17th and  ("phosus"  into time  the  is  PH  amongst o t h e r  a p o i s o n and a magic  intelligence.  fact,  associate  given  300 y e a r s ago has been u s e d a s , aphrodisiac,  In  to  room  in  and during  gas g i v e n  off  s o l u t i o n . When in the  same  temperature  (r107) . The P H infrared  3  s p e c t r u m has been w e l l  (rl09-r113)  (r82,r114,r102) forbidden  and t h e m i c r o w a v e  where o b s e r v a t i o n  transitions  ground s t a t e many h i g h e r The P D extensively  3  has  rotational order  constants  distortion  that  of  of  in  both  the  regions  many n o r m a l as w e l l  led to a complete determination  PH . 3  as w e l l as q u a r t i c  as of  and  constants.  s p e c t r u m has a l s o as  studied  been s t u d i e d a l t h o u g h not  Allowed  submillimeter  wave  as  1 37 transitions Helminger  in  the  ground v i b r a t i o n a l  and Gordy  (r1 14)  in  s t a t e were r e p o r t e d  1969. L a t e r  forbidden t r a n s i t i o n s  of  state  was made by Chu and Oka  (r82).  (r25)  increased  measured f o r b i d d e n  number of  the  ground  1974 an  study  the  in  in  In  and p r e s e n t e d a s i m u l t a n e o u s a n a l y s i s and a l l o w e d  rotational  This analysis re-analysis (see  also  b a s e d on t h e  the  section  and Tanaka  The P T fundamental cursory  except  established 3  vibration  e x a m i n a t i o n of  in  phosphines, lies  that  the  of  Their  like  well  frequencies the  arsine  the  above  they  only  to  1  in  the of  b a s e d on  to  "standard"  s t u d i e d and o n l y  the  have been r e p o r t e d spectrum  partially  virtually deuterated  transitions,  These R branches are  in a  (r98).  the  difficulty namely R  t o be f o u n d  at  100 GHz, and a l t h o u g h s t r o n g , a r e  3  the  been  constrained the  PD  1 .  counterparts, of  a  in Chapter  was  t o p p h o s p h i n e s have been  their  present  the  analysis  infrared  For  state.  allowed  in p r a c t i c e  not  forbidden  fundamental v i b r a t i o n s  "fitting"  literature.  (AJ=1).  frequencies  the  have been o b s e r v e d  i n o b t a i n i n g a whole c l a s s  branches  all  transitions  and we s h a l l  "Failures  in Chapter  spectrum i s  The a s y m m e t r i c ignored  four  (r115).  o f f e n d i n g terms  parameters  of  Helms and G o r d y and would have a l s o  unsatisfactory z e r o the  1977 Helms and Gordy  ground v i b r a t i o n a l  f o r b i d d e n and  the  initial  vibrational  formalism established  transitions  s p e c t r u m of  approach of  the  entitled  Finally,  vibration-rotation  by K i j i m a  of  proved u n s a t i s f a c t o r y  Reductions").  infrared  data  by  beyond  1 38 the  frequency In  for  PH D.  m e a s u r e d two o t h e r for  transitions  that  course  of  most  Later,  2  transitions  for  of  spectrometers.  1951 L o o m i s and S t r a n d b e r g  transition (r116)  capability  of  the  PD H. 2  they  present  in  (r99)  1953, S i r v e t z  transitions  They a l s o  for  al  was t o  investigate  (r117).  the  study in  (the  the  these  2  s p e c t r u m on t h e  cases  higher  (r117)  only  transitions branches  than  they  of  100 K H z ) .  r e s o l u t i o n measurements. i n c r e a s e d the to  number of  9 and s i n c e a l l  (AJ=0)  they the  report  these  could only rotational  m o l e c u l e R b r a n c h e s were s t i l l they  for  this  splittings In  a  few  our own measurements have been r e p l a c e d  c o m b i n a t i o n s of  fit  needed a  was a v a i l a b l e  order  here  The p u r p o s e  d e u t e r i u m q u a d r u p o l e moment p r o d u c e s  PH D  special  instrument  the  deuterium  2  resolution  other  measured  q u a d r u p o l e moment i n PH D and c o n s e q u e n t l y higher  a v a l u e of  Still,  PH D 2  al  rotational  determine  were Q  linear - to  "solve"  needed. F u r t h e r ,  one q u a r t i c  by  K u k o l i c h et  transitions  constants  new  2  some t r a n s i t i o n s  2  experiment  PH D and t h r e e  to a s s i g n . During  PH D were r e p o r t e d by K u k o l i c h e t their  one  and Weston  reported several  were u n a b l e work  reported  this  from  constant,  their  D ,, T t  J K  defined  in a p r o l a t e  subsequent data is  analyses  straightforward  this  case  we g e t  p r o b l e m w i t h the data  set  was  basis,  just  are  at  all  to c a l c u l a t e  a very  4 . 3 0 ± .04 MHz. Our done i n o b l a t e D  different  analysis  of  too s m a l l .  J K  in a p r o l a t e  v a l u e of  K u k o l i c h et This  bases  al  but  basis.  In  3.64 MHz. The is  that  f o r c e d them t o  their  constrain  it  1 39 to  zero various  of  the  case  parameters  they  the parameters  c o n s i d e r e d as as we s h a l l for  parameters  the  1.  consider  report  the  sextic  solution.  in p r e c i s e l y constants  this  been  the  same way  we r e p o r t  later  tops.  of  for  this  investigation  Sodium h y d r o x i d e p e l l e t s ,  nitrogen'the until  bubbled o f f ,  the  was p r e p a r e d by  white phosphorus i n a sodium h y d r o x i d e  p h o s p h o r u s were d e - g a s s e d i n a g l a s s  flame  have  In  PREPARATION OF PHOSPHINE GAS  reduction  mm. of  significance  uncertain.  should only  parameters  phosphine asymmetric  The p h o s p h i n e the  physical  could determine  they  fitting  making t h e  H 0 / D 0 and 2  2  vacuum l i n e .  m i x t u r e was h e a t e d w i t h a  phosphorus melted  (m.p.  was p a s s e d t h r o u g h a water  condensed with l i q u i d  nitrogen.  white 150  semi-luminous  40°C). trap  Under  The p h o s p h i n e  and was  1 40  2.  PREDICTION OF THE ASYMMETRIC TOP PHOSPHINE  Estimates asymmetric as  in  the  average  of  the  rotational  parameters  top phosphine s p e c t r a arsine  asymmetric  rotational  were made i n  top study.  constants  lengths  parameters  (r82).  From a p r e l i m i n a r y  r  z  force  field  c o r r e s p o n d i n g symmetric  top data  alphas  Adding t h i s  were c a l c u l a t e d .  rotational empirical one p e r  constants rotational  of  calculated  where bond Chu and Oka  derived  from  contribution  to  "guesstimates"  usually  by-product  t h e p r e d i c t i o n of  of  the  of  the  the  of  r  z  the  t o much b e t t e r  initial  the asymmetric  Certainly  d u r i n g the  errors  in  the q u a r t i c  contribution  frequency  predictions  relative  rotational However,  constant  once t h e  established, substantially the  point  the harmonic p a r t s  gave us  constants  tensor  same way  than  cent.  A useful was  above  the  A , B and C were z z z inertia  were t h e  the  The z e r o  by d i a g o n a l i z i n g t h e moment of and a n g l e s  for  SPECTRA  the  initial  contribution  rotational excellent  search times.  "empirical" quartics  v a l u e s as  the  not  the  constants  It  the case  a s s i g n m e n t s made e a r l y  the  transition  errors  A , 0  in  the  significant. B , 0  predictions  to  made f r o m t h e  and C  were  0  shortened  the  force  hyperfine  study  were o f t e n  watch  field  increased.  In  arsine,  splitting.  i n p h o s p h i n e and c e r t a i n l y the  constants.  was a l s o c o m f o r t i n g t o  "converge"  in  study  f o r Q and R b r a n c h e s ,  were not  number of measured l i n e s  a s s i g n m e n t s were e a s i l y T h i s was  to  quartic  field  top q u a r t i c  searches  to  force  here  b a s e d on how  141 close  subsequent q u a r t i c s  came t o  the  force  field  predictions. The p h o s p h i n e s t u d y identical are  that  to in  structure that  the  the  discernible PD  asymmetric  "c"  3.  are  4. no  The  differences  hyperfine resolution  and  r e - a n a l y z e d w i t h no new  for  and " c "  an o b l a t e  2  the  AsH D, 2  type  PH D  is  2  a  prolate  transitions.  asymmetric  top with  Also  "b"  and  (Tables  high  range  PH D and  s u p p o r t e d by t h e  section,  relative  number of  measured  increased to  and 5 . 2 ) . R branches  T h e s e new of  the  PD H. 2  frequency  intensities  2  J=0—>1.  varies  and  The  from z e r o  assignments  discussed in  and S t a r k  PH D  transitions  The t r a n s i t i o n fits,  rotational  24 f o r  type  from 9-264 GHz and J  14 f o r  2  the  has been  5.1  frequency  frequencies 19 f o r  study  frequencies  2  SPECTRA  2  present  PD H  include  to  is  2  is  virtually  transitions.  transition  new  PD H  Chapter  there  simply  with " a "  is  our e x p e r i m e n t a l  OBSERVED PH D AND P D H  In  17 f o r  of  As was t h e c a s e  rotor  2  within  spectrum i s  3  AsD H, type  study  the phosphine case  measurements.  like  arsine  p r e s e n t e d here  effects  the  as  next  discussed  below. Not  having a t r a n s i t i o n  s u p e r i m p o s e d on t h e assignment difficult  of than  the the  dependent h y p e r f i n e  rotational  asymmetric arsine  transitions  made  structure the  t o p p h o s p h i n e s p e c t r a much more  asymmetric  top s p e c t r a  assignment.  1 42 Whenever p o s s i b l e t h e n , subjective  estimate  convincingly for  of  a s s i g n m e n t s were made from a relative  from an a n a l y s i s  of  a AM=0 Q b r a n c h t r a n s i t i o n  Stark  s h i f t e d components v a r y  (e1.43) will  a plot  give  of  a straight  c o u r s e when t h e differ  frequency  in J  it  line  choices often  of  intensities Stark  the  for  Also  2  of  the  to  more For  example,  intensities  from  of  equation  a component v e r s u s  correct  assignment  suffices  shifts.  relative  as M .  shift  or  M  2  a s s i g n m e n t . Of  for  a  transition  simply count  the  Stark  lobes. The  initial  phase  was v e r y  similar  to  asymmetric  constant  at  allowing  the  previously (r116) then  the  were  numbers, low J  initial  Low  p h a s e of  two  "fit".  F i x i n g one  rotational  roughly,  used in  that  the  for  transitions  preceding section  each  fitting  of  parameters  This  is  transitions  with  intermediate  for  values  perhaps  J  were  were These  transitions of  assigned high J  found.  the 3  the  illustrated  by  quantum  including  be r e - a s s i g n e d when  and  species  transitions.  the J  initially  sometimes had t o  to v a r y ,  isotopic  b a s e d on p r e d i c t i o n s made from f i t s  lines,  arsine  be r e a s o n a b l e  order  fits.  the  further  to only  were on t h e  study  rotational  constants  The s u b s e q u e n t  t u r n e d out  the  top  f r e q u e n c y Q b r a n c h e s were m e a s u r e d  p r e d i c t e d v a l u e of  other  values  transitions fact  phosphine asymmetric  spectrometers.  used to p r e d i c t ,  whose J  the  assigned transitions  predictions  the  the  top study.  u s i n g the U . B . C .  of  only  143 The s e c o n d p h a s e of frequency Jet  R branches.  Propulsion  branch  lines.  identify  shape t o  0  o o  ->1i1  low  one S t a r k field  led  J,  lobes.  to  of  PD H.  transition  unfortunately  that  the  greater  were e n c o u n t e r e d  inconceivable  that  search  and i t  continually lobed,  low  miss J,  the  the  so g r e a t  Finally line  it  of  and t h e  size  the  assigning  the only two R  was  had an  suggesting that  feature  was  expected  It  was  off  than  our  we c o u l d searches. but  it  in a d d i t i o n were  it  an  The  double  could  impurity  coefficient.  the  lobed  double It  was  not  the  absorption  same.  a  transitions  farther  that  for  this  Q branch  impurity;  0—>1 were one and t h e  the  goes down as M  ±10 GHz s e a r c h .  that  of  characteristic  for  case  and  were one t o o many  other  an u n r e a l i s t i c  this  ±10 GHz had  re-investigated,  seemed i n e v i t a b l e  in  transition  on s u c c e s s i v e  likely  to  In  moment, a n o t h e r  improbable  R b r a n c h was  w o u l d have  the  this  line  intensity it  out  easy  establishing  within  lobes  prediction  was  t o any  line  that in  be a t t r i b u t e d was  this  we were l o o k i n g  than  in  intensity  Ruling  J=0—>1 f o r  0—>1 t r a n s i t i o n  region  two  are  the  J=0—>1 R  anticipation  showed t h e  where l o b e  AM=0, J=0—>1 t r a n s i t i o n .  intensity  For  2  goes up,  the  t o measure  patterns.  However,  some h i g h  were done a t  identical  some d i f f i c u l t y  pattern  as  lobe  line.  branch Stark  transition  Stark  R branch candidate  This  but  t o measure  We were a b l e  distinctive  transition  suitable Stark  see  zero  above p a t t e r n  was  The AM=0, J=0—>1 R b r a n c h e s  to  the  study  T h e s e measurements  Laboratory.  from t h e i r  one e x p e c t s  this  mystery  then  1 44  suggested,  and l a t e r  calculation,  that  c o n f i r m e d by measurement  the  s e c o n d l o b e was due t o  weak AM=±1 t r a n s i t i o n . line  This  came as a s u r p r i s e  wave r e g i o n d u r i n g t h e  for  were so weak as  because  waveguide  frequencies microwave  to  experimental  this in  p h a s e of  absorption c e l l s  (see  the  cell so  we s t i l l  u s e d a low  (X band c e l l )  the  the  AM=±1  at  Not a l l Stark  effect.  next  lower  one mode o f transitions.  waveguide  absorption  length  waves  the  transitions,  assignment  p r e d i c t i o n and s u b s e q u e n t measurement  transition  the  Later  for  is  the  modes, p a r t i c u l a r l y  modes r e s p o n s i b l e  became more p r e v a l e n t . the  other  the  AM=±1  our  when we went t o m i l l i m e t e r  p r o p a g a t i o n of  perpendicular  frequency  This  as  for  p r o p a g a t i o n d o m i n a t e d p r o d u c i n g o n l y AM=0 However,  centimeter  used at  2)  wave  s t u d y was  same o r d e r  Chapter  supposed  millimeter  this  w a v e l e n g t h and hence  configuration  the  be n o n - e x i s t e n t .  had d i m e n s i o n s on t h e  radiation  for  experience  initial  transitions the  result  and  was c o n f i r m e d by  of  the  second 0 — > 1  175.8 G H z . transitions In  section  these  c o u l d be a s s i g n e d by  cases  the  frequency  fit  established  the  assignment.  intensity discussed  or in  1 45 Table Observed Transition  PD H 2  5. 1  s p e c t r u m i n MHz  Frequency(Unc)  Obs-Calc  12005.030(0.030)  0.030  1 1 o 0oo  175822.284(0.020)  0.027  111  158754.114(0.020)  -0.027  1  -  1 1  1  0 1  _  _  1  0 0  U  1 o  1  -  29073.210(0.050)*  0 1  0.093  36003.720(0.030)  -0.057  330 3 2  111544.084(0.050)  -0.059  32 2 - 3 , 2  19415.190(0.050  0.018  4« i _ - 4 3 i  112616.065(0.030)  0.022  4ao"4 1  1 14000.480(0.100)  221 2i1 -  -  2  3  -0.087  53 3~5 3  24079.480(0.050)^  0.031  63 4 ^2 4  9049.330(0.030)  0.020  28759.350(0.050)*  0.045  2  9 5 5-945  9 8 2~97 2  1166-11  263757.276(0.030)  -0.000  29833.830(0.030)  -0.025  5 6  13  7 7  -13  6 7  14  7 7  -14  6 8  254588.040(0.030)  0.000  14  7 8  -14  6 8  11861.050(0.050)  0.001  t  of  0.023  ref(r116)  '*' part  30531.330(0.050)*  unassigned t r a n s i t i o n s this  '**'  as  ref(r116)  assigned  as  work above  — Numbers uncertainties  of  in in  except  this  parentheses units  of  the  transition are last  was  estimated  re-measured.  measurement  significant  figures.  1 46 Table  5.2  O b s e r v e d PH D s p e c t r u m 2  Transiti-on  i n MHz  F r e q u e n c y (Unc)  Obs-Calc  loi  40561.870(0.050)  k  - 0 . 124  1 10 " 111  6024.645(0.005)  k  -0.005  In  "  172524.933(0.015)  0.0  loi  " Ooo  2,2  - 2 2  35142.100(0.030)  2,,  " 2,2  18070.910(0.035)  -0.038  3,2  " 3,3  36093.670(0.030)  0.015  3,3  " 30 3  28158.500(0.030)  3  "  2  0  2  0.022  k  0.002  k  112977.568(0.050)  3,2  0.056  4,«  - 4 «  20815.334(0.005)  k  0.000  5,  " 5 5  14246.690(0.0l0)  k  -0.005  "  19611.360(0.100)  5  0  5  2  0  3  5 2 tt  6,6  " 6 6  6  ~  n  2  V, 9 10,  2  "  -1 0  3 8  1 0  -1 0  2 9  11 2 14 15 16 17  3 >  3 f  3  3 f  5536.382(0.003)  0 7  3 7  256864.972(0.015)  -0.001  2 8  112617.504(0.030)  0.006  20754.570(0.050)*  1 9  , -  14  ,3-  15  ,  16  ,5-  17 I9  2 f  2 f  2 f  13259.560(0.030)  -0.050  ,2  34520.190(0.030)  0.023  ,3  22821 . 9 0 0 ( 0 . 0 5 0 ) *  -0.057  ,4  14504.990(0.040)  -0.009  ,5  8916.190(0.060)  0.044  , 6  34207.180(0.050)  -0.003  1  2 f  3  f  0.221  0  \  > **  and  'k'  ref(r117)  -  0.001  k  -0.053  , 01 "1" 2  0.015  30957.770(0.030)  9,8  10  -0.003  k  36331.040(0.040)  2  "  8  1 0  6 5  " 7  7  7  9121.376(0.005)  0  0.068  Numbers  uncertainties  as in in  in Table  5.1  parentheses  are  units  of  the  last  estimated  measurement  significant  figures.  1 47 Table R e - a n a l y s i s of  5.3  t h e m e a s u r e d PD  3  spectrum  i n MHz  +  Transition  Frequency(Unc)  Normal R o t a t i o n a l J+K-J, J=  1  Obs-Calc  Transitions  K<-K r e f (r1 1 4)  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  Forbidden Rotational J<-J,  K=*l<-±2  Transitions  ref(r82)  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=9  93498.65(0.02)  0.02  10  93265.70(0.02)  0.05  11  93009.29(0.02)  -0.08  J<-J,  K=0<-3  ref(r25)  1 48 Table Transition  5.3  continued  Frequency(Unc)  Obs-Calc  1 2  92729.42(0.02)  -0.04  13  92425.42(0.02)  0.04  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  -J,  K=±1<-±4  f  Numbers  uncertainties  ref(r25)  in parentheses in u n i t s  of  the  are last  estimated  measurement  significant  figures.  149 4.  ANALYSIS OF THE PHOSPHINE  The p a r t i a l l y  deuterated  SPECTRA  phosphine s p e c t r a  a S reduction  hamiltonian  method  Even t h o u g h PH D and P D H a r e  (r12).  prolate  2  asymmetric  representation cursory  rotors  deuterated  phosphine, PD ,  in  reference  same manner as since  the  must be  (e1 .21)  (r25)  in  data  the  set  AsD  the  and a f i r s t  parameters  here  listed  order the  where a i s  frequencies  the  r e p o r t e d by Chu and Oka  in Table  See  the  later  Reductions" not  for  section  of  7  the  parameter  approach  h . 3  The  were o b t a i n e d by  in  equation.  the For  their  "Failures why  precisely  complete  secular  we t o o k  entitled  an e x p l a n a t i o n  satisfactory.  the  5.4  uncertainty.  (r82)  in  earlier .  perturbation  were w e i g h t e d  experimental  Both  same v a l u e of  t h e c o m p l e t e d i a g o n a l i z a t i o n of experimental  re-analyzed  fit.  The  unsatisfactorily  |K j = 3 l e v e l s  global  tops.  were  study d e s c r i b e d  3  includes  led to e s s e n t i a l l y  empirical  is  and  the  spectrum  3  (r25,r82,r114)  included in  diagonalization  is  and 5.2)  The t o t a l l y  analyzed  7  5.1  of  symmetric  study.  references  3  (Tables  t o make  parameters  oblate  and.  oblate  r  be a b l e  the a r s i n e  in  h  the  to  iterative  oblate  a 111  in  reported  Here  of  uncertainties  d e t e r m i n e d as  to  empirical  tops with those  experimental  the  in order  the  u s i n g an  2  respectively  was c h o s e n  c o m p a r i s o n s of  asymmetric  {e1.15-e1.18)  were f i t  this  fits the  by  The 1/a  2  30 GHz comb  quoted  of  the  previous  analysis  1 50 experimental  errors.  However,  r e p o r t e d by Helms and Gordy 1 in  10  w h i c h we c o n s i d e r  8  substituted, 5.3.  somewhat  The a l l o w e d  each given For  PD  asymmetric  arsine  d i s c u s s i o n in asymmetric  coefficients separated  of  w i t h our d a t a  reduction)  was  harmonic determine  field  linear  D j and H^ t e r m s .  the  prediction the  order  the  D  q u o t e d from t h e The c h o i c e fitting  T  — in  terms  force in  these  this  least  linear  into  be w i t h i n  squares  parameters  is  from  the we  constants of  Dj  with  allows  combinations  a s s u m i n g Hj the  of  in  this  their  identical  to  and  is  rotational  error  the  be  the  order  of  the uncertainty  standard  sextic that  (and of  in  errors  fits.  and i n t e r p r e t a t i o n  the  (S  s h o u l d be on t h e the  the  that  values  our c a s e we e x p e c t to  and  Empirically  constants  way  obtain  to determine  6).  