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Chemical applications of pulsed and steady-state nuclear magnetic resonance Shaw, Keith Newman 1971

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11157  CHEMICAL APPLICATIONS OF PULSED AND STEADY-STATE NUCLEAR MAGNETIC  RESONANCE  ' by K e i t h N. 'Shaw, 'B.Sc.  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  " i n the Department  Chemistry  We a c c e p t t h i s t h e s i s as conforming required  THE  standard  UNIVERSITY OF BRITISH COLUMBIA  t o the  In p r e s e n t i n g an the  advanced degree at Library  I further for  this thesis  shall  the  f u l f i l m e n t of  University  of  agree that p e r m i s s i o n f o r  by  his  of  this thesis  representatives.  be  for f i n a n c i a l gain  CH€hA\ S T K - Y  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r ; 8, C a n a d a  the  It i s understood  Columbia  shall  the  requirements  Columbia,  for reference  extensive  g r a n t e d by  permission.  Department o f  British  make i t f r e e l y a v a i l a b l e  s c h o l a r l y p u r p o s e s may  written  in partial  that  not  and  copying of Head o f my  be  I agree  that  study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  .  ABSTRACT  A g e n e r a l m a t r i x f o r m u l a t i o n f o r the e f f e c t s o f chemical exchange p r o c e s s e s  i n h i g h r e s o l u t i o n n u c l e a r magnetic resonance  has been developed  which a l l o w s a c o n c i s e d e s c r i p t i o n and e f f i c i e n t  numerical  c a l c u l a t i o n o f exchange m o d i f i e d l i n e s h a p e s f o r an a r b i t r a r y  number o f s p i n - s i t e s .  A complete d e s c r i p t i o n o f a l l r a t e p r o c e s s e s i s  c o n t a i n e d i n a s i n g l e m a t r i x , and both f i r s t -  and second-order  systems (as d e s c r i b e d i n terms o f a s p i n d e n s i t y m a t r i x ) through  (NMR)  specific  forms f o r a s p i n - s i t e f r e q u e n c y  spin  a r e accommodated  matrix.  The h i n d e r e d r o t a t i o n about t h e N-C bond i n s u b s t i t u t e d amides has been s t u d i e d on a comparative  basis using free energies o f  a c t i v a t i o n d e r i v e d from f i r s t - o r d e r r a t e c o n s t a n t s o b t a i n e d by complete analyses o f d i g i t a l  lineshape data.  has been.used f o r r o u t i n e i t e r a t i v e  A v e r s a t i l e FORTRAN computer program lineshape f i t t i n g  and i t has been  shown t h a t , even w i t h the p r e c i s i o n now a t t a i n a b l e u s i n g t h i s the most r e l i a b l e k i n e t i c parameter i s the f r e e energy Huckel  H-MO  technique,  of activation.  and s e m i - e m p i r i c a l SCF-LCAO-MO c a l c u l a t i o n s have been used  i n a d e s c r i p t i o n o f the e l e c t r o n i c f a c t o r s d e t e r m i n i n g t h e b a r r i e r s to h i n d e r e d r o t a t i o n and the charge  d i s t r i b u t i o n s i n t h e amides s t u d i e d  experimentally. A p p l i c a t i o n o f the F o u r i e r t r a n s f o r m i n h i g h r e s o l u t i o n has  NMR  a l s o been c o n s i d e r e d i n d e t a i l , w i t h p a r t i c u l a r emphasis upon  q u a n t i t a t i v e l i n e s h a p e s t u d i e s u s i n g t h e d a t a a v a i l a b l e from p u l s e d NMR experiments.  simple  The advantages o f the p u l s e method w i t h  d a t a a c q u i s i t i o n have been examined, and t h e n u m e r i c a l  digital  computations  and  corrective efficient  factors  i n v o l v e d have been i n c o r p o r a t e d  computer program.  i n t o a general  and  TABLE OF CONTENTS  Abstract Acknowledgements Chapter 1  Introduction  Chapter 2  Theory  2.1  Bloch equations  2.2  M o d i f i e d Bloch equations  2.3  Saturation  2.4  Zero s a t u r a t i o n  2.5  Chemical exchange i n f i r s t - o r d e r s p i n systems  2-6  ^ h ^ ^ i c a ^ exch^o^e i n se^ond'-crder s p i n systems  Chapter 3  effects limit  Instrumentation  3.1  FT-1064 computer-spectrometer unit  3.2  R f - p u l s e gate  Chapter 4  interface  E x p e r i m e n t a t i o n and c a l c u l a t i o n s  4.1  Hindered r o t a t i o n  i n substituted  4.1.1  N.,N-dimethyl  carbamyl  chloride  4.1.2  N,N-dimethyl  carbamyl  bromide  4.1.3  Methyl N,N-dimethyl  4.1.4  NjN-dimethyl  4.1.5  Formamide  4.2  Hindered r o t a t i o n calculations  4.3  S e m i - e m p i r i c a l SCF-LCAO-MO  amides  carbamate  carbamyl  flouride  i n amides., Huckel  MO  calculations  Chapter 5  F o u r i e r Transform  Applications  5.1  Basic  formulation  5.2  Resonance  5.3  F i n i t e complex f o u r i e r t r a n s f o r m  168  5.4  Phase  177  5.5  Signal  5.6  High i - e s o l u t i o n  condition  163  corrections zero c o r r e c t i o n  189  NMR  Appendix 1  Lorentzian  Appendix 2  Gaussian F o u r i e r T r a n s f o r m  References  155  191  F o u r i e r Transform  pair  pair  LIST OF TABLES F o l l o w i n g page  2.1  B a s i s (eigen) f u n c t i o n s f o r ABX ( J and c o r r e s p o n d i n g energy l e v e l s .  = 0) s p i n  system 51  2.2  Transition  f r e q u e n c i e s f o r ABX ( J ^ g = 0) s p i n system  51  2.3  B a s i s f u n c t i o n s f o r g e n e r a l ABX s p i n system and s p i n t r a n s i t i o n operator.  67  2.4  Transitions  71  4.1  K i n e t i c d a t a f o r N,N~dimethyl  carbamyl c h l o r i d e , neat  4.2  K i n e t i c d a t a f o r N,N-dimethyl solution  carbamyl c h l o r i d e ,  f o r AB-part o f g e n e r a l ABX spectrum  carbamyl c h l o r i d e  CCli  100  (  103  4.3  N,N-dimethyl  4.4  K i n e t i c data f o r N,N-dimethyi  4.5  K i n e t i c d a t a f o r methyl N,N-dimethyl  4.6  K i n e t i c d a t a f o r N,N-dimethyl  4.7  Spectral  4.8  K i n e t i c d a t a f o r formamide,  4.9  Huckel MO d a t a f o r h i n d e r e d r o t a t i o n i n s u b s t i t u t e d N,N-dimethyl amides  135  4.10  Parameters f o r CNDO/2 SCF-LCAO-MO c a l c u l a t i o n s  145  4.11  CNDO/2 MO d a t a f o r formamide  146  parameters f o r  a c t i v a t i o n parameters carbamyl bromide carbamate  carbamyl f l u o r i d e  103 107 109 116  N-formamide, acetone s o l u t i o n 121 acetone s o l u t i o n  125  LIST OF FIGURES Following page 2.1  Motion of an isolated nuclear magnetic dipole  13  2.2  Nuclear spin isochromats in the rotating frame of reference  17  2.3  Two-site exchange absorption mode lineshapes  25  2.4  Saturation effects for two-site chemical exchange system  30  Modified Lorentzian component spectral lines and resultant absorption mode exchange lineshapes, k < ft  41  Combined absorption and dispersion functions and resultant absorption mode exchange lineshapes, k < ft  41  Modified Lorentzian component functions and resultant absorption mode exchange lineshapes, k > ft  45  2.7  AB-part of f i r s t order ABX NMR spectrum  52  2.8  Intramolecular exchange lineshapes for AB-part of a first-order ABX spin system  59  2.9  X-part of first-order ABX NMR system  61  2.10  Intramolecular exchange lineshapes for X-part of a first-order ABX spin system 61  2.11  AB-part of general ABX spectrum  72  2.12  X-part of general ABX spectrum  75  3.1  FT-1064 computer sweep control  82  3.2  FT-1064 control sequence  82  3.3  Spectrometer-computer interface unit  84  3.4  Differential amplifier circuit  85  2.5 (a)  2.5 (b)  2.6  3.5  (a)  R f - p u l s e gate c i r c u i t  90  (b)  Control dc-pulse generator c i r c u i t  92  3.6  R f - p u l s e gate o p e r a t i o n a l c h a r a c t e r i s t i c s  92  4.1  Lineshape f i t s  4.2  A r r h e n i u s p l o t s f o r N,N-dimethyl carbamyl n e a t l i q u i d and CCli, s o l u t i o n  4.3  f o r N,N-dimethyl carbamyl c h l o r i d e  100  chloride, 103  V a r i a t i o n o f a c t i v a t i o n parameters f o r h i n d e r e d r o t a t i o n i n N,N-dimethyl carbamyl c h l o r i d e  103  A c t i v a t i o n parameters o b t a i n e d from complete l i n e s h a p e a n a l y s e s f o r N,N-dimethyl carbamyl chloride  105  4.5  Lineshape f i t s  f o r N,N-dimethyl carbamyl bromide  107  4.6  Arrhenius plot  f o r N,N-dimethyl carbamyl bromide  107  4.7  Arrhenius plot CHC13 s o l u t i o n  f o r methyl N,N-dimethyl  4.4  4.8  carbamate, 109  Temperature dependence o f c h e m i c a l s h i f t s f o r N,N-dimethyl carbamyl f l u o r i d e , CCli» s o l u t i o n  116  4.9  Lineshape f i t s  f o r N,N-dimethyl carbamyl f l u o r i d e  116  4.10  Arrhenius plot  f o r N,N-dimethyl  117  4.11  Temperature dependence o f c h e m i c a l s h i f t s f o r N - f o r m a m i d e , acetone s o l u t i o n  122  D e n s i t y m a t r i x l i n e s h a p e f o r N-formamide ABCX s p i n system w i t h i n t r a m o l e c u l a r exchange  125  carbamyl f l u o r i d e  15  4.12  15  4.13  Arrhenius p l o t  4.14  C o r r e l a t i o n o f f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n w i t h H u c k e l MO d i f f e r e n t i a l TT-energy  136  E f f e c t s o f v a r i a b l e c a r b o n y l oxygen Huckel t h e o r y Coulomb i n t e g r a l f o r methyl N,N-dimethyl carbamate  137  C o r r e l a t i o n o f f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n w i t h t h e group e l e c t r o - n e g a t i v i t y o f the X - s u b s t i t u e n t i n N,N-dimethyl amides  139  4.15  4.16  f o r N - f o r m a m i d e , acetone s o l u t i o n 15  126  4.17  C o r r e l a t i o n o f f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n w i t h t h e N-C ir-bond o r d e r o b t a i n e d from Huckel MO c a l c u l a t i o n s .140  4.18  Huckel MO e l e c t r o n i c charge d e n s i t y maps f o r the (a) unconjugated and (b) c o n j u g a t e d s t a t e s o f formamide  141  Huckel MO e l e c t r o n i c charge d e n s i t y maps f o r t h e (a) unconjugated and (b) c o n j u g a t e d s t a t e s o f carbamyl f l u o r i d e  141  S t r u c t u r e o f formamide used i n CNDO/2 SCF-LCAO-MO calculations •  143  4.19  4.20  4.21  T o t a l energy d i f f e r e n c e s f o r formamide p l a n a r ground s t a t e and v a r i a b l e geometry h i n d e r e d r o t a t i o n t r a n s i t i o n state 149  4.22  S t r u c t u r e o f carbamyl f l u o r i d e used i n CNDO/2 SCF-LCAO-MO c a l c u l a t i o n s  .152  5.1  P u l s e d mode NMR resonance c o n d i t i o n s  165  5.2  F i n i t e Fourier transform c h a r a c t e r i s t i c s f o r (a) L o r e n t z i a n l i n e s h a p e system; and (b) Gaussian l i n e s h a p e system  174  F i l t e r and r f r e f e r e n c e phase L o r e n t z i a n l i n e s h a p e system  182  5.3  5.4  5.5  5.6  5.7  5.8  corrections f o r a  L i n e a r f r e q u e n c y dependent phase L o r e n t z i a n l i n e s h a p e system Amplitude f u n c t i o n and phase L o r e n t z i a n l i n e s h a p e system  correction f o r a 185  correction f o r a •  R e s o l u t i o n enhancement, m o d i f i e d L o r e n t z i a n shapes  186  line187  Numerical F o u r i e r t r a n s f o r m d i s t o r t i o n due t o nonzero average s i g n a l l e v e l  190  F r e e i n d u c t i o n decay and f i n i t e F o u r i e r t r a n s f o r m spectrum f o r d i m e t h y l n i t r o s a m i n e  192  ACKNOWLEDGEMENTS  S i n c e r e thanks  are due t o Dr. L. W. Reeves under whose  s u p e r v i s i o n t h i s work was accomplished. Thanks are a l s o due t o R. Hobson, E. A l l a n Shaddick  and R. C.  f o r t h e i r c o o p e r a t i o n d u r i n g the course o f t h i s work.  Assistance  from the Chemistzy  Department t e c h n i c a l s t a f f and t h e  Computing C e n t r e p e r s o n n e l i s a l s o g r a t e f u l l y  acknowledged.  CHAPTER I.  INTRODUCTION  In g e n e r a l , the l i n e s h a p e s and widths o f n u c l e a r magnetic r e s o nance  (NMR) s p e c t r a l  processes  l i n e s are p a r t i c u l a r l y  s e n s i t i v e t o time-dependent  o c c u r r i n g w i t h i n a n u c l e a r s p i n system.  Thus, f o r over a de-  cade NMR has been a p p l i e d , t o ' v a r y i n g l e v e l s o f s o p h i s t i c a t i o n , t o the study o f c h e m i c a l r a t e p r o c e s s e s . determination  of p r e c i s e rate constants  meters f o r m o l e c u l a r p r o c e s s e s u l a r isomerism fined f i r s t to  and chemical  such  method a l l o w s the  and a s s o c i a t e d a c t i v a t i o n  exchange which may be d e s c r i b e d by w e l l de-  order r a t e constants.  A q u a n t i t a t i v e study  leads  directly processes  a l s o may l e a d i n d i r e c t l y t o fundamental i n f o r m a t i o n on the e l e c t r o n i c  s t r u c t u r e s o f the m o l e c u l a r The lies  para-  as h i n d e r e d i n t e r n a l r o t a t i o n , molec-  the e v a l u a t i o n o f p o s s i b l e mechanisms f o r these m o l e c u l a r  and  significance  systems and t r a n s i t i o n s t a t e s i n v o l v e d .  o f the NMR  i n i t s inherent a b i l i t y  cesses  from measurements on chemical  methods.  method i n chemical k i n e t i c s t u d i e s  t o provide data f o r r e l a t i v e l y  mation b e i n g u n a t t a i n a b l e through cal  This experimental  fast  pro-  systems a t e q u i l i b r i u m , t h i s  infor-  the a p p l i c a t i o n o f c o n v e n t i o n a l  chemi-  Two fundamental c h a r a c t e r i s t i c s o f NMR  a l l o w such  measure-  ments : (i)  the c h a r a c t e r i s t i c time relatively  s c a l e s i n magnetic resonance are  slow f o r m i c r o s c o p i c p r o c e s s e s .  t h a t m o l e c u l a r motions g i v e r i s e effects  i n a n u c l e a r s p i n system.  I t i s w e l l known  to generalized relaxation F l u c t u a t i n g magnetic  fields  associated  with l a t t i c e motions h a v i n g frequency  c o r r e s p o n d i n g t o the may  resonant spin p r e c e s s i o n a l  induce t r a n s i t i o n s between the  leading  components frequencies  n u c l e a r energy  to s p i n - l a t t i c e r e l a x a t i o n p r o c e s s e s and  levels observable  spectral effects.  When m o l e c u l a r systems are  studied  viscous solutions,  these random l a t t i c e f i e l d  fluctuations  show o n l y as o f the  s m a l l time-average e f f e c t s and  nuclear spins  c o n s i d e r e d to be the  c o r r e s p o n d i n g l y weak.  local  be  c o n s i d e r e d to be  m o l e c u l a r e l e c t r o n i c chemical spin coupling  rapid,  associated  with These  with  the  indirect  absence o f n u c l e a r  spinelectric  A g a i n , i f these f l u c t u a t i o n s  a time-average magnetic f i e l d  each n u c l e a r s p i n r e s u l t i n g i n a s i n g l e resonance line.  be  condition  nuclear spins  s h i e l d i n g and  mechanisms, i n the  quadrupolar i n t e r a c t i o n s . sufficiently  Under t h i s  l o c a l magnetic f i e l d s becomes dominant.  f i e l d s may  non  interaction  with t h e i r l a t t i c e environment may  time-dependent i n t e r a c t i o n o f the  fluctuating  the  as  acts  are on  spectral  Very slow f l u c t u a t i o n s , however, a l l o w each s p i n  p r e c e s s at a f r e q u e n c y c h a r a c t e r i s t i c o f i t s c h e m i c a l  to  envir-  onment r e s u l t i n g i n a spectrum o f resonance m u l t i p l e t s . the  intermediate region  l o c a l magnetic f i e l d and that  width o f the the  ( f o r a diamagnetic s p i n  fluctuations  resonance s p e c t r a l  o v e r a l l form of an  c a l l y dependent upon the magnetic f i e l d  determine the lines.  observed NMR  as  the  lineshape  Thus i t i s seen  spectrum i s  frequencies associated  fluctuations  system)  In  compared with the  with  critilocal  difference  3.  i n p r e c e s s i o n a l f r e q u e n c i e s o f n u c l e a r s p i n s i n s p e c i f i c environmental field  l o c a l magnetic f i e l d s .  S i n c e the e n v i r o n m e n t a l  d i f f e r e n c e s a r e o f t e n v e r y s m a l l , r e l a t i v e l y slow mol-  e c u l a r motions g i v e r i s e NMR s p e c t r o s c o p y  to observable s p e c t r a l e f f e c t s i n  and hence r a t e p r o c e s s e s h a v i n g  first-order  r a t e c o n s t a n t s i n t h e range 10 ^ t o 10 ^ s e c ^ may be s t u d i e d , (ii)  i n t h e a p p l i c a t i o n o f NMR i t i s p o s s i b l e t o c o n t r o l t h e mot i o n o f the e f f e c t i v e n u c l e a r magnetic d i p o l e moments through the magnitude and form o f t h e impressed and,  thereby,  s p i n system.  r f magnetic  field  a l s o t h e mode o f o b s e r v a t i o n f o r a g i v e n n u c l e a r In t h i s manner, both  NMR methods may be used chemical rate process  t o extend  studies.  type o f magnetic resonance  s t e a d y - s t a t e and p u l s e d t h e range and a c c u r a c y o f  Moreover, by u s i n g a s p e c i f i c  d e t e c t i o n , a l l phase i n f o r m a t i o n  r e l a t i n g t o the n u c l e a r motion under c o n s i d e r a t i o n i s r e t a i n e d . This f e a t u r e allows d e t a i l e d  correlation  time a n a l y s i s o f the  n u c l e a r and m o l e c u l a r motions f o r a s p i n system and l e a d s , under normal e x p e r i m e n t a l  conditions, to a direct  between the s t e a d y - s t a t e and p u l s e d mode responses onant s p i n system through The fer  Fourier transform  study o f c h e m i c a l r a t e p r o c e s s e s  of resonant  correspondence of a res-  analysis.  i s concerned  with the t r a n s -  n u c l e a r s p i n s between d i f f e r e n t magnetic environments  t h a t a g i v e n s p i n i s under the a c t i o n o f an e f f e c t i v e f l u c t u a t i n g magnetic f i e l d .  Such a t r a n s f e r p r o c e s s may be d e s c r i b e d  such  local  i n terms o f a  g e n e r a l i z e d t r a n s v e r s e s p i n r e l a x a t i o n mechanism, the t r a n s v e r s e component of  the e f f e c t i v e n u c l e a r m a g n e t i z a t i o n  giving rise  t o the resonance s i g -  4.  nal observed under normal NMR experimental conditions.  Although the  basic concept of transverse spin relaxation in terms of phase r e l a t i o n ships between individual nuclear spins (or spin isochromats) i s physic a l l y simple, an exact theoretical analysis is often subtle and complicated.  Analytical formulations for steady-state NMR lineshapes modi-  fied by nuclear spin transfer effects are necessarily complicated.even in the simplest cases., and the literature i n this f i e l d lias become quite extensive.  The importance of spin relaxation effects was > recognized  in the o r i g i n a l NMR work of Bloembergen et a l . * , who considered magnetic moments under the action of random fluctuating magnetic f i e l d s ; and 2  concurrently by Bloch , i n the introduction of his phenomenological equations of motion for a nuclear spin system incorporating a transverse '" . _ ^ T nor- ^ o . i ,3,4 spin relaxation  time,  ''  x  1  1  -"-333,  ouLoivSKy  e l a i . * presented the  f i r s t formulation for a magnetic resonance lineshape i n the presence of transfer processes within a first-order (weakly coupled) nuclear spin system.  This semi-classical treatment was based upon the Bloch equations,  and the only spectral effect, i n addition to those due to natural relaxation processes, i s considered due to random modulation of the nuclear spin precessional frequencies.  A similar description for  such a nuclear spin system, as suggested by Halm and Maxwell , was i n 5  troduced by McConnell^.  This formulation allows a simplified intro-  duction of spin transfer probabilities d i r e c t l y into the phenomenological equations, as proposed by Bloch, through terms analogous to those describing transverse spin relaxation.  These terms, involving a charact-  e r i s t i c correlation time, are an inherent part of a general stochastic theory of magnetic resonance lineshapes as exemplified by the more  sophisticated  theories  7  o f M a r k o f f i a n random m o d u l a t i o n developed  by  8 9  Anderson  and  Kubo  .  The  manner i s a l s o c o n s i s t e n t  introduction  w i t h the b a s i c  of t r a n s f e r e f f e c t s i n t h i s concept, i m p l i c i t  in  the  phenomenological Bloch e q u a t i o n s , t h a t n u c l e a r magnetic d i p o l a r motion associated  with generalized  that  i s o l a t e d n u c l e a r s p i n i n an  o f an  r e l a x a t i o n e f f e c t s may  be  a p p l i e d magnetic f i e l d .  simple uncoupled n u c l e a r s p i n system, i t i s p o s s i b l e cit  .quantum mechanical d e s c r i p t i o n  complete s p i n H a m i l t o n i a n , i n the field,  c o n t a i n s only  separations. included first  secular  absence o f a p e r t u r b i n g  terms r e p r e s e n t i n g  additional secular  order perturbation  theory.  i n the  For  matrix f o r m a l i s m ^  c o u p l e d s p i n system term d e s c r i b i n g  ( i n a non-viscous  a general  that  f l u c t u a t i n g magnetic f i e l d s  anism may levels.  be  l i q u i d ) i s due This  eigen-states  Alexander"^ ^ .  has  been developed by  Through the  t h i s work shows t h a t  the  spiri  rigorously  to the  term may  i n the  de-  mechanics in a non-secular be  consid-  s p i n system  spin-spin  so  mech-  Kaplan  '  of s p i n  and  transfer  extended  d e f i n i t i o n of a s p i n t r a n s f e r  by  operator,  d e s c r i p t i o n of s p i n t r a n s f e r e f f e c t s i s  s i s t e n t with a generalized  be  t r a n s i t i o n s between n u c l e a r energy  A g e n e r a l quantum m e c h a n i c a l f o r m u l a t i o n 14 15  e f f e c t s i n NMR  may  rmr.loar  complication  a s s o c i a t e d " w i t h the  e f f e c t i v e i n inducing  the  level  coupling  c o u p l e d  basic  coupling.  ered to r e p r e s e n t a m i x i n g o f b a s i s  expli-  term i n accordance w i t h normal  The  indirect spin-spin  an  a  r f magnetic  i n terms o f n o n - e q u i l i b r i u m quantum s t a t i s t i c a l  density  For  n u c l e a r energy  system, however, magnetic r e l a x a t i o n p r o c e s s e s can be s c r i b e d only  to avoid  on  of s p i n t r a n s f e r e f f e c t s since  Furthermore, i n c e r t a i n cases s p i n - s p i n  through an  superimposed  spin relaxation  t h e o r y d e v e l o p e d by  con20-2; Bloch  and  takes  a form completely  analogous t o t h a t used i n t h e s t o c h a s t i c 23 24  theories.  More r e c e n t l y , Johnson  25  '  and Binsch  t e r n a t i v e quantum mechanical f o r m u l a t i o n s processes for  f o rintra-molecular transfer  i n which t h e time dependence o f t h e average d e n s i t y  a l l p o s s i b l e molecular  i •  have p r o p o s e d a l -  matrix  s p i n c o n f i g u r a t i o n s i s d e s c r i b e d by a  . 11,26 operator  . „  Liouville  P u l s e d mode NMR d i f f e r s from the s t e a d y - s t a t e mode i n t h a t a l a r g e p u l s e d r f magnetic f i e l d system t o r o t a t e t h e thermal  i s a p p l i e d t o a resonant  nuclear spin  equilibrium resultant nuclear  i n t o a plane p e r p e n d i c u l a r t o t h e s t a t i c a p p l i e d magnetic F o l l o w i n g t h e p u l s e , n u c l e a r s p i n isochromats plane  and dephase due t o magnetic f i e l d  transverse r e l a x a t i o n processes, magnetization signal.  precess  magnetization field.  i n the t r a n s v e r s e  inhomogeneity and g e n e r a l  t h e t r a n s v e r s e component o f the n u c l e a r  g i v i n g r i s e t o t h e observed  NMR f r e e i n d u c t i o n decay  Thus t h e form o f t h i s decay i s a l s o s t r o n g l y dependent upon 27  nuclear spin transfer effects.  In 1961, Woessner  response o f t h e n u c l e a r m a g n e t i z a t i o n magnetic f i e l d  considered the  d u r i n g and f o l l o w i n g a p u l s e d  f o r an uncoupled s p i n system i n terms o f the m o d i f i e d  Bloch e q u a t i o n s ^ p r e v i o u s l y mentioned.  The form o f t h e p r e d i c t e d  f r e e i n d u c t i o n decay as m o d i f i e d by s p i n t r a n s f e r e f f e c t s was then 28 q u a l i t a t i v e l y v e r i f i e d e x p e r i m e n t a l l y by Reeves and Wells . However, a s i n g l e p u l s e experiment can o n l y g i v e i n f o r m a t i o n e q u i v a l e n t t o t h a t obtained  i n s t e a d y - s t a t e s t u d i e s o f chemical  rate processes.  On t h e  o t h e r hand, p u l s e d mode NMR becomes a v e r s a t i l e and independent method when m u l t i - p u l s e sequences a r e used t o produce c o r r e s p o n d i n g  s p i n echo  29 trains  .  Such p u l s e sequences e f f e c t i v e l y  magnetic f i e l d  inhomogeneity, which n o r m a l l y  remove t h e i n f l u e n c e o f l i m i t s t h e range o f r a t e s  7.  t h a t may  be measured by NMR  techniques.  C a r r and  Purcell  30  and Meiboom  31 and G i l l ematic  have developed  s p e c i f i c m u l t i - p u l s e sequences t o reduce  e r r o r s i n the d e t e r m i n a t i o n  o f t r a n s v e r s e r e l a x a t i o n times  c o n t r i b u t i o n s from s p i n t r a n s f e r p r o c e s s e s . first  Analytical  s p i n echo amplitudes Powles and  with of  Luz  by  Carr-Purcell 33 Bloom, Reeves and W e l l s  f o r simple u n c o u p l e d  s p i n systems i n  a s t o c h a s t i c t h e o r y based upon a c l a s s i c a l  the accumulated phases o f s p i n i s o c h r o m a t s . 35 36  o b t a i n e d by A l l e r h a n d and tions.  The  Gutowsky  '  statistical  formalism  '  .  accordance  averaging  S i m i l a r r e s u l t s were  u s i n g the m o d i f i e d  a n a l y s i s has been extended t o i n c l u d e c o u p l e d 37 38  using Alexander's  a  f o r m u l a t i o n s f o r the decay o f  have been developed 34  Strange  with 32  and Meiboom  r e p o r t e d the measurement o f s p i n t r a n s f e r r a t e s u s i n g such  p u l s e sequence.  and  In 1963,  syst-  B l o c h equas p i n systems  About the same time, Gutowsky, V o i d  39 Wells developed a m a t r i x f o r m u l a t i o n based upon the Anderson40 41 Weiss and Banwell-Primas t r e a t m e n t s o f g e n e r a l s p i n systems. and  In the study tage  o f the p u l s e d NMR  rate constants  of chemical  c o n s i d e r a b l y h i g h e r than those  an i n h e r e n t l a c k o f s e l e c t i v i t y transitions usually  observed  spectrum o f l i q u i d s .  a s s o c i a t e d with  an  Due  The  normally  advan-  a c c e s s i b l e to  p u l s e method, however, shows  f o r the v a r i o u s homonuclear s p i n  i n the s t e a d y - s t a t e h i g h - r e s o l u t i o n t o the wide f r e q u e n c y  i n t e n s e p u l s e d r f magnetic f i e l d ,  i n t e r a c t s with a l l n u c l e i Although  a fundamental  method i s an e x t e n s i o n to the measurement o f  q u a n t i t a t i v e s t e a d y - s t a t e methods.  NMR  rate processes,  distribution this  field  of. a g i v e n s p e c i e s i n a m o l e c u l a r  i t i s p o s s i b l e to o b t a i n s e l e c t i v i t y  system.  i n a v e r y simple  spin  42 system  u s i n g c o n v e n t i o n a l s p i n echo t e c h n i q u e s ,  general  s t u d i e s of  8.  transverse relaxation properties  of i n d i v i d u a l nuclear spin  transitions 43-  are most r e a d i l y measured u s i n g the phenomenon of r o t a r y  spin  echoes  45 Comparable s t u d i e s  i n the  s t e a d y - s t a t e mode use  double resonance  46-48 techniques  . In a d d i t i o n ,  quantitative accessible  studies  these double resonance t e c h n i q u e s  of,much slower r a t e p r o c e s s e s than those n o r m a l l y  to NMR methods.  Hindered i n t e r n a l r o t a t i o n nitrites i n that  and  the magnitude and  system.  i n m o l e c u l a r systems such as  n i t r o s a m i n e s i s a p r o c e s s of c o n s i d e r a b l e c h e m i c a l  expected to be the  form o f the  d i r e c t l y r e l a t e d to the  This  r o t a t i o n may  be  ,~ -1-  Tn,~  4 -  Ml,tn  m  4 - 1 , ~ ,1 o  nkm.n  o -v- ^  o f such m o l e c u l a r r a t e p r o c e s s e s , u s u a l l y e t i c techniques.  the N-C  importance i n the  theories  o r i g i n a l l y postulated o f the  had  group.  structure  been i n t e r p r e t e d * 5  A l t h o u g h the  is  IR  intra-  a f i r s t - o r d e r rate  1„  <- , . n +- ,-, A  +• ^  rti.Jinr  to c h e m i c a l  hindered r o t a t i o n 49 by  of  by  inaccessible  other molecules of b i o l o g i c a l s i g n i f i c a n c e tide functional  U n o l  Of p a r t i c u l a r i n t e r e s t i s the  bond o f amides as  interest  detailed electronic structure  c o n s i d e r e d i n terms o f an  n,.tl  amides,  p o t e n t i a l b a r r i e r to r o t a t i o n  molecular nuclear spin t r a n s f e r process, described „„„  allow  Pauling  of p r o t e i n s ^ containing  the  kinabout  * , being and  of  many  common pep-  spectrum o f N-methylacetamide  i n terms o f such a r o t a t i o n , the 52  the NMR spectrum o f N,N-dimethylformamide by  Phillips  analysis  i n 1955  of  demon-  s t r a t e d u n e q u i v o c a l l y the  e x i s t e n c e o f h i n d e r e d r o t a t i o n i n such a 53-57 system. As d e s c r i b e d i n the reviews a v a i l a b l e , the d e t e r m i n a t i o n o f h i n d e r e d i n t e r n a l r o t a t i o n r a t e s and the c o r r e s p o n d i n g p o t e n t i a l 58 barriers,  i n accordance w i t h f i r s t - o r d e r a b s o l u t e r a t e  substituted  amides has  become one  o f the  major t o p i c s  theory o f NMR  , in kinetic  9.  studies. exist  To  date, however, v e r y few  f o r these systems.  of c h e m i c a l l y due  i n part  Moreover, a comprehensive study o f a s e r i e s  r e l a t e d compounds has  to the n e c e s s i t y  d i t i o n s and  i n part  t o the  f e r e n t NMR  methods a p p l i e d must be  isticated kinetic  ' "  been c o n s i d e r e d s t a t e and  systematic  errors inherent  to the d e t e r m i n a t i o n of  ~.Even though the  mode NMR  are  systematic  o f such k i n e t i c d a t a i s e x e m p l i f i e d  obtained  (by the  fitting  search g r o u p s ^ initial  o f t o t a l NMR  by  development o f NMR  i n and  the  without due  Of n e c e s s i t y ,  methods f o r k i n e t i c s t u d i e s ,  of t h e i r v a l i d i t y  the  chemical applied  electronic struct-  i n t e r - r e l a t i o n s h i p s between a s e r i e s o f r e l a t e d m o l e c u l a r  systems, f o r example, the  s u b s t i t u t e d amides.  For  t h i s reason,  purpose o f the p r e s e n t t h e s i s i s to develop c o n s i s t e n t t h e o r e t i c a l and NMR  in  in p a r t i c u l a r studies.  Chemists are b a s i c a l l y i n t e r e s t e d i n the u r e s o f and  results  molecular  Furthermore, approximate methods have been  consideration  the  independent r e -  systems were chosen f o r s i m p l i c i t y o f a n a l y s i s r a t h e r than significance.  have  steady-  consistent  by  soph-  quantitative  Nonetheless,  lineshapes)  f o r N,N-dimethylformamide.  dif-  Only v e r y  r e s u l t s obtained using  validity  con-  errors involved  seldom c o n s i s t e n t .  is  results  i n the  c a r e f u l l y considered.  i n some d e t a i l * ^ , the  pulsed  This  operating  e x p e r i m e n t a l care been e x e r c i s e d  a n a l y s i s been a p p l i e d  data  y e t been r e p o r t e d .  l a c k of d e t a i l e d a n a l y s i s o f the  In a d d i t i o n , the  sufficient  not  of optimum i n s t r u m e n t a l  obtained.  r e c e n t l y has  r e l i a b l e k i n e t i c measurements  in substituted  and v e r s a t i l e  experimental techniques f o r e f f e c i e n t a n a l y s i s  k i n e t i c data i n a systematic amides.  With the  the  of  study o f h i n d e r e d i n t e r n a l r o t a t i o n advent o f f i e l d - f r e q u e n c y  locked  s t e a d y - s t a t e spectrometers  and d i g i t a l  now p o s s i b l e t o o b t a i n r e l i a b l e cessed  d a t a a c q u i s i t i o n systems, i t i s  s p e c t r a l d a t a t h a t may be r a p i d l y  t o g i v e the parameters o f chemical  are developed,  interest.  General  and p l o t t e d u s i n g a h i g h speed computer.  The  equations  c o n s i s t e n t w i t h a p r e s c r i b e d p h y s i c a l model f o r a g i v e n  r a t e p r o c e s s , and the r e s u l t a n t NMR s p e c t r a a r e n u m e r i c a l l y  digital  pro-  Experimental  computed  lineshapes i n  form can then be compared d i r e c t l y w i t h t h e o r e t i c a l  spectra.  w e l l known u n c o u p l e d AB s p i n system, and the r e l a t e d ABX s p i n  system, i s c o n s i d e r e d  i n d e t a i l t o a l l o w t h e development o f a more gen  e r a l model a p p l i c a b l e t o s t e a d y - s t a t e and p u l s e d mode a n a l y s e s comparison o f d i f f e r e n t t h e o r e t i c a l Under normal e x p e r i m e n t a l p u l s e d mode responses o f a Fourier transformation^  treatments. c o n d i t i o n s , t h e s t e a d y - s t a t e and  n u c l e a r s p i n system a r e r e l a t e d by a To d a t e ,  have not been a p p l i e d t o any e x t e n t c o p y ^ '^and  and a  Fourier transform  techniques  i n h i g h - r e s o l u t i o n NMR  spectros-  t h e emphasis has been upon the enhancement o f s i g n a l - t o 71  noise r a t i o  .  In view o f i n c r e a s e d i n t e r e s t  i n lineshape s t u d i e s ,  a t h e o r e t i c a l a n a l y s i s has been made t o a s c e r t a i n t h e p o s s i b l e advantage  o f such a t r a n s f o r m a t i o n i n t h e a n a l y s i s o f p u l s e d mode NMR  Towards t h i s end, t h e n e c e s s a r y  computer programs  computations have been developed. transform eristics tail.  A spectrometer  s t u d i e s has a l s o been developed o f the c r i t i c a l  circuitry  f o r r a p i d iiumerical system f o r F o u r i e r  and the o p e r a t i o n a l c h a r a c t -  i n v o l v e d has been c o n s i d e r e d  With the a v a i l a b i l i t y o f d i g i t a l  data.  data a c q u i s i t i o n  a c c u r a t e d a t a may be accumulated v e r y r a p i d l y i n a most  i n de-  systems, convenient  form and i t i s shown t h a t t h i s g e n e r a l method i s extremely  versatile  11.  and  does n o t i n c l u d e  many o f the i n h e r e n t d i s a d v a n t a g e s o f the c o r r e -  sponding normal s t e a d y - s t a t e methods.  The e x t e n s i o n o f d i g i t a l  tech-  n i q u e s t o m u l t i - p u l s e NMR p r o c e d u r e s a l s o has obvious advantages and may l e a d  to a s i g n i f i c a n t reduction  associated  i n the l a r g e  w i t h t h i s method a t the p r e s e n t  systematic  time.  In t h i s t h e s i s , t h e b a r r i e r s t o i n t e r n a l r o t a t i o n of s u b s t i t u t e d  amides have been c o n s i d e r e d u s i n g r o u t i n e  computer programs d e v e l o p e d .  that p r e v i o u s l y  available.  This  i n a series  total  shape f i t t i n g based upon the p h y s i c a l models, t h e o r e t i c a l and  errors  line-  expressions  d a t a has been c o r r e l a t e d  In a d d i t i o n ,  with  molecular o r b i t a l c a l c u l a t i o n s  72-74 applying  the H u c k e l  and complete n e g l e c t  of differential  overlap  75-77 (CNDO)  approximations have been used t o i n t e r p r e t the measured  k i n e t i c parameters i n terms o f m o l e c u l a r e l e c t r o n i c s t r u c t u r e .  In  p a r t i c u l a r , the p o t e n t i a l b a r r i e r t o h i n d e r e d r o t a t i o n and o t h e r tral) properties  o f the p a r e n t  compound formamide have been  i n d e t a i l f o r comparison, where p o s s i b l e , orbital and  calculations.  the r e l a t e d  been s t u d i e d  The e l e c t r o n i c s t r u c t u r e  compounds a c e t y l  to obtain  with a l t e r n a t i v e  (spec-  considered molecular  o f carbamyl f l u o r i d e  f l u o r i d e and a c e t a l d e h y d e have  also  i n f o r m a t i o n on the g e n e r a l a p p l i c a b i l i t y o f the  aforementioned approximate MO methods.  12.  CHAPTER 2.  THEORY  2.1  Bloch  To  Equations.  a l l o w a c o n s i s t e n t development o f the t h e o r y o f n u c l e a r mag-  n e t i c resonance e f f e c t s  due t o n u c l e a r s p i n t r a n s f e r , a c l a s s i c a l  model w i l l be c o n s i d e r e d s p i n system.  t o d e s c r i b e the dynamics o f a g e n e r a l  As the Bloch equations  nuclear  f o r such a system a r e fundamental  to a l l f o l l o w i n g d i s c u s s i o n s , the b a s i s f o r these briefly  vector  equations  w i l l be  formulated. The motion o f an i s o l a t e d n u c l e a r magnetic d i p o l e , u, i n a time  independent u n i f o r m  magnetic f i e l d , II . i s d e s c r i b e d i n a f i x e d • —o  r  frame  o f r e f e r e n c e Oxyz by  % dt  = yxYH ~ ~°  (2.1.1)  where y i s the n u c l e a r gyromagnetic r a t i o .  Assuming t h e magnetic  to be d i r e c t e d along the z - a x i s so t h a t H  = kH , (2.1.1) becomes  -i/£  ui .yu-*k ,  =  0  co » 0  YH  0  field  (2.1.2)  where k i s the u n i t v e c t o r i n the z - d i r e c t i o n and co i s the Larmor f r e — o quency  ( r a d s . s e c *) .  This equation  o f motion i s c o n s i s t e n t w i t h  quantum  78 mechanical concepts  i n t h a t i t i s r e a d i l y shown  by the e x p e c t a t i o n v a l u e The f i n e d by  t h a t u may be r e p l a c e d  <u>.  e f f e c t o f a c i r c u l a r l y p o l a r i z e d magnetic f i e l d , H ( t ) , de-  13.  H  on  a  x  nuclear  = ILcosoJt. H ft) 1 ' y  in  F i g . 2.1a.  ation  shown,  field  vectors In  the  That  nuclear and  i s ,  normal  polarized  with  H (t)  as  produced  in  a  terms  of  (2.1,3)  a vector  particular  geometrical  to  about  tends  precess  frequencies  general  equation  the  model  H(t),  coil.  is  defined  H (t)  1  as  configurmagnetic  <s>^ a n d OJ^ ,  respect-  (2.1.2), (2.1.4)  conditions,  field,  simple  in  w > 0  = yH,.  = 2H coso)t,  x  the  angular  the  1  experimental  magnetic  for  dipole  co  Under  represented  with  accordance  = -I-Lsinwt, i >  J  K  d i p o l e may be  shown  ively.  (t)  associated  with  a  linearly  by:  =0,  A field  (2.1.5)  of  this  form,  however,may  be  con-  79 sidered of  in  which  terms  is  of  two  equivalent  The independent  component to  that  described  complicated overall magnetic  fields,  circularly  H  by  magnetic and  H  polarized fields Eq.  one  (2.1.3).  dipolar motion  is  ,  simplified  by  due  to  the  considering  "1  * O  80 the  system  fectively of  y_ i n  follows  in the  the  a rotating time  reference  dependence  rotating  frame  from  ,  associated reference  Ouvz,  with is  H^.  defined  to. e l i m i n a t e If  the  by  (dy/dt)  ef-  time  dependence ,  i t  that  = where  of  frame  OJ ( > 0 ) Eqs.  is  + the  (2.1.2)  angular  and  , u? = - u y k r  frequency  (2.1.6)  it  is  of  shown  the  rotating  that  frame.  (2-1.6) Therefore,  F i g . 2.1  M o t i o n o f an i s o l a t e d n u c l e a r magnetic  dipole  14.  (i^)  = /*: * (w-u))k  r  ,  r  0  (2.1.7)  and t h i s e q u a t i o n o f motion may be e x p r e s s e d i n the form  where H i s an e f f e c t i v e magnetic f i e l d -eff . r  "^eff  =  W  o  r  " r ' W  ^  m  a  ^ ^  e  a s s u m e c  ^  t  n  a  t  i n the z - d i r e c t i o n and  the magnetic f i e l d v e c t o r I-I  i s a l o n g the u - a x i s o f the r o t a t i n g frame and t h a t co^ = co, c f . Eq.  (2.1.3)  Under these c o n d i t i o n s , the n u c l e a r magnetic d i p o l e p r e c e s s e s about a r e s u l t a n t time-independent magnetic f i e l d  H:  H = H , i + H __k, — 1— eff— as shown i n V i a .  (2.1.9)  2.1b. with an anuijlar freouenr.v Q ci vp.n hv  JL  *  [U-u^f+l^]*  (2.1.10)  The n u c l e a r magnetic resonance c o n d i t i o n may now be d e f i n e d as t h a t c o r r e s p o n d i n g t o a maximum time-independent p e r t u r b a t i o n o f the n u c l e a r d i p o l a r motion by t h e a p p l i e d magnetic f i e l d ,  H(t) , t h i s  con-  d i t i o n b e i n g d e f i n e d by  co = co = co . r o Thus, (JJ may be r e f e r r e d t o as the resonance q  i s o l a t e d nuclear spin.  v  (2.1.11) J  (Larmor) f r e q u e n c y f o r an  A l s o , i n accordance w i t h Eqs. (2.1.8, 10, 11):  and thus the c o n d i t i o n on co d e f i n i n g resonance a l s o determines the  15.  s t e a d y - s t a t e c o n d i t i o n f o r the n u c l e a r system i n t h e r o t a t i n g of  reference.  That i s , under t h i s  c o n d i t i o n , an i n i t i a l l y i n - p h a s e  r e l a t i o n s h i p between y_ and H^, c o r r e s p o n d i n g t o maximum i s time-independent.  frame  interaction,  At e x a c t resonance, the r e s u l t a n t magnetic  i n the r o t a t i n g r e f e r e n c e frame i s s i m p l y  field  and hence the magnetic  d i p o l e p r e c e s s e s about the u - a x i s a t the f r e q u e n c y OJ^.  In t h e f i x e d  r e f e r e n c e frame, t h i s resonance p r e c e s s i o n a l motion appears as a slow n u t a t i o n superimposed upon t h e Larmor p r e c e s s i o n a l motion w i t h a c o r r e s p o n d i n g change in. d i p o l e o r i e n t a t i o n , as d e f i n e d by the angle 3 i n Fig.  2.1b.  Under s t e a d y - s t a t e n u c l e a r magnetic resonance c o n d i t i o n s , a spectral  line of f i n i t e  magnetic f i e l d s  width i s observed.  The e f f e c t s o f c o n t r i b u t i n g  at. t h e s i t e o f a s p e c i f i c , n u c l e a r s p i n w i t h i n  a. g i v e n  s p i n system, i n t h e absence o f an a p p l i e d o s c i l l a t o r y magnetic  field,  may be c o n s i d e r e d i n terms o f : (i)  static field  (ii)  s p i n - s p i n r e l a x a t i o n , and  (iii) A local  spin-lattice  s t a t i c magnetic f i e l d  inhomogeneity,  relaxation. f o r a g i v e n n u c l e a r s p i n i s determined  by the m o l e c u l a r e l e c t r o n i c environment, g i v i n g r i s e shift  e f f e c t , and by the s m a l l e r inhomogeneity a s s o c i a t e d w i t h the ap-  p l i e d magnetic f i e l d , cy  H^.  These f i e l d s  give r i s e  d i s t r i b u t i o n and an a s s o c i a t e d s p e c t r a l  t o a resonance f r e q u e n -  l i n e broadening.  form o f the inhomogeneity b r o a d e n i n g i s d i f f i c u l t but  to a b a s i c chemical  to define  The e x a c t explicitly,  a common assumption made i s t h a t the resonance f r e q u e n c y d i s t r i b u -  t i o n may be d e s c r i b e d by a L o r e n t z i a n l i n e s h a p e f u n c t i o n , f ( w ) , o f t h e  16.  general  form  (2.1.12)  where ca i s the independent nance f r e q u e n c y and  f r e q u e n c y v a r i a b l e and oi^ i s the mean r e s o -  f o r a given spectral  5 i s a l i n e - w i d t h parameter.  line;  A i s a n o r m a l i z a t i o n constant  The c h a r a c t e r i s t i c  l i n e - w i d t h at h a l f -  maximum, A/2, i s then g i v e n s i m p l y as 2/£. An a l t e r n a t i v e f u n c t i o n i s that corresponding t o a Gaussian  lineshape  f r e q u e n c y d i s t r i b u t i o n as  d e s c r i b e d by the l i n e s h a p e f u n c t i o n  $Cw) with a corresponding The  =  A  e*p [ - ^ ( w - u ) , ) * ' ]  (2.1.13)  l i n e - w i d t h 2(2£n2) / £ .  l i f e t i m e w i t h i n a n u c l e a r Zeeman energy  s i d e r e d t o be l i m i t e d due t o simultaneous  mutually  l e v e l may be con-  induced  transitions  between a d j a c e n t resonant n u c l e a r s p i n s , these t r a n s i t i o n s b e i n g i n duced through accordance process  a time-dependent magnetic d i p o l a r i n t e r a c t i o n .  w i t h the quantum mechanical  uncertainty principle,  l e a d s t o an e f f e c t i v e broadening  responding the induced  t o a f i n i t e width transitions, this  o f the Zeeman energy  f o r the s p e c t r a l  line  l i f e t i m e broadening  a Lorentzian lineshape function.  n  cor-  b e i n g d e s c r i b e d by  A characteristic spin-spin relaxline-width i n  m  w i t h Eq. (2.1.12),  such a  associated with  a t i o n time, T 2 0 ' a y then be d e f i n e d t o d e s c r i b e t h i s accordance  In  w i t h £ = T„ . 2o  t o as the n a t u r a l t r a n s v e r s e s p i n r e l a x a t i o n  T_ i s normally 2o time.  referred  I t s h o u l d be noted  t h a t , i n more g e n e r a l terms, a c h a r a c t e r i s t i c time d e f i n e d i n t h i s manner e f f e c t i v e l y d e s c r i b e s the time-dependent phase  relationships  17.  between p r e c e s s i n g  resonant nuclear spins  Thus the concept o f a g e n e r a l i z e d  within  transverse  a given s p i n  system.  s p i n r e l a x a t i o n time i s  a v e r s a t i l e means o f d e s c r i b i n g many r e l a x a t i o n p r o c e s s e s g i v i n g t o s p i n dephasing e f f e c t s .  In c o n t r a s t  c e s s e s showing a c o n s e r v a t i o n  o f energy w i t h i n  l a t t i c e r e l a x a t i o n processes involve the  spin  ibrium the  to spin-spin  relaxation  a s p i n system,  pro-  spin-  a net t r a n s f e r o f energy between  system and i t s l a t t i c e environment t o m a i n t a i n thermal  conditions  rise  f o r the complete s p i n - l a t t i c e system.  i n t e r a c t i o n between the s p i n system and i t s l a t t i c e  Also,  equildue t o  environment,  f l u c t u a t i n g e n v i r o n m e n t a l magnetic f i e l d s may induce t r a n s i t i o n s between n u c l e a r Zeeman energy l e v e l s l e a d i n g l i n e b r o a d e n i n g which may be d e s c r i b e d  t o an a d d i t i o n a l  by a s p i n - l a t t i c e  spectral  relaxation  time, T,. x Under the assumption t h a t lineshape i s described spin relaxation  the complete resonance  by a L o r e n t z i a n  function,  a total  spectral transverse  time, T^, may be d e f i n e d by  A-  =  J_  +  i  +  J,  +  JL.  (2.1.14)  •k  where T^ and £ a r e r e l a x a t i o n times d e s c r i b i n g and  a d d i t i o n a l transverse  the  study o f l i q u i d  field  inhomogeneities  spin relaxation processes, r e s p e c t i v e l y .  In  systems, m o t i o n a l a v e r a g i n g determines a g e n e r a l  condition:  i n the absence o f s p i n - l a t t i c e r e l a x a t i o n mechanisms due t o paramagn e t i c species  and n u c l e a r q u a d r u p c l a r i n t e r a c t i o n s .  18.  To a l l o w a d i r e c t o u t l i n e d above, i t i s now  a p p l i c a t i o n o f the s e m i - c l a s s i c a l  c o n s i d e r e d t h a t a macroscopic  concepts  nuclear spin  2 29 isochromat due  '  , M(co  , <{>) , may  be d e f i n e d as the r e s u l t a n t  to a l l s p i n s h a v i n g a resonance  time-dependent phase angle (j).  for  such an isochromat  frequency CO and an a s s o c i a t e d q  In t h i s manner, the resonance  d i s t r i b u t i o n f o r a. g i v e n s p e c t r a l e r t i e s o f a macroscopic  field  frequency  l i n e i s simply r e l a t e d to the  n u c l e a r s p i n system.  