introduced  constants  ),  value c a l c u l a t e d  Chapter  the  to  i n each c a s e ,  H-, c o n t r i b u t i o n a n d / o r of  were  terms cannot  n  In  rotational  in  ref.(r114)  concerning  ) (n=1,2,3...)  field  we o b t a i n  rotational  higher)  2  constants,  The u n c e r t a i n t y  sextic  (e1.15  Given  the  Table  we were a b l e  rotational  then determine  in  200 K H z .  c o m b i n a t i o n s of  the Dj  constants  ( J  (see  us t o e s t i m a t e  small.  of  given  s h o u l d be a p p a r e n t  set.  f i x e d at  force  in  the p r e c e d i n g c h a p t e r  diagonal  rotational  reported  From e q u a t i o n  of  and we  errors  uncertainty  it  transitions  c l a i m an a c c u r a c y  the  top spectra  tops,  individual  forbidden  they  arbitrarily,  o n l y J=0—>1 R b r a n c h e s . similar  (r25)  transitions  3  the  too o p t i m i s t i c  an e x p e r i m e n t a l  the  for  the  151 arsine  study.  thought  of  These h i g h e r  only  determination  as  of  The r e s u l t s Obs-Calc's,  that  fitting  order  constants  of  the  fits  are  given  is  the  best  fit  f r e q u e n c y minus  experimental  frequencies.(Tables  Weston  in t h i s  the  otherwise  these  indicated  fits.  5.1  the and  in Tables  5.1  Sirvetz  uncertainty  The t r a n s i t i o n s  be  better  5.4.  of  The  the  tables  of  5.2) to  definitely  r e p o r t e d by S i r v e t z  transitions  t h e measurements of  final  in  in Table  we were a b l e  unassigned l i n e s  quoted experimental the  listed  study  ( r 1 1 6 ) . Some of  r e - m e a s u r e d as  are  to  enable  constants.  frequency,  assign a l l  that  are  the q u a r t i c  experimental  Finally,  parameters  have  and  been  and 5 . 2 ,  but  and W e s t o n ,  with  their  0.050 MHz, were u s e d  that  Sirvetz  and Weston  c o u l d not a s s i g n were " h i g h "  J  large  in phosphine  extrapolations  available  Sirvetz  centrifugal  from t h e  lower  were not  good enough t o e n a b l e  transitions. high J  In  J  distortion transitions  this  transitions  previously  study  the  has a l l o w e d  unassigned  transitions.  in  to  Because  these  identification  of  transitions.  the  and Weston  them t o a s s i g n  us t o d e f i n i t e l y  of  higher  many assign  J  other these  1 52 Table Empirical  5.4  s p e c t r o s c o p i c parameters PH D  PHD  2  of  various  phosphines PD  2  3  X  MHz  129843.14(4)  93917.79(2)  Y  MHz  89279.34(2)  81910.92(2)  69471.09(11)  Z  MHz  83250.52(2)  64841 . 8 0 ( 3 )  58974.42(11)  KHz  2726.04''*  KHz  -3475.8(23)  D.  1612.75  t  1020.4(68)  L  D JK K  d  KHz  1873.54(56)  -352(19)  KHz  - 2 2 6 . 7 ( 1 .7)  -233.4(3)  KHz  2  449.1(6)  H,  Hz  H JK  Hz  •346( 18)  H  Hz  1 16(20)  K  Hz  67(20)  h,  Hz  H  KJ  L  3  JJJK  171(19) 140(23)  -1290(150)  -4.2(8) 4.63(3)  Hz  0.013(3)  least  f igures.  221(31 )  Hz  — Numbers the  1026.9(22)  -326.20(17)  -71(4)  fHarmonic Force  of  -1312.28(46)  -52(3)  Hz h  -794.3(64)  Field  value  in parentheses  squares  fits  are  in u n i t s  the of  standard the  last  deviations significant  1 53 G.  CHAPTER In  field  6:  this  FORCE F I E L D S , chapter  descriptions  we s h a l l  of  the  p h o s p h i n e and a r s i n e . immediately First  get  we g e t  constants  excellent  2  in  the  fits  to  the  other  the  third, the  From t h e  force  values  for  to  the  for A s D ) .  R  3  spectral  empirical  force  field  estimation  of  the  determination  of  zero point  constants  sort  obtained  (r43).  Still  from XH  force  these  resolution  3  constants  force  these  of  the  the  values  Second, necessary  in  alphas  structures  are  various  allows from  also  useful  in  hydrides.  be made. harmonic As e a r l y  force as  spectra.  the  field  the  studies  1930's  frequencies  these  hydride  frequencies  1 9 4 0 ' s and 1 9 5 0 ' s  had a l l  force  Woodward has  b a s e d on h i g h e r  constants  and  estimated.  vibrational  3  for  contribution  parameters  averaged  calculating  and X D studies  4 and 5 (D^  be r e m o v e d .  from v i b r a t i o n a l  p r o b l e m of  Subsequent  "better"  of  were c a l c u l a t e d  c o n s i d e r e d the constants  of  group V h y d r i d e s .  f r o m low  information. distortion  their  themselves  will  T h e r e have been t e n s  constants  to  c a n be  comparing bonding parameters  done on t h e  allows  harmonic p a r t  which e q u i l i b r i u m s t r u c t u r e s  this  the q u a r t i c  we  d i s t o r t i o n d i p o l e moments;  evaluation  C o m p a r i s o n s of  of  C o n s t r a i n i n g to  parameters  the  analysis  in chapters  data  force  namely ammonia,  pieces  we can c a l c u l a t e  of  The p o t e n t i a l  the harmonic  field  important  we c o u l d not d e t e r m i n e  2  for  look at  group V h y d r i d e s ,  three very  P H D and P D H and D  from t h e  STRUCTURES AND DIPOLE MOMENTS  resolution  force alone reported studies.  been o b t a i n e d f r o m  just  1 54 the  vibrational  Mills  (r118)  frequencies  of X H  p u b l i s h e d a method of  coupling constants  and q u a r t i c  along with v i b r a t i o n a l  iterative  least for  method t h a t analysis  squares  constants  and q u a r t i c  frequencies  the  For  harmonic  the  from X H  XH  molecules  3  incomplete constants  constants  R  Alti  et  (r98).  over  earlier  Still  studies  in c a l c u l a t i o n s  of  the X T  3  data  constants  in  d i d however  et  al  force  (r77)  for this  their  another  vibrational enabled  a definitive  include  only  constants  were  to  coupling  include  the  r e p o r t e d by De  vibrational  frequencies  For  completeness  extensive  the  for  data  improvement  d i d not of  study  molecules.  study  include force  any  include  quartic  constants.  "anharmonicity"  " s p e c t r o s c o p i c " masses. T h i s  we  by  1966. They a l s o d i d not  to account  below.  field  was a marked  constants.  refinement  attempt  further  study  this  coupling  species  3  is  and XH„  3  d i d not  used o n l y  in  force  Coriolis  the X T  and f u r t h e r m o r e  what one m i g h t c a l l discussed  and t h e  that  should a l s o mention here Shimanouchi  were a b l e  They a l s o  frequencies  XH  The q u a r t i c  values)  inaccurate.  vibrational  they  the  an  also  constants  to present  of  into  It  Coriolis  force  1960  Coriolis  (see  along with  of  fields  species.  3  (no D  al  Including  1964 (r76)  force  and X D  3  in  harmonic  In  distortion  the method).  the  in a c a l c u l a t i o n  Duncan and M i l l s of  procedure  a d e s c r i p t i o n of  presented here.  including  frequencies  fitting  we have u s e d f o r  species.  3  centrifugal  constants  (r8l,r1l9)  and X D  3  They  using is  1 55 T h e r e have a l s o studies terms  of  group V h y d r i d e s .  as w e l l  potential. force  Of  field  and t h e (rl03)  been s e v e r a l  as q u a d r a t i c special  studies  studies  of  and l a t e r  interest  of  N H , PH 3  3  by Hoy e t  why a f u r t h e r since  the  study  study  parameters  of  it  3  is  first al  is  (the  value presented  latter  deuterated  calculate  the  new  field  force  accurate  per  study  A second reason fact  that  almost  anharmonic  studies,  anharmonic  effects  empirical  show t h a t  to  (r120)  in  1967  in  been g r e a t l y symmetric  reasons  the of  period the.  improved.  tops NH  thesis)  fields  give  accuracy  have  A l s o the q u a r t i c  of  this  the data  is  thus  that  first  a new f o r c e  every has  study  of  of  this  Dennison is  (the by  from w h i c h then  the  to  for  a  set. field  to d a t e ,  "attempted"  3  just  increase  reason  PD  we now have a much more  increased data for  More  and  3  constants  thesis)  available  The  in v i b r a t i o n  relations  al  force  First,  the  in. t h i s  constants.  and g r e a t l y  1961  ammonias, p h o s p h i n e s and a r s i n e s  cent  force  on t h e  here.  the  two p r e s e n t e d as p a r t fifty  we w i l l  literature  for  R  become a v a i l a b l e .  more t h a n  the  D  anharmonic  1972 (r121 ) .  warranted  values  partially  in  the  the in  perhaps necessary  importantly  recently  are  field  and h i g h e r  included in  by M o r i n o e t  t o them has  of  cubic  us h e r e  Duncan and M i l l s  available  latter  to  case  force  p h o s p h i n e by K u c h i t s u  ammonia,  and A s H  this  terms a r e  W i t h s u c h an e x t e n s i v e of  In  anharmonic  study  including  to account  frequencies (r79).  In  an u n r e a s o n a b l e  is  the  for  using  the  found  next  thing  the section  t o do and  in  156 that  force  constants  obtained using this  method  are  unsatisfactory.  1.  AN EXAMINATION OF STANDARD ANHARMONICITY TECHNIQUES  A principal force  fields  calculation or  is  the  frequencies section)  in  "correct" effects a  "true"  look  at  made i n  "accidental"  of  the  contributions  of  shifts  (discussed  empirical the  molecular  final force  how c o r r e c t i o n s past.  data  for  force  the  In  from  methods p r o m o t e d by p r e v i o u s  of  this  this  authors  the  course  when to  w o u l d be c l o s e r  section  we  due t o a n h a r m o n i c e f f e c t s  The r e s u l t  next  extra-harmonic  constants  field.  in  one would p r e f e r  these  are  vibrational  - terms we of  constants  effects  sort  the parameters  potential  parameters  generally  frequencies felt  are  t o have t h e  have been  s h o u l d be t h a t  the  are  found to  the  experimental  c o n t r i b u t i o n and so i t  is  anharmonic  have been a p p l i e d . The  corrections  only  to  largest  these  to  shall  be  unsatisfactory. The v i b r a t i o n a l  the  oscillator  frequency  to  used in  from  this  resonance  force  harmonic  parameters  harmonic a p p r o x i m a t i o n . I d e a l l y  so t h a t  the  of  c o n t r i b u t i o n s of  contributions  harmonic  the  empirical  the harmonic  due t o F e r m i  the  the c a l c u l a t i o n  significant  The l a r g e s t  terms  calculating  in the  outside  and t h e  anharmonic ignore  concern  include  approximation. generally  APPLIED TO VIBRATIONAL FREQUENCIES  whether  interactions  CORRECTION  anharmonic  frequencies  that  anharmonic  1 57 contribution  to the q u a r t i c  is  generally  assumed t o be l e s s  than  contribution  to the v i b r a t i o n a l  frequencies  constants without  are generally  correction.  As we s h a l l sum r u l e s  to  there  or i n v a r i a n c e  parameters.  If  contributions anharmonic  corrections  used t o determine  in various  with a v i b r a t i o n  correction  experimentally  Now r e c a l l fundamental  by s y m m e t r i c  these  parameters.  usually  e  2  or " n o r m a l "  due t o a n h a r m o n i c i t y . measured f r e q u e n c y = u(l-2x  vibrations  to  magnitudes of the anharmonic  - x w(v+l/2)  for a pyramidal  in  t h e n be b a s e d on t h e c r i t e r i o n  o f quantum number v i s  v  parameters  starts  m o l e c u l e where t h e e n e r g y  = CJ(V+1/2)  relations,  contribution  An o b j e c t i o n  experimental  where u> i s t h e " h a r m o n i c " the  anharmonic  of a n h a r m o n i c i t y  example o f a d i a t o m i c  "trace"  point.  the anharmonic  parameters. will  calculations  amongst t h e " h a r m o n i c "  use t h e s e  the r e l a t i v e  Consideration  E(v)  various  relations  to estimate  in other  contributions  exist  we c a n t h e n  relations  field  a n d so t h e s e  o u t t o be an i m p o r t a n t  we assume n e g l i g i b l e  some p a r a m e t e r s  invariance  turns  constants  the anharmonic  used i n f o r c e  This  see l a t e r  and C o r i o l i s c o u p l i n g  XY  associated  ( r l 2 2 pg 274)  +  [6.1]  frequency  a n d x^co i s  The " f u n d a m e n t a l " o r  is E(1)-E(0),  )  3  with the  or [6.2]  molecule  there  are four  ( 2 A , + 2E) w h i c h c a n be c h a r a c t e r i z e d  and a n t i s y m m e t r i c  bond s t r e t c h e s  and a n g l e  1 58 bends. three  Furthermore for  the A,  vibrations. the  E  there  s p e c i e s and t h r e e  For  argument's  the  E species  account  for  and F  correspond to  M  the  respectively.  is  The F „ 3  is  3  angle  similarity  molecules  it  are  of  E  species  now c o n s i d e r  only  bond s t r e t c h i n g constant If  the  the  6.5  that  constants  is  to  for  bond s t r e t c h In  past  case  that  bond s t r e t c h e s  constants.  3 f t  of  XY  3  using  and a n g l e  bends way.  It  in  force  appear This  F  anharmonicity  d i s c u s s i o n to note  generally  3 3  and  because  in pyramidal  interaction  3 q  is  in a " d i a t o m i c - l i k e "  later  the d i a g o n a l  F  constant  and CJ,  this  bond s t r e t c h e s  do not  F  bending  force  bonds and a n g l e s  all  and  f l 4  bond s t r e t c h i n g  interaction  "off-diagonal"  unfortunately  with respect the  the  ,F  f r e q u e n c i e s may be a c c o u n t e d f o r  instructive  that  3 3  and a n g l e  mixes the  essentially  all  F  3  has been assumed i n t h e  (e6.2):  might prove  that  the  constants,  co and co„. The c o n s t a n t s  i n d e p e n d e n t and e a c h behave  Table  force  constants  bend i n c h a r a c t e r .  the v i b r a t i o n a l  equation  3 force  two v i b r a t i o n s  t h e n co  essentially  in  for  sake we s h a l l  the  bending motions.  small  the  six d i s t i n c t  vibrations. For  angle  are  to  be  should  and a n g l e  bends c a n n o t  potential  function,  small  suggest  be t r e a t e d  as  independent. If  we t a k e  "diatomic"  for  the  oscillates,  a Morse P o t e n t i a l  in which  then  x  is  our found  e be p r o p o r t i o n a l can w r i t e  t o co ( r l 2 2 pg 2 7 4 , r 1 2 3 pg 101)  equation  (e6.2)  for  the E  vibrations  so t h a t  we  to  159  V  =  3  where t h e  CJ  3  (  1  3  x^ a r e  s h o u l d be u s e f u l fundamental  V „  isotope in  this  197 r 7 9 . )  two  in  six  needed. F i r s t ,  not  independent.  pyramidal  XY  species  unknowns. the w's  for  the  is  (M+3m ) u  t h e mass of  (Zeta)  Coriolis  197, see a l s o  coupling constants  experimentally (e6.3)  (rl24)).  then  For symmetry  the  In  c o u l d be  parallel  suggested that  /  1  symmetric  this  for  [6.4]  relation  harmonic  way  if  the  of  v -v 2  equation  harmonic  field  determination.  vibrations  where by  no 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  since  (ref.  determined  force A,  are  2  the anharmonic c o r r e c t i o n s  c o u l d be u s e d i n t h e  there are  rule  coupling constants  c o u l d be e s t i m a t e d and t h e p r o p e r  frequencies  relations  J  frequencies page  page  produces  species  t h e X a t o m . As a s e c o n d  Coriolis  on h i s  further  7^  between t h e  and t h e  "slightly  pg232,r78)  Dennison produced a r e l a t i o n  (r79)  the  (e6.3)  product  TT  data  (r79).  isotopic  V H ]  u  (M+3m )  where M i s  (e6.3)  footnote  (r37  The  Dennison proposed  3  two  two  The R e d l i c h - T e l l e r  r  isotopic Given  equation  Clearly  E type v i b r a t i o n s  3  (e6.3).  D e n n i s o n , see  isotopic  are  g  so t h a t  and X D  3  from x .  d i s c u s s i o n we have a d o p t e d t h e  a p p r o a c h of  4 equations  XH  [6.3]  )  different  independent  for  a  solving equation  more l o g i c a l " For  a  solving equation  method of  for  CJ<, ( 1 + X C J  =  new c o n s t a n t s  E frequencies  following  (Actually  )  3  x^ r e p r e s e n t  parameters  the  +X ( J  1 and b o t h v i b r a t i o n s  are  Dennison  160 "essentially" corrections  v.  bond s t r e t c h e s  in  corresponding product  rule  M c C o n a g h i e and N i e l s e n r e l a t i o n -for the Shaffer  A,  (r125)  (actually  $  to  to  be s e t  in  1 2  anharmonicity  for  (r97)  the  of  to  from  to produce a  second  theoretical  times)  Hayne and N i e l s e n  was  results $  1 3  small  (rl26)  of  Later  assumption that  the  those  the  vibrations.  u s i n g the  further  notation  (see  the A,  were a b l e  vibrations  w i t h the  zero  equal  would t h e n be c a l c u l a t e d  2  of  the  s h o u l d be p r o p o r t i o n a l l y  The v a l u e of co  2  that  enough  for  misprints) . 8  The above method has been by technique the  by w h i c h a n h a r m o n i c  vibrational  however,  is  Dennison  (r79  constants  right  unsatisfactory. his  pg.  is In  not  of  s h o u l d be - s i g n s .  8  c o r r e c t e d as they  their  define  rule  for  2  the  relating  Coriolis  frequencies  with other  it  seems t h a t  last  equation  Furthermore  the  on h i s  term M ' / 2  can b e t t e r  ammonia-like  to  coupling for  the  relations  the  of  E  given  + s i g n s on  the  on D e n n i s o n ' s page  relations page  for  199 a r e  by  197  the not  needs t o  be  s h o u l d r e a d ju'm/m'. A l s o t h e C be c a l c u l a t e d  where - b = $ + $ = I 2  have been made  second r e l a t i o n  of M c C o n a g h i e and N i e l s e n  C=(2-b )/(3b -2) 2  First  coupling constants  The r e l a t i o n  that  the  popular  T h i s method as u s e d up t o now  consistent  particular  hand s i d e  Coriolis  197)  t h e most  corrections  t o masses and h a r m o n i c  vibrations Dennison.  frequencies.  far  3  a  molecules.)  /Ul  from  )-\  (the  zeta  sum  161 obviously  equivalent  p r o d u c e d by S h a f f e r Shaffer's  invariance  This  3  where m  is  47r c w?.  Unlike  I.  result  last  relation m  2  point  X.(l-$.)  1  easy  equation  the  the  16. A l s o  if  we  Dennison's relation  it  appears  from b o t h X H  which is  only  and i t  axis  (r127  equ.  of  relation  of  suggested that  that  (e6.5)  A preliminary above c o r r e c t i o n s corrections in  the  large  amount  quartic quite  of  of  field these  "other"  distortion  significantly  anharmonic  E  was made of  i n c l u d i n g the  force  feature  the  by t a k i n g v a r i o u s  and t h e n  harmonic  important  study  of  replace  fits fits  kinds of  constants. worse  In  M e a l and  if  correction  the c o r r e c t i o n  III-6)  D e n n i s o n and  t h e above method of  for  "trace"  1  in preserving  equation  one  is  Dennison's  "second"  vibrations. the  effects  of  combinations  of  that  data,  they  namely  cases  t h a n when we u s e d  the  the  frequencies  described later.  all  Polo  anharmonicity  "harmonized"  was  from  [6.5]  interested  equation  in  3  data  w i t h the  use  atoms and where X. =  equations  trace  is  and X D  3  requires  consistent  off  "second" the  equations  = constant  1  1  t h e mass of  to v e r i f y  identical  of M e a l and P o l o  M c C o n a g h i e and N i e l s e n is  his  include C o r i o l i s data  one s p e c i e s .  2  (r125),  to Dennison's  9  supposedly  formalism to d e r i v e  we need t o contrast  to  An  included a Coriolis fits  and  were  uncorrected  frequencies.  Certainly anharmonicity  t h e above correction  result is  suggests that  suspect.  this  Furthermore,  method of one m i g h t  1 62 ask  why  this  this  p r o b l e m has  problem i s  in  this  not  been n o t i c e d b e f o r e ?  case  noticeable  extensive  and a c c u r a t e l y  measured d a t a  prior  say,  vibrational  to,  1960 o n l y  in c a l c u l a t i n g vibrational  there  of  rules,  are  (r43)).  fields.  frequencies  determination product  force  one  only  For  the  of X H  6 force  for  3  these and X D  validity  of  be a s c e r t a i n e d .  