The  equation of  H ( t ) ) i s g i v e n i n accordance  w i t h Eqs.  prop-  motion  i n the r o t a t i n g frame o f r e f e r e n c e w i t h  a n g u l a r frequency co ( c o r r e s p o n d i n g to t h a t o f the a p p l i e d magnetic  magnetization  an  oscillatory  (2.1.8) and  (2.1.9) as  =  dt  where u, V,  Xl£l*,4>>**]-^i  -  12,  iVj, \z  -^CMx-Mojk Ti  have been d e f i n e d as the C a r t e s i a n components of the  isochromat M(x,  (j)), as shown i n F i g . 2.2,  and x i s now  the  v a r i a b l e , d e f i n e d as x = co - co , w h i l e M i s the thermal o o value of M  (2.1.16)  .  The  t o t a l e f f e c t i v e magnetic  e q u a t i o n i s 11 = H i  + (x/y)k_, c f . Eq.  field  (2.1.9).  e q u a t i o n s o f motion  to oo > OO . q  I t s h o u l d be  (2.1.16),  -°° < x <  noted 00  such  the component  are g i v e n as  AM' +  JL u.  -  X-U.  - >cV • = +  dt ^  From Eq.  equilibrium  H appearing i n t h i s  t h a t the v a r i a b l e x has been d e f i n e d on the i n t e r v a l t h a t x > 0 corresponds  independent  -  u i i ^  i_  T*.  V  o  NU •  =  -  =  - ^ ( ^ - ^ o )  U3  ±  (2.1.17)  19.  where O J ^ = yH^,  c f . Eq.  (2.1.4).  These e q u a t i o n s  the f o r m u l a t i o n o f n u c l e a r s p i n t r a n s f e r e f f e c t s  2.2  M o d i f i e d Bloch  Nuclear w i l l be  spin transfer effects  considered i n i t i a l l y  this  the b a s i s f o r  i n the f o l l o w i n g s e c t i o n .  Equations.  s p i n system may  be  i n an uncoupled  AB  spin  i n terms o f s t o c h a s t i c p r i n c i p l e s  a simple p h y s i c a l model f o r such foi'  form  effects.  The  system  to  s t e a d y - s t a t e NMR  develop  spectrum  r e p r e s e n t e d as shown below:  A  B  0 'A  l  Site-A, a s s o c i a t e d with a d i s t i n c t chemical  s h i f t , may  w i t h a resonance  B  e l e c t r o n i c magnetic environment  be c o n s i d e r e d i n terms o f a mean s p i n isochromat,  frequency  x^ = -Q r e l a t i v e  the normal r o t a t i n g  M ,  t o the average f r e q u e n c y a j : Q  ft, in  and  frame o f r e f e r e n c e . At any  (2.2.1)  g i v e n time, the  t i o n a l p o p u l a t i o n of n u c l e a r spins a s s o c i a t e d with s i t e - A , p  frac-  defines A.  the magnitude o f the s p i n isochromat e f f e c t s , M^  and M^  precess  M^.  independently  In the absence o f s p i n  transfer  i n the a p p l i e d magnetic f i e l d  H .  Due  to a r e v e r s i b l e m o l e c u l a r p r o c e s s  described s p i n may  by be  f i r s t - o r d e r rate constants c o n s i d e r e d to be  a random s t a t i s t i c a l this  (such as a h i n d e r e d  (i)  The  as  a l l spins  k  ,  t r a n s f e r r e d between the  manner.  t r a n s f e r p r o c e s s are  and  basic  rotation)  a resonant s i t e s A and  assumptions made i n  nuclear B in  describing  follows: remain i n s i t e - A w i t h a mean l i f e t i m e x. A.  until  an  i n s t a n t a n e o u s t r a n s f e r takes p l a c e  site-B.  Precessional  are n e g l e c t e d and type has  no  as  e f f e c t s i n the transfer  f o r any k. per A.  transfer interval  i n t o a s i t e o f the  o b s e r v a b l e e f f e c t on  transfer into a different site (ii)  the  i s considered;  u n i t time f o r t r a n s f e r i n t o s i t e - B . t h i s prob-  -  -  to the f r a c t i o n a l  p^;  l i f e t i m e x^  i s independent of the  r e l a x a t i o n times T,. and IA (iv)  only  spin i n s i t e - A , there i s a constant p r o b a b i l i t y  population the  same  s p i n system,  a b i l i t y being inversely proportional  (iii)  into  associated  spin  T„ ; A  2k'  n u c l e a r s p i n i s o c h r o m a t s r e l a x i n d e p e n d e n t l y except for spin transfer e f f e c t s .  Under these assumptions, the c o n s t a n t k^  are  simply r e l a t e d k  and  site  l i f e t i m e x^ and  the  corresponding  by  = x~\  A  (2.2.2)  i n accordance w i t h the P k A  A  rate  = p k B  p r i n c i p l e of d e t a i l e d b a l a n c e  B  .  81  (2.2.3)  with  PA  As the observed  +  =  ]  •  -  NMR s i g n a l i s due t o t h e t r a n s v e r s e component o f t h e  nuclear magnetization magnetization  P B  (and f o r a n a l y t i c s i m p l i c i t y ) a complex  transverse  f o r s i t e - A , G^, i s d e f i n e d i n t h e r o t a t i n g frame o f  reference by  G  A  =I A U  as shown i n F i g . 2.2. may now be e x p r e s s e d and  +  i v  A  l>  ( 2  The B l o c h e q u a t i o n s  ; 2  4 )  f o r the s p i n isochromat M  i n complex form i n accordance w i t h Eqs. (2.1.16)  (2.1.17) as  cfct  (2.2.5)  1  where e. =  ioo 1 -p  and io = w  A  A  - fl. Assuming n e g l i g i b l e s a t u r a t i o n  O  and q u a s i - s t e a d y - s t a t e c o n d i t i o n s :  w  1  ->• 0, M . ->- M ZA  J.  and Eq. (2.2.5)  OA  reduces t o  <U*k + with M  = P^M >  a n c  q  A  ^  U  A  + t w ) &  A  =  - U U 3  i  P  a  / ^  0  (2-2.6)  t h e t h e r m a l e q u i l i b r i u m z-component magneti-  z a t i o n f o r t h e complete n u c l e a r s p i n system. to the s i t e - B magnetization,  G = G  A s i m i l a r equation applies  so t h a t f o r t h e complete s p i n system:  A  + G . B  (2.2.7)  Following McConnell^, due  to nuclear  such t h a t  -k.G A  from s i t e - A The  A.  to  time-dependence  spin transfer  defines  and k  time-dependence  (2.2.6),  the  is  incorporated  ito , c f .  •—^— 2A  verse relaxation  + k  -  time,  t o be s i t e + i u = r^  w i t h r^  =  + k^.  pressed  i n the matrix  _  X  T^,  Cx  of  constant  for  magnetization  this  transfer.  the normal Bloch e q u a t i o n s ,  Eq.  (2.2.5).  site-A  Eqs.  magnetization:  Assuming the  i n the  it  absence of  follows  + ft) ,• x = to -  OJ  total  spin  trans-  transfer  that (2.2.10)  Q  modified Bloch equations  +  s  l u),  ~ ~  t e r m ioo M P may be r e f e r r e d Q  an a p p l i e d o s c i l l a t o r y  conditions,(dG/dt)  transverse  by:  may now be  ex-  form  ..15: The  magnetization  magnetization  independent,  These  of  of motion f o r  as d e f i n e d  + i (x  transverse  may now be d e s c r i b e d  rate  into  give a modified equation  site-B  the  transfer  the f i r s t - o r d e r  with a.  processes,  of  is  for  =  processes  the r a t e  and a c o u p l e d e q u a t i o n  of  R-G = —  1  P  follows  = - iw.M P, 1 o—  (2.2.11)  °~  t o as a d r i v i n g  magnetic f i e l d :  = 0 , and i t  1  term,  to^ = 0 .  from Eq.  as i n  Under  (2.2.11)  the  absence  steady-state that (2.2.12)  v  23.  or  explicitly,  PA  Gc  4  6  The  total  complex t r a n s v e r s e m a g n e t i z a t i o n  Eq.  (2.2.12) as a function, o f the independent v a r i a b l e x:  6  may now be d e r i v e d from  + lC  whe re  6  -  C  =  and  (2.2.13)  The  NMR a b s o r p t i o n mode corresponds  verse nuclear magnetization  i n quadrature  t o t h e component t r a n s -  phase with  the a p p l i e d o s c i l -  l a t o r y magnetic f i e l d v e c t o r , H , as shown i n r i g . 2.2.  Thus the ab-  s o r p t i o n mode l i n e s h a p e , V ( x ) , i s g i v e n i n accordance w i t h Eq. (2.2.13) as the imaginary  p a r t o f G (x), v i z . ,  (2.2.14)  6 4 Z  C  Z  This lineshape function applies to a general-population  t w o - s i t e un-  coupled AB n u c l e a r s p i n t r a n s f e r system under normal s t e a d y - s t a t e conditions.  The term w^M  expresses  NMR  t h e l i n e a r dependence o f s i g n a l  24.  amplitude upon the and  may  be  considered The  considerable lineshape  magnitude o f H  , i n the  as a n o r m a l i z a t i o n  two-site  equal p o p u l a t i o n  c h e m i c a l i n t e r e s t , and  f u n c t i o n V(x)  takes the  absence o f s a t u r a t i o n e f f e c t s ,  factor. system  (p^ = p  = 0.5)  i n accordance with Eq.  i s of  (2.2.14),  the  s i m p l i f i e d form  (2.2.15) where  6 • ^ ( l TzvTz  and  A i s the n o r m a l i z a t i o n  of the  f u n c t i o n V(x)  x  For k << ft, x' NMR  1  2ft.  As  a condition  2  -  = ± ft, and  the  I t i s now i n terms  l c ) ^ - ^ +  2k ) 2  determined  rad.  1 5  (2.2.15),  From Eq.  -1 sec."  positions  (2.2.16)  i n t h i s l i m i t o f v e r y slow s p i n t r a n s f e r , s p e c t r a l l i n e s s e p a r a t e d by  r a t e of s p i n t r a n s f e r i n c r e a s e s ,  = 0 and  the  as  f o r s p e c t r a l l i n e c o a l e s c e n c e may  x'  The  maxima are  = ±(ft  a  constant.  spectrum c o n s i s t s o f two  shift  +  /2 k -  convenient t o c o n s i d e r  general  be  x'  the  chemical  decreases and  defined  the  thus  as  ft.  (2.2.17)  nuclear  spin transfer effects  of (i)  slow s p i n t r a n s f e r :  (ii)  intermediate  (iii)  f a s t spin transfer:  exchange l i n e s h a p e s  k < 0.2  k -  spin transfer: k > 5  for a representative  ft, ft,  ft. equal p o p u l a t i o n  system,  25.  as g i v e n i n accordance w i t h Eq. (2.2.15), are shown i n F i g . 2.3 f o r a range o f v a l u e s  o f the c h a r a c t e r i s t i c parameter k/Q,.  For c l a r i t y ,  these  l i n e s h a p e s have b e e n - n o r m a l i z e d , "through the c o n s t a n t A, to an a r b i t r a r y maximum independent o f k. chemical  s h i f t between s i t e s  I t i s important  i n the absence o f exchange, 2fi, d e f i n e s  the o v e r a l l range o f the r a t e c o n s t a n t NMR  lineshape analyses.  t o note t h a t t h e  k t h a t may be determined  from  A l s o , the g e n e r a l p o p u l a t i o n a b s o r p t i o n mode  l i n e s h a p e d e r i v e d f o r the t w o - s i t e exchange system, Eq. (2.2,14), i s c o n s i s t e n t with The  87 t h a t g i v e n by Gutowsky and Holm ".  validity  o f the phenomenological B l o c h e q u a t i o n s has 20  been c o n s i d e r e d modified  Bloch equations  transfer effects matrix  i n detail  and, under the assumptions o u t l i n e d above, may be extended t o d e s c r i b e n u c l e a r s p i n  i n a g e n e r a l f i r s t - o r d e r NMR exchange system.  A  f o r m u l a t i o n f o r an n - s i t e system, based upon the equations o f  motion d e r i v e d above f o r the simple  t w o - s i t e exchange system,  allows  a c o n c i s e d e s c r i p t i o n o f g e n e r a l t r a n s f e r e f f e c t s and a l s o leads t o expressions  which a r e r e a d i l y adapted t o e f f i c i e n t  computer c a l c u -  lations . Initially, all  frequency  i t i s assumed t h a t exchange p r o c e s s e s  modulate  d i f f e r e n c e s , |to^ - w_. | , a s s o c i a t e d with n d i s t i n c t  s i t e s o f d i f f e r i n g Larmor f r e q u e n c y , a t r a n s f e r of nuclear magnetization  OK .  Again  f o l l o w i n g McConnell^,  may be d e s c r i b e d f o r t h e £-mode  i n the form  (2.2.18)  k/fl = 0.032  0.16  0.64  3.2  0.5  P  A  =  0.65  Two-site exchange a b s o r p t i o n mode l i n e s h a p e s  26.  cf. Eq. (2.2.8).  The indices r e f e r to s i t e and the double indices r e f e r  to f i r s t order transfer rate constants between s i t e s such that k.. £. 1  i j  represents the rate of transfer of £-mode magnetization from s i t e i to_ site j . for  This form of notation i s important i n defining matrix elements  a m u l t i - s i t e exchange process.  The time-dependence of the u-mode  magnetization associated with s i t e - i may now be expressed i n the Bloch form  cit  k  l  u  with w^ = co -  i  1  1  (2.2.19)  . As usual, co i s the angular frequency of the o s c i l -  latory radio frequency f i e l d defining the normal rotating frame of reference. <o. i s defined as the sit.e-i resonance ' l  frenuencv. and T_. 2i  is  the corresponding spin-spin relaxation time including a contribution from magnetic f i e l d inhomogeneity. the system i s then given by chromats being independent  u =  The t o t a l u-mode magnetization for Z u^,  the s i t e magnetization iso-  except f o r transfer e f f e c t s .  Considering  this magnetization as a column vector, u, Eq. (2.2.19) may be expressed i n matrix form to include a l l s i t e s , v i z . , dec oft with  -t-  = T^ + j£, where  fc-.UL "  =  W.V "  =  0  (2.2.20) 7  i s a diagonal n x n matrix with general  element T • and K i s the transfer rate matrix, n x n, with elements 2i = '  K = U  gV^  and  ^  = -V-^y  , L*L  C - - ) 2  2  21  27.  S i m i l a r l y , w i s a diagonal frequency  d e v i a t i o n matrix., n x n w i t h  elements w^ and V i s the v-mode m a g n e t i z a t i o n v e c t o r w i t h V..  elements  The o f f - d i a g o n a l elements o f the r a t e m a t r i x K are i n d i v i d u a l  first  o r d e r r a t e c o n s t a n t s f o r each p a i r o f s i t e s ,  and the d i a g o n a l  elements are the sums o f r a t e c o n s t a n t s from each s i t e - i .  Thus,  the d i a g o n a l elements r e p r e s e n t the o v e r a l l r a t e p r o c e s s e s  forn  sites.  A l s o , any column has a zero sum, c o n s i s t e n t w i t h  balance  of rate processes,  -  +  S i m i l a r matrix equations tization  detailed  0  can be w r i t t e n f o r the v- and z-mode magne-  transfers invoked  i n a chemical  exchange p r o c e s s :  (2.2.22)  where  = yH^,  =  + JC  w i t h T^ a d i a g o n a l s p i n - l a t t i c e  t i o n m a t r i x , n x n w i t h elements T.., T,. b e i n g the s p i n - l a t t i c e  li a t i o n time for  forsite-i;  each s i t e  li  b  relax-  1  P i s a. column v e c t o r o f the f r a c t i o n a l p o p u l a t i o n  such t h a t  Z p. = 1 and M . = M p. where M .  .  oi  1  o i  e q u i l i b r i u m v a l u e f o r the z-mode m a g n e t i z a t i o n coupled matrix equations magnetization  relaxa-  (2.2.20),  (2.2.22),  i s g i v e n e x p l i c i t l y by  i s the thermal  o  in site-i.  S o l v i n g the  t h e • s t e a d y - s t a t e V-mode  28.  L-  with  .  _  §2.-il-Ti •  Z ^-  (2.2.23)  /-  Here I i s the u n i t n x n m a t r i x and e  and .e^ a r e the i n v e r s e m a t r i c e s  c o r r e s p o n d i n g t o R^ and R^, r e s p e c t i v e l y . For an a r b i t r a r y r e f e r e n c e f r e q u e n c y to , the independent f r e q u e n c y v a r i a b l e x may be d e f i n e d as x = to - w  w. i  = x  such t h a t f o r s i t e i  -ft. I  (2.2.24) ft.  a.  = to.  -  l  where ft. i s the chemical l  to  o  s h i f t w i t h r e s p e c t t o to . o r  now d e f i n e s the s t e a d y - s t a t e NMR  The v e c t o r V —  a b s o r p t i o n mode spectrum as a f u n c t i o n  o f x, v i z . ,  tt  Vtt) = ^ V fcO = 1=1 L  I-i  (2.2.25)  The elements o f the m a t r i x w i n Eq. (2.2.23) a r e now d e f i n e d by Eq. (2.2.24) and I i s the row v e c t o r w i t h each o f n elements equal t o unity. The c o r r e s p o n d i n g m a t r i x e q u a t i o n s and  z-mode m a g n e t i z a t i o n s  u.  ~  f o r the s t e a d y - s t a t e u-  are now g i v e n i n terms o f V as  g . z  w. V (2.2.26)  29.  This formulation nuclear  allows  magnetization  a compact and v e r s a t i l e d e s c r i p t i o n of  t r a n s f e r e f f e c t s i n a f i r s t - o r d e r NMR 46  i n c l u d i n g those a s s o c i a t e d with  spin-lattice relaxation  general  spectrum, 48  '  and  nuclear  spin transfer saturation.  2.3  Saturation  Effects.  Under normal s t e a d y - s t a t e S' may  be  d e f i n e d f o r a g i v e n NMR  conditions, a saturation factor  s p e c t r a l l i n e i n the  /  form  1  S  =  /  ±-  ,  0<  S  ^1  1 + CJ^TIU Now,  s i n c e i n the p r e s e n c e of chemical  exchange s a t u r a t i o n e f f e c t s are  J  .I  r i o  1  • „  r?~  i n that part defined  as  ^-i  are c o n t a i n e d  _i-  j-i—  r>  —  S'  =  I +  o  ? •>  Wi  _i  ,  i  —  C - - ) 2  3  1  2 it  i s c o n v e n i e n t t o r e p l a c e co^ by  an e q u i v a l e n t parameter 3 d e f i n e d  i n terms o f an average s a t u r a t i o n f a c t o r S'  where the p a r t d e f i n e d an average v a l u e i n the  as  i n terms of the r e l a x a t i o n times T^  for a l l sites.  In t h i s way,  Eq.  and  T^ i s  (2.3.1) i s w r i t t e n  form  S'  =  I  +  g  £ . 2  (2.3.2)  A complete a n a l y s i s o f s a t u r a t i o n  e f f e c t s would i n c l u d e  the i n t e r 83  r e l a t i o n s h i p w i t h the d e v i a t i o n  from s t e a d y - s t a t e  conditions  Under normal e x p e r i m e n t a l c o n d i t i o n s , however, l o c k e d - f i e l d spectrometers allow conditions  a very  good a p p r o x i m a t i o n t o the r e q u i r e d  and the above f o r m u l a t i o n The  slow passage  i s adequate.  e f f e c t s o f s a t u r a t i o n on the NMR a b s o r p t i o n  shape have been s t u d i e d  f o r a p a r t i c u l a r two-site  mode  line-  (A- and B - s i t e s )  c h e m i c a l exchange system d e s c r i b e d  by the s p e c t r a l p a r a m e t e r s : p. = 0.5, A 20 = \0. - 0 \ = 10.0 Hz and T „ . = T„_ = 0.64 sec. A l s o , as i s u s u a l A B 2A 2B f o r l i q u i d systems, i t i s assumed t h a t T.. = T_.. The s p i n - s p i n r e l a x XA A a t i o n t i m e , T , chosen c o r r e s p o n d s t o a L o r e n t z i a n f u l l width a t h a l f U  1  1  O A  i-r\.  maximum o f 0.5 Hz and 20, i s t h e c h e m i c a l s h i f t between s i t e s absence o f exchange, c f . Eq. (2.2.24). numerical a n a l y s i s , using  i n the  By means o f an i t e r a t i v e  Eqs. (2.2.23) and (2.2.25), v a l u e s  o f the  s a t u r a t i o n parameters 3 and S' were determined c o r r e s p o n d i n g t o a prescribed  mean d e v i a t i o n o f t h e l i n e s h a p e  t h a t f o r the r e f e r e n c e  function, V(x),  l i m i t o f zero s a t u r a t i o n  centage d e v i a t i o n , AV, was d e f i n e d  (3=0).  from A mean per-  f o r N d a t a p o i n t s as  .100 u=i i n which V\ (x) and \\ (x) a r e the r e f e r e n c e function values, x^.  and s a t u r a t e d  lineshape  r e s p e c t i v e l y , c o r r e s p o n d i n g t o the f r e q u e n c y  value  The r e s u l t s o f such an a n a l y s i s a r e shown i n F i g . 2.4 f o r  AV = 2 %  and 5% as p l o t s o f 3 and S' as a f u n c t i o n o f l o g j ( k / 2 Q ) . 0  Figure  2.4  S a t u r a t i o n parameters f o r t w o - s i t e chemical exchange system.  I t i s seen t h a t s a t u r a t i o n e f f e c t s f o r a f i x e d magnitude o f the i r r a d i a t i o n r f magnetic f i e l d , v e r y slow spectral AV H  1  9.4  l i n e s o f minimal width  (Y  x 10  , are most severe  (k << ft) and v e r y f a s t  = 5% the v a l u e s of H^ NMR  2H  h  = 2.7  x 10*  f o r k/2ft = 0.01  mgauss, r e s p e c t i v e l y .  2  s p e c t r a l frequency  of  (k >> ft) exchange c o r r e s p o n d i n g (0.5 Hz).  rads s e c "  i n the l i m i t s  1  to  For a mean d e v i a t i o n and  1.0  are computed f o r  g a u s s ) t o be  0.56  - 1  These f i e l d s  x IO"  correspond  d i s t r i b u t i o n o f o n l y 0.002 and 0.04  and  2  to a  Hz.  However,  84 Grunwald et a l .  have shown e x p e r i m e n t a l l y t h a t H^  fields  o f magnitude g r e a t e r than those c a l c u l a t e d above g i v e d i s t o r t i o n due  to s a t u r a t i o n f o r chemical  combination  s t a t e passage c o n d i t i o n s and r f power l e v e l s As H^  a general broadening  of s p e c t r a l  apparent  field  A 5% d e v i a t i o n as  (k <<  (~ ft r a d . s e c . ) - 1  line intensity  to  considered However, (k -  ft)  g r e a t e r than t h a t c a u s i n g ft)  is s t i l l  In the i n t e r m e d i a t e exchange r e g i o n , the  S t r i c t l y , a spectral  observed  leads t o an i n c r e a s e d  twenty times  i n the slow exchange r e g i o n  l i n e s are o f maximal width sity.  l i n e s and  i n the i n t e r m e d i a t e exchange r e g i o n  with a magnitude a p p r o x i m a t e l y  (see F i g . 2.4).  steady-  determine  l e a d t o an e r r o r i n a f i t t e d k v a l u e o f 5-10%.  an i r r a d i a t i o n  distortion  of  This  i s increased, s a t u r a t i o n gives r i s e  f i r s t - o r d e r r a t e c o n s t a n t , k.  above may  (H )  order  negligible  exchange systems.  implies that, i n a c t u a l f a c t a complicated  saturation effects.  an  acceptable spectral  and hence minimal  inten-  is linearly proportional  t o Hj o n l y i n the absence of s a t u r a t i o n e f f e c t s , but i t i s seen t h a t the i n t e n s i t y and hence s i g n a l - t o - n o i s e r a t i o of a r e c o r d e d may  be  lineshape  i n c r e a s e d t o l e v e l s a c c e p t a b l e f o r the measurement o f r e l i a b l e  32.  data f o r lineshape f i t t i n g by  increasing H  without  adverse  f o l l o w i n g an e x p e r i m e n t a l  s a t u r a t i o n i n the slow exchange l i m i t .  d i s t o r t i o n due t o s a t u r a t i o n , linearity  Relative H  check f o r n e g l i g i b l e field  strengths  c o r r e s p o n d i n g t o n e g l i g i b l e s a t u r a t i o n d i s t o r t i o n may then be e s t i m a t e d over a complete range o f the c h a r a c t e r i s t i c parameter k/2fl from F i g . 2.4, and hence r e l i a b l e  s p e c t r a f o r complete l i n e s h a p e f i t t i n g  tained, i n effect,  i n the l i m i t  2.4  o f zero  may be ob-  saturation.  Zero S a t u r a t i o n L i m i t . '  For a g e n e r a l f i r s t - o r d e r NMR n - s i t e exchange system, the zero s a t u r a t i o n l i m i t may be d e f i n e d by w  ->• 0 (to = yti^) and thus the  m o d i f i e d Bloch e q u a t i o n d e s c r i b i n g t h e s t e a d y - s t a t e z-mode m a g n e t i z a t i o n is  g i v e n i n m a t r i x form i n accordance  the i n v e r s e o f the m a t r i x R .  with  change, R  =  i n each s i t e  £i»  a n c  *  n  e  n  for  e  M_ = M P_. z  (2.2.22) as  In the absence o f c h e m i c a l ex-  That  i s , the z-mode m a g n e t i z a t i o n  i s d i r e c t l y p r o p o r t i o n a l t o the s i t e f r a c t i o n a l p o p u l a t i o n ,  p..*, and i s independent dition  c  w i t h Eo  of s p i n - l a t t i c e relaxation effects.  i s shown t o be v a l i d a l s o i n t h e presence  This  o f chemical  con-  exchange  a l l exchange r a t e s and i s c o n s i d e r e d i n d e t a i l f o r a g e n e r a l two-  s i t e system a t a l a t e r p o i n t . The m o d i f i e d Bloch e q u a t i o n s d e s c r i b i n g c h e m i c a l i n the l i m i t  o f zero s a t u r a t i o n reduce  g i v e n i n accordance  exchange  t o two c o u p l e d m a t r i x  w i t h Eqs. (2.2.20) and (2.2.22) :  equations  3ft  '~  ~  =  (2.4.1)  •from which the s t e a d y - s t a t e v-mode m a g n e t i z a t i o n  V  =  -  ui  ±  M  0  C  i s g i v e n e x p l i c i t l y by  | P  (2.4.2)  with  ik Again, V(x)  =  +-  W . JL2.. w  the s t e a d y - s t a t e NMR a b s o r p t i o n mode spectrum i s determined by  as d e f i n e d i n Eq. (2.2.25).  (2,4 = 2) show? t h a t the NMR  In the l i m i t  o f zero s a t u r a t i o n , Eq.  ahsomtion intensity  i s linearly related to  the e f f e c t i v e magnitude o f the i r r a d i a t i n g r f magnetic f i e l d , i s independent o f s p i n - l a t t i c e r e l a x a t i o n e f f e c t s exchange r a t e s . this  Within  , and  f o r a l l chemical  the l i m i t s d i s c u s s e d i n the p r e c e d i n g s e c t i o n ,  i s the e x p r e s s i o n d e f i n i n g t h e l i n e s h a p e f u n c t i o n V(x) n o r m a l l y  used i n the a n a l y s i s o f s t e a d y - s t a t e NMR Although  data.  the a b s o r p t i o n mode l i n e s h a p e f u n c t i o n , V ( x ) , i s  g i v e n e x p l i c i t l y by Eq. (2.4.2) i n terms o f the r e a l m a t r i x £ , the independent v a r i a b l e x i s c o n t a i n e d  i n the frequency  w and hence an e v a l u a t i o n o f V(x) over  a specified  d e v i a t i o n matrix  frequency  r e q u i r e s an i n v e r s i o n o f the m a t r i x £ f o r each v a l u e  o f x.  range An a l t e r -  n a t i v e f o r m u l a t i o n i n the l i m i t o f zero s a t u r a t i o n based upon Eq. (2.4.1) a l l o w s a much s i m p l i f i e d magnetization,  calculation of V(x).  A complex  transverse  G, may be d e f i n e d f o r an n - s i t e exchange system i n  34.  v e c t o r form as  u + iV ,  where the component V w i t h the j - s i t e . tization  (2.4.3)  d e s c r i b e s the v-mode m a g n e t i z a t i o n a s s o c i a t e d  From Eq. (2.4.1), the s t e a d y - s t a t e t r a n s v e r s e magne-  i s now g i v e n by  1^ and  +  LW  -1  . P  the c o r r e s p o n d i n g complex l i n e s h a p e f u n c t i o n G(x) f o l l o w s as  -1  ?  where I_ i s as d e f i n e d p r e v i o u s l y , c f . Eq. (2.2.25).  (2.4.4)  The a b s o r p t i o n  mode l i n e s h a p e f u n c t i o n , V ( x ) , i s simply the imaginary p a r t o f G ( x ) . The  independent  v a r i a b l e x appears  now o n l y i n the d i a g o n a l m a t r i x  vv, and thus i t i s p o s s i b l e t o t r a n s f o r m the m a t r i x a completely G(x)  [R  + iw] i n t o  d i a g o n a l form hence a l l o w i n g a ready e v a l u a t i o n o f  f o l l o w i n g a s i n g l e matrix d i a g o n a l i z a t i o n .  Now c o n s i d e r the  m a t r i x R d e f i n e d by  R = 1  and r e l a t e d t o R  by  + K - ifi  [R  (2.4.5)  + iw] = R + i x j ^ .  In these e q u a t i o n s , £  is a n x n  d i a g o n a l m a t r i x w i t h elements Q,y the c h e m i c a l  shift for  the j - s i t e ,  Eq. (2.2.24),  In terms  of  and I, i s the u n i t n x n m a t r i x .  the d i a g o n a l i z e d m a t r i x J\ c o r r e s p o n d i n g t o R, Eq. (2.4.4) may be  expressed  i n the form  (k iH)  l- 1  = - t u ^ M. I. S 0  - i. 5  (2.4.6)  35.  where S i s the matrix which diagonalizes R., v i z . ,  A = Sf -R-S  (2.4.7)  1  Following t h i s transformation, the evaluation of inverse matrix ments i s reduced to the determination of the matrix  ele-  of r e c i p r o c a l diagonal elements  ./V + ixj^ , and the complex function G (x) i s defined i n  component form by  Gurt  = - c u i ^ g  I:  (2-4.8)  where Xj_ i s t h e J - t h diagonal element of the matrix iV.  S_.^ and  (S  are the j J - t h elements of the transformation matrices S. and S~ ,  )^  re-  1  spectively.  - 1  As the matrix elements involved are a l l complex, the  computation of each s p e c t r a l data point defined by G(x) requires a number of operations i n complex arithmetic on a d i g i t a l computer. These operations are r e l a t i v e l y slow and hence the absorption mode lineshape function, V(x), i s f i n a l l y given i n terms of operations i n r e a l arithmetic only by defining a matrix J5 as the r e a l part of the matrix S/j^ A + ix_l] ~  1  1  .  .That i s , i n accordance with Eq . (2.4.6),  V(x) = A I/FvP, in which A i s a normalization constant, and the matrix elements B., jk  are given by  where a..^ and b j j ^  (and Xj and x|) are the r e a l and imaginary  parts  36.  o f S.  J  • (S *) ,  (and X ), r e s p e c t i v e l y ; d  K  independent  i s d e f i n e d in. terms o f the  v a r i a b l e x as  d-j - L^x Analogous to Eq.  (2.4.8), V(x)  V(x)  =  A computer program based  i s now  IB  AEI.  p  upon Eqs.  +  t  (fj_ + *>)  g i v e n as  .  (2.4.9)  (2.4.7) and  (2.4.9) has been  v e l o p e d and has been shown to g i v e r e l i a b l e and e f f i c i e n t total  lineshape analyses f o r m u l t i - s i t e Although  specific details  iterative  exchange systems.  w i l l be d i s c u s s e d i n a f o l l o w i n g  c h a p t e r d e a l i n g w i t h e x p e r i m e n t a l a p p l i c a t i o n s , one eral iterative  de-  l i n e s h a p e f i t t i n g procedure  aspect o f a gen-  s h o u l d be o u t l i n e d at t h i s  j.i. point. two  In the  sites  (m + 1 )  i and j i s determined k.  m  +  = k.  1  il  and thus  i t e r a t i o n , the r e f i n e d r a t e constant  L 1  i n accordance  ^m+1  as Ak..  m +  iJ  i l  w i t h Eq.  = T  2  = T  2  +  (K K  +  f o r any  m  (2.4.5),  + AK)  m + 1  -  -  i£  i£  where the elements o f the m a t r i x Aj( are d e f i n e d i n terms o f the mental r a t e constants Z\k  , c f . Eq.  (2.2.21).  Now,  a t i o n , t r a n s f o r m a t i o n m a t r i c e s have been determined Eq.  (2.4.7) such t h a t  A  m  =  (S- ) -R 'S 1  m  m  m  .  i n the m * t  1  increiter-  i n accordance  with  37.  Thus i n the  (m+l) ' t  1  i t e r a t i o n , i t f o l l o w s t h a t a m a t r i x A' d e t e r  mined by  A  (S-'f-lf^-S  =  1  i s approximately  diagonal.  The m a t r i x A/ may now be d i a g o n a l i z e d  e x a c t l y w i t h c o n s i d e r a b l e r e d u c t i o n i n computation time. i t e r a t i o n t r a n s f o r m a t i o n m a t r i c e s , s"" * and ( S ) 1 1  - 1  ! l 1 +  The  (m+3.)  th  * a r e r e a d i l y ob-  t a i n e d as  A  =  m + 1  = with ,S  m + 1  = S -T  and  m  ,  .  T -A'-T _1  Cr ) 1  (S" ) "* 1  ,m+l „m+l  complex m a t r i c e s A  m + 1  ,j |  1  1  -R  m + 1  -s  m + 1  =T -(S~ ) . 1  Having d e f i n e d t h e  1 m  , , _i.m+l' . n  and (S )  m  . t h i s manner, t h e compu-  t a t i o n o f t h e l i n e s h a p e f o r the r a t e c o n s t a n t s k™ (2.4.9).  a p p l i c a t i o n o f Eq. The  r e d u c e s t o an  +±  complex m a t r i x f o r m u l a t i o n o f c h e m i c a l  i s most r e a d i l y i l l u s t r a t e d by an a n a l y s i s system i n the l i m i t o f z e r o s a t u r a t i o n .  exchange  processes  o f the s i m p l e t w o - s i t e  Such a system may be c o n s i d -  ered w i t h exchange s i t e s A and B d e f i n e d by r e l a t i v e c h e m i c a l  shifts  ( r a d s . sec"-') fl = - f l andfl^= fl i n terms o f t h e independent  frequency  A  v a r i a b l e x w i t h CO = q  populations p  (co^ + Wg), c f . Eq. 2 . 2 . 2 4 ) ,  and f r a c t i o n a l  and p . The r e l a x a t i o n , r a t e and c h e m i c a l  are then d e t e r m i n e d a s :  shift  site  matrices  38.  .OL  0  ft  In t h e l i m i t o f zero s a t u r a t i o n , t h e z-mode m a g n e t i z a t i o n i s g i v e n by Eq.  (2.2.22) as  K li-Xi-i with  *A  5  k,  16  tip"  1  k,  A  A  and  D =  1  k T  .  k,  i-6  U s i n g Eq. 2.2.3), i t i s r e a d i l y shown t h a t the m a t r i x p r o d u c t J^'T^-P_ reduces t o P_ i n t h i s l i m i t , thus showing t h e v a l i d i t y o f t h e m o d i f i e d B l o c h e q u a t i o n s as expressed i n Eq. (2.4.1).  For a n a l y t i c s i m p l i c i t y  i t w i l l now be assumed t h a t t h e s p i n - s p i n r e l a x a t i o n times f o r t h e two s i t e s a r e t h e same so t h a t t h e m a t r i x _I where 2 Eq.  i s the 2 x 2 u n i t m a t r i x .  becomes a s c a l a r m a t r i x , . v i z . , The exchange m a t r i x R, d e f i n e d by  ( 2 . 4 . 5 ) , may now be e x p r e s s e d i n t h e form (2.4.10)  39.  and  the d i a g o n a l m a t r i x A. c o r r e s p o n d i n g  A = S~  to  K - i f l j i s now  K - i f l •S. If,  initially,  an equal p o p u l a t i o n system i s c o n s i d e r e d ,  d i a g o n a l elements o f the m a t r i x .A above are given  X =  k ± a  a =  w i t h k = k^ = k^.  The  1 S  ivjicrc  pl+  [k  3..  2  ,  = (u ± i f l ) / k .  j ,X = 1, 2 and s!\  i n accordance w i t h  (2.4.12).  (2.4.6) now  GcLi.) - - C u i N l 1  0  I.  S.  non-orthogonal.  the c o n d i t i o n  =1  the complex conjugate  The  rion-Hermitean,  the column v e c t o r s  are complex and  t h a t a l l n o r m a l i z a t i o n constants  dance with Eq.  o f S_. .  I t should  have been combined i n the  a l s o be factor  complex l i n e s h a p e f u n c t i o n , G ( x ) , i n a c c o r takes the  A  +  form  ( - i + i * ) l -1  The parameter a, as d e f i n e d i n Eq. for  K. - i£j, or R., i s  elements X are i n g e n e r a l complex and  S*.S.. + S * S , . .11 3 3 h 3-3  i n Eq.  as  (2.4.12)  s;  As the m a t r i x  These v e c t o r s are n o r m a l i z e d  k/2a  are g i v e n  1  d e f i n i n g the t r a n s f o r m a t i o n m a t r i x  noted  (2.4.11)  - fl ]^ ,  2  the  by  a s s o c i a t e d transformation matrices  =  the m a t r i x  for  d e f i n e d by  k > fl and k < fl, r e s p e c t i v e l y .  (2.4.11),  5  _i.  P  i s r e a l or pure  Thus, from Eq.  (2.4.13),  (2.4.13)  imaginary the  complex  40.  lineshape f u n c t i o n f o r r a t e constants expressed  with A = - u M , 1 o n  U^.Ve) )  +  r = =• + k T  and e = ft  2  &w  -  +  t w o - s i t e chemical elements  2  .  .. 4  14)  The a s s o c i a t e d  i n t h e range ft^k < °° i s  _u±kU  (2.4.14) and (2.4.15) allow  the NMR a b s o r p t i o n  - k  (2  2  lineshape f u n c t i o n f o r r a t e constants  (2.4.15)  a l u c i d and c o n c i s e d e s c r i p t i o n o f  l i n e s h a p e f u n c t i o n V(x) exchange system.  f o r the equal  population  In general, the diagonal  matrix  determine t h e p o s i t i o n s and l i n o - w i d t h s o f t h e component,  spectral lines.  ^ ft i s  i n terms o f a r e a l parameter e as  a«> . u U ^ e )  Eqs.  i n t h e range 0 £ k  lines  and t h e m a t r i x S. determines the i n t e n s i t i e s  Thus, f o r O ^ k ^ f t ,  ponents c o r r e s p o n d i n g  o f these  Eq. (2.4.14) d e f i n e s a b s o r p t i o n mode com-  to modified  L o r e n t z i a n l i n e s d e s c r i b e d by the  lineshape function  Vc*)  = A .  (r+k)+4>c  , CrA)~^X (  (2.4.16)  In t h e absence o f exchange, k = 0 (r =  , E =ft), i t i s shown t h a t 2 the spectrum c o n s i s t s o f pure L o r e n t z i a n l i n e s c e n t e r e d at x = ±ft with  f u l l widths at h a l f maximum o f 2/T^, c o n s i s t e n t w i t h  definitions  o f 2ft as the chemical  parameter d e s c r i b i n g s p e c t r a l  s h i f t between s i t e s  t h e normal  and T^ as the  l i n e - w i d t h i n t h e absence o f exchange  8? effects  In t h e presence o f exchange, t h e s p e c t r a l  components may  41.  be  c o n s i d e r e d i n terms o f a s u p e r p o s i t i o n o f pure L o r e n t z i a n  s o r p t i o n and  a s s o c i a t e d d i s p e r s i o n f u n c t i o n s , the degree o f  b e i n g determined by the f a c t o r k/e for  i n Eq.  c,  the component c e n t r e d at x  V  (2.4.16).  (x) i s g i v e n  For  abmixing  example,  by  (2.4.17)  Thus i t i s shown t h a t the  L o r e n t z i a n type  w i d t h a t half-maximum o f 2 ( ^  component has  + k), linearly  a full  line-  i n c r e a s i n g w i t h k,  and  2 a p o s i t i o n e < Q, such t h a t the with  i n c r e a s i n g k.  A s i m i l a r f u n c t i o n V_(x)  ponent c e n t r e d at x = -e.  2.5a,  the m o d i f i e d  component s e p a r a t i o n , 2e,  f o r the com-  These b a s i c l i n e s h a p e c h a r a c t e r i s t i c s  L o r e n t z i a n component s p e c t r a l l i n e s , V  V _ ( x ) , are shown as dashed l i n e s ; V ( x ) , i s shown as a f u l l bined  i s obtained  decreases  a b s o r p t i o n and  line,  and  the r e s u l t a n t l i n e s h a p e f u n c t i o n ,  c f . F i g . 2.3.  In F i g . 2.5b,  d i s p e r s i o n f u n c t i o n s , with  the component l i n e at x = e b e i n g r e s p e c t i v e l y , i n Eq.  and  (x)  are  g i v e n by  the com-  c o n t r i b u t i o n s from  the f i r s t  and second terms,  (2.4.17), are shown as the dashed l i n e s .  the computed l i n e s h a p e s have been n o r m a l i z e d ,  Again,  through the f a c t o r A,  to  an a r b i t r a r y maximum independent o f k. E x p e r i m e n t a l l y , use o f an r f p h a s e - s e n s i t i v e d e t e c t o r the o b s e r v a t i o n o f an NMR of  s i g n a l p r o p o r t i o n a l to a s p e c i f i c  the t r a n s v e r s e n u c l e a r m a g n e t i z a t i o n ,  The  component  G ( x ) , as an o s c i l l a t o r y  a r l y p o l a r i z e d component at the i r r a d i a t i o n frequency l a b o r a t o r y frame o f r e f e r e n c e .  allows  to i n t h e  d i r e c t i o n of a s p e c i f i c  line-  fixed  component  k/ft = 0 . 6 4  Figure  2.5(a)  0.80  M o d i f i e d L o r e n t z i a n component s p e c t r a l l i n e s and r e s u l t a n t a b s o r p t i o n mode exchange l i n e s h a p e s .  Figure  2.5(b)  Combined a b s o r p t i o n and d i s p e r s i o n f u n c t i o n s and r e s u l t a n t a b s o r p t i o n mode exchange l i n e s h a p e s .  of transverse magnetization  i s d e f i n e d by t h e r e l a t i v e phase angle  between t h e d e t e c t o r r e f e r e n c e r f f i e l d v e c t o r H —r f i e l d vector  and t h e r f i r r a d i a t i o n  d e f i n i n g t h e u - a x i s o f t h e normal r o t a t i n g frame o f  r e f e r e n c e , c f . F i g . 2.2. Thus i n t h e absence o f exchange e f f f e c t s , by s e t t i n g cj> = 90  t h e s p e c i f i c component c o r r e s p o n d i n g  m a g n e t i z a t i o n v e c t o r V i s observed.  to. t h e v-mode  The a b s o r p t i o n mode spectrum con-  (assuming a s i m p l e 1/T -type s p i n - s p i n  s i s t s o f pure L o r e n t z i a n l i n e s  r e l a x a t i o n mechanism) c e n t r e d a t x = ±ft o r ,  w = w  ± ft. The g e n e r a l  d i s p e r s i o n mode spectrum i s d e s c r i b e d by a l i n e s h a p e f u n c t i o n u ( x ) g i v e n i n a c c o r d a n c e w i t h Eq. (2.4.16) as  (2.4.18)  where u^ ( x ) , analogous t o V (x) i n Eq. (2.4.17), i s  (2.4.19)  T h i s form o f spectrum i s observed  by s e t t i n g cj> = 0.  The f i r s t term  i n Eq.(2.4.19) d e s c r i b e s t h e d i s p e r s i o n mode l i n e c e n t r e d a t x = +ft i n t h e l i m i t o f no exchange. for  I n t h e presence o f chemical  exchange  k < ft, t h e m i x i n g o f normal a b s o r p t i o n and d i s p e r s i o n mode NMR  s i g n a l s a s s o c i a t e d w i t h t h e v- and u-mode components, r e s p e c t i v e l y , of the transverse magnetization  i n t h e r o t a t i n g frame o f r e f e r e n c e  may now be c o n s i d e r e d q u a l i t a t i v e l y i n terms o f t h e b e h a v i o u r t r a n s v e r s e components o f t h e i n d i v i d u a l n u c l e a r s p i n M(x,  cj)). I n t h e absence o f c h e m i c a l  time d e s c r i b i n g t h e isochromat  o f the  isochromats,  exchange, T^ i s t h e r e l a x a t i o n  dephasing and c o n s e q u e n t i a l  spectral  l i n e - w i d t h due to l o c a l magnetic  field  inhomogeneity  g i v i n g r i s e to  d i s t r i b u t i o n s o f Larmor f r e q u e n c i e s c e n t r e d a t t h e A - and B - s i t e resonance f r e q u e n c i e s , x = ±ft. a coherent e f f e c t . the  individual  sites  T h i s type o f isochromat dephasing i s  In the p r e s e n c e o f chemical exchange, however,  s p i n isochromats a r e i n v o l v e d i n a t r a n s f e r between  c o r r e s p o n d i n g to d i s t i n c t  s h i f t s ) a t random times.  l o c a l magnetic  w i t h a fundamental  fluctuating  local  - 1  .  each  magnetic  f r e q u e n c y component i n t h e a s s o c i a t e d  quency d i s t r i b u t i o n o f k r a d s . s e c . the  (chemical  C o n v e r s e l y , i t may be c o n s i d e r e d t h a t  s p i n isochromat e x p e r i e n c e s a randomly field  fields  fre-  T h i s random p r o c e s s a l t e r s  form o f isochromat dephasing, which may now be d e s c r i b e d i n term 33 85  of  a probability function  '  f o r t h e isochromat r e l a t i v e phase  d i s t r i b u t i o n , and the time average  effect  f o r a l l isochromats i s  o b s e r v e d as an e f f e c t i v e m i x i n g o f the normal v e r s e m a g n e t i z a t i o n s as d e s c r i b e d by Eqs. s h o u l d be emphasized  u- and v-mode t r a n s -  (2.4.16) and (2.4.18).  It  t h a t t h e f u n c t i o n V ( x ) by d e f i n i t i o n d e s c r i b e s  the  spectrum  as o b s e r v e d i n t h e p r e s e n c e o f exchange p r o c e s s e s f o r  the  phase s e n s i t i v e d e t e c t o r r e l a t i v e phase (J> = 90  , and t h i s  will  always be r e f e r r e d t o as t h e a b s o r p t i o n mode s i g n a l .  Now, i f the  fundamental  local  field  frequency a s s o c i a t e d with the f l u c t u a t i n g  due t o a  chemical exchange p r o c e s s becomes comparable  f r e q u e n c y d i f f e r e n c e between exchange s i t e s , the  magnetic to the  2ft, t h e m o d i f i c a t i o n o f  b a s i c T^ isochromat dephasing i s expected t o be most  significant  T h i s i s a c t u a l l y o b s e r v e d i n t h e form o f maximal b r o a d e n i n g and c o a l e s c e n c e o f t h e component s p e c t r a l  lines  f o r r a t e constants  k - ft. F o r r a t e c o n s t a n t s k - ft, i t f o l l o w s from Eq. (2.4.16) t h a t  r  = A  V(x)  l  +  +  (2.4.20) Thus i t i s seen t h a t t h e l i n e s h a p e  c h a r a c t e r i s t i c s are strongly  dependent upon t h e second term, i n a d d i t i o n t o the b r o a d e n i n g des c r i b e d by the parameter r , due t o the f a c t o r k/e f a c t o r that absorption shift  describes  .  t o r a t e constant f o r a p a r t i c u l a r chemical  d i f f e r e n c e between exchange s i t e s i n the r e g i o n  about  component  F o r k = ft (e = 0) the second term i n Eq. (2.4.20)  now makes no e f f e c t i v e c o n t r i b u t i o n  to the l i n e s h a p e  f u n c t i o n V(x)  thus the spectrum c o n s i s t s o f a s i n g l e L o r e n t z i a n  x = 0 with f u l l w i d t h at half-maximum o f sity  It i s this  the w e l l known s e n s i t i v i t y o f the observed  mode l i n e s h a p e  l i n e coalescence.  and  00  line  c e n t r e d at  2r = 2\\  + ft] and i n t e n 2 f a c t o r 2Ar. Q u a l i t a t i v e l y , t h i s i s c o n s i s t e n t w i t h t h e s i m p l e  isochromat. model f o r exchange e f f e c t s p r e v i o u s l y  discussed.  As the  r a t e o f isochromat t r a n s f e r between exchange s i t e s i n c r e a s e s , o r conversely, ating  as the fundamental frequency a s s o c i a t e d  l o c a l magnetic f i e l d  w i t h the f l u c t u -  f o r any isochromat i n c r e a s e s ,  t h e time  average c o n t r i b u t i o n from t h e normal d i s p e r s i v e u-mode m a g n e t i z a t i o n i s maximized and then e f f e c t i v e l y c a n c e l l e d out f o r t h e p a r t i c u l a r c o n d i t i o n k = ft. T h i s  c a n c e l l a t i o n c o i n c i d e s w i t h an exact  o f t h e component l i n e s e p a r a t i o n , c o n s t a n t s k > ft t h e r e from a d i s p e r s i v e type  2e, t o zero.  Also,  averaging  f o r a l l rate  i s no e f f e c t i v e c o n t r i b u t i o n t o the l i n e s h a p e function.  45.  For r a t e c o n s t a n t s k > ft, t h e a b s o r p t i o n mode l i n e s h a p e f u n c t i o n i s g i v e n i n accordance  w i t h Eq. (2.4.15) as  (2.4.21)  with r = —  + k,  ft ) .  a = (k 2  2 2  That  i s , t h e spectrum  consists of  2 a modified Lorentzian line  c e n t r e d at x = 0 which may be d e s c r i b e d as  the s u p e r p o s i t i o n o f two L o r e n t z i a n components, t h a t g i v i n g a p o s i t i v e c o n t r i b u t i o n t o V(x) b e i n g d e f i n e d as  w i t h a f u l l width A ( l + k/a).  a t half-maximum o f 2 ( r - a) and an i n t e n s i t y  The parameters r and a determine  a l i n e w i d t h which de-  creases w i t h i n c r e a s i n g k and a maximum v a l u e f o r t h e i n t e n s i t y o f 2A.  factor  factor  The o t h e r component d e f i n e d i n Eq. (2.4.21) g i v e s a n e g a t i v e  c o n t r i b u t i o n and i s a L o r e n t z i a n f u n c t i o n w i t h l i n e - w i d t h 2 ( r + a ) , increasing  l i n e a r l y with k, and i n t e n s i t y f a c t o r A(l  - k/a).  Again,  f o r k - ft t h e l i n e s h a p e i s s t r o n g l y dependent upon t h e f a c t o r k/o; (as a -»- 0) and the o b s e r v e d  l i n e s h a p e i s p a r t i c u l a r l y s e n s i t i v e to  the r a t e c o n s t a n t k i n t h i s r e g i o n .  I n t h e l i m i t o f v e r y f a s t ex-  change (k » ft) the component V (x) becomes dominant and t h e a b s o r p t i o n mode spectrum 2/12  a n <  *  a  n  intensities  c o n s i s t s o f a s i n g l e L o r e n t z i a n l i n e with l i n e - w i d t h  intensity  factor  2A, c o n s i s t e n t w i t h t h e f a c t t h a t t h e  a s s o c i a t e d w i t h the component l i n e s a t x = ±ft i n the  l i m i t o f slow exchange have now combined i n the s i n g l e at t h e mean Larmor frequency.  line  centred  These l i n e s h a p e c h a r a c t e r i s t i c s a r e  shown i n F i g . 2.6 f o r two v a l u e s o f k/ft > 1 where a g a i n t h e computed  (a)  (b)  k/n  Figure  2.6  =3.2  M o d i f i e d L o r e n t z i a n component f u n c t i o n s and r e s u l t a n t a b s o r p t i o n mode exchange lineshapes.  46.  lineshapes.have of  k.  are  In  Fig.  s h o w n as  been 2.6a,  dashed  shown  as  a full  shown  to  be  (2.4.13)  A  leads  ^ p.^, to  f o r ft ^ k  Lorentzian functions  component  lines; In  and the  arbitrary  resultant  F i g . 2.6b,  maximum  lineshape  the  functions  for  the  independent V  (x)  and V _ ( x )  function  V (x)  V(x)  and V ( x )  is  are  equivalent.  matrix  treatment  two-site  exchange  complex lineshape  system  more  general  using  functions  for  Eqs.  unequal (2.4.10)  and  0 J ~ k *S fl:  j U+4)-Mf )t + U-4> W i - ^ ) f  few and  the  nearly  A similar p  an  line.  very  population,  normalized to  A  <  0 0  :  l  Ir-nt) +  i\r+<£ ) + i X  U.4-.2.3) where  with In  the  k  = %(k  this  case  compared of  the  characteristic  + k ) , 2fl = w  A  g  the  parameters  directly with  equal  parameters  parameters  the  population  now  determine  -  B  e  o)  A  and  both  defined  =  fl  a  remain  analogous  system.  are  It  &  -  ,  and  component  w  =  (to  complex, but  parameters is  by  used  interesting line  to  position  in  + to  they the  note  may  ). be  analysis that  (imaginary  these part)  and  line-width  ( r e a l p a r t ) f o r a l l r a t e constants  equal p o p u l a t i o n  k, whereas i n the  case the r e a l parameters e and a determined o n l y  line p o s i t i o n or line-width, respectively.  In the l i m i t o f slow ex-  change, f o r a g e n e r a l p o p u l a t i o n t w o - s i t e exchange system analogous to t h a t a l r e a d y d i s c u s s e d , i t i s shown from Eq. (2.4.22) t h a t the a b s o r p t i o n mode l i n e s h a p e f u n c t i o n s i m p l i f i e s  \)(^  - A{  f  -V-7-U-^  ft  A  to t h e form  )—-JC  1(2.4.24)  H i us, i n the absence o f exchange (k = 0 ) , the spectrum c o n s i s t s o f pure L o r e n t z i a n factors p  lines  c e n t r e d at x = -Q and x = +0, w i t h  intensity  and (1 - p ) = p , r e s p e c t i v e l y , and f u l l widths a t h a l f -  A  A  maximum o f Z/'T^'  -*-  nt  n  e  D  l i m i t o f v e r y f a s t exchange t h e second term  i n Eq. (2.2.23) becomes dominant and t h e a b s o r p t i o n mode spectrum consists o f a single Lorentzian  c e n t r e d at x = (1 - 2p )ft w i t h A  unequal p o p u l a t i o n  line  a l i n e - w i d t h 2/T . /.  