Certainly  extensive  of  study  this  any  that  unsatisfactory, "uncorrected" surprising  later,  always b e t t e r  than  these  directly  reason very  for  this  fits  of  results,  to  were u s e d  the  unique  (recall  there  data,  see  the  Mills.  i n t h e mid 6 0 ' s , (r76) data,  one v e r y  are  contradict is  constants  small,  important  on t h e  aspect  of  and 6  fits  they  available to  order the  the of  some  was they is  used  the  then  including  frequencies  were  were made;  what we see h e r e . probably  the  we s h o u l d f i n d  D e n n i s o n . It that  in  in which  where no s u c h c o r r e c t i o n s  The a n h a r m o n i c c o r r e c t i o n s  frequencies  and so  with 6 equations  vibrational  contradiction  and q u a r t i c  2  Woodward  The  t o be f o u n d i n  s m a l l and p o o r l y d e t e r m i n e d c o r r o b o r a t i n g d a t a  Coriolis  are  " h a r m o n i z i n g " scheme co.uld not  second e q u a t i o n  anharmonic c o r r e c t i o n s  work  allowed  "harmonizing" procedure  their  the  the 8  even more so p e r h a p s b e c a u s e  to note  results  3  of  Duncan and M i l l s  this  cases  sort  - 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 indication  In  t h e A and E v i b r a t i o n s  6 independent p i e c e s of  set.  reason  from our  frequencies  constants  e a c h of  calculations  unknowns t h e  In  arises  The  set  the of  t o Duncan and vibrational  a few p e r c e n t ,  h a r m o n i z i n g method i s  but that  1 63  the  frequencies  product  rule.  implicit the  "fiddled"  Now t h e  the  consistent  since  particular  the  they  rule  invariance  is  field  field  to  the  may be i n c o r r e c t . data  poorly  produce a b e t t e r  over-all  vibrational  frequencies  fit  data  the  not  Finally  frequencies  experimental  is  are  constants,  then  introduce  harmonic  if  so what  data  set  are  field.  small  that  fit  the  this  Coriolis the  experimental  d e t e r m i n e d and  better!  i m p o r t a n c e of  relation  even t h o u g h the  unreasonable  will  the  force  frequencies  accordingly  should  it  calculations  should produce reasonable  and q u a r t i c  and q u a r t i c  an  obey  to produce a frequency  with a p a r t i c u l a r  coupling constants  Coriolis  is  with u n t i l  force  corrections  force  corrections  product  harmonic  h a r m o n i z i n g does  exactly Also  in  are  the  weighted  harmonized  - certainly This  the  discussion  "weighting"  - a problem c o n s i d e r e d in the  of  next  sect ion. If not  the  h a r m o n i z i n g method does not  work? One c a n c e r t a i n l y  approximation made but are  first  is  argue  not a p p l i c a b l e  there  is  c o n s i d e r i n g , even  if  relation  one u s e s  trace  estimate  t h e co's from t h e  v's  we use  frequency  if  £ constant  v.  If  the  in  In  it  diatomic be  t h e method we method  order  to  experimentally  the is  initial  "corrected"  $ constants.  b e g i n t o p r o d u c e r e a s o n a b l e to's  fundamental  the  why d o e s  argument w i l l  flaw  of M e a l and P o l o .  d e t e r m i n e d and a n h a r m o n i c  in a C o r i o l i s  the  and t h a t  a fundamental  w i t h the  inherent  that  work,  T h i s method w i l l  only  anharmonicity  less  than  anharmonicity  in a in a  $ were  1 64 greater give  than  in a  anharmonicity  experimentally orders  constants correct the  this  harmonized $ c o n s t a n t s .  relative  few  v then  of  it  is  It  the  is  frequencies  difficult  found t h a t  frequencies  are  d e t e r m i n e d than  s h o u l d seem u n r e a s o n a b l e  the  s t a n d a r d anharmonic  force  fields  evaluated  t h e method of D e n n i s o n s h o u l d be r e g a r d e d as If  one  is  corrections the  still  are  method A r n e t t  ethylene  in which  the  bond s t r e t c h  the  diatomic  satisfied. of  the  every  any a l t e r n a t e  include  the  "second" advantage  XT  of 3  of  calculations  all  harmonized using previous  study)  solutions?  One c o u l d  the a n h a r m o n i c i t y  assumed a l l  and v a r i e d  admit  this  the is  proposed  that  t o be made f o r  was  in  within  product  rule  was  arbitrary  and,  in  light  unsatisfactory.  d o e s manage t o a v o i d many o f  frequency  becomes a n o t h e r 3  use  for  frequency  the Dennison method. T h i s  u s i n g the X T  independently.  that  they  vibrational  equation  completes  (r128)  a solution  to  correction  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 ,  objections  Coriolis  and C r a w f o r d J r .  frequency  a  i n making a n h a r m o n i c  a p p r o x i m a t i o n so t h a t  is  usually  approximate.  interested  T h i s method t h e y  There the  (essentially only  there  from data  but  This  method due t o D e n n i s o n . We s h o u l d now s u g g e s t previous  to the  to attempt  using C o r i o l i s constants.  against  used  to estimate  v parameters  $ or  magnitude b e t t e r  and so i t  argument  in  method would be b e t t e r  data  is  product that  both the  Also a consistency  data.  of  it  is  simply  In  this  rule.  A  allows  E and A , treatment  to  case  the  further parallel  vibrations is  maintained  1 65 in  that  we o n l y  equation  (e6.3)  frequencies. corrections ideas.  roots.  A preliminary to  the  the  XT  several  data,  3  frequencies is  our  in  anharmonic  u s i n g the  errors  this  6.3 and 6 . 4 ,  same a c c u r a c y  discussed in  initial  the  as  the  XH  diagonal  section,  effects.  3  and X D  constants  (see  a p p r o x i m a t i o n may not  be  due  to shift  always the  earlier applicable  molecules.  d i s c u s s i o n we s h a l l  this  diatomic  re-write  approximation.  equation  (e6.3)  in  form v  i  where K.  is  the  = (1/2*)  force  r  K • -, 1 /2 1  constant  u' = u- (1 2 x l *i -  and  3  any  approximation; given  force  real  not  could  Finally,  a  result.  are  h e r e w o u l d mask  next  "diatomic"  the d i a t o m i c  above  s o l u t i o n of  rationalizing  6.2,  We s h o u l d now i n v e s t i g a t e this  t o h a r m o n i z e our  o u r e x a m p l e s , p r o d u c e d no  reasons  the  like  was made on t h e  t h e co's i n v o l v e d  "large"  the o f f  to ammonia-like  For  in order  frequencies  and mask a n h a r m o n i c  m a g n i t u d e of discussion)  equations  c o n t r i b u t i o n . Also small perturbations  resonance,  suspect  study  see T a b l e s  and c e r t a i n l y  anharmonic Fermi  of  equation which,  There are  rules  fundamental  known t o anywhere n e a r data,  a s s u m p t i o n s of  and p r o d u c t  The c a l c u l a t i o n  quadratic  First  use t h e  '  [6.6]  and  .co. ) i i  [6.7]  the  1 66  w  as  is  the  anharmonic  rK.ni/2 1  = (1/2TT)  i  diatomic  [6.8]  reduced mass.  corrections  are  "small"  The a s s u m p t i o n t h a t has a l s o  the  been made.  From  equations  (e6.6-e6.8) i t m i g h t seem r e a s o n a b l e t o c o n s i d e r isotope 1 ratios • „ . ^ — ~ . I f we make t h e a s s u m p t i o n t h a t t h e isotope 2 ^  the  force  constants  frequency  are  ratios  isotopically  depend o n l y  can be c a l c u l a t e d . sort rule.  is  that  This  from t h e  we  experimental  corrections. vibrations exact i s  implicitly  introduce  product  product  rule  frequencies.  approximation  on r a t i o s  begins  product  XH  rule  (u u,) 3  (co a>„ ) 3  From e q u a t i o n  3  to  rule  on t h e  In  other  fail  and X D .  at  of  Redlich  m  (8)  r  D  m  D  M'  turns  from which  the  of  this product  deviate  same o r d e r  as  the  the  level  of  the A,  and T e l l e r  c,  diatomic the  anharmonic  and  vibrations (r37  v  2  the  pg 2 3 2 ,  -,1/2  u  r78)  [6.9]  (M+3m )  H  the  to  these  (M+3m ) H'  above  out  words  the  For  3  H  of  the  an a p p r o x i m a t e  As an example we c o n s i d e r  for  then  A problem with c a l c u l a t i o n s  approximate  exact  invariant  D  "diatomic"  approximation product  rule  is (W CJ„) 3  (u ««) 3  We can now c o n s t r u c t  m  h  D  =  the  ^  D  (M+m ) H  (M+m )  [  D  following  table.  6  '  1  0  ]  167  NH /ND 3  Exact  Product  Rule  Experimental Diatomic  Product  From t h e above diatomic simply  the  validity  large  the  diatomic  for  1 .875  1 .939  1 .974  s h o u l d be a p p a r e n t  vibration  as  3  the over  equal  to  the  c o u l d p e r h a p s be made  approximation arsine  is  in  the case  we s h a l l  the  to  our  fundamental  The p r o b l e m w i t h ammonia w i l l  due t o  inversion,  plane  of  the  much more a n h a r m o n i c  of  conclude  not a p p l i c a b l e to  for  seems more  in general  corrections  through the  that  improvement  frequencies  approximation  partially  atom p a s s e s  no r e a l  instance  frequencies.  be seen as  the  1 .908  An argument  anharmonic  vibrational  2  1 .880  t h a n ammonia, but  p r o b l e m of  v  1 .851  the d i a t o m i c  reduced mass,  "diatomic" that  of  3  1 .962  fundamental  frequencies.  ASH /ASD  3  1 .917  numbers i t  the  3  1 .844  approximation o f f e r s  taking  harmonic  Rule  PH /PD  3  where t h e  hydrogens,  later  nitrogen making  than would o r d i n a r i l y  the be  case. The above  ratio  "spectroscopically"  method c a n defined  result  in  r e d u c e d masses  (e6.7)  used to d e f i n e  " s p e c t r o s c o p i c " masses. That  masses so t h a t  the  agrees  w i t h the  instance  m  Sverdlov  et  carry  over  Q  product  experimentally  equals al  "exact"  2.126  (r129)  these  "new"  rule  pg. 4 3 f f . ) .  one  frequency  determined  i n s t e a d of  is,  which  ratio.  masses t o d i f f e r e n t  is  defines  ratio For  2.014 a . m . u .  The hope  are  that  (see we can  problems.  This  1 68 procedure above.  is  equivalent  I n s t e a d of  equation  (e6.6)  isotopic  force  w o u l d be  in  Shimanouchi of  force  exact  of  et  (r77).  al  rule  also  In  s t e p towards  the  was  satisfied.  This  case  various  This  is  section  45ff.).  some of  To t h i s  His  the  on  the  problems  constants.  His  problems  also  of  the  order  the  in  terms  Nakata to  taken  other  r  more s a t i s f y i n g  et  root  al  (r53)  structures). than  the  first the  methods: 'amplitude  stretching  frequency  amplitude'  of  a  see  method assumes a  modes by the  anharmonic  (r130,  i n t r o d u c e d by  anharmonic'  'anharmonic  expressed  Curtis  w i t h b o t h b e n d i n g and  "mixing"  application is  the  therefore  the  i n t r o d u c e d two  anharmonic',  'conventional  phenomenological higher  Curtis  the  same as  (i.e.  has  the  similar  ratios  such t h a t  the  work of  Curtis  end he has  correction  (see  t h e method of  anharmonicity  accommodate  amplitudes  e6.7  defined  isotopes  b e i n g made on e a c h v i b r a t i o n  corrections  is  they  corrections  force  define  an e q u a t i o n  same s h o r t c o m i n g s  'conventional  diatomic-like  given  masses d i s c u s s e d above and  pg.  solving  anharmonic'.  masses,  approximation).  (r129)  above m e t h o d s .  case  their  we s h o u l d c o n s i d e r  Sverdlov  one c a l l e d  this  constants.  Before concluding t h i s corrections  method d i s c u s s e d  " n o r m a l " masses and  for  from t h e  the. d i a t o m i c  In  force  spectroscopic  should suffer of  constants.  diatomic  spectroscopic  one c o u l d use  terms  product  the  defining  constants  method of  to  off  to  diagonal  method  uses  frequency  mean for  square an  interesting  This approach  methods we have  of  described  169 previously  as a n h a r m o n i c c o n t r i b u t i o n s  from v a r i o u s  higher  order  studied systematically.  this  iterative  fit.  method h o w e v e r ,  in  At  it  some l e v e l  s i m p l y do a " c o m p l e t e "  study.  Even t h o u g h t h e methods of direction  to warrant  various new  that  anharmonic  fitting  whether  this  adopting his  would a l s o appear  can  of  Curtis  in order  the  force  error.  Finally  the  C u r t i s m e t h o d , as he  corrections potential studied,  is  his  used.  suggests  method p a r a m e t e r i z e s  the  demand t o o many  or  arise  just  fitting within  that  harmonic for  "error"  the  cases  more t h a n  he  the  corrections.  Where t h e n does t h i s  d i s c u s s i o n l e a v e us?  that  none of  anharmonicity  even  when " c o r r e c t e d " ,  molecules  for  anharmonicity  d e p e n d i n g on t h e  This  It  correctly  inconsistency  his  in  enough  study.  p r o b l e m would t h e n  one l a r g e  of  a step  large  present  to account  found t h a t  were d i f f e r e n t  function  anharmonic  there  least  field  represent  we were d e t e r m i n i n g s o m e t h i n g " r e a l "  the  be  one would do  would n e c e s s a r i l y  and t h e  be  sophistication  anharmonic  methods i n  can  from a  s t e p d o e s not a p p e a r  effects  parameters  potential  can be a r g u e d t h a t  to  right  frequencies  obtained empirically  better  the  the  the  These c o n t r i b u t i o n s  p a r a m e t e r i z e d and e a s i l y squares  terms  to  of  the  this  previous are  study.  applicable  Furthermore  anharmonic  contributions correctly  anharmonic  force  field  study  much more c o m p l i c a t e d t h a n  It  would  correction  to the  t o do  requires  t o be d o n e ; w h i c h  u s i n g an e l a b o r a t e  schemes,  ammonia-like  in order  really  seem  is  the an  not  that  approximation  1 70 scheme,  in the  spirit  of  Curtis'  need t o  be d e v e l o p e d .  It  would seem t h e n  if  one wants t o  better  do an a n h a r m o n i c  obvious  vibrational  making of  for  instance, too.  the v i b r a t i o n a l data  anharmonic  was  larger  if to  corrections constants  the  have  is,  had  rather  to  other  relevant distortion  anharmonic  arbitrary  to  anharmonicity  correct  only  and l e a v e  a r g u m e n t s c o u l d be made  it  a general that  the  thesis.  have been made and t h e  would  the that  frequencies  seem  rule.  s c o p e of  this  This  and q u a r t i c  parameters  beyond the  one  the v i b r a t i o n a l  s h o u l d now be a p p a r e n t  in  study.  being,  for  Even  in other  study p r e s e n t e d  o u r answer  corrections  seems r a t h e r  contribution  are  that  still  t h e most damning c o n c e r n  should a l l  t o make t h i s  anharmonicity  to  that  frequencies  than  unreasonable It  It  uncorrected.  the  field  field  anharmonic  frequencies,  contributions  other  force  C o r i o l i s coupling constants  constants  would  anharmonic c o r r e c t i o n s  s o l u t i o n addresses perhaps  r e g a r d i n g the  data,  include  method, that  effects  the harmonic  force  No a n h a r m o n i c i t y  final  c a n be e x p e c t e d t o c o n t a i n  of  "harmonic"  force  small c o n t r i b u t i o n s  due  anharmonicity. Finally,  we c a n n o t  say  after  this  about  anharmonic  there anything data  we c a n  and q u a r t i c  reasonable  say?  force  lengthy  constant  field  i n c l u d i n g the  d i s c u s s i o n on what  frequency  Perhaps.  distortion  harmonic  be done w i t h o u t  rather  corrections,  Enough C o r i o l i s data  studies  exist  (see  fundamental  is  coupling  for  section  vibration  6.3)  tc  171 frequencies estimate  in  the  quartic  the  data  in  quartic  alone.  calculation  would e x p e c t  of  If  in  this  this  data  of  peculiarity: all  higher  than  was  and q u a r t i c  v. 3  than  In  all  for  the  series  as  cases  XH , 3  the XD  for  predicted  symmetric  same a s  for  probably  the p r e d i c t e d  3  the  the  effects  is  from t h e 1  of  the  The.  small. Coriolis  the  actual  prediction  bending  significant for  ammonia were  whereas  for  A surprising from t h e  Coriolis vibration  was much  differences  for  the  larger  scaled  r e d u c e d mass f o r  The d i f f e r e n c e s  in  suggests  stretching  frequency  this  calculated  stretch  not  lower.  and t h e of  In  frequencies.  ±30 c m "  the a n t i s y m m e t r i c  and X T .  and  vibrational  b e n d i n g modes  were a l l  root  these  the  harmonic  the v a l u e s p r e d i c t e d  square 3  of  frequencies  the measured f r e q u e n c y  roughly  the C o r i o l i s  the measured f r e q u e n c i e s  data  prove  bending frequencies  predicted  found in  might  then these  the  in the  they  it  and t h a t  t o be w i t h i n  following  p h o s p h i n e and a r s i n e result  case  were a l l  The e r r o r  the  small  i n the  alone  w i t h the  but  be u s e d  non-harmonic e f f e c t s  represent  v-i was p r o p o r t i o n a l l y  vibrations  the  were t h e  The b e n d i n g f r e q u e n c i e s  and  be t h e most a n h a r m o n i c .  are  calculation  to  method c o u l d not  the c a l c u l a t i o n s  anharmonicity  measurements.  to  data  might b e t t e r  and q u a r t i c  this  us  from C o r i o l i s  frequencies  hoped t h a t  cancel  frequencies  the  is  and q u a r t i c  frequencies.  that  Certainly  anharmonic c o r r e c t i o n  it  judiciously  result  frequencies  i n d i c a t i n g which  data  Coriolis  This procedure allows  vibrational  a s a method of useful  fits.  the  series  for  1 72 X=N were 2 3 6 , 289,  191,  145,  157 c m  - 1  113 c m " .  be  in character  to  effects  modes and most  2.  1  frequencies  can p r o b a b l y ,  i g n o r e d . Furthermore  anharmonic  4 8 9 , 3 3 0 , 279 c m " ; X=As,  The s u g g e s t i o n h e r e m i g h t  anharmonic c o r r e c t i o n s bending  ; X=P,  1  are  these more  important  are  to. a f i r s t  results  important  if  that  be  might  primarily approximation,  suggest  i n bond  t h o s e modes a r e  that  stretching  antisymmetric.  FURTHER CONSIDERATIONS WHEN MAKING A HARMONIC  FORCE  F I E L D CALCULATION It field  is  alway.s d e s i r a b l e  calculation  may " p e r t u r b "  t o be aware of  empirical  un-representative q u e s t i o n of section  is  specific  through the Fermi  symmetry  More  constants.  Certainly  one  we s h a l l  due t o F e r m i  the  degenerate"  overtone  borrows  for  levels  force  that  the  same e n e r g y , In  from t h e  calculations the  added  t h e X atom  of  pg 2 1 5 , 2 1 6 ) .  field in  the  and t h e  in pyramidal XY  intensity  have  are  Two more  molecules.  3  when two v i b r a t i o n a l  the  (r37  in  general  effects.  of  that  previous  resonance  inversion  Y atoms  have n e a r l y  the  the  force  effects  result  now c o n s i d e r  f u n d a m e n t a l and one o v e r t o n e ,  importantly  vibrational  extra-harmonic  r e s o n a n c e may o c c u r  species  weaker  of  can  as d i s c u s s e d i n  i n t r o d u c e d by t h e  plane  "accidentally the  that  one example of  perturbations  extra-harmonic  parameters  anharmonicity  anharmonicity  usually  force  examples t h a t  frequency  when making any h a r m o n i c  levels,  same  that such  is  are  cases  fundamental. these  zero approximation  1 73 nearly  the  with the level  same e n e r g y  greater  w i t h the  Table  lower  s u g g e s t i n g the  Except  ammonia we n o t i c e s p e c i e s XY this  e n e r g y moves t o  fundamental v i b r a t i o n s 6.1.  all  3  scaling  each other  e n e r g y moves t o h i g h e r  As e v i d e n c e on o u r  "repel"  for  the  that  scale  factor  v  the  the  least  the  resonances present  in  molecular Furthermore  approximately,  instance  species according to equation  3  Fermi  especially  for  at  For  with a diatomic approximation. given XY  of  same f a c t o r .  consistent,  level  frequency.  i n d e e d " s m a l l " , we  the v i b r a t i o n s by a b o u t  is  lower  vibrations  2  the  f r e q u e n c y and  effects  are  so t h a t  v W v  (e6.6)  for  2  a  s h o u l d be  1 /2 given the  r o u g h l y by t h e  valence  ratio  of  (f^/f^)  where  7  bond s t r e t c h i n g and b e n d i n g f o r c e  r e s p e c t i v e l y , see T a b l e 6 . 6 . A l s o - / \ > f ° X Y and X Y , s i m i l a r l y s h o u l d be r o u g h l y t h e v  3  x  v  about  r  t  large  individual various  all  r o u g h l y the  Fermi  resonance  s h o u l d be l e s s frequency.  o  .  v  2  for  s u g g e s t any F e r m i than one p e r c e n t  the  worry  perturbing  resonance of  The  the  same s u g g e s t s we n e e d not interactions  are  species of t h e  ( n ' / u )  fundamentals. A l s o the d e v i a t i o n s  ratios  w  ratio  1 /2  s q u a r e r o o t of t h e d i a t o m i c r e d u c e d masses f a c t that these frequency r a t i o s (excepting are  and f^  constants  3  present)  f^  of  these  frequency  vibrational  shift  1 74 Table Various  frequency  ratios  of  6.1  ammonia, p h o s p h i n e , a r s i n e  and  st i b i n e NH  ND  3  NT  3  3  PH  PD  3  PT  3  3  AsH  3  AsD  3  AsT  SbH  3  3  SbD  3  iii  3.51  3.24  3.07  2.34  2.31  2.24  2. 33 2.31  2. 27  2.42  2.42  iii.  2.12  2.15  2.19  2.08  2.11  2.10  2. 1 32.14  2. 13 2.28  2.30  vn  rf  r  i  1/2  3.42  2.18  2.06  f v  v /v 2  z  / b / a^1/2 (y /M )  n/vu  NHf/NDa  1 .38  1 .27  1 . 34  1 .37  1 .37  NH3/NT3  1 .66  1 .45  1 .58  1 .63  1 .63  ND3/NT3  1 .20  1.14  1.17  1.19  1.19  PH3/PD3  1 .38  1 .36  1 .37  1 .39  1 .39  PH3/PT3  1 .66  1 .59  1 .66  1 .67  1 .68  PD3/PT3  1 .20  1.17  1.21  1 .20  1.21  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 / A s T  1.21  1.19  1 .22  1.21  1 .22  AsH /AsD 3  3  3  3  Recall:/2=  1.414  /3=  1.732  v/3/v/2= 1.22 5 NOTE: are  The v a l u e s  from T a b l e  of  6.6  the  valence  force  constants  f  f  and f  1 75 A rather frequency similar this  ratio  ratios  c a n be  involving involving  a symmetric  t h e m o t i o n of  similar  to  feature  of  is  v  2  other  Table  the  frequencies.  vibration  effective  atoms the  (XY )  and i n v e r s i o n  3  energy  below  the  barrier  of  t h e o =1  effective  the  weight  effects,  to decrease the  hydrogen  isotopically  point  in  inversion  is  vibration  t h a n one m i g h t  frequency  will  that is  isotopes  v  2  sit  3  during is  lowered.  times during  the  plane  of  Now r e c a l l our X Y  the Y  that  XD ,  is  Lowering  be " c l o s e r " of  the  species  3  222f).  the  to  the the  lower  potential  s h o u l d be a t from  an  This  even  general  Furthermore  wells,  since  because  the anharmonic 3  the  the  p r o p o r t i o n a l l y higher  potential  order X H ,  vibration  2  much more a n h a r m o n i c .  expect  one s h o u l d e x p e c t  barrier  the  considerations.  invariant  the  to  is  has v  that  The r e s u l t  measured f r e q u e n c y  frequency  anharmonic lower  the  barrier.  barrier  this  (r79)  barrier  H e r z b e r g r37 page  the  vibration  is  for a l l  2  inversion  means t h a t  and  certain  t h e u> v i b r a t i o n  t o p of  experienced for  lower  (see  means t h a t  "anharmonic"  vibration  be e a s i e r .  vibration  2  barrier  breathing,  be c l o s e r  will  for is  inversion  t h e X atom w i l l  2  A reason  from  vibration  "somehow"  T h i s c a n be v i s u a l i z e d by n o t i n g a t  any  different  i n v e r s i o n makes t h e  A possible  u> v i b r a t i o n  that  2  m o t i o n . Dennison  more a n h a r m o n i c . 2  is  t h e co  t h e atoms d u r i n g t h i s  somehow t h e  a> t h e  that  b e n d i n g , or  inversion  suggested that  6.1  significantly  f o u n d when one r e c a l l s  essentially that  striking  XT . 3  Finally  t o b e g i n w i t h the more  of  in  zero  corrections the  lower  noticeable  1 76  these  perturbations  biggest  It  is  not  of  Table  from t h e expect also  scaling  ratios  be n e c e s s a r y .  constants  of  from d a t a  the  other  is  for  the  by  certainly  for  "inversion"  unreliable  v =1  arsine,  the  studies,  on t h i s  primary  where t h e  constants  it  is  the  f^  should s t i l l  problem w i l l  these smaller  be made i n  the  lead  that  next  of the  the  that of  and  will ignoring  to  p h o s p h i n e and  harmonic  be r e p r e s e n t a t i v e .  for  constants  Whereas  i n ammonia m i g h t  are  thesis  T h i s means  constant.  for  these  calculated  methods; t h i s  hoped t h a t  to  to  w i t h each other  initio"  c o n c e r n s of effects  here  should  constants  this  force  for  estimated  "correct"  vibration.  2  "ab  effects  results,  is  constants  fields  i n comparing the  with those c a l c u l a t e d  these  force  has been made t o  group V h y d r i d e s c a l c u l a t e d  case  by  the  b u t one  the q u a r t i c  p u r p o s e s of  be on h a r m o n i c  the  of  how c o r r e c t i o n s  especially  So,  some c a r e must be t a k e n  distortion  not c l e a r  where no e f f o r t  be t h e  frequencies  It  in  lowering 2  the q u a r t i c  added a n h a r m o n i c i t y  This  a q u a s i u> c o u l d be  to  tops.  main e m p h a s i s w i l l  the  due t o  c o u l d be made,  asymmetric  be i n ammonia f o l l o w e d  o b v i o u s how one c o u l d c o r r e c t  Certainly  corrections  the  6.1.  immediately  barrier.  to  i n p h o s p h i n e and a r s i n e .  added a n h a r m o n i c i t y  inversion  the  corrections  experience  this  be and hence we s h o u l d e x p e c t  anharmonic c o r r e c t i o n s  much s m a l l e r the  will  the  force final  Further section.  field force  comments  177 3.  HARMONIC FORCE F I E L D REFINEMENTS: AMMONIA,  PHOSPHINE  AND ARSINE The h i g h e s t derivatives  symmetry  (excepting  available)  is  coordinates  Cs.  In  may be A'  XHTD f o r  this  taken  Block:  S 5 5  Block:  S S  where t h e change  change  which  case  the  following  2  set  yet of  symmetry  3  = l(A/3,+A/3 +A^ )  2  7  =  2  76"  (  2  A  r  i "  A  3  2 " ^ 3  r  )  =- l(2A/3,-A/3 -A/3 )  4  7  2  3  =^(Ar -Ar )  g  2  3  ^ItA^-A^)  6  = Ar(X-Y^),  and  A/3. = A/3(Y .-X-Y, ) , w i t h X t h e ]  H,  symmetry  Cs g r o u p . F o r  the  D or  T.  the heavy  K  atom and Y r e p r e s e n t i n g in  is  isotopic  = ^ ( A r T+Ar +Ar )  1  1  species  various  no d a t a  i n a bond l e n g t h Ar^  i n a bond a n g l e  the  9  3  5 A"  common t o a l l  A'  and A" the  represent  symmetric  top  case  t h e symmetry d e s i g n a t i o n becomes 2E + 2A, and S^ above becomes S-, , S.- becomes S , S. becomes S. and S becomes 3a b 3D 4 4a 6 o u  4b*  S  F  o  ^  r  coordinates S-.  9  ,  S^,  e  s  y  m  for  and S.  The d e f i n i t i o n  matrix the  m  e  t  r  ^  the , S.,  of  c  A,  tops S  and S  1  totally are  the  symmetry  w h i c h when m u l t i p l i e d  "symmetrized" B matrix  (r43)).  r  of  2  are  symmetric symmetry  the  species  and  coordinates  coordinates  defines  by t h e W i l s o n equation  symmetry  used  also  B matrix  e1.45  (see  the to  the  gives  Woodward  U  1 78 represent  the  Each weighted  doubly degenerate  input as  of  following  the  inverse  effects.  1% t o  = 1%.  scheme  the  that was  of  determined;  anharmonicity  other  The d a t a are  presented  calculated  used  in  fit"  fits.  6.3  the  All  The  fits  we  constants quartic  weighting  that  is  input  v,  data  for were made.  various  and 6.4  Here  uncertainties  parameter,  the  constructing  values.  The  arbitrary  no c o r r e c t i o n s  6.2,  (r8l).  The z e t a  non-harmonic e f f e c t s  in Tables  "best  t y p e of  for  were c h o s e n  ±0.02.  somewhat  The  to account  been a s s i g n e d  "polarize"  were e m p i r i c a l l y or  ±0.01 to  this  no p a r t i c u l a r  found to  an a t t e m p t  frequencies.  of  and  uncertainty.  A l d o u s and M i l l s  have a l s o  The a d v a n t a g e  is  o r D,  constants  the  The u n c e r t a i n t i e s  assigned uncertainties  distortion of  of  of  are  s u g g e s t i o n s of  a s s i g n an e r r o r are  square  uncertainties  extra-harmonic  species.  datum was a s s i g n e d an u n c e r t a i n t y  the  assignments  E  force  along with  fields the  were done u s i n g  the  iterative  approach reviewed  by A l d o u s and M i l l s  (r8l)  and  discussed  in d e t a i l  ( r 1 1 9 ) . The d e r i v e d  force  fields  in  the  6.5. Table  internal  The more 6.6.  by Gans  symmetry  intuitive  (r43  pg  193)  coordinate valence  basis  force  are  fields  given are  in  given  Table in  1 79 Table Harmonic NH  Obs V 1  v  2  "3  D  JK  D  K  $3  S«  study ND  Obs  Ref  O  of  ammonia NT  3  3  -c  Ref  1 4  131 2014  6  1 32  657  25  1 33  2  1 32  o-c  Obs  3337  -74  131 2421  950  -7  748  3444  -53  2564  1 0  "  2185  -7  1191  5  "  1000  5.91  0.01  135  2.598  -0.072  -10.49  0.25  "  -4.472  0.15  2 5 . 21  J  field  3  O-C  1 627  D  force  6.2  0.18  134  - 4 6 . 61 - 0 . 4 4 2 6 . 93  0.02  0.042  0.003  131  -0.255  0.002  "  20  -6.16*'"  1 1  -2.59  0.133  0.009  131  -0.327  0.003  137  -0.012  ND H 2  o-c  Obs  o-c  0.11  10.03  -0.04  -0.43  -14.79  -0.40  1 0.95  0.25  6.49  0.34  d,  4.181  0.77  3.38  -0.06  d  0.1324  0.0003  -1 .59  0.03  1 5.82  J  D  JK  D  K  -23.95  2  Asymmetric  top data  f  force  Units:  Harmonic jv,cm  _ 1  ,  D's  from  field  MHz and  ref(r!05)  value $'s  no  units.  1 36  t  -0.383  2  D  98  0.228^  N H; D  Obs  Ref  1 33  Harmonic PH  force  Table  6.3  field  study PD  3  Obs  o-c  Ref  2321  -28  1 1 21 682  992  -4  "3  2327  -23  "»  1118  -3  phosphine PT  3  O-C  Ref  O-C  1398  7  2  623  13  1 1 21 693  4  1401  -1  1 1 3 803  4  668  10  -0.3  728  115  3.94  -0.03  1 02 1 .020  -0.026  0.491^  -5.18  -0.06  -1.312  0.008  -0.611^  4.13  -0.04  1 .027  -0.002  0.457* *  $3  0.017  -0.007  1 1 2 0.056  -0.009  115  0. 103 *  $4  -0.442  -0.013  1 1 3 -0.463  -0.012  "  0.495* *  $24  0.526  0.004  5,3  0.004 t  D  J  D  JK  D  K  ?i  ?i  1  0.501 *  1  1  0.03  0.062*^*  PD H  2  Obs  D  JK  D  K  d, d  2  o-c  2.72*1*  J  Obs 1 .62* " 1  0.06  -0.794  0.04  1 .874  0.06  -0.353  -0.06  -0.227  0.006  -0.233  0.003  0.449  0.002  -0.326  0.004  P D , P D H and PDH quartic constants of t h i s work. t Harmonic f o r c e f i e l d v a l u e U n i t s : i > , c n r , D ' s MHz and $'s no u n i t s . 3  o-c  -3.476  2  part  1  0.512* * 0.062  2  1  Ref  1  PH D  D  3  Obs  1 13  Obs  of  2  were  o b t a i n e d as  98 II  H  H  181 Table Harmonic AsH Obs V 1  v  2  3  D  JK  D  K  $3  > i  2«  AsD Ref  Obs  of  arsine AsT  3  O -c  Ref 97  -24  92  1 523  2  907  -6  95  660  6  -23  92  1 529  -1  IT  -7  95  714  -1  II  -0.01  93  0.742  - o . 01 1  2.930  J  study  2115  999  D  field  3  o-c  21 26  v  force  6.4  -3.724  -0.026  3.357  -0.004  -0.013  -0.002  92  -0.446  -0.03  95  0.502  -0.009  it  -0.940  it  II  0. 008  J  A  JK  A  K  6  J  5  K  1 256  6  553  1 2  590  4  0.369* 0. 008  96  0.049*  0.485*  0.493*  0.507*  0.504*  AsD H 2  Obs  o-c  Obs  o--c  1 .236  -0.02  1.131  -0 .02 0 .007  0.002  -1 .876  -2.630  -0.005  2. 173  0. 126  0.004  -0.407  -o .003  1.994  0.03  0.980  0 .008  A s D , A s D H and A s D H q u a r t i c c o n s t a n t s a s p a r t of t h i s w o r k . t Harmonic f o r c e f i e l d v a l u e U n i t s : C j C i n " , D ' s MHz and $'s no u n i t s . 2  1  98  -1  1 258  2.230  3  Ref  -0.423*  AsH  A  o-c  0.344*  0.833* -0.04  Obs  3  2  -0 .02  were  obtained  it  n  it  182 Table Force  constants NH  F  (internal  symmetry  PH  3  coordinates) AsH  3  3  6 .732(106)  3. 194(20)  2 . 682(17)  mdyn  0 .560(30)  0. 156(10)  0.1 14(8)  2 2  mdyn A  0 .439(8)  0. 594(4)  0.575(3)  3 3  mdyn A"  6 .671(104)  3. 165(18)  2.705(19)  -0 . 1 6 1 ( 2 6 )  - 0 . 036(10)  -0.005(10)  0 .636(3)  0. 7 1 0 ( 4 )  1  F  mdyn A"  6.5  F «  mdyn  F  mdyn A  3  n n  1  1  Table  6.6  Force constants NH  PH  3  0.699(3)  (valence) AsH  3  3  f  mdyn A"  1  3 . 175(31 )  2.697(31)  f  r , mdyn Arr  6 .691(250)  1  0 .020(70)  0.010(13)  -0 . 0 0 8 ( 1 2 )  0 .571(14)  0.671(7)  0.638(5)  -0 . 0 3 9 ( 3 )  -0.031(2)  h  mdyn A  <3<r mdyn f  ' r f  -0 . 0 6 6 ( 3 )  A  mdyn  0 .240(19)  0 .064(7)  0.040(6)  mdyn  0 .080(70)  0 .028(17)  0.035(16)  G i v e n : F , 1=f + 2f , r rr F F  F  F  =f  2  2  3  «=f r / 3 ' ~ r/3 f  3 3  =  f  r/3  ,+2t  r/3  = f - f rr , r  F (, « = f Qf  P0'  F o r T a b l e s 6.5 and 6.6 t h e numbers i n p a r e n t h e s e s a r e t h e s t a n d a r d d e v i a t i o n s of t h e l e a s t s q u a r e s f i t s i n u n i t s o f t h e l e a s t s i g n i f i c a n t f i g u r e s . See t h e t e x t f o r a d i s c u s s i o n of t h e w e i g h t i n g o f t h e v a r i o u s i n p u t d a t a .  183 When c o m p a r i n g t h e various the  hydrides  general  trend  much h i g h e r relatively  of  than only  valence  Table  is  the  those  6.6  constants  we see t h a t ,  potential  of  slightly  force  constants  not  as  ionization potentials  also  show t h i s  P,  As a r e  9.81  volts  of  14.53,  also  10.484,  a similar  CH„ , S i H „  and GeH„  The e x c e p t i o n t o ammonia. that the  ammonia i s  harmonic  "spring"  of  symmetric gave  the  v  vibration  This extra inversion  force  these  m o t i o n of  group V h y d r i d e  force  "first  lowers  order"  simplistic,  constant  is  directly  v  ~ the  Table  then  the  f^  if  it  it  the  6.1.  In  ratios  to  last  that  section  were due t o  the  expected.  traced  we c a n e x p l a i n to  constant.  were c o r r e c t  the  to  the  from c o m p a r i s o n s of  recourse  this  appear  o p e n i n g and  the  tentatively  potential  for  parameter,  than o t h e r w i s e  that  f^  vibration  2  now would be t h a t  that  N,  parameters  would  related  same v  anomalous  without  to connote  because  for  is  constants,  explanation  for  (r73)).  is  s m a l l . The f^  ammonia. We see now,  a n o m a l o u s ammonia r e s u l t s order"  was  such  (i.P.'s  respectively  constants  b e i n g more a n h a r m o n i c  anharmonicity  effectively  of  This  parameters  trend  are  are  arsine.  generalization  force  2  f^,  3.  unusually  anomalous r a t i o s  suggested that  A."first  above  bending v i b r a t i o n  was  2  the  turn  of  related  for  ammonia  found in v a r i o u s  in Chapter  t h e H-N-H a n g l e ,  it  other  t r e n d was  B a s e d on t h e o t h e r  £p f o r  closing  as  of  in  those  is  Recall  a surprising result  than  the  except  p h o s p h i n e which higher  of  the  anharmonicity. inversion The  explanation  term is  too  we s h o u l d be a b l e  to  184 f i n d a f^  that  would p r o d u c e r e a s o n a b l e  found in Table vibrational constants F  2 2  .  It  the  ammonia v a l u e s  found t h a t  produced frequency the  ammonia f o r c e  required  (r138).  people c a l c u l a t i n g as  these  4.  section  rotational  field This  to  2 2  the  force  but  varying  0 . 8 - 1 . 0 mdyn A ,  ratios.  and t h e  Clearly  implications  should c e r t a i n l y  force  1.4  constants include  the  effects  b a s e d on more  of  be of  using "ab  part  we d e f i n e d t h e  the  sum of  two  of  equation e1.33. ^  interest  Here  (e1.50)  a  top  r  is  xxxz  and e q u a t i o n necessity  by A l i e v  and Watson  in Table  calculated  It  xxxz  by  6.7  so-called  part.  of  calculated  1.49.  (r139).  u s i n g the ^  Analytical  a force  from  field  forms  below  I = 2 B ( 2 - B /B ) x x' z 3  1  small for  analysis  Estimates  of  r  are  r  "a" xxxz  are 2  a l o n g w i t h an u p p e r bound  (r140)  w o u l d seem t h a t  "true"  methods  distortion  A. A A  T  to  inversion.  c o n t r i b u t i o n can be e s t i m a t e d  bypass the  given  work  inversion  initio"  symmetric  terms,  and v i b r a t i o n a l  The r o t a t i o n a l  given  6.5  sort  of  approaching expectations  and s t i b i n e  methods do n o t  d i p o l e moment as  that  Table  the  ESTIMATION OF THE DISTORTION DIPOLE MOMENT  In  matrix  of  of  predictions  increasing F  ratios  phosphine, arsine  on t h e is  As an e x e r c i s e ,  f r e q u e n c i e s were made f i x i n g a l l  to  was  6.1.  ratios  value quite  the  1/2  ZP~.  min  CET  1  CJI_ ..TIT  max  upper bound e s t i m a t e  well.  [6.12]  approximates  the  1 85  Table  6.7  Estimation  (MHz) of r xxxz  | T  NH  3  ND PH PD  In the  0.78  0.83  2.29  2.45  0.59  0.63  determinable  the  obtained  as  of  these  from q u a l i t a t i v e  absolute  intensity  compounds a r e  but  to  be an  assumption in  light  of  is  of  (r141) .  The  are  infra-red  the  from t h e  have  6.8  not  are  fully isotopic  (rl29,r143) r  | 3 M / 9 Q | -,  [6.13]  "1 of  isotopic  arsine  are  derivative  estimated of  e x p e c t e d t o be r e a s o n a b l e  the  3  9M/9Q'S in T a b l e  9M/9Q'S for  H  arsine  intensity  and P H  3  to  where  recorded spectra  calculated  rule  derivatives  NH  examination  the  The  |9M/9Q|  The d i p o l e  9M /9Q-, ,  from i n f r a - r e d  (r142)  values.  invariance  contribution  d i p o l e moment d e r i v a t i v e s  perturbations  deuterated  of  have been m e a s u r e d f o r  been p u b l i s h e d . As a r e s u l t given  vibrational  moment we need v a l u e s  signs  intensity  3.11  to c a l c u l a t e  measurements, relative  2.92  3  These v a l u e s ,  Bound  2.76  3  distortion  1=3,4.  Upper  |  Z  2.72  3  order  X  11.37  3  AsD  X  11.26  3  AsH  X  similarity  of  by a s s u m i n g ,  phosphine. (say  This  to w i t h i n  p h o s p h i n e and a r s i n e  25%)  (example:  186 force  constants,  structural  parameters....).  Table I SM/BQ-L |  6.8 (amu  A)  1=3 NH ND  3  3  PH  3  PD  3  AsH  3  AsD  3  The v i b r a t i o n a l estimated ©  . are rot  given  0.5402  0.1367  0.3951  0.8304  0.5209  0.6043  0.3741  0.7589  0.4655  0.5457  0.3326  contribution  in Table  c o r r e s p o n d t o whether signs  0.1857  using equation 3  1=4  e1.35. 6.9.  the  to  the  Values  dipole  of  ©  The two v a l u e s  3M/9Q'S  have t h e  V  I  moment along  B  of  was  with  © „ VIB X T T  same or  opposite  respectively. Table  Estimation  of  the to  6.9  rotational the  0  ND  3  3  PH PD  3  3  AsH AsD  3  3  5  Debye)  and v i b r a t i o n a l  distortion I ROTI  NH  (X10  dipole  contributions  moment I VIBI 0  7.48  5.35,4.71  3.32  2.90,2.59  5.02  3.51,2.02  2.04  1.89,1.12  3.31  2.94,1.74  1.35  1.53,0.91  1 87 The v a r i o u s used  in  these  moments of  calculations  equilibrium structures these  inertial  v,  of  inertia  were c a l c u l a t e d Table  parameters  and r o t a t i o n a l  6.13.  a l o n g w i t h the  3  PH =0.55 D ( r 9 9 ) , M 3  AsH =0.22 D (r99)  z  quantities. percent.  The e r r o r  This  " a " matrix  usually  within  derivative  authors the  a's  of  quadratic  (r141)  t o be a b o u t  uncertainties  optimistically  of  the  on t h e  arsine of  is  than  a  products  10 p e r c e n t .  of  of  5 and  the  in  our  the  "isotopic" is  q u o t e d by  hard  V  I  B  's  to  the  s h o u l d mean  and 0  T  few  constants  latter  This  0 Q 'S R  an  the  3M/9Q'S i s  various  order  other  s h o u l d come f r o m  p h o s p h i n e . The e f f e c t the  are  The main u n c e r t a i n t i e s  assumption that  of  the  in  M,  "a's"  reproduce d i s t o r t i o n  2 percent.  The a c c u r a c y  ,r145),  s h o u l d be l e s s  d i p o l e moment e s t i m a t e s  3 M / 9 Q ' S and t h e  quantify.  the  elements  1 or  1 0  when c o m p a r e d t o  seems r e a s o n a b l e • a s  various  distortion  in  of  uncertainties  ) and t h e  3  e x p e c t e d t o be n e g l i g i b l e  the  The u n c e r t a i n t i e s  and M , ( M „ N H = 1 . 4 7 D ( r 1 0 5 , r 1 4 4  3  using  constants  that  is  13 p e r c e n t  respectively. The o n l y is  in  (r39)  the  1 0  In  relative  have  ammonia  two  signs  of  the  distortion  the  t e r m s have t h e  r144 t h e  contribution subsequent  in  to  authors  the  estimates  of  r  they  estimates  infer  same s i g n and t h e  ignore  distortion  dipole  two c o n t r i b u t i o n s .  