exchange system the s p e c t r a l l i n e i n t h i s  i s p o s i t i o n e d away from t h e mean frequency the resonance frequency change s i t e . and  That i s , f o r the  corresponding  x = %C°^  ftg)  =  0 towards  to the l a r g e r p o p u l a t i o n ex-  A r e f o r m u l a t i o n o f the imaginary  (2.4.23) shows t h a t t h e s e g e n e r a l  +  limit  equations  p a r t s o f Eqs.  (2.4.22)  a r e e q u i v a l e n t to t h e  e x p r e s s i o n p r e v i o u s l y d e r i v e d , c f . Eq. (2.2.14), f o r a t w o - s i t e exchange system.  48.  2.5  F i r s t - O r d e r J-Coupling.  In p r e v i o u s s e c t i o n s , a - m u l t i - s i t e n u c l e a r s p i n system has been c o n s i d e r e d i n terms o f d i s t i n c t Larmor f r e q u e n c y U K , an i m p l i c i t associated with  assumption  a p a r t i c u l a r molecular  corresponding r e l a t i v e  chemical s h i f t  exchange on s p i n - s p i n m u l t i p l e t s  roximate  analytical  t h a t each s i t e i s  fl^.  The  effect of  s p e c t r a was  co-workers'\  cases^*'  ^6,87^  j_  C O U  chemical  recognized i n  and v a r i o u s app-  e x p r e s s i o n s have been developed  e t i c data f o r s p e c i f i c f i r s t - o r d e r NMR  being  differing  e l e c t r o n i c environment, and  i n NMR  the o r i g i n a l work o f Gutowsky and  sites of  exchange  to d e r i v e k i n -  pij g n  j_  general  na  exchange system, however, i s r e a d i l y i n c l u d e d i n a  m a t r i x f o r m u l a t i o n of exchange p r o c e s s e s based a s t i c model as a l r e a d y  upon a s i m p l e s t o c h -  developed.  A general spin Hamiltonian absence o f exchange p r o c e s s e s may  be  f o r a system o f N-spins considered i n the  i n the  form  (2.5.1)  whore i t i s assumed t h a t a l l terms i n t h i s H a m i l t o n i a n in  energy  u n i t s h = h/2iT, h b e i n g P l a n c k ' s  constant.  are  expressed  A term  de-  s c r i b i n g the i n t e r a c t i o n o f the o b s e r v i n g r f magnetic f i e l d w i t h n u c l e a r s p i n system i s not n e c e s s a r y is  the  i n the f o l l o w i n g a n a l y s i s . }-{  a g e n e r a l i z e d Zeeman term  yi  0  =  -  W  ^  D  X^i  (2.5.2)  l =i w i t h a r e f e r e n c e Larmor f r e q u e n c y  w  rad. s e c T  1  d e f i n e d as  co  = yH  o  - A,  A > 0.  (2.5.3)  o  As u s u a l , I ^ i s the z-component s p i n a n g u l a r momentum o p e r a t o r f o r the i ^ - s p i n  and H  i s the magnitude o f the s t a t i c magnetic  field  a p p l i e d i n the p o s i t i v e z - d i r e c t i o n o f the C a r t e s i a n r o t a t i n g f i x e d r e f e r e n c e frames. netic ratio system  f o r the resonant  and A i s a frequency  d e t e r m i n i n g the o r i g i n cf.  Eq.  trary  In Eq.  (2.2.24).  (2.5.3), y  mental  i s the n u c l e a r gyromag-  spins i n a general hetero-nuclear s h i f t parameter,  f o r the independent  d e f i n e d to be  positive,  frequency v a r i a b l e  T h i s form o f f r e q u e n c y v a r i a b l e a l l o w s an  choice o f frequency o r i g i n  g e n e r a l NMR  and  a simplified  analysis  s p e c t r a , w h i l e b e i n g d i r e c t l y r e l a t e d t o the  irradiating  and  arbi-  of  experi-  K Q i s the c h e m i c a l s h i f t  frequency co.  x,  term  M  -  ^  i n which fi^ i s the c h e m i c a l s h i f t  l  ^  i  (2.5.4)  f o r the i ^ - s p i n ,  or  r e l a t i v e to t h e r e f e r e n c e Larmor frequency to , c f . Eq. In  t h i s manner, ft^ may  be  c o n s i d e r e d as the s i t e - i  in  terms o f the independent  terms o f the o b s e r v a b l e  v a r i a b l e x. may  site-i, (2.2.24).  Larmor  frequency  d e f i n i t i o n o f Ji^  The  in  be r e l a t e d t o the concept o f a  88 p o s i t i v e e l e c t r o n i c s h i e l d i n g parameter time-independent  magnetic f i e l d  R  such  t h a t }i  a  M  = yH  as d e s c r i b e d by  Z a . l ..  j  X  i n the z - d i r e c t i o n : H.  =  an i n c r e a s e d  the p a r t i c u l a r  local  (1 - a.)H 1  I n c r e a s e d s h i e l d i n g o f the i  th  1  0  -spin,  Z1  c r e a s e i n the Larmor f r e q u e n c y r  d e s c r i b i n g the  B  O  In  o^,  v a l u e , thus  corresponds  to a de-  a t c o n s t a n t magnetic f i e l d  case f o r which A = 0  (to = yH ) , a l l s i t e o ' o  H . Larmor  ,  50.  f r e q u e n c i e s are such t h a t to. ^ 1  shifts  0  co  and hence a l l r e l a t i v e  chemical  are n e g a t i v e . The p a r t i a l H a m i l t o n i a n  may  be r e f e r r e d to as a  first-  o r d e r c o u p l i n g term and i s g i v e n as  >C  -£  where J._. i s the i n d i r e c t the i ^ ' 1  the  In  and j ^ - s p i n s .  I l**L  ( - ^ 2  Z  ( s c a l a r ) s p i n - s p i n c o u p l i n g constant between Thus , K ^  i s a second-order  c o u p l i n g term of  form  the ^ - r e p r e s e n t a t i o n , w i t h b a s i s f u n c t i o n s as simple p r o d u c t s  e i g e n - f u n c t i o n s o f the s p i n o p e r a t o r I , the term  }i ,  2 '  to  5  o f f - d i a g o n a l elements  o r d e r NMR  spectrum  may  i n the s p i n H a m i l t o n i a n m a t r i x "H_. A  be d e f i n e d by the c o n d i t i o n J . . « 13  as the m a t r i x ^  ments may  corresponds  J  R  and,  i s diagonal i n  of  |Q. 1  first- Q.\  i  3  a l l off-diagonal matrix  1  ele-  be n e g l e c t e d i n a f i r s t - o r d e r d e t e r m i n a t i o n o f s p e c t r a l  t r a n s i t i o n f r e q u e n c i e s and i n t e n s i t i e s .  The H a m i l t o n i a n term )/j ^ , 2  however, has been shown to g i v e r i s e to a s p i n - s p i n mechanism which may  relaxation  be d e s c r i b e d by B l o c h type equations f o r weakly  89 c o u p l e d n u c l e a r s p i n systems  .  The e f f e c t o f such a r e l a x a t i o n  mechanism i n a d d i t i o n to chemical exchange p r o c e s s e s i s not i n c l u d e d at  this point. In  for  the I ^ - r e p r e s e n t a t i o n , the e f f e c t i v e s p i n H a m i l t o n i a n  a weakly coupled  W -  Ho  (first-order) spin +  X  + n  Xj  ( 1 )  system, (2.5.7)  i s d i a g o n a l and a l l terms con-cspond to s e c u l a r e n e r g i e s f i r s t - o r d e r spin transitions.  The energy  defining  E^ associated with the  b a s i s f u n c t i o n ( a l s o an e i g e n - f u n c t i o n o f ^ ' j cf)^ i s now g i v e n as  E  £  =  -  a  o l  m  ~l i\i  U  n  +  i, ij £i £j c  i  J  fJ  where I . cj)- = m..cj) , and <*>. = IT ^.. . Z X  responding  Kr  Xs  JC th  X  to the i  ^  -part,  f o r s p i n l-h,  =  |  m  '  m  ( 2  -  5 > 8 )  I . i s the spin operator cor-  X/X  Z X  , o f the p r o d u c t  = +h> -h f o r  =  f u n c t i o n cf)^.  Tlius,  ct, 3 , where a and 3 a r e the  s p i n e i g e n - f u n c t i o n s d e f i n e d by I ^|a> = +%|a>,  ^ j j ^  z  =  _  ^|3 . >  Spin systems commonly o c c u r r i n g i n s t u d i e s o f m u l t i - s i t e exchange p r o c e s s e s , e s p e c i a l l y h i n d e r e d i n t e r n a l r o t a t i o n , may be a n a l y z e d as f i r s t - o r d e r ABX and ABX„ l-h  systems w i t h J  6  As  t h e simple ABX system i n c l u d e s a l l t h e s a l i e n t  g e n e r a l a n a l y s i s , t h i s p a r t i c u l a r system w i l l some d e t a i l .  -H = -%[ zK 1  with  l  +  T  zB zx]  chemical s h i f t s  +  :  i n accordance +  ^ z A " ZB] T  features of a  f o r an ABX ( J ^ g = 0)  w i t h Eq. (2.5.7) as +  V X Z  d e f i n e d as : io = co - ft, co„ = co + ft and A o B o A  co = co - fl where co = ^(co. + co ) and hence A = YH X o X o A B n  c f . Eq. (2.5.3).  = 0.  now be c o n s i d e r e d i n  The f i r s t - o r d e r s p i n H a m i l t o n i a n  s p i n system may be expressed  n  AB  Simple  representation for this  - %(co. + co ) , o A B n  p r o d u c t b a s i s f u n c t i o n s , <j>^, i n t h e I 3-spin  (I =h) system a r e g i v e n i n T a b l e Z.l. ,  where i n g e n e r a l cj)^ = ^ ^ C ^ g C ^ ^ J by d e f i n i t i o n , t h e s e f u n c t i o n s a r c a l s o eigen-functions o f the spin Hamiltonian energy  7^'.  'Die c o r r e s p o n d i n g  l e v e l s , as d e r i v e d from Eq. (2.5.8), are a l s o g i v e n i n T a b l e 2  TABLE 2.1 BASIS  (EIGEN) FUNCTIONS FOR ABX ( J  I = y  A  B  NUCLEI, AND CORRESPONDING ENERGY LEVELS  B a s i s f u n c t i o n , (j) 1  = 0) SPIN SYSTEM,  Energy l e v e l , E^ 3  acta  a3a  - ~ OJ  3on  4  w  o  0  4 %  +  1  1  y  — 2  -  wo -  J  1  -j  n  '  Z  A  \fix n +  Y. J  +  3a3 I n  333  J  = 4 C BX ± JAX) ; J  ±  2 ° U  J  A r  J  B  X  > o  a  "A + y-  O  1  M_  T  j % + « - j J.  +  1  a33  1.  1 _  7  %  I T +  7  J  +  TABLE  TRANSITION  FREQUENCIES  FOR  2.2  Transition Type  Transition Number a  1  A  2  A  (J^g = 0)  ABX  Energy  S P I N  SYSTEM  levels  Frequency  4 4  (1,3), (2,4) (5,7), (6,8)  3  (1,2), (3,4)  4  (5,6), (7,8)  n  T  A X  J  A X  B X  j  "4  J  J  BX  1  X  (1,5)  2  X  (3,7)  3  X  (2,6)  -ft^+j_  4  Y  (4,8)  -,a,+.T. "A  These t r a n s i t i o n numbers correspond  t o those  used  i n F i g s . 2.7 and 2.9. The  bracket  and  corresponding  Table  ±  J  =  1  (£,m) r e f e r s t o energy l e v e l s E£ and E , m  eigen-functions  1.  (  J  B X  1  J  AX>'  J  B X '  J  A X  >  0  <f>^ and <j) , g i v e n i n m  i  x  52.  The  a l l o w e d t r a n s i t i o n f r e q u e n c i e s , i n terms o f the independent  and <j>m c o r r e s p o n d i n g  x, are l i s t e d i n Table 2.2. The e i g e n f u n c t i o n s to  the energy  l e v e l s d e t e r m i n i n g these s p e c t r a l  c a t e d by the b r a c k e t s  (I,  m).  t r a n s i t i o n s are i n d i -  By d e f i n i t i o n ft > 0 > ft. > ft , and i t v  A  D  i s assumed i n i t i a l l y t h a t J In  DA  variable  > J  A A  > 0 such t h a t J  —  A  = ^ ( J  D  V  DA  ± J . ) > 0. v  AA  the absence o f exchange, f o r a m o l e c u l a r system without a p r e f e r r e d  conformation, a l l t r a n s i t i o n i n t e n s i t i e s spectrum  i n the AB-part  are e q u a l , as a r e those i n the X-part.  o f the NMR  In a more g e n e r a l case,  these i n t e n s i t i e s may be determined by t h e f r a c t i o n a l p o p u l a t i o n s o f a number o f p o s s i b l e  conformations.  The AB-part itions  1-4  rr  o y  \T\  P"i  o f the ABX spectrum  c o n s i s t s o f the f o u r t r a n s -  (each o f which i s doubly degenerate ^Ii  1 1c  as  v>o T_i^r->-i nrr duis t o tl^s X~ s jp i n  = 0) as shown clct^oXjTiincc  fciXx*  e f f e c t i v e Larmor f r e q u e n c i e s x = -ft ± % J ^ and x = ft ± %Jg^> which may in  t u r n be c o n s i d e r e d to d e f i n e f o u r d i s t i n c t  sites  are analogous  to the two environmental  x = ± ft i n the absence o f J - c o u p l i n g . change e f f e c t s , a b a s i c environmental  exchange s i t e s .  s i t e s w i t h Larmor f r e q u e n c i e s  In a g e n e r a l d e s c r i p t i o n o f ex(chemical s h i f t ) s i t e w i t h an  a s s o c i a t e d X - s p i n s t a t e may now be r e f e r r e d t o as s p i n s i t e Larmor frequency x_. , c f . Eq. (2.2.24).  - j with  In t h i s manner x.. may be con-  s i d e r e d as an e f f e c t i v e chemical s h i f t . d e f i n e d above, s p i n s i t e - 1 corresponds  Such  Thus f o r t h e J - c o u p l i n g s to a b a s i c A - s i t e a s s o c i a t e d w i t h  an a X-spiri s t a t e , as shown by t h e e i g e n - f u n c t i o n s c o r r e s p o n d i n g to the energy  l e v e l s d e f i n i n g the Larmor ( t r a n s i t i o n ) frequency x  = -ft-%J 1  (aaa,  Baa) and (aga, 33a),  c f . Tables  J - c o u p l i n g i n a f i r s t - o r d e r NMR  : A A  2.1 and 2.2. Thus i t i s seen t h a t  spectrum  f o r a s p i n system w i t h  chemical  a4J2 A X  2 AX J  a-4J 2 BX  J  '2 8X  J  J  k» a  2+  2 -f  J  J  (b) A  X  a  8  a  2  ?  k23-  K  •a-i|J| AX  - o + | | J  A  X  n-|J  |  Q + i2JJ B X  BX  k -• -J 1  F i g u r e 2.7  i  J  AB-part o f f i r s t - o r d e r ABX  spectrum  exchange p r o c e s s e s , s i m p l y i n c r e a s e s the number of p o s s i b l e exchange sites  to be  c o n s i d e r e d i n a d e s c r i p t i o n o f the system by  modified  Bloch equations.  ..  C o n s i s t e n t w i t h the s e m i - c l a s s i c a l v e c t o r model p r e v i o u s l y d i s c u s s e d , the n u c l e a r m a g n e t i z a t i o n a s s o c i a t e d w i t h a g e n e r a l s p i n s i t e - j may  be c o n s i d e r e d i n terms o f a s p i n isochromat  M(x^ , (()_.) w i t h  a c h a r a c t e r i s t i c Larmor frequency x.. and a r e l a t i v e phase (with r e s p e c t to the u - a x i s o f the normal r o t a t i n g frame o f r e f e r e n c e uvz ) cj)_. , c f . Fig.  2.2.  The  f r a c t i o n a l p o p u l a t i o n of s p i n s i t e - j , p.., determines  magnitude of M(x_. , <})_.).  In the presence  o f a chemical exchange p r o c e s s  d e f i n e d i n terms o f a f i r s t - o r d e r r a t e constant k s e c . " , the s p i n  iso-  1  chromat i s i n v o l v e d i n a random t r a n s f e r between s p i n s i t e - j site-i  d e s c r i b e d by  it  be  may  tuating  a r a t e c o n s t a n t k...  (2.2.18).  and  of k rad. sec." . 1  spin  Conversely,  e x p e r i e n c e s a randomly  l o c a l magnetic f i e l d w i t h a fundamental frequency  assumption  fluc-  component i n The b a s i c  f o r a s t o c h a s t i c d e s c r i p t i o n o f such an exchange p r o c e s s i n  a f i r s t - o r d e r n u c l e a r s p i n system may (i)  c f . Eq.  c o n s i d e r e d t h a t t h i s isochromat  the a s s o c i a t e d frequency d i s t r i b u t i o n  the  be summarized as f o l l o w s :  the isochromat M(x_. , (}>_.) remains i n s p i n s i t e - j w i t h a mean l i f e t i m e T . u n t i l a random i n s t a n t a n e o u s  t r a n s f e r to a d i f f e r e n t  J  s i t e takes p l a c e , such t h a t p r e c e s s i o n a l e f f e c t s i n the i n t e r v a l may (ii)  the s i t e  be  neglected;  l i f e t i m e x.. i s independent  o f the a s s o c i a t e d s p i n - s p i n  and s p i n - l a t t i c e r e l a x a t i o n times, T^^ (iii)  transfer  i n d i v i d u a l s p i n isochromats transfer effects;  and  and T^.. , r e s p e c t i v e l y ;  r e l a x independently  except f o r s i t e  (iv)  f o r the s p i n isochromat  M(x^ , cb^) i n s p i n s i t e - i  constant p r o b a b i l i t y per u n i t site-i,  time, kj^>  f°  r  t h e r e i s a.  transfer  into  t h i s p r o b a b i l i t y b e i n g i n v e r s e l y p r o p o r t i o n a l to  the  f r a c t i o n a l s i t e p o p u l a t i o n p_. . Under these assumptions, a s i t e c o n s t a n t s , k_.^,  a  r  e  lifetime. T  simply r e l a t e d  £ k.. li  and  associated rate  as  = T. , 1  (2.5.10)  _ 1  where i ( i / j ) i n c l u d e s a l l a l l o w e d t r a n s f e r s i t e s  connected w i t h 82  site-j.  Also in.accordance with  the p r i n c i p l e o f d e t a i l e d  for  the r a t e p r o c e s s e s , s i t e p o p u l a t i o n and  two  spin sites  = p.k.'.  (2.5.11)  These g e n e r a l i z e d r e l a t i o n s h i p s may  and  be  compared w i t h t h o s e f o r the  t w o - s i t e exchange system i n i t i a l l y  d i s c u s s e d , c f . Eqs.  (2.2.2)  (2.2.3). Exchange e f f e c t s  « may  r a t e c o n s t a n t s f o r any  satisfy p.k. .  simple  balance  now  i n a f i r s t - o r d e r ABX  (J  AB  = 0) NMR  be d e s c r i b e d i n terms o f t h i s s t o c h a s t i c model.  i n t r a m o l e c u l a r exchange p r o c e s s e s t a t i o n ) w i l l be assumed to be  spectrum  Only g e n e r a l  ( f o r example, h i n d e r e d i n t e r n a l r o -  discussed e x p l i c i t l y .  The b a s i c exchange p r o c e s s i s  a t r a n s f e r of n u c l e a r m a g n e t i z a t i o n between s p i n - s i t e s  d i s t i n g u i s h e d by  c h e m i c a l s h i f t s f o r the A-  r e s p e c t i v e l y , c f . Eq.  (2.5.9).  This process  and B - s p i n s , - ft and  + ft,  i s d e f i n e d i n terms o f a  s i n g l e f i r s t - o r d e r r a t e c o n s t a n t , k s e c . " , which i s n e c e s s a r i l y  de-  fined  The  1  as a reduced  r a t e c o n s t a n t f o r a g e n e r a l exchange system.  AB-part  o f the ABX spectrum  i s shown i n F i g . 2.7a f o r t h e s p i n  system  i n t h e absence o f exchange (k = 0) w i t h c o u p l i n g c o n s t a n t s J^^, > J In F i g . 2.7a, t h e b a s i c A and B environmental  s i t e s are i n d i c a t e d  > 0 along  w i t h the X - s p i n ' s t a t e c o r r e s p o n d i n g t o each AB-exchange s p i n - s i t e . A l l p o s s i b l e t r a n s f e r s between the f o u r AB s p i n - s i t e s may be r e p r e s e n t e d by:  (2.5.12)  the forward  t r a n s f e r s o n l y , o f t h e p a i r s d e f i n e d by Eq. (2.5.11) b e i n g  shown f o r c l a r i t y .  C o n s i s t e n t w i t h assumption  ( i ) above, i t i s t o be  assumed t h a t the X - s p i n s t a t e i s unchanged i n a s i t e t r a n s f e r and hence the a l l o w e d t r a n s f e r s a r e 1 - 3 r a t e constants k ^  ^31)  and k^^  ^42-''  interval  and 2 - 4 as d e f i n e d by t h e This i -  s  r e a d i l y seen from a  comparison  o f t h e s p i n e i g e n - f u n c t i o n s , c f . Table2.1 a s s o c i a t e d w i t h  the doubly  degenerate  E EE  transitions  1 and 3:  ^A^B^X  ^A^B^X  aaa  aaa  3aa  a3a  a 3a  3aa  63a  33a  (2.5.13)  where 1 i s a f i r s t - o r d e r A - s p i n t r a n s i t i o n as d e f i n e d by the u s u a l transition  I*|3eta> = |aaa>  operator I*:  3 i s a B-spin t r a n s i t i o n b a s i c A and functions is  t o be  and thus  B environmental  concerned  the t r a n s f e r  sites  have an a  X-spin p a r t . 1-2  Similarly,  as d e f i n e d i s between  as r e q u i r e d .  compared w i t h the t r a n s f e r s  i n X-spin  I*|63a> = |a3a>.  and  spin  A l s o , a l l product  T h i s type o f s i t e and  2-4  eigen-  transfer  involving  a change  state. The  s i m p l e s t o c h a s t i c model c o n s i d e r e d above i s c o n s i s t e n t  w i t h a quantum mechanical  treatment  o f exchange p r o c e s s e s .  I t may  be  assumed t h a t the m o l e c u l a r  system f o l l o w i n g a n u c l e a r s p i n - s i t e t r a n s -  f e r has  o f the same form  a spin Hamiltonian  t h a t - t h i s Hamiltonian  differs  as i t had  o n l y i n t h a t the n o n - e q u i v a l e n t  B - s p i n s have i n t e r c h a n g e d magnetic p r o p e r t i e s . be  initially,  5  by  A-  and  An o p e r a t o r may  d e f i n e d * ' * ^ t o d e s c r i b e the change o f t o t a l n u c l e a r s p i n  under exchange, as determined  and  now  state,  s p i n e i g e n - f u n c t i o n s w i t h i n the  o f s e p a r a b i l i t y of a wave-function  limits  f o r the complete m o l e c u l a r - s p i n  20 system  .  For an i n t r a m o l e c u l a r exchange p r o c e s s , t h e n u c l e a r  s t a t e o f an i n d i v i d u a l m o l e c u l e determined  following' a s p i n - s i t e t r a n s f e r i s  c o m p l e t e l y by the o r i g i n a l  fundamental d i f f e r e n c e between such chemical  exchange p r o c e s s  i n which  state.  a process  This i s actually and  the s t a t e o f an i n d i v i d u a l  t r a n s f e r depends upon the o r i g i n a l  site  e c u l e and  a l s o upon the s t a t e o f the i n t e r a c t i n g m o l e c u l e  and  P|a3£ > = |3a£  >.  molecule  state of this  be  Thus, i t i s seen  d e f i n e d by P|aa^  mol17  involved  s p i n - s i t e t r a n s f e r i n the f i r s t - o r d e r  system, an exchange o p e r a t o r , P, may  the  an i n t e r m o l e c u l a r  following  F o r an i n t r a m o l e c u l a r AB  spin  > =  ABX  |aa£  that t h i s operator defines  >  e x a c t l y the t r a n s i t i o n s  i n v o l v e d i n the a l l o w e d  site  d e f i n e d i n terms o f the s t o c h a s t i c model above and Eq.  (2.5.13) f o r the e i g c n - f u n c t i o n s h a v i n g Spin-sites  1 and  associated with  Table  Thus the t r a n s f e r 1 - 2  characteristics b a s i c Ato  and  transitions  sites.  i n d u c e d by  magnetic f i e l d  transition  specifically  B-spin  as  represented i n  X-part.  2 d i f f e r o n l y i n t h e X - s p i n s t a t e , the  functions I.  an a  transfer,  2 being  does not  (aa3,  and  w i t h t h e X - s p i n o f the ABX  333),  spectral  the s t o c h a s t i c exchange  t r a n s f e r s , however,  the i n t e r a c t i o n  (a33,  c o n t r i b u t e to NMR  associated with Such s i t e  3a3)  eigen  o f a randomly  correspond  fluctuating  s p i n system l e a d i n g to  X-spin  90 lifetime if  limiting  as d e s c r i b e d by  the X - s p i n has  a quadrupole  randomly f l u c t u a t i n g  a c o r r e l a t i o n time  moment  electric, f i e l d  similar transfers  1 - 2. and  2-3,  In  addition  ( 1 > 1 ) , the i n t e r a c t i o n o f a  g r a d i e n t may  r e l a x a t i o n mechanism w i t h a d i f f e r e n t  .  lead  correlation  to a  similar  time, x^.  the c o r r e l a t i o n  time  X  £  For may  the  be  c o n s i d e r e d t o d e f i n e r a t e c o n s t a n t s : k,„ = k „ , = x . A l s o , such 12 34 c a mechanism may l e a d t o e f f e c t i v e t r a n s f e r s 1 - 4 and 2 -3-, and i n t h i s case the r a t e c o n s t a n t s k, . and k„_ would be d e f i n e d as a sum « 14 23 - 1  of  c o n t r i b u t i o n s from.exchange and  overall spectral eters  effect  In G(x) sites  of these a d d i t i o n a l  c h a r a c t e r i s t i c s would be  ( T ^ ) '  X-spin t r a n s i t i o n processes.  1  and  ( T ^ ) "  accordance  determining  expected  w i t h Eq.  i n the  on  NMR  t o depend upon the param-  . (2.4.6),  the s t e a d y - s t a t e NMR  i s expressed  G(x)  1  spin r e l a x a t i o n processes  The  a complex  spectrum  lineshape function  associated with n  spin-  form  = A I_-S-[A + (1/T  2  +  ixU]" -S-^P 5  (2.5.14)  58.  i n which A. i s the d i a g o n a l m a t r i x c o r r e s p o n d i n g to an n x n m a t r i x J< - iQ  , the d i a g o n a l m a t r i x Q b e i n g d e f i n e d by the s p i n - s i t e  f r e q u e n c i e s x_. i n terms o f the independent has  been i m p l i c i t l y  ( l / T ^ ) ^ , with  a l l o w e d by  f r e q u e n c y v a r i a b l e x.  assumed t h a t the s p i n - s p i n r e l a x a t i o n time  same f o r a l l s p i n - s i t e s matrix  Larmor  i n d e f i n i n g the r e l a x a t i o n m a t r i x  1^ the n x n u n i t m a t r i x .  It  i s the  as a s c a l a r  This s i m p l i f i c a t i o n i s  assuming t h a t the s p i n - s p i n r e l a x a t i o n time  is  independent  9] of s p i n - s i t e  and t h a t a l l n o n - s e c u l a r r e l a x a t i o n p r o c e s s e s  s c r i b e d by o f f - d i a g o n a l elements ditional  i n T^,  are n e g l i g i b l e .  c o n s i d e r e d at t h i s p o i n t , these p r o c e s s e s bv o f f - d i a g o n a l acteristies  elements  Although  t r a n s f e r are not  ad-  o f the ABX  ( J  A  G  1  (2.5.10).  corresponds  Thus the K and  equal p o p u l a t i o n exchange system  with  i n the  transfer  as o r d e r e d i n F i g . 2.7a  k IC  =  0  0  0  Q m a t r i c e s f o r an > J  k  are g i v e n e x p l i c i t l y  0  0 -I  0  k  0  -k  0  k  noted  to the i n v e r s e s i t e -  coupling constants J  -k  not  I t i s to be  > AA  DA  spin-sites  char-  a 4 x 4 rate  (2.2.21) w i t h k^ ^ = 0 f o r any  a l l o w e d i n terms o f the s t o c h a s t i c exchange model.  l i f e t i m e T T , c f . Eq.  further  Lineshaue  = 0) spectrum  o f exchange are c o n c i s e l y d e s c r i b e d through  t h a t the d i a g o n a l m a t r i x element k „  and  are e q u i v a l e n t l y d e s c r i b e d  i n the r e l a x a t i o n m a t r i x T..  f o r the AB-part  m a t r i x K d e f i n e d by Eq.  and  de-  r e l a x a t i o n p r o c e s s e s a s s o c i a t e d w i t h the J-coup l e d X - s p i n  c o n s i d e r e d above as c o n t r i b u t i n g t o s p i n - s i t e  presence  ', as  as  0,  59.  and 0  0  0  0  0 0  0  0  0  0  0  • (-O^^^yJzt (2.5.15)  The  vectors  P_ and I_ i n Eq. (2.5.14) a r e d e f i n e d by f o u r e q u a l  0.25 and 1.0, r e s p e c t i v e l y . 'The e x t e n s i o n population  population efficient  to a more g e n e r a l  system o n l y r e q u i r e s t h e e v a l u a t i o n o f K. m a t r i x  accordance w i t h  numerical  Equation  a n a l y s i s o f the NMR  (2.5.14) a l l o w s  a b s o r p t i o n mode  f u n c t i o n , V ( x ) , g i v e n as t h e r e a l p a r t o f G ( x ) , o f computer programs f o r r a p i d i t e r a t i v e f u n c t i o n to experimental  data.  the s i t e -  a very  lineshape  and the development  fittings  Absorption  unequal  elements i n  Eqs. (2.2.21) and (2.5.11), c o n s i s t e n t w i t h  v e c t o r , P_, elements.  elements  of this  theoretical  mode l i n e s h a p e s have been  c a l c u l a t e d f o r t h e AB-part o f an ABX spectrum d e f i n e d by t h e a r b i t r a r y parameters Q •- 4.0 Hz, J . _ = 0, J . = +2.0, J v  AD  AA  D  V  = +5.0 Hz, T_ = 0.64 s e c . Z  DA  (0.5 Hz f u l l - w i d t h a t half-maximum) and a r e shown i n F i g . 2.8a f o r r a t e constants  i n t h e range 0 v< k  have been n o r m a l i z e d  of fast  spectrum reduces to a d o u b l e t . 2  - 1  .  The l i n e s h a p e  exchange  It is  (k >> ft, t h i s p a r t o f the  T h i s f e a t u r e i s c h a r a c t e r i s t i c o f an  s p i n system and i s c o n s i s t e n t w i t h  the expectation  exchange l i m i t b a s i c A and B e n v i r o n m e n t a l s i t e s become As  functions  to a maximum i n t e n s i t y independent o f k.  seen t h a t i n t h e l i m i t  A X  4 200 s e c .  that i n t h i s equivalent.  shown i n F i g . 2.7a, f o r J.,, and J o f the same s i g n , the r e s u l t a n t ' AX BX ^~  lines  b  n  v  a r e c e n t r e d a t x = ±%J w i t h J = % ( J „ „ + J . „ ) . + + BX AX  These p o s i t i o n s r  Figure. 2.8  I n t r a m o l e c u l a r exchange l i n e s h a p e s f o r the AB-part o f a f i r s t - o r d e r ABX s p i n system.  60.  may  be m o d i f i e d , however, by  oi and A  u .  temperature  dependent chemical  n  B  .  Now  f o r an ABX  (J  = 0) s p i n system w i t h J  AD  d i f f e r e n t s i g n , the f i r s t - o r d e r s p i n H a m i l t o n i a n may accordance  w i t h Eq.  zA  +  J  J  v  zB  ^Ax'VzX  zX  I  l e v e l s , as determined "* ° ^AX r  r  from those  ^  n  be e x p r e s s e d i n  ^able  X  jI . — I [ zA zB  X zX  (2.5.16)  0 and J„, > 0. BX r  u s i n g Eq.  Thus the  rate  corresponding r  (2.5.8) are g i v e n by  l i s t e d i n T a b l e 2.  The  AB-part  o f the spectrum  substituting  case, the i n t r a m o l e c u l a r AB  exchange p r o c e s s  k  o o -k  o  V  1 and  AA (k - fl) and  and J  n  v  the  o  as  k  o  o  k/  (2.5.17)  T h i s m a t r i x d e f i n e s the a b s o r p t i o n mode l i n e s h a p e s shown i n F i g . Hz  2  Therefore, i n  i s d e s c r i b e d by  o  -IL  o -k  - - 2.0  spin  and k ^ ^ j moreover, the r a t e m a t r i x JC  c o n s t a n t s d e s i g n a t e d as  -ji  for this  i n which the o r d e r e d t r a n s i t i o n s  f o r an equal p o p u l a t i o n system i s g i v e n e x p l i c i t l y  for J  b  J the a l l o w e d t r a n s i t i o n s are s i m i l a r l y d e r i v e d  correspond t o 3 and a X - s p i n s t a t e s , r e s p e c t i v e l y .  this  of  DA  I  system i s shown i n F i g . 2.7b, now  and J„  BX zB zX  where i t i s assumed t h a t J , . < AX  ~^AX^  AA  (2.5.9) as  i . + i _ + i  energy  shifts,  = + 5.0  Hz.  In the r e g i o n o f  2.8b  coalescence  DA  i n the f a s t exchange l i m i t ,  f o r a g i v e n r a t e constant k  61.  d e s c r i b i n g t h e i n t r a m o l e c u l a r exchange p r o c e s s , these distinctly  lineshapes are  d i f f e r e n t from those p r e v i o u s l y d i s c u s s e d , c f . F i g . 2.8a.  Hence the r e l a t i v e s i g n s o f t h e c o u p l i n g c o n s t a n t s J be  determined  spectrum; through  directly  NMR spectrum Of  additional information i s r e a d i l y  available  study o f the temperature dependence o f a f i r s t - o r d e r  (or p a r t o f ) f o r a s p i n system undergoing  c o u r s e , such  may  from l i n e s h a p e f i t t i n g o f an exchange m o d i f i e d  i n general, this  a simple  and  information i s equivalent  o b t a i n e d u s i n g t h e well-known double  chemical  (or complementary) t o t h a t  resonance t e c h n i q u e s .  This  shape method, however, may allow the d e t e r m i n a t i o n o f r e l a t i v e of c o u p l i n g c o n s t a n t s  f o r systems t o which double  resonance  are i n a p p l i c a b l e o r o n l y a p p l i e d w i t h g r e a t d i f f i c u l t y . sennenc.fi  '  o f  t h e  d i f f e r e n t  r e l a t i v e  "  1  sultant x = ±  ' -----  lines  hJ  o f  <;ion<;  i n the f a s t  , where J  =  k(J  - \j, \) ,  nv  v  for Jg^ >  J  i s  o f an ABX  (J^g  K and Q , c o m p l e t e l y  analogous t o  A g e n e r a l X-part  spectrum  as shown i n F i g . 2.9a i n t h e  ) ; the corresponding  (k »  J_),  a b s o r p t i o n mode l i n e s h a p e s  AX.  f o r t h e a r b i t r a r y ABX s p e c t r a l parameters d e f i n e d above = 2 . 0 Hz) a r e shown i n F i g . 2.10a. system, i t i s t o be noted  the s p i n - s i t e s  ( Q - 4.0 Hz,  F o r an equal p o p u l a t i o n  t h a t the exchange p r o c e s s  terms o f the s i n g l e m a t r i x element Fig..2.9a with  r e -  0) spectrum  =  with J  -J  t h e  "  (k = 0) and i n the f a s t exchange l i m i t  DA  t h a t  BX  shown i n F i g . 2.7b.  s  (2.5.15) and (2.5.17).  > 0 may be r e p r e s e n t e d  = %(J  techniques  A f u r t h e r con-  absence o f exchange  —  signs  spectrum a r e now c e n t r e d a t  i n t h e X-part  are a l s o d e s c r i b e d by 4 x 4 m a t r i c e s those g i v e n i n Eqs.  a n d  line-  AX  DA  Exchange e f f e c t s  a  J  - AX *  exchange l i m i t  exchange.  ~  i s defined i n  = -k, as shown i n  involved corresponding  t o t h e AB s p i n  (a)  aa  AB  1  ap  pa 2  PP 4  k 23  -a k  X  »J_  AB  pa 1  n-J.  aa  PP  2  -n-J.  3  -n+J.  - a  k»J.  F i g u r e 2.9  X-part of  ap  f i r s t - o r d e r ABX  spectrum  62.  s t a t e s a3  and 3a.  spectrum f o r J  Similarly, < 0 and J  F i g . 2.9b shows the g e n e r a l > 0.  In t h i s  l i m i t may be d e f i n e d by k >> J , w i t h J +' + !  = h(J ^ BX  (fl = 4.0 Hz, J  A. A  exchange The  +  nv  J  corresponding  case the f a s t  X-part  AX  1  1  = - 2.0 Hz) a b s o r p t i o n mode  lineshapes  are shown i n F i g . 2.10b, and a g a i n , t h e r e l a t i v e s i g n s o f t h e c o u p l i n g constants  and J g  g i v e r i s e to marked d i f f e r e n c e s i n these  shapes a l l o w i n g a simple A lineshape perimental  sign determination  from e x p e r i m e n t a l  a n a l y s i s o f both the AB- and X-parts  ABX spectrum a l l o w s  a check on the i n t e r n a l  the f i t t i n g p r o c e d u r e used to o b t a i n r a t e c o n s t a n t s .  linespectra.  o f an ex-  consistency of A l s o , t h e coa-  lescence  c o n d i t i o n f o r the AB-part o f the spectrum i s d e f i n e d by t h e  chemical  s h i f t d i f f e r e n c e 29., whereas the c o r r e s p o n d i n g  the X-oart  i s d e f i n e d bv the narameters J  = J*>f.T_.„ - J...1 and  '  J  +  = ^(Jgx  I^AX^ ^  +  o r  and J g y , r e s p e c t i v e l y . a lineshape  AX'  " " BX  ^  e  s  a  m  e  a n c  condition for  * different  r e l a t i v e signs of  Thus, i f J _ (or J ) and 0, d i f f e r s i g n i f i c a n t l y , +  a n a l y s i s o f b o t h p a r t s o f t h e spectrum allows  determination  o f r a t e constants  an a c c u r a t e  over an extended temperature range  l e a d i n g t o more r e l i a b l e a c t i v a t i o n parameters f o r the i n t r a m o l e c u l a r exchange p r o c e s s 2.6  involved.  Second-Order  J-Coupling.  In a g e n e r a l spin Hamiltonian to r e p r e s e n t  n u c l e a r s p i n system, the n o n - s e c u l a r  describing indirect  p a r t o f the  s p i n - s p i n c o u p l i n g may be  a mixing o f b a s i c s p i n e i g e n - s t a t e s .  considered  As mixed s t a t e s are  11 92 most c o n v e n i e n t l y effects  d e s c r i b e d by a d e n s i t y m a t r i x  i n a second-order  '  , spin transfer  ( t i g h t l y - c o u p l e d ) s p i n system can be r i g o r -  o u s l y d e s c r i b e d o n l y i n terms o f q u a n t u m - s t a t i s t i c a l mechanics''^ u s i n g  63.  the s p i n d e n s i t y m a t r i x formalism"'' '  ^.  4  An i n t r a m o l e c u l a r c h e m i c a l  exchange p r o c e s s w i l l be c o n s i d e r e d i n terms o f a d e n s i t y m a t r i x f o r an ABX s p i n system t o develop  a more g e n e r a l model f o r exchange and t o show  a c o r r e l a t i o n w i t h the s e m i - c l a s s i c a l m o d i f i e d Bloch e q u a t i o n s a l r e a d y presented f o r a f i r s t - o r d e r  (J^B  s  ^  =  P  xn  system.  The s p i n H a m i l t o n i a n f o r a g e n e r a l ABX s p i n system i n a f i x e d frame o f r e f e r e n c e may be expressed i n the form  Xo  X ^  +  -to I . + I _ + I zA  zB  .  CD  +  X  +  H  T  + ft I  zX  tx) T  . - I zA zB  +  VzX  + J I .1 + J I .1 + J I I + % J ( l t l " + I . I * ) (2.6.1) zA zB AX zA zX BX zB zX A B A B A V  v  n v  n  v  v  where J i s the AB i n d i r e c t  1  s p i n - s p i n c o u p l i n g c o n s t a n t and a l l o t h e r  parameters have been p r e v i o u s l y d e f i n e d , c f . Eqs. (2.5.1) and (2.5.9). For s i m p l i c i t y ,  the r e l a t i v e  chemical s h i f t s f o r the A- and B-spins  have been d e f i n e d as ft^ = -ft and ftg = +ft, c f . Eq. (2.5.4).  In a more  g e n e r a l f o r m u l a t i o n , the Zeeman and chemical s h i f t terms i n Eq. (2.6.1) would be e x p r e s s e d as  • o ' ' ^-flL = " (W  Q  -  ft) I .  zA  + I  zB D  + I zX V  ft I . + ft„I _ + ft I A zA B zB X zX A  (2.6.2) w i t h ft = % ( f t . + ft ) . n  A  B  In t h e S c h r o d i n g e r r e p r e s e n t a t i o n w i t h b a s i s f u n c t i o n s cj)^, an N - s p i n  system i s d e s c r i b e d by t h e wave f u n c t i o n iKt)  = S  ^ ( t ) ^ ,  (2.6.3)  64.  and  t h e time-dependent expansion  average  coefficients  spin density m a t r i x ^ , £, with  P  j £  (t)  = «J> [p U > = j  £  A s i m p l i f i e d Schrodinger  C j  d e f i n e an ensemble-  elements  (t)c*(t).  e q u a t i o n o f motion  (2.6.4) 92  f o r this  density matrix  i s now o b t a i n e d as  -^^-A^  1  '"  5)  (2 6  21 by adding p h e n o m e n o l o g i c a l  terms  to d e f i n e s p i n - s p i n and s p i n - l a t t i c e  r e l a x a t i o n , such t h a t £ ° i s the ensemble thermal e q u i l i b r i u m d e n s i t y matrix. The H a m i l t o n i a n i s g i v e n i n Eq. (2.6.5) i n m a t r i x form f o r the chosen b a s i s {<J>^}f i n e d by a c o r r e l a t i o n  A random i n t r a m o l e c u l a r s p i n t r a n s f e r , as detime  T.  may  h o  d e s c r i b e d by an  i n the d e n s i t y matrix equation o f m o t i o n ^  te^m  a d d i t i o n a l  ^  X  (2.6.6)  where E i s a s p i n o p e r a t o r d e f i n e d under t h e assumptions t h a t f o l l o w i n g s p i n t r a n s f e r the s p i n H a m i l t o n i a n has t h e same form and t h a t none q u i v a l e n t s p i n s have s i m p l y Eq.  i n t e r c h a n g e d magnetic p r o p e r t i e s .  (2.6.6), x may be c o n s i d e r e d as t h e average  In  time between s p i n t r a n s -  f e r s and hence a f i r s t - o r d e r r a t e c o n s t a n t f o r the t r a n s f e r  process  may be d e f i n e d as k = x ~ . 1  Under t h e a c t i o n o f an o b s e r v i n g r f f i e l d ,  H_ , t h e s p i n  system i s most c o n v e n i e n t l y c o n s i d e r e d i n an i n t e r a c t i o n r e p r e s e n t a t i o n corresponding disqussed.  t o the c l a s s i c a l  r o t a t i n g r e f e r e n c e frame p r e v i o u s l y  The e q u a t i o n o f motion f o r the d e n s i t y m a t r i x i n such a  65.  r e p r e s e n t a t i o n , and i n the system  shows a minimal  i s g i v e n i n accordance  l i m i t of  zero s a t u r a t i o n such t h a t the  d e v i a t i o n from thermal e q u i l i b r i u m with  Eqs. ( 2 . 6 . 1 ) ,  (2.6.5)  and  spin  conditions,  (2.6.6)  in  com-  ponent form as  -  k  (2.6.7)  where  = yH^  and x i s the independent  d e f i n e d as x = co - U , c f . Eq. q  (2.2.24).  frequency v a r i a b l e In the h i g h  (rad. s e c . " ) 1  temperature  approximation''"^, an e v a l u a t i o n o f the m a t r i x element d e s c r i b i n g interaction  o f the s p i n system w i t h the i r r a d i a t i n g  (2.6.7) i s s i m p l i f i e d  by d e f i n i n g a r e a l  rf field  c o n s t a n t C such  3  i n Eq.  that  t u » < < | > ^ [ / ' , X ^ t I x 6 + I c / ] l ^ > " -cCCfA^tAj 1  the  (2-6.8)  where C = ca co^/2kT, k i n t h i s i n s t a n c e b e i n g t h e Boltzmann constant, o  The  component quantum number M  i s g i v e n by M.  0  t h a t i s , f o r the b a s i s f u n c t i o n cb^: The  = E m ., p  XV  .-v  I +  A/X  ^~ |  1^  c f . Eq.  + 1^  (2.5.8);  ^^p-  =  e x p e c t a t i o n v a l u e o f the complex t r a n s v e r s e n u c l e a r mag-  n e t i z a t i o n , G_, as d e f i n e d i n Eq.  (2.4.3) and  shown i n F i g . 2.2,  is  now  +  g i v e n d i r e c t l y i n terms o f the s p i n system t r a n s i t i o n o p e r a t o r I  and  the f r e q u e n c y v a r i a b l e x as G(x)  where A may  = A<I > = Tri&'l}  be t r e a t e d as a n o r m a l i z a t i o n c o n s t a n t and  the t r a c e of the m a t r i x p r o d u c t j ^ - ^ elements p.~ 1 through Eq.  +  Tr{...}•denotes  o v e r the s p i n s t a t e s .  are i n g e n e r a l complex''""'" and show an i m p l i c i t  The  matrix  x'dependence  +  (2.6.7).  The  I  elements o f the m a t r i x termines  (2.6.9)  +  t r a n s i t i o n operator I  *{4>^1 and  i n the b a s i s  d e f i n e s the non-zero hence e f f e c t i v e l y  the d e n s i t y m a t r i x elements d e f i n i n g G(x)  i n Eq.  de-  (2.6.9).  C o n s i s t e n t w i t h the normal s e l e c t i o n r u l e f o r magnetic d i p o l a r t r a n s itions,  the t o t a l ABX  s p i n system  (I = h s p i n s ) o p e r a t o r may  f i n e d as I*" = I* + 1^ + I * , such t h a t  lV =&  M  £^  o f the p r o d u c t  transitions  (2.6.8)  (2.5.8), c o r r e s p o n d i n g to the  In t h i s manner, i t i s p o s s i b l e to a s s i g n  to a s p e c i f i c s p i n w i t h i n a g i v e n s p i n system i n the  order J-coupling l i m i t .  I t s h o u l d be noted  transitions of f i n i t e  first-  at t h i s p o i n t , however,  t h a t the o p e r a t o r d e f i n e d above does not determine  the  combination  i n t e n s i t y normally associated with a t i g h t l y  c o u p l e d n u c l e a r s p i n system. determined  Eq.  by t h i s o p e r a t o r d i f f e r i n o n l y one p a r t  f u n c t i o n cj)^, c f . Eq.  component o p e r a t o r i t .  £  de-  £' j  J  The b a s i s f u n c t i o n s connected  <j>, c f .  M + 1  be  These combination  o n l y by g e n e r a l i z i n g the o p e r a t o r 1^  t r a n s i t i o n s may to connect a l l  be  6 7 .  b a s i s f u n c t i o n <JK and cjj^ w i t h  - M_. = + 1 .  <|>£ f o r an ABX  and the c o r r e s p o n d i n g upper d i a g o n a l  (I = h)  s p i n system  The p r o d u c t b a s i s f u n c t i o n s  I_ m a t r i x are shown i n T a b l e 2.3., the elements  determining  +  t r a n s i t i o n s being i n parentheses.  combination  F o r . t h e b a s i s f u n c t i o n s o r d e r e d as  shown i n T a b l e 2 . 3 the m a t r i x elements  I * ^ and  I*^ define A-spin trans-  i t i o n s i n the f i r s t - o r d e r l i m i t as a l r e a d y d e s c r i b e d f o r an ABX s p i n system,  c f . Table 2 . 2 .  The  f i n e d by the m a t r i x elements the d e n s i t y m a t r i x elements  (J^g = 0 )  c o r r e s p o n d i n g B - s p i n t r a n s i t i o n s are  +  +  and  1 ^  P'21'  ^  Ij4>  P31'  P42  A  N  C  *  n  d a n c e with Eq.  a c c o r  de-  ( 2 . 6 . 9 ) ,  d e s c r i b e an AB-part  /°43  of  +  a g e n e r a l ABX  spectrum.  Thus the I_ m a t r i x i n the f a c t o r e d form  c a t e d i n T a b l e 2 . 3 shows t h a t the ABX terms o f s e t s o f d e n s i t y m a t r i x  =  (*? ti*)  A|  I/ ,* +^  + ^ 4 +yC?3 4 ) +  7  |3  *  o f motion  d e n s i t y matrix  w i t h i n each o f these s e t s are now o f the form  be c o n s i d e r e d i n  elements  where p.. = p.. f o r the Hermitean elements  t r a n s i t i o n s may  g i v e n i n Eq.  11  l^+/ ST ^Y 78) /  p.  The  equations  now  be c o n s i d e r e d spin-sites  f o r the coupled i n e q u i v a l e n t A- and  B-  An example o f such a p r o c e s s i s the  h i n d e r e d r o t a t i o n i n a m o l e c u l a r system h a v i n g two g i v i n g r i s e t o the AB-spin  /  ( 2 . 6 . 7 ) .  to c o r r e s p o n d t o a t r a n s f e r o f n u c l e a r m a g n e t i z a t i o n between  s p i n s , -ft and ft, r e s p e c t i v e l y .  +  d e n s i t y matrix  d e f i n e d by coupled  A b a s i c i n t r a m o l e c u l a r exchange p r o c e s s may  d i s t i n g u i s h e d by chemical s h i f t s  indi-  possible  conformations  inequivalence, t h i s r o t a t i o n being defined i n  terms o f a s i n g l e f i r s t - o r d e r r a t e c o n s t a n t , k s e c . . 1  p r o d u c t b a s i s f u n c t i o n s <$>^, the o p e r a t o r E g i v e n i n Eq.  For the ( 2 . 6 . 6 )  chosen and  Table  2.3  B a s i s F u n c t i o n s f o r ABX I = h Nuclear  <j>£  aaa  aBa  3  Baa 1  Systems  Spins  BBa  aaB  1 2  Spin  1  aBB 1  2  BaB 1  2  BBB  1  2  3  2  2  Matrix i n Basis ( ^ J " -  Spin T r a n s i t i o n Operator  1  o 0  i  i l  0  o  .CD  (1)  o  o  8  o  o  CD  o  o  0  68.  d e s c r i b i n g the above exchange p r o c e s s may  be  defined  f o r an equal p o p u l a t i o n  by  E|aa£ > = |aa£ >, x  E|a3£ > =  x  A p p l i c a t i o n o f t h i s exchange o p e r a t o r conditions  g i v e n by s e t s o f coupled matrix  A.  i n Eq.  (2.6.7) shown t h a t under  (-^- £ = 0) the d e n s i t y m a t r i x  b i n g an AB-part o f the ABX  (2.6.11)  |Ba£>  .A.  steady-state  system  elements  descri-  spectrum i n the p r e s e n c e of exchange  equations  which may  be  expressed  i n the  are general  form:  R* -G'  iCP  (2.6.12)  such t h a t  V4  U  l[*+/iZ.+£(3"+T }c)] A  r+i,  X  - k  X  [*-^+^(T-KJ  1  6 ) C  )J  1  and  u  A-  _ k  1  - - i d X  1  (2.6.13)  where r = ^  + k.  I t i s assumed t h a t s p i n - s p i n r e l a x a t i o n e f f e c t s i n  2 the absence o f exchange are d e f i n e d by The  the s i n g l e r e l a x a t i o n  time T ,  c f . Eq.  (2.6.5).  above m a t r i x e q u a t i o n i s e q u i v a l e n t to t h o s e  viously  d e r i v e d i n terms o f s e m i - c l a s s i c a l m o d i f i e d B l o c h e q u a t i o n s f o r  the s i m p l e t w o - s i t e exchange system, c f . Eq. exchange system i n the f i r s t In the t i g h t l y  coupled  (J  (2.2.12),  and  s p i n system, l i n e a r  be  s i t e magnetizations  to those a s s o c i a t e d w i t h t h e  x  nt n  e  analogous  first-order  2.4.4).  combinations  o f the v e c t o r elements Gj may  M( j, ^j) i  a multi-site  o r d e r J - c o u p l i n g l i m i t , ( c f . Eq. ± 0) ABX  pre-  considered to define transverse s p i n -  l i m i t , c f . F i g . 2.2.  That  isochromats  is, a  spin-site  m a g n e t i z a t i o n v e c t o r G_ i s d e f i n e d by D-G  = -iCQ  (2.6.14)  where J) i s the d i a g o n a l m a t r i x  c o r r e s p o n d i n g to R/ .  elements o f the m a t r i x J) determine characteristics Q determine sites.  the p o s i t i o n s  for specific spin-sites  the o v e r a l l  spectral  I t s h o u l d be n o t e d t h a t , through  a l i z a t i o n procedure  (2.6.4), t h i s  d e f i n e d i n Eq.  these  o f the g e n e r a l  diagonal matrix i s  consisting of eigen-functions (2.6.1).  The  above  diagon-  i s a l s o d i r e c t l y r e l a t e d to t h a t a l r e a d y used i n  the s i m p l i f i e d f i r s t - o r d e r o f the AB-part  associated with  the d e f i n i t i o n  e q u i v a l e n t to t h a t o b t a i n e d i n a b a s i s {^^} o f the s p i n H a m i l t o n i a n  and g e n e r a l l i n e s h a p e  and the elements o f the v e c t o r  intensities  s p i n d e n s i t y m a t r i x element i n Eq.  In t h i s manner, the  spectrum  as Gj =  and  G^ =  elements G^  and  G^  limit,  c f . Eq.  (2.4.6).  The  detailed  a s s o c i a t e d w i t h the v e c t o r elements ma  >'  n  o  w  be  determined  (defining spin-sites  1 and  form  designated  i n terms o f the v e c t o r 3) c o r r e s p o n d i n g to  the  70.  diagonal, matrix elements  D  and  given from Eq.  = r + i x + %(J + J ) + D  with D_ = h 4(ft  ,  +  D„ = r + i x + 3 2  (2.6.13) as  Js(J+J)-D +  - k ) + J ( J - 4ik) + J_(J_ - 4ft) ^ and J 2  +  = %GJg  In accordance with Eq. (2.6.14), a p a r t i a l lineshape function may  ± J^)  X  now  be  expressed i n the form  3>  +  V  (2.6.15)  where the normalization constant A includes C and the complex Q_ vector elements corresponding to and  = b + ie.  by equating Eqs.  and G^ have been defined as  = a + id  These i n t e n s i t y vector elements are now (2.6.13) and  (r + k) + i x + J + hJ  +  determined  (2.6.15), that i s  = (a + i d ) { r + i [ x + h(J + J ) +  DJ }  +  + (b + i e ) { r + i [ x + % ( J + J ) + D J }. +  Thus the p a r t i a l lineshape corresponding to s p i n - s i t e s 1 and 3 i s given i n terms of G^(x) and G^(x), where  1  with 3_ = (%J - ik)/D .  X  As usual, the absorption mode spectrum  scribed by the imaginary parts of G^(x) and G^(x).  i s de-  In the absence of  71.  exchange (k = 0 ) , Eq. (2.6.16) shows t h a t G (x) d e f i n e s a L o r e n t z i a n a b s o r p t i o n mode l i n e w i t h a f u l l - w i d t h a t half-maximum o f 2/T  rad.  sec."  c e n t r e d a t x^ = - % ( J + J ) - D_ w i t h a r e l a t i v e i n t e n s i t y o f 1 - J/2D . +  T h i s component l i n e i s r e p r e s e n t e d as s p i n s i t e - 1 i n F i g . 2.11 showing t h e complete A B - p a r t o f t h e ABX spectrum, where i t has been assumed that  >  > J > 0.  As t h e independent f r e q u e n c y v a r i a b l e x and  the common r e l a x a t i o n time  form p a r t o f t h e d i a g o n a l elements o f  the m a t r i x R' i n Eq. (2.6.12), t h i s m a t r i x may be r e f o r m u l a t e d t o g i v e  R + (i + i x U l G ' = -iCP 2 where _I. i s the u n i t m a t r i x .  1  ,  (2.6.17)  Spectral characteristics associated with  exchange a r e now d e f i n e d i n terms o f a d i a g o n a l m a t r i x A  corresponding  to R, where i n accordance w i t h Eq. (2.6.14)  [A + d L  The  + ix)i]G  =  - iCQ.  (2.6.18)  2  elements o f t h e m a t r i x yV, A.., and t h e c o r r e s p o n d i n g  intensity fac-  t o r s Q^. a r e g i v e n i n T a b l e 2.4 f o r a l l s p i n s i t e s - j i n t h e AB-part o f an ABX  spectrum.  