i n t r o d u c e d a model from w h i c h  the  ref  ambiguity  the  incorrect.  al  that  in  effects  of  vibrational  d i p o l e moment and are  Oka e t  their  188 rotational other.  The c o n t e n t i o n  rotational moment  is  (r82).  of the  |0  that  T 7 T T  J  c o n t r i b u t i o n s to give  s t r e n g t h of  known l i n e  suggest  that  one a n o t h e r .  i n the A s D In  this  3  case  7.2  of  distortion of  3  of  case  and dipole  the  PH  t o a sum of  |©  R O  estimate  compared w i t h  two c o n t r i b u t i o n s  enhance  a d i s t o r t i o n d i p o l e moment  our  TI  germane s p e c t r u m a l s o  the  of  3  5  absorption lines the  each  x 10 " D made by Chu and Oka  be a t t r i b u t e d  difference  sensitivity  final  Furthermore a s u b j e c t i v e  the AsD  strengths  c a l c u l a t e d as t h e the  4.9.  the  experimental estimate  d i p o l e moment of  in Table  b o r r o w i n g enhance  we can add t h e v i b r a t i o n a l  T h i s v a l u e can o n l y  the  below  intensity  s u p p o r t e d by t h e  distortion  and  and v i b r a t i o n a l  the  two e f f e c t s  spectrometer.  w o u l d be  189 •5.  STRUCTURE OF AMMONIA, PHOSPHINE AND ARSINE  The r e s u l t s are  given  are  the  of a z e r o p o i n t  in Tables 6.10,  /3 i s The r  From t h e  the  B  rather  <XMX where M = N,  empirical For  easy.  For  "r"  symmetric  equation e1.49.  evaluated  For  the  structures  B  zero point  3  case  B  constants  0  the  there  to  the  calculation  a constants  section  the  using  then the  (e1.58).  iterative tabulated B  were not enough r o t a t i o n a l  from a l i n e a r  The  fit  of  r  z  the  values.  z  constants  e x t r a p o l a t i o n of  T h i s bond a n g l e was t h e n u s e d w i t h t h e constant  in  the c a l c u l a t i o n  Equilibrium structures diatomic  using  c o n s t a n t s were values  the  effective  were c a l c u l a t e d  z  was  Script  Kivelson-Wilson  B  equal  b o t h t h e bond a n g l e and bond l e n g t h . The bond  a n g l e was e s t i m a t e d  rotational  way. were  and t h e n c o r r e c t e d f o r  were d e t e r m i n e d by a d i r e c t  to determine  (r54).  of  The  following  tops Watson's  average  T  to hydrogen,  i n the  cases  the  r  As and X = H, D and T .  rotational  0  and  D  empirical constants  The " r ' s "  g e o m e t r i c a l parameters  the NT  r  H  a more c o m p l e t e d i s c u s s i o n see  c a l c u l a t e d harmonic p a r t s  various  these  the asymmetric  These ground s t a t e  converted to  In  tops the  (e1.52,e1.53).  There r ,  constants,  c o n t r i b u t i o n to give  parameters  P,  determination  respectively.  were e s t i m a t e d  rotational  C o n s t a n t s were f i r s t  1.6.  bond l e n g t h s  c o n s t a n t s and so f o r  0  small  r  structures  z  calculated. the  and 6 . 1 2 .  n i t r o g e n , p h o s p h o r u s or a r s e n i c  d e u t e r i u m and t r i t i u m angle  6.11  structural  approximation  (see  of  0  and /3  H  lone NT  D  3  r . T  were e s t i m a t e d u s i n g  the  e q u a t i o n e 1 . 5 9 and T a b l e  6.13).  1 90 Since  the c o r r e c t i o n s  equilibrium changes  structure  i n the r  give  case  mainly  Although t h i s z  with c a u t i o n .  where t h e a n h a r m o n i c cubic  situation  ' s with d i f f e r e n t  the  linear  Y are nearly J  extrapolation  a r e on t h e o r d e r  of  (e1.59)  are taken  of t h e r e l i a b i l i t y  p h o s p h i n e and a r s i n e  ammonia r tritiated  z  m  pair  where t h e l i n e a r  to  length  (r49)  i s more ±0.05  °.  of the d i a t o m i c  of  realistic. As an  approximation of  of  1.013,  calculated  closer  combinations of  information  from the  F o r t h e c o m b i n a t i o n s r^r^,  is probably  systems v i b r a t e  the  an a c c u r a c y  from t h e p o s s i b l e p a i r  r_.r one h a s r values D T e  structure  since  we c a n compare t h e e q u i l i b r i u m  ammonia d a t a .  g  (r54).  s o u r c e s of random  i n t o account  ' s where we have e x t r a  The r  f o r the  terms  the e q u i l i b r i u m s t r u c t u r e s  calculated  respectively.  higher  i n t h e bond  i s p e r h a p s good t o a b o u t  in estimating  only  f o r the bending motion  When a l l  0.001 A f o r bond l e n g t h s  structures  is allowed  1 0 " " A . The bond a n g l e  error  indication  extrapolation  t h e same t h e c o r r e c t i o n  errors  and s y s t e m a t i c  The bond a n g l e  isotopic  s h o u l d be s m a l l .  i s around 0 . 0 1 ° .  best  the  structures  the  with smaller  standard error  at  the  h a s n o t been e s t a b l i s h e d  The s t a n d a r d d e v i a t i o n parameters  than  Strictly  potential  terms  to give  the e q u i l i b r i u m  t h e e q u i l i b r i u m bond a n g l e  contains  B  structures  z  a r e much l a r g e r  parameters  s h o u l d be t r e a t e d to  to the r  r r H  and  1.012 and 1.010 A from t h e r r D  T  f a v o u r e d as l a r g e r  average r e d u c e d mass  t o t h e b o t t o m of t h e p o t e n t i a l  extrapolation  T  of the d i a t o m i c  wells  approximation  191 works ±0.002  better.  The r  bond l e n g t h s . s h o u l d be r e l i a b l e  g  A. Table r  z  H  0 =1O7.O39( 16)° H  Empirical ND H  223 217 . I 1  i u  160  r  s  6.10  (MHz)  of  ammonia  structure  r =1.02304(7)A  2  r =1.02030(10)A  a  208 .10  1 U  T  /3 = 107. 0 8 5 ( 1 9 ) °  0 = 1 07 . 1 0 ( 3 )  D  B  ( N) r =1.01876(22)A*  D  Script  1 1 2510 .58  B  0  970. 8  -44  160  1 60 206 .58  1 58 304. 5  -1 16  1 1 2514 .03  1 1 1 804. 8  300  290  285 549. 7  198  2 0 8 . 10  i n  1 92 1 94 .18  192 194. 04  192 189 . 1 1 189  1 40 795 .26  140  1 40 787 .32  NH  ND  r  s  3  3  NT a ' f  3  290  141. 97  8 2 7 . 70  1 38.35  666. 0  -126  1 39 750. 3  1 23  293 240. 3  -76  298  1 1. 7 I0  186  726 .36  185 091 . 7  1 54 1 73 . 3 8 °  1 52 375. 6  6  1 05 565 . 3 7  1 04 546. 5  0  b  d  r e f ( r l 0 5 ) ,b r e f Calculated  (r134),c  ref(r135),  from a l i n e a r  d  extrapolation  text) calculated  Standard  assuming the  deviation  of  the  angle  fit  is  0  -180  ref(r136) of  /3  U  M  '*'  O-C  219  290  a  t  223 219 .16  112 521 . 1 3  1 25 . 3 8  o  223 2 2 2 . 80  NH D 2  to  T  154 M H z . .  and  (see JJ  1 92 Table r  6.11  structure  z  of  . r =1.42774(9)A H  Empirical PD H  93 9 1 7 . 7 9  III S  2  r  phosphine  r =1.42373(13)A  H  0 =93. 2 8 6 ( 1 2 ) °  (MHz)  Q  /3 =93.332{ 1 8 ) ° D  Script  B  B  0  O-C  93 9 1 8 . 4 3  93 9 1 6 . 1 9  92 9 7 9 . 7  81 9 1 0 . 9 2  81 9 1 2 . 5 0  81 9 1 0 . 1 7  81  171.8  111  64 8 4 1 . 8 0  64 8 4 3 . 0 7  64 8 4 3 . 5 6  64 4 2 5 . 3  -5  129 8 4 6 . 4 8  129 8 4 5 . 3 3  128 3 9 7 . 0  58  a  54  PH D  129 8 4 3 . 1 4  IIl S  89 2 7 9 . 3 4  89 2 8 3 . 5 6  89 2 7 7 . 6 0  88 4 7 1 . 9  -190  83 2 5 0 . 5 2  83 2 5 8 . 6 6  83 2 5 0 . 9 3  82 5 9 2 . 7  -168  131 9 6 6 . 5  -11  116 3 4 6 . 9  16  68 9 0 5 . 0  43  58 5 7 7 . 2  83  2  r  PH  3  133 4 8 0 . 1 3  a  b  117 4 8 8 . 6 9 PD  3  69 4 7 1 . 0 9  a  58 9 7 4 . 4 2 a this  work,  b  ref(rl02)  Standard d e v i a t i o n  of  the  fit  is  96 M H z . .  1 93 Table r H  r  = 1  6.12  structure  .52765(8)A  (MHz) of  r  D  =  0 =91 . 9 4 1 ( 9 ) ° H  5  Empirical  91.982 ( 1 4 ) °  5  6  Script  B  B  0  o-c  133.06  77  131.20  z 76 417 .8  24  71 525.64  71  527.90  71 526.01  70 909 .0  60  55 5 1 1 . 8 8  55  511.12  55 511.73  55  110  582.13  1 10 581.32  109 425 .5  77  Ii A r  1.52337(11 )A  77  AsD H 2  arsine  131.53  a  171 .6  -38  AsH D  1 1 05 8 1 . 6 7  IIl A  75 133.53  75  142.48  75 136.64  74 51 1. 1  -58  72 548.71  72  551.19  72 5 4 5 . 17  7 1 9 8 0 . 57  -56  11 1 282 .7  -54  1 03878 .0  -39  2  r  AsH  3  1 1 24 7 0 . 6 1  a  b  • 1 048 8 4 . 0 7 AsD  3  57 4 7 7 . 6 0  a  52 6 4 1 . 9 8 a  this  work,  b  ref(r93)  Standard deviation  of  the  fit  is  65 M H z . .  1 55  57 045 .9  1 6  52 287 .4  33  1 94 Table Equilibrium  structures  of  6.13  ammonia, p h o s p h i n e and r  e  0  (A)  e  arsine  (°)  Ammonia This  work  Benedict Helminger  1.011 (2)  107.21(11)  1.0116(8)  106.99(2)  1.0135(4)  107.14(9)  7  and P l y l e r et  al  (1957)  (1974)*  Phosphine This  work  Kijima  1.4 1 3 (2 )  93.45(9)  1 .41175(50)  93.421(60)  1.4115 (6)  93.32 (2)  1.41154(50)  93.36(8)  1 . 51 2 ( 2 )  92.08(7)  1-511060(14)  92.0690(14)  1 .5108(4)  92.083(43)  5  and Tanaka  Helms and G o r d y Chu and Oka  (1981)t (1977)ft  (1974)ff  9  8  Arsine This  work  Carlotti  8  et  al  (1 9 8 3 ) f  O l s o n , Maki ( 1 9 7 5 ) t and Sams  tt  tt  * T h i s s t r u c t u r e i s an a v e r a g e 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 o n l y be c o n s i d e r e d as an a p p r o x i m a t e r structure. g  t T h e s e e q u i l i b r i u m s t r u c t u r e s a r e e s t i m a t e d by e m p i r i c a l l y d e t e r m i n i n g t h e a c o n s t a n 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 b a n d s . R e c a l l t h a t i n 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 i m a t e the e q u i l i b r i u m s t r u c t u r e . t t T h e s e two s t r u c t u r e s f o r t h i s m o l e c u l e a r e n o t i n d e p e n d e n t as t h e l a t e r a u t h o r s use i n t h e i r structural d e t e r m i n a t i o n r e s u l t s o b t a i n e d by t h e e a r l i e r a u t h o r s . — Numbers i n p a r e n t h e s e s a r e of t h e l e a s t s q u a r e s f i t s i n u n i t s f igures.  the s t a n d a r d d e v i a t i o n s of t h e l a s t s i g n i f i c a n t  1 94 Table Equilibrium  structures  6.13  o f ammonia, p h o s p h i n e and a r s i n e r  e  (A)  0  e  (°)  Ammonia This  work  Benedict Helminger  1.011 (2)  107.21(11 )  1.0116(8)  106.99(2)  1.0135(4)  107.14(9)  1.413 (2)  93.45(9)  1.41175(50)  93.421(60)  1 .4115 ( 6 )  93.32 (2)  1.41154(50)  93.36(8)  1.512 ( 2 )  92.08(7)  1.511060(14)  92.0690(14)  1.5108(4)  92.083(43)  7  and P l y l e r et a l  (1957)  (1974)*  Phosphine This  work  Kijima  5  and T a n a k a  Helms and Gordy Chu and Oka  (1981)t  (I977)ft  (1974)ff  9  8  Arsine This  work  Carlotti  8  et a l  (I983)f  O l s o n , Maki ( 1 9 7 5 ) t t t and Sams  tt  * T h i s s t r u c t u r e i s an a v e r a g e 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 c a n o n l y be c o n s i d e r e d a s an a p p r o x i m a t e r structure. g  t T h e s e e q u i l i b r i u m s t r u c t u r e s a r e e s t i m a t e d by e m p i r i c a l l y , d e t e r m i n i n g t h e a c o n s t a n t s from a n a l y s i s o f 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 b a n d s . R e c a l l t h a t i n 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 i m a t e the e q u i l i b r i u m s t r u c t u r e . t t T h e s e two s t r u c t u r e s f o r t h i s m o l e c u l e a r e not i n d e p e n d e n t a s t h e l a t e r a u t h o r s use i n t h e i r structural d e t e r m i n a t i o n r e s u l t s o b t a i n e d by t h e e a r l i e r a u t h o r s . — Numbers i n p a r e n t h e s e s a r e t h e s t a n d a r d d e v i a t i o n s of t h e l e a s t s q u a r e s f i t s i n . u n i t s of t h e l a s t s i g n i f i c a n t f igures.  195 6. DISCUSSION OF STRUCTURAL From T a b l e  6.13 we see  d e t e r m i n e d as p a r t results values the  of  of  previous  this  previous  are  results.  s h o u l d be aware of  that work  studies.  presented here  significantly  seems u n d e r s t o o d t h a t interpreted  determined,  but  rather  equilibrium rotational  For  the  equals  the  the  the  number of  structural of  where  parameters  rotational  constant  number of  rotational are  In  this  extrapolation  technique  that  study  within  3  A,  better that  even  though the  (10""-10~  the  earlier  5  A).  In  studies  have q u o t e d bond l e n g t h s accuracy.  order  constants  of  atomic  are  reproduce stated  error.  parameters  then the  the high  precision  without  regard to  physical  we have  seen t h a t  the  when a p p l i e d t o  p r e c i s i o n of  r ,  light that  of  this  T h e s e comments a r e  and r  D  consistent  T  to  for only  numbers was much  linear  extrapolation  to a p r e c i s i o n  directed  linear  one s h o u l d s u g g e s t  used the  and a n g l e s  r  H  these  not  accurately  its  structural  ammonia p r o d u c e d e q u i l i b r i u m s t r u c t u r e s 10"  than  quote  will  o f t e n quoted to  constants  significance.  the  length  as a p a r a m e t e r  the  case  accurate  bond l e n g t h s  as a p h y s i c a l  the  c o m p a r i s o n s we  of  these  the  seem t h a t  what p e o p l e .mean when t h e y t o an a c c u r a c y  structures  with  less  When making t h e s e  radii.  t o be  equilibrium  would a l s o  bond l e n g t h s  meant  the  compare w e l l  It  molecular It  PARAMETERS  to  the  work  beyond of  Helms  and G o r d y and Chu and Oka on p h o s p h i n e . The s t u d i e s empirically  d e n o t e d by t h e  determined values  f  for  in Table many of  6.13 the  a  have constants  196 and so no f o r c e and no l i n e a r part  of  the  presented they is  field  give  extrapolation  alphas  in  (to  is  these  to  studies  similar  are  A.  order  This  is  Benedict  3  the  from w h i c h that  From t h e  very  high p r e c i s i o n  one i s o t o p i c  parameters  their  they  of  calculated  d i s a g r e e d by  work the  probably  reliable  to  results.  Furthermore  work,  studies.  the  the  accuracy  structures  w i t h an e m p h a s i s on t h e  e s t a b l i s h e d as isotopic  only  on ammonia  they  molecules  are  calculated  rather  than  they for  z  NH  3  structural  .04°.  authors  we q u o t e  that  for  structures,  from f o u r  from one or  or  the  are our  p r e s e n t e d as p a r t r  3  of  above d i s c u s s i o n we s h o u l d c o n c l u d e q u o t e d by p r e v i o u s  more  ammonia = 1 0 "  equilibrium  equilibrium structures  this  consistent  work  It  isotopic  a constants  .0016 A and  but  species.  on d i f f e r e n t  we f o u n d f o r  t o make good e s t i m a t e s  parameters  parameters  c o n c l u s i o n one draws from t h e In  a's)  The s t r u c t u r a l  calculations  (r131).  the  anharmonic  from o n l y  the c o n s i s t e n c y  and P l y l e r  were a b l e and ND  of  for  of  the  of  s p e c i e s would produce s t r u c t u r a l on t h e  harmonic p a r t  correct  required.  have been c a l c u l a t e d  suggested that  the  are  of well  five  two as  in  earlier  1 97 H.  CHAPTER In  of  7:  this  chapter  rotational  presented in  INTERESTING EXTRAS  the  in  we s h a l l  spectra this  that  thesis.  literature  review  are  at  they  r a t i o n a l i z i n g group V hydride  The f i r s t Gordy It  for  was  later  ammonia the  their  (r24)  has  s p e c t r u m of  the  quasi-spherical methods have  the  (field level  This  "first  following,  T  K  L  in  terms  energy  of  the  is is  levels.  particularly hydrides  (H X 2  to  apparent  for  where X=0, S,  of  Tarrago  the  this  other All  Se,  r  these empirically  rationalize  the  to  determinable  the  in  w i t h was d e v i s e d  power of  easily  Te or X H  3  the  Reductions."  series the  "acceleration"  light  in  see,  of  to deal  of  the  analysis  discussed further  momentum o p e r a t o r s ,  The need f o r  of  As we s h a l l  not a  "Failures  convergence  angular  of  evaluating  Z  The s e c o n d method we w i s h "accelerate"  and  study  recently  3  method"  entitled  them.  (r25,r26).  their  their  s u c h as O P F .  of  y  in  Very  data a l o n e .  X  use  in  it  u s e d by Helms  in  spectra  approximation necessary data,  represented  scheme as p e r h a p s u s e f u l  tops  frequency  frequency  parameter. section  (r115).  3  i n common a way  of  available  and Tanaka  rotational  symmetric  free)  why we d i d not  and M a r g o l i s  PD  method  top spectra  of O l s o n as  suggested a s i m i l a r of  well  on p h o s p h i n e and a r s i n e  and by K i j i m a  the a n a l y s i s  from  symmetric  that  u s e d by P o y n t e r  infra-red  (r146)  studies  with the  analysis  have been u s e d p r i m a r i l y  to e s t a b l i s h  method i s  variance  T h e s e methods a r e  and s i n c e  s h o u l d be n e c e s s a r y  two methods of  to  expansion, rotational is  distorted where X=N, P,  As  198 or XH„ where X=C, S i  and Ge)  where o f t e n  phenomenological parameters the  our power  necessary  to  number of  measured f r e q u e n c i e s .  distortion  "fit"  of  constants  sucessively  higher  find ourselves  in  spectrum i s  are  large  degree the  of  a d d i n g a new f i t t i n g  point.  In  order  acceleration standard  the  power  series  constructing (P.A.'s). studies  inversion  of  s p e c t r u m of extensively  Rotational  the  necessary earlier  Spectra."  the  spirit  Fraction  above of  in order  in a l a t e r  There  it is  will  to d e f i n e  the  data  the  latter  the  the  method  alternative  convergence levels  of  by  or Pade a p p r o x i m a n t s g r o u p s have made  H S  (r149)  2  entitled  the F i t t i n g  be f o u n d t h a t  unsatisfactory. 1,  will  uniquely see  "unity"  and  the  The method i s  section  will  As we s h a l l  explain  In  energy  Approximants in  studies  use.  the  (r150).  Watson and C h a p t e r  studies  reductions  ammonia  into  This  thesis.  (r148),  2  e a c h new  we  of  1 and has been t h e  approach v a r i o u s H 0  with  convergence  accelerate  fractions,  (r147),  3  this  made t o  rational  PH  Rational  used in  for  incorporated  in Chapter  the  the  to converge  situation  expression for  Using this  of  somewhat  are  when  as  embarrassing p o s i t i o n  schemes have been  attempts  model  momentum o p e r a t o r ,  parameter  used through-out  of  same o r d e r  case,  f o r m a l i s m of Watson and o t h e r s .  analysis  method,  this  angular  to a v o i d t h i s  f o r m a l i s m was p r e s e n t e d of  In  number  series  on t h e  and slow  somewhat  almost  the  reviewed  "On t h e of the  method  Reductions,  be shown t o  correlations  in  be  the a p p r o x i m a n t s t h e need f o r  use  these  these reported  199  between  1.  various  "Pade"  parameters  FAILURES OF THE  Two of rather  the  in  is  symmetric  top reduction  limit.  both cases  for  the  presented  specific  reduction  In  undefined in is  the  constants  constants. are  limits.  the  In  terms  the  inappropriate.  For m o l e c u l e s  limiting  use of  parameters others; that  of  this  reflects  symmetric  Chapter S  asymmetric  1).  top l i m i t  Strow  reductions  quantitative  all  method of  both reductions final  that  are  top  responsible  these  rotational  reduction  close  being very  to  results  of  the  this  s o l v e d with the  representations determining  For  becomes  these in  some  much l a r g e r  than  "reduction"  can mean  c a n become v e r y  the  to  that  large.  S reduction  produced e q u a l l y  presented  (right  the  and g i v e s  inappropriate  studies  representations  representation  arbitrarily.  spherical  the problem w i t h the A r e d u c t i o n was  in various  three  the  (r 151 ) has made a c o m p a r i s o n o f  and r e p r e s e n t a t i o n s . thesis  and so t h e  Furthermore  elements tops  cases  contribution  parameter.  o f f - d i a g o n a l matrix  the  the  the  and  1/(V-W) where V and W a r e  reductions  a given degree  particular  For  these  top A  top l i m i t  transformations  limiting  equal  1 become  The a s y m m e t r i c  symmetric  unitary  d e f i n e d as  cases  studies.  