Thus i t i s seen t h a t i n t h e absence o f exchange, the  d e n s i t y m a t r i x elements P^'  ^24  a n c  * ^34 d e t e r m i n e  t  n  e  spectral  c h a r a c t e r i s t i c s f o r t h e s p i n s i t e s 1 - 4 , t h i s p a r t o f t h e spectrum c o n s i s t i n g o f a t y p i c a l AB q u a r t e t .  I t i s i n t e r e s t i n g t o note t h e  c o r r e l a t i o n between these p a r t i c u l a r d e n s i t y m a t r i x elements and t h e energy l e v e l s d e f i n i n g t h e degenerate AB s p i n s i t e s f o r a f i r s t - o r d e r (J  A B  = 0 ) ABX  s p i n system as l i s t e d i n Table 2.2.  For t h e type o f i n t r a m o l e c u l a r exchange p r o c e s s under cons i d e r a t i o n f o r an ABX s p i n system, t h e exchange o p e r a t o r E d e t e r m i n e s  1  Table  2.4  A B - P a r t o f an ABX  Spectrum  M a t r i x Element X.  Intensity Factor Q  3  %(J + J ) + D + -h(J  - J ) + +  h(J  D  +  + J ) - D_ +  i(J - J J  =2(J  -  d  J  +  - D  1- Jt_  -%(J + J J + E  1 -  3  1 +  3  1 +  3  1 -  3,  i  - y  1 + j  j  %(J  - J ) - E_  1 + Y  h(J  + J ) - E  1 " Y,  +  +  +  4 (ft  2  - k )  %[^4(ft  2  - k ) + J ( J ± 4 i k ) + J _ ( J _ + 4ft)  2  2  \hJ ± i k ] / D  ^ BX J  + J ( J ±. 4 i k ) + J _ ( J _ - 4ft)  *  J  ±  AX)  ,  Y  ±  = [hJ  ± ik]/E  ±  t h e independence o f t h e two q u a r t e t s f o r m i n g t h e A B - p a r t o f t h e spectrum.  Thus t h e form o f t h e m a t r i c e s i n Eq. ( 2 . 6 . 1 3 ) , as d e f i n e d  by t h e s p i n t r a n s i t i o n o p e r a t o r I  +  (and E)  allows  a relatively  simple  a n a l y t i c a l f o r m u l a t i o n o f t h e a b s o r p t i o n mode l i n e s h a p e f o r a component AB q u a r t e t . G^(x)  From Eq. (2.6.16) and s i m i l a r e x p r e s s i o n s f o r  and G ^ ( x ) , t h i s  l i n e s h a p e i s o b t a i n e d f o r an e q u a l p o p u l a t i o n  exchange system as  (2.6.19)  where  B The  .  ±  JW/(^ k) ± +  independent f r e q u e n c y  ifj  v a r i a b l e w has been d e f i n e d , t o s i m p l i f y t h e  above e x p r e s s i o n , as w = x + hJ  f o r t h e AB q u a r t e t c e n t r e d on x = - % J  +  J  c f . T a b l e 2.4. A l s o , i n t h i s manner Eq. (2.6.19) may be a p p l i e d i n genera'l f o r a s i m i l a r i n t r a m o l e c u l a r exchange p r o c e s s s p i n system.  i n an a r b i t r a r y AB  Such an a n a l y t i c a l e x p r e s s i o n f o r t h e a b s o r p t i o n mode  l i n e s h a p e a l l o w s a v e r y e f f i c i e n t i t e r a t i v e comparison w i t h data.  experimental  A l s o , i t i s r e a d i l y seen t h a t Eq. (2.6.19) reduces t o t h e l i n e -  shape e q u a t i o n p r e v i o u s l y d e r i v e d f o r t h e s i m p l e two s i t e u n c o u p l e d AB s p i n s y s t e m , c f . Eq. (2.2.15) w i t h x =. w, from s e m i - c l a s s i c a l Bloch equations.  modified  U s i n g Eq. (2.6.19) and a s i m i l a r e x p r e s s i o n f o r t h e  AB q u a r t e t c e n t r e d a t x = hJ , +  l i n e s h a p e s have been c a l c u l a t e d f o r a  range o f t h e p a r a m e t e r k/ft and a r e shown i n F i g . 2.11. These a b s o r p t i o n  73.  mode l i n e s h a p e s are d e f i n e d by the parameters 9 = 4.0 Hz, J = 4.2 Hz, J  A  Y  AA  = 10.0  Hz, J  13 A  = 15.0  Hz and T = 0.64 sec. 2  (0.5 Hz f u l l - w i d t h a t  half-maximum), and may be compared w i t h those f o r the f i r s t - o r d e r ABX s p i n system shown i n F i g .  2.8a.  A b a s i c i n t r a m o l e c u l a r exchange p r o c e s s has been c o n s i d e r e d i n terms o f a t r a n s f e r o f n u c l e a r m a g n e t i z a t i o n between s p i n - s i t e s d i s t i n g u i s h e d by d i s t i n c t Amay  and B - s p i n s .  c h e m i c a l s h i f t s f o r the c o u p l e d i n e q u i v a l e n t  The o p e r a t o r d e s c r i b i n g t h i s p r o c e s s , c f . Eq. ( 2 . 6 . 6 ) ,  be g e n e r a l i z e d t o i n c l u d e an unequal p o p u l a t i o n exchange system.  An example,of t h i s more g e n e r a l exchange p r o c e s s i s the h i n d e r e d r o t a t i o n i n a m o l e c u l a r system h a v i n g two p o s s i b l e c o n f o r m a t i o n s , one o f which i s p r e f e r r e d .  The exchange terms i n the component e q u a t i o n  o f motion f o r t h e d e n s i t v  m a t r i x . F,n . f2.6.71 . mav be c o n s i d e r e d i n t h e  form k^<<j)_. |EpE|(J)^> - k ^ p ^ ,  where k^ i s the p r o b a b i l i t y f o r a t r a n s f e r  from a b a s i c A - s p i n s i t e w i t h a f r a c t i o n a l p o p u l a t i o n p . a m a t r i x e q u a t i o n o f the g e n e r a l form g i v e n i n Eq. s i t y m a t r i x elements p ^ and  I n terms o f  ( 2 . 6 . 1 2 ) , t h e den-  a r e now d e f i n e d b y  PA =  r  - \-^K 1  where r = — A T  & +  i[x-^.(w  + k w i t h p. + p A A B r  2  r  D  & J (  )]  (2.6.20)  fe  = 1 and p.k. = p k . A A B B r  -IC  r  n  n  This matrix  f o r m u l a t i o n f o r an unequal p o p u l a t i o n system a l l o w s a much s i m p l e r c a l c u l a t i o n o f l i n e s h a p e s , f o r the p a r t i c u l a r exchange p r o c e s s c o n s i d e r a t i o n , t h a n an a l t e r n a t i v e p r o c e d u r e  under  p r o p o s e d by J o h n s o n  23  The p a r t i a l l i n e s h a p e a s s o c i a t e d w i t h t h e above d e n s i t y m a t r i x  elements  i s g i v e n by G^(x) and G (x) i n Eq. (2.6.16) where t h e parameters D and $  a r e now d e f i n e d as  3. and J. -  w i t h AJ = J  - (P J  +  A  B X  iV + P  -  AT -  J B  A X  )>  A-p J&>  2  =  k  k  + A  k  a B  n  d  A  P = P  A  - P B  The  complete A B - p a r t spectrum may now be d e f i n e d i n terms o f the m a t r i x elements X^ and c o r r e s p o n d i n g  i n t e n s i t y f a c t o r s Q.. , c f . Eq. ( 2 . 6 . 1 8 ) ,  as g i v e n i n T a b l e 2.4 w i t h m o d i f i e d p a r a m e t e r s :  3>. E  7. L  =  ±  l [ 4 - ( ^ - ^ ) + T ( T ^ 4 - : k ) 4- T_(T_ + 4vro.) + 4-1 k (LT-X, Lljh)j  J  V  ±  .-  ±. C.L — A T  %  - A.pa2v  [ J j f c l k , +- A T ^ A.p <_OvJ  /  (2.6.21)  I t has been assumed t h a t a l l J - c o u p l i n g c o n s t a n t s s p i n system have t h e same s i g n .  For the p a r t i c u l a r  i n t h e ABX  exchange p r o c e s s  c o n s i d e r e d , t h e form o f t h e AB-part spectrum may be c r i t i c a l l y dependent upon t h e r e l a t i v e s i g n s o f t h e s e c o u p l i n g c o n s t a n t s  due t o terms such  as ±4ikJ and ±4J ft i n t h e c h a r a c t e r i s t i c parameters D the r e l a t i v e s i g n s o f J AX  and J  D  V  +  and E .  Thus  w i t h r e s p e c t t o t h e AB c o u p l i n g  BX  c o n s t a n t J d e t e r m i n e t h e form o f the A B - p a r t spectrum i n a manner s i m i l a r t o t h a t f o r a f i r s t - o r d e r ABX system as i l l u s t r a t e d i n F i g .  2.8,  and a complete l i n e s h a p e f i t t o e x p e r i m e n t a l d a t a may  a l l o w a.  simple d e t e r m i n a t i o n of the r e l a t i v e signs o f a l l c o u p l i n g constants f o r a g e n e r a l ABX  s p i n system.  The X - p a r t o f an ABX  spectrum has been shown to be d e t e r -  mined by the s i x s p i n d e n s i t y m a t r i x elements P^^> and p^g  c f . Eq.  3  w i t h combination  (2.6.10).  The  elements p^j  $26'  shows t h a t ( 2 , 7) and  chosen b a s i s f u n c t i o n s (J). , cj>, (f>. and <))_ are i n v a r i a n t r  o  4  and p  4o  2  The  under t h e  ex-  o  change o p e r a t o r E as d e f i n e d i n Eqs. it  ^37  levels  spectrum, c f . Table  (3,6) are not i n c l u d e d i n t h i s l i m i t .  1  ^36'  and p ^ a r e a s s o c i a t e d  t r a n s i t i o n s , and comparison w i t h t h e energy  d e t e r m i n i n g t h e X - p a r t o f a f i r s t - o r d e r ABX  p  ^27'  (2.6.6) and  a r e d i r e c t l y r e l a t e d , t h r o u g h Eq.  (2.6.11), and hence  (2.6.14), t o the t r a n s -  v e r s e m a g n e t i z a t i o n a s s o c i a t e d w i t h s p i n - s i t e s 1 and 6 as shown i n F i g . 2.12.  I n the p r e s e n c e  o f exchange, t h e s e s p i n - s i t e s  to s i m p l e L o r e n t z i a n l i n e s d e t e r m i n e d  from Eqs.  (2.6.7) and  correspond (2.6.14)  as GcAiL)  =  &  -  c  La)  LA  Ik  ik \  The  +  ^ U+*fi*-T )  remaining s p i n - s i t e magnetizations  m a t r i x R i n Eq.  (2.6.16), t h a t i s  +  a r e determined  ( 2  by a 4 x 4  .  6 < 2 2 )  k = 0.1. sec  76.  iZ  -i  iZ z  X  + XiJh  0  IZ  7~ (2.6.23)  where  [_R +  + i (x + ft )}JJG_'  P_* c o r r e s p o n d i n g  to combination  = -iCPJ .  The elements o£ t h e v e c t o r  transitions  (2 and 5 i n F i g . 2.12) a r e  s e t equal t o zero t o a l l o w a c o n s i s t e n t a p p l i c a t i o n o f t h e g e n e r a l i z e d I  +  operator i n the e v a l u a t i o n o f the general s p i n d e n s i t y m a t r i x  o f motion term g i v e n i n Eq. (2.6.8).  The l i n e s h a p e f u n r*+- i  f~\ T \  n  equation  f^r~\  A,  s c r i b i n g t h i s p o r t i o n o f the ABX spectrum i s most r e a d i l y e v a l u a t e d by c o n s i d e r i n g t h e above e q u a t i o n i n t h e g e n e r a l forms g i v e n i n Eqs.(2.6.16) and  (2.6.17).  R through  That i s , a d i a g o n a l m a t r i x A_ i s d e r i v e d from t h e m a t r i x  a s i m i l a r i t y t r a n s f o r m a t i o n as d e s c r i b e d i n d e t a i l f o r the  f i r s t - o r d e r ABX s p i n system, c f . Eq. (2.5.14).  For a t i g h t l y  coupled  s p i n system, however, t h e v e c t o r I_ i n Eq. (2.5.14) must t a k e the same form as P_ t o a l l o w f o r t h e e f f e c t o f c o m b i n a t i o n o v e r a l l lineshape.  t r a n s i t i o n s on t h e  Computed l i n e s h a p e s f o r t h e X-part o f an e q u a l  p o p u l a t i o n ABX s p i n system a r e shown i n F i g . 2.12 f o r t h e parameters chosen t o d e f i n e the A B - p a r t spectrum, c f . F i g . 2.11.  Again,  these  l i n e s h a p e s may be compared w i t h those f o r t h e f i r s t - o r d e r ABX s y s t e m , c f . F i g . 2.10a.  I t i s t o be noted t h a t f o r p a r t i c u l a r c a s e s , f o r  example, when  ,  »  J i n a h e t e r o n u c l e a r system, i t i s p o s s i b l e  77.  t o t r e a t the X-part approximation, and  spectrum as t h a t i n the f i r s t - o r d e r l i m i t .  In t h i s  the o v e r a l l l i n e s h a p e i s s i m p l y o b t a i n e d i n terms o f  (x) g i v e n i n Eq.  (2.6.22) and a l i n e s h a p e f o r an e f f e c t i v e  s i t e exchange system w i t h s p i n - s i t e f r e q u e n c i e s d e f i n e d i n Table x , = -fl 3 x  - J  -  and x  = -fl  4  x  + J  -  , c f . Eq. n  b a s i c condition f o r t h i s approximation,  (2.2.15) w i t h fl = J  The  o f c o u r s e , corresponds  to a  complete n e g l e c t o f the o f f - d i a g o n a l elements i n J i n the m a t r i x In g e n e r a l , the e f f e c t o f d i f f e r e n t r e l a t i v e s i g n s o f J .  v  R.  and J  AA  very  two2 as  .  <-  G^(x)  is DA  s i m i l a r t o t h a t i l l u s t r a t e d f o r the f i r s t - o r d e r case i n F i g .  2.10b. A d e t a i l e d a n a l y s i s o f a g e n e r a l ABX  s p i n system has  allowed  a c o n s i s t e n t development o f the p h y s i c a l model and m a t r i x f o r m u l a t i o n r e q u i r e d f o r a d e s c r i p t i o n o f i n t r a m o l e c u l a r exchange e f f e c t s i n a t i g h t l y c o u p l e d s p i n system. may  Although  a n a l y t i c a l lineshape  equations  be d e r i v e d i n p a r t f o r more c o m p l i c a t e d s p i n s y s t e m s , a formu-  l a t i o n o f exchange e f f e c t s a p p l i c a b l e to the n u m e r i c a l the l i n e s h a p e f u n c t i o n s G(x) N - s p i n system may  now  computation o f  and V(x) f o r a g e n e r a l t i g h t l y  coupled  be d e s c r i b e d i n terms o f the p r o p e r t i e s o f the  d e n s i t y m a t r i x model c o n s i d e r e d above.  In g e n e r a l , h a v i n g  determined  the d e n s i t y m a t r i x elements r e q u i r e d i n the d e s c r i p t i o n o f an s p e c t r u m (or p a r t o f one) I  i n Eq.  through  the g e n e r a l i z e d t r a n s i t i o n  NMR operator  ( 2 . 6 . 9 ) , t h e s e elements must be e v a l u a t e d i n a c c o r d a n c e w i t h  the component e q u a t i o n o f m o t i o n , Eq.  (2.6.7).  I n the chosen b a s i s  {(j)^}, t h e e q u a t i o n o f motion term i n v o l v i n g the independent  frequency  v a r i a b l e i n an i n t e r a c t i o n r e p r e s e n t a t i o n i s s i m p l y Kf  ix«(.  j  |[p  Z  I  z  i  l l *  a  >  = -ixp  j £  (2.6.24)  78.  The  e f f e c t i v e spin Hamiltonian  s i d e r e d i n t h e form }(^ + Xj"^ Eq.  The p a r t X ^  (2.6.1).  i n t h i s r e p r e s e n t a t i o n may now be con> analogous t o t h a t d e f i n e d i n  +  + Xj^  i s d i a g o n a l i n t h e b a s i s i4> } and £  hence i t f o l l o w s t h a t  with the Hamiltonian  m a t r i x element }i ^  ti  Hji. = -  d e f i n e d by  M  dl*-J*  +  r  where  = £i^'  7iji£ - X ' j j  d e f i n e s a f i r s t - o r d e r t r a n s i t i o n energy and hence the s p i n  m  c:|r  . ^ l - (2.5.8).  (2.6.26)  T^^JLl^J-i  c  I t i s t o be n o t e d t h a t t h e term  d e n s i t y m a t r i x element d e f i n e s d i r e c t l y a s p i n - s i t e i n t h i s l i m i t , as p r e v i o u s l y considered  f o r p a r t i c u l a r s p i n systems.  Such a r e l a t i o n s h i p 92  i s defined i n general through the d e f i n i t i o n o f a L i o u v i l l e operator  ',  such t h a t t h e elements o f t h e d e n s i t y m a t r i x jo may be c o n s i d e r e d as components o f a v e c t o r i n a s u i t a b l y d e f i n e d v e c t o r space. 41 v a t i o n super-operator  used by B a n w e l l and Primas  The d e r i -  i n a direct calcu-  l a t i o n " o f NMR s p e c t r a i s i n f a c t i d e n t i c a l t o t h e L i o u v i l l e 92 d e f i n e d by Fano . As t h e s e c o n d - o r d e r c o u p l i n g H a m i l t o n i a n  operator (2) term,Xj  mixes f u n c t i o n s i n t h e b a s i s ify^} and c o r r e s p o n d s t o t h e o f f - d i a g o n a l p a r t o f t h e complete s p i n H a m i l t o n i a n , s p i n d e n s i t y m a t r i x elements.  t h i s term l e a d s t o a m i x i n g o f  Thus i t f o l l o w s t h a t  <*.|Cp,WJ ]|V = x £ p 2)  where X j  <j>£ = <J>^, and X j  r  <f>j = <j>j , •  exchange e f f e c t s i s e v a l u a t e d as  j r  - X ^ , ,  (2-6.27)  S i m i l a r l y , t h e term d e f i n i n g  k«j). |EpE|(j> > = k p . „ £  where E<J) - <j)^ . £  M  determined  £1I  ,  (2.6.28)  C o n s t r u c t i o n o f a v e c t o r G_' w i t h elements  by t h e t r a n s i t i o n o p e r a t o r I  as  and a l s o i n d e x i n g v e c t o r s  associated with the e f f e c t i v e density matrix s u b s c r i p t c o n t r a c t i o n , s p e c i f i c s p i n H a m i l t o n i a n m a t r i x elements and t h e b a s i s f u n c t i o n s connected by t h e exchange o p e r a t o r E a l l o w s t h e m a t r i x R i n Eq.(2.6.14) t o be e f f i c i e n t l y assembled on a computer.  A l l elements o f t h i s  m a t r i x a r e n u m e r i c a l l y e v a l u a t e d i n terms o f t h e normal s p i n H a m i l t o n i a n elements d e f i n e d f o r a s p e c i f i c s p i n system i n t h e b a s i s by making use o f Eqs. (2.6.24) - (2.6.28).  The computation  a b s o r p t i o n mode l i n e s h a p e f u n c t i o n V ( x ) then reduces  ify^),  of the  to a p p l i c a t i o n s  o f t h e b a s i c e q u a t i o n s g i v e n i n Eq. (2.4.6) and ( 2 . 4 . 9 ) .  The o v e r -  a l l d i m e n s i o n a l i t y o f t h e m a t r i x R f o r an i n t r a m o l e c u l a r exchange process  i s determined  f i n e d by I  +  by the number o f d e n s i t y m a t r i x elements de-  i n Eq. ( 2 . 6 . 9 ) , t h a t i s , t h e number o f a l l o w e d and com-  b i n a t i o n t r a n s i t i o n s f o r a g i v e n s p i n system.  As t h e d i a g o n a l i z a t i o n  o f t h e complex m a t r i x R and t h e i n v e r s i o n o f t h e a s s o c i a t e d complex t r a n s f o r m a t i o n m a t r i x J5 a r e t h e time d e t e r m i n i n g  computational  pro-  c e d u r e s , a l l f a c t o r i z a t i o n s o f t h e s e m a t r i c e s as d e f i n e d by t h e form o f t h e H a m i l t o n i a n f o r a s p e c i f i c s p i n system s h o u l d be t a k e n account.  A computer program GENLIN has been developed  into  on t h e b a s i s  o f t h e s i m p l i f i e d d e n s i t y m a t r i x f o r m u l a t i o n o u t l i n e d above, and w i l l be d e s c r i b e d i n more d e t a i l as r e l a t e d t o the a n a l y s i s o f a p a r t i c u l a r 4 - s p i n system i n an e x p e r i m e n t a l s e c t i o n o f t h i s  thesis.  I t i s t o be n o t e d t h a t , d u r i n g the c o u r s e o f t h e work 93  described here, Binsch  i n d e p e n d e n t l y proposed  a similar  method  n u m e r i c a l a n a l y s i s f o r t i g h t l y c o u p l e d s p i n systems based upon a more f o r m a l t h e o r y i n the L i o u v i l l e r e p r e s e n t a t i o n .  81.  CHAPTER 3  INSTRUMENTATION  3.1  .-  FT-1064 Computer I n t e r f a c e  Complete l i n e s h a p e a n a l y s e s  o f chemical  exchange  processes  u s i n g NMR i n v o l v e s e t s o f d a t a p o i n t s d e f i n i n g t h e e x p e r i m e n t a l l y recorded  steady-state s p e c t r a , a s e t o f d i s c r e t e data p o i n t s  being  f i t t e d t o a t h e o r e t i c a l lineshape equation to obtain a s p e c i f i c rate constant. frequency  The t e d i o u s manual c o n v e r s i o n o f t h e r e c o r d e d and c o r r e s p o n d i n g  amplitude  an e l e c t r o n i c s i g n a l s a m p l i n g  data to d i g i t a l  v a l u e s may be e l i m i n a t e d by u s i n g  d e v i c e and an a n a l o g - t o - d i g i t a l (A-D)  c o n v e r t e r l i n k e d t o a s m a l l computer w i t h a m a g n e t i c memory core t o s t o r e the d i g i t a l i n f o r m a t i o n . memory l o c a t i o n s ( c h a n n e l s ) frequency  Of c o u r s e , t h e computer must scan i t s  i n s y n c h r o n i z a t i o n with the spectrometer  ( o r f i e l d ) sweep so t h a t t h e s i g n a l a m p l i t u d e  data p o i n t i n  any g i v e n memory l o c a t i o n may be a c c u r a t e l y a s s i g n e d a f r e q u e n c y d e r i v e d from two c a l i b r a t i o n f r e q u e n c i e s .  value  The d i g i t a l i n f o r m a t i o n may  be t r a n s f e r r e d t o a f u l l - s c a l e d i g i t a l computer, t o a l l o w an e f f i c i e n t i t e r a t i v e l i n e s h a p e f i t , v i a an i n c r e m e n t a l magnetic tape o r a d i r e c t line i f this i s available.  The FABRITEK FT-1064 computer has been u s e d ,  "in c o n j u n c t i o n w i t h the spectrometer-computer i n t e r f a c e u n i t d e s c r i b e d b e l o w , t o g i v e an a u t o m a t i c  d i g i t i z a t i o n and f i t t i n g o f NMR  lineshape  data. The  i n t e r f a c e u n i t has been d e s i g n e d  to allow a general  appli-  82.  c a t i o n o f the FT-1064 computer w i t h  only simple modifications  of  s p e c t r o m e t e r c i r c u i t r y . I n i t i a l l y i t w i l l be assumed t h a t t h e NMR signal-to-noise  r a t i o i s such t h a t a s i n g l e s c a n o f t h e s t e a d y - s t a t e  spectrum i s s u f f i c i e n t t o o b t a i n complete l i n e s h a p e a n a l y s i s .  data o f a q u a l i t y warranting a  A b l o c k diagram showing the b a s i c  o f t h e i n t e r f a c e i s g i v e n i n F i g . 3.1, and t h e a s s o c i a t e d sequence may be r e p r e s e n t e d as shown i n F i g . 3.2.  units  timing  control  I n the s i n g l e scan  mode, t h e s p e c t r o m e t e r sweep .mechanism i s used under normal o p e r a t i n g conditions  and t h e NMR s i g n a l a t t h e o u t p u t o f the s p e c t r o m e t e r  r e c o r d e r a m p l i f i e r i s f e d c o n t i n u o u s l y t o a h i g h impedance d i f f e r e n t i a l a m p l i f i e r , A, and t h e n t o t h e s a m p l i n g A-D SW/1, o f t h e FT-1064 computer.  input  converter,  T h i s d i f f e r e n t i a l a m p l i f i e r a c t s as a  b u f f e r between t h e s a m p l i n g d e v i c e and t h e s p e c t r o m e t e r and a l s o  allows  a v e r s a t i l e g a i n and dc l e v e l c o n t r o l independent o f t h e normal  spectro-  meter c o n t r o l s .  A t time t  Q  the control  b i s t a b l e m u l t i v i b r a t o r , B,  changes t o i t s a c t i v e s t a t e under t h e a c t i o n o f t h e manual sweep t r i g g e r , T.  T h i s m u l t i v i b r a t o r i s u s e d as a m a s t e r c o n t r o l and as a t r i g g e r  b u f f e r , t h e m u l t i v i b r a t o r s t a t e a t any time b e i n g shown by t h e i n d i c a t o r I.  At t h i s time, t  Q  , a t r i g g e r p u l s e from t h e c o n t r o l  bistable  i n i t i a t e s t h e FT-1064 channel sweep and s i m u l t a n e o u s l y a p o s i t i v e p u l s e i s ac-coupled t o the input emitter  f o l l o w e r F l acts  of the spectrometer recorder a m p l i f i e r .  as a b u f f e r between t h e c o n t r o l  the FT-1064 sweep u n i t which has a r e l a t i v e l y low i n p u t monostable m u l t i v i b r a t o r M s u p p l i e s s u f f i c i e n t to give a well-defined c h a r t , t h i s marker a c c u r a t e l y  The  b i s t a b l e and impedance.  The  a 5 msec p u l s e o f an a m p l i t u d e  v e r t i c a l marker on t h e r e c o r d e r  d e t e r m i n i n g the p o s i t i o n o f t h e i n i t i a l  CD  S V V E E R  —  R E S E T  ^  R E C O R D E R I N P U T  F2  A S W E E P O U T P U T  F i g . 3.1  D I G I T A L R E A D O U T  FT-1064 computer sweep c o n t r o l  R E C O R D E R O U T P U T  I  Sweep trigger  Reset trigger  Bistable control voltage  i i  i i  i  i i  i  i  I 1  FT-1064 aate -  FT-1064 sweep voltage  i  Marker pulse i i i i—,  F i g . 3.2  FT-1064 c o n t r o l sequence  i  ,  1  1 1 1 1  83.  d i g i t i z e d d a t a p o i n t as t h e sweep times used c o r r e s p o n d t o 50 msec 2 sec per channel.  A 5 msec p u l s e has been found t o g i v e a f a s t  r i s i n g marker w i t h i n the normal system.  I n t h e time i n t e r v a l t  response 0  time l i m i t o f t h e r e c o r d e r  t o t i the c o n t r o l  b i s t a b l e remains i n  the a c t i v e s t a t e , t h e FT-1064 gate v o l t a g e i s p o s i t i v e and t h e l i n e a r c h a n n e l sweep v o l t a g e changes as shown i n F i g . 3.2.  The a n a l o g  NMR  s i g n a l i s sampled and d i g i t i z e d and a l s o r e c o r d e d on t h e s p e c t r o m e t e r c h a r t o v e r t h i s time i n t e r v a l , t h e d i g i t a l s i g n a l a m p l i t u d e d a t a b e i n g c o n t i n u o u s l y s t o r e d i n t h e computer magnetic t h e FT-1064 channel sweep i s complete  c o r e memory.  At time  ti,  and a p o s i t i v e p u l s e d e r i v e d from  the t r a i l i n g edge o f t h e g a t e , i n t h e i n v e r s i o n and d i f f e r e n t i a t i o n u n i t F2, t r i g g e r s t h e monostable m u l t i v i b r a t o r , M, t o g i v e a second marker which determines  the p o s i t i o n o f t h e f i n a l d i g i t i z e d d a t a p o i n t .  c a l i b r a t i o n f r e q u e n c i e s c o r r e s p o n d i n g t o the i n i t i a l  and f i n a l  The  digitized  d a t a p o i n t s a r e now o b t a i n e d by d i r e c t measurement o f t h e o b s e r v i n g f r e quency r e l a t i v e t o t h e NMR f i e l d l o c k i n g f r e q u e n c y by m a t c h i n g c o r d e r pen p o s i t i o n w i t h t h e c h a r t markers.  the r e -  I n t h i s manner t h e s e  f r e q u e n c i e s a r e r e a d i l y d e t e r m i n e d t o ah a c c u r a c y o f ± 0.1 Hz, which r e q u i r e s a c o u n t e r g a t e time o f a t l e a s t 10 sees and i n t u r n p r e v e n t s an a c c u r a t e f r e q u e n c y measurement d u r i n g a spectrum s c a n , and a r e a l s o shown t o be r e p r o d u c i b l e w i t h i n t h e s e l i m i t s .  F o r an N-channel sweep,  t h e f r e q u e n c y a s s o c i a t e d w i t h channel n and s i g n a l a m p l i t u d e V g i v e n by  = w  0  n  i s simply  + n . Aw,with t h e f r e q u e n c y i n c r e m e n t Aw d e f i n e d as  Aw = ( w i o ) / ( N - l ) where w - w  Q  and Wi a r e t h e c a l i b r a t i o n f r e q u e n c i e s c o r -  r e s p o n d i n g t o t h e markers s e t a t times t  0  and t j ,  respectively.  The  c o n t r o l b i s t a b l e i n i t s a c t i v e s t a t e p r e v e n t s an i n a d v e r t a n t computer  84.  sweep, and p r e c e d i n g a new sweep t h i s b i s t a b l e s t a t e must be changed under t h e a c t i o n o f t h e manual r e s e t t r i g g e r T at a time t , as shown 2  i n Fig.3.2. another  The s p e c t r o m e t e r  s i n g l e scan  r e c o r d e r may now be r e p o s i t i o n e d f o r  and t h e FT-1064 channel  sweep i s i n i t i a t e d  by t h e manual sweep t r i g g e r , T, a t time t$.  again  I n t h i s way t h e s p e c t r o -  meter may be swept i n an a r b i t r a r y d i r e c t i o n and d i g i t i z e d d a t a may be s t o r e d i n independent s e c t i o n s o f t h e computer memory c o r e . The  d e t a i l e d c i r c u i t diagram f o r t h e i n t e r f a c e u n i t i s g i v e n  i n F i g . 3.3, and t h e p e r t i n e n t c o n t r o l waveforms and v o l t a g e s a r e i n c l u d e d on t h i s diagram.  The manual t r i g g e r c i r c u i t s c e n t r e d on t h e  2N3646 h i g h speed NPN s i l i c o n s w i t c h i n g t r a n s i s t o r s TI and T2 a r e p a r t i c u l a r l y s i m p l e and have been shown t o be c o m p l e t e l y r e l i a b l e i n operation.  W i t h s w i t c h SI open, T I i s b i a s e d i n t o c o n d u c t i o n and t h e  c o l l e c t o r v o l t a g e i s h e l d a t +0.4V w h i l e t h e base c a p a c i t o r C i s x  charged t o +4.5V.  When t h e s w i t c h S I i s c l o s e d , t h e c a p a c i t o r C i i s  d i s c h a r g e d and T I i s c u t - o f f as t h e base v o l t a g e f a l l s t o n e a r OV.  The  c o l l e c t o r v o l t a g e r i s e s and t h e p o s i t i v e g o i n g o u t p u t t r i g g e r p u l s e f o l l o w i n g d i f f e r e n t i a t i o n has an a m p l i t u d e 0.1 usee.  o f ^ 2V w i t h a r i s e time o f  When S I i s reopened, C i r e c h a r g e s  w i t h a time c o n s t a n t  RiCi  and a s p u r i o u s t r i g g e r waveform ( n o r m a l l y a s s o c i a t e d w i t h s w i t c h bounce is eliminated.  The e m i t t e r - c o u p l e d c o n t r o l b i s t a b l e m u l t i v i b r a t o r , T3  and T 4 , i s o f c o n v e n t i o n a l d e s i g n and uses a s i n g l e MC-715 MRTL d u a l t r i p l e - i n p u t gate w i t h a s w i t c h i n g t h r e s h o l d o f +0.7V and s w i t c h i n g times o f t h e o r d e r o f 0.1 usee. c o n t r o l b i s t a b l e correspond respectively.  The q u i e s c e n t and a c t i v e s t a t e s o f t h e  t o T4 c o l l e c t o r v o l t a g e s o f +0.2 amd +4.2V,  A s i m i l a r MRTL c i r c u i t , T7-T9, i s used f o r t h e monostabl  —o 220  3900 Rl  Ll  640  1000:  1000 S  0.001  Tl  560  47  5601  r 560 r : 560  i . T3  SWEEP  s, |J H  •20 CI  MC 715  T  B  2N3646  10K<  6  4  1 T  Y  W  0.01  Fl  f^P\ 560  560  560  560  ^  ^00V  2N3646  F i g . 3.3  -  I  C2  0 0 0 1  j  6  2N3566  640  0  ?220  T  1 Q K  47  S2  R2 470 K  2200  NPUT  RESET  T  FT-1064 SWEEP TRIGGER O  0.2-  ,  .  r  -o +6V  ^t-gXJQ  ^  20-  1000:  T5  GATE  T10  |  0 001  FT-1064  T2  6800;  2N3646  1  2700;  T4  ^6800  "2.4  CONTROL  • 0.2 >  •0.40.001  BISTABLE  4.2  0  3900  +6V  10K  2N3646 1N914  T7  T8  T9  MC-715  F2  Spectrometer-computer i n t e r f a c e  unit  0.1 MARKER PULSE  85.  m u l t i v i b r a t o r , t h e output pulse constant  R2C2.  width being  The i n d i c a t o r c i r c u i t  determined by t h e time  consists of a series  lamp, L I , and s i n g l e t r a n s i s t o r , T10, such t h a t t h i s biased in  into conduction  a n d t h e lamp i s o n when t h e c o n t r o l b i s t a b l e i s  circuitry  for the d i f f e r e n t i a l  a m p l i f i e r , A i n F i g . 3.1,  shown i n F i g . 3.4 a n d i s b a s e d u p o n P h i l b r i c k  types  1009 a n d 1301.  which features  isolation  an FET i n p u t s t a g e  t o provide  o f t h e computer sampling  even though d c - c o u p l i n g frequency simple  signal  dc zero  This  c o n t r o l through a voltage  s w i t c h S I i n p o s i t i o n 1.  The o v e r a l l  an u n d i s t o r t e d low a  summing a t o n e i n p u t o f  a maximum g a i n o f 10 w i t h  linearity  o f t h e a m p l i f i e r was  r a n g e -5 t o +5V a n d was f o u n d t o b e  The o u t p u t o f t h i s  F T - 1 0 6 4 SD/1 s a m p l i n g  system,  The 1301 o p e r a t i o n a l a m p l i f i e r i s u s e d as  checked over t h e output voltage  the  complete  i n p u t impedance a l s o a l l o w s  a v a r i a b l e gain i n v e r t i n g dc a m p l i f i e r with  b e t t e r t h a n 0.4%.  allows e s s e n t i a l l y  u n i t from t h e spectrometer  The h i g h  amplifier.  unit  a n i n p u t i m p e d a n c e o f 20M  i s retained t o provide  response.  level  differential  operational amplifiers,  The 1009 i s a g e n e r a l p u r p o s e l o w - n o i s e  f o r t h eu n i t y gain c o n f i g u r a t i o n used.  the  transistor i s  i t s active state. The  is  connected  a m p l i f i e r may b e f e d d i r e c t l y t o  A-D c o n v e r t e r w i t h  an i n p u t i m p e d a n c e o f 5K.  An o s c i l l o s c o p e d i s p l a y o f t h e computer memory c o n t e n t s o v e r 1024  channels  is  a v a i l a b l e and the d i g i t a l  a p a r t i c u l a r memory l o c a t i o n i s p l a y by an i n t e n s i f i e d s p o t . c o n v e n i e n t method o f  selecting  be f i t t e d t o a t h e o r e t i c a l  data p e r t a i n i n g  i n d i c a t e d on t h e a n a l o g s i g n a l  This d i g i t a l  dis-  d i s p l a y a l l o w s a most  a s m a l l e r number o f d a t a p o i n t s  lineshape,  to  the a s s o c i a t e d  frequency  to  86.  b e i n g d i r e c t l y r e l a t e d to t h e channel previously described.  (memory l o c a t i o n )  In g e n e r a l , t h e d i g i t a l  data i s  number as transfered  from the memory c o r e to an i n c r e m e n t a l magnetic tape u s i n g a FABRITEK FT-282 c o n t r o l  u n i t w i t h a KENNEDY Model 1400 tape  recorder.  A d i g i t i z e d s i g n a l a m p l i t u d e v a l u e i s s t o r e d and read o u t as a word c o n s i s t i n g o f two b y t e s , each b y t e b e i n g s i x b i n a r y b i t s . T h i s d i g i t a l word f o r m , however, i s o n l y c o m p a t i b l e w i t h IBM 360 a r i t h m e t i c f o l l o w i n g a c o n v e r s i o n to the s t a n d a r d 18 b i t s f o r system.  this  Thus a mixed FORTRAN-IV/SYMBOLIC program GET was d e v e l o p e d  to a l l o w the e f f i c i e n t  r e a d i n g o f t h e magnetic tape i n t h e form o f  independent o r d e r e d b y t e s ,  the u n p a c k i n g o f s e l e c t e d  two b y t e words  and t h e s e t t i n g up o f an a r r a y o f e q u i v a l e n t t h r e e byte words f o r i t e r a t i v e lineshape f i t t i n g .  the  A l t h o u g h 1024 d a t a p o i n t s were a v a i l a b l e  from the FT-1064 computer, an optimum minimal number o f d a t a p o i n t s were a c t u a l l y used f o r the l i n e s h a p e f i t t i n g . s i t e exchange s p e c t r a ,  For t w o - and f o u r -  i t was found t h a t the a c c u r a c y o f the f i t t i n g  was not i n c r e a s e d above 64 o r 128 d a t a p o i n t s .  Thus t h e s e d a t a p o i n t s  were s e l e c t e d a t o p t i m i z e d i n t e r v a l s o v e r the f r e q u e n c y range o f d i g i t i z e d d a t a by the program GET and the t o t a l  the  CPU t i m e r e q u i r e d  f o r a c o m p l e t e l y a u t o m a t i c l i n e s h a p e f i t to the taped d i g i t a l  data  was o f t h e o r d e r o f 10 sees p e r spectrum u s i n g t h e IBM 360/67 s y s t e m . The complete i n t e r f a c e satisfactory  r e s u l t s w i t h both a JE0LC0 C-60 and a VARIAN HA-100  spectrometer,  o n l y the a d d i t i o n o f t h e two r e c o r d e r  shown i n F i g . 3.1 1064  u n i t d e s c r i b e d above has been used w i t h most  computer.  inter-connections  b e i n g r e q u i r e d to l i n k the s p e c t r o m e t e r  to the FT-  87.  I f the s i g n a l - t o - n o i s e  r a t i o f o r a s i n g l e scan spectrum  i n s u f f i c i e n t to a l l o w a r e l i a b l e l i n e s h a p e a n a l y s i s , i t i s to operate  the c o m p u t e r - s p e c t r o m e t e r  t i v e mode.  necessary  system i n a m u l t i - s c a n a c c u m u l a -  In t h i s case the i n t e r n a l sweep f r e q u e n c y o s c i l l a t o r  the NMR s p e c t r o m e t e r  i s r e p l a c e d by an e x t e r n a l  by t h e dc v o l t a g e a v a i l a b l e , as shown i n F i g . 3 . 2 .  the d i g i t a l  sweep  Thus the normal  r e c o r d e r sweep mechanism i s d i s a b l e d , but the i n t e r n a l f i e l d Once the r e q u i r e d s i g n a l - t o - n o i s e  of  voltage c o n t r o l l e d  o s c i l l a t o r d r i v e n i n s y n c h r o n i z a t i o n w i t h the FT-1064 channel  is retained.  is  ratio is  lock  attained  d a t a i s used as p r e v i o u s l y d e s c r i b e d , and a permanent  c h a r t r e c o r d i n g i s o b t a i n e d by f e e d i n g t h e computer memory a n a l o g o u t p u t to the s p e c t r o m e t e r  r e c o r d e r o p e r a t i n g i n i t s normal l i n e a r  sweep mode u s i n g the above i n t e r f a c e  u n i t t o t r i g g e r the a n a l o g  r e a d - o u t and to p l a c e the f r e q u e n c y c a l i b r a t i o n m a r k e r s .  88.  3.2  Rf-pulse  gate  An i d e a l p u l s e d NMR  spectrometer  would c o n s i s t o f a t r a n s -  m i t t e r s u p p l y i n g a high i n t e n s i t y r f pulse over a very s h o r t  time  i n t e r v a l , as compared w i t h the n u c l e a r r e l a x a t i o n times i n v o l v e d , t o r o t a t e the n u c l e a r m a g n e t i z a t i o n Fig.  2,1;  through  a w e l l - d e f i n e d angle 3, c f .  and a r e c e i v e r t h a t would i n s t a n t l y r e c o r d the f r e e i n d u c -  t i o n decay s i g n a l f o l l o w i n g the p u l s e w i t h o u t d i s t o r t i o n o r the addition of noise. constant amplitude  In p r a c t i c e , i t i s d i f f i c u l t  t o produce a l a r g e  r f magnetic f i e l d a s s o c i a t e d w i t h an a c c u r a t e l y  t i m e d p u l s e h a v i n g a c l o s e l y c o n t r o l l e d w i d t h and m i n i m a l f a l l times.  rise  and  A l s o , i t i s o n l y p o s s i b l e t o have a low n o i s e (so t h a t  o v e r a l l s e n s i t i v i t y i s l i m i t e d by the t h e r m a l n o i s e i n the sample c o i l ) l i n e a r r e c e i v e r w i t h a minimal f o l l o w i n g the r f p u l s e .  time f o r r e c o v e r y from o v e r l o a d  In a d d i t i o n , an r f phase coherent  system i s  29 r e q u i r e d t o r e t a i n the n u c l e a r s p i n i s o c h r o m a t  phase i n f o r m a t i o n  i n h e r e n t i n the d e t e c t e d f r e e i n d u c t i o n decay o r s p i n echo, and t o be a b l e t o a c c u r a t e l y d e t e r m i n e resonance c o n d i t i o n s and the form o f the s p i n i s o c h r o m a t  motion under the a c t i o n o f s p e c i f i c  o f r f p u l s e s i n a g e n e r a l m u l t i - p u l s e sequence. d e t e c t i o n scheme has  detailed types  An r f phase s e n s i t i v e  a l s o been shown t o p r o v i d e a c c u r a t e l y l i n e a r  d e t e c t i o n o f weak s i g n a l s and t o a l l o w f u l l usage o f p o s t - d e t e c t i o n 120 121 integration ' . In h i g h r e s o l u t i o n NMR a p p l i c a t i o n s , the o v e r a l l s t a b i l i t y o f the s p e c t r o m e t e r system i s c r i t i c a l . F o l l o w i n g the 29 30 o r i g i n a l p u l s e d NMR e x p e r i m e n t s o f Hahn and C a r r and P u r c e l l , a number o f s p e c t r o m e t e r 110  systems have been d e s c r i b e d i n the  literature  However, F o u r i e r t r a n s f o r m a p p l i c a t i o n s and the s t u d y  of  89.  c h e m i c a l exchange u s i n g m u l t i - p u l s e sequences r e q u i r e p a r t i c u l a r l y s o p h i s t i c a t e d i n s t r u m e n t a t i o n . . Indeed, the l a r g e s y s t e m a t i c e r r o r s apparent i n the k i n e t i c parameters may  be l a r g e l y due One  d e r i v e d from p u l s e d mode NMR  t o the l a c k o f such  o f the most c r i t i c a l  meter i s the r f - p u l s e g a t e .  data  apparatus.  component u n i t s i n a p u l s e s p e c t r o -  T h i s gate i s r e q u i r e d t o p r o v i d e p u l s e s  o f c o n s t a n t w i d t h and a m p l i t u d e w i t h a v e r y h i g h r f s u p p r e s s i o n i n the time i n t e r v a l  f o l l o w i n g each' p u l s e .  The  r f p u l s e ( o n - p e r i o d ) t o the s t e a d y r f  r a t i o o f the a m p l i t u d e o f an  feedthrough  r e f e r r e d t o as the gate r e j e c t i o n r a t i o , a.  The  ( o f f - p e r i o d ) may  be  r f f e e d t h r o u g h may  be  c o n s i d e r e d t o be n e g l i g i b l e as l o n g as the resonance S , c f . Eq.  ( 2 . 3 . 1 ) , i s i n the range 0.99  1  0.99  corresponds  < S'  < 1.0.  saturation The  factor  factor  t o a s a t u r a t i o n e f f e c t d i s t o r t i o n of the free induc-  t i o n decay o f about 1% o v e r t h e e n t i r e measurement time i n t e r v a l . the iT/2-pulse w i d t h i s assumed t o be 0.01  T 2 , where T2 i s t h e  total  t r a n s v e r s e r e l a x a t i o n t i m e , t h e r e s t r i c t i o n on the s p i n - l a t t i c e  relaxa-  t i o n time T i s a t i s f y i n g t h e above s a t u r a t i o n c o n d i t i o n i s T i < 4 x a T2 ^*  .  IO"  I n l i q u i d s T i - T 2 , and t h i s i m p l i e s t h a t the minimum  r e j e c t i o n .ratio required i s a = 2 x 10 , 5  r e s o l u t i o n NMR i n l i q u i d s T 0.01  If  T2 c o r r e s p o n d s  2  ^ 0.6  t o 6 msecs.  o r 110 dB.  For normal h i g h  s e c , and hence a p u l s e w i d t h o f I n s t u d y i n g c h e m i c a l exchange p r o -  c e s s e s , however, the e f f e c t i v e t r a n s v e r s e r e l a x a t i o n time may  be  s i g n i f i c a n t l y d e c r e a s e d and hence a iT/2-pulse w i d t h o f the o r d e r o f micro-seconds  i s desirable.  In F o u r i e r transform a p p l i c a t i o n s l a r g e  phase e r r o r s t h a t are d i f f i c u l t through  t o q u a n t i t a t i v e l y c o r r e c t may  arise  the g e n e r a l i n a b i l i t y t o measure t h e f r e e i n d u c t i o n decay  6  90.  signal  f o r times t -+ o .  To m i n i m i s e t h e s e e r r o r s  i t i s necessary to  use r e l a t i v e l y s h o r t i T / 2 - p u l s e s and a r e c e i v e r w i t h a f a s t from o v e r l o a d .  Thus g e n e r a l  pulse spectrometer  specifications  used f o r c h e m i c a l  f o r an r f - p u l s e g a t e i n a  exchange s t u d i e s  t i o n F o u r i e r t r a n s f o r m a p p l i c a t i o n s may be g i v e n a s : w i d t h = 10-20 a  = 10  (120  6  ysec,  r i s e and f a l l  t i m e = 0.1  and h i g h r e s o l u iT/2-pulse  y s e c and r e j e c t i o n  t h e above s p e c i f i c a t i o n s  i n the l i t e r a t u r e  a r e t h o s e o f Blume ^  t h a t appear to meet  and Lowe and T a r r  •j I C  The Blume g a t e has been adapted by C l a r k T h i s c i r c u i t uses an e x p e n s i v e  1  ful  o f 0.01  electronic  r f severely  construction.  and c o -  7077 p l a n a r t r i o d e as a  restricts  cathode-to-plate  r f feedthrough, with  rf  r a t i o f o l l o w e d by a m u l t i - s t a g e  The g a t e d e s c r i b e d here has  d e v e l o p e d to a v o i d the use o f s p e c i a l  components,  s t r u c t i o n o r a g a t e d h i g h power r f a m p l i f i e r . been d e s i g n e d to be a s i m p l e and a d a p t a b l e  been  an e l a b o r a t e c o n -  In a d d i t i o n , i t  has  u n i t t h a t may be r e a d i l y  l i n k e d w i t h s t a n d a r d NMR and r f i n s t r u m e n t a t i o n w h i l e m e e t i n g specifications  care-  Lowe and T a r r use a t r a n s i s t o r i s e d  g a t e w i t h a r e l a t i v e l y low r e j e c t i o n gated h i g h power r f a m p l i f i e r .  pp  and Bloom  g r o u n d e d - g r i d r f a m p l i f i e r so t h a t the v e r y low capacitance  ratio  dB).  The o n l y g a t e systems  workers.  recovery  the  o u t l i n e d above.  The d e t a i l e d c i r c u i t diagram f o r the r f p u l s e g a t e o p e r a t i n g a t 10MH  Z  i s g i v e n i n F i g . 3 . 5 ( a ) , and t h e p e r t i n e n t  waveforms a r e i n c l u d e d on t h i s d i a g r a m .  v o l t a g e l e v e l s and  The dc s u p p l y v o l t a g e s  were  chosen t o match t h o s e a v a i l a b l e from the s t a b i l i z e d power s u p p l y f o r 125  t h e TEKTRONIX 160 s e r i e s p u l s e g e n e r a t o r s  used t o s e t  up g e n e r a l  multiple  F i g . 3.6 (a)  R f - p u l s e gate  91.  and/or r e p e t i t i v e p u l s e sequences.  Of c o u r s e , a l l power l e a d s a r e  h e a v i l y f i l t e r e d t o p r e v e n t r f leakage between u n i t s o f the complete pulse spectrometer.  The 6GY6, V I , i s o p e r a t e d as a low g a i n c l a s s A  gated r f a m p l i f i e r w i t h a tuned i n p u t and an untuned o u t p u t .  This  vacuum tube i s a sharp c u t - o f f pentode w i t h d u a l c o n t r o l g r i d s , g i and g3,  and low i n t e r - e l e c t r o d e c a p a c i t a n c e s .  c a r e f u l l y s h i e l d e d through the o n l y m e c h a n i c a l  The tuned g r i d c i r c u i t r y i s  t h e vacuum tube s o c k e t , b u t t h i s i s a c t u a l l y  r f s h i e l d i n g used.  VI i s b i a s e d i n t o c u t - o f f i n  the gate q u i e s c e n t s t a t e ( o f f - p e r i o d ) by a dc b i a s Vg3 = -12V, and t h e r f feedthrough  f o r t h i s s t a g e f o r the maximum i n p u t v o l t a g e o f 0.8V p-p  i s l e s s than 15 mV p-p.  T h i s feedthrough  i s due m a i n l y t o t h e g r i d - t o -  p l a t e c a p a c i t a n c e which has a minimum v a l u e o f 0.026 pF.  When t h e  gate system i s d r i v e n i n t o the a c t i v e s t a t e ( o n - p e r i o d ) by a p o s i t i v e c o n t r o l p u l s e a p p l i e d t o t h e second c o n t r o l g r i d , g 3 , t h e v o l t a g e Vgs i s h e l d a t n e a r the cathode p o t e n t i a l by t h e l i m i t i n g d i o d e D l and the 6GY6 a c t s as an a m p l i f i e r w i t h Vgj = -0.4V.  This a m p l i f i e r gives a  maximum output o f 12V p-p and hence t h e r e j e c t i o n r a t i o f o r t h i s stage, o f the gate system i s o f t h e o r d e r o f 1 0  3  (60 d B ) .  first  I t s h o u l d be  n o t e d h e r e t h a t as t h e c a p a c i t a n c e between t h e two c o n t r o l g r i d s i s l e s s than 1 pF, e x c e l l e n t r f i s o l a t i o n o f t h e c o n t r o l p u l s e i s very simply a t t a i n e d . to-cathode  circuitry  The low p l a t e l o a d r e s i s t a n c e , R l , and p l a t e -  c a p a c i t a n c e o f 6pF determine  the r i s e and f a l l t i m e s f o r t h e  r f p u l s e and a s s o c i a t e d p e d e s t a l i n t h e stage o u t p u t c i r c u i t as l e s s than 0.2 usee.  The 6CY5, V2, a c t s as a p u l s e  inverter  92.  to g i v e an u n d i s t o r t e d g a t i n g waveform f o r the 6 J C 6 , V 3 , o u t p u t amplifier.  This tetrode  has a low g r i d - t o - p l a t e c a p a c i t a n c e  and r f f e e d t h r o u g h i s m i n i m i s e d i n t h i s s t a g e by u s i n g the c o n f i g u r a t i o n shown i n F i g . 3 . 5 ( a ) .  s t a g e and a l s o d e t e r m i n e s  (maximum) i n the a c t i v e s t a t e .  rf  f e e d t h r o u g h i n the  the 6JC6 i n p u t v o l t a g e o f 4V p-p  The 6JC6 i s o p e r a t e d as a h i g h g a i n  c l a s s C a m p l i f i e r and the c o n t r o l g r i d b i a s maximum r e j e c t i o n  pF)  particular  As t h i s s t a g e i s u n t u n e d , the  g a i n i s l e s s than u n i t y which f u r t h e r reduces quiescent  (0.03  i s a d j u s t e d to g i v e a  r a t i o and minimum p u l s e r i s e and f a l l  are f a c i l i t a t e d by t h e low-Q p l a t e c i r c u i t .  t i m e s , which  A g a i n a low g r i d - t o - p l a t e 3  capacitance (72 d B ) .  o f 0.02  pF l e a d s to a s t a g e r e j e c t i o n  r a t i o o f 4 x 10  The r f g a t e o u t p u t i s o b t a i n e d t h r o u g h an i n d u c t i v e l i n k ,  o u t p u t impedance b e i n g a p p r o x . 100 are i l l u s t r a t e d i n F i g . 3.6  fi.  The o u t p u t p u l s e  characteristics  and may be summarised as f o l l o w s :  v o l t a g e <: 20V p - p , r f f e e d t h r o u g h < 15 yV p - p , r i s e - t i m e < 0.4 fall  time < 0.1  ysec.  the  output usee and  The f e e d t h r o u g h v o l t a g e was measured u s i n g a h i g h  g a i n l i n e a r r f a m p l i f i e r w i t h an i n p u t impedance o f 75 n a t 10MHz and a d e t e c t i o n t h r e s h o l d o f 2yV p - p , and hence the o v e r a l l r e j e c t i o n  ratio  f o r the g a t e c i r c u i t may be c o n s i d e r e d to be a c c u r a t e l y d e t e r m i n e d g r e a t e r than 120 dB.  as  The o p e r a t i n g f r e q u e n c y o f the r f g a t e d e s c r i b e d  above i s r e a d i l y changed to any f r e q u e n c y up t o 20 MHz by v a r y i n g o n l y Cl and C 2 , and the o u t p u t p u l s e c h a r a c t e r i s t i c s  a r e shown t o be w i t h i n  the 10MHz l i m i t s g i v e n above. The r f g a t e c o n t r o l monostable m u l t i v i b r a t o r c i r c u i t i s g i v e n in Fig. 3.5(b).  This pulse generator  was d e v e l o p e d as an i n t e g r a l  F i g . 3.6  (b)  Control dc-pulse  generator  Gate c o n t r o l waveform and o u t p u t r f (0.5  ysec/cm)  Output r f p u l s e ( 0 . 5  Fig.  3.6  pulse.  ysec/cm,  R f - p u l s e gate o p e r a t i o n a l  5V/cm)  characteristics.  p a r t o f t h e r f gate system t o a t t a i n t h e r f p u l s e r i s e and f a l l previously  s p e c i f i e d , a TEKTRONIX 161 p u l s e g e n e r a t o r b e i n g  i n t h i s r e s p e c t , and t o s i m p l i f y t h e r f i s o l a t i o n o f t h i s circuitry.  The c a t h o d e - c o u p l e d  times  inadequate  control  m u l t i v i b r a t o r u s i n g t h e 6DJ8 double 123  t r i o d e , VI and V2, was d e s i g n e d the c r i t i c a l  gating pulse f a l l  124  f o l l o w i n g w e l l known p r i n c i p l e s time b e i n g m i n i m i s e d  '  through a c a r e f u l  s t u d y o f t h e m u l t i v i b r a t o r c h a r a c t e r i s t i c s , t h e use o f t h e p l a t e c a t c h i n g d i o d e D l and i s o l a t i o n o f t h e m u l t i v i b r a t o r c i r c u i t r y by a cathode f o l l o w e r , V3.  