in Chapter  undefined in  reductions contain  rotational  above  REDUCTIONS  reductions  unwieldy  i n the  A and  reductions  in  so t h a t  and r e d u c t i o n c o u l d be made  (see  a  this  handed s y s t e m s )  good f i t s  in  in  choice  of  200 We s h a l l section  now c o n s i d e r  1.2c t h a t  symmetric  to s i x t h degree  top hamiltonian  contained  elements,  that  series  J and J  operators,  z  should also but  this  plus  the  hamiltonian x  /4 ( a c t u a l l y  J  H  (  J  2  )  3  of Watson's  R  J K  -  z  (  J  2  D  V  )  j  H  +  ( J  2  K J  J  ) 2  argument),  w i t h c o e f f i c i e n t h' . 3  in a standard 2  25 ( r 1 2 )  was d e s c r i b e d t h a t  t o be r e - w r i t t e n  H  d e g r e e we  equation  the present  transformation  x  +  to s i x t h  - B  V  +  J  H  J  K  2  J  reduced  - D J «  2  K  z  allowed  z  form.  +  K V  + h (J*+J )  [7.1]  6  3  (in  XXX z  -B )J '  2  matrix  o f f the d i a g o n a l  s i x o f f the d i a g o n a l  a unitary  = B J'+(B  r  three  o m i s s i o n does n o t a f f e c t  Further,  H  r  and e  terms  from  e x p a n s i o n s of t h e d i a g o n a l  elements  include  finally  diagonal  plus  with c o e f f i c i e n t  K)  Recall  t h e p r e - r e d u c t i o n C^y  symmetric  were power  tops.  where ( r 1 2 ) B  x  D H  2  =  with  H  K J "  r  3  0  z  B  =  B  , H  = D^-130£  R  K J  *  '  K~ ^  =  ^  '  H  K  z"  7  *  2  '  = Hj-2|  a  =  H  + K  1  4  J  D  *  =  , H '  H  3  =  D  j "  J  R  H  1  6  *  '  D  J K =  = HJ +18£ R  D  J K "  1  2  6  *  >  ,  **~*  [  7  2  ]  2  - i 6 ( " " ) z If  '  [  7  -  3  ]  x  we c o u n t  t h e number o f p a r a m e t e r s  transformation  we see t h e r e  constants  T  plus  are eleven  (or £ ) . A f t e r  :  before  the u n i t a r y  10 " p r i m e d "  the u n i t a r y  XXXZ  transformation constants. eigenvalues  there  Since  are only  a unitary  10 p a r a m e t e r s ,  transformation  of a h a m i l t o n i a n ,  the energy  the unprimed  cannot levels  affect  the  and hence  201 transition  frequencies  parameters.  depend on o n l y  T h i s a l s o means t h a t  independent parameters combinations of.the constants  makes the of  £ (and hence  number of  S reduction  Gordy  in  course  reduce  their  xxxz  )  to  that  it  In  and M a r g o l i s hamiltonian  approach e m p i r i c a l l y values  values  dependent p r o b l e m of  of  on t h e the  general  will other  10. The  for  to  the  asymmetric  as e q u a l work  top  to  and  some  on ammonia we  h'  to  3  zero.  and o t h e r s  label  Using  who have  £ and t h e n  infer  the  adopted  using  . The p r o b l e m h e r e  T XXXZ  in t h i s  used.  This  is  way  is  are  identical  that  the  tops  (see  terms.  of  (e7.3)  These r e d u c t i o n  The above  the  (1.2b)).  determined  from i n f o r m a t i o n o t h e r  frequency measurements.  to  both the A  section  empirically  £ d e f i n e d by e q u a t i o n  be c a l c u l a b l e  advantage  " f o l l o w i n g " Helms and G o r d y  asymmetric  find  reduce  1  r e d u c t i o n dependent  terms w i l l free)  xxxz  to  p o s s i b l e . Helms  H'  calculated  reduction  we w i l l  be e q u a l  are  determine  they  in order  reduction  limit.  a later  authors  arbitrary.  standard  r e d u c t i o n dependent parameters  and S r e d u c t i o n s In  r  the  w i t h the  by s e t t i n g  their  the  top  (r24)  reductions  that  agrees  to define  constant.  these  that  10  linear  completely  required  reductions  appear  (e7.3)  the  10  only  (pre-reduction)  indeterminate  above  equation  Which  is  see  symmetric  other  (r25,r26)  Poynter  is  the  indeterminable that  r  we can  parameters  reduction  Of  empirically.  11 symmetry a l l o w e d  (e7.2)  s  this  we can d e t e r m i n e  we c h o o s e t o d e t e r m i n e  From e q u a t i o n s  10 i n d e p e n d e n t  plus  £'s some  dependent than  argument w i l l  (field also  202 be t r u e "true"  for  all  primed parameters  pre-reduction  frequency  data  We a r e to  the  re-fit  the  parameters  in a p o s i t i o n  p u b l i s h e d PD  3  The r e a s o n was  of  different  All  the  although others,  for  all  the  so  it  spectrum to a standard  reduced  to a v o i d  are  "fit"  remain  (the  un-primed c o n s t a n t s  of  then very  the  to  this  dipole  closely  £ is  the  very  as  sets  of  of  all  PD  3  the  small  the  (e7.2)) is,  that  is  For  parameters  reduction  argument  is  that  in equation of  particular  we c h o o s e t o do e x p e r i m e n t s  on  molecules  t o enhance  (rl46)  has  we t r y  to  (see  to  the  the  forbidden  section  our  borrowing  rotational  1.4a).  suggested that fit  rotational  in  (e7.3)  is  in order  This  is  B^-B^  £=0.  pre-reduction  large.  one  standard  (that  standard  than  to choose  equation  denominator  £ rather  the  better  difficult  by t h e  the  for  relating  equivalent,  A problem with t h i s  intensities  that  "true"  given  top l i m i t  limit  Tarrago ~ B'  data  The a d v a n t a g e  moment c o n t r i b u t i o n  transition  B'  type  z e r o making  concern here near  this  parameters.  spherical  goes  re-fitting  and we c a n make the a p p r o x i m a t i o n  molecules  fitting  By  p r o b l e m of  analytically  reduction  are  the  species.  limited  should s t i l l  most m o l e c u l e s  why'it  same d e f i n i t i o n  another.  large)  from  decided  over  for  determinable  reason  reduction  that  not  the  was  isotopic  reductions  some m i g h t  to  reductions.  s p e c t r u m we would have t h e parameters  are  in general  alone.  now  hamiltonian.  parameters  so t h a t  spectra  for to  molecules terms  up t o  where fourth  203 degree: the  B'  B',  A  L  three-off  hamiltonian course,  D',  U  D ',  J l \  and  I\  the d i a g o n a l  Ir  implicit  in  this  XXXZ  equations  (e7.1)-(e7.3)  method t o  fit  t o B',  primed s e x t i c  with only  compared w e l l hamiltonian  D',  D ',  six  f i x e d at  (r152).  with a f i t  in  matrix.  reduction it  Of  is  given an  by  identical  and £ w i t h a l l  zero.  Using his  reasonable of  method  fit his  the  to method  he c o u l d p r o d u c e a f i t  using a standard  9 parameters  his  unitary  The a d v a n t a g e  parameters  requiring  the  D'  to produce a very  data  3  the  whole  and t h e r e f o r e  B',  constants  T a r r a g o was a b l e p u b l i s h e d OPF  method i s for  includes  terms e x p l i c i t l y  r  responsible  Tarrago  I. XXXZ  and t h e n d i a g o n a l i z e s t h e  transformation  that  D'  - i.e.  is  that  reduced sextic  terms  were  included. The r e a s o n T a r r a g o ' s method works clear. MHz)  For  the  near  term  contribution parameters. account  spherical  t o p s where B'  £ can become v e r y to  In  the these  f o r much of  large  unprimed s t a n d a r d cases the  so w e l l  fitting  sextic  to  z  ~ B^  s h o u l d be  (OPF  and be t h e reduction  3  AB'  dominant sextic  £ should allow  contribution  to  =214  us  to  the  frequencies. Tarrago's "other" the OPF  MHz)  is  essentially  the  r e d u c t i o n s mentioned p r e v i o u s l y . 3  spectrum to  usefulness sets.  reduction  of  some r e d u c t i o n s  Since AsD a study  fewer p a r a m e t e r s  3  is  over  a l s o a near  was made u s i n g t h e  same as  His  the  ability  probably  to  fit  reflects  the  others  for  some d a t a  spherical  top  (AB'  = 4800  above a p p r o a c h i n hopes of  204 fitting  this  spectrum to a lower  improvements parameters identical It  in  was  the  fits  required. A similar  the  instructive  £ term. Given  rotational  we can c o n s t r u c t  constants  the  of  following Table  Estimation *est  PD  3  (e7.3), 6.10,  the Table  6.11  and  contribution  of  £ to  h  % cont.  3  Ref.  39  111  53.81  80  94  that  the h  3  heavier, is  small  that  for  contribution  more s p h e r i c a l  ammonia, h 3  (r24)  assume h NH  3  of  3  to  This  3  6.7  is  seen  where f o l l o w i n g  of  parameters  the  should  the  (r25,r26)  the  u s i n g the  study  they a value  with the  of  value  this  same s y m b o l s  in d i f f e r e n t  by  method of  11.26 M H z . . We s u s p e c t  t o be a r e s u l t  "similar"  in  be z e r o and c a l c u l a t e  of  It  work  c o u l d make a  0.44 MHz i n d i s a g r e e m e n t  in Table  hydrides,  becomes r e a s o n a b l e .  to h .  6.12  7.1  86.3  for  6.7  table.  43. 1  disagreement represent  Tables  o u t l i n e d by Helms and G o r d y  implicitly  to c o n s i d e r  equation  (Hz)  and M a r g o l i s  presented  gave  3  this  significant  for  the  ( H 2 )  be a p p a r e n t  O l s o n as  PD  78  that  Poynter  of  4.63  assumption also  same number of  3.6  3  We see  r  of  33.3  3  AsH  study  No  results.  m a g n i t u d e of  PH  parameters.  were f o u n d and t h e  s h o u l d now p r o v e  and t h e  number o f  to  reductions.  For  205 a more d e t a i l e d A  3  is  equal  to  Finally,  f o r m u l a t i o n of  h  see  3  ref.(rl39)  where  their  h /2. 3  through-out  this  thesis,  has  T  been  XXXZ  l a b e l e d as  not  determinable  frequency  measurements.  determine  T  from s t a t i c  After  this  electric  field  free  d i s c u s s i o n on how not  one m i g h t ask how one m i g h t .  One way  is  to to  XXXZ  measure  the  distortion  (e1.33)  that  the  d i p o l e moment. R e c a l l  rotational  contribution  d i p o l e moment d e p e n d s on T  to  from the  equation  distortion  . T h i s d i p o l e moment c a n  be  XXXZ  determined  from a b s o l u t e  from S t a r k  shift  (e1.44).  determine  the  higher  resulting expansion. discussed  into  limits  that  separate  electric  calculation  is  a vibrational  from t h e  for  the  behaviour  might h e l p  the  type  of  in  of  is  or  fields)  the borrowing  difficult  to  rotational,  accelerating  the next  reductions the  m o l e c u l e s we a r e  i n a slow c o n v e r g e n c e Ways of  the  justifying  degree d i s t o r t i o n c o n s t a n t s  in  (e1.37)  (e1.36).  insight  various  earlier  with t h i s  static  — a c o n t r i b u t i o n which  and h a r d t o  contribution This  (e1.35)  measurements  (using  d i p o l e moment c o n t a i n s  contribution  the  measurements  The d i f f i c u l t y  distortion  intensity  section.  of  this  c o n c e r n made  interested  can be q u i t e  our power  in  series  convergence  are  large  in,  206 2.  ON THE USE OF RATIONAL F I T T I N G OF ROTATIONAL  A general power  series  series  is  degree  necessary  to w i t h i n light  to  include  their  is  include  that  all  higher  degree  data  The d i f f i c u l t y increases  degree  are  for  the  in'turn  or  effects  bad enough f o r  linear  power  J  c a n be  symmetric  tops,  to higher  degree d r a m a t i c a l l y  where we f i t  parameters  we must d e t e r m i n e  terms.  symmetric  parameters  other  The c h o i c e  of  tops,  to  when t h e  the  only  new  where even This  problem  fitting J,  but  b e c a u s e of  all  the  to  fix  J,K  "determine"  m i n i m i z i n g of  is  going new  cross some  c o n s t r a i n e d to to zero  for  series,  number of  often  data higher  the  are  of  data  increases  to  to  when t h e s e . " e x t r a "  w i t h two v a r i a b l e  which parameters and e r r o r  quite  parameters  one v a r i a b l e  in order  parameters  made t h r o u g h t r i a l  empirical  m o l e c u l e s where t h e in only  for  h i g h degree  may be s i g n i f i c a n t .  series  frequencies  The p r o b l e m  transitions,  are  usual  i n t r o d u c i n g terms  aggravated  equations  For  spectral  means we need b i g g e r  especially  "higher"  distortion  that  infinite The  contributions  number of  further  unavailable,  available  is  This  is  to  t o as h i g h a  degree e f f e c t s  distortion  the  sets.  we sometimes must go t o v e r y  The p r o b l e m i s are  the  uncertainty.  (where h i g h e r  we must d e t e r m i n e . sets.  data  parameters  for  spectra  truncate  finite  empirical  to account  significant  frequencies.  molecular  where t o  experimental  molecules  large)  is  t o accommodate t h e  procedure as  SPECTRA  c o n c e r n when f i t t i n g  expressions  so as  FRACTION APPROXIMANTS IN THE  zero.  inevitably  parameter  207 correlations  and s t a n d a r d d e v i a t i o n s .  procedures are  necessary  interpretation  of  and t h e r e  is  constants, of  fitting  no a priori  over  others,  out  other  this  is  empirical  finally  of  a fit.  order  fit  How much of  data  in order  expansions with c e r t a i n  needed.  parameters  The p r o b l e m t h e n ,  solution  to  spectral  frequencies  should l i k e  the  of  to maintain  interpretation In  p r o b l e m of  for  the  is  slowly  light  convergence.  Of  science,  methods by w h i c h  convergence  of  special  slowly  discusses a family transformations acceleration.  Of  transformations  of  origin  First  in is  series  for  determine.  concerned  especially  with  in  we c a n a c c e l e r a t e  computer  the  sequences.  non-linear  sequence-to-sequence  relevance  found to e f f e c t to  Pade a p p r o x i m a n t s . As we s h a l l  Shanks  see  as  (rl53)  s u c h an  spectroscopists  known c o l l e c t i v e l y  the  we  converging  which are  a  physical  often  importance,  we  we would l i k e  we e m p i r i c a l l y  is  data  however  s o m e t h i n g new  of  all  new  m o l e c u l e s and s e c o n d ,  parameters one  If,  converging  some d e g r e e  numerical analysis  are  twofold.  interpret  to a c c o m p l i s h .  the p h y s i c a l  then c l e a r l y  effect  parameters  some s e m b l a n c e  of  certain  the  to p r e d i c t  wish to  fitting  nebulous  a problem  trying  error  physical  leaving  Of c o u r s e  z e r o s h o u l d seem r e a s o n a b l e .  these  the  rather  for  constrained to retain  and  c a n be t o m o d i f y how we  constants.  to  is  reason  d e p e n d s on what we a r e  we w i s h s i m p l y  then h i g h e r  high degree  parameters  physical  c o n s t r a i n i n g parameters  the  If  the  because at  These t r i a l  set  of  rational "fractions  or  these  Pade  is  the  approximants  208  c o u l d prove  useful  in a c c e l e r a t i n g  converging angular  the  momentum power  convergence  series  of  slowly  frequency  expressions. The u s e f u l n e s s method of  of  rational  accelerating  established theory,  of  in  the convergence  such f i e l d s  computer  science,  continued f r a c t i o n s .  theoretical rotational still  accounts  of  spectroscopy  quite  as  approximate  heuristic of,  of  series theory,  liquids  (r154).  rational  However  fraction  their  representation Much l a t e r  of  Shanks, for,  concerning  a function in  by  1955 ( r 1 5 3 ) ,  gave  and showed many of there  the  has been a  of  more c o m p l e t e d i s c u s s i o n of b e i n g u s e d one  and books of  is  functions  to get  referred  works  it  technique  a feel is  to  by T a y l o r  the  to  the  advantages  to  the  series.  following  for  For  a  f i n d i n g the  limit".  textbooks  what a Pade a p p r o x i m a n t first  to c o n s i d e r  roots  a p p r o x i m a t i o n known as A i t k i n ' s  "extrapolation  a  (r156-r160).  instructive  for  the  how and where Pade a p p r o x i m a n t s  c o l l e c t e d papers  order  Pade,  growing  representation  successive  are  rational  standard  numerical  in  in  name from H e n r i  p u b l i s h e d an a r t i c l e  motivation  it  study  approximants  i n Pade a p p r o x i m a n t s a s an a l t e r n a t i v e  and why  field  They have been u s e d i n c h e m i s t r y  P a d e ' s method. Since then  In  well  phenomena and t h e  interest  are  is  new.  1892 ( r 1 5 5 )  fractions.  a p p r o x i m a n t s as a  scattering  critical  Pade a p p r o x i m a n t s d e r i v e who i n  fraction  Here  of S  2  is  a  equations  by  process,  or  we c h o o s e a g e n e r a l  form  209 for  an e q u a t i o n  t o be s o l v e d by  iteration  g(x)=0 It  is  o f t e n more c o n v e n i e n t  to w r i t e  this  as  x=G(x) where of  G is  the  a generator  iterances.  that  This  is  gives  the  usually  functional  expressed in  dependence  recursion  form  G ( x  where  x  L  process.  i  represents For  the  L  values  true  root  ) = X  L+1  of  x p r o d u c e d by some  iterative  r, G(r)=r  0  A p p l y i n g the linear  Taylor  t e r m one  e x p a n s i o n and t r u n c a t i n g  after  the  has G(x )  = G(r)  L  + (x -r)  G'(r)  L  or r  and i n  the  X  L+1  =  same way we a l s o r  Dividing  "  these  last  - x  L  G  '  (  r  (  r  _  X  L  )  have  = G'(r)  two e q u a t i o n s X  )  (r-x _ ) L  and s o l v i n g  L+1 L-1 " X  1  X  L  for  r  yields  210 or  where A x  = *  L  equivalent (x ,Ax ) T  +  (x  i_i  Li  familiar  8  any  than  often  Li  points  axis  and  system  Aitken that  Equation  (r162).  In  (e7.4) this  produces values  some r e a l  root  than  the  of  approximation to  for  which  equation  approximations  infinity.  is For  series  to  three  (e7.5)  "extrapolate"  =  i§0  to A  the  take  equation the  successive  real  root  iterants.  a good a p p r o x i m a t i o n  generated  e x p a n s i o n of  convergent  and c l o s e r  equation  L  equal  t h e n any  is  our  of  are  series.  L = 0 , 1 , 2 , . ..}  L  A  where g ( x )  of  given  we c a n  them i n t o  Pade a p p r o x i m a n t s we c o n s i d e r  (A :  a power  last (e7.4)  "geometric"  defining  sums of  the  the  x each  last,  "better"  took  is  case,  and p r o d u c e a we s i m p l y  is  I  process to  (e7.4)  t h r o u g h the  and p u t  In  Given  (x,Ax)  equation  iterants  called  closer  on a  line  of  successive  if  Series  ,)  T  f o r m u l a of  2  closer  three  (e7.4)  1  a straight  t o Ax = 0 ( r 1 6 1 ) .  some i t e r a t i v e is  This derivation  L >  ,,Ax  T  Li  extrapolating  which  - x  1  to drawing  and  r  L  L  a  i  x  in  by t h e  function  partial g(x).  l  [  the  series, real  limit  5  from i n f o r m a t i o n  2  a s L goes  successive  value  successive  and A i t k e n ' s  the  successive  of  the  A 's L  7  will  function  inherent  (e7.4) in  we  these  5  ]  to be g(x).  expansion c o e f f i c i e n t s process  -  can three  of  21 1 coefficients  to higher A  B  =  order  L-1 L AA  r  = A  . - A .  T  T  Usually  Taylor  expansion c o e f f i c i e n t s  a ,  and a .  0  a,  equal  In  2  to  this  f o r m i n g the  ratio  " V ^ L - I AA L-1  A A  T  where A A  by  case  in  if they the  [7.6]  we knew t h r e e  successive  would be t h e above  ratio  first  three:  L w o u l d be  one.  When we know more t h a n have more A ' s  three  to e x t r a p o l a t e  L  we have a g e n e r a l  AA  this  general  we case  (r!53,rl58),  AA AA  L-M+1  AA  coefficients  L-1  L-M  AA  f r o m and i n  form a p p r o x i m a n t  L-M  Taylor  L-1 T  L AA  I  AA  L+1  AA  L-1  L+M- 1  B.  M,L  AA AA  AA  AA  L-M  L-1  AA.  L-M+1  AA AA  AA  L-1  T  L+1  L+M-1  [7.7] where s p e c i a l denominator  definitions  vanishes.  are  i n v o k e d whenever  the  212 Substituting  { e l .1)  the  definition  and e x p a n d i n g t h e  written  as  rational  the  ratio  A  determinants  of  (e7.5)  T  into  we f i n d B  two p o l y n o m i a l s ;  that  M  equation can  L  is,  as  be  a  fraction. P B  Where P ( x ) L  polynomial  P  of  L  of  '  /  L  M  ]  =  degree at  Pa  a  (x)  T  Q^TT  a p o l y n o m i a l of  =a?0  where t h e (r158).  is  M,L  [  d  Q  M  7  degree  most M,  n  f ' ^  in  =  1  at  the  variable  ^  +  s t a n d a r d n o r m a l i z a t i o n q =1  and Q  [  been  a  M  x  ^  has  0  The c o e f f i c i e n t s  most L,  8  7  '  9  ]  adopted  p  and q „ c a n be c a l c u l a t e d from p i n an e n t i r e l y e q u i v a l e n t way f r o m a  equation  (e7.7) A  or  <>  where O means  "terms  definition  the  series  of  order".  