This f o l l o w e r configuration  t i v e capacitance at the output p l a t e  reduces  the e f f e c -  (V2) o f t h e m u l t i v i b r a t o r , t h i s  c a p a c i t a n c e b e i n g the major f a c t o r d e f i n i n g t h e o u t p u t p u l s e f a l l The p u l s e - w i d t h , t  w  time.  , i s governed by t h e time c o n s t a n t R i C i and t h e g r i d  b i a s f o r V I , and hence a v e r s a t i l e w i d t h c o n t r o l i s o b t a i n e d the v a r i a t i o n o f b o t h o f these parameters by R l and R2. ponent v a l u e s shown, t  w  may be v a r i e d  through  F o r the com-  from 5 t o 100 usees and may be  r e a d i l y a d j u s t e d t o w i t h i n 1% o f a s p e c i f i e d v a l u e .  The i n h e r e n t p u l s e  w i d t h j i t t e r i s measured as l e s s than 0.05% o f the p u l s e - w i d t h .  The  o u t p u t p u l s e a m p l i t u d e may be v a r i e d , through R3, between 1 and 30V. The gate c o n t r o l p u l s e i s shown i n F i g . 3 . 6 , and t h e dc p u l s e r i s e and fall 0.1  times a r e measured under normal r f . gate o p e r a t i n g c o n d i t i o n s as and 0.05 u s e e , r e s p e c t i v e l y .  These s w i t c h i n g times a l l o w an r f  p u l s e t o be d e f i n e d i n terms o f an i n t e g r a l number o f r f c y c l e s , and a l s o a l l o w t h e development o f a c o m p l e t e l y c o h e r e n t p u l s e system.  In  such a system, the p u l s e gate waveform i s l o c k e d t o the r f waveform and hence a sequence o f r f p u l s e s w i t h an i d e n t i c a l phase may be generated.  The c o n t r o l p u l s e g e n e r a t o r i s t r i g g e r e d by a 6V p o s i t i v e  94.  p u l s e a t the c o n t r o l g r i d o f V I .  A l l additional timing triggers  d e r i v e d from the low impedance p u l s e o u t p u t s  A  In the complete p u l s e s p e c t r o m e t e r ,  and B  are  i n F i g . 3.6(b).  the p u l s e from the r f  gate i s used as the i n p u t t o a c o n v e n t i o n a l h i g h power r f a m p l i f i e r u s i n g push-push frequency o f 10, 20 o r 40 MHz)  d o u b l e r s (to g i v e a f i n a l o p e r a t i n g  and p u s h - p u l l c l a s s C a m p l i f i e r s .  d i s t o r t i o n i s minimised  frequency  Rf p u l s e shape  by u s i n g o n l y low-Q tuned c i r c u i t s .  With a  s t a b i l i z e d h i g h v o l t a g e power s u p p l y f o r the f i n a l r f a m p l i f i e r , up 1.6  kW  (400 V p-p  i n t o a 100 Q, r e s i s t i v e load) i s a v a i l a b l e i n a p u l s e  sequence w i t h a minimum p u l s e i n t e r v a l p u l s e w i d t h f o r 'H NMR and t h i s c o r r e s p o n d s sample volume.  ( y ^ = 2.7  x 10  4  o f 60 usees. -A t y p i c a l TT/2r a d . sec  1  gauss *) i s 10 u s e e s ,  to an H i m a g n e t i c f i e l d o f 5.8  I t s h o u l d be n o t e d  sample tube i s n o r m a l l y 5 mm  gauss over  the  t h a t , f o r the V a r i a n V-4300 c r o s s e d -  c o i l probe used,, the t r a n s m i t t e r c o i l i s 25 mm  Although  to  i n diameter w h i l e  with a corresponding  the c r o s s e d - c o i l system a t t e n u a t e s  8 mm  the  receiver coil.  the a v a i l a b l e r f power i n  a p u l s e , i t does g i v e a h i g h homogeneity H i f i e l d o v e r the sample volume w h i c h i s u s u a l l y more i m p o r t a n t  i n high r e s o l u t i o n a p p l i c a t i o n s .  In a d d i t i o n , when t h i s c o i l system i s used i n c o n j u n c t i o n w i t h a 125 m o d i f i e d LEL l i n e a r r f a m p l i f i e r  , the time f o r r e c o v e r y from o v e r -  l o a d f o l l o w i n g an r f p u l s e i s t y p i c a l l y l e s s than 5 usees when the probe i s c o r r e c t l y b a l a n c e d . scheme uses a HP  An improved r f p h a s e - s e n s i t i v e d e t e c t i o n  10514B b r o a d band m i x e r , w i t h i n p u t s from the 125  s i g n a l a m p l i f i e r and a s i m p l i f i e d r e f e r e n c e a m p l i f i e r  LEL  , f o l l o w e d by  a low o u t p u t impedance dc a m p l i f i e r which d r i v e s a l l r e c o r d i n g d e v i c e s  \  95.  i n c l u d i n g t h e FABRITEK FT-1064 computer.  The o v e r a l l l i n e a r i t y o f the  r e c e i v e r system has been checked o v e r t h e i n p u t range o f 10 uV t o 10 mV p-p a t a g a i n o f 1 0  k  t o be b e t t e r than 2%.  96.  CHAPTER 4 EXPERIMENTATION AND  4.1  H i n d e r e d r o t a t i o n i n amides.  4.1.1  N,N-dimethyl  carbamyl  T h i s compound was r o t a t i o n about the N-C t i o n i n the NCO  CALCULATIONS  chloride.  chosen f o r an i n i t i a l  s t u d y o f the h i n d e r e d  bond i n amides, due t o the e l e c t r o n  bond system, u s i n g the s t e a d y - s t a t e NMR  c r i b e d i n d e t a i l i n Chapter 2 o f t h i s t h e s i s .  delocalisa-  methods des-  A l t h o u g h a number o f  s u b s t i t u t e d amides have been s t u d i e d , the parameters d e r i v e d from the NMR  d a t a show a l a r g e v a r i a n c e and a s e m i - q u a n t i t a t i v e c o r r e l a t i o n o f  the  data a v a i l a b l e i s not p o s s i b l e .  The o n l y example o f an N-C  t i o n a l b a r r i e r t h a t has been s t u d i e d u s i n g complete NMR  rota-  lineshape  a n a l y s e s i n independent l a b o r a t o r i e s , and f o r which the k i n e t i c p a r a meters are a p p a r e n t l y c o n s i s t e n t w i t h i n the c a l c u l a t e d e x p e r i m e n t a l e r r o r s , i s t h a t f o r N,N-dimethyl is  formamide  ^,61^  f o r t u i t o u s i n view o f the f a c t t h a t one o f the models used ^  t h i s f o u r - s i t e exchange p r o c e s s i s i n c o r r e c t . o b t a i n e d here f o r N,N-dimethyl for  result for  Thus the k i n e t i c d a t a  carbamyl c h l o r i d e (DMCC) i s i n t e n d e d  comparison w i t h t h a t a v a i l a b l e from a r e c e n t independent s t u d y  u s i n g a complete l i n e s h a p e a n a l y s i s ^ to  However, t h i s  and s p i n echo  s u b s t a n t i a t e the g e n e r a l v a l i d i t y and r e l i a b i l i t y  o f NMR  studies o f the a p p l i c a t i o n  methods t o t h e s t u d y o f h i n d e r e d r o t a t i o n . DMCC was o b t a i n e d from K § K L a b o r a t o r i e s Co. L t d . , and  p u r i f i e d by a d o u b l e vacuum d i s t i l l a t i o n  and s t o r e d o v e r m o l e c u l a r  was  sieves. compound.  E x p e r i m e n t a l and l i t e r a t u r e p h y s i c a l c o n s t a n t s agreed f o r t h i s As DMCC i s a l i q u i d  (b.p. = 165°C a t 760 mm) t h i s amide was  s t u d i e d n e a t and i n t h e n o n - p o l a r s o l v e n t carbon t e t r a c h l o r i d e  (6 mole % ) .  In p r e p a r i n g t h e NMR samples, 2 mole % o f t e t r a m e t h y l s i l a n e (TMS) was added t o p r o v i d e a s t a b l e f i e l d - f r e q u e n c y l o c k s i g n a l and 2 mole % o f dioxane was a l s o added as a r e f e r e n c e peak as i t i s c o n v e n i e n t l y p l a c e d w i t h r e s p e c t t o t h e DMCC spectrum and has a temperature independent l i n e w i d t h .  The samples .were t h o r o u g h l y degassed by t h e u s u a l  freeze-pump-thaw c y c l e and were s e a l e d i n vacuo i n t h i n - w a l l e d NMR tubes o f 5 mm o.d.. The NMR s p e c t r a were o b t a i n e d on a JEOLCO JNM-C-60H s p e c t r o meter used i n t h e i n t e r n a l l o c k mode and equipped w i t h t h e JES-VT-2 v a r i a b l e temperature c o n t r o l l e r .  The t e m p e r a t u r e i s n o r m a l l y m o n i t o r e d  by a thermocouple p l a c e d n e a r the sample i n the t e m p e r a t u r e c o n t r o l gas stream.  As a t e m p e r a t u r e measurement i s c r i t i c a l i n q u a n t i t a t i v e  k i n e t i c s t u d i e s , e s p e c i a l l y when i t i s o n l y p o s s i b l e t o a p p l y complete l i n e s h a p e a n a l y s e s o v e r a r e l a t i v e l y s m a l l t e m p e r a t u r e range, t h e temp e r a t u r e was c a l i b r a t e d w i t h a second thermocouple immersed i n t h e same volume o f sample c o n t a i n e d i n an open n o n - s p i n n i n g NMR tube and a l s o by r e p l a c i n g the sample w i t h t h e e t h y l e n e g l y c o l sample s u p p l i e d by V a r i a n . In a d d i t i o n , t h e temperature was checked b e f o r e and a f t e r a number o f s p e c t r a were r e c o r d e d .  Two c o n c l u s i o n s were i m m e d i a t e l y e v i d e n t .  While t h e use o f t h e n o n - s p i n n i n g sample does n o t g i v e a r e l i a b l e measurement o f t h e s p i n n i n g sample t e m p e r a t u r e , t h e i n s t r u m e n t thermoc o u p l e t e m p e r a t u r e r e a d i n g can be a c c u r a t e l y c o r r e l a t e d w i t h t h a t g i v e n from t h e e t h y l e n e g l y c o l c h e m i c a l s h i f t - t e m p e r a t u r e e q u a t i o n r e p o r t e d by  Van  Geet  ment was  .  As the a b s o l u t e e r r o r i n the sample temperature measure-  estimated  t e m p e r a t u r e was  t o be ± 0.3°C, i t was  c o n s t a n t w i t h i n these  a l s o shown t h a t the sample  l i m i t s o v e r the r e c o r d i n g time  f o r a number o f s p e c t r a at a g i v e n c o n t r o l l e r temperature s e t t i n g . Thus, i n g e n e r a l , i t i s p o s s i b l e t o use the i n s t r u m e n t  thermocouple  measurements c a l i b r a t e d a t f o u r temperatures w i t h the g l y c o l sample t o reduce the time i n v o l v e d w i t h c o n t i n u a l sample r e p l a c e m e n t . r a t e s as low as 0.05  Hz sec  1  were used t o ensure a good  Sweep  approximation  t o slow passage c o n d i t i o n s and s p e c t r a were checked f o r p o s s i b l e d i s t o r t i o n due  t o s a t u r a t i o n , c f . s e c t i o n 2.3.  At each t e m p e r a t u r e ,  field  c u r v a t u r e and homogeneity and r f phase adjustments were made t o ensure r e p r o d u c t i o n o f the t r u e s p e c t r a l l i n e s h a p e . a t half-maximum o f the r e f e r e n c e peak was 0.5  Hz.  The  resultant linewidth  h e l d w i t h i n the range 0.3  M u l t i p l e s p e c t r a were o b t a i n e d at each t e m p e r a t u r e and  -  any  s p e c t r a t h a t were n o t p r e c i s e l y r e p r o d u c i b l e were d i s c a r d e d . The h i n d e r e d  r o t a t i o n i n DMCC may  p o p u l a t i o n t w o - s i t e exchange p r o c e s s  be c o n s i d e r e d  as an  equal  and hence the s t e a d y - s t a t e  a b s o r p t i o n mode l i n e s h a p e f u n c t i o n , V ( x ) , i s g i v e n by Eq.  NMR  (2.2.15) i n  accordance w i t h the s t o c h a s t i c model d e s c r i b e d i n d e t a i l i n s e c t i o n The  experimental  amplitude  s p e c t r a l data i s converted  to a series of N  2.2.  digital  v a l u e s , V (XJ[) , and a s s o c i a t e d , f r e q u e n c i e s , x^ . These f r e -  q u e n c i e s are d e r i v e d from two  c a l i b r a t i o n f r e q u e n c i e s measured d i r e c t l y  as the d i f f e r e n c e between the sweep ( o b s e r v i n g ) f i e l d lock frequency,  t o an a c c u r a c y  o f ± 0.1  frequency  Hz.  and the  A FORTRAN-IV computer  program PLONK has been d e v e l o p e d to a l l o w an e f f i c i e n t i t e r a t i v e shape a n a l y s i s o f the d i g i t i z e d d a t a .  The  fixed  line-  i t e r a t i v e f i t t i n g i s based upon  99.  a simple but e f f e c t i v e search routine which rapidly converges  to a  b e s t - f i t parameter value through a series of optimised parameter increments, the b e s t - f i t value corresponding to a minimum i n the sum of 127 squares of deviations S  =  , S, where  N Z i=l  [VCxi)  - VCxJ]  2  and V(xi) i s the t h e o r e t i c a l lineshape amplitude f o r x=x^.  The  line-  shape function i s normalised to the experimental data, V ( x ^ ) , through the constant A i n Eq. (2.2.15).  The parameters to be considered i n a  general lineshape analysis are the chemical s h i f t between exchange s i t e s 2ft, cf. Eq. (2.2.1), the transverse relaxation time T  2  describing the  r e s i d u a l linewidth i n the absence of exchange and the f i r s t order rate constant k defining the exchange process.  T?_ i s defined i n terms of the  linewidth of the reference peak, where i t i s i m p l i c i t l y assumed that the magnetic f i e l d homogeneity i s described by a Lorentzian frequency dist r i b u t i o n , and i s considered to be a fixed parameter.  A standard non-  128 l i n e a r least squares regression analysis parameters was  using ft and k as variable  shown to be u n r e l i a b l e , e s p e c i a l l y f o r data obtained  above the coalescence temperature,  and this i s probably due to the  unusual forms of the p a r t i a l derivatives required, [ —  j Xi  and (  ) Xi  Therefore, the program PLONK takes k as the variable parameter f o r a s p e c i f i c ft value, with the option of being able to vary ft i n a welldefined manner to determine the coupled parameter values corresponding to a minimum S value. analysis to determine  This appears to be the most s a t i s f a c t o r y form of a temperature  dependence of the chemical s h i f t ,  100.  29.  F o r N = 20-60, t h e CPU time i n v o l v e d f o r each k i t e r a t i v e f i t i s  o f t h e o r d e r o f 4 sees on an IBM 360/67 system, and t h e s e t t i n g up o f a l l a r r a y s f o r c o n t r o l o f a CALCOMP p l o t t e r t a k e s l e s s than 1 s e c . Typical lineshape f i t s  f o r DMCC (neat) u s i n g t h e computer program  d e s c r i b e d above and a c o n s t a n t c h e m i c a l s h i f t 29, = 6.80 Hz a t 60 MHz a r e shown i n F i g . 4.1, and t h e r a t e c o n s t a n t s o b t a i n e d by a v e r a g i n g t h e k v a l u e s from t h e f i t t i n g o f m u l t i p l e s p e c t r a a t a l l t e m p e r a t u r e s a r e summarised i n T a b l e 4.1.  I t s h o u l d be n o t e d t h a t t h e mean  d e v i a t i o n , c a l c u l a t e d as  1 i  N  £ i=l  [V'CxiO  - V(x )]/V(x ), i  i  f o r t h e s p e c t r a shown i n F i g . 4.1 i s 1.5%. The A r r h e n i u s p l o t f o r DMCC (neat) i s shown i n F i g . 4.2. The a c t i v a t i o n energy, E , f o r t h e h i n d e r e d r o t a t i o n i s d e t e r m i n e d as a  17.64  ± 0.52 k c a l s . mole  1  from a l e a s t squares f i t  to the usual  129 equation k  =  A exp(-E /RT), a  (4.1.1)  where T i s t h e a b s o l u t e t e m p e r a t u r e , A i s t h e f r e q u e n c y f a c t o r f o r t h e r a t e p r o c e s s and R i s t h e u n i v e r s a l gas c o n s t a n t (1.986 c a l . deg  1  Both E  a  mole  and A a r e assumed t o be t e m p e r a t u r e i n d e p e n d e n t , and t h e a c t i v a -  t i o n energy i s d e f i n e d as t h e d i f f e r e n c e between t h e average e n e r g i e s o f t h e ground s t a t e and t h e a c t i v a t e d t r a n s i t i o n s t a t e i n v o l v e d i n t h e hindered r o t a t i o n . shows t h a t  A thermodynamical  formulation of reaction  rates  130  TABLE 4.1  K i n e t i c d a t a f o r N,N-dimethyl carbamyl c h l o r i d e (neat)  Temp.°C  k sec  28.7  4.32  29.2  4.52  33.0  6.35  33.2  6.35  35.8  8.16  70  ii  A  n  A  39.4  11.46  40.5  10.55  41.5  13.59  46.8  23.00  48.4  25.99  50.8  32.02  E  a  =  17.64 ± 0.52 k c a l s .  mole"  at 25°C  AH  #  =  17.05 k c a l s . m o l e "  AS  #  =  0.8 ± 1.6 c a l d e g " m o l e "  AG  #  =  16.82 k c a l s . m o l e "  1  1  1  1  1  F i g . 4.1  Lineshape f i t s  f o r N,N~dimethyl  s i t e e q u a l p o p u l a t i o n exchange  carbamyl c h l o r i d e , system.  two-  \  101.  k  =  -j- exp (-AG /RT) #  4  ^--^  k#  4 1 2  where fe i s t h e Boltzmann c o n s t a n t , n i s P l a n c k ' s c o n s t a n t and A G i s the f r e e energy o f a c t i v a t i o n .  K  may be i n t e r p r e t e d  as an e q u i l i b r i u m  c o n s t a n t f o r t h e i n t e r c h a n g e between ground and a c t i v a t e d s t a t e s , and may t h e r e f o r e be d e f i n e d i n terms o f t h e i n c r e a s e i n energy i n f o r m i n g the t r a n s i t i o n s t a t e , AE*, by t h e g e n e r a l  ^ Thus i t f o l l o w s  d , „# AE Ink =  relationship ,A  .  T  (4.1.3)  that E  =  a  AE*  +  RT  and t h e e n t h a l p y o f a c t i v a t i o n , AH , may now be d e f i n e d as AH* where AV  =  AE* - PAV  #  ,  i s t h e i n c r e a s e i n volume on g o i n g i n t o t h e t r a n s i t i o n s t a t e .  For t h e u n i m o l e c u l a r h i n d e r e d r o t a t i o n p r o c e s s , AV  = 0 and i n t h i s  p a r t i c u l a r case AH* may be c a l c u l a t e d d i r e c t l y from the e x p e r i m e n t a l a c t i v a t i o n energy a t a g i v e n temperature AH* Substitution  for E  a  =  E  &  - RT  as .  (4.1.4)  i n Eq. (4.1.1) now a l l o w s a d e t e r m i n a t i o n o f t h e  e n t r o p y o f a c t i v a t i o n , AS , i n accordance w i t h Eq. (4.1.2) as  102.  AS  with The  #  AG a c t i v a t i o n parameters  #  =  R[ln ^  =  AH  #  - 1],  - TAS  (4.1.5)  (4.1.6)  #  c a l c u l a t e d f o r DMCC ( n e a t ) a t 25°C a r e  i n c l u d e d i n T a b l e 4.1 and a r e i n v e r y good agreement w i t h t h o s e o b t a i n e d by Neuman e t a l . ^ 16.8  k c a l s mole  1  , AS  ( E = 16.9 ± 0 . 5 k c a l s . mole g  = -1.6 c a l s . deg  1  mole  l  l  , AG^ =  ) . I t has p r e v i o u s l y  , . 59, 60, 64 . . . • *i • u + •. been n o t e d ' t h a t as i n c r e a s i n g c a r e i s t a k e n m o b t a i n i n g k i n e t i c d a t a u s i n g NMR methods, t h e v a l u e s o f E and A t e n d t o i n c r e a s e ' a b  w h i l e t h e f r e e energy  o f a c t i v a t i o n remains n e a r l y c o n s t a n t .  p r e s e n t e d here e x e m p l i f y ; N.N-dimethvl  the  amides the ent.ronv  47 r e l a t i v e l y small  trends.  The d a t a  For hindered r o t a t i o n s i n  of activations  i s exnected t o be  59 '  . As t h i s e n t r o p y change i n l i q u i d s i s most  probably a s s o c i a t e d with a d i f f e r e n c e i n the s o l v a t i o n s t r u c t u r e s f o r t h e ground and t r a n s i t i o n s t a t e s i t i s p r e d i c t e d t h a t t h e r e w i l l be a s m a l l i n c r e a s e i n e n t r o p y on f o r m a t i o n o f t h e t r a n s i t i o n  state,  i n t h a t t h e d i p o l e moment f o r t h i s s t a t e i s n o r m a l l y s m a l l e r t h a n t h a t f o r t h e p l a n a r ground s t a t e .  W i t h i n t h e e r r o r l i m i t s g i v e n above, and  c a l c u l a t e d as t h e s t a t i s t i c a l 95% c o n f i d e n c e l i m i t s , t h i s i s shown t o be t h e case f o r DMCC s t u d i e d as a neat l i q u i d .  I n comparison, t h e  e n t r o p y o f a c t i v a t i o n d e r i v e d from s p i n - e c h o d a t a i s ~ -10.5 c a l s . deg.  1  mole  1  and t h e c o r r e s p o n d i n g a c t i v a t i o n energy  i s 14.0 ± 0.9 k c a l s . mole under t h e assumption  I t i s t o be n o t e d , however, t h a t  1  t h a t AS  f o r DMCC (neat)  =0.  t h e a c t i v a t i o n energy may be  c a l c u l a t e d from t h e s p i n - e c h o d a t a a t t h e c o a l e s c e n c e  temperature^  103.  t o be 17.5 k c a l s mole v a l u e p r e s e n t e d above. determined  , i n e x c e l l e n t agreement w i t h t h e s t e a d y - s t a t e I n a d d i t i o n , t h e f r e e energy  of activation i s  as AG* = 16.6 k c a l s m o l e , c f . 16.8 k c a l s mole" T a b l e 4.1. - 1  1  S o l u t e - s o l v e n t i n t e r a c t i o n s f o r t h e s t r o n g l y p o l a r DMCC molecule  (p - 4.08D  may be expected  t o i n f l u e n c e t h e magnitude o f  t h e p o t e n t i a l b a r r i e r f o r h i n d e r e d r o t a t i o n i n t h i s amide. was  a l s o s t u d i e d as a d i l u t e  vent CCl4 t o minimise from t o t a l  Thus DMCC  (6 mole %) s o l u t i o n i n the n o n - p o l a r  such i n t e r a c t i o n s .  sol-  The k i n e t i c d a t a o b t a i n e d  lineshape analyses, using a constant chemical  shift  29. = 7.1 Hz a t 60 MHz a r e summarised i n T a b l e 4.2. The c o r r e s p o n d i n g A r r h e n i u s p l o t i s shown i n F i g . 4.2 and t h e a c t i v a t i o n energy i s determined  as 17.05  ± 0.47 k c a l s mole . 1  o n l y a s m a l l s o l v e n t dependence.  Thus t h i s p a r a m e t e r shows  The e n t r o p y o f a c t i v a t i o n i s a g a i n  v e r y s m a l l b u t f o r m a l l y n e g a t i v e , AS  = -0.6 ± 1 . 5 c a l s deg  1  mole  A l t h o u g h the h i n d e r e d r o t a t i o n i n t h i s p a r t i c u l a r amide i s shown t o be n e a r l y independent  o f s o l u t e c o n c e n t r a t i o n , t h e form o f t h e  ' a n i s o t r o p y o f t h e c a r b o n y l group  132 133 ' , which  magnetic  i s assumed t o be t h e  dominant mechanism g i v i n g r i s e t o t h e c h e m i c a l s h i f t between t h e Nmethyl  groups c i s and t r a n s t o t h i s f u n c t i o n a l group,  s i g n i f i c a n t l y from t h a t i n t h e n e a t l i q u i d system.  i s changed  T h i s change i s  p r e s u m a b l y due t o t h e break-down o f a s p e c i f i c s o l u t e - s o l v e n t i n t e r action. In o r d e r t o c o n s i d e r the i n t e r - r e l a t i o n s h i p s between d e r i v e d a c t i v a t i o n parameters,  a l l data a v a i l a b l e f o r the hindered r o t a t i o n i n  DMCC have been c o r r e l a t e d i n T a b l e 4.3.  The AG*,  have been c a l c u l a t e d f o r a f i x e d t e m p e r a t u r e  AH* and AS* v a l u e s  o f 25°C.  The o v e r a l l  TABLE 4.2  K i n e t i c d a t a f o r N,N-dimethyl carbamyl  chloride  (6% C C l ^ s o l u t i o n )  0.5  E  .  0.44  5.9  0.52  15.2  1.25  30.5  6.30  34.5  9.45  39.3  13.54  44.0  18.83  44.6  18.85  48.8  31.58  49.5  33.62  54.5  51.05  54.8  55.55  58.0  93.42  63.2  137.10  a  AH  #  =  17.05 ± 0.47 k c a l s . m o l e "  =  16.46 k c a l s . m o l e "  1  a t 25°C  AS* =  -0.6 ± 1.5 c a l . d e g "  AG* =  16.64 k c a l s .  mole"  1  1  1  mole"  1  TABLE 4.3  N,N-dimethyl carbamyl c h l o r i d e a c t i v a t i o n parameters  ref.  E  a  17,.64 + 0,,52  16,.82  17,,05 + 0,,52  a  17,.05 + 0.,47  16.,64  16.,46 + 0.,47  b  16,.9  + 0.,5  16,,8  16,.3  + 0,,5  -1.,6 + 2,.0  b  17,.7  + 0,.9  16,.3  17,.1  + 0,,9  2,,6 + 3,,2  c  14,.0  + 0.,9  16,,6  13,,4  + 0,.9  -10,,6 + 2.,7  c  9,.7  + 0.,5  16,.6  9,,1  + 0,,5  -25,,3 + 2..8  c  8,.6  + 1.,7  16.,6  8,,0  + 1.,7  -28..9 + 5,,5  d  7.,3  + 0.,5  16,.4  6,,7  + 0..5  -27,, 1 + 2,,4  e  6,,8  + 0.,2  16,.2  6,.2  + 0,,2  -33,,6 + 1,,0  f  8..65 + 0.,88  16,,3  8,,06 + 0,,88  a  T h i s work  b  R.C. Neuman, D.N. Roark and V. Jonas  c  A. A l l e r h a r d and H.S. Gutowsky J.Chem.Phys. 41_, 2115, 1964  d  M.T.  e  J.C. Woodbrey and M.T.  f  AG  AH*  #  AS''  !  0,,8 + 1,,6 -0.,6  1,,5  -27,,5 + 3,,0  JACS 89^, 3412, 1967  Rogers and J.C. Woodbrey J.Phys.Chem. 66, 540, 1962 Rogers  JACS 84_, 13, 1962  "E. Krakower, Ph.D. T h e s i s , UBC, 1967.  \ s  \  F i g . 4.2  A r r h e n i u s p l o t s f o r N,N-dimethyl  o  6% C C l ^ s o l u t i o n  ©  neat  liquid  carbamyl  chloride.  20  a,b a . b  o  «> 16 o  E.  rd u  *  <  12  c,f  8  d.  4  30  -20  10  AS" cals.deg. mole~ 1  Fig.  4.3  0  5  1  V a r i a t i o n o f a c t i v a t i o n parameters f o r h i n d e r e d r o t a t i o n i n N,N-dimet.hyl carbamyl  chloride.  104.  v a r i a t i o n i n AH*  and AS*,  as o b t a i n e d u s i n g d i f f e r e n t methods o f  a n a l y s i s and a l s o d i f f e r e n t s o l v e n t s and s o l u t e c o n c e n t r a t i o n s , i s shown g r a p h i c a l l y i n F i g . 4.4.  The  l a r g e v a r i a t i o n i n these parameters  f o r i d e n t i c a l c h e m i c a l systems i s due  t o r e l a t i v e l y s m a l l e r r o r s i n the  e x p e r i m e n t a l d e t e r m i n a t i o n o f s p e c i f i c r a t e c o n s t a n t s and/or temperatures.  Although  t h e s e e r r o r s may  g i v e r i s e t o o n l y a s m a l l change i n  the s l o p e of the corresponding A r r h e n i u s p l o t over the range f o r a p a r t i c u l a r s t u d y , t h i s t e m p e r a t u r e small  temperature  range i s u s u a l l y v e r y  (20-80°C) and hence r e l a t i v e l y l a r g e v a r i a t i o n s i n t h e E  and  A  cL  v a l u e s determined  from Eq.  (4.1.1) soon become e v i d e n t .  This of  l e a d s t o c o r r e s p o n d i n g v a r i a t i o n s i n AH  and AS  Eqs.  I n g e n e r a l , as shown i n  (4.1.4) and  4. F i g . 4.4, 47,  64  (4.1.5), r e s p e c t i v e l y .  higher E  course  i n accordance w i t h  (All ) v a l u e s are a s s o c i a t e d w i t h h i g h e r A v a l u e s cl  , and v i c e v e r s a .  Except  f o r a s i n g l e p o i n t , however, a  linear  V  #  c o r r e l a t i o n between AH  •  #  and AS  f r e e energy o f a c t i v a t i o n , AG  i s o b t a i n e d and t h i s i m p l i e s t h a t t h e , i s approximately  i n v a r i a n t to small  s y s t e m a t i c e r r o r s i n h e r e n t i n the d e t e r m i n a t i o n o f E (4.1*6).  and A, c f . Eq.  Thus i t becomes e v i d e n t t h a t t h e o n l y s i g n i f i c a n t p a r a m e t e r 64  o b t a i n e d from NMR The  d a t a may  i n d e e d be the f r e e energy o f a c t i v a t i o n  a c t i v a t i o n parameters  f o r DMCC d e t e r m i n e d  l i n e s h a p e a n a l y s e s are shown i n d e t a i l i n F i g . 4.5,  by complete  t h i s method o f  a n a l y s i s g i v i n g t h e most r e l i a b l e and h i g h e s t p r e c i s i o n e s t i m a t e s these parameters.  The  d a t a o b t a i n e d i n i n d e p e n d e n t s t u d i e s are  s i s t e n t w e l l w i t h i n the c a l c u l a t e d e r r o r l i m i t s , but i t i s s t i l l p o s s i b l e t o r e l i a b l y compare t h e e n t h a l p i e s o f a c t i v a t i o n f o r the neat amide and the amide i n a C C H  solution.  of  connot  determined  Again the f r e e  105.  energy o f a c t i v a t i o n becomes t h e most s i g n i f i c a n t p a r a m e t e r , and i s d e t e r m i n e d f o r AS k c a l s mole . 1  = 0, c f . Eq. (4.1.6) and F i g . 4.4, as AG  = 16.65  I n t h i s p a r t i c u l a r case i t i s i n t e r e s t i n g t o note t h a t  the s l o p e o f t h e l i n e a r p l o t shown i n F i g . 4.5 d e f i n e s  a characteristic  t e m p e r a t u r e o f 258°K, which i s t o be.compared w i t h t h a t o f 298°K a t which AH  and AS  may be c o n s i d e r e d  have been c a l c u l a t e d .  This temperature d i f f e r e n c e  t o i n d i c a t e a s m a l l dependence o f AG* on s o l v e n t and  t h u s may be used t o s e m i - q u a n t i t a t i v e l y d e s c r i b e s o l u t e - s o l v e n t a c t i o n s f o r a s e r i e s o f s u b s t i t u t e d amides.  For the hindered  inter-  rotation  i n an amide, a f r e e energy change i s e x p e c t e d t o be d e f i n e d p r i m a r i l y by i n t r a m o l e c u l a r e f f e c t s and hence i s much s i m p l e r t o i n t e r p r e t t h a n the energy o f a c t i v a t i o n w h i c h i s s e n s i t i v e t o i n t e r m o l e c u l a r e f f e c t s . Also,there  i s a general  tendency i n r a t e p r o c e s s e s i n s o l u t i o n f o r 134'  enthalpy  and e n t r o p y changes t o compensate each o t h e r  135 '  so that,  the e f f e c t i v e change i n f r e e energy i s r e d u c e d and becomes l e s s s e n s i tive to external effects. preted  T h i s compensation e f f e c t i s s i m p l y  i n terms o f a s o l u t e - s o l v e n t i n t e r a c t i o n .  inter-  Any i n t e r a c t i o n t h a t  leads t o a stronger b i n d i n g o f solvent molecules t o a s o l u t e molecule lowers the enthalpy  o f t h e system, and a l s o , by r e s t r i c t i n g t h e f r e e -  dom o f motion o f b o t h s o l u t e and s o l v e n t m o l e c u l e s t h i s i n t e r a c t i o n lowers the entropy.  T h i s c h a r a c t e r i s t i c i s shown by t h e parameters  p r e s e n t e d i n F i g . 4.4.  I n t h i s p a r t i c u l a r c a s e , i t may be  considered  t h a t a d e c r e a s e i n e n t r o p y f o r t h e p l a n a r ground s t a t e ( r e l a t i v e t o that f o r the hindered  r o t a t i o n t r a n s i t i o n s t a t e ) o f DMCC as a neat  l i q u i d , due t o a d i p o l a r s o l u t e - s o l u t e  (solvent) i n t e r a c t i o n , leads t o a  I  1  -4  1  :  1  -2  1  1  0  1  2  1  1  1  A  A S cals.deg" mole F i g . 4.4  A c t i v a t i o n parameters o b t a i n e d by complete l i n e s h a p e a n a l y s i s f o r N,N-dimethyl  e  6% C C l t, s o l u t i o n  o  neat l i q u i d  carbamylchloride  106.  s m a l l p o s i t i v e AS  v a l u e as compared w i t h a commensurate n e g a t i v e  As" v a l u e f o r a d i l u t e CCli, s o l u t i o n i n which such an i n t e r a c t i o n i s presumably m i n i m i s e d .  In g e n e r a l , although d i f f e r e n t  solvents  and s u b s t i t u e n t s i n f l u e n c e AH* and AS* i n a complex manner, t h e p a r t i a l compensation e f f e c t i s o f such a form t h a t t h e i r  influence  If  on AG  i s much s i m p l e r i n form.  NMR l i n e s h a p e d a t a  Until, the present steady-state  * supplemented by s p i n - e c h o and d o u b l e resonance a r e  d a t a o f comparable p r e c i s i o n o v e r an extended t e m p e r a t u r e r a n g e , i t appears t h a t t h e f r e e energy o f a c t i v a t i o n i s t h e o n l y that w i l l  allow quantitative correlations f o r a series of substituted  N,N-dimethyl  4.1.2  parameter  amides.  N,N-dimethyl  carbamyl  N,N-dimethyl  bromide  carbamyl bromide  h a l o - s u b s t i t u t e d N,N-dimethyl  (DMCB) was chosen as a  amide i n a s e r i e s t o a l l o w a c o n s i s t e n t  s t u d y o f s u b s t i t u e n t e f f e c t s on t h e h i n d e r e d r o t a t i o n about t h e amide N-C bond. DMCB was p r e p a r e d by s a t u r a t i n g ~10 gms o f N,N-dimethyl 136 carbamyl c h l o r i d e  (DMCC)  by b u b b l i n g a m i x t u r e o f . H B r and N  t h r o u g h t h e n e a t l i q u i d c h l o r i d e k e p t a t 0°C. any HC1 o r C l  2  The N  2  2  gases  s t r e a m removed  formed and s i g n i f i c a n t l y i n c r e a s e d t h e y i e l d o f t h e  c a r b a m y l bromide.  The bromide was p u r i f i e d by m u l t i p l e  distillations  at ~10 mm, t h e b o i l i n g p o i n t o f t h e f i n a l p r o d u c t b e i n g 63°C a t t h i s pressure.  The p r o d u c t was i d e n t i f i e d by NMR and was shown t o c o n t a i n  ~2% o f t h e carbamyl c h l o r i d e , t h i s b e i n g v e r i f i e d by t h e e l e m e n t a l analysis:  107.  C  H  Calculated  23.68  3.95  Found  24.54  4.30  Br  Br + C l  52.63 -  54.11  T h i s l i q u i d amide was s t u d i e d neat as s o l u t e - s o l v e n t i n t e r a c t i o n s were shown t o have o n l y a s m a l l e f f e c t on t h e h i n d e r e d r o t a t i o n i n t h e s i m i l a r DMCC system, and on t h e a c t i v a t i o n of  interest at this point  parameter  ( A G ) , as d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n .  The NMR sample was p r e p a r e d as p r e v i o u s l y o u t l i n e d .  Complete  shape a n a l y s e s f o r t h i s t w o - s i t e e q u a l p o p u l a t i o n exchange  system,  u s i n g the i t e r a t i v e f i t t i n g program PLONK, gave t h e r e s u l t s i n T a b l e 4.4. T y p i c a l l i n e s h a p e f i t s are shown i n Fig.. 4.5. f i t s were n o t improved  summarised These  ( w i t h i n the r a t e c o n s t a n t e r r o r bounds) by  c o n s i d e r i n g a t e m p e r a t u r e dependent and thus the c i s - t r a n s N-methyl 5.4  line-  c h e m i c a l s h i f t 2ft, c f . Eq. ( 2 . 2 . 1 ) ,  c h e m i c a l s h i f t may be assumed t o be  + 0.2 Hz a t 60 MHz o v e r t h e t e m p e r a t u r e range -15°C t o 70°C.  As  t h i s c h e m i c a l s h i f t i s reduced by 20% and 32% from t h o s e f o r DMCC and the  p a r e n t amide DMF (N,N-dimethyl formamide,  2ft = 8.1 H z ) , r e s p e c t i v e l y ,  c o n s i d e r a b l e magnetic a n i s o t r o p y a s s o c i a t e d w i t h the C-Br bonding system i s i n d i c a t e d . compensates  T h i s a n i s o t r o p y i s o f a form t h a t  partially  t h a t due t o t h e C = 0 system, and t h i s compensation  effect  i s a l s o shown by the C - C l bonding system i n t h a t t h i s c h e m i c a l s h i f t for  DMCC i s reduced by 16% from t h a t f o r DMF.  However, i t i s t o be  n o t e d t h a t t h e s e c h e m i c a l s h i f t v a r i a t i o n s are due i n p a r t t o i n t e r 133 m o l e c u l a r i n t e r a c t i o n s and e l e c t r i c f i e l d e f f e c t s  137 '  . The  a c t i v a t i o n parameters c a l c u l a t e d f o r DMCB a t 25°C a r e g i v e n i n T a b l e 4.4,  TABLE 4.4 K i n e t i c d a t a f o r N,N-dimethyl carbamyl bromide (neat) Temp. °C  k sec"  -10.2  0.62  -  2.0  2.06  -  1.0  2.14  7.2  3.60  7.3  3.52  11.5  5.52  15.7  7.35  16.2  7.28  16.4  8.67  19.0  11.2  19.4  14.0  19.7  15.0  24.0  20.9  24.0  21.5  32.5  33.7  36.6  60.2  48.8  125.5  55.0  161.9  58.7  300.3  61.6  464.1  continued/...  TABLE 4.4 c o n t i n u e d  E  a  =  15.25 ± 0.36 k c a l s . mole  AH* =  14.66 k c a l s . m o l e "  AS* =  -3.3 ± 1.2 c a l s . deg"  AG* =  15.66 k c a l s . m o l e "  1  1  1  a t 25°C 1  mole  T = 24.0°C k = 20.9 s e c " F i g . 4.5  Lineshape f i t s  36.6 60.2 •  1  for"N,N-dime.thyl carbamyl bromide,  e q u a l p o p u l a t i o n exchange system.  two-site  H C  2.8 -  3  ^ 2.  H C 3  2.0 o CD  g  1.2  0.4  0.4-  2.8  3.2 1/T°K  F i g . 4.6  3.6  x10  4.0  3  A r r h e n i u s p l o t f o r N,N-dimethyl carbamyl bromide  108.  the energy o f a c t i v a t i o n f o r t h e h i n d e r e d r o t a t i o n b e i n g d e t e r m i n e d from t h e A r r h e n i u s p l o t shown i n F i g . 4.6.as 15.25 ± 0.36 mole . 1  mole  1  kcals.  A g a i n t h e e n t r o p y o f a c t i v a t i o n AS* = -5.3 + 1.2 c a l s . deg. i s d e t e r m i n e d t o be r e l a t i v e l y s m a l l , as  1  expected f o r the type  o f r a t e p r o c e s s under c o n s i d e r a t i o n .  4.1.3 M e t h y l N,N-dimethyl  carbamate  The m e t h y l e s t e r o f N,N-dimethyl c a r b a m i c a c i d , m e t h y l N,N-dimethyl carbamate  (DMCO), was  s t u d i e d t o a l l o w a comparison o f  the 0-methyl group w i t h h a l o g e n s and pseudo-halogens as s u b s t i t u e n t s i n an amide s e r i e s . I f h i n d e r e d r o t a t i o n about t h e N-C  bond i s p r e s e n t i n t h i s  carbamate, t h e r a t e p r o c e s s may be a n a l y s e d i n terms o f a s i m p l e twos i t e equal population  exchange system as t h i s compound i s n o t e x p e c t e d  t o have a p r e f e r r e d c o n f o r m a t i o n and the s p i n - s p i n c o u p l i n g the N-methyl and 0-methyl p r o t o n s w i l l be n e g l i g i b l e .  between  However, Drago  138 et a l equivalent  have r e p o r t e d  t h e N-methyl groups i n t h i s carbamate t o be  i n t h e n e a t l i q u i d and i n a number o f s o l u t i o n s a t a l l  t e m p e r a t u r e s down t o -46°C, showing t h a t t h e b a r r i e r t o r o t a t i o n about the N-C bond i s i n h e r e n t l y low, t h a t i s l e s s t h a n about 8 k c a l s . mole 139 A l a t e r s t u d y by L u s t i g e t a l i s increased inequivalence  1  showed t h a t t h i s b a r r i e r t o r o t a t i o n  i n c h l o r o f o r m s o l u t i o n s and t h a t a c i s - t r a n s N-methyl i s o b s e r v e d a t ~-25°C, t h e a s s o c i a t e d c h e m i c a l s h i f t  b e i n g dependent upon s o l u t e c o n c e n t r a t i o n .  The i n c r e a s e d n i t r o g e n  p a i r d e l o c a l i s a t i o n i n a CHCI3 s o l u t i o n , l e a d i n g t o an N-C r o t a t i o n  lone  109.  o b s e r v a b l e by NMR, i s presumably due t o a hydrogen-bonding  interaction  between t h e c a r b o n y l oxygen l o n e p a i r e l e c t r o n s and the s o l v e n t m o l e c u l e s which would tend t o p r e f e r e n t i a l l y s t a b i l i z e t h e amide +  resonance  form r e p r e s e n t e d as N = C = 0.  a 10 mole %  CHCI3  Thus DMCO i s s t u d i e d here as  s o l u t i o n t o o b t a i n an e s t i m a t e o f the e f f e c t o f  such a s o l v e n t i n t e r a c t i o n on t h e N - C = 0 bond system  and t o a l l o w  a c o r r e l a t i o n w i t h a s i m p l e m o l e c u l a r o r b i t a l model developed later section. chemical s h i f t  A 10% s o l u t i o n c o r r e s p o n d s  in a  t o a near maximal N-methyl  (2ft) o f o n l y 1.8 ± 0.1 Hz a t 60 MHz and a t -30°C.  DMCO was p r e p a r e d t h r o u g h the r e a c t i o n o f d i m e t h y l a m i n e w i t h methyl The  c h l o r o - f o r m a t e as d e s c r i b e d by Hartman and Brethen^'''.  l i q u i d p r o d u c t was p u r i f i e d by d i s t i l l a t i o n s a t ~40 mm and s t o r e d  over m o l e c u l a r s i e v e s , the f i n a l product having a b o i l i n g p o i n t o f 58°C a t t h i s p r e s s u r e , c f . l i t . was  v a l u e 56.5 - 5 7 ° C  1 4 1  p r e p a r e d as p r e v i o u s l y d e s c r i b e d f o r DMCC, w i t h  .  The NMR sample  Spectrograde  d e u t e r o c h l o r o f o r m which h a d been t h o r o u g h l y d r i e d o v e r m o l e c u l a r s i e v e s . Owing t o t h e v e r y s m a l l c h e m i c a l s h i f t 2ft, c f . Eq. ( 2 . 2 . 1 ) , the temperature  range f o r t h e k i n e t i c s t u d y i s l i m i t e d t o 25°C.  f o r e , t o o b t a i n r e l i a b l e r a t e c o n s t a n t s from complete  lineshape analyses,  a l a r g e number o f s p e c t r a were f i t t e d at. each t e m p e r a t u r e program PLONK, as d e s c r i b e d i n s e c t i o n 4.1.1. be t e m p e r a t u r e  independent  limits attainable.  There-  using the  The s h i f t 2ft was shown t o  (± 0.1 Hz) w i t h i n the l i n e s h a p e f i t t i n g  The averaged  error  r a t e c o n s t a n t s , k s e c " , are given i n 1  T a b l e 4.5 and the c o r r e s p o n d i n g A r r h e n i u s p l o t i s shown i n F i g . 4.8. The  energy o f a c t i v a t i o n , E , i s d e t e r m i n e d ;  as 15.18  ± 0.58 k c a l s m o l e "  1  TABLE 4.5 K i n e t i c d a t a f o r N,N-dimethyl carbamate (10% C H C I 3  solution)  Temp. °C  k sec  -23.9  0.25  -22.5  0.41  -16.0  0.76  -14.6  1.05  -14.4  1.15  - 9.3  1.36  - 9.3  1.44  - 8.9  1.57  - 8.3  1.68  - 5.4  2.23  - 5.2  2.46  - 2.7  3.18  - 2.2  3.42  - 1.1  4. 18  - 1.0  4.43  1.5  5.36  E  =  15.18 ± 0.58  =  14.54 k c a l s . m o l e "  k c a l s . mole  1  cl  AH  #  AS* = AG* =  a t 25°C  1  -2.0 ± 2.2 c a l s . d e g " mole 15.19 k c a l s . m o l e " 1  1  F i g . 4.7  Arrhenius p l o t f o r methyl N N-dimethyl 3  (10% CHC13  solution)  carbamate  110.  and  thus i t i s shown t h a t  the  carbonyl-CHCI3 i n t e r a c t i o n increases  the b a r r i e r t o h i n d e r e d r o t a t i o n f o r DMCO by about 8 k c a l s . mole This implies N-C  that,  i n g e n e r a l , the magnitude o f E g ( o r AG ) f o r t h e  r o t a t i o n may be a s e n s i t i v e measure o f i n t e r r a o l e c u l a r  interactions  w i t h an amide C = 0 group, when t h e s e p a r a m e t e r s are d e r i v e d t h r o u g h c o m p l e t e NMR l i n e s h a p e a n a l y s e s . d e t e r m i n e d from the deg  mole  1  implies  solute-solvent  f o r the  that  structure  i s very nearly  CHCI3  i n v a r i a n t t o the  change i n  amide system i n f o r m i n g the  r a t e p r o c e s s under c o n s i d e r a t i o n .  o f the  =-2.0 ± 2.2 c a l s  c a r b o n y l - CHCI3 i n t e r a c t i o n and hence  the  c h e m i c a l s h i f t 2ft i s d i f f i c u l t modification  entropy o f a c t i v a t i o n  above k i n e t i c d a t a f o r DMCO, AS  o v e r a l l d i p o l a r c h a r a c t e r o f the state  The s m a l l  transition  The magnitude o f t h e  t o i n t e r p r e t as i t w i l l depend upon a  form o f the C = 0 m a g n e t i c a n i s o t r o p y due t o t h e  h y d r o g e n - b o n d i n g and a l s o t h e m a g n e t i c a n i s o t r o p y a s s o c i a t e d  with 137  the  4.1.4  C-OCH3  group, i n a d d i t i o n t o s o l v e n t  N,N-dimethyl carbamyl  and e l e c t r i c f i e l d  effects  (DMCF) was s t u d i e d  as a  fluoride  N,N-dimethyl c a r b a m y l f l u o r i d e  f u r t h e r member o f a s e r i e s o f h a l o - s u b s t i t u t e d  N,N-dimethyl amides  showing h i n d e r e d r o t a t i o n about the N - C bond. i s w e l l known as a s p e c i f i c enzyme i n h i b i t o r " ' ' m a t i o n i s a v a i l a b l e on i t s e l e c t r o n i c s t r u c t u r e  A l t h o u g h t h i s compound 4 2 ,  very l i t t l e  as compared w i t h  inforother  144 amides o f i m p o r t a n c e i n b i o c h e m i c a l systems pound a l l o w s a d i r e c t c o m p a r i s o n o f the orbital calculations  . A s t u d y o f t h i s com-  data a v a i l a b l e  from m o l e c u l a r  f o r N,N-dimethyl formamide (DMF) and DMCF and f o r  111.  the p a r e n t  compounds formamide and carbamyl f l u o r i d e .  Carbamyl  fluoride  145 i s s t a b l e o n l y a t r e l a t i v e l y low t e m p e r a t u r e s  and thus i t i s n o t  p o s s i b l e t o e x p e r i m e n t a l l y determine the b a r r i e r t o hindered r o t a t i o n f o r t h i s p a r t i c u l a r amide. DMCF was p r e p a r e d dimethyl  by a s i m p l e exchange r e a c t i o n between N,N-  carbamyl bromide (DMCB) and AgF, u s i n g  CH3CN  as a s o l v e n t .  The  y i e l d was low (~ 1 0 % ) , however, and an improved method o f p r e p a r a t i o n 146 u t i l i z e s SbF3 as a f l u o r i n a . t i n g agent  .  The p r o d u c t  was p u r i f i e d by  vacuum d i s t i l l a t i o n s a t ~10mm (bp = 35°C), and a l s o by a d i s t i l l a t i o n at 760 mm (bp = 129°C) u s i n g a s p i n n i n g - b a n d  column, and gave an  elemental a n a l y s i s :  Calculated Found  C  H  F  39.58  6.59  20.88  %  39.81 6.58 20.68 % DMCF was s t u d i e d as a 16 mole % CCli* s o l u t i o n t o m i n i m i s e 147  s o l u t e - s o l v e n t i n t e r a c t i o n s f o r t h i s h i g h l y p o l a r compound (u = 4.02D The  NMR sample was p r e p a r e d  i n t h e manner p r e v i o u s l y d e s c r i b e d and as  u s u a l d i o x a n e was used as t h e r e f e r e n c e peak d e f i n i n g t h e t r a n s v e r s e r e l a x a t i o n t i m e , T 2 , i n t h e absence o f c h e m i c a l an  A 3 B 3 X  spectrum, t h e m e t h y l group  A3  exchange.  DMCF g i v e s  being trans t o the carbonyl  oxygen atom and r e s o n a t i n g a t a lower f i e l d t h a n t h e B3 group.  This  s p e c t r a l assignment i s c o n s i s t e n t w i t h t h e g e n e r a l NMR c h e m i c a l  shift  c h a r a c t e r i s t i c s a s s o c i a t e d w i t h the magnetic a n i s o t r o p y o f the carbonyl 132,133,148 _ i i•^ . ^1 -, l , r 1 9 „ group coupling constants  . I n a d d i t i o n , t h e unequal  H -  F spm-spm  a r e d e t e r m i n e d i n t h e absence o f exchange  - 20°C)  ).  112  as  |1 j f  AX  I  1  =  0.3 + 0.05 and -  1  normal t r a n s c o u p l i n g  | - J , 1, |  BX  V  ( J g ^ ) being  = 0.8 + 0.05 Hz, c o n s i s t e n t w i t h a ' 152 g r e a t e r than a c i s c o u p l i n g  .  This  assignment i s s i m i l a r t o t h a t made f o r N,N-dimethyl f o r m a m i d e ^ , i n = 0 . 3 5 and J g ^ = 0.60 Hz.  which case  These c o u p l i n g s may be com-  p a r e d w i t h t h o s e f o r t h e r e l a t e d a c e t y l compounds CH CF0 3  (Jj_,p  7.6 Hz)  =  and CH CH0 ( J j ^ = 2.85 H z ) , showing t h e e f f e c t o f t h e i n t e r p o s e d N atom 3  and  t h e N-C d o u b l e bond c h a r a c t e r on t h e s e i n d i r e c t s p i n - s p i n  The  chemical  s h i f t between t h e A and B m e t h y l groups (20) i n DMCF i s  v e r y s m a l l , s o l v e n t and c o n c e n t r a t i o n dependent. mole % C C l  dependent, and a l s o t e m p e r a t u r e  F o r example, t h e v a l u e s o f 20 f o r t h e n e a t amide and a 16 l(  s o l u t i o n a t ^-20°C a r e 1.2 + 0.1 and 2.20 + 0.05 Hz,  r e s p e c t i v e l y , a t 100 MHz. CC1,,  couplings  The c o r r e s p o n d i n g  s o l u t i o n i s 1.75 + 0.05 Hz.  s h i f t a t 30.5°C f o r t h e  T h i s p a r t i c u l a r s h i f t was a l s o  measured a t 220 MHz and was shown t o be a c c u r a t e l y c o n s i s t e n t w i t h the o v e r a l l assignment made f o r t h e A B - t r a n s i t i o n s . chemical  The m e t h y l  proton  s h i f t i s p r e d o m i n a n t l y due t o t h e combined e f f e c t o f t h e  m a g n e t i c s u s c e p t i b i l i t y a n i s o t r o p i e s o f t h e C-F and C=0 bond s y s t e m s . The V e l a t i v e l y s m a l l 20 v a l u e t h u s i n d i c a t e s t h a t t h e forms o f t h e s u s c e p t i b i l i t y t e n s o r s a s s o c i a t e d w i t h t h e e l e c t r o n i c charge d i s t r i b u t i o n s l o c a l i s e d i n t h e s e two bonds a r e such t h a t t h e two m e t h y l r e g i o n s have v e r y s i m i l a r c h a r a c t e r i s t i c s .  shield!  T h i s i n t u r n may i m p l y  d e l o c a l i s a t i o n o f f l u o r i n e 2 ^ e l e c t r o n s , as r e p r e s e n t e d  that  by t h e  r e s o n a n c e form N - C = F , i s s i g n i f i c a n t i n t h e ground s t a t e o f DMCF. 1  0 Such a charge d i s t r i b u t i o n would l e a d t o an a n i s o t r o p y  o f t h e C-F bond  w h i c h i s n o t a x i a l l y symmetric and i s comparable t o t h a t noi-mally  113.  a s s o c i a t e d w i t h t h e C=0 system.  The temperature dependence o f 29 may  be a s c r i b e d t o a t e m p e r a t u r e dependent  intermolecular dipole-dipole  a s s o c i a t i o n which a f f e c t s t h e magnetic a n i s o t r o p i e s o f t h e C=0 and/or C-F g r o u p s , and hence i t becomes apparent t h a t t h e r e d u c t i o n o f s o l u t e - s o l v e n t i n t e r a c t i o n s i n a CCli+ s o l u t i o n may i n c r e a s e t h e p r o b a b i l i t y o f s o l u t e - s o l u t e i n t e r a c t i o n s between t h e p o l a r DMCF molecules. Although chemical  s h i f t s a r e g e n e r a l l y dependent  solvent i n t e r a c t i o n s , i n d i r e c t spin-spin couplings  upon d e t a i l e d  are u s u a l l y very  n e a r l y independent o f t h e s e i n t e r m o l e c u l a r e f f e c t s .  I n t h e case o f an  amide such as DMCF, however, such i n t e r a c t i o n s may s t a b i l i z e t h e +  g r o u n d - s t a t e resonance form  N = C - F  l e a d i n g t o enhanced  couplings  0. between t h e m e t h y l p r o t o n s and t h e f l u o r i i i e n u c l e a r s p i n due t o t h e i n c r e a s e d N-C double-bond c h a r a c t e r . mentally,  i n that  This i s a c t u a l l y observed e x p e r i -  | j | = 1.1 ± 0.1 Hz f o r neat DMCF.  T h i s i s an  i n c r e a s e o f 0.3Hz as compared w i t h t h e v a l u e f o r t h e DMCF/CCli, s o l u t i o n , showing t h a t t h e CCli* s o l v e n t does break down s p e c i f i c s o l u t e - s o l u t e i n t e r a c t i o n s t o some e x t e n t .  As t h e f o u r - b o n d s p i n - s p i n c o u p l i n g  between t h e A 3 and B 3 methyl groups i n N,N-dimethyl  amides i s n e g l i g i b l e  the H NMR spectrum f o r t h e DMCF s p i n system may be a n a l y s e d 1  i n terms  o f an AB-part o f a f i r s t - o r d e r ABX ( J ^ g = 0) spectrum as d e s c r i b e d i n d e t a i l i n s e c t i o n 2.5 couplings experimental  o f t h i s t h e s i s and i l l u s t r a t e d i n F i g . 2.7. The  and J g ^ a r e shown t o be t e m p e r a t u r e independent ( w i t h i n e r r o r ) , as t h e averaged c o u p l i n g i n t h e l i m i t o f v e r y f a s t  114.  exchange (k»ft) i s d e t e r m i n e d as 0.6 calculated coupling J lineshapes  = ^^AX  +  * ^BX^  + 0.1  Hz c o r r e s p o n d i n g  "  - 0.10  Hz.  to a  The  observed  i n the p r e s e n c e o f exchange unambiguously l e a d t o  c o n c l u s i o n t h a t the r e l a t i v e s i g n s o f t h e s e c o u p l i n g s  the  are the same,  as i m p l i c i t l y assumed i n the above d e f i n i t i o n o f the parameter J . c h a r a c t e r i s t i c ABF i g . 2.7  p a r t o f a f i r s t - o r d e r ABX  (a) f o r J  K  > J  V  DA  t o an A X 2  Y  spectrum i s shown i n  > 0, where f o r k»ft  the spectrum r e d u c e s  AX  form w i t h an e f f e c t i v e c o u p l i n g J .  I n such an a n a l y s i s f o r  DMCF i t has been assumed t h a t t h e two-bond s p i n - s p i n c o u p l i n g s t h e m e t h y l p r o t o n s and the amide A l t h o u g h a li  -  1  The  { N}  14  N  n u c l e a r s p i n are  double resonance s t u d y * *^ o f  14  between  negligible.  N-substituted  amides i n d i c a t e d t h a t t h e s e c o u p l i n g s were indeed n e g l i g i b l y s m a l l , t h e 151 152 d a t a a v a i l a b l e from f u r t h e r s t u d i e s u s i n g N - amides ' leads to c a l c u l a t e d N c o u p l i n g c o n s t a n t s J . and J. , o f 0.7 and 0.8 Hz, r e s • NA NB p e c t i v e l y . N e v e r t h e l e s s , i t may be assumed t h a t the m e t h y l p r o t o n s 15  1 4  X 1  IT  r  are c o m p l e t e l y  d e c o u p l e d by q u a d r u p o l a r  ll,  N  nuclear  spin-lattice  153 relaxation  . I t s h o u l d be n o t e d , however, t h a t the X - p a r t o f  complete A B X spectrum may 3  3  show e f f e c t s due  t h r e e - b o n d c o u p l i n g b e i n g enhanced by the N-C character.  The  '"'N  c a l c u l a t e d as 10.4 may the  - H coupling J 152 X  Hz  H  lk  due  this  p a r t i a l d o u b l e bond  f o r N,N-dimethyl formamide i s  14  N  -  1 9  F  coupling  For a s c a l a r c o u p l i n g o f t h i s magnitude,  q u a d r u p o l a r r e l a x a t i o n mechanism may  spin decoupling  t o the c o u p l i n g  , and hence the c o r r e s p o n d i n g  be o f the o r d e r o f 15 Hz.  the  give only a p a r t i a l spin-  t o the t e m p e r a t u r e dependence o f the  molecular  r e o r i e n t a t i o n g i v i n g r i s e t o the time-dependent e l e c t r i c f i e l d  gradient  115.  at t h e "*N n u c l e u s . 1  The  f i r s t - o r d e r r a t e c o n s t a n t s , k, d e s c r i b i n g t h e h i n d e r e d  r o t a t i o n i n DMCF have been d e t e r m i n e d u s i n g t o t a l l i n e s h a p e  analyses  i n terms o f t h e s t o c h a s t i c model f o r such a m u l t i - s i t e exchange developed  i n s e c t i o n 2.5.  The a b s o r p t i o n mode l i n e s h a p e f u n c t i o n , V ( x ) ,  i s t h e r e a l p a r t o f t h e complex f u n c t i o n G(x) The  g i v e n i n Eq. (2.5.14).  d i a g o n a l m a t r i x A and a s s o c i a t e d t r a n s f o r m a t i o n m a t r i c e s  appearing  process  i n t h i s e x p r e s s i o n a r e d e r i v e d from a 4 x 4 m a t r i x  d e f i n i n g t h e s p e c i f i c r a t e process  and S_  1  [K_ - i f l j  f o r t h e equal p o p u l a t i o n DMCF  exchange system and t h e Larmor f r e q u e n c i e s f o r t h e f o u r s p i n - s i t e s t o be c o n s i d e r e d  ( c f . F i g . 2.7).  have been g i v e n e x p l i c i t l y i n Eq. (2.5.15). program GPLONK was developed  >  The m a t r i c e s r e q u i r e d f o r  > 0  A FORTRAN-IV computer  t o allow a r a p i d numerical  calculation of  g e n e r a l m u l t i - s i t e exchange l i n e s h a p e s based upon t h e component e x p r e s s i o n s g i v e n as Eqs. (2.4.8) and ( 2 . 4 . 9 ) .  Representative  intra-  m o l e c u l a r exchange l i n e s h a p e s f o r t h e A B - p a r t o f a f i r s t - o r d e r ABX s p i n system, as c a l c u l a t e d u s i n g GPLONK, a r e shown i n F i g . 2 . 8 .  The CPU time  i n v o l v e d on an IBM 360/67 system f o r each 600 p o i n t s p e c t r u m , i n c l u d i n g t h e s e t t i n g - u p o f c o n t r o l a r r a y s f o r a CALCOMP p l o t t e r , i s l e s s  than  4 sees which e x e m p l i f i e s t h e advantages o f a p p l y i n g a m a t r i x f o r m u l a t i o n , and t h e s p e c i f i c component form p r e v i o u s l y d e s c r i b e d i n d e t a i l , t o lineshape calculations.  The program GPLONK, i n c o n j u n c t i o n w i t h a  s u b r o u t i n e GFITT, a l s o a l l o w s an e f f i c i e n t a u t o m a t i c  iterative  of a t h e o r e t i c a l lineshape to d i g i t i z e d experimental  data.  r o u t i n e GFITT i s based upon t h e s i m p l e s e a r c h t h r o u g h parameter increments  fitting  The sub-  optimised  p r e v i o u s l y d e s c r i b e d , c f . s e c t i o n 4.1.1, where t h e  1.16.  m a t r i x K_ i s changed  i t e r a t i v e l y i n accordance w i t h t h e g e n e r a l p r o c e d u r e  o u t l i n e d i n s e c t i o n 2.4 f o l l o w i n g Eq. ( 2 . 4 . 9 ) . M u l t i p l e s p e c t r a were o b t a i n e d f o r t h e DMCF C C l ^ s o l u t i o n a t each o f n i n e t e m p e r a t u r e s o v e r t h e range -15.2 t o 78.2°C u s i n g a FABRITEK FT-1064 computer w i t h t h e s p e c t r o m e t e r - c o m p u t e r d e s c r i b e d i n s e c t i o n 3.1.  A typical  interface  sampled and d i g i t i z e d  spectrum f o r  DMCF i s shown i n F i g . 3.5. Because o f t h e v e r y s m a l l m e t h y l c h e m i c a l s h i f t 20 f o r DMCF, and hence NMR l i n e s h a p e f r e q u e n c y i n t e r v a l , difficult  it is  t o o b t a i n a c c u r a t e l y r e p r o d u c i b l e s t e a d y - s t a t e s p e c t r a even  w i t h t h e o v e r a l l s t a b i l i t y a v a i l a b l e from a f i e l d - f r e q u e n c y l o c k e d spectrometer.  Some t y p i c a l  l i n e s h a p e f i t s a r e shown i n F i g . 4.9 and t h e  average r a t e c o n s t a n t s o b t a i n e d from m u l t i p l e f i t s a t each t e m p e r a t u r e are  summarised  i n T a b l e 4.6.  chemical s h i f t s w  The t e m p e r a t u r e dependences o f t h e  and O J ^ , c f . Eq. ( 2 . 2 . 2 4 ) , d e r i v e d from t h e s e l i n e -  shape f i t s a r e shown i n F i g . 4.8 a l o n g w i t h t h e mean f r e q u e n c y d e f i n i n g t h e r e f e r e n c e zero f o r t h e independent v a r i a b l e x.  A l l  f r e q u e n c i e s a r e measured w i t h r e s p e c t t o t h a t f o r TMS a t c o n s t a n t  field.  The v a r i a t i o n i n methyl c h e m i c a l s h i f t s i s presumably due t o a s t e r e o specific the  i n t e r m o l e c u l a r i n t e r a c t i o n which may l e a d t o a s m a l l change i n  form o f the a n i s o t r o p i c s h i e l d i n g a s s o c i a t e d w i t h t h e C=0 and C-F  groups i n DMCF.  Thus t h e c h e m i c a l s h i f t 20 v a r i e s from 2.2 t o 1.4 Hz  o v e r t h e t e m p e r a t u r e range -15.2 t o 78.2°C.  The c o u p l i n g c o n s t a n t s J .  and J g ^ were assumed t o be t e m p e r a t u r e i n d e p e n d e n t , and t h e c h e m i c a l s h i f t s c o r r e s p o n d i n g t o minimum e r r o r m u l t i p l e l i n e s h a p e f i t s were found t o be c o n s i s t e n t t o w i t h i n + 0.05 Hz. fitting  Thus t h i s  lineshape  p r o c e d u r e may be used t o d e t e r m i n e s m a l l c h e m i c a l s h i f t s  and/or  Y  TABLE 4.6 K i n e t i c d a t a f o r N,N-dimethyl carbamyl f l u o r i d e (16 mole % i n C C K )  temp., °C  k, s e c  error  12.2  0.076  4.8  15.6  0.159  4.2  30.2  '0.550  3.7  1  32.5  0.692  4.1  37.6  1.99  2.2  44.9  2.29  1.2  46.5  2.57  1.3  53.8  5.11  1.3  62.7  9.55  3.9  78.2-  26.3 E  a  log  =18.3+0.6 kcals. 1 ( )  4.6 mole  - 1  A = 12.9  AH* = 17.7 + 0.6 k c a l s .  mole"  1  AS* = -1.4 + 2.1 c a l . d e g " . m o l e " 1  AG* = 18.2 ± 0.6 k c a l s .  mole"  1  1  Hz  /  /  / / /  /a  -292  /  . / /  - 293/  /.  '  '  /  0  /  t  9 o/  •  /  a  /  294H  295  ,0 /  / /  / /  /o  -296  /  /  / 2C1  -40  2.2  1.75.  1.4 T  0  40  Hz  80  T°C F i g . 4.8  Temperature dependence o f c h e m i c a l s h i f t s f o r N,N-dimethyl carbamyl f l u o r i d e  62.7 9.55  F i g . 4.9  T o t a l l i n e s h a p e f i t s f o r N,N-dimethyl carbamyl fluoride  117.  coupling  constants t o t h i s p r e c i s i o n  '  under normal NMR s p e c t r a l r e s o l u t i o n .  , which c o u l d n o t be o b t a i n e d  I t i s t o be noted t h a t t h e e r r o r s  ( c f . s e c t i o n 4.1.1) f o r t h e DMCF f i t t e d l i n e s h a p e s a r e maximal i n t h e l i m i t s o f v e r y s l o w and v e r y f a s t exchange. d i f f i c u l t y i n recording range o f o n l y 5Kz, reproducible  T h i s may be due t o t h e  the s t e a d y - s t a t e lineshapes over a frequency  b u t i n a c t u a l f a c t t h e s e l i n e s h a p e s were  within the error l i m i t s involved.  exchange l i m i t , t h e r e s i d u a l l i n e w i d t h s  accurately  A l s o , i n t h e v e r y slow  were g r e a t e r t h a n t h e r e f e r e n c e  w i d t h s and hence a d d i t i o n a l t r a n s v e r s e r e l a x a t i o n e f f e c t s may have been o b s e r v e d i n t h i s p a r t i c u l a r amide.  Such e f f e c t s may a r i s e from  t r o p i c n u c l e a r d i p o l e - d i p o l e , q u a d r u p o l a r and c h e m i c a l s h i f t  interactions  t h r o u g h i n c o m p l e t e a v e r a g i n g by m o l e c u l a r r e o r i e n t a t i o n * ^ ' . a d d i t i o n , enhanced c r o s s - r e l a x a t i o n s  associated  with these  t e n s o r i n t e r a c t i o n s may g i v e r i s e t o d i f f e r e n t i a l l i n e w i d t h s absence o f h i n d e r e d r o t a t i o n i n DMCF. not been i n c l u d e d  aniso-  In  anisotropic, i n the  Such r e l a x a t i o n p r o c e s s e s have  i n the a n a l y s i s reported here.  The A r r h e n i u s p l o t  f o r DMCF i s most s a t i s f a c t o r y , however, and i s shown i n F i g . 4.10. The  a c t i v a t i o n energy i s d e t e r m i n e d as 18.3 ± 0.6 k c a l s . mole . l  f r e e energy o f a c t i v a t i o n i s t h e n c a l c u l a t e d a t 25°C as AG +0.6 deg  k c a l s . mole , w i t h 1  1  an e n t r o p y o f a c t i v a t i o n AS  The  =18.2  = -1.4 + 2.1 c a l .  m o l e w h i c h i s o f a magnitude t y p i c a l o f t h e h i n d e r e d r o t a t i o n s i n 1  a l l o f the substituted  N,N-dimethyl amides s t u d i e d  here.  -1.6 1  2.3  3.0  32  1/T°Kx10 F i g . 4.10  2 A ~ ~  3J6~  3  A r r h e n i u s p l o t f o r N,N-dimethyl carbamyl f l u o r i d e  118.  4.1.5.  Formamide Although  the hindered  some s e m i - q u a n t i t a t i v e k i n e t i c d a t a were" a v a i l a b l e  for  r o t a t i o n i n formamide*^^ *^^ , a t o t a l l i n e s h a p e a n a l y s i s ,  )  had not been r e p o r t e d and hence t h i s p a r e n t  amide was  studied to obtain  r e l i a b l e a c t i v a t i o n p a r a m e t e r s f o r c o m p a r i s o n w i t h t h o s e f o r the s u b s t i t u t e d N,N-dimethyl amide s e r i e s a l r e a d y d i s c u s s e d .  I t i s also  o f i n t e r e s t t o compare the e x p e r i m e n t a l l y d e t e r m i n e d r o t a t i o n b a r r i e r s i n formamide and N,N-dimethyl formamide (DMF) c o r r e l a t i o n w i t h the corresponding  to allow a f u r t h e r  s e r i e s o f p a r e n t amides, w h i c h a r e  u s u a l l y e x p e r i m e n t a l l y i n a c c e s s i b l e b u t a r e c e r t a i n l y more f o r m o l e c u l a r o r b i t a l c a l c u l a t i o n s b e i n g the s i m p l e s t c o n t a i n i n g t h e N-OO  bond system.  convenient  molecular  Of c o u r s e , the b a r r i e r i n formamide  i s o f fundamental i m p o r t a n c e as a r e f e r e n c e p o i n t f o r a l l d e s c r i p t i o n s o f s u b s t i t u e n t e f f e c t s on the h i n d e r e d r o t a t i o n about t h e N-C  bonds i n  amides and o f t h e b o n d i n g c h a r a c t e r i s t i c s i n t h e s e compounds.  I n the  c o u r s e o f the work d e s c r i b e d h e r e , an i n d e p e n d e n t NMR was  r e p o r t e d * ^ and the a d d i t i o n a l d a t a now  lineshape  a v a i l a b l e may  study  be used t o  a s c e r t a i n the a p p l i c a b i l i t y o f t h e g e n e r a l l i n e s h a p e f i t t i n g p r o c e d u r e t o a more c o m p l i c a t e d t i g h t l y - c o u p l e d n u c l e a r s p i n system.  As  already  d e s c r i b e d i n d e t a i l i n s e c t i o n 2.6,  the a n a l y s i s o f c h e m i c a l  i n a t i g h t l y - c o u p l e d (second-order)  s p i n system r e q u i r e s a complete  spin density matrix  treatment.  In c o n t r a s t t o DMF, NMR  s t u d y o f the h i n d e r e d  polar  ll|  N  nucleus  exchange  t h e r e are-two c o m p l i c a t i n g f a c t o r s i n an  r o t a t i o n i n formamide.  F i r s t l y , t h e quadru-  leads t o very broad resonances f o r the  directly  119.  bonded amino p r o t o n s e l i m i n a t i n g a l l c h e m i c a l these protons*^*, enriched  15  s h i f t information  and hence i t i s n e c e s s a r y t o use  r  N - f o r m a m i d e sample  158  an  for  isotopically  or double r f - i r r a d i a t i o n  159  The  former s i m p l i f i c a t i o n i s p r e f e r a b l e f o r a q u a n t i t a t i v e lineshape as the d i s t o r t i o n i n h e r e n t evaluate.  i n s p i n - s p i n decoupling  is difficult  study to  S e c o n d l y , the i n t e r m o l e c u l a r exchange o f t h e b a s i c amino  p r o t o n s o f formamide gives a d d i t i o n a l s p i n t r a n s f e r e f f e c t s i n the steady-state  s p e c t r u m and the i n c l u s i o n o f such e f f e c t s i n a  a n a l y s i s l e a d s t o a l a r g e i n c r e a s e i n the computer c o r e and required f o r numerical  calculations.  i n t e r m o l e c u l a r exchange p r o c e s s e s s i m u l t a n e o u s l y  rotation)  and  from l i n e s h a p e  but as t h e l a t t e r p r o c e s s i s not o f c u r r e n t i n t e r e s t i t may  u  time  In p r i n c i p l e i t i s p o s s i b l e to  d e t e r m i n e t h e r a t e s o f the i n t r a m o l e c u l a r ( h i n d e r e d  ressed  lineshape  be  fits, supp-  o v e r t h e t e m p e r a t u r e range f o r an i n t r a m o l e c u l a r exchange «  •  '  study  158  by u s i n g a c e t o n e as a s o l v e n t 1 5  N - e n r i c h e d (98 atom %) formamide was  Sharp and  Dohme  solution.  The  t i o n and was  spectrum.  c o m m e r c i a l formamide was  was  A separate  shape s t a n d a r d standard  s t u d i e d as a 10 mole % a c e t o n e used w i t h o u t  d r i e d w i t h the s o l v e n t o v e r m o l e c u l a r  d i s i l o x a n e (^2%) locking.  and was  obtained  from Merck (spectro-grade)  further purifica-  sieves.  Hexamethyl-  added t o the NMR  sample f o r f i e l d - f r e q u e n c y  s i n g l e l i n e was  not r e q u i r e d f o r a r e f e r e n c e  as the a c e t o n e peak i s a v a i l a b l e and  i s a l s o present  i n the C - p a r t o f t h e ABCX  line-  a convenient 15  N-formamide  l\  1  That i s , i n the absence o f i n t e r m o l e c u l a r exchange t h e r e  f o u r l i n e s that, have w i d t h s t h a t are i n v a r i a n t t o exchange e f f e c t s ,  are  120.  cf.  s e c t i o n 2.5, and hence t h e s e l i n e s may be used t o e s t i m a t e t h e  l i n e w i d t h a s s o c i a t e d w i t h m a g n e t i c f i e l d i n h o m o g e n e i t y and slow i n t e r molecular  exchange which may be p r e s e n t  NMR sample was t h o r o u g h l y  at higher temperatures.  The  degassed by t h e u s u a l freeze-pump-thaw c y c l e  and was s e a l e d i n vacuo i n t h i n w a l l e d t u b e s o f 5 mm o.d.. The 100  Ml NMR s p e c t r a were o b t a i n e d a t 100 MHz w i t h a V a r i a n HA-  spectrometer  equipped w i t h a V-6031 v a r i a b l e - t e m p e r a t u r e  temperature c o n t r o l l e r . erature f o r the N  2  The V-6031 c o n t r o l l e r m a i n t a i n s  probe and  a p r e s e t temp-  gas h e a t i n g o r c o o l i n g t h e sample, and t h e gas  t e m p e r a t u r e i s sensed a t about 5 cm from t h e sample volume i n t h e spectrometer  receiver c o i l .  Thus t h e sample t e m p e r a t u r e was measured  b e f o r e and a f t e r r e c o r d i n g a spectrum u s i n g a s t a n d a r d methanol f o r g l y c o l ) sample and i n t e r p o l a t i o n o f t h e -OH c h e m i c a l  s h i f t s obtained i n  126 HA-mode u s i n g t h e d a t a o f Van Geet i s estimated  . The t e m p e r a t u r e thus d e t e r m i n e d  t o be a c c u r a t e t o ± 0.3°C and was shown t o be s t a b l e ,  w i t h i n t h e s e l i m i t s , o v e r a p e r i o d o f about 20 mins. were r e c o r d e d  a t sweep r a t e s o f 0.02 o r 0.05 Hz s e c  s t e a d y - s t a t e c o n d i t i o n s , and w i t h a low a m p l i t u d e  1  The s p e c t r a t o best  r ffield  to m i n i m i s e l i n e s h a p e d i s t o r t i o n due t o s a t u r a t i o n e f f e c t s .  approximate  (y 0.02 mgauss) A t each  t e m p e r a t u r e , t o ensure r e p r o d u c i b i l i t y , a t l e a s t f o u r s p e c t r a were d i g i t i s e d and s t o r e d u s i n g a F a b r i t e k FT-1064 computer i n t e r f a c e d w i t h the HA-100 as d e s c r i b e d i n s e c t i o n 3.1. A l l f r e q u e n c i e s were measured, u s i n g t h e V-4315 c o u n t e r , w i t h r e f e r e n c e t o HMDS as t h e i n t e r n a l An optimum H -15  0  f i e l d homogeneity was m a i n t a i n e d  t o 83°C c o r r e s p o n d i n g  standard.  o v e r t h e t e m p e r a t u r e range  t o a 0.4 Hz r e s o l u t i o n ( T 2 = 0.8 sec.)  and a  121.  r e f e r e n c e lineshape that best approximated a symmetrical L o r e n t z i a n shape.  Wilmad PS-S05 5 mm t h i n - w a l l e d sample t u b e s were used t o a t t a i n  r e p r o d u c i b l e s p i n n i n g c h a r a c t e r i s t i c s and f i e l d homogeneity  control.  The h i n d e r e d r o t a t i o n i n N-.formamide may be c o n s i d e r e d i n 15  terms o f a mutual s p i n t r a n s f e r o f t h e A- and B- s p i n s i n an ABCX n u c l e a r s p i n system, f o r which t h e b a s i c e q u a l p o p u l a t i o n c h e m i c a l l y s h i f t e d s i t e s a r e p r e d o m i n a n t l y due t o t h e d i a m a g n e t i c s u s c e p t i b i l i t y a n i s o t r o p y o f t h e c a r b o n y l group.  Consistent with p r i o r studies within  t h i s s e c t i o n , t h e amino B- p r o t o n w h i c h i s c i s t o t h e c a r b o n y l oxygen i s a s s i g n e d t o h i g h - f i e l d o f t h e A-proton, and t h u s t h e C- s p i n i s t h e f o r m y l p r o t o n w h i c h i s t o l o w - f i e l d r e l a t i v e t o b o t h t h e A- and Bprotons.  Owing t o t h e s p e c i f i c i n t e r a c t i o n o f t h e a c e t o n e s o l v e n t  m o l e c u l e w i t h t h e s o l u t e m o l e c u l e , i t i s t o be e x p e c t e d t h a t t h e amino p r o t o n s w i l l have c h e m i c a l s h i f t s showing s i g n i f i c a n t t e m p e r a t u r e dependences,  while spin-spin couplings are usually less s e n s i t i v e to  intermolecular interactions. to  Therefore, i n i t i a l l y , i t i s necessary  c a r e f u l l y a n a l y s e t h e ABCX spectrum i n t h e absence o f exchange  e f f e c t s t o d e t e r m i n e t h e form o f t h e t e m p e r a t u r e dependences, for  the s p e c t r a l parameters.  i f any,  The s p e c t r a were a n a l y s e d u s i n g a  computer program NMRFIT, which i s a much m o d i f i e d v e r s i o n o f 162 LAOCOON to  giving a very e f f i c i e n t i t e r a t i v e f i t o f spectral  parameters  t r a n s i t i o n f r e q u e n c e s , by a s s i g n i n g a l l 24 l i n e s o v e r t h e tempera-  t u r e range -12.4 t o 30.2°C.  The r e s u l t a n t c h e m i c a l s h i f t s and  c o u p l i n g s a r e g i v e n i n T a b l e 4.7, a l o n g w i t h t h o s e o b t a i n e d from t h e 158 e a r l i e r s t u d y o f an acetone s o l u t i o n o f t h e same c o n c e n t r a t i o n  T a b l e 4.7 S p e c t r a l parameters f o r N - f o r m a m i d e 15  \  J  AB  J  AC  J  BC  J  AX  J  BX  J  cx  ref.  12.4  -4.9  4.9  -84.6  2.8  1.8  13,.5  90,.1  87,.5  15,.1  -5.0  -5.7  5.7  -90.3  2.8  1.7  13..5  90.,2  88..3  15..2  7.0  7.0  -97.4  2.8  •1.7  13..5  90..0  87,.6  15,.3  1 ?  -10.3  10.3  -118.8  2.8  1.6  13..5  90..0  87..6  15..5  II  -10.7  10.7  -118.8  2.9  1.7  13.,5  91..0  88.,0  16..4  (158)  -12.1  12.1  -97.0  2.8  1.7  13..4  89.,7  86.,4  16.,4  (152)  -18.5  18.5  -69.8  2.6  1.6  13.,5  89.. 3  87, i  15..6  (160)  -9.9  9.9  -102.5  2.7  1.7  13..5  --  (159)  3.0 30.2  :  10 mole % acetone s o l u t i o n , -12.4 t o 30.2°C 35 mole % acetone s o l u t i o n , 30°C 14.1 % d i g l y m e s o l u t i o n , 25°C 22.2% acetone  solution  A l l c h e m i c a l s h i f t s r e f e r t o 100 MHz NMR.  --  t h i s wo t!  122.  , . . . . ,. . . . • 152,159,160 and those from more r e c e n t s t u d i e s u s i n g s i m i l a r s o l v e n t s +  Only r e l a t i v e l y s m a l l v a r i a t i o n s i n t h e v a l u e s o f the c o u p l i n g constants  a r e e v i d e n t f o r d i f f e r e n t c o n c e n t r a t i o n s o f formamide i n  acetone and d i g l y m e as s o l v e n t , and a r e even s m a l l e r o v e r t h e temperature interval  o f 42°C f o r t h e 10 mole% acetone s o l u t i o n . . Thus a l l  coupling constants  a r e assumed t o be t e m p e r a t u r e independent i n t h e  following lineshape a n a l y s i s .  As i s n o r m a l , t h e t r a n s c o u p l i n g J g ^  i s much g r e a t e r t h a n t h e c i s c o u p l i n g J^Q> b o t h b e i n g enhanced due t o t h e p a r t i a l double bond c h a r a c t e r o f t h e i n t e r p o s e d N-C bond. chemical  s h i f t s ft., ft and ft , however, show r e l a t i v e l y n  A  D  large variations  L  w i t h s o l u t e c o n c e n t r a t i o n and t e m p e r a t u r e . frequency  The  I n T a b l e 4.7, t h e r e f e r e n c e variable x for a  d e f i n i n g t h e independent f r e q u e n c y  l i n e s h a p e a n a l y s i s i s t a k e n as t h e average o f t h e s h i f t s co. and U L , A b cf-. Eq. ( 2 . 2 . 2 4 ) , and hence ft„ < ft. < 0 < ft . The s h i f t s co. , co and D  co  L  are measured r e l a t i v e  approximately  D  D  A  and UL, i s shown t o be  to hexamethyl-disiloxane  c o n s t a n t w h i l e co and co A.  Lt  (ft = co -to ) have A.  D  A  A.  relatively  U  s t r o n g t e m p e r a t u r e dependences as i l l u s t r a t e d i n F i g . 4.11. chemical  s h i f t s d e r i v e d u s i n g NMRFIT were r e p r o d u c i b l e t o w i t h i n + 0.2 Hz  and as t h e v a r i a t i o n o v e r t h e t e m p e r a t u r e i n t e r v a l linear,  As t h e  -12.4 t o 30.2°C was  as d e p i c t e d by t h e open p o i n t s i n F i g . 4.11, i t was  that a l i n e a r  assumed  e x t r a p o l a t i o n c o u l d be used t o d e t e r m i n e t h e c h e m i c a l  s h i f t s used i n a l i n e s h a p e f i t a t any t e m p e r a t u r e up t o 85°C. actual lineshape f i t s  showed t h a t the v a r i a t i o n s o f a l l c h e m i c a l  o v e r t h i s t e m p e r a t u r e range a r e i n d e e d s h i f t s corresponding  The  linear,  shifts  i n that the chemical  t o t h e minimum e r r o r f i t s a r e shown as t h e c l o s e d  -820  -20  0  20  '  60  100  TEMP °C F i g . 4.11  Temperature dependence o f t h e c h e m i c a l s h i f t s f o r N-formamide, 10% acetone s o l u t i o n 5  123.  points i n Fig. The  4.11.  a b s o r p t i o n mode s t e a d y - s t a t e NMR  l i n e s h a p e s f o r the  ABC-  p a r t o f the ABCX N - f o r m a m i d e system as m o d i f i e d by mutual t r a n s f e r 15  o f t h e A- and B - s p i n s have been c a l c u l a t e d u s i n g a FORTRAN-IV computer program GENLIN.  The  t h e o r y f o r t h e s e c a l c u l a t i o n s has been d i s c u s s e d  i n d e t a i l i n s e c t i o n 2.6 m a t r i x e q u a t i o n Eq.  and the computer program i s based upon t h e  ( 2 . 6 . 1 7 ) , w h i c h i s d e r i v e d from t h e g e n e r a l com-  ponent e q u a t i o n o f m o t i o n f o r the s p i n d e n s i t y m a t r i x , c f . Eq. The  (2.6.7).  s p i n t r a n s i t i o n o p e r a t o r m a t r i x , I , i n the b a s i s o f s i m p l e  product  +  f u n c t i o n s f o r the f o u r s p i n system i s v e r y s i m i l a r t o t h a t g i v e n e x p l i c i t l y f o r t h e ABX  ( o r ABC)  m a t r i x g i v e n i n T a b l e 2.3  s p i n system i n T a b l e 2.3.  In f a c t  the  forms a b l o c k d i a g o n a l p a r t o f the complete  ABCX t r a n s i t i o n m a t r i x which d e f i n e s 30 t r a n s i t i o n s , i n c l u d i n g a l l o w e d and " c o m b i n a t i o n "  t r a n s i t i o n s , i n the ABC-part o f t h e ABCX  spectrum and hence the fundamental m a t r i x R , 1  30 x 30 m a t r i x w i t h complex elements. the l a r g e number o f n u m e r i c a l  c f . Eq.  To s i m p l i f y t h e c a l c u l a t i o n f o r  s p e c t r a l data p o i n t s (M000) r e q u i r e d i n  an i t e r a t i v e l i n e s h a p e f i t t i n g p r o c e d u r e f o r such a s p e c t r u m , the m a t r i x R' i s t r a n s f o r m e d frequency  (2.6.12), i s a  complicated  t o d i a g o n a l form and  the  dependent p a r t i s r e t a i n e d as the s c a l a r m a t r i x x l _ , I_ b e i n g  t h e 30 x 30 u n i t m a t r i x .  I n comparison w i t h Eq.  complex l i n e s h a p e f u n c t i o n G(x)  (2.4.6) g i v i n g the  f o r a f i r s t - o r d e r s p i n system, the  d i a g o n a l m a t r i x Q i s d e r i v e d from the m a t r i x R g i v e n i n Eq. where Q i s now  an o f f - d i a g o n a l m a t r i x .  The  (2.4.5)  elements o f the m a t r i x ft 92  a r e f o r m a l l y d e f i n e d by t h e L i o u v i l l e H a m i l t o n i a n  operator  93 , but  124;  a r e v e r y s i m p l y d e t e r m i n e d u s i n g a s i m p l e a l g o r i t h m based upon Eq.  (2.6.27).  On d i a g o n a l i s a t i o n , i n the absence o f exchange e f f e c t s ,  t h i s m a t r i x d i r e c t l y d e t e r m i n e s the s e c o n d - o r d e r t r a n s i t i o n and  a s s o c i a t e d complex i n t e n s i t i e s .  frequencies  I t i s t o be n o t e d t h a t t h e d i a g o - .  n a l elements o f the m a t r i x ft are f i r s t - o r d e r a p p r o x i m a t i o n s  to  the  t r a n s i t i o n f r e q u e n c i e s , and i n d e e d the o n l y change i n g o i n g t o a s e c o n d - o r d e r a n a l y s i s o f exchange e f f e c t s u s i n g t h i s m a t r i x i s i n the d e f i n i t i o n , o f the s p i n - s i t e f r e q u e n c y (2.4.5).  formulation  m a t r i x , ft, i n Eq,  F o l l o w i n g the s i m i l a r i t y t r a n s f o r m a t i o n as g i v e n i n Eq.  (2.4.7) , the n u m e r i c a l  l i n e s h a p e c o m p u t a t i o n i s reduced t o  simple  m u l t i p l i c a t i o n s i n r e a l a r i t h m e t i c as p r e v i o u s l y d e s c r i b e d , c f . Eq. (2.4.8) .  The  computer program GENLIN i s c o m p l i c a t e d  d e x i n g and s o r t i n g t o maximise the c o m p u t a t i o n a l d e t a i l s need not be d i s c u s s e d at t h i s p o i n t .  by i n v o l v e d i n -  e f f i c i e n c y and  The  such  advantage o f u s i n g  the m a t r i x f o r m u l a t i o n d e s c r i b e d as p a r t o f t h i s work, however, i s e x e m p l i f i e d by comparison w i t h the c o m p u t a t i o n a l independent s t u d y o f  15  N-formamide" "^^.  The  1  CPU  methods used i n the t i m e quoted f o r the  c o m p u t a t i o n o f a s i n g l e s p e c t r a l d a t a p o i n t f o r a 4 - s p i n system u s i n g an IBM  360/75 computer i s 0.3  f o r a 1000 numerical  s e c o n d * ^ , and hence the t i m e i n v o l v e d 1  p o i n t spectrum as r e q u i r e d f o r a r e l i a b l e v i s u a l comparison w i t h e x p e r i m e n t a l  data i s 5 minutes.  or The  CPU  time  f o r the c a l c u l a t i o n o f the same spectrum u s i n g t h e program GENLIN on IBM  360/75 computer i s l e s s t h a n 10 s e c o n d s , r e p r e s e n t i n g a r e d u c t i o n  i n t i m e by a f a c t o r o f about t h i r t y .  That i s , i n a p p r o x i m a t e l y  minutes i t i s p o s s i b l e t o o b t a i n a c o m p l e t e l y  automatic  2  numerical  an  125.  i t e r a t i v e l i n e s h a p e f i t o v e r any p r e s c r i b e d f r e q u e n c y  range and t o a  p r e c i s i o n o f the o r d e r o f 1% f o r the f i r s t - o r d e r r a t e d e f i n i n g the i n t r a m o l e c u l a r exchange.  The  constant  s u b r o u t i n e GFITT used i n  c o n j u n c t i o n w i t h GENLIN f o r the i t e r a t i v e f i t t i n g has been p r e v i o u s l y d e s c i b e d , c f . s e c t i o n 4.1.1. The  average r a t e c o n s t a n t s  f i t s t o b o t h t h e AB-  and C- p a r t s o f the  t h e t e m p e r a t u r e range 43.3  a r e shown i n F i g . 4.12. o b t a i n e d a t 60 MHz and  ^-formamide spectrum o v e r  to these p a r t i c u l a r r a t e  These l i n e s h a p e s may 160^  shifts  1  from m u l t i p l e l i n e s h a p e  t o 83.0°C are l i s t e d i n T a b l e 4.8  complete l i n e s h a p e s c o r r e s p o n d i n g  chemical  obtained  ^  g  e  f£  e c t  f  Q  t  and  constants  be compared w i t h ]  i  e  the  those  t e m p e r a t u r e dependent  i s c l e a r l y shown i n F i g . 4.12.  A l s o , from  t h e form o f the minimum e r r o r l i n e s h a p e f i t s , i t i s r e a d i l y  deter-  mined t h a t the r e l a t i v e s i g n s o f the c o u p l i n g c o n s t a n t s J ^ , and are the same.  The  l i n e s h a p e s i n F i g . 4.12  c a l c u l a t e d f o r the c l o s e l y r e l a t e d ABX  may  be compared w i t h  s p i n system, F i g s . 2.8  showing the e f f e c t s o f t h e r e l a t i v e s i g n s o f the c o u p l i n g and Jg^-  The  those  and  2.10  constants  e f f e c t o f the c o u p l i n g s w i t h the same s i g n i s  e s p e c i a l l y o b v i o u s i n the C - p a r t o f the spectrum.  This sign deter152  m m a t i o n i s c o n s i s t e n t w i t h a double resonance s t u d y  .  The  upper  t e m p e r a t u r e l i m i t f o r the s t u d y o f i n t r a m o l e c u l a r exchange i s d e t e r mined as t h a t at which t h e a d d i t i o n a l i n t e r m o l e c u l a r exchange e f f e c t s become s i g n i f i c a n t . process  That i s , the l i n e w i d t h a s s o c i a t e d w i t h t h e  i n c r e a s e s t o 0.8  Hz at 83°C and t h i s was  considered  latter  t o be  the  maximum a l l o w a b l e c o n t r i b u t i o n f o r the o v e r a l l l i n e s h a p e f i t s not t o be  TABLE 4.8 K i n e t i c d a t a f o r formamide (10% acetone s o l u t i o n ) Temp°C  k sec *  43.3  3.63  45.5  4.68  47.5  .  6.46  49.0  6.76  52.2  9.55  54.6  10.50  55.0  10.00  ^y.4  15.90  60.0  19.10  67.0  33.90  77.4  69.10  77.6  75.90  78.0  85.10  83.0  120.00  E  a  =  19.27 + 0.37 k c a l s . m o l e "  AH* =  18.68 k c a l s . m o l e "  AS  3.0+1.2  1  =  AG* =  1  1  a t 25°C  c a l s . d e g " mole  17.77 k c a l s . m o l e "  1  1  1  126.  a f f e c t e d by t h i s a p p r o x i m a t i o n w i t h i n the o v e r a l l f i t t i n g e r r o r attainable. may  The  f o u r narrow l i n e s i n the C-part o f the spectrum w h i c h  be used t o m o n i t o r t h i s l i n e w i d t h  i n F i g . 4.12.  The  the  At  J  = 17.8  t i o n AS may  ± 0.4  k c a l s . mole' + 1.2  1  6  cal.deg.  °:  AG*  1 5 9  mole  1  = 17.8  1  with  c a l s . deg.  be compared w i t h  solution (22.2%) 2.4  k c a l s . mole  1  which i s  the  :  AG  ± 0.2  •'mole . 1  These a c t i v a t i o n p a r a m e t e r s  t h o s e o b t a i n e d f o r a ^N-formamide acetone 1  #  , and  1  = 17.0  + 1.9  for a  15  kcals.mole"  1  and  AS  = -5.4  #  N-formamide diglyme s o l u t i o n  kcals.mole"  1  and  AS*  = 4.0  + 0.7  f o r the  l i n e s h a p e and  double r e s o n a n c e s t u d i e s , r e s p e c t i v e l y ,  process.  o f a c t i v a t i o n are p o s i t i v e and  negative they  as i s t o be e x p e c t e d f o r an i n t e r n a l r o t a t i o n  A p o s i t i v e e n t r o p y c o r r e s p o n d s t o a more o r d e r e d ground  s t a t e , f o r the s t a t e may  (14.1%) 1  entropies  are r e l a t i v e l y small  +  cals.deg" .  Although the  1  as  a corresponding entropy of a c t i v a -  mole .  with  activa-  consistent  25°C, the f r e e energy o f a c t i v a t i o n i s d e t e r m i n e d  =3.0  now  + 0.4  and  e a r l i e s t e s t i m a t e f o r t h i s p a r a m e t e r * * ^ , namely 18 + 3 k c a l s .  mole . AG*  c o n t r i b u t i o n are c l e a r l y shown  A r r h e n i u s p l o t i s shown i n F i g . 4.13  t i o n energy o b t a i n e d i s 19.3 with  limits  be  formamide-acetone complex, and  e x p e c t e d to have the  an i n c r e a s e d  solute-solvent  as the p l a n a r ground  l a r g e s t d i p o l e moment t h i s i s  consistent  e l e c t r o s t a t i c i n t e r a c t i o n leading  to  a more s t a b l e complex i n s o l u t i o n w i t h reduced degrees o f freedom. However, t h e r e are two  possible pyramidal t r a n s i t i o n states f o r  the  formamide h i n d e r e d r o t a t i o n , and w h i l e t h e s e s t a t e s have v e r y d i f f e r e n t d i p o l a r c h a r a c t e r t h e y have n e a r l y f o r an i s o l a t e d m o l e c u l e . i t may  i d e n t i c a l e n e r g i e s , as  calculated  Thus i f s o l v a t i o n p l a y s an i m p o r t a n t r o l e ,  w e l l be t h a t the more d i p o l a r t r a n s i t i o n s t a t e o f the  two  F i g . 4.13  A r r h e n i u s p l o t f o r N - f o r m a m i d e , 10% acetone s o l u t i o n 15  127.  p o s s i b l e i s s t a b i l i s e d by s o l v a t i o n l e a d i n g t o a d e c r e a s e d a negative entropy o f a c t i v a t i o n .  e n t r o p y and  The d i f f e r e n t t r a n s i t i o n s t a t e s  w i l l be f u r t h e r d i s c u s s e d i n a. f o l l o w i n g s e c t i o n ,  In view o f the  l a r g e v a r i a t i o n s i n a c t i v a t i o n parameters n o r m a l l y a s s o c i a t e d w i t h i n d e p e n d e n t NMR d e t e r m i n a t i o n s , c f . T a b l e 4.3, i t i s g r a t i f y i n g t o see t h e agreement on b o t h t h e a c t i v a t i o n energy and f r e e energy o f a c t i v a t i o n f o r formamide as d e t e r m i n e d  using total  lineshape analyses.  Of c o u r s e t h i s i s t h e f i r s t c o m p a r i s o n o f r e s u l t s from complete d e n s i t y m a t r i x l i n e s h a p e s t u d i e s and t h e agreement may be somewhat f o r t u i t o u s i n t h a t d i f f e r e n t s o l v e n t s and s l i g h t l y d i f f e r e n t s o l u t e c o n c e n t r a t i o n s were u s e d , a l t h o u g h t h e s o l v e n t s do have s i m i l a r cteristics.  N e v e r t h e l e s s , i n g e n e r a l , i t may be e x p e c t e d  chara-  t h a t more  r e l i a b l e r a t e c o n s t a n t d e t e r m i n a t i o n s a r e p o s s i b l e f o r more comp l i c a t e d t i g h t l y - c o u p l e d s p i n systems.  I t i s well recognised that the  most a c c u r a t e k i n e t i c d a t a are o b t a i n e d f o r a s i m p l e t w o - s i t e exchange system i n the r e g i o n o f coalescence.  As t h e r e a r e a number o f analogous  r e g i o n s f o r a more c o m p l i c a t e d s p i n system s p a n n i n g r a t e c o n s t a n t s , i t i s t o be e x p e c t e d  a l a r g e r range o f  t h a t more r e l i a b l e p r i m a r y  may be o b t a i n e d by l i n e s h a p e a n a l y s i s o v e r an i n c r e a s e d range.  data  temperature  Thus w i t h t h e advent o f e f f i c i e n t computer programs, such as  GENLIN d e s c r i b e d i n t h i s work, t o h a n d l e more c o m p l i c a t e d s p i n systems i n an i t e r a t i v e NMR l i n e s h a p e f i t t i n g p r o c e d u r e ,  a wider v a r i e t y o f  c h e m i c a l systems o f c u r r e n t i n t e r e s t showing k i n e t i c e f f e c t s may be p a r t i c u l a r l y amenable t o s t u d y u s i n g t h e NMR  technique.  128.  4.2  Hindered The  N-C  r o t a t i o n i n amides, Huckel  systematic experimental  MO c a l c u l a t i o n s  study o f h i n d e r e d r o t a t i o n about the  bond i n a s e r i e s o f N,N-dimethyl amides o f the form N-C-X  with  6 d i f f e r e n t s u b s t i t u e n t s X, as d e s c r i b e d i n the p r e c e d i n g s e c t i o n s , has .shown t h a t the f r e e energy o f a c t i v a t i o n f o r the r o t a t i o n p r o c e s s i s n e a r l y independent o f i n t e r - m o l e c u l a r i n t e r a c t i o n s . energy may  Thus t h i s f r e e  be c o n s i d e r e d as the a c t i v a t i o n parameter c h a r a c t e r i s i n g  the i n t r a m o l e c u l a r r a t e p r o c e s s . systematic experimental  In a d d i t i o n , i t i s apparent t h a t  e r r o r s i n h e r e n t i n the d e t e r m i n a t i o n o f s p e c i f i c  r a t e c o n s t a n t s u s i n g s t e a d y - s t a t e NMR  l e a d t o minimal  e s t i m a t i o n o f t h i s p a r t i c u l a r a c t i v a t i o n parameter.  e r r o r i n the In terms o f the  u s u a l a b s o l u t e r a t e t h e o r y , the e f f e c t o f m o l e c u l a r s t r u c t u r a l s u b s t i t u e n t changes on a r a t e p r o c e s s  or  i s r e f l e c t e d by the f r e e energy  129 of a c t i v a t i o n  and hence t h i s thermodynamic parameter may  be  c o r r e l a t e d w i t h the changes i n the t o t a l m o l e c u l a r e n e r g y , as c a l c u lated using molecular o r b i t a l  (MO)  s t r u c t u r a l o r s u b s t i t u e n t changes.  theory, corresponding The  t o the  e a r l y work o f Woodbrey and  87 Rogers  shows t h a t the b a r r i e r t o h i n d e r e d r o t a t i o n i n amides i s  s t r o n g l y i n f l u e n c e d by the s u b s t i t u e n t X.  In t h i s work the b a r r i e r s  were d e t e r m i n e d u s i n g o n l y approximate a n a l y s e s o f s t e a d y - s t a t e d a t a , however, and the a c t i v a t i o n e n e r g i e s were d i s c u s s e d  NMR  qualitatively  i n terms o f the e f f e c t o f a g i v e n s u b s t i t u e n t on the b o n d - o r d e r o f the N-C  bond.  Now  t h a t r e l i a b l e v a l u e s o f the f r e e e n e r g i e s o f a c t i v a t i o n  are a v a i l a b l e , a s e m i - q u a n t i t a t i v e c o r r e l a t i o n w i t h d a t a a v a i l a b l e from MO  calculations i s feasible.  Thus a s i m p l e model may  be  developed,  129  w i t h i n the framework o f a l i n e a r c o m b i n a t i o n o f atomic o r b i t a l s  (LCAO)  72 MO  theory  , t o d e s c r i b e the i n t e r a c t i o n s between the c a r b o n y l group,  t h e n i t r o g e n l o n e p a i r e l e c t r o n s and the s u b s t i t u e n t X i n a s u b s t i t u t e d amide.  A c a l c u l a t i o n o f the energy change ( e l e c t r o n i c o n l y i n the  s i m p l e s t model) on r o t a t i o n about the N-C  bond and d i r e c t  comparison  w i t h the e x p e r i m e n t a l d a t a then a l l o w s an e v a l u a t i o n o f t h e g e n e r a l v a l i d i t y of the  model and p o s s i b l y a s i m p l e s e m i - q u a n t i t a t i v e p h y s i c a l  d e s c r i p t i o n o f the h i n d e r e d r o t a t i o n and the e f f e c t s o f d i f f e r e n t s u b s t i t u e n t s on the b a r r i e r t o r o t a t i o n .  A l t h o u g h a number o f c a l -  c u l a t i o n s based upon s i m p l e H u c k e l m o l e c u l a r o r b i t a l been r e p o r t e d f o r a m i d e s * ^  4  168^  ^  e  (HMO)  t h e o r y have  o n l y c a l c u l a t i o n o f t h i s type f o r 169  a s e r i e s o f r e l a t e d compounds i s t h a t r e p o r t e d by Sand Strom a s e r i e s of s u b s t i t u t e d N,N-dimethylthioamides.  ;  the f r e e e n e r g i e s o f a c t i v a t i o n o b t a i n e d from NMR crude c o r r e l a t i o n w i t h N-C 170 w-Huckel method  In the l a t t e r  study,  s p e c t r a show o n l y a  ir-bond o r d e r s c a l c u l a t e d u s i n g a m o d i f i e d  171 '  , w h i l e a s l i g h t l y improved  correlation i s  o b t a i n e d w i t h t h e c o r r e s p o n d i n g l o s s i n u - e l e c t r o n energy which when the dimethylamino group about the N-C  for  occurs  group i s r o t a t e d w i t h r e s p e c t t o t h e t h i o c a r b o n y l  bond.  I n t h i s s e c t i o n , a s i m p l e i r - e l e c t r o n HMO  model  i s used t o c a l c u l a t e b o t h bond o r d e r s and e l e c t r o n d e l o c a l i s a t i o n e n e r g i e s f o r a s e r i e s o f s u b s t i t u t e d N,N-dimethyl amides. The  a p p l i c a t i o n o f quantum-mechanical methods t o the  cal-  c u l a t i o n o f m o l e c u l a r p r o p e r t i e s f o r systems o f c h e m i c a l i n t e r e s t a t t h i s time s t i l l  i n v o l v e s a number o f a p p r o x i m a t i o n s , and  consequently  t h e emphasis has been p l a c e d upon c o r r e c t n u m e r i c a l agreement w i t h experimental data.  In t h e MO  a p p r o x i m a t i o n , e l e c t r o n s are a s s i g n e d t o  130.  m o l e c u l a r o r b i t a l s which a r e most c o n v e n i e n t l y chosen t o be one e l e c t r o n wavefunctions atomic  expressed  o r b i t a l s , c|> .  as l i n e a r c o m b i n a t i o n s  That i s , t h e LCA0-M0  TII. = V  3  1  o f a f i n i t e number o f i s g i v e n as  C. <$> IV  (4.2.1)  u  u  172 where t h e c o e f f i c i e n t s C j a r e d e t e r m i n e d by a v a r i a t i o n a l  calculation  Through n e g l e c t o f a l l t w o - e l e c t r o n i n t e r a c t i o n s , t h e t o t a l  molecular  U  wave-function  may be c o n s i d e r e d as a s i m p l e p r o d u c t  m o l e c u l a r o r b i t a l s so t h a t t h e t o t a l e l e c t r o n i c ground s t a t e ,  energy, f o r a s i n g l e t  I n t h e s i m p l e s t LCA0-M0 t h e o r y , an e l e c t r o n  may be c o n s i d e r e d t c occupy a o n e - e l e c t r o n MO i n a t o a l l o t h e r e l e c t r o n s and n u c l e i the energy a s s o c i a t e d w i t h t h e MO H a m i l t o n i a n , H.  I n t h i s manner  , E•, i s d e t e r m i n e d by an e f f e c t i v e  I n accordance w i t h t h e v a r i a t i o n a l p r i n c i p l e , f o r t h i s  I v  [H - 6 E.] C. yv yv 3 jy J  =  Ej are given i n  (4.2.2)  0  i s t h e H a m i l t o n i a n m a t r i x element f o r t h e p a r t i c u l a r b a s i s  AO's <f>y and <J). V  orthonormal  c e n t r a l .. f i e l d due  i n a given molecule.  H a m i l t o n i a n , t h e e i g e n - f u n c t i o n s i>. and e i g e n - v a l u e s 172 terms o f a s e t o f s i m u l t a n e o u s e q u a t i o n s :  v  one-electron  i s g i v e n as t h e sum o f t h e e n e r g i e s f o r e l e c t r o n s i n t h e  occupied molecular o r b i t a l s .  where H^  o f these  I t may be assumed t h a t the b a s i s f u n c t i o n s form an  s e t and t h a t a l l e f f e c t s o f AO o v e r l a p a r e d e s c r i b e d by t h e  o f f - d i a g o n a l m a t r i x elements ^ ( y * v ) , H  v  and hence 6  U V  i s the Kronecker  184 delta function.  Inclusion o f overlap  d i s t r i b u t i o n i n a molecule  leads t o a d i f f e r e n t  charge  b u t has o n l y a s m a l l e f f e c t on d e l o c a l i s a t i o n  131.  e n e r g i e s and t o t a l ir-encrgies.  Thus a l t h o u g h i t i s more r i g o r o u s t o  i n c l u d e o v e r l a p , i n g e n e r a l no s i g n i f i c a n t improvement o c c u r s i n the r e s u l t s from s i m p l e H u c k e l MO eigenvalues  calculations.  I n i t i a l l y a l l o f the  a r e c a l c u l a t e d u s i n g the s e c u l a r d e t e r m i n a n t a l  III  and t h e n the e x p a n s i o n the s i m u l t a n e o u s d e f i n e d by H ^  - 6  El =  0  (4.2.3)  ,  coefficients C j  equation:  V  are determined  equations Eq..(4.2.2) w i t h constant  and the s p e c i f i c e i g e n v a l u e E j .  as s o l u t i o n s o f coefficients  G i v e n an e x p l i c i t  form  f o r the o n e - e l e c t r o n e f f e c t i v e H a m i l t o n i a n o p e r a t o r , the m a t r i x elements H  u v  the  may  be c a l c u l a t e d i n the chosen b a s i s o f AO's,  o v e r l a p i n t e g r a l s may  be e v a l u a t e d .  {<J>), U  and  similarly  However, i n such a s i m p l i f i e d  form o f LCAO-MO t h e o r y t h e r e i s l i t t l e p o i n t i n d e f i n i n g the  Hamiltonian  o p e r a t o r e x p l i c i t l y and hence the m a t r i x e l e m e n t s a r e d e f i n e d o n l y i n terms o f e m p i r i c a l p a r a m e t e r s . d e s c r i b e d by the i n t e g r a l s S  The  i n c l u s i o n o f o r b i t a l o v e r l a p as  i s not c o n s i s t e n t w i t h the c o m p l e t e  n e g l e c t o f a l l e l e c t r o n - e l e c t r o n and e l e c t r o n - n u c l e u s i n t e r a c t i o n s ,  and  hence i t must be assumed t h a t such i n t e r a c t i o n s are i n c l u d e d i n the e s t i m a t i o n of the e m p i r i c a l parameters, c o m p a r i s o n o f the r e s u l t a n t MO  w h i c h are o b t a i n e d from  data with s p e c i f i c experimental q u a n t i t i e s  o r by d i r e c t c o m p a r i s o n w i t h w e l l d e f i n e d m o l e c u l a r p r o p e r t i e s . f u r t h e r s i m p l i f i c a t i o n may molecules  be c o n s i d e r e d i n the s t u d y o f i r - c o n j u g a t e d  such as amides, i n t h a t t o t h e l e v e l o f a p p r o x i m a t i o n  c o n s i d e r a t i o n t h e i r - e l e c t r o n s may O-electrons.  A  under  be assumed t o be i n d e p e n d e n t o f the  T h a t i s , the a - e l e c t r o n s are c o n s i d e r e d t o form a l o c a l i s e d  132.  