by Q  L + M + 1  This  )  [7.10]  is  the  Pade a p p r o x i m a n t  (e7.5)  (e7.l0) a  of  [L/M]  expansion  equation  = 0(x  " TTT^T  x  (r158).  to  powers  = Pi  0  a2 aiq +a q +  a  L L  x we  = Po  0  +  1  a  +  L-l 1  +  0  q  a  i  = p  2  +  L  +  q  i  a  +  +  a  °  q  L  L - M + l S M  =  =  Taylor  both s i d e s of  ai+a qi  a  the  Multiplying  and e q u a t i n g  M  standard  P  L °  2  of  find,  213  a L+M,+a L+M-1 q i w i t h a^ = 0" i f Clearly determined,  the  point  that  fraction will two  i<0 and q . = 0 i f  number of  L+M+1  known T a y l o r  = 0  +  (as  Pade c o e f f i c i e n t s  q =1),  must be e q u a l  0  expansion c o e f f i c i e n t s .  the  transformation  P (x)  / Q (x)  L  is  M  j>M.  of  This  A(x)  to  to  be  the  is  an  (e7.5)  number  of  important  to a  rational  a one t o one m a p p i n g . T h i s  c o n c e r n us a g a i n when we i n v e s t i g a t e  point  approximants  in  variables. As an example of  how u s e f u l  these  convergence  acceleration  methods c a n be we c o n s i d e r  the  For  x=1 t h i s  series  In  the L e i b n i t z  series  sums t o  4 This  is  slow  to converge.  digits  a very  (that  million  the  that  the  information  order  On t h e  the T a y l o r  case  we  very  for  rr/4 b e c a u s e  it  is  to get  TT/4 c o r r e c t  to  eight  less  hand,  accuracy  terms of  this  7  and e q u a t i o n  digit  few  than  other  series  5  i g n o r i n g terms  series  first  3  In  same e i g h t  the  TT/4 ( r 1 5 3 ) .  poor a l g o r i t h m  is  terms!  terms of to  [7.11]  this  than  10~ )  from j u s t (e7.6) (r153). series  series  requires  7  the  first  have  5  10  we can c a l c u l a t e It  s h o u l d be  TT/4  clear  p o s s e s s much more  would have  us  believe.  214 Unfortunately, accurate  in  analytical  frequency  Taylor  coefficients  s p e c t r o s c o p y we do not as y e t  methods of  series  are  fitting  series.  These e m p i r i c a l  effects  due t o the c o n v e r g e n c e series  observed frequencies Taylor  so i t  These  determined e m p i r i c a l l y  squares  the T a y l o r  rotational  expansion c o e f f i c i e n t s .  usually of  p r o d u c i n g the  is  not  by  least  to t r u n c a t e d  coefficients  will  and t r u n c a t i o n clear  have  that  power  contain  p r o b l e m s of  f o r m i n g Pade  approximants with these  Taylor  will  produce  anything  would seem t h e n a b e t t e r  idea to  beneficial.  It  t o a Pade a p p r o x i m a n t , series,  rather  and t a k e a d v a n t a g e  coefficients  than a t r u n c a t e d  of  these  fit  Taylor  special properties  of  Pade a p p r o x i m a n t s . To see how Pade a p p r o x i m a n t s c o u l d be u s e d i n p r o c e d u r e s we s h a l l that  consider  some n o v e l  d u r i n g some measurement  given  by t h e  functional f(x)  process yields  is  = (1  often  God-given f u n c t i o n a l  + 4X)  Given  parameter,  we m i g h t  series  x,  in x.  For  d e t e r m i n i n g the and a  2  in  the  a data  this first  observables  [7.12]  1 / 2  case,  form i s  observations.  phenomenon  form  = 1 + 2x - 2 x Now s u p p o s e , as  physical  fitting  try  2  + 4x  we a r e  responsible  set to  3  -  [7.13]  unaware for  this  data  one  t o a power  d i s c u s s i o n we w i l l  concentrate  three  parameters  power  series  this  our  as a f u n c t i o n of fit  that  on a , 0  a,  215  where f  T  The  f  (x)  = a  stands  for  "truncated  [1/1]  + a,x  0  2  p  U )  series".  to equation - a.  1  [7.14]  :  Taylor  Pade a p p r o x i m a n t a  f  + a x  (7.14)  is  a " 2  [7.15]  = 1 -  To p r o d u c e a d a t a values  of  x=0.1  uncertainties as  the  least  to  1.0  to a l l  inverse squares  set  steps  of  data  of  of  this  uncertainty.  f  and f  fitting  T  "true"  0.01  to  p  Taylor  (e7.12)  0 . 1 . We a s s i g n  the  square  below a l o n g w i t h t h e  in  we use e q u a t i o n  and w e i g h t  this  for arbitrary  each  datum  The r e s u l t s  data  expansion  set  are  of  shown  coefficients  (el.13). Table C o m p a r i s o n of fraction  truncated  fp  in  Taylor  fitting  data  7.2 series  f  versus  T  p r o d u c e d from e q u a t i o n  1.037(7)  1.022(4)  1 .000  1.583(30)  1.715(23)  2.000  -0.389(25)  -0.714(35)  -2.000  0  a, a v  2  7.12  Truth  T a  Pade  2.28  2  where t h e  numbers i n p a r e n t h e s e s  the  least  significant  the  squares  of  the  figures  deviations.  0.63  0.000  are  standard deviations  and x  2  is  the  weighted  in  sum of  216  The r e s u l t s that  in  this  in Table  case  we have  t w o f o l d p r o b l e m of and t h e In  the  physical first  Pade f i t because better model is  the  that  for  the  convergence  is  2  for  data.  roughly  higher  equation  is,  This  s p e c t r o s c o p y as we l i k e  the  various  expansion  As a f u r t h e r approximants over [1/1]  If  [1/I]  In  order  know t h e  to  f ( x )  insight  first  ratio  making point Pade  just  to a t t a c h  the  Taylor  to equation  we  for  the  faster, it  of fit  a  better  Table are  7.1  better  values  (see  what we would want  physical  meaning  to  rational Taylor  Truncated Taylor  f(x)  advantages series  of  Pade  we n o t i c e  (e7.13)  the  is  find  = 1 + 2x - 2 x  three  parameters.  T h i s would be  analytical  course,  into  truncated  form t h e  the  the  expansion  converges  important  seem  parameters.  Pade a p p r o x i m a n t  we expand t h i s  series  fit.  would  addresses  smaller  effects  of  It  fitting  it  "true"  of  for  the  Taylor  order  the  the  4 times  parameters to  of  since  The o t h e r  d e t e r m i n e d and c l o s e r e7.13).  of  truncated  empirical  encouraging.  f o u n d a method t h a t  Pade a p p r o x i m a n t ,  the  the  are  interpretation x  accounts for  the  place  than  7.2  2  + 2x  [7.17]  3  fraction  we o n l y  needed  to  coefficients, = 1 + 2x - 2 x  2  [7.18]  217 Comparing e q u a t i o n s Pade a p p r o x i m a n t only  = 2x ,  4x ,  roughly  3  see  this  in  differs  whereas  3  (e7.l7)  a general  (e7.l8)  from t h e  the  t w i c e the  and  truncated error  way  of  we n o t i c e  true  expansion  Taylor  the  that  series  (e7.l3) differs  Pade f r a c t i o n .  by p a r t i a l l y  the  expanding  by by =  We c a n  equation  (e7.15)  al 3  a1 - a —x a. x  f  = a  p  0  + a,x  + a x 2  +  2  1 1  The b e a u t y  of  the  finite  number of  higher  degree  implicit In  in  general  parameters  are  known a s a " s e r i e s  [N-1,N]  Pade a p p r o x i m a n t s  With a l l is  their  for  around that  must  for  the  of are  Also  the  for  even  for  components  most  the  this  sort  type  [N/N]  u p p e r and l o w e r of  powerful,  so-called of  and  limits would  of  certainly  Pade a p p r o x i m a n t s ,  are  probably  t h o u g h Pade a p p r o x i m a n t s it  is  see an e x t e n s i v e  be t a k e n  account  a specific  Stieltjes",  Limits  advantages  a l o n g time  there  a  predictions.  First,  we c a n  Second,  or  use n o t more common? T h e r e a r e  this.  to  degree  where L=M, the  (r163).  series,  for  with only  fraction.  those  respectively(r158,r164).  that  "higher"  the most e f f i c i e n t ,  approximants  be u s e f u l  is  we c a n a t t e m p t  w i t h the  rational  Pade a p p r o x i m a n t s diagonal  Pade a p p r o x i m a n t  effects  the  2  certain  only  in  the. l a s t  interest  functions  when c o n s t r u c t i n g  in t h e i r  for  which  approximants.  why  two  have  then  answers been  10-15 y e a r s application. great For  care instance,  218 if  instead  the  of  equation  functional  truncated than  the  (e7.12)  form e  Taylor  then  series  to  We see If  that  for  1  trouble.  in  Also,  x  (for  but  rather  for  that  these  see  have  Stieltjes  all  negative are  (rl58);  well  the real  no p o l e s  and l o w e r  of  e  the  2 the  [1/1]  f u n c t i o n does n o t . is  Pade  1.29  of  T h i s does  for  for  reference  their  (r165)  some  approximant  a poor a l g o r i t h m  equation  Gammel  Pade  x=2 we w o u l d have  than  the  (e7.13).  series In  series For  p o l e s of axis (rl58).  limits  for  series the in  not e  for  (r158))  applicability techniques  in  for  rr/4 and t h e  series  e x a m p l e s of  series  successive series  of  this  type  it  words,  for  can be on  positive  terms  of  w h i c h Pade a p p r o x i m a n t s  Pade a p p r o x i m a n t s a r e  other  Also,  for  these  s i g n s . These are  most a p p r o p r i a t e . that  [1/1]  fit  poles.)  by e q u a t i o n opposite  exact  to determine  we must be wary of (see  the  Pade a p p r o x i m a n t has a p o l e .  trying  Now we r e - c o n s i d e r given  [1/1]  the  x  from  2 + x T^-TL  =  x greater  instance  regions,  handling  x  a Pade a p p r o x i m a n t  all  various  i e  ]  the v i c i n i t y  c h a n g e s . s i g n whereas mean t h a t  I  x=2 t h i s  we were e m p i r i c a l l y  coefficients  /  of  set  produced a b e t t e r  e x p a n s i o n of  r ,  a data  some v a l u e s  C o n s i d e r the  the T a y l o r  [  for  would have  Pade f r a c t i o n .  approximant  we had b u i l t  are shown  the x  there  a l o n g w i t h the w e l l d e f i n e d  mentioned e a r l i e r ,  e s t a b l i s h e d convergence  series  properties  of  this  (r158).  sort  What  is  upper have  219 particularly evidence rotors this  encouraging  be t h e  case  rather  than  better  choice  that  there  then c e r t a i n l y  for  Taylor  empirical  for  The r e a s o n  for  this,  is  how s u c c e s s i v e  series  Taylor  series  series  to a value  extrapolate Finally very  for  rotational  e x p a n s i o n s may e m u l a t e  applicable  the to  frequencies  discussion  linear  Ultimately  we s h o u l d l i k e  variables.  As a s t e p  The function  in  symmetric  "standard"  in  We have  represent  of  many  top  Taylor  series.  in a and  to  the  Taylor  then "truth".  as  their  power series  advantages. only  the of  generally  rotational  one  fitting  direction  variable. method i n N  we now c o n s i d e r  the  case.  representation  by a one v a r i a b l e  seen  a  Stieltjes,  now i s  a similar  that  terms  frequencies  terms  with  Pade a p p r o x i m a n t s may be  m o l e c u l e s where  can be w r i t t e n  two v a r i a b l e  'till  the  w i t h a Pade  closer  series  w h i c h Pade a p p r o x i m a n t s have Still,  i n d e e d be  to converge,  has been s u g g e s t e d t h a t  appropriate  series for  it  the  fractions  been c o n c e r n e d  better  seen,  If  functions.  a p p r o x i m a n t we a n a l y z e h e l p the  Pade  then can a t r u n c a t e d  we have  diatomic  (r123).  one v a r i a b l e .  a Pade a p p r o x i m a n t c a n o f t e n i n one v a r i a b l e  order  would  we have of  for  diatomics  fitting  functions  considerable  corrections  series  d i s c u s s i o n so f a r  schemes f o r  function  is  sign with successive  truncated  the  fitting  that  s u g g e s t i n g the d i s t o r t i o n  a l s o change  In  is  of  a one  Pade a p p r o x i m a n t  variable is  as  the  ratio  220 of  one v a r i a b l e  p o l y n o m i a l s whose c o e f f i c i e n t s  by c o n s t r a i n i n g fraction  the  to agree  expansion a f t e r  w i t h the  Taylor  t o as h i g h a degree as p o s s i b l e is  not  might  surprising then, try  to  division  of  (see  equation  the  a p p r o x i m a n t s as  t o as h i g h a d e g r e e as p o s s i b l e w i t h  case  "diagonal"  expansion. Furthermore,  t h e most e f f i c i e n t  since  in  and d e n o m i n a t o r a r e  equal. This  is  o r d e r s of  ratios  of  the the  approximants are  a p p r o x i m a n t s where t h e  we  division  L=M a p p r o x i m a n t s , we s h o u l d i n i t i a l l y  two v a r i a b l e  It  that  the  should agree  variable  function  two v a r i a b l e s ,  p o l y n o m i a l s whose e x p a n s i o n a f t e r  Taylor  rational  (e7.10)).  two v a r i a b l e  relevant  obtained  the  e x p a n s i o n of  on g o i n g t o  form two v a r i a b l e  are  one  the consider  the  t h e method of  numerator Chisholm  (r166) . The d i f f i c u l t y have t e r m s  of  denominator for  the  w i t h the C h i s h o l m approximant  similar  introduces  degree  no l o n g e r  one t o  coefficient  one.  This  approximant  fit  the  various  solve  used to this  (n66).  studies  two v a r i a b l e s  by c o n s t r a i n i n g  "cross"  to  immediately  obvious  K in a symmetrical  that  For  the  solution  T h i s means  will  the  come up sort  frequencies. t h a t was  we need t r e a t  is later  of  Chisholm  symmetrical  the c o e f f i c i e n t s  symmetric  to  and  where t h i s  rotational  zero.  that  to Chisholm c o e f f i c i e n t  p r o b l e m i n a way  terms  into  indeterminacy  other  is  numerator  an i n d e t e r m i n a c y  when we r e v i e w v a r i o u s  for  the  approximant c o e f f i c i e n t s  mapping from T a y l o r  chose to  in  is  tops  it  the p a r a m e t e r s  way and so we s h o u l d e x p e c t  of  is  not  J  and  different  221 "reductions"  might prove  Now we s h a l l fraction  re-consider  literature  was  convergence  fractions  of  in  of  in  The f i r s t  fitting  variable  rational  (r138) fit  deviation  truncated  for  the  of  power  their  represent  spectroscopic  (rl68).  using  spectra  their  Mizushima  study  series  they  used a  terms  similar  two  i n powers for  This  of  two o r d e r s  frequencies  rational  in J  and K — t h e  fraction even  to  (r170). fitting  better  than  the  the (see  set  was  standard  of m a g n i t u d e )  were f i t  Costain  the  At  for  error  same d a t a in  J  the  c o u l d not a c c o u n t  improvement  of  to C h i s h o l m ) .  an u p d a t e d s t u d y ) . (an  also  rational  to account  (i.e.  (see  the  was by Young and  with similar  ammonia  series  data  expansions  experimental  f o r m of  the  the  rational  spectroscopy.  spectrum to w i t h i n  fit  the  "exponential"  rotational  at  and d e n o m i n a t o r  much b e t t e r  of  logarithms  that  In  s t a n d a r d power  inversion  found to  rotational  s p e c t r u m of the  attempt  fraction  and K i n n u m e r a t o r  time  of  u s i n g Pade a p p r o x i m a n t s t o a c c e l e r a t e  1978 ( r 1 5 0 ) .  inversion  history  Pade a p p r o x i m a n t s t o  Young i n  Spirko  brief  standard perturbation  r l 6 4 and r 1 6 9 ) .  ammonia  the  p r o b a b l y due t o M . M i z u s h i m a  interested  the  (r167).  a p p r o x i m a n t s as a p p l i e d t o  The i n t r o d u c t i o n  was  better  when  the  standard  so-called  Young and Young f o u n d f u n c t i o n appeared  the  exponential  to  Costain  function. Later, hamiltonian. to  the  in  1981, B e l o v  That  eigenvalue  is,  et  rather  equation  al  (r147)  i n t r o d u c e d a Pade  than w r i t i n g  they  Pade a p p r o x i m a n t s  w r o t e Pade a p p r o x i m a n t s  to  222 the  hamiltonian.  method of  method p h o s p h i n e was  and most  The r e s u l t was an  recently  H 0  of  various  indication  account  to  that  all  complaint — they correlations  all  in  s h o u l d be c l e a r  that  studies  fractions  is  that  there  2  suffered high,  similar  studies  better  frequencies. from the  essentially of  terms  The p r o b l e m w i t h t h e  mapping f r o m T a y l o r fraction  could  the  unity,  for  argument  in both  previous  no a t t e m p t  coefficient  spectroscopic rational one  rational  coefficient.  Other available. Lutterodt  two v a r i a b l e  approximant  formulations  are  For  we have t h e  formulations  of  instance  (r171 ) and K a r l s s o n  both these  m e t h o d s , as  by w r i t i n g  the  in  and W a l l i n  t h e method of  two v a r i a b l e  (r172).  In  Chisholm,  a p p r o x i m a n t as  the  polynomials. Ultimately,  we s h o u l d e x p e c t  variable  approximant  the  polynomials,  but  it  heuristic  to  end up as  ratio  of  would be more s a t i s f y i n g motivation  behind these  using  we b e g i n  ratio  variable  stronger  for  numerator  t o make a one t o  to Chisholm  with  this  C h i s h o l m - l i k e two v a r i a b l e is  same  numerators  d e n o m i n a t o r s . The r e a s o n  use t h e s e  H S  fraction  from C h i s h o l m ' s i n d e t e r m i n a c y  and d e n o m i n a t o r . studies  rational  spectroscopic  a p p r o x i m a n t s where we f i n d  the  more c o m p l e x .  s t u d i e d , and l a t e r  some p a r a m e t e r s  the  same as  somewhat  fractions  found very  amongst  those  rational  these  the  (r148).  2  these  rotational  Unfortunately,  some of  method i s  Young and Y o u n g , a l t h o u g h  With t h e i r (r149)  The r e s u l t i n g  two  of  any  two two  variable  t o have a  approximants,  223 similar  t o what we had e a r l i e r  As an example of the  s u g g e s t i o n of  Taylor  the  a more p h y s i c a l  Tuyl  (r173).  one v a r i a b l e  case.  a p p r o a c h we c o n s i d e r  The s t a n d a r d two  variable  expansion  <  f  Tuyl  for  re-writes  x  =a§O0io  ' ^  a  aB  ^  '  [7  19]  as f(x,y)  * ,  (  x  )  y*  [7.20]  3  with  Y  The  idea  "one"  is  series  2  this  e x a m p l e , we d e f i n e  given  and of  series  standard  [1/1]  technique  the  y  2  of  the  do not  i n t o account  quite variable  indeterminacy  problem.  is  that  total  where  the  approximant of  coefficient  we d e a l  the x  3  x y 2  coefficients  ^p( )  term, 3  y.  [1/1]  a  terms;  x  (e7.21).  power  know t h e  and y  x and the  equation  2  with  powers of  highest  we need t o  f o u r t h degree  two  one  1,1 . T u y l a p p r o x i m a n t a s  (e7.20)  so t h a t  into  u s i n g one  a p p r o x i m a n t we f i n d t h e  y is  take  we a r e  of y and n o t  equation  by t h e  we form t h i s x  of  x and powers  a p p r o x i m a n t of are  now t h a t  we no l o n g e r have an  The d i s a d v a n t a g e  For  two v a r i a b l e  so we can use t h e  Furthermore  approximants,  of  [7.21 ]  a  Pade a p p r o x i m a n t s . T h i s a p p r o a c h i s  effective.  powers  = ij a „ x a=0 aB  t o decompose the  variable  variable  B(x),  of  When  x  is  Taylor 2 2  .  However,  t e r m s of  we  third  224 degree  in  x and y .  reasonable account  thing  for  In  specific  to do,  third  but  cases  this  generally  degree e f f e c t s  may be a  we would l i k e  to  b e f o r e we go on t o  fourth  degree. The above of  difficulty  a more g e n e r a l  approximant designed  p r o b l e m w i t h many of  schemes. That  for  calculating  functions the  various  been t h a t  we have  knowledge  of  where  these  application of  this  is  to  these  variable  methods e x i s t  Taylor not  for  The r e s u l t  has  require  coefficients.  satisfactory  which T a y l o r  as  For  often  coefficients  rigid analytical  is  flexible  m e t h o d , one t h a t  relies  The m o t i v a t i o n  for  this  = ?Q L  (xy) [-a L  any a r b i t r a r y  equation  (el .22)  approximantist variable  it  For  cases  a  L L  i  +  of  become f i n i t e .  L  of  f  x  L  where  the  the  re-writing  (e7.l9), a ^  The a r d e n t  j  +  L  as  y* ] [ 7 . 2 2 ]  summations  what t o d o : g i v e n terms or m o r e ,  series  in  Pade  one v a r i a b l e  truncated  separation".  comes from  1  f(x,y)  three  w i t h the a p p r o p r i a t e  s u g g e s t a more  equation  now knows p r e c i s e l y series,  two  on " s u c c e s s i v e  series,  truncation  truncated  replace  tempted to  new t e c h n i q u e  s t a n d a r d two v a r i a b l e  the  example  determine.  