bonding  system which does n o t v a r y s i g n i f i c a n t l y i n form w i t h s t r u c t u r a l  o r s u b s t i t u e n t changes.  T h i s i s p r o b a b l y a good a p p r o x i m a t i o n f o r t h e  s u b s t i t u t e d N,N-dimethyl amides w i t h a s i n g l e v a r i a b l e s u b s t i t u e n t X and a s t r u c t u r a l change c o r r e s p o n d i n g t o a s i m p l e r o t a t i o n o f t h e dimethylamino  group about t h e N-C bond.  The d i f f e r e n t i a l  energy, AF.,  c o r r e s p o n d i n g t o s t r u c t u r a l o r s u b s t i t u e n t changes may now be c o n s i d e r e d i n the p a r t i t i o n e d  form  j AE = ( A E + A E ) + A E 0  where AE  =  n  ,  (4.2.4)  £ E , t h e summation b e i n g o v e r t h e o c c u p i e d TT- m o l e c u l a r j 3  o r b i t a l e n e r g i e s , and A E i n t e r a c t i o n energy. the bonding  U  n  r e p r e s e n t s any change i n t h e non-bonding  Thus i n c o n j u g a t e d m o l e c u l e s ,  for simplicity,  c h a r a c t e r i s t i c s and energy v a r i a t i o n s due t o s t r u c t u r a l  and s u b s t i t u e n t changes may now be d e s c r i b e d i n terms o f a f r - e l e c t r o n o n l y LCAO-MO model. bonding  From c a l c u l a t i o n s b a s e d upon t h i s model, t h e  c h a r a c t e r i s t i c s a r e d e s c r i b e d i n terms o f d e l o c a l i s e d  o r b i t a l s d e f i n e d as l i n e a r c o m b i n a t i o n s TT  d i f f e r e n t i a l energy becomes AE The  o f 2P  molecular  - AO's (Tr-type) , and t h e  n  + AE  i n accordance  w i t h Eq. (4.2.4)  f o l l o w i n g s i m p l e H u c k e l c a l c u l a t i o n s a r e based upon t h e  much s i m p l i f i e d Tr-only LCAO-MO model w i t h t h e a i m o f c h e c k i n g t h e g e n e r a l a p p l i c a b i l i t y o f such a model t o a d e s c r i p t i o n o f t h e h i n d e r e d r o t a t i o n i n s u b s t i t u t e d amides.  Attempts  have been made t o i n c o r p o r a t e 173 ]74  e l e c t r o n r e p u l s i o n i n t o simple Huckel theory o f a p p r o x i m a t i o n such a p r o c e d u r e  '  , but at t h i s  level  i sd i f f i c u l t to j u s t i f y i nthat a  r e d e f i n i t i o n o f the p u r e l y e m p i r i c a l parameters  i sinvolved.  Therefore  133.  a n o n - i t e r a t i v e H u c k e l c a l c u l a t i o n i s used i n t h i s s t u d y w i t h t h e e f f e c t i v e H a m i l t o n i a n m a t r i x elements f o r s u b s t i t u t e d amides b e i n g d e f i n e d by e m p i r i c a l Coulomb i n t e g r a l s , a , and r e s o n a n c e i n t e g r a l s , 3 A  R  Y  72  ,  L» — A  where  a  =  x  a°  +  h  x  3 °  and  (4.2.5) e  c-x  =  k  C-X  3  °  As o n l y r e l a t i v e e n e r g i e s a r e s i g n i f i c a n t , t h e s t a n d a r d Coulomb i n t e g r a l a° may be t a k e n as t h e energy r e f e r e n c e z e r o and then t h e m a t r i x elements a  v  and 3  X  RY  a r e d e f i n e d i n terms o f t h e s t a n d a r d resonance  C—A  i n t e g r a l , 3 ° < o , w h i c h i s u s u a l l y t a k e n as t h a t f o r t h e C-C bond i n benzene  1 0  .  I t i s t o be noted t h a t t h e p a r a m e t e r 3  r e f e r s t o N-C,  RY  C —A  C=0 and C-X Tr-type bonds i n t h e s u b s t i t u t e d amides >N-C-X, a l l o t h e r 0 o f f - d i a g o n a l H a m i l t o n i a n m a t r i x elements b e i n g d e f i n e d t o be z e r o . I n d e s c r i b i n g t h e energy v a r i a t i o n f o r h i n d e r e d r o t a t i o n about t h e N-C bond, due t o d e l o c a l i s a t i o n o f t h e f o r m a l l o n e p a i r e l e c t r o n s on t h e sp - h y b r i d i s e d N atom, t h e energy f o r t h e p l a n a r ground s t a t e i s g i v e n as a sum o f  TT-MO  energies.  In the r o t a t i o n t r a n s i t i o n s t a t e , the d i -  methylamino group i s r o t a t e d i n t o a p l a n e p e r p e n d i c u l a r t o t h a t o f t h e c a r b o n y l group and hence t h e c o n j u g a t i o n between t h e amino and C-X :  0  groups i s b r o k e n down  as t h e symmetry r e l a t i o n s h i p n e c e s s a r y f o r  ir-bonding between t h e s e two groups no l o n g e r e x i s t s . m o l e c u l a r energy must be c o n s i d e r e d as a sum o f C-X  IT-MO  I n t h i s case t h e energies f o r the  group and a c o n t r i b u t i o n from non-bonding e l e c t r o n s on t h e N atom,  6 E , n  c f . Eq. ( 4 . 2 . 4 ) .  Thus t h e d i f f e r e n t i a l  7T-energy a s s o c i a t e d w i t h t h e  134.  h i n d e r e d r o t a t i o n i s g i v e n as  AE  =  E^N-C-X) - {E^CC-X) + 2 h ] , M  TT  ||  1/  0  o  (4.2.6)  IN  where t h e e n e r g i e s a r e g i v e n i n terms o f the H a m i l t o n i a n m a t r i x :  k  k  "  C-N 0  0  | 1 1  0  C-N  0  . c=o  k  c  k  c=o  k  c-x  k  k  k  0 k  under  In c o n t r a s t  (4.2.7)  0  o  i n u n i t s o f 3°, c f . Eq. ( 4 . 2 . 5 ) .  for the substituents  c-x  x  The o n l y non-bonding energy i s t h a t  consideration.  t o an ab  i n i t i o c a l c u l a t i o n , t h e v a l u e s chosen  f o r t h e e m p i r i c a l p a r a m e t e r s h^, and k^_^ are o f p r i m a r y i m p o r t a n c e  since  t h e y a l o n e may d e t e r m i n e t h e r e s u l t s o f a s i m p l e H u c k e l MO c a l c u l a t i o n a n d ^ t h e r e f o r e r e p r e s e n t t h e e s s e n t i a l c h a r a c t e r i s t i c s o f the s i m p l i f i e d model b e i n g used h e r e .  A Coulomb r e s o n a n c e i n t e g r a l i s e x p e c t e d t o be  d i r e c t l y r e l a t e d t o b o t h t h e i o n i s a t i o n p o t e n t i a l and e l e c t r o n  affinity  f o r a g i v e n atom i n a m o l e c u l e , and hence t o t h e a t o m i c e l e c t r o 17 6 -17 S negativity  .  The c o n c e p t o f v a r i a b l e e l e c t r o - n e g a t i v i t y h a s been 179 180  i n t r o d u c e d by J a f f e and  coworkers  '  , and t h i s a l l o w s t h e r i g o r o u s  d e f i n i t i o n o f an o r b i t a l e l e c t r o - n e g a t i v i t y w h i c h may be c o n s i d e r e d as a measure o f t h e a t t r a c t i n g power o f an atom, as i t e x i s t s i n a m o l e c u l e , t o w a r d an e l e c t r o n  i n a s p e c i f i c t y p e o f AO, t h a t  i sfor a s p e c i f i c  135.  Thus t h e i T - o r b i t a l e l e c t r o n e g a t i v i t i e s f o r  atomic valence s t a t e . sp - h y b r i d i s e d 2  N,C and 0 atoms have been u s e d t o f i x t h e h^, p a r a m e t e r s  as 0.7, 0.0 and 1.4, r e s p e c t i v e l y .  These p a r a m e t e r s may be compared  181 with "standard" values  .  The p a r a m e t e r h ^ has a l s o been m o d i f i e d t o  a l l o w f o r t h e h y p e r c o n j u g a t i v e e f f e c t o f t h e N-methyl groups f o r t h e amides under c o n s i d e r a t i o n .  For the general substituent  X i n the  N,N-dimethyl amides, i t has been assumed t h a t a l l m u l t i - a t o m i c subs t i t u e n t groups may a d e q u a t e l y be r e p r e s e n t e d as p s e u d o - a t o m s * ^ ' * ^ and 2  are  therefore characterised  8  by a s i n g l e h ^ p a r a m e t e r d e r i v e d t h r o u g h  a l i n e a r r e l a t i o n s h i p w i t h t h e c o r r e s p o n d i n g group e l e c t r o - n e g a t i v i t i e s . The s e t o f h  Y  p a r a m e t e r s used i n t h e H u c k e l MO c a l c u l a t i o n s  i s given  i n T a b l e 4.9 a l o n g w i t h r e f e r e n c e s t o t h e method f o r c a l c u l a t i o n o f t h e group e l e c t r o n e g a t i v i t i e s , namely t h e use o f a v a r i a b l e negativity  electro-  f o r t h e c e n t r a l atom i n a group and e q u a l i s a t i o n  of electro-  179 negativity  i n a l l bonds.  p a r a m e t e r k^  , characterises  The r e s o n a n c e i n t e g r a l , and hence t h e t h e 7T-bonding between t h e C atom and t h e  atom o r pseudo-atom X and i s e x p e c t e d t o be d i r e c t l y r e l a t e d t o t h e bond l e n g t h y{C-X)  1 8 5 j 1  ^  6  t h e o v e r l a p i n t e g r a l between 2 P - A 0 ' s * . 8 7  o  r  Z  The k -,_ v a l u e s d e r i v e d t h r o u g h a l i n e a r c o r r e l a t i o n w i t h bond l e n g t h s 188 (  and  x  corresponding overlap i n t e g r a l s  f o r t h e carbamyl f l u o r i d e  structure  shown i n F i g . 4.22 a r e 0 . 9 0 ( 0 . 9 0 ) , 0.80(0.86) and 0.60(0.52) f o r t h e C=0, N-C and C-F bonds, r e s p e c t i v e l y .  As i t i s d i f f i c u l t t o e s t i m a t e  o v e r l a p i n t e g r a l s f o r pseudo-atoms, t h e s i m i l a r i t y o f t h e above s e t s o f parameters i n d i c a t e s  t h a t t h i s d i f f i c u l t y may be a v o i d e d by u s i n g t h e  s i m p l e r c o r r e l a t i o n w i t h bond l e n g t h s , and hence t h e s e t o f k„  v  para-  meters g i v e n i n T a b l e 4.9 has been d e t e r m i n e d i n t h i s way from t h e  T a b l e 4.9 HuckelMO data f o r hindered r o t a t i o n i n substituted N,N-dimethyl  £  X  \  '  k  C-X  H a)  CN  a) F  aj  Br  b) OCH  3  c)  NCS  AG #  *Vc  P  C-0  Pj-O  %  0, .759  21.0  0.,766  0,.516  0..753  1 .58  0. .737  20.6  0,.752  0,.511  0 .731  1 .59  .728  18.2  0 .746  0 .508  0 .723  1 .60  4. .6  0 .55  3..95  4 .0  0. .60  3 .00  1 .4  0 .40  0 .715  16.8  0,.737  0 .502  0 .704  1 .58  2. .80  0,.8  0 .32  0 .710  15.7  0,,735  0,.500  0 .694  1,.61  2. .68  0. 65  0. 50  0. 639  (5.8)  0. 685  0.473  0.613  1. 65  3 .91  3..85  0 .39  0 .745  (21.4)  0,.757  0 .512  0,.739  1 .59  4,.15  4 .50  0 .54  0 . 737 (20.2)  0,.752  0,.511  0,.731  1 .59  4,.42  4.,90  0,.54  0,.739  (20.4)  0.,753  0,.511  0,.734  1,.59  3,.05  1,,55  0,.56  0,.686  (12.6)  0.,717  0,.492  0..674  1 .63  2 .61  0..80  0,.60  0,.621  (3.0)  0.,670  0,.465  0,,597  1 .66  2,.40  0.,70  0,.60  0,,609  (1.0)  0,,661  0,.460  0,.583  1 .67  c) SCN  +  E  4 .17  a) Cl  A  amides  °.  c) N  3  a) NCO b) NH  2  NMe  b) 2  K, = 0.7 N  C-•N  k^ = 0.0 C  C-•0  k  0  0. 8 - 0.9  = 1.4  ^ i n u n i t s o f 3 <o, see text. a)  W. Gordy and W.J.O. Thomas  J . Chem. Phys. 24_, 439, 1956  b)  J . E . Huheey  J . Phys. Chem. 69_, 3284, 1965  c)  J . E . Huheey  J . Phys. Chem. 70, 2086, 1966  136.  s t r u c t u r e s a v a i l a b l e f o r amides o r t h e r e l a t e d a c e t y l compounds H3C-C-X n 0  and f o r m y l compounds H-C-X. 11  C  The  d i f f e r e n t i a l e n e r g i e s . AE , f o r h i n d e r e d r o t a t i o n i n sub-  s t i t u t e d N,N-dimethyl amides a r e g i v e n i n T a b l e 4.9 a l o n g w i t h t h e N-C TT-bond o r d e r s f o r t h e p l a n a r ground s t a t e s and t h e C-0 TP-bond o r d e r s G T f o r t h e ground and r o t a t i o n t r a n s i t i o n s t a t e s , vely.  PQ-Q  a n <  ^ ^C 0' -  R  E  S  P  E  C  T  ^  _  These bond o r d e r s a r e d e f i n e d i n terms o f t h e LCAO-MO e x p a n s i o n  c o e f f i c i e n t s as  : p  =  2  I C. C.  ,  (4.2.8)  where <f> and cj) a r e 2P -AO's on bonded atoms and t h e summation i s o v e r \i  v  Zi  the o c c u p i e d rr m o l e c u l a r o r b i t a l s .  A c o r r e s p o n d i n g TT-charge d e n s i t y  i s defined as:  p l  W  =  2  T, C. C. j 1U JU  ,  (4.2.9) J  and t h e s e d e n s i t i e s f o r t h e N atom i n t h e amide ground s t a t e s , q^, a r e a l s o l i s t e d i n T a b l e 4.9. As t h e NMR spectrum  f o r N,N-dimethyl u r e a shows a s i n g l e  sharp peak f o r t h e N-methyl p r o t o n s down t o -118°C, t h e f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n i n t h i s amide may be e s t i m a t e d as 3 k c a l . mole  1  a t 298°K.  The c o r r e l a t i o n diagram f o r f r e e energy o f a c t i v a t i o n ,  # AG  , and H u c k e l MO d i f f e r e n t i a l fr-energy, AE^, g i v e n as F i g . 4.14 i n c l u d e s  the e s t i m a t e d energy f o r X=NH p r e l i m i n a r y v a l u e o f AG carbamyl  cyanide  (X=CN).  2  as an e x p e r i m e n t a l p o i n t and a l s o a  = 20.6 + 0.8 k c a l . m o l e "  1  f o r N,N-dimethyl  The c o r r e l a t i o n o b t a i n e d i s e x c e l l e n t , w h i c h  X  22 H  H  18  3 v C  3  SCN CN  \ / N--C  C  X  N  NC5  0  F Cl  o E  Br OCHc  u  i  3  NCO  <  m  6  o'  OCH< NHN(CH ) 3  0.60  0/70  0.65  0.75  AE Fig.  4.14  C o r r e l a t i o n o f f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n w i t h H u c k e l M0 d i f f e r e n t i a l u-energy  :  137.  shows t h a t the method o f p a r a m e t e r i s a t i o n i s c o n s i s t e n t f o r such a s e r i e s o f compounds and  a l s o i n d i c a t e s t h a t the change i n Tr-energy on  b r e a k i n g down the c o n j u g a t i o n through the N-C  bond may  f a c t o r i n determining  r o t a t i o n about t h i s bond.  the b a r r i e r t o h i n d e r e d  be a dominant  I t i s o f some i n t e r e s t t o c o n s i d e r i n more d e t a i l the c a l c u l a t e d d i f f e r e n t i a l Tr-energy f o r m e t h y l N,N-dimethyl carbamate (X=0CHg), as e x p e r i m e n t a l l y i t was  found t h a t t h e b a r r i e r t o h i n d e r e d  i n c r e a s e d by a t l e a s t 8 k c a l s . mole c f . s e c t i o n 4.1.3.  With t h e h  Y  1  w i t h c h l o r o f o r m as a s o l v e n t ,  and k  values  l i s t e d i n Table  AE ' i s c a l c u l a t e d as 0.639 3° w h i c h c o r r e s p o n d s t o a AG 5.8  k c a l . mole , as shown by the open p o i n t i n F i g . 4.14, l  c o n s i s t e n t w i t h the e x p e r i m e n t a l  rotation  results.  Now  value  of  which i s  i f i t i s postulated  t h a t t h e c h l o r o f o r m forms a s t e r e o - s p e c i f i c hydrogen-bond w i t h carbonyl  4.9,  the  lone p a i r a - e l e c t r o n s , t h i s i n t e r a c t i o n would m o d i f y the  Coulomb i n t e g r a l f o r the c a r b o n y l oxygen atom and hence t h e H u c k e l p a r a m e t e r h^.  Such a s o l u t e - s o l v e n t i n t e r a c t i o n would be e x p e c t e d  to  i n c r e a s e the c o n t r i b u t i o n from the resonance form I I shown below i n the carbamate ground s t a t e , and  thus t h i s i n t e r a c t i o n i s d e s c r i b e d  an i n c r e a s e i n t h e magnitude o f h .  N-C-X  <—>-  N=C-X  (ID  (i)  The  e f f e c t o f a v a r i a b l e h^, w i t h a l l o t h e r p a r a m e t e r s as l i s t e d i n  T a b l e 4.9,  i s shown i n F i g . 4.15;  and  i t i s seen t h a t t h e h.  value  by  H  3 x C  /  HC  N / N-l-C  0 C H  3  w 0  3  0.721  I \  o  Ox  JO  o' /  0\  0.70 'O \  o'o'  .O  LU  <  /  /  /  /  OO  ^  0.66P  h / 0  0.64  /  o /  / / /  0.62-  F i g . 4.15  /  o'  /  /  o  1.4  c=o  RN-C  0.68  1.8 0.70  2.2 0.72  2.6 0.74  0.24  0.32  0.40  0.48  h  0  RN-C ROO  E f f e c t s o f v a r i a b l e c a r b o n y l oxygen H u c k e l t h e o r y Coulomb i n t e g r a l , h , f o r m e t h y l N,N-dimethyl carbamate c a l c u l a t i o n  138.  corresponding  t o AE  = 0.705 3 ° and hence AG  TT  #  =15.2  shown by the c l o s e d p o i n t f o r X=0CH3 i n F i g . 4.14, t o 2.6.  Thus a l t h o u g h  creased  the s i m p l e H u c k e l  kcal.mole must be  , as  increased  model p r e d i c t s an i n -  TT-MO  b a r r i e r t o r o t a t i o n f o r the s p e c i f i c i n t e r a c t i o n a t t h e  carbonyl  oxygen, the change i n the h^ p a r a m e t e r r e q u i r e d t o r e p r o d u c e the magn i t u d e o f t h e i n c r e a s e i n AG*, 4.1.3, i s r e l a t i v e l y l a r g e . r o t a t i o n about the N-C  as d e t e r m i n e d e x p e r i m e n t a l l y i n s e c t i o n I t i s a l s o p o s s i b l e t h a t the b a r r i e r to  bond may  be  a f f e c t e d by the c h l o r o f o r m  forming  hydrogen bond w i t h the methoxy oxygen lone p a i r o - e l e c t r o n s . i n t e r a c t i o n may  Such  a  an  l e a d t o an i n c r e a s e d c o n t r i b u t i o n from the r e s o n a n c e form  I I I shown b e l o w , and  i s d e s c r i b e d i n terms o f t h e s i m p l e H u c k e l  "1 />  TT-MO  +  • N-G-X  -*—*.  II 0  N=G=0 X  .  .  .  (.4.2.11)  (III)  model by an i n c r e a s e i n t h e magnitude o f h., f o r X=OCH . Thus AE is • X TT c a l c u l a t e d as 0.702 3 ° f o r h = 1.95, as compared w i t h 0.65 i n T a b l e 4.9, 3  J  5  A  corresponding  t o AG  = 15 k c a l s . m o l e . . 1  An  i n c r e a s e i n the b a r r i e r  to  r o t a t i o n i s p r e d i c t e d as a consequence o f the s p e c i f i c s o l u t e - s o l v e n t i n t e r a c t i o n a t the methoxy oxygen atom, but a g a i n t h e change i n the parameter r e q u i r e d i s r e l a t i v e l y l a r g e . in h  Q  r e q u i r e d t o g i v e a c a l c u l a t e d AE^ ^  corresponding  increment i n h , Y  I n so f a r t h a t t h e 0.71  h  Y  increment  3 ° i s much l e s s t h a n the  a s p e c i f i c i n t e r a c t i o n at the  carbonyl  atom i s p r e d i c t e d t o have a more s i g n i f i c a n t e f f e c t on the b a r r i e r t o rotation.  T h i s i s c o n s i s t e n t w i t h the i n t e r a c t i o n e f f e c t b e i n g  f e r r e d t h r o u g h the c a r b o n y l Tr-bond, t h i s bond b e i n g d e s c r i b e d  by  trans-  139.  k _  = 0.9 as compared w i t h k  „„  =0.5.  Of c o u r s e , i f t h e i n t e r a c t i o n  between s o l u t e m o l e c u l e and 0 atom i s o n l y d e s c r i b e d by a r e l a t i v e l y l a r g e change i n h^, t h e v a l u e o f k^_^ s h o u l d  a l s o be v a r i e d and thus t h e  commensurate change i n h^ w i l l be r e d u c e d . From t h e c o r r e l a t i o n diagram g i v e n i n F i g . 4.14 the f r e e of a c t i v a t i o n f o r hindered  r o t a t i o n i n N,N-dimethyl amides w i t h pseudo-  h a l o g e n s u b s t i t u e n t s X = SCN, NCS, N 3 and NCO, and a l s o i n t h e trical  symme-  u r e a (X = N ( C H ) 2 ) may be p r e d i c t e d from t h e c a l -  tetramethyl  c u l a t e d AE  energies  3  values.  These AG  values  a r e i n c l u d e d i n T a b l e 4.9 and  TT  are shown as open p o i n t s i n F i g . 4.14,  The AG  - AE  correlation  i n d i c a t e s t h a t as t h e e l e c t r o - n e g a t i v i t y o f t h e s u b s t i t u e n t the b a r r i e r t o h i n d e r e d w i t h an i n c r e a s e d  r o t a t i o n also increases.  increases,  This i s consistent  c o n t r i b u t i o n from t h e resonance form IV shown below  c o r r e s p o n d i n g t o a d e c r e a s e d b a r r i e r , as a d e l o c a l i s a t i o n o f u - e l e c t r o n s  —  T*  (4.2.12)  N  0^  0 (IV)  on t h e atom o r pseudo-atom s u b s t i t u e n t group X i s p o s s i b l e f o r a l l o f the s u b s t i t u e n t s c o n s i d e r e d there  i s no s i m p l e  here.  As shown i n F i g . 4.16, however,  c o r r e l a t i o n between t h e f r e e e n e r g i e s  of activation  AG , as measured o r p r e d i c t e d i n accordance w i t h t h e H u c k e l w i t h t h e atomic o r group e l e c t r o - n e g a t i v i t i e s £ . Y  attempted u s i n g r e l i a b l e e x p e r i m e n t a l  data.  model,  Thus the use o f even  a s i m p l i f i e d LCA0-M0 model f o r t h e d e s c r i p t i o n o f h i n d e r e d amides i s j u s t i f i e d i f a s e m i - q u a n t i t a t i v e  TT-MO  rotations i n  c o r r e l a t i o n i s t o be The s e l f - c o n s i s t e n c y o f t h e  X  22 -  ©  SCN CN N NCS  / ©/ /  I  3  / /  I  18  1  '  F  /  I  Cl  QJ  Br  O  E  OCH.  14  u  NCO  /  I /  #  /  t  I  10  / /  6  OCIi NH.  2 N(CH )  /  3  2.4  3.2  T  r  4.0  1  r-  4.8  X F i g . 4.16  C o r r e l a t i o n o f f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n w i t h group e l e c t r o - n e g a t i v i t y , e , o f t h e X - s u b s t i t u e n t i n N,N-dimethyl amides'  2  140.  Huckel  c a l c u l a t i o n i s i l l u s t r a t e d b y the t r e n d s i n the ir-bond o r d e r s  and ir-charge d e n s i t i e s  l i s t e d i n T a b l e 4.9. As shown i n F i g . 4.17,  an e x c e l l e n t c o r r e l a t i o n i s a l s o o b t a i n e d between AG order P J ^ Q -  and t h e N-C Tf-bond  The c a l c u l a t e d b o n d - o r d e r s p ^ ^ a r e c o n s i s t e n t l y h i g h e r  than  Q  t h o s e f o r the c a r b o n y l bond i n t h e amido  ground s t a t e ,  but t h i s  VQ-Q>  i s o n l y a consequence o f t h e p a r t i c u l a r p a r a m e t e r i s a t i o n chosen f o r t h e N atom and does n o t a f f e c t any o f the r e s u l t s p r e s e n t e d f o r the  relative  e f f e c t s o f the X s u b s t i t u e n t s . As t h e s i m p l e H u c k e l  model d e s c r i b e d above g i v e s a con-  TT-MO  s i s t e n t e s t i m a t e o f the d i f f e r e n t i a l energy AE^ f o r t h e ground and h i n d e r e d r o t a t i o n t r a n s i t i o n s t a t e s f o r s u b s t i t u t e d N,N-dimethyl amides, such a model may be u s e f u l i n a more g e n e r a l d e s c r i p t i o n o f TT-bonding w i t h i n a conjugated  amide system.  I n g e n e r a l terms, t h e d e t a i l s o f an  e l e c t r o n i c charge d i s t r i b u t i o n f o r any g i v e n m o l e c u l e the expansion  c o e f f i c i e n t s f o r the  LCAO  are contained i n  molecular o r b i t a l s , c f .  191 Eq. ( 4 . 2 . 1 ) . The p r o b a b i l i t y d e n s i t y , D, a t a p o i n t i n space a s s o c i a t e d w i t h an e l e c t r o n i n a m o l e c u l a r o r b i t a l ii. i s and 3  hence i n accordance  w i t h Eq.  D  where p Eqs.  and p  =  2  3 3  (4.2.1):  P  4>*<> j  +  I  I  P.J>*<J>  >  ( 4 - 2 . 1 3 )  a r e defined f o r doubly occupied o r b i t a l s i n  (4.2.8) and ( 4 . 2 . 9 ) , r e s p e c t i v e l y .  Thus the bond o r d e r p  i s the  predominant f a c t o r d e t e r m i n i n g the d i s t r i b u t i o n o f T r - e l e c t r o n charge i n t h e space between the bonded atoms w i t h 2P -A0's c|> and cj>^ . Z  an approximate  Although  a n a l y s i s i s p o s s i b l e u s i n g bond o r d e r s , an e l e c t r o n i c  X  22 HoC 3 HC  \ N--C /  3  18  / x  X  SCN CN  ¥  o  N  NCS  F  O  E  i  Cl  9  Br  ©  #  3  .OCH,  O  9  <] NCO 1  n  6  OCHo  NH  ©  2 N(CH ) 3  0.66  0.70  0.74  0.78  P MN-C F i g . 4.17  C o r r e l a t i o n o f f r e e energy o f a c t i v a t i o n f o r h i n d e r e d r o t a t i o n w i t h N-C TT-bond o r d e r , p , o b t a i n e d from H u c k e l MO c a l c u l a t i o n s .  2  141.  192 193 charge d e n s i t y map  '  i s t h e most e x p l i c i t and complete  means o f  showing t h e form o f t h e 'rr-charge d i s t r i b u t i o n i n a c o n j u g a t e d system.  amide  A FORTRAN-IV computer program CNTR has been d e v e l o p e d t o  a u t o m a t i c a l l y p l o t c o n t o u r s f o r c o n s t a n t n-charge d e n s i t i e s by d i r e c t . . 194 numerical c a l c u l a t i o n o f the p r o b a b i l i t y d e n s i t y D u s i n g S l a t e r type atomic o r b i t a l s . sum  A t o t a l charge d e n s i t y i s o b t a i n e d as a n o r m a l i s e d  o f c o n t r i b u t i o n s from a l l o f t h e o c c u p i e d m o l e c u l a r o r b i t a l s .  The  H u c k e l u-MO e l e c t r o n i c charge d e n s i t y maps o b t a i n e d f o r (a) an unconj u g a t e d s t a t e and (b) t h e c o n j u g a t e d ground s t a t e o f formamide ( o r N,N-dimethyl formamide) as d e f i n e d by t h e m o l e c u l a r o r b i t a l s u s i n g t h e parameters  generated  l i s t e d i n T a b l e 4.9 a r e shown i n F i g . 4.18, f o r t h e o  p l a n e and d i s p l a c e d by 0.6 A.  Comparison o f t h e s e d e n s i t y maps shows  the r e d i s t r i b u t i o n o f charge i n b o t h t h e N-C and C=0 b o n d i n g  regions  i n t h e c o n j u g a t e d ground s t a t e , t h e f o r m a l l o n e p a i r e l e c t r o n d e n s i t y f o r t h e N atom b e i n g i l l u s t r a t e d i n F i g . 4.18 ( a ) .  The change i n t h e  Q  T-bond o r d e r P Q _ Q on f o r m i n g t h e u n - c o n j u g a t e d T i t i o n s t a t e , as g i v e n by VQ-Q>  1 S  hindered r o t a t i o n  trans-  r e f l e c t e d i n t h e charge d e n s i t y maps  as t h e double bond c h a r a c t e r i n c r e a s e s i n t h e t r a n s i t i o n s t a t e and t h i s i s shown t o c o r r e s p o n d t o an a d d i t i o n a l d e l o c a l i s a t i o n o f charge the c a r b o n y l oxygen atom.  from  This i s a l s o c o n s i s t e n t w i t h a decreased  c o n t r i b u t i o n from t h e resonance  form I I g i v e n i n ( 4 . 2 . 1 0 ) .  Similar  d e n s i t y maps f o r carbamyl  f l u o r i d e a r e shown i n F i g . 4.19 f o r t h e geo-  m e t r y g i v e n i n F i g . 4.22.  In t h e c o n j u g a t e d ground s t a t e f o r t h i s  p a r t i c u l a r m o l e c u l e t h e Tr-charge d e n s i t y i s c l o s e t o b e i n g  symmetrical  7  /  •  '  s  / / //,-;. (a)  V V >  I ' M v \\ \  \ \  i \V 1  1  1/  /  w  N\S  V 1 \  V  N  \  /  / /  ,  7 /  ^  —  \\ \ \  N s  ©  x  v  „  ^  ^  'V~"^N ^ —  \  'In;  i  / / /  \  \ \ \  (b)  M  ® :> !/;  . v\--v////  F i g . 4.18 H u c k e l MO e l e c t r o n i c charge d e n s i t y maps f o r t h e (a) u n - c o n j u g a t e d and (b) c o n j u g a t e d s t a t e s o f formamide  / /  /  f  .  N.  V  ,  I  ,  '  /  ------ '-xV \ / \  /' (a)  ft// • / I  •^  >  x \ ( V  V  -  / ,  x  i ' i ; : 111!'  \  \  "  \ \  \  \  ^  /  /'III  \\\^:-''// \  >.  V \  N  if -  "--^  x  ' /  / .  III,''(?){],,))  I f 1  i  ','  1 :  / /  /  {  GO  '' ' I l l  \\ s  P i g . 4.19  H u c k e l MO e l e c t r o n i c charge d e n s i t y maps f o r t h e (a) u n - c o n j u g a t e d and (b) c o n j u g a t e d s t a t e s o f carbamyl f l u o r i d e  142.  with  r e s p e c t t o the N-C  factor leading  bond and  t h i s feature  t o v e r y s i m i l a r NMR  magnetic a n i s o t r o p i c s  o f the C-F  may  be  the  shielding regions, and  C=0  due  c o n t o u r s f o r the  shows' the  f o r formamide and  maps are c o n s i s t e n t  i n F i g s . 4.18(b) and  4.19(b) N-C  carbamyl f l u o r i d e , r e s p e c t i v e l y .  with  dia-  Comparison o f  s i g n i f i c a n t l y d i f f e r e n t charge d i s t r i b u t i o n s i n the  bond r e g i o n density  same charge d e n s i t y  to the  g r o u p s , f o r the m e t h y l groups  i n N j N - d i m e t h y l carbamyl f l u o r i d e , c f . s e c t i o n 4.1.4. the  dominant  an i n c r e a s e d  These  bond o r d e r p^ ^ and  a  c o r r e s p o n d i n g l y h i g h e r b a r r i e r f o r h i n d e r e d r o t a t i o n i n formamide, c f , Table  4.3  4.9.  S e m i - e m p i r i c a l SCF-LCAO-MO c a l c u l a t i o n s The  containing  electronic structure  o f formamide, the  the amide group N-C=0, i s o f b a s i c  troscopic  study of a s e r i e s of s u b s t i t u t e d  amides.  o r b i t a l c a l c u l a t i o n f o r formamide g i v e s a b a s i s  interest. has  The  The  amide  speclinkage  hence a d e t a i l e d m o l e c u l a r f o r the development o f a  model f o r t h e s e more c o m p l i c a t e d systems o f b i o l o g i c a l h i n d e r e d r o t a t i o n about the N-C  been s t u d i e d  in substituted  molecule  i m p o r t a n c e i n any  i s a l s o c h a r a c t e r i s t i c o f the p o l y p e p t i d e s and  configurational  simplest  bond i n  e x p e r i m e n t a l l y i n s e c t i o n 4.1.5, and  amides have been e x t e n s i v e l y  discussed i n sections  4.1.1  to 4.1.4  and  15  N-formamide  similar  s t u d i e d by NMR  rotations  methods as  as summarised i n a r e c e n t  195 review  .  However, v e r y l i t t l e a t t e n t i o n has  d e t e r m i n a t i o n o f the quantitative  been p a i d t o a t h e o r e t i c a l  form o f the t r a n s i t i o n s t a t e s o r t o a semi-  c o r r e l a t i o n o f the measured f r e e e n e r g i e s o f a c t i v a t i o n  w i t h d a t a a v a i l a b l e from a m o l e c u l a r o r b i t a l d e s c r i p t i o n o f  substituted  143.  amides. for  Empirical  formamide*^ '  of substituted this thesis;  e s t i m a t e s o f t h e b a r r i e r t o r o t a t i o n have been made and simple. H u c k e l "rr-MO c a l c u l a t i o n s f o r a s e r i e s  amides have been o u t l i n e d i n t h e p r e c e d i n g s e c t i o n o f b u t v e r y r e c e n t l y , C h r i s t e n s e n e t a l . have p u b l i s h e d a 196  f u l l ab i n i t i o m o l e c u l a r o r b i t a l s t u d y o f formamide and  hindered r o t a t i o n t r a n s i t i o n states.  i n t e r e s t t o supplement t h e s e r e s u l t s w i t h  i n t h e ground  Thus i t i s o f p a r t i c u l a r a detailed  semi-empirical  SCF-LCAO-MO c a l c u l a t i o n f o r t h e h i n d e r e d r o t a t i o n i n formamide, i n t h e 7 S 76 CNDO/2 a p p r o x i m a t i o n " ' . As p r e l i m i n a r y  CNDO/2 c a l c u l a t i o n s showed t h a t t h e ground197  s t a t e geometry p r o p o s e d by C o s t a i n ding planar configuration, and  was l e s s s t a b l e t h a n a c o r r e s p o n 198  the basic  structure  d e t e r m i n e d by K u r l a n d  shown i n F i g . 4.20, i s used i n t h e f o l l o w i n g c a l c u l a t i o n s .  i n c l u d i n g d - f u n c t i o n s i n a G a u s s i a n type o r b i t a l b a s i s 196 i n i t i o calculation  By  s e t , t h e ab  i n d i c a t e s t h a t t h e n o n - p l a n a r s t r u c t u r e may be  more s t a b l e , b u t t h e v e r y s m a l l result.  ,  energy d i f f e r e n c e p r e c l u d e s a  The geometry o f t h e N-C"^  conclusive  group i s f i x e d i n t h e m o l e c u l a r  x y - p l a n e i n F i g . 4.20, w h i l e a v a r i a b l e geometry f o r t h e amino group may be d e f i n e d  by t h e HNH a n g l e 2a, t h e d i h e d r a l  a n g l e 6 and t h e a n g l e <j)  d e t e r m i n i n g t h e r o t a t i o n about t h e N-C bond. A l l bond l e n g t h s i n ° °198 F i g . 4.20 a r e g i v e n i n A, and t h e N-H-bond l e n g t h i s f i x e d a t 0.995 A To d a t e , t h e most r e l i a b l e MO c a l c u l a t i o n s a r e based upon 199 a f u l l Hartree-Fock s e l f - c o n s i s t e n t f i e l d electrons  (SCF) t h e o r y  i n a g i v e n m o l e c u l a r s y s t e m , an u n l i m i t e d  w a v e f u n c t i o n s and an e x p l i c i t f o r m u l a t i o n Roothaan^^^ c o n s i d e r e d a f i n i t e b a s i s  , involving a l l  basis  s e t o f atomic  of a l l electron  interactions.  s e t o f orthonormal atomic o r b i t a l s ,  4.20  S t r u c t u r e o f formamide used i n CNDO/2 SCF-LCAO-MO calculations  144.  (fy, and m o d i f i e d t h e f u l l SCF t h e o r y t o g i v e a more t r a c t a b l e f o r m u l a t i o n based upon a l i n e a r c o m b i n a t i o n  o f atomic o r b i t a l s  (LCAO) i n an  approximate d e s c r i p t i o n o f m o l e c u l a r o r b i t a l s , c f . Eq. ( 4 . 2 . 1 ) . a p p l i c a t i o n o f t h e Roothaan method, however, i s s t i l l computational  l i m i t e d by  c o m p l e x i t y and hence, i n g e n e r a l , i t i s n e c e s s a r y t o  consider f u r t h e r approximations The  The  w i t h i n t h e c o n t e x t o f t h e SCF t h e o r y .  CNDO method, as r e c e n t l y developed  e m p i r i c a l SCF-LCAO-MO t h e o r y . w h i c h  by P o p l e and coworkers,  i s a semi-  t r e a t s only valence e l e c t r o n s  e x p l i c i t l y and s i m p l i f i e s t h e b a s i c Roothaan e q u a t i o n by use o f t h e complete n e g l e c t o f d i f f e r e n t i a l o v e r l a p (CNDO) a p p r o x i m a t i o n . Roothaan SCF e q u a t i o n then reduces  -  t o t h e form  P.P.  (A  where F_ i s t h e Fock H a m i l t o n i a n m a t r i x and MO e x p a n s i o n  The  i s t h e column v e c t o r o f  c o e f f i c i e n t s C ^ as g i v e n i n Eq. ( 4 . 2 . 1 ) .  e i g e n - v a l u e c o r r e s p o n d i n g t o t h e MO I J K .  % 1 ~l  E^ i s t h e energy  O v e r l a p between atomic  orbitals  i s only i n c l u d e d i n the e v a l u a t i o n o f c e r t a i n bonding i n t e g r a l s i n v o l v e d i n t h i s approximate SCF t h e o r y , b u t i n t e g r a l s d e s c r i b i n g t w o - e l e c t r o n r e p u l s i o n s a r e i n c l u d e d i n t h e Fock m a t r i x elements F . The fundamental yv assumptions used i n t h e d e r i v a t i o n o f t h e Fock m a t r i x elements have been d e s c r i b e d i n d e t a i l by Pople  75  76 , and i n t h e CNDO/2 a p p r o x i m a t i o n  t h e s e m a t r i x elements a r e g i v e n as  F  yy  = - h(i  + A ) +  y  y-  [(P L V  AA  ^(P*yy - 1 )\]AA Y 1 J  145.  and F  yv f o r valence  = ktf°. + 3°) ^ A  S  -hp  yv •  -^yv  Y»„, AB  J.  ,  (4.3.2)  y+v  '  ^  J  s h e l l a t o m i c ( S l a t e r ) o r b i t a l s cj) and <j)^ on atoms A and  respectively.  In the above f o r m u l a t i o n , one-  B,  and t w o - c e n t r e e l e c t r o n  r e p u l s i o n i n t e g r a l s are r e t a i n e d w h i l e p e n e t r a t i o n i n t e g r a l s are excluded.  The  non-zero two  electron integrals, y ^  and Y^g>  are  only  dependent upon the atoms w i t h which the o r b i t a l s (j) and <j>^ are a s s o c i a t e d and not upon the t y p e o f a t o m i c o r b i t a l .  Thus the a t o m i c i n t e g r a l s y^g  r e p r e s e n t an average i n t e r a c t i o n between an e l e c t r o n i n a v a l e n c e AO atom A and  another e l e c t r o n i n a valence  o r b i t a l on atom B, and  on  such 201  i n t e g r a l s are r e a d i l y e v a l u a t e d u s i n g f o r m u l a e d e r i v e d by Roothaan The p  charge d e n s i t i e s and bond-orders i n Eq. , are those p r e v i o u s l y d e f i n e d i n Eqs. i s the t o t a l v a l e n c e  P  (4.2.7) and  and  (4.2.8);  and  e l e c t r o n d e n s i t y f o r atom A:  AA jj.P =  W  '  ( 4  where Z. i s the core charge f o r atom A. A b  a t o m i c o r b i t a l s <j>  ( 4 . 3 . 2 ) , namely p  and 4>^ and may  S  yv  ' 3  3 )  i s the o v e r l a p i n t e g r a l f o r  be e v a l u a t e d u s i n g the a n a l y t i c ] 88  expressions  d e r i v e d by M u l l i k e n e t a l  for Slater orbitals.  The 76  s e m i - e m p i r i c a l b o n d i n g p a r a m e t e r s , 3^, f o r 2s- and 2p-  and the p a r a m e t e r s h(l^  atomic o r b i t a l s d e t e r m i n i n g  +  the atomic core  A )  Hamiltonian  75 m a t r i x elements T a b l e 4.10  as used i n the f o l l o w i n g c a l c u l a t i o n s are l i s t e d i n  a l o n g w i t h the S l a t e r exponents C .  A FORTRAN-IV computer  program s i m i l a r t o t h a t a v a i l a b l e from the Quantum C h e m i s t r y Program  Table 4.10 Parameters f o r CNDO/2 c a l c u l a t i o n s  H 1.20 h{i v  h(i  -3A  +  s p  A )  s  7.186  + A )  p  9.0  N  0  F  1.625  1.95  2.275  2.60  14.051  19.316  25.390  32.272  eV  5.572  7. 275  9.111  11.08  eV  39.0  eV  21.0  25.0  31.0  146.  Exchange has been developed  f o r use on an IBM 360 system and a l l r e s u l t s  r e p o r t e d here have been o b t a i n e d u s i n g t h i s program. I n i t i a l l y , t h e h i n d e r e d r o t a t i o n was c o n s i d e r e d as a s i m p l e 1^8 r o t a t i o n o f t h e p l a n a r NH  group w i t h f i x e d N-H bond l e n g t h and 2a = 119° "  2  so t h a t t h e t r a n s i t i o n s t a t e c o r r e s p o n d s ( s t r u c t u r e I I ) i n F i g . 4.20.  t o 6 = 0° and cb = 90°  The p l a n a r ground s t a t e i s t h e n d e f i n e d  by 6 = 0° and cf> = 0° ( s t r u c t u r e I) , and t h e a s s o c i a t e d SCF-LCAO-MO i s summarised i n T a b l e 4.11.  The n e t atomic  e l e c t r o n i c charge (a + IT)  f o r t h e N atom, q^, i s shown t o i n c r e a s e on f o r m i n g t h e t r a n s i t i o n while the corresponding  data  charge f o r t h e 0 atom, q^, d e c r e a s e s .  state;  These  v a r i a t i o n s a r e a l s o r e f l e c t e d i n t h e n e t atomic TT-charges which a r e deTT  OCC  TT  f i n e d as q. = Z. - 2 £ A A ^ TT A w i t h TT-core charge Z. A n  s t a t e ) and C.  C.  , where cb i s a 2P atomic o r b i t a l on atom u z ( o r a 2P o r b i t a l f o r N i n t h e t r a n s i t i o n y IU  i s the a s s o c i a t e d expansion  c o e f f i c i e n t f o r the i t h  1U TT  occupied orthonormal  LCAO m o l e c u l a r o r b i t a l . TT  i s approximately  T a b l e 4.11 shows t h a t q^  IT  c o n s t a n t w h i l e q^ and q^ d i f f e r w i d e l y i n t h e ground  and t r a n s i t i o n s t a t e s .  T h i s i s c o n s i s t e n t w i t h an i n c r e a s e d c o n t r i b u t i o n  from t h e resonance form N = C - 0 f o r t h e Tr-system i n t h e p l a n a r ground s t a t e , i n a s i m p l i f i e d d e s c r i p t i o n o f t h e amide g r o u p , c f . s e c t i o n 4.2. occ TT  The  N-C TT-bond o r d e r , g i v e n as p., _ = 2 N-C  E i  C. IU  C. w i t h <j) and <f> iv u v  2P  z  atomic o r b i t a l s on t h e N and C atoms, i s c a l c u l a t e d as 0.482 i n t h e grounds t a t e and hence r e p r e s e n t s a s i g n i f i c a n t d e l o c a l i s a t i o n o f t h e N TT-electrons. For and as 0.63 u s i n g using the Kurland In t h e t r a n s i t i o n  c o m p a r i s o n , t h i s b o n d - o r d e r i s d e t e r m i n e d as 0.41 203 an a l t e r n a t i v e e m p i r i c a l e q u a t i o n p r o p o s e d by 202 Gordy b o n d - l e n g t h d a t a i n a f o r m u l a g i v e n by P a u l i n g , s t a t e , c o n j u g a t i o n between t h e N P ^ - o r b i t a l and t h e  T a b l e 4.1:. CNDO/2 MO Data f o r Formamide StrUCture  TT q  N  I  q  C  0 = 0°, r  A ' O O O N  C-0.292) II  III -0.236 IV  %  q  r ~ n ' + 0 . 1 8 9  (+0.561)  (-0.477)  C  q  O  P  N-C  P  C-0  E  e  a) ,-. E  total  ,, m  +0.229  -0.418  0.482  0.845  -79.5883  -39.3033  5.92 (4.15)  +0.206  -0.263  0.272  0.931 -79.5031  -39.2718  2.96  +0.229-0.419  0.483  0.845-79.5730  -39.3040  3.95  +0.212  -0.283  0.282  0.924  -79.9207  -39.2S83  1.45  +0.213  -0.284  0.284  0.925  -79.6962  -39.2879  4.24  2a = 119°, <j) = 90°  +0.355 6 = 0°,  O  a) „  T T T f T T  2 a - 119°, cb = 0°  / n " r ? ? >  0 = 0% -0.313  q  TT  -0.262  +0.037  2a = 117.5°,<J> = 0°  +0.364  -0.335  +0.190  9 = 55°, 2a = 110°, • cb = 90° -0.260  V  +0.335  -0.273  6 = 55°, 2a = 110°, -0.249  +0.343  -0.261  -<j>. =-90° --  a)  E l e c t r o n i c o n l y energy, E , and t o t a l energy, E  b)  D i p o l e moment components :ln Debyes  s  , given  i n atomic u n i t s  147.  C  =  0  IT-system i s not p o s s i b l e and  decreases to  P^_Q  0 . 2 7 2  ( t h i s bond-  o r d e r r e p r e s e n t i n g an e l e c t r o n i c charge o v e r l a p between the c a r b o n y l TT-system and t h e N h y b r i d a-bonding  system) w i t h a c o r r e s p o n d i n g i n c r e a s e  TT  i n P ( - _ Q J c o n s i s t e n t w i t h an i n c r e a s e d c o n t r i b u t i o n from the form N -  C = 0 i n this state.  I n g e n e r a l , e l e c t r o n i c charge  resonance distribu204  t i o n s are s u c c e s s f u l l y p r e d i c t e d i n the i t i s of  CNDO/2  approximation  and  thus  i n t e r e s t t o compare the above g r o u n d - s t a t e d a t a w i t h t h a t 196  a v a i l a b l e from t h e ab i n i t i o • c a l c u l a t i o n given i n parentheses i n Table 4 . 1 1 . i n good agreement, a l t h o u g h the i n magnitude;  w i t h o u t d - f u n c t i o n s , as  The r e l a t i v e n e t a t o m i c charges a r e  CNDO/2  v a l u e s are c o n s i s t e n t l y  smaller  b u t i t i s t o be n o t e d t h a t , on i n c l u s i o n o f d - f u n c t i o n s  t h e r e a r e s i g n i f i c a n t changes and the ab i n i t i o q^ v a l u e i s g i v e n as - 0 . 5 8 4  w h i l e q^ and q^' become +  0 . 4 9 8  and  - 0 . 4 7 9 ,  respectively,  i l l u s t r a t i n g the s e n s i t i v i t y t o the chosen b a s i s s e t o f a t o m i c o r b i t a l s . The n e t c h a r g e s f o r the amino hydrogen C  atoms c i s and t r a n s t o the t  c a r b o n y l oxygen i n s t r u c t u r e I , q^ and q , a r e c a l c u l a t e d as + 0 . 1 2 4 ,  r e s p e c t i v e l y , w h i c h may  calculation: +  0.177  and  + 0 . 1 3 1  be compared w i t h t h o s e from the ab  + 0 . 1 6 0 .  and initio  Thus t h e s e charge d e n s i t i e s a r e s i g n i -  i i c a n t l y d i f f e r e n t i n the ground s t a t e and b o t h c a l c u l a t i o n s i n d i c a t e c . t t h a t q^ i s more p o s i t i v e t h a n q^. T h e r e f o r e , f o r an i s o l a t e d formamide m o l e c u l e , i n the absence  o f through-space  e f f e c t s from the m a g n e t i c a l l y  a n i s o t r o p i c C = 0 and C - H bonds the c i s p r o t o n i s l e s s s h i e l d e d hence would r e s o n a t e t o low f i e l d ment, assuming  o f the t r a n s p r o t o n i n an N M R e x p e r i -  the d i a m a g n e t i c c o n t r i b u t i o n t o the p r o t o n c h e m i c a l s h i f t  to be predominant.  In view o f the c u r r e n t i n t e r e s t i n t h e s e c h e m i c a l  205  shifts  and  this i s a particularly interesting  result.  148.  The nitude  complete CNDO/2 ground s t a t e d i p o l e moment has a mag-  | y | = 3.92 D and i s o r i e n t e d a t 3 = 41° w i t h  respect  to the  198 N - C bond, i n good agremement w i t h t h e e x p e r i m e n t a l d a t a  :  | y | = 3.71 + 0.06 D and 3 = 39.6°, whereas t h e ab i n i t i o c a l c u l a t i o n leads t o a higher  d i p o l e magnitude and an i n c r e a s e d  o r i e n t a t i o n angle.  I n t h e CNDO/2. a p p r o x i m a t i o n , d i p o l e i n t e g r a l s i n v o l v i n g t h e p r o d u c t o f two  a t o m i c o r b i t a l s on t h e same atom a r e r e t a i n e d i n t h e c a l c u l a t i o n o f 204  a molecular e l e c t r i c dipole charges a t n u c l e a r c r i b e d by y 1  centres  = 1.90 and y  x  . Thus t h e d i p o l e a s s o c i a t e d w i t h n e t f o r t h e ground s t a t e o f formamide i s des= 1.42 D, and t h e c o n t r i b u t i o n due t o  y  asymmetry o f e l e c t r o n i c charge about t h e s e c e n t r e s y^ = 1.15 D.  g i v e s y _ = -1.06 and  T h i s shows t h e i m p o r t a n c e o f i n c l u d i n g t h e asymmetry  c o n t r i b u t i o n , which i s e s s e n t i a l l y a h y b r i d i s a t i o n t e r m , i n t h e d i p o l e c a l c u l a t i o n f o r a p o l a r molecule. The  b a r r i e r t o h i n d e r e d r o t a t i o n i n formamide i s s i m p l y t h e  d i f f e r e n c e i n t o t a l energy ( i n c l u d i n g n u c l e a r ground and t r a n s i t i o n s t a t e s , t c a l c u l a t e d as 19.8 k c a l . mole  r e p u l s i o n s ) between t h e  - E , and f o r s t r u c t u r e s I and I I i s g'  . T h i s energy may be compared d i r e c t l y  w i t h t h e f r e e energy o f a c t i v a t i o n f o r t h i s r o t a t i o n , ^G , d e t e r m i n e d by a t o t a l l i n e s h a p e  a n a l y s i s o f NMR d a t a f o r d i l u t e s o l u t i o n s o f  if  N -formamide: 15  AG  = 1 8 . 0 + 0 . 4 k c a l . mole  (10 mole % i n acetone) a t  25°C, c f . s e c t i o n 4.1.5] and a l s o w i t h 19.9 k c a l . mole i n i t i o c a l c u l a t i o n * ^ and 20.1 k c a l . mole 9  CNDO c a l c u l a t i o n by Scheraga and coworkers energies derived  obtained 206  1  1  from t h e ab  i n an independent  . Of c o u r s e , t h e t o t a l  i n t h e CNDO/2 a p p r o x i m a t i o n a r e f a r removed from t h e  H a r t r e e - F o c k SCF l i m i t and hence, i n such a semi - e m p i r i c a l SCF  149.  c a l c u l a t i o n , i t must be assumed t h a t t h e i n h e r e n t e r r o r s i n v o l v e d i n an energy c a l c u l a t i o n c a n c e l o u t i n t h e energy d i f f e r e n c e t a k e n f o r any two  c o n f i g u r a t i o n s o f a given molecule.  Such an assumption appears t o  be v i n d i c a t e d i n t h e c l o s e agreement o f the t h e o r e t i c a l t  - E g  v a l u e s quoted above, i n t h a t a CNDO/2 t o t a l energy c o r r e s p o n d s t o o n l y about 25% o f t h a t i n v o l v e d i n an ab i n i t i o c a l c u l a t i o n . In g e n e r a l , g e o m e t r i c a l  changes i n a m o l e c u l e may o c c u r i n  the p r o c e s s o f i n t e r n a l r o t a t i o n ; and i n p a r t i c u l a r , a change i n h y b r i d i s a t i o n a t t h e N atom i n formamide may be e x p e c t e d t o have a s i g n i f i c a n t e f f e c t upon t h e b a r r i e r t o r o t a t i o n about t h e N-C bond. Such a change i n h y b r i d i s a t i o n would be a s s o c i a t e d w i t h t h e breakdown o f c o n j u g a t i o n between t h e f o r m a l N l o n e - p a i r u - e l e c t r o n s c a r b o n y l ir-system i n t h e t r a n s i t i o n s t a t e . changes were c o n s i d e r e d  by C h r i s t e n s e n  r e s t r i c t e d to a planar 9 = 0 ° group.  and t h e  Although d e t a i l e d 196  et a l .  geometrical  , a l l c a l c u l a t i o n s were  ( i n F i g . 4.20) c o n f i g u r a t i o n f o r t h e NH  2  A CNDO/2 energy m i n i m i s a t i o n , t h r o u g h a change i n bond a n g l e  o n l y , was checked by showing t h a t t h e minimum energy c o n f i g u r a t i o n f o r a p l a n a r ground s t a t e c o r r e s p o n d s t o 2a = 117.5° ( s t r u c t u r e I I I ) , which 198 i s i n good agreement w i t h t h e e x p e r i m e n t a l  v a l u e o f 119°  . The  SCF-LCAO-MO d a t a f o r t h i s s t r u c t u r e i s summarised i n T a b l e 4.11,  and i t  i s seen t h a t t h e t o t a l energy, E , i s lowered by o n l y 0.44 k c a l . mole  1  from t h a t f o r s t r u c t u r e I . The t r a n s i t i o n s t a t e geometry about t h e n i t r o g e n atom was t h e n c o n s i d e r e d  f o r <j) = 90° ( F i g . 4.20) i n terms o f v a r i a t i o n s  o f the d i h e d r a l angle 6 i n t h e range 25 - 75° and the HNH bond a n g l e 2a i n t h e range 100 - 120°. The c a l c u l a t e d t o t a l energy d i f f e r e n c e s , E  = E , a r e p l o t t e d i n F i g . 4.21,  where i t i s shown t h a t t h e minimum  o  6  18  l  o  2 16  *0  o  S  o •wo*  u  3 75  UJ 111  !  12  /  65  o  /  '  /  r /  f  /  / /  '  5  --o~~-  10 8  i  .  .o'  1  90  F i g . 4.21  100  2a  110  120  T o t a l energy d i f f e r e n c e s f o r the formamide p l a n a r ground s t a t e and v a r i a b l e geometry t r a n s i t i o n s t a t e  150.  energy c o n f i g u r a t i o n (structure  IV).  