a p p r o x i m a n t s one  For  two  d e f i n e d approximants that  variable  f(x,y)  an  a p p r o x i m a n t s were  coefficients.  data d i c t a t e s  As an a l t e r n a t i v e  the  these  analytical  Taylor  rigidly  availability  we c a n  is,  r i g i d c o m b i n a t i o n s of  spectroscopic the  w i t h t h e T u y l method i s  Pade  contains  any  one  simply fraction. fewer  than  225 three  terms,  since  a Pade a p p r o x i m a n t c a n n o t  shall  simply  leave  these  this  way have  only  one v a r i a b l e  many a d v a n t a g e s .  d e t e r m i n i n g the  various  well  First,  these  to  i n c l u d e a new p a r a m e t e r  values  of  i  and j  (e7.22)  we d e t e r m i n e this  quite all  this  as  i n which  library,  sub-series.  flexible  in a f i t .  the  Finally,  the degrees  of  are say  From  sub-series  of  simply  call, that  approximants  various  we  the  new a p p r o x i m a n t  these  the  using  with  approximants  a^j  in  in  (unlike  t e r m b e l o n g s and t h e n  from a Pade a p p r o x i m a n t replaces  no p r o b l e m  s u i t e d t o c o m p u t e r s . As an e x a m p l e ,  wanted  equation  b e c a u s e we a r e  Pade c o e f f i c i e n t s Also,  we  Approximants b u i l t  a p p r o x i m a n t s we have  Chisholm approximants). especially  terms a l o n e .  be f o r m e d ,  are  sub-series  are  independent. We have  fractions  suggested that  c o u l d be u s e f u l  Pade a p p r o x i m a n t s or  as  fitting  rationalizing  spectroscopic data.  expected that  the  c o u l d be b e t t e r approximant. some of they  the  are  For  NH , 3  at  PH ,  realize  3  series  in  the  2  for  molecules  Taylor  we have  Furthermore,  s u i t e d to  we have  and H 0 2  p r o m i s e of  spectra  rational  a new m e t h o d , a method of  fractions  to  Finally,  we have  separation,  Pade  that a  earlier  fractions  gone a w r y .  successive  needs of  seen where  rational have  the  is  reviewed  a p p r o x i m a t i o n schemes and f o u n d well  it  expression  "one v a r i a b l e "  two v a r i a b l e s  u s i n g two v a r i a b l e  H S  linear  r e p l a c e d by a s t a n d a r d  not p a r t i c u l a r l y  attempts  For  s t a n d a r d one v a r i a b l e  previous  .spectroscopist.  functions  rational  fit to  introduced as a  starting  point  for  studies  of  this  sort  in  the  future.  BIBLIOGRAPHY  l".  H.M. H a n s o n , J .  Mol. Spect.  2.  J.K.G.  3.  K.  4.  M.R. A l i e v , S o v . P h y s . J E T P L e t t . J_4,417 A l i e v and V . M . M i k h a y l o v , J . M o l . S p e c t .  5.  I.  6.  T . Oka i n " M o l e c u l a r ( E d . N . K . Rao) V o l . 2  7.  J.K.G.  8.  S . C . Wang,  9.  G.W. K i n g , R.M. H a i n e r , 1943  10.  H.A. Kramers, G.P.  11.  C H . Townes and A . L . Schawlow i n S p e c t r o s c o p y " , M c G r a w - H i l l 1955  12.  J . K . G . Watson i n " V i b r a t i o n a l S p e c t r a and S t r u c t u r e " ( e d . J . D u r i g ) V o l . 6 Pg.1 E l s e v i e r , Amsterdam 1977  13.  J.K.G.  14.  D. Papousek and M.R. A l i e v i n " M o l e c u l a r Vibrational-Rotational Spectra", Elsevier, 1 982  15.  I.  Ozier, J .  16.  P.  Yi,  17.  A . J . D o r n e y and J . K . G . 1 972  18.  A.J.  Watson,  Fox, Phys.  J.  Watson,  Lett.  , J.  Kirschner  1971 ; M.R. 4 9 , 1 8 1974  2 7 , 1 3 2 9 1971  j_9,465-487 1970  3 4 , 2 4 3 1929 P.C.  Ittmann,  Cross,  J . C h e m . Phys 11,27  Z.Physik  Mol. Spect.  Mol. Spect.  Ozier,  1971  S p e c t r o s c o p y : Modern R e s e a r c h " , Pg 2 2 9 , A c a d e m i c P r e s s 1976  M o l . Phys. Rev.  4 0 , 5 3 6 - 5 4 4 1971  22,1329  Lett.  Rev.  Phy.  Watson  I.  Mol. Spect.  Rev.  O z i e r , Phys.  2 3 , 2 8 7 1967  5 3 , 5 5 3 1929  "Microwave  5_5,498 1975 Amsterdam  5 3 , 3 3 6 - 3 4 5 1974  C H . Anderson, Phys. Watson,  and J . K . G .  J.  Watson,  Rev.  165,92 1968  Mol. Spect. J.  42,135  Mol. Spect.  1973 19.  K.  Fox and I.  20.  C. H o l t ,  Ozier, J .  M.C.L.  Gerry  Chem.  and I.  5 3 , 1 7 9 1 - 1 8 0 5 1975 227  Phys.  5 2 , 5 0 4 4 1970  O z i e r , Can. J .  Phys.  47,347  228 21.  J.K.G.  22.  G. W i n n e w i s s e r ,  23.  C P . S l i c h t e r in ' P r i n c i p l e s H a r p e r and Row 1963 P q . 2 3 5  24.  R.L.  Poynter,  25.  D.A.  Helms and W.  Gordy,  J.  Mol. Spect.  6 6 , 2 0 6 - 2 1 8 1977  26.  D.A.  Helms and W.  Gordy,  J.  Mol. Spect.  69,473-481 1978  27.  H . H . N i e l s e n and D.M. D e n n i s o n , P h y . see a l s o r e f r 1 2 .  28.  J.M.  29.  A . R . Edmonds, " A n g u l a r Momentum i n Quantum M e c h a n i c s " P r i n c t o n U.P. P r i n c t o n N.J. 1960  30.  H . G . C a s i m i r i n 'On t h e i n t e r a c t i o n between a t o m i c n u c l e i and e l e c t r o n s ' T e y l e r s Tweede G e n o o t s c h a p , E . F . Bohn H a a r l e m 1936  31.  W. G o r d y and R . L . Cook i n " M i c r o w a v e M o l e c u l a r S p e c t r a " , W i l e y - i n t e r s c i e n c e , New York 1970  32.  H.P.  33.  R.P. Feynman, R.B. L e i g h t o n , M. Sands i n " T h e Feynman L e c t u r e s On P h y s i c s V o l 3" A d d i s o n - W e s l e y 1965  34.  J.K.G.  35.  I.  36.  M. M i z u s h i m a and P.  37.  G. H e r z b e r g " M o l e c u l a r S p e c t r a V o l 2 " , D.Van N o s t r a n d 1966  38.  Watson,  Reid  in  Benz,  J.  Chem.  J.  Chem.  J.S.  Phys.  56,2944 1972  of M a g n e t i c  "The Atomic N u c l e u s " ,  A . Bauder  Watson, 1979  Meerts,  2J_,705 1953  C r o s s , R.M.  12,210 1944  Resonance' 4 8 , 4 0 1 - 4 1 8 1983  Rev.  72,  J.  Oka, J .  Hainer  J.  93,  Chem.  and M o l e c u l a r  and G.W.  King, J .  T . O k a , F . O . S h i m i z a , T . S h i m i z a and J . K . G . A s t r o p h y s i c J . 165, L15 1971  40.  J.K.G.  41.  S.J.  Watson,  J.  Mol. Spect.  G o l d e n and E . B .  Wilson  50,281  Jr.,  J.  Spect.  Phys. 164 1982  Phys. Structure  Chem.  39.  1972  Mol.  Chem.  Mol. Spect.  Venkateswarlu,  1101 1947  P e n g u i n Books  and H s . H . G u n t h a r d , J .  M. Takami and T .  O z i e r and W.L.  P.C.  4_8,4517 1968  M a r g o l i s , M o l . Phys.  2J_,156-164 1966  70,5376  Phys.  Phys.  Watson,  1974 Chem.  Phys.  16,669  229 1 948 42.  J . L . Duncan i n " M o l e c u l a r S p e c t r o s c o p y V o l B u r l i n g t o n H o u s e , B i l l i n g s and Sons 1975  43.  L . A . Woodward i n " I n t r o d u c t i o n To The T h e o r y of M o l e c u l a r V i b r a t i o n s and V i b r a t i o n a l S p e c t r o s c o p y " , O x f o r d 1972  44.  S.J. Cyvin in "Molecular S t r u c t u r e s E l s e v i e r , Amsterdam 1972 ( g r e e n )  45.  S.J. C y v i n i n " M o l e c u l a r S t r u c t u r e s and Mean A m p l i t u d e s " E l s e v i e r , Amsterdam 1968 ( p i n k )  46.  E. B r i g h t W i l s o n J r . , J . C . D e c i u s and P . A . " M o l e c u l a r V i b r a t i o n s " M c G r a w - H i l l 1955  Cross  47.  G.O.  Mol.  S0rensen,  35,489-490  G.  Hagen and S . J .  3",  Chem Soc  and V i b r a t i o n s "  Cyvin,  J.  Square in Spect.  1970  48.  J.H.  M e a l and S.R.  49.  A . G . R o b i e t t e i n " M o l e c u l a r S t r u c t u r e By D i f f r a c t i o n M e t h o d s - V o l 1" The C h e m i c a l S o c i e t y 1973  50.  D.  Kivelson  Polo,  and E . B .  J.  Chem.  Wilson  Jr.,  Phys.  J.  2 4 , 1 1 1 9 1956  Chem.  Phys.  2 0 , 1575  1952 51.  I.M. K.N.  M i l l s i n " M o l e c u l a r S p e c t r o s c o p y : Modern Rao and C.W. Mathews E d s . A c a d e m i c P r e s s  52.  D.R. H e r s c h b a c h and V.W. 8 , 4 5 8 - 4 6 3 1961  53.  M. N a k a t a , S. Yamamoto, T . M o l . S p e c t . J_00,143 1983  54.  T.  55.  H . F . S c h a e f e r " T h e E l e c t r o n i c S t r u c t u r e of Atoms and M o l e c u l e s " A d d i s o n - W e s l e y , R e a d i n g M a s s . 1972  56.  I. O z i e r , R. L e e s 5 4 , 1 0 9 4 1976  57.  A N S I , 1 9 7 0 . G r a p h i c s y m b o l s f o r e l e c t r i c a l and e l e c t r o n i c s d i a g r a m s , Monograph Y 3 2 . 2 ( A m e r i c a n N a t i o n a l S t a n d a r d s I n s t i t u t e , New Y o r k )  58.  C. H o l t ,  59.  MOS-5 Handbook, M i c r o w a v e E a s t Syracuse N.Y.  Oka and Y.  Phd.  Morino, J .  Laurie,  J.  Chem.  Fukugama and K.  Mol.  and M . C . L .  Thesis U.B.C.  Spect.  Gerry,  Research" 1972  Phys. Kuchitsu,  8,300-314  Can. J .  of  J.  1962  Phys  1975  Systems  Inc.,  One A d l e r  Dr.,  230 60.  A . F . Harvey 1963  in  61.  R. Varma and L.W. H r u b e s h i n ' C h e m i c a l A n a l y s i s M i c r o w a v e R o t a t i o n a l S p e c t r o s c o p y , PP. 2 1 7 - 2 5 6 , A c a d e m i c P r e s s , L o n d o n , 1979  62.  H.M.  63.  H.W. K a t t e n b e r g , W. 4 4 , 4 2 5 1972  64.  I. O z i e r , M . C . L . G e r r y , D a t a J_0, 1 085 1 981  65.  I.  O z i e r , A. Rosenberg, Can. J .  66.  R.  Fox,  67.  K.T.  68.  W.A. K r e i n e r , 6 6 , 4 6 6 2 1977  69.  W.A. K r e i n e r , B . J . A 1 5 , 2 2 9 8 1977  70.  I. O z i e r , R.M. 6 5 , 1 7 9 5 1976  71.  J.  72.  J . M o r e t - B a i l l y , L. S p e c t . 25,355 1965  73.  Handbook of 1975-6  74.  J.K.G.  75.  D.L.  76.  J . L . Duncan and I . A . 2 0 , 5 2 3 - 5 4 6 1964  77.  T . S h i m a n o u c h i , I. Nakagawa, J . J . M o l . S p e c t . J_9,78-107 1966  78.  O. R e d l i c h ,  79.  D.M. D e n n i s o n , Rev.  80.  W.H. S h a f f e r , 5 6 , 8 9 5 1939  Pickett,  Appl.  Phys.  Hecht  "Microwave  Rev.  , J.  E n g i n e e r i n g " , Academic  O p t . J_9,2745-2749  A.G. Robiette,  5,355  G r a y and A .  Orr,  J.  U.  Phys.  Ref.  5J_, 1882 1973  M.C.L.  Phys.  Gerry,  J.  Chem.  j_5,344 1965  Gartier,  Montagutelli, J .  J.  Mills,  Rev.  Phys-.  Mol. Spect.  Robiette,  Z.Physik  Chem.  Anderson, T. Oka, Phys.  Mol. Spect.  H.H.  Spect.  1960  Anderson, T . Oka, J .  J.  Mol.  Chem.  Phys.  C h e m i s t r y and P h y s i c s ,  Watson,  J.  A 6 , 9 0 7 1972  Lees,  Moret-Bailly,  by.  1981  Gabes and A . Oskum,  Mol. Spect.  U.  Press  5 6 t h e d . CRC  Mol. Press  7 4 , 4 8 3 - 4 8 5 1979  M o l . Phys. 37,1901-1920 Spectrochimica Acta Hiraishi  and M.  Ishii,  Chem B28,371 1935 Mod.'Phys  V2,  175 1940  N i e l s e n and L . H .  Thomas, P h y s .  Rev.  231 81.  J . A l d o u s and I . M . M i l l s , 1073-1091 1962  82.  F.Y.  83.  P. P u l a y , 1978  84.  R.K. Can.  85.  I. O z i e r , P o n - n y i n g Y i , A . Khosha and N . F . P h y s . R e v . L e t t . 2 4 , 642 1970  86.  W.  87.  R.H.  Chu and T .  Oka, J .  W. M e y e r ,  Spectrochimica  Chem. P h y s .  J.E.  Boggs,  K a g a n n , I. O z i e r , G . A . J . P h y s . 5 7 , 5 9 3 1.979  Itano,  J.  Kagann,  64,3487  Chem. P h y s . I.  Acta.  6 0 , 4 6 1 2 1974  J.  Chem. P h y s .  McRae and M . C . L .  72,  4941  O z i e r and M . C . L .  18,  68,5077  Gerry,  Ramsey,  1980 Gerry,  J.  Chem.  Phys.  1976  88.  ' T o x i c and H a z a r d o u s I n d u s t r i a l C h e m i c a l S a f e t y M a n u a l ' , I n t e r . T e c h . I n f o . I n s t . 1977: a l s o N.I. Sax , ' D a n g e r o u s P r o p e r t i e s of Industrial M a t e r i a l s ' , R e i n h o l d . 1968  89.  J . W . M e l l o r i n "A C o m p r e h e n s i v e T r e a t i s e On I n o r g a n i c and T h e o r e t i c a l C h e m i s t r y V o l . IX" W i l l i a m s C l o w e s and Sons 1970  90.  G.S. B l e v i n s , 1955  A.W.Jache,  91.  P. H e l m i n g e r , A 3 , 1 2 2 1971  E.L.  92.  Wm.B. O l s o n , A . G . M a k i , 5 5 , 2 5 2 - 2 7 0 1975  93.  A.V. Burenin, V.P. M e l ' n i k o v and S . M . 1 982  94.  P.K.L.  95.  K. S a r k a , D. Papousek 3_7, 1-19 1 971  96.  M. C a r l o t t i , G. D i L o n a r d o and L. S p e c t . J_02,310-319 1983  Fusina,  97.  V . M . M c C o n a g h i e and H . H .  Phys.  Yin  and K . N .  W.  Beeson  Gordy  Jr.,  R.L.  W.  , Phys. Gordy,  Sams,  J.  Phys.  Mol.  Kazakov, A . F . Krupnov, Shapin, J . Mol. Spect. Rao,  J.  Mol. Spect.  and K . N .  Rao,  Nielsen,  Rev.  J.  97,684 Rev.  Spect. A.A. 94,253-263  29,486-491 Mol. J.  Rev.  1969  Spect. Mol. 75,633  1 949 98.  G . D e A l t i , G . C o s t a , and V . A c t a 2 0 , 9 6 5 1964  Gallasso,  Spectrochimica  232 99.  C C . L o o m i s and M.W.P. 1951  100.  J . E . D r a k e and C . R i d d l e , 14,McGraw H i l l 1972  101.  R.C  102.  S.P. Belov, A . V . Burenin, L.I. G e r s h t e i n , A . F . Krupnov, V . N . M a r k o v , A . V . M a s l o v s k y and S . M . S h a p i n , J . M o l . S p e c t . 8 6 , 1 8 4 - 1 9 2 1981  103.  Y . M o r i n o , K. K u c h i t s u and S. A c t a 2 4 A , 3 3 5 1968  104.  R.M. G a r v e y , F . C . De L u c i a Phys. 3 J _ , 2 6 5 - 2 8 7 1 9 7 6  105.  E.A.  1 982  Rondeau,  J.  Strandberg,  Phys.  Inorganic  Rev.  Synthesis  5,798 X I I I pg  Chem. E n g . D a t a J_1_, 124 1966  and J . W .  Cohen and H.M. P i c k e t t ,  Mol. Spect.  93,83-100  107.  R. Klement i n "Handbook o f P r e p a r a t i v e Inorganic Chemistry V o l . 1 " , 2 n d . E d . B r a u e r , Academic P r e s s 1 9 6 3  108.  J.  109.  J . M . Hoffman, H.H. N i e l s e n and K . N . Rao, Fur E l e c t r o c h e m i e £ 4 , 6 0 6 1 9 6 0  110.  A . G . Maki,  New S c i e n t i s t  R.L.  1973  111.  P.K.L.  112.  A. B a l d a c c a ,  Sams and W.B.  V.M. Devi  G. T a r r a g o ,  M o l . Spect.  and K . N . R a o , J .  Chem.  Phys.  5J_, 1 9 9 - 2 0 7 1 9 7 4 M o l . Spect.  M. Dang-Nhu a n d A . G o l d m a n , J .  P.  Helminger  115.  K.  Kijima  116.  M.H. S i r v e t z  and W. G o r d y ,  and R . E .  S.  L.  and T . T a n a k a ,  1 953  Kukolich,  9_4,393 1 9 8 2  I.M.  J.  1981  114.  118.  Olson,  Zeitschrift  1980  88,311-322  117.  5 5 , 4 4 8 8 1971  74,769 1977  Y i n and K . N . R a o , J .  8j_,179-206  113.  Phys.  Cederberg, Mol.  S.G. Kukolich,  58,4502  Chem.  J.  Spectrochimica  106.  Emsley,  J.  Yamamoto,  Mills,  Phys.  J.  Rev.  Jr.,  J.  Schaum and A . M u r r a y ,  Spectrochimica  188,100 1969  M o l . Spect.  Weston  Mol. Spect.  8J3,62 1981  Chem. J.  Phys.  21,898  Mol. Spect.  Acta j_6,35 1 9 6 0  233 119.  P.  Gans  in  " V i b r a t i n g M o l e c u l e s " Chapman and H a l l  120.  K.  Kuchitsu, J .  121.  A . R . H o y , I.M. M i l l s 2_4, 1 265-1 290 1972  122.  L. P a u l i n g and E . B . W i l s o n J r . , " I n t r o d u c t i o n M e c h a n i c s " M c G r a w - H i l l 1935  Mol. Spect.  7,399-409 1961  and G . S t r e y ,  123. G . H e r z b e r g " M o l e c u l a r S p e c t r a V o l 1", D.Van N o s t r a n d 1950  Mol.  Shaffer,  J.  Chem.  126. W.H.  H a y n i e and H . H .  Phys.  Phys. To Quantum  and M o l e c u l a r  124. M. J o h n s o n and D . M . D e n n i s o n , P h y s . 125. W.H.  1971  9,607  Nielsen, J .  Rev.  Structure  4 8 , 8 6 8 1935  1941  Chem.  Phys.  21 , 1839  1 953 127.  J.H.  M e a l and S.R.  128.  R.L.  Arnett  Polo,  and B . L .  J.  Chem.  Phys.  Crawford J r . ,  J.  2 4 , 1 1 2 6 1956  Chem.  Phys.  18,  118 1950 129. L . M . S v e r d l o v , M.A. Kovner and E . P . K r a i n o v i n " V i b r a t i o n a l S p e c t r a of P o l y a t o m i c M o l e c u l e s " , John W i l e y 1974 130.  E.C.Curtis,  J.  131. W.S. B e n e d i c t 1 957 132.  L.H.  Jones,  133.  K . N . Rao, McDowell,  134.  E.A.  Mol. Spect. and E . K .  W.W.  389-410 1963  J_4,279-291 and 292-307 1964  Plyler,  B r i m and K . N .  Can. J .  Phys 3 5 , 1 2 3 5  Rao,  Mol.  J.  W.W. B r i m , J . M . H o f f m a n , L . H . J o n e s , J . M o l . S p e c t . 7,362-383 1961  Cohen and H.M.  Pickett,  J.  Mol. Phys.  135. H e l m i n g e r , F. De L u c i a and W. G o r d y , J . 2 9 , 9 4 - 9 7 1971 136.  Spect.  P. H e l m i n g e r , F . C . P.A. S t a a t s , Phys.  5 0 , 727 1983  Mol.  De L u c i a , W. G o r d y , H.W. R e v . A 9 , 1 2 - 1 6 1974  R.S.  Spect. Morgan and  137. W.W. B r i m Phd T h e s i s , O h i o S t a t e U n i v e r s i t y , 1960 ; a n d W.W B r i m , H.H N i e l s e n and K . N . Rao i n " T h e M o l e c u l a r S p e c t r o s c o p y S y m p o s i u m " , O . S . U . 1960 138. V . S p i r k o , J . M o l . S p e c t . J_0J_,30-47 1983; a l s o , W.T. Weeks, K . T . Hecht and D.M. D e n n i s o n , J . M o l . S p e c t .  234  8,30-57  1962  139.  M.R.  Aliev  and J . K . G .  Watson,  J.  Mol. Spect.  6J_,29 1 9 7 6  140.  M.R.  Aliev  and J . K . G .  Watson,  J.  Mol. Spect.  74,282-293  141.  D.C.  McKean and P . N .  142.  I.M.  Mills  143.  B.  Crawford,  144.  K.  Shimoda,  1979  2J_,181-189  145.  M.D.  , Pure A p p l . J. Y.  1980  Marshall  85,322-326  Schatz,  Chem.  Chem.  Chem.  Phys.  Ueda and J .  and J . S .  J.  Phys.  325-344  20,977  J.  1956  1965  1952  Iwahora,  Muenter,  24,316  Appl. Mol.  Phys.  Spect.  1981  146.  G . T a r r a g o P r i v a t e C o m m u n i c a t i o n To I.  Ozier  147.  S . P . B e l o v , A . V . B u r e n i n , O . L . P o l a n s k y and S . M . Shapin, J . Mol. Spect. 9 0 , 5 7 9 - 5 8 9 1 9 8 1 ; also A.V. B u r e n i n , O . L . P o l y a n s k y and S . M . S h a p i n , O p t . Spektrosk. 5 3 , 6 6 6 - 6 7 2 1982  148.  A.V. B u r e n i n , T . M . F e v r a l ' s k i k h , E.N. K a r y a k i n , O.L. P o l y a n s k y and S . M . S h a p i n , J . M o l . S p e c t . 1 0 0 , 1 8 2 - 1 9 2 1 983  149.  A.V. Burenin, O.L. Polyansky Spektrosk. 5 4 , 4 3 6 - 4 4 1 1983  150.  L . D . G . Young and A . T . Y o u n g , J . Q u a n t . Radiat. Transfer 2 0 , 5 3 3 - 5 3 7 1978  151.  L.L.  Strow,  152.  R.H.  Kagann,  J.  Mol.  I.  Spect.  and S . M .  9_7,9-28  Ozier, M.C.L.  Shapin,  Opt.  Spectrosc.  1983  Gerry,  J.  Mol.  Spect.  7J_,28l-298  1978  153.  D.  J.  154.  R.H.  155.  H. P a d e , S c i e n t i f i c t r a n s a c t i o n s superieure in P a r i s 1892  156.  ' T h e Pade A p p r o x i m a n t i n T h e o r e t i c a l P h y s i c s ' , G . A . Baker J r . and J . L . Gammel e d i t o r s , Academic P r e s s 1 9 7 0  157.  ' P a d e A p p r o x i m a n t s and t h e i r a p p l i c a t i o n s ' , G r a v e s - M o r r i s E d . , Academic p r e s s 1 9 7 3  Shanks, Ree  Math,  and W.G.  and P h y s . ,  Hoover,  J.  34,  Chem. of  1 1955 Phys. the  J_2, 9 3 9 1 9 6 4 ecole  normale  P.R.  235 158. G . A . Baker J r . , 'Essentials A c a d e m i c P r e s s 1975  of Pade A p p r o x i m a n t s ' ,  159.  ' P a d e and R a t i o n a l F r a c t i o n A p p r o x i m a t i o n ' , E . B . and R . S . V a r g a E d . , Academic p r e s s 1977  160.  ' L e c t u r e N o t e s i n P h y s i c s : Pade A p p r o x i m a n t s Method and i t s A p p l i c a t i o n t o M e c h a n i c s ' , H. Cabannes E d . S p r i n g e r - V e r l a g 1976  161.  R . C . Johnson  162.  R.L. Burden, J . D . F a i r e s , A . C . Reynolds i n A n a l y s i s ' P r i n d l e , Weber and S c h m i d t 1981  163.  P.R.  Graves-Morris  164.  J.L.  Gammel r e f ( r l 5 9 )  pg 365  165.  J.L.  Gammel r e f ( r 1 6 0 )  pg 141  166.  J . S . R . C h i s h o l m , M a t h , o f C o m p u t a t i o n 2 7 , 841 1973 see a l s o : R . H . J o n e s and G . J . M a k i n s o n , J . I n s t . Maths A p p l i e s J _ 3 , 299-310 1974 J . S . R . C h i s h o l m a n d J . M c E w a n , P r o c . R. S o c . L o n d . A . 3 3 6 , 421-452 1974 J . S . R . C h i s h o l m r e f ( r 1 6 0 ) pg 34  167.  A . K . Common and P.R. G r a v e s - M o r r i s , A p p l i c . J_3229-232 1974  ref(rl57)  Saff  pg 53  ref(r160)  'Numerical  pg 55  J.  Inst.  Math.  168. M. M i z u s h i m a , J . M a t h . P h y s . J _ 2 , 2216 1971 : and i n ' T h e T h e o r y o f R o t a t i n g D i a t o m i c M o l e c u l e s ' John W i l e y 1 975 169.  P . M . Morse and H. F e s h b a c h , ' M e t h o d s o f T h e o r e t i c a l P h y s i c s V o l II' pg 1008 f f McGraw H i l l 1953  170.  C C . Costain,  171.  C H . Lutterodt,  172.  J.  Karlsson  173. A . H . T u y l  Phys. J.  R e v . 8 2 , 108 1951 Math.  Anal.  and H. W a l l i n  ref(rl60)  pg 225  Applic.  ref(r159)  5 3 , 89 1976  pg 83  

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