c l o s e l y c o r r e s p o n d s t o 6 = 55° and 2a = 110°  A l l calculated  e n e r g i e s a r e seen t o l i e on w e l l  d e f i n e d energy c u r v e s which a r e n o t , i n a about t h e minimum energy p o i n t . structure  i s 10.4 k c a l . mole  state considered (structure  1  however, n e c e s s a r i l y  symmetric  The t o t a l energy f o r  lower than that  this  for the i n i t i a l  I I ) , and t h e b a r r i e r t o r o t a t i o n  p o n d i n g t o t h e m i n i m a l energy m o l e c u l a r c o n f i g u r a t i o n s ,  transition corres-  as d e t e r m i n e d  i n t h e CNDO/2 a p p r o x i m a t i o n , i s now g i v e n as 9.86 k c a l . mole ,  This  J  e n e r g y i s a p p r o x i m a t e l y one h a l f o f t h e e x p e r i m e n t a l l y d e t e r m i n e d b a r r i e r , and t h e above c a l c u l a t i o n i l l u s t r a t e s t h e n e c e s s i t y f o r c a r e f u l geometrical optimisation  before comparison with experimental  data i s attempted. The  above c a l c u l a t i o n o f t h e b a r r i e r t o h i n d e r e d r o t a t i o n i n  formamide, and t h e d e s c r i p t i o n states., a p p l i e s  o f t h e a s s o c i a t e d ground and t r a n s i t i o n  t o an i s o l a t e d m o l e c u l e ; i n d e e d , t h e c h a r a c t e r i s t i c s  of the e l e c t r i c d i p o l e  moment f o r t h i s m o l e c u l e , as d e t e r m i n e d  m e n t a l l y from measurements i n t h e gas phase, have been r e p r o d u c e d by t h e a p p r o x i m a t e SCF-LCAO-MO model. available  f o r t h e i n t e r n a l r o t a t i o n energy,  satisfactorily  The e x p e r i m e n t a l d a t a  however, a r e o b t a i n e d from  measurements i n s o l u t i o n and i t i s e x p e c t e d t h a t s o l u t e - s o l u t e solute-solvent  interactions  for the polar  and  formamide m o l e c u l e w i l l  a complex e f f e c t upon t h e h i n d e r e d r o t a t i o n . only possible  experi-  have  Thus, i n g e n e r a l , i t i s  t o compare t h e o r e t i c a l and e x p e r i m e n t a l energy d a t a f o r  a series of substituted an e x t r a p o l a t i o n  amides a t t h e same c o n c e n t r a t i o n ( o r t h r o u g h  to i n f i n i t e d i l u t i o n ) i n a solvent that  minimises  151.  intermolecular interactions.  Such a c o r r e l a t i o n o f r e l a t i v e  w i l l be p r e s e n t e d i n d e t a i l e l s e w h e r e .  energies  At t h i s p o i n t , i t i s o f i n t e r e s t  to consider  f u r t h e r the c a l c u l a t e d d i p o l e moments f o r the p a r e n t amide--  formamide.  As  IV has  shown i n T a b l e 4.11,  a magnitude o f 1.45  l a r g e c o n t r i b u t i o n i s due charge d i s t r i b u t i o n  D and  the t o t a l d i p o l e moment f o r s t r u c t u r e  i s o r i e n t e d at 6° t o the N-C  bond.  A  t o the asymmetry o f the N l o n e - p a i r e l e c t r o n i c  ( a s s o c i a t e d w i t h an s p - t y p e h y b r i d o r b i t a l ) 3  i s i n o p p o s i t i o n t o t h a t from the c a r b o n y l  system and  and  thus l e a d s t o a  d e c r e a s e d t o t a l moment as compared w i t h t h a t f o r the p l a n a r ground s t a t e , s t r u c t u r e I (or I I I ) . Therefore,  i f the s o l u t e - s o l v e n t i n t e r a c t i o n i s  e l e c t r o s t a t i c i n form and n o n - s p e c i f i c , the s o l v a t i o n s t a b i l i s a t i o n energy i s d e c r e a s e d i n the t r a n s i t i o n s t a t e and hence the e n t h a l p y  of  a c t i v a t i o n AI-i' ( o r A r r h e n i u s  than  f  AG  #  corresponding  a c t i v a t i o n energy E ) w i l l be g r e a t e r a  t o a p o s i t i v e e n t r o p y o f a c t i v a t i o n AS  a c t i v a t i o n p a r a m e t e r s AH*  and AS*  are s e n s i t i v e t o  #  .  That i s , the  intermolecular  # i n t e r a c t i o n s , where AG i n t e r a c t i o n s due  i s p r e d o m i n a n t l y d e t e r m i n e d by  to an e f f e c t i v e compensation o f AH  thermodynamic r e l a t i o n s h i p AG*  = AH*  - TAS*.  intramolecular  and  So f a r , i t has  assumed t h a t the amino hydrogens are c i s t o the c a r b o n y l t r a n s i t i o n s t a t e , <£> = 90°  i n F i g . 4.20.  s i m i l a r to that already d e s c r i b e d , the c a r b o n y l  oxygen (ct = - 90°)  f i g u r a t i o n again  IV.  As shown i n T a b l e 4.11,  i n the been  oxygen i n the  A variational calculation,  f o r t h e s e hydrogen atoms t r a n s  to  shows t h a t the minimum energy con-  c o r r e s p o n d s t o 0 = 55°  the t o t a l energy i s o n l y 0.25  AS  k c a l . mole  and 1  2a = 110°  ( s t r u c t u r e V)  and  higher than that f o r s t r u c t u r e  a l l n e t a t o m i c c h a r g e s and  a r e v e r y s i m i l a r to those f o r s t r u c t u r e IV, but  bond-orders  the c a l c u l a t e d d i p o l e  152.  moment now has a magnitude N-C bond.  o f 4.24 D and i s o r i e n t e d  In t h i s c o n f i g u r a t i o n ,  i s increased  t h e s o l v a t i o n s t a b i l i s a t i o n energy  i n t h e t r a n s i t i o n s t a t e and i f t h e d i f f e r e n t i a l  energy exceeds 0.25 k c a l . mole the p r e f e r r e d  #  i s negative.  solvation  this p a r t i c u l a r configuration  1  t r a n s i t i o n state f o r a solvated  i n t h i s case A S solvent  a t 70° t o t h e  formamide  becomes  m o l e c u l e , and  A description of stereo-specific  i n t e r a c t i o n s may r e q u i r e  solute-  a more e l a b o r a t e c o m p u t a t i o n a l model,  but t h e r e l a t i v e ease w i t h which t h e CNDO/2 SCF a p p r o x i m a t i o n can be a p p l i e d may a l l o w  a r e l i a b l e and comprehensive d e s c r i p t i o n o f t h e form  of these i n t e r m o l e c u l a r  i n t e r a c t i o n s and t h e i r e f f e c t upon t h e  h i n d e r e d r o t a t i o n on formamide, which i s o f fundamental i m p o r t a n c e i n t h e s t u d y o f the more complex b i o l o g i c a l systems c o n t a i n i n g linkage.  t h e amide  However, i t remains t o be a b l e t o d e t e r m i n e t h e a c t i v a t i o n #  p a r a m e t e r s AH  #  and AS  with  s u f f i c i e n t p r e c i s i o n t o w a r r a n t such  d e t a i l e d c a l c u l a t i o n s f o r comparison w i t h e x p e r i m e n t a l d a t a . To a l l o w  a comparison w i t h  f o r the hindered r o t a t i o n s  the experimental data a v a i l a b l e  i n N,N-dimethyl  formamide  and N,N-dimethyl  carbamyl f l u o r i d e , c a l c u l a t i o n s i n t h e CNDO/2 a p p r o x i m a t i o n s i m i l a r t o t h o s e d e s c r i b e d above f o r formamide p a r e n t amide carbamyl f l u o r i d e .  have a l s o been c a r r i e d o u t f o r t h e  As t h i s p a r t i c u l a r amide i s u n s t a b l e a t  room temperature a s t r u c t u r e has n o t d e t e r m i n e d from the  microwave  145 spectrum  , and hence the b a s i c  structure  shown i n F i g . 4.22 has been 20  d e r i v e d by comparison o f the s t r u c t u r a l d a t a a v a i l a b l e f o r a c e t a l d e h y d e 208 198 acetyl fluoride and formamide . The NCF bond a n g l e o f 112.5° a l s o c o r r e s p o n d s t o a minimum energy c o n f i g u r a t i o n CNDO/2  approximation.  as d e t e r m i n e d i n t h e  F i g . 4.22  S t r u c t u r e f o r carbamyl f l u o r i d e used i n CNDO/2 SCF-LCAO-MO calculations  153.  The  minimum energy p l a n a r ground s t a t e c o n f i g u r a t i o n  ponds t o 2a = 118° Rotation state  and  the  o f the p l a n a r NH  (cb = 90°)  t o t a l energy i s g i v e n as -66.3134  b a r r i e r t o r o t a t i o n o f 13.2  structure  hence a  1  f o r a s i m i l a r r o t a t i o n i n formamide, c f . The  t o t a l e n e r g i e s f o r the m i n i m a l energy  c o r r e s p o n d i n g ' t o (j) = 90°  as -66.3046 and  and  k c a l s . mole , which i s s i g n i f i c a n t l y lower  I I i n T a b l e 4.11.  configurations  a.u.,  group t o form a h i n d e r e d r o t a t i o n t r a n s i t i o n  2  g i v e s a t o t a l energy o f -66.2922 a.u.  than that c a l c u l a t e d  corres-  -66.3062 a.u.,  and  cb = -90°  respectively.  are then  Thus the  calculated  t r a n s i t i o n state  i n which the amino hydrogen atoms e c l i p s e the  f l u o r i n e atom i n carbamyl  f l u o r i d e ( cb = -90°), f o r w h i c h 0 = 57.5° and  2a = 107.5°, i s more  s t a b l e t h a n the i s therefore  a l t e r n a t i v e t r a n s i t i o n s t a t e by  the  1.1  k c a l . mole  most p r o b a b l e t r a n s i t i o n s t a t e i n s o l u t i o n .  s t a b i l i s a t i o n of t h i s p a r t i c u l a r non-planar c o n f i g u r a t i o n associated  with  atom.  alternative t r a n s i t i o n state structure  The  by 6 = 60°  and  The  1 J  w h i c h i s t o be  fluorine  (cj) = 90°)  is  defined  b a r r i e r to r o t a t i o n , corresponding to  the minimum energy t r a n s i t i o n s t a t e , i s t h e n c a l c u l a t e d kcals.mole  The  i s probably  the non-bonding i n t e r a c t i o n s i n v o l v i n g the  2a = 107.5°.  and  1  compared w i t h  as AG  =5.12  t h a t g i v e n above f o r formamide:  # AG  =9.86 kcals.mole . l  groups i n s u b s t i t u t e d  Assuming t h a t the N,N-dimethyl and  amides a l l o w  mental d a t a f o r h i n d e r e d r o t a t i o n s  a d i r e c t c o m p a r i s o n o f the about the N-C  bond, the  o f a c t i v a t i o n f o r carbamyl f l u o r i d e i s e s t i m a t e d as 15.4 mole  The  CNDO/2 c a l c u l a t i o n s t h e r e f o r e  f o r t h e s e a c t i v a t i o n e n e r g i e s , but 1.9  amino  w h i l e the e x p e r i m e n t a l r a t i o i s 1.2.  free  + 0.8  p r e d i c t the c o r r e c t  the r a t i o o f p r e d i c t e d  experienergy kcals. order  energies i s  N e v e r t h e l e s s i t appears  that  154.  MO  c a l c u l a t i o n s a t the  CNDO/2  l e v e l o f approximation  may be v e r y u s e f u l  i n d e t a i l e d s e m i - q u a n t i t a t i v e d e s c r i p t i o n s o f the o v e r a l l bonding c h a r a c t e r i s t i c s f o r s u b s t i t u t e d amides, and i n c o r r e l a t i o n s w i t h e x p e r i m e n t a l l y determined d i f f e r e n t i a l energies. e l e c t r o n s are considered  As a l l v a l e n c e  and b o t h e l e c t r o n and n u c l e a r r e p u l s i o n s a r e  included i n the c a l c u l a t i o n o f molecular  e n e r g i e s , a l l o w i n g an e s t i m a -  t i o n o f non-bonding i n t e r a c t i o n e n e r g i e s , the  CNDO/2  c a l c u l a t i o n s have  more p h y s i c a l s i g n i f i c a n c e than those u s i n g t h e s i m p l e H u c k e l model p r e v i o u s l y d i s c u s s e d .  TT-MO  CHAPTER 5  FOURIER TRANSFORM APPLICATIONS  5.1  Basic  Formulation.  The d u a l i t y o f the p u l s e d and s t e a d y - s t a t e NMR expressed  methods i s  by a F o u r i e r t r a n s f o r m r e l a t i o n s h i p between t h e r e s p e c t i v e  r e s p o n s e f u n c t i o n s d e s c r i b i n g t h e n u c l e a r magnetic system under r e s o nance c o n d i t i o n s .  Indeed, w i t h i n t h e r e s t r i c t i o n t h a t a n u c l e a r s p i n  system may be t r e a t e d as a l i n e a r system, i t can be shown t h a t t h e f r e e i n d u c t i o n decay a s s o c i a t e d w i t h t h e t r a n v e r s e n u c l e a r magnetiz a t i o n as o b s e r v e d i n the p u l s e mode i s a l s o determined, d i r e c t l y as the i n v e r s e F o u r i e r t r a n s f o r m o f the u n s a t u r a t e d  s t e a d y - s t a t e spectrum  d e f i n e d by the same component o f n u c l e a r m a g n e t i z a t i o n .  I n 1954, Kubo  94 and T o m i t a  d e r i v e d an e x p r e s s i o n f o r t h e f r e q u e n c y  dependent mag-  n e t i c s u s c e p t i b i l i t y o f a g e n e r a l n u c l e a r s p i n system i n terms o f quantum s t a t i s t i c a l mechanics.  T h i s t h e o r y p r o v i d e s a g e n e r a l phys-  i c a l b a s i s f o r t h e F o u r i e r t r a n s f o r m c a l c u l a t i o n o f resonance  line-  shapes from an a u t o c o r r e l a t i o n f u n c t i o n ( o r r e l a x a t i o n f u n c t i o n ) f o r n u c l e a r m a g n e t i z a t i o n , and may be c o n s i d e r e d as a g e n e r a l i z a t i o n o f 40 the s t o c h a s t i c model d e v e l o p e d by Anderson and Weiss  .  Lowe and  67 Norberg NMR  demonstrated the above d u a l i t y e x p e r i m e n t a l l y t h r o u g h an 69  s t u d y i n the s o l i d s t a t e , and i n 1967, E r n s t and  Anderson  demonstrated the a p p l i c a t i o n o f the g e n e r a l F o u r i e r t r a n s f o r m concept t o h i g h r e s o l u t i o n NMR.  156.  A s i m p l e model, b a s e d upon s e m i - c l a s s i c a l concepts d i s c u s s e d i n t h i s t h e s i s , may  be d e v e l o p e d t o a l l o w a. c o n c i s e formu-  l a t i o n of F o u r i e r t r a n s f o r m methods as a p p l i e d t o g e n e r a l n u c l e a r s p i n systems. the n u m e r i c a l  systems i n c l u d i n g second-  A f i r s t - o r d e r spectrum may  be c o n s i d e r e d  o f n s p e c t r a l l i n e s c e n t r e d a t resonance f r e q u e n c i e s OK o v e r a l l l i n e s h a p e f u n c t i o n , F (co), i s e x p r e s s e d  F(co) = zVf. where A. I  first-order  Such a f o r m u l a t i o n forms an adequate b a s i s f o r  a n a l y s i s o f more c o m p l i c a t e d  order J-coupling.  already  i n terms  such t h a t  i n the form  (co),  (5.1.1)  i s an i n t e n s i t y n o r m a l i z a t i o n f a c t o r a s s o c i a t e d w i t h t h e i t h J  2 l i n e d e f i n e d by t h e L o r e n t z i a n f u n c t i o n f ^ (co) = c f . Eq.  the  (2.1.12).  2-1  [1 + T^(co - O L ) ]  TV.; i s the t o t a l t r a n s v e r s e r e l a x a t i o n t i m e ,  ,  and  f o r the i t h s p e c t r a l i i n e . As the normal NMR i t i s convenient  d e t e c t i o n schemes i n v o l v e r f phase s e n s i t i v e d e t e c t i o n , t o a g a i n d e f i n e an independent v a r i a b l e x by x = co - to  such t h a t the f u n c t i o n f. (x) i s d e f i n e d on the i n t e r v a l -°° < x <  00  in a  I  r o t a t i n g frame o f r e f e r e n c e X U)  In t h i s way co  O  Eq.  as  ,  (5.1.2)  the i t h l i n e p o s i t i o n r e l a t i v e to the r e f e r e n c e  i s g i v e n as Q. = co. - co , c o n s i s t e n t w i t h p r e v i o u s 1 1 o' r  frequency  d e f i n i t i o n s , cf.  (2.2.24). I n g e n e r a l , the i n v e r s e F o u r i e r t r a n s f o r m f ( t ) i s d e f i n e d i n  t h e t i m e domain by +00  f(t) = i _ J f ( ) x  exp(itx)dx  (5.1.3)  157.  The  t r a n s f o r m p a i r may be c o n s i d e r e d i n complex spaces and f ( t ) , i n  g e n e r a l complex, may then be r e a d i l y e v a l u a t e d u s i n g t h e c o n t o u r  inte-  g r a t i o n i n Appendix 1 as V  t}  =  2T~ 2i  E  X  P  [ _ (  Y  _ i f i  2i  i  ) t ]  t 1 = B. exp(- 7f.— ) [cosft. t + i s i n f t . t ] , B. = — 2i f o r O ^ t < °°.  (5.1.4)  2i  The o b s e r v a b l e f r e e i n d u c t i o n decay d e s c r i b e d by t h e '  f u n c t i o n f ( t ) i s n e c e s s a r i l y d e f i n e d by t h e r e a l p a r t o n l y . i n accordance w i t h Eq. ( 5 . 1 . 4 ) , f ^ ( t ) c o r r e s p o n d i n g  Therefore,  to a single  L o r e n t z i a n s p e c t r a l l i n e , f ^ ( x ) , c o n s i s t s o f an o s c i l l a t o r y decay w i t h a f u n d a m e n t a l f r e q u e n c y ft. and a c h a r a c t e r i s t i c time c o n s t a n t T„. . As 95 the F o u r i e r t r a n s f o r m a t i o n i s l i n e a r  , i t f o l l o w s from Eq. (5.1.4)  t h a t t h e o v e r a l l r e a l decay c o r r e s p o n d i n g d e s c r i b e d by S(t) =  Z B. cosft.t  t o F(x)  (Eq. (5.1.1)) i s  exp("f )  {  s  i  s  )  That i s , i n g e n e r a l t h e f u n c t i o n S ( t ) r e p r e s e n t s a modulated o s c i l l a t o r y decay w i t h t h e m o d u l a t i o n s |ft^ ±ft_.| .  d e t e r m i n e d by a l l f r e q u e n c i e s  The d e t a i l e d f o r m o f an observed  f r e e i n d u c t i o n decay  a l s o depends upon t h e r e l a t i v e phases o f t h e i r r a d i a t i n g and r e f e r ence r f m a g n e t i c f i e l d s i n v o l v e d i n a g i v e n phase s e n s i t i v e d e t e c t i o n scheme.  This aspect The  i s considered f u r t h e r at a l a t e r p o i n t .  normal a b s o r p t i o n mode s p e c t r u m i s d e f i n e d by F ( x ) .  d i s p e r s i o n mode spectrum i s then d e f i n e d by a c o r r e s p o n d i n g H(x)  where  A  function  158.  n H(x) = E A h . ( x ) i=l  =• ^kiJLjtt-Aj) c f . Eq. (5.1.1).  (5.1.6)  As shown i n Appendix 1, t h e i n v e r s e F o u r i e r t r a n s f o r m  o f h (x) i s g i v e n as h. ( t ) = i B.exp[-( ~~ - i«.)t] i i 2i  (5.1.7)  A complex l i n e s h a p e f u n c t i o n ' g ( x ) may now be d e f i n e d i n the f r e q u e n c y domain as  g(x) = f ( x ) - i h ( x ) , such t h a t t h e c o r r e s p o n d i n g  (5.1.8)  complex i n v e r s e F o u r i e r t r a n s f o r m i s g i v e n  from Eqs. (5.1.4) and (5.1.7) as  g(t) =  B exp(-|-') [cosftt + i s i n f t t ] .  (5.1.9)  2  C o n v e r s e l y , t h e F o u r i e r t r a n s f o r m o f g ( t ) , g ( x ) , i s g i v e n by CO  g(x) = J  g(t) exp(-itx)dt =  T  _ _ M _ _ _  o  ^  f o r g ( t ) d e f i n e d on t h e i n t e r v a l 0$t G(x)  (5.1.10)  < .  A complete s p e c t r a l f u n c t i o n  m  may now be c o n s i d e r e d i n t h e form 1  n  G(x) = Z A [ f (x) - i h (x)]= E A 1  1  1  i=l 1  ± 1  +VT  Z L  (5.1.11)  (/- A ) T  I  T h i s f u n c t i o n i s e q u i v a l e n t t o t h a t p r e v i o u s l y d e f i n e d i n terms o f a g e n e r a l complex t r a n s v e r s e n u c l e a r m a g n e t i z a t i o n , G_(x, cj)), i n t h e n o r mal r o t a t i n g frame o f r e f e r e n c e Ouvz, c f . Eq. (2.4.3) and F i g . 2.2. I n d e f i n i n g the n u c l e a r isochromat  component G_(x, <j>), the u - a x i s  corresponds  159.  to  t h e r e a l a x i s o f t h e complex p l a n e and hence t h e phase angle <j) i s  referred to this axis.  A l s o , i n a s t e a d y - s t a t e NMR e x p e r i m e n t , t h e  i r r a d i a t i n g f i e l d vector Eq.  i s assumed t o be a l o n g t h e u - a x i s , c f .  ( 2 . 1 . 9 ) , and t h u s t h e a b s o r p t i o n mode l i n e s h a p e f u n c t i o n V ( x )  d e s c r i b e s t h e v-component m a g n e t i z a t i o n . ever, the e f f e c t i v e f i e l d v e c t o r  I n p u l s e d mode NMR, how-  i s considered  t o be a l o n g t h e  v - a x i s o f t h e normal r o t a t i n g frame, such t h a t a t t i m e t = 0 ( f o l l o w i n g a TT/2 p u l s e , $ = TT/2 i n F i g . 2.1) a l l i s o c h r o m a t s i n the d i r e c t i o n o f t h e u - a x i s .  are aligned  This allows a c o n s i s t e n t d e s c r i p t i o n  o f i s o c h r o m a t d e p h a s i n g i n terms o f t h e phase angle 6 f o r t h e f r e e i n d u c t i o n decay as d e t e r m i n e d by t h e u-component Therefore,  magnetization.  i n t h e complex p l a n e Ouv, t h e f u n c t i o n G(x) i n Eq. (5.1.11)  i s t o be m o d i f i e d b y a phase f a c t o r e x p ( i T r / 2 ) , g i v i n g  G(x) =  I A . ^ X )  + if.(x)],  so t h a t i n a c c o r d a n c e w i t h Eq. (2.4.3) u^(x) A^f to  (x).  (5.1.12)  = A^hu (x) and V\ (x) =  A l s o , i t i s t o be n o t e d t h a t t h e r e s p o n s e s ( t ) c o r r e s p o n d s  an i r r a d i a t i n g r f f i e l d H_(t) d e f i n e d i n a f i x e d frame o f r e f e r e n c e  as H ( t ) = 2 H c o s ( w t + TT/2), c f . Eq. (2.1.5). 1  Thus i t i s seen t h a t  the a b s o r p t i o n mode l i n e s h a p e f u n c t i o n f ( x ) d e s c r i b i n g t h e v-component nuclear magnetization,  a s s o c i a t e d w i t h the imaginary  a x i s o f a complex  p l a n e i n t h e normal r o t a t i n g frame i s g i v e n as t h e r e a l p a r t o f t h e complex F o u r i e r t r a n s f o r m o f t h e f r e e i n d u c t i o n decay f u n c t i o n s ( t ) . I t i s now n e c e s s a r y  t o c o n s i d e r t h e p h y s i c a l consequences  o f t a k i n g the r e a l part o n l y , s ( t ) , o f the causal f u n c t i o n g(t) i n the t i m e domain, c f . Eq. (5.1.9).  I n accordance w i t h t h e g e n e r a l  t r a n s f o r m a t i o n g i v e n i n Eq. (5.1.10),  integral  the Fourier transform o f s ( t ) i s  160.  (5.1.12) with s(t) = ^  exp(- £ 2  )cosftt 2  and  (5.1.13)  —  g (x) +  u  T  —  z  ( % ?^2j  Thus i t i s shown f o r s ( t ) d e f i n e d by the s i n g l e f r e q u e n c y ft, t h a t F o u r i e r t r a n s f o r m a t i o n d e f i n e s two  l i n e s h a p e f u n c t i o n s c e n t r e d symmet-  r i c a l l y about the r e f e r e n c e f r e q u e n c y f _ ( x ) (as d e f i n e d i n Eq.  CO  q  (x = 0) .  That i s , f (x) +  (5.1.8)) d e t e r m i n e L o r e n t z i a n l i n e s  at x = ft and x = - f t , r e s p e c t i v e l y .  the  and  centred  T h i s i s an example o f the  Faltung  95 property mining NMR  o f a F o u r i e r t r a n s f o r m a t i o n and i s o f i m p o r t a n c e i n d e t e r -  the f r e q u e n c y  , i n a pulsed  experiment. The  Eq.  of the i r r a d i a t i n g r f f i e l d , co = w  F o u r i e r t r a n s f o r m o f g ( t ) i s g i v e n i n accordance w i t h  (5.1.10) as 00  g(x) =  ^ g ( t ) cos t x - i s i n t x dt o  and hence a c o s i n e F o u r i e r t r a n s f o r m i s d e f i n e d  S U) C  =  by  ^ g ( t ) c o s t x dt  (5.1.14)  o The  c o s i n e t r a n s f o r m o f the r e a l f u n c t i o n s ( t ) = ^ e x p ( - t / T 2 c f . Eq. ( 5 . 1 . 9 ) , i s r e a d i l y shown t o be  )cosftt,  161.  g (x) c  = [1 + T ( x + f t ) ] 2  2  [1  _ 1 +  +  T^Cx-ft) ]" 2  1  Comparison w i t h Eq. (5.1.13) shows t h a t t h i s e x p r e s s i o n  corresponds  to f (x) + f _ (x) .  That i s , i n g e n e r a l , t h e c o s i n e F o u r i e r t r a n s f o r m  o f s ( t ) determines  t h e a b s o r p t i o n mode l i n e s h a p e f u n c t i o n .  a s i n e t r a n s f o r m determines The  Similarly,  t h e d i s p e r s i o n mode f u n c t i o n , h (x) + h ( x ) .  complex F o u r i e r t r a n s f o r m o f s ( t ) d e f i n e s b o t h t h e ab-  s o r p t i o n and d i s p e r s i o n mode l i n e s h a p e f u n c t i o n s i n t h e f r e q u e n c y domain as t h e r e a l and i m a g i n a r y p a r t s o f t h e f u n c t i o n g ( x ) , r e s p e c t i v e l y . 96 For a g e n e r a l l i n e a r p h y s i c a l system, t h e Kramers-Kronig r e l a t e t h e r e a l and i m a g i n a r y  relations  p a r t s o f the frequency response f u n c t i o n s 97  g(x).  These r e l a t i o n s a r e e q u i v a l e n t t o a H i l b e r t t r a n s f o r m p a i r  in  t h a t t h e r e a l and i m a g i n a r y p a r t s , f ( x ) and h ( x ) , a r e r e l a t e d b v -+00  iu) - A  +00  V*')  , I  W  )  „ __1  tl+'iAt'  (5.1.15)  — 00 — OO where t h e i m p r o p e r i n t e g r a l s a r e c o n s i d e r e d as p r i n c i p a l v a l u e s . These 97 t r a n s f o r m s may be compared w i t h t h o s e d i s c u s s e d by Abragam . To a v o i d the improper i n t e g r a l s i n Eq. ( 5 . 1 . 1 5 ) , t h e H i l b e r t t r a n s f o r m h ( x ) may 98 be c o n s i d e r e d i n t h e form  OO  L + ' < L * o  dt  (5.1.16)  o  F o r the L o r e n t z i a n l i n e s h a p e f u n c t i o n f ( x ' ) g i v e n i n Eq. ( 5 . 1 . 2 ) , i t follows that 00  This i n t e g r a l represents a sine Fourier transform f o r a causal f u n c t i o n ,  162.  and hence h ( x ) =  at)  [l+T^t<-tft.f  ]  + T (* ^} [ 1_ + x  +  U+^y*-] .  Thus t h e H i l b e r t t r a n s f o r m o f f ( x ) determines the d i s p e r s i o n mode l i n e s h a p e  f u n c t i o n s h .(x) as g i v e n i n Eq. (5.1.13). ±  96 Now b y d e f i n i t i o n the  , h ( x ) i n Eq. (5.1.15) i s a l s o g i v e n by  convolution h(x) = - e ( x ) * f ( x )  (5.1.17)  where e ( x ) = (TTX) * and * denotes t h e p r i n c i p a l v a l u e o f a c o n v o l u t i o n integral.  T h i s c o n v o l u t i o n i s r e a d i l y e v a l u a t e d i n d i r e c t l y by u s i n g  F o u r i e r t r a n s f o r m a t i o n s , i n t h a t i t i s w e l l known t h a t a c o n v o l u t i o n i n the f r e q u e n c y  domain c o r r e s p o n d s t o a s i m p l e m u l t i p l i c a t i o n i n t h e time  domain  h(t) = - e(t) f ( t ) . The i n v e r s e F o u r i e r t r a n s f o r m e ( t ) i s g i v e n b y a c o n t o u r  i n t e g r a t i o n as  e(t) = -i, t > 0  (5.1.18)  and hence i t f o l l o w s t h a t h ( t ) = i v - ^ i — exp - (^ 2"  - ift)t } which defines  2  the d i s p e r s i o n mode f u n c t i o n g i v e n i n Eq. (5.1.6). shown t h a t f ( x ) = e ( x ) * h ( x ) .  I ti s similarly  W i t h t h e advent o f f a s t n u m e r i c a l  Fourier  t r a n s f o r m a l g o r i t h m s , i t i s p o s s i b l e t o e x t r a c t a c c u r a t e a b s o r p t i o n and 99 d i s p e r s i o n mode l i n e s h a p e s from a s t e a d y - s t a t e spectrum r f phase c h a r a c t e r i s t i c s .  This phasing  with  f e a t u r e i s o f course  arbitrary  an i n h e r e n t  p r o p e r t y o f p u l s e d mode NMR d a t a i n t h e form o f t h e f r e e i n d u c t i o n decay f o l l o w i n g a TT/2 p u l s e .  163.  5.2  X  Resonance C o n d i t i o n .  In p u l s e d mode NMR, the i r r a d i a t i n g r f m a g n e t i c f i e l d , H_ , i s applied for a f i n i t e width t  w  time o n l y , i n t h e form o f a p u l s e o f a m p l i t u d e  w i t h a frequency  to = to rads s e c . .  A and  The f r e e i n d u c t i o n decay  - 1  a s s o c i a t e d w i t h a component t r a n s v e r s e n u c l e a r m a g n e t i z a t i o n may then be observed  i n t h e absence o f an r f magnetic f i e l d .  Such a p u l s e may be  c o n s i d e r e d as one o f a sequence w i t h a r e p e t i t i o n p e r i o d T , and may be d e f i n e d i n a f i x e d frame o f r e f e r e n c e by  P(t) ^  P(t)  = Acosto t o  0 ^ t < t w  =0  t < t iT . w  (5.2.1) J  may now be c o n s i d e r e d i n terms o f a complex F o u r i e r s e r i e s T "°° ito i f c (to)e w i t h t h e F o u r i e r c o e f f i c i e n t s c (to) = — j P ( t ) e  itot 'dt.  f  P (t) =  i a >  n  1 U  n  F o r to ^ to , i t i s shown t h a t f o r t h e fundamental f r e q u e n c y  A I such t h a t the f r e q u e n c y  siviCwui^fw).  1  to^ =  ^  2TT/T  (5.2.2)  spectrum a s s o c i a t e d w i t h t h e p u l s e c o n s i s t s o f  d i s c r e t e f r e q u e n c i e s to d e f i n e d i n terms o f to_ as n f to = to + nto,., n o f The  n = ±1, ± 2 , ••• .  frequencies corresponding  to as the upper s i d e b a n d  v  (5.2.3) J  t o n = 1, 2 , ••• a r e c o n v e n i e n t l y r e f e r r e d  s p e c t r u m , and t h o s e c o r r e s p o n d i n g  to n = -1, - 2 ,  1  ••• t h e n form t h e lower s i d e b a n d spectrum.  The power c o r r e s p o n d i n g t o 2  each component f r e q u e n c y i s g i v e n b y c (to) , maximum power a t to = O J  Q  2 being  (At /2T) .  Thus r f power i n p u l s e d mode NMR i s d i s t r i b u t e d o v e r  a s p e c i f i c f r e q u e n c y range  (symmetric about C O ) w h i c h may i n c l u d e a Q  l a r g e range o f resonance f r e q u e n c i e s  . T h i s f r e q u e n c y range i s d e t e r  mined by t h e p u l s e w i d t h o n l y , and from Eq. (5.2.2) i t i s seen t h a t t h e first  zero c o e f f i c i e n t i s d e f i n e d by t h e c o n d i t i o n naut = TT, t h a t i s . ' f w  to = 2ir/t .  F o r a 5usec. p u l s e , t h i s c o r r e s p o n d s t o a f r e q u e n c y d i s t r i -  b u t i o n about W  q  o f ± 200 kHz and t h e r f power may be c o n s i d e r e d t o be  c o n s t a n t o v e r a range ± 2 kHz.  I n g e n e r a l , i f w (Hz) i s t h e f r e q u e n c y  w i d t h o f a resonance spectrum t o be i r r a d i a t e d , t h e p u l s e w i d t h i s def i n e d as t  - 0.01 w  1  sec.  I n a d d i t i o n , t h e o v e r a l l r e s o l u t i o n and  w  '  r e p r o d u c i b i l i t y o f t h e resonance spectrum o b t a i n e d through a F o u r i e r t r a n s f o r m a t i o n o f t h e r e s p o n s e f u n c t i o n s ( t ) f o r a n u c l e a r s p i n system f o l l o w i n g an r f p u l s e a r e d e t e r m i n e d by t h e s p a c i n g o f t h e d i s c r e t e i r r a d i a t i n g f r e q u e n c i e s to , t h a t i s , T  -  1  HZ.  The form o f t h e above p u l s e f r e q u e n c y d i s t r i b u t i o n d e t e r m i n e s the  c e n t r e f r e q u e n c y to t o be used i n a g i v e n p u l s e d NMR experiment.  As an r f phase s e n s i t i v e d e t e c t o r i s n o r m a l l y r e f e r e n c e d t o to , t h e resonance  spectrum  d e f i n e d i n terms o f 'the independent f r e q u e n c y  variable x i s considered f o r 0 a resonance f r e q u e n c y  x <  0 0  only.  For a spectral l i n e with  > to , t h e d e t e c t e d component f r e e  decay s i g n a l may be assumed t o have t h e form s (t)  = B. exp (-t/T„.) cosft. t 1  w i t h £h = ah - W the  q  .  induction  £k  I n a c c o r d a n c e w i t h Eq. ( 5 . 1 . 1 3 ) , t h e r e a l p a r t o f  F o u r i e r t r a n s f o r m o f s ( t ) d e f i n e s L o r e n t z i a n l i n e s d e s c r i b e d by  l i n e s h a p e f u n c t i o n s g i v e n as  1  165.  f. (x)  =  1 +  1 +  £  (x-ftj  = _  ( x )  .  2  ± 1 + T, (x+ft.) 2 i 2  2  (5.2.4)  On the i n t e r v a l 0 < x < °°, the c o n t r i b u t i o n s from f. (x) and f. fx.) a t 2 x = ft^ are 1 and  1 + 4T  2-1 ft^ , r e s p e c t i v e l y . Thus a g e n e r a l c o n d i t i o n  f o r n e g l i g i b l e d i s t o r t i o n o f the l i n e s h a p e f ^ f. i s T„ft. >> l 2 I  1.  The  +  d i s t o r t i o n a l e f f e c t s due  f ^ _ a r e shown i n F i g . 5.1  due  to a c o n t r i b u t i o n from  to the t r a n s f o r m f u n c t i o n  i s to be n o t e d t h a t a 0.25%  c o n t r i b u t i o n a t x = ft. from f . l . i to a freouenc.Y s h i f t ft. = LOT.* rad- sec.." . An a d d i t i o n a l 1  1  d i s t o r t i o n may  O^T^'  f o r v a r y i n g v a l u e s o f the parameter  corresponds 1  lineshane  2  a r i s e through  d e t e c t o r r f r e f e r e n c e phase  adjustment.  T h i s l e a d s t o the r e a l p a r t o f the F o u r i e r t r a n s f o r m g(x) b e i n g a mixt u r e o f a b s o r p t i o n and d i s p e r s i o n mode f u n c t i o n s , and t h e e f f e c t o f a dispersion function lu  (analogous  to f ^ _ ) i s more severe i n t h a t the 2 2 -1 c o n t r i b u t i o n at x = ft. i s 2T„ft. 1 + 4T^ft. . In t h i s c a s e , a 0.25% l 2 l 2 l c o n t r i b u t i o n from h. corresponds lr  t o fti = 2 0 0 T * . l 2  T h i s form o f d i s -  t o r t i o n , however, i s e l i m i n a t e d by a phase c o r r e c t i o n as d e s c r i b e d i n the f o l l o w i n g s e c t i o n .  The  centre frequency  ui^ may  be d e c r e a s e d  i n c r e a s e the s h i f t from r e s o n a n c e , ft., and hence d e c r e a s e l  the t r a n s -  f o r m o v e r l a p d i s t o r t i o n f o r a g i v e n s p e c t r a l l i n e ; but i n so the f r e q u e n c y  doing,  a s s o c i a t e d w i t h the r e s p o n s e s ( t ) i s i n c r e a s e d .  m e n t a l l y , the f r e e i n d u c t i o n decay s i g n a l i s d i g i t i z e d u s i n g a technique  and the maximum f r e q u e n c y  component t h a t may  to  Experisampling  be a n a l y z e d w i t h  n.'T i  2  = o  50  100  400  F i g u r e 5.1  P u l s e d mode NMR  resonance c o n d i t i o n s  166.  N samples o v e r t h e p u l s e i n t e i ' v a l x i s N/2x ^9*100^ x > N / 2 T are a p p l i e d to the sampling the frequency  higher  frequencies  d e v i c e , they a r e down-converted i n t o  range 0 < x < N / 2 T and a r e then e f f e c t i v e as t h e lower  f r e q u e n c i e s x' = |x - n N / x | ,  1 0 1  w i t h n = 1, 2,  Following a Fourier  t r a n s f o r m a t i o n o f s (t) t h e s e f r e q u e n c i e s may g i v e r i s e t o s p u r i o u s t r a l l i n e s on t h e i n t e r v a l 0 < x < N/2x. t a t i v e l i n e s h a p e from t h e f u n c t i o n s ( t ) , CO  spec-  In order to o b t a i n a quantii t i s required i n general that  be s e t w i t h r e s p e c t t o a l l resonance f r e q u e n c i e s uh such t h a t t h e  q  f r e q u e n c i e s ft^ a r e m i n i m i z e d . o v e r l a p d i s t o r t i o n must be  Simultaneously,  minimized  the transform f u n c t i o n  a c c o r d i n g t o t h e above c o n d i t i o n s  on t h e p a r a m e t e r Q/T^. I n a g e n e r a l p u l s e d mode NMR e x p e r i m e n t ,  the t o t a l  spectrum  to be o b t a i n e d as a F o u r i e r t r a n s f o r m o f S ( t ) . c f . Eq. ( 5 . 1 . 5 ) . must to one s i d e o f t h e c e n t r e f r e q u e n c y  CO . q  lip.  That i s , f o r x > 0 i t i s r e -  q u i r e d t h a t a l l resonance f r e q u e n c i e s co^ be such t h a t ah > to (ft^ > 0 ) . In  t h i s manner o n l y t h e upper s i d e b a n d  frequency  spectrum a s s o c i a t e d  w i t h t h e r f p u l s e induces resonance e f f e c t s i n a g i v e n n u c l e a r s p i n system.  The r e a s o n  f o r t h i s g e n e r a l resonance c o n d i t i o n i s shown by  c o n s i d e r i n g two resonance f r e q u e n c i e s oo. and oo. such t h a t to. < co < co. . i l 1 o 1 The a c t u a l s e p a r a t i o n o f s p e c t r a l l i n e s i s ' to. - co., o r ft. + I f t . I where ft. = co. - to < 0. 3 J o  The r e s p o n s e S ( t ) , however, i s d e f i n e d i n terms o f r  v  the f r e q u e n c i e s ft^ and |ft^ | and hence t h e F o u r i e r t r a n s f o r m shows specl i n e s c e n t r e d a t x = ft^ and x = |ft_. j w i t h a f a l s e s e p a r a t i o n  tral 1  ft.  l  -  1  ft.II 11  j »  and i n t h e p a r t i c u l a r case f o r which j  r e c t frequency  r  ordering.  1  1  Ift.I  i  > ft. an i n c o r -  The r e s p o n s e S (t) b e i n g d e f i n e d i n terms o f  the f r e q u e n c i e s ft^ and [ft.. | may be c o n s i d e r e d t o c o r r e s p o n d  to the  167.  resonance e f f e c t s i n d u c e d by b o t h t h e upper and lower s i d e b a n d f r e q u e n c y s p e c t r a a s s o c i a t e d w i t h the r f p u l s e . I n q u a n t i t a t i v e l i n e s h a p e s t u d i e s , once t h e g e n e r a l form o f the NMR  spectrum has been d e t e r m i n e d , i t i s o f t e n p o s s i b l e t o s e t t h e  c e n t r e f r e q u e n c y t o m i n i m i z e the o f f - r e s o n a n c e s h i f t £L ular spectral l i n e of i n t e r e s t .  In t h i s way,  f o r the p a r t i c -  t h e l o w e s t f r e q u e n c y com-  ponents i n S ( t ) d e f i n e the l i n e s h a p e f u n c t i o n f (x) and t h e s e are the components most r e l i a b l y a n a l y z e d f o r any g i v e n number o f sampled d a t a p o i n t s , N.  168.  5.3  F i n i t e Complex F o u r i e r T r a n s f o r m .  Experimentally,  a f u n c t i o n s ( t ) i n the time domain may  be  d e t e r m i n e d f o r a f i n i t e number o f d a t a p o i n t s o n l y and hence the complex Fourier transform  f u n c t i o n g(x) i s d e t e r m i n e d f o r an  number o f d a t a p o i n t s . cuss a b a s i c f i n i t e  Thus, at t h i s p o i n t , i t i s n e c e s s a r y to  complex F o u r i e r  0  T and may  t. £ T, be  i t may  be  considered  f i n i t e o n l y on the i n -  t o be p e r i o d i c w i t h a p e r i o d  e x p r e s s e d i n terms o f a g e n e r a l  f i n i t e F o u r i e r s e r i e s as  s(t) = E A exp(ix t)  where  (5.3.1)  i s a complex c o e f f i c i e n t  ponent w i t h f r e q u e n c y x . n  The  corresponding  t o the harmonic com-  fundamental f r e q u e n c y a s s o c i a t e d  the p e r i o d i c f u n c t i o n s ( t ) i s x^, d e f i n e d as x^ = The x  f i n i t e number o f harmonic f r e q u e n c i e s = n-2-rr/T, and  from Eq.  dis-  transformation.  I f the f u n c t i o n s ( t ) i s e s s e n t i a l l y terval  equivalent  2TT/T  rad.  x^ are t h e n g i v e n  with  sec. . - 1  by  (5.3.1) i t f o l l o w s t h a t  s(t) = I A exp(i^pt).  (5.3.2)  Thus i t i s seen t h a t the h i g h e s t harmonic o f the fundamental f r e q u e n c y x^ c o n s i d e r e d  i n the f i n i t e s e r i e s a p p r o x i m a t i o n t o a F o u r i e r a n a l y s i s  i s d e f i n e d by N.  I t may  p o i n t s on the i n t e r v a l k  now  0  t  d a t a p o i n t i s t ^ = kT/P,  i e n t s A^  are now  g i v e n by  be assumed t h a t s ( t ) i s sampled by P d a t a T so t h a t t h e time c o r r e s p o n d i n g where k = 0, 1,  • • •, P - l .  a f i n i t e Fourier transform  The  o f the  to  the  coeffic-  form  P-J.  A  =  —  ^  S Jj_  (-c  _EL^.n)  (5.3.3)  169  for  P even, where  i s the f u n c t i o n s ( t ) v a l u e at t ^ .  s h i p i s v e r i f i e d by c o n s i d e r i n g  This  an i n t e g r a l f o r m o f Eq.  relation-  (5.3.1):  o  The  f i n i t e complex i n v e r s e F o u r i e r t r a n s f o r m o f  i s g i v e n i n a c c o r d a n c e w i t h Eq. tt  d e f i n e d i n Eq.  (5.3.2) as  2iik  s, = E.A exp ( i - n - n ) . k n^j n  (5.3.3)  (5.3.4)  p  I t i s t o be n o t e d t h a t the c o e f f i c i e n t A -n  i s g i v e n as A* n  and hence an  b  e x p l i c i t c a l c u l a t i o n f o r n < 0 i s r e d u n d a n t and s^ i s e q u i v a l e n t l y defined  as s  *•-• 2-rrk = E A exp(i-p-n)  k  (5.3.5)  n  *\= o  for  N even.  The  c o e f f i c i e n t s A^ d e f i n e t h e f i n i t e complex F o u r i e r  transform f u n c t i o n corresponding Eq.  t o g ( x ) as p r e v i o u s l y c o n s i d e r e d , c f .  ( 5 . 1 . 8 ) , i n terms o f f u n c t i o n v a l u e s g ( x ) a t t h e d i s c r e t e f r e -  quencies  x^ s e p a r a t e d by the fundamental f r e q u e n c y  x^.  That i s , the  complex l i n e s h a p e f u n c t i o n g ( x ) i s d e f i n e d on t h e normal f r e q u e n c y i n terval 0 ^ x <  r  a  b y g ( x ) n b  The expressed  -  =  A. n  F o u r i e r c o e f f i c i e n t A^ as g i v e n i n Eq.  i n t h e form  (5.3.3) may  be  170.  . A n  1 — p  =  ,2'rrkn. )] p  „ r ,2Trkn. . • . S. I c o s f ) - 1 sm k p  £  L  k = ( }  (5.3.6)  and hence the r e a l p a r t o f t h i s complex c o e f f i c i e n t , a a  1  =  -  .  P  n  P^  „ .  1  E k=0  ,2iTkn,  S| cos  (  )  .  as  (5 3  7)  •  P  Thus i t i s seen t h a t a  , i s given n'  i s d e t e r m i n e d as the f i n i t e F o u r i e r c o s i n e  trans-  n form o f s. , and  this  c o e f f i c i e n t i s equivalent  f u n c t i o n g (x) i s d e f i n e d i n Eq. form may  a l s o be  considered  (5.1.4).  to g (x) where the  A f i n i t e Fourier sine  to d e t e r m i n e the c o e f f i c i e n t b  general trans-  as the  ima-  n ginary part of A .  A s i m p l e c o r r e l a t i o n between t h e F o u r i e r  ents d e f i n i n g s^ i n Eq.  (5.3.5) and  m e t r i c F o u r i e r s e r i e s i s shown hv  coeffici-  those i n a standard f i n i t e  r.onsi Heri nc  s_  i n the  trigono-  form  k  .  s  =  a  +  r  ,2Trkn  .  ,  .  .  ,2Trkn  .-,  l a cos ( n) + b sin ( n) J k 0 i n P n p n=l 2P„* „ ,2TTkn. , , 2 P* „ . ,2-rrkn. where a = - E S cos - — ) and b = E S sm - — ) . i t then f o l l o w s n P k p n P k p A  L  v  c  k = 0  that A  n  = %(a n v  k = Q  - i b ) and A = % a , showing t h a t a n o o n 6  l e n t to the f u n c t i o n s f ( x ) and h ( x ) shape f u n c t i o n g ( x ) , i n t h a t a In g e n e r a l  d e f i n i n g the g e n e r a l  = f ( x ) and b  = h(x  are  equivan  complex  then a c c o r d i n g  line-  ).  s t r i c t l y f i n i t e s p e c t r a l bandwidth.  s i g n a l s ( t ) o f d u r a t i o n T may  b a n d w i d t h IV and  and b n  t e r m s , i t i s not p o s s i b l e to have a s i g n a l  f i n i t e time d u r a t i o n and a given  J J  x  be assumed t o have an  of  However, effective  to the s a m p l i n g theorem i n the time  do-  main**^, t h i s s i g n a l f u n c t i o n i s d e t e r m i n e d f o r a l l 0 ^ t ^ T t o a good a p p r o x i m a t i o n by 2W  * sec.  This  i t s values  a t 2T1V >>  1 sampling points separated  by  i m p l i e s t h a t the f u n c t i o n s ( t ) i s w e l l - d e f i n e d i f the  171.  sampling frequency f  i s such t h a t 2W < f  g  Hz.  The number o f sampled .  d a t a p o i n t s , P, o b t a i n e d o v e r the time i n t e r v a l T i s d e t e r m i n e d by the number o f memory l o c a t i o n s a v a i l a b l e f o r t h e d i g i t a l A-D c o n v e r s i o n o f the sampled analog s i g n a l d a t a . f r e q u e n c y u s u a l l y corresponds  d a t a f o l l o w i n g an  The maximum s a m p l i n g  t o t h e minimum t i m e , t , r e q u i r e d i n t h e  A-D c o n v e r s i o n p r o c e s s g i v i n g d a t a t o a s p e c i f i e d a c c u r a c y bits),  (number o f  as the s a m p l i n g d e v i c e may o p e r a t e o v e r a time i n t e r v a l o f 1  usee, o r l e s s .  The s a m p l i n g f r e q u e n c y may be e f f e c t i v e l y i n c r e a s e d , 107  however, by u s i n g a sample-and-hold  device or "boxcar" i n t e g r a t o r  determine d a t a p o i n t s a t a time s e p a r a t i o n t  < t .  to  The s a m p l i n g t h e -  orem i n t h e f r e q u e n c y d o m a i n * ^ s t a t e s t h a t t h e f r e q u e n c y  spectrum  c o r r e s p o n d i n g t o a s i g n a l s (t) o f d u r a t i o n T s e c . i s c o m p l e t e l y d e t e r mined by a m p l i t u d e v a l u e s at a s e r i e s o f p o i n t s s e p a r a t e d by l / T H z T h i s s e p a r a t i o n f r e q u e n c y i s j u s t t h e fundamental  ;  f r e q u e n c y x^, and  hence the s a m p l i n g theorem p r o v i d e s a b a s i s f o r u s i n g t h e f i n i t e F o u r i e r s e r i e s g i v e n i n Eq. (5.3.2) t o d e f i n e t h e f u n c t i o n s ( t ) , where t h e coefficients  ( a m p l i t u d e s ) A^ a r e d e t e r m i n e d o n l y f o r t h e d i s c r e t e  f r e q u e n c i e s x^ = nx^.  I n a d d i t i o n , t h e s a m p l i n g i n t e r v a l i n t h e time  domain has been determined  as 2W * s e c . and t h e maximum f r e q u e n c y x > 0  i n c l u d e d i n t h e f i n i t e F o u r i e r s e r i e s g i v e n i n Eq. (5.3.2) i s N/T Hz. Thus f o r P »  1 i t f o l l o w s t h a t N i s d e t e r m i n e d as N = P/2.  That i s ,  the maximum f r e q u e n c y t h a t may be a n a l y z e d f o r t h e sampled f u n c t i o n s(t)  as d e f i n e d by P d a t a p o i n t s i s t h e lowered f r e q u e n c y f  The f r e q u e n c y range o f i n t e r e s t becomes 0  = P/2T Hz.  x £ P/2T H z , as i n a c c o r -  dance w i t h t h e s a m p l i n g theorem i t i s i n v a l i d t o c a l c u l a t e A^ f o r n > P/2 u s i n g Eq. (5.3.5).  The h i g h e s t f r e q u e n c y t h a t may be a n a l y z e d  172.  i s t h e r e f o r e d e t e r m i n e d d i r e c t l y by the s a m p l i n g f r e q u e n c y f , now.def i n e d as f s  = 2f . m  I n p u l s e d mode NMR  using Fourier transform  techniques,  the  s i g n a l f u n c t i o n s ( t ) i s tire f r e e i n d u c t i o n decay f o l l o w i n g a Tr/2-pulse and T i s e q u i v a l e n t t o the p u l s e r e p e t i t i o n p e r i o d T , as d i s c u s s e d  in  the p r e c e d i n g  be  section.  A Fourier transform  f r e q u e n c y spectrum may  o b t a i n e d by n u m e r i c a l l y e v a l u a t i n g the complex a m p l i t u d e s A^ as g i v e n i n Eq.  (5.3.3).  directly,  T h i s i s a time-consuming p r o c e s s i n t h a t P  o p e r a t i o n s , each o f w h i c h may  be  considered  as a complex m u l t i p l i c a t i o n 108  f o l l o w e d by an a d d i t i o n , are r e q u i r e d f o r each A .  C o o l e y and Tukey  ,  however, have d e v e l o p e d an a l g o r i t h m f o r a b i n a r y a r i t h m e t i c c a l c u l a t i o n that allows a very s i g n i f i c a n t r e d u c t i o n i n computational ital  conrouter.  The  t i m e on a d i g -  a l g o r i t h m renlar.es an a r r a v o f comnlex numbers o f  l e n g t h P = 2 , r an i n t e g e r , by i t s complex F o u r i e r t r a n s f o r m .  The 108  t o t a l t i m e i n v o l v e d c o r r e s p o n d s to l e s s than 2P£og2P o p e r a t i o n s as d e f i n e d above, w i t h o u t for  r e q u i r i n g more d a t a s t o r a g e  the i n p u t d a t a a r r a y .  The  l a r g e s t a r r a y t h a t may  depends upon the computer memory: mum  i s P -- 8192.  ,  than i s r e q u i r e d be  transformed  f o r a 32,768 word memory, t h e maxi-  I t i s t o be n o t e d t h a t f o r the f i n i t e  t r a n s f o r m p a i r as d e s c r i b e d by Eqs.  (5.3.3) and  Fourier  (5.3.5), t h i s  algor-  i t h m g e n e r a t e s a number o f l i n e a r l y independent s p e c t r a l a m p l i t u d e s equal  to the number o f i n p u t d a t a p o i n t s .  However, i t i s r e a d i l y  shown t h a t f o r r e a l i n p u t d a t a , the a l g o r i t h m o u t p u t d a t a i s symmetric about n = P/2  i n that A  = A* n  p o i n t s d e f i n e d by n = 0, 1,  .  T h i s i s c o n s i s t e n t w i t h the  data  P-n • • •, P/2  spectrum o v e r the f r e q u e n c y range o f  determining interest.  the  transform  173.  A FORTRAN computer program LGTRN has been w r i t t e n and checked f o r general Fourier transform  a p p l i c a t i o n s i n NMR..  t h e IBM SHARE l i b r a r y s u b r o u t i n e  PK FORT t o c a r r y o u t t h e b a s i c F o u r i e r  t r a n s f o r m a t i o n u s i n g t h e Cooley-Tukey a l g o r i t h m . normalized,  transformed  and t h e n s e p a r a t e d  and d i s p e r s i o n mode s p e c t r a l d a t a s e t s . processed  T h i s program uses  The r e a l i n p u t d a t a i s  into re-normalized  absorption  These d a t a s e t s may be f u r t h e r  u s i n g r e a l a r i t h m e t i c t o o b t a i n phase c o r r e c t e d s p e c t r a , en69  hanced r e s o l u t i o n and f i l t e r e d s p e c t r a  w i t h improved s i g n a l - t o - n o i s e  r a t i o s , as d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n .  Full plotting  options  a r e a v a i l a b l e f o r a CALCOMP p l o t t e r and s p e c t r a l d a t a may be p r e s e n t e d o v e r any s p e c i f i e d f r e q u e n c y  r a n g e , w i t h i n t h e l i m i t s 0 and P/2T Hz.  The b a s i c o u t p u t d a t a p o i n t s e p a r a t i o n i s d e f i n e d as l / T Hz, and hence a s u b r o u t i n e SBHARM has been d e v e l o p e d t o a l l o w t h e c a l c u l a t i o n o f sub-harmonic d a t a p o i n t s .  This c o n s i s t s o f extending  the e f f e c t i v e  time i n t e r v a l f o r t h e i n p u t data and hence t h e number o f d a t a p o i n t s P, r e a r r a n g i n g t h e r e a l i n p u t d a t a as the two p a r t s o f a complex a r r a y , t r a n s f o r m i n g t h i s a r r a y and t h e n a s s e m b l i n g s p e c i f i c frequency  range.  t h e s p e c t r a l data over a  I n t h i s manner, a c c u r a t e F o u r i e r  transform  s p e c t r a may be o b t a i n e d f o r sub-harmonic m u l t i p l i c i t i e s o f two, f o u r and  e i g h t on a computer w i t h a 32K memory.  the s a m p l i n g  I n h i g h r e s o l u t i o n NMR,  i n t e r v a l T, as n o r m a l l y d e t e r m i n e d by t h e s p i n - s p i n r e -  l a x a t i o n time 1 , may be 2 - 5 s e e s , and hence t h e fundamental quency i s 0.5 - 0.2 Hz.  Data p o i n t s a t t h i s f r e q u e n c y  r e q u i r e d t o define a c c u r a t e l y frequency  fre-  separation are  p o s i t i o n s and i n t e n s i t i e s f o r  s p e c t r a l l i n e s w i t h f u l l - w i d t h s a t half-maximum o f t h e o r d e r o f 0.5 Hz. T h i s i s n o t p o s s i b l e , i n g e n e r a l , and s i g n i f i c a n t d i s t o r t i o n s a r e  174.  apparent i n the r e s u l t a n t F o u r i e r transform s p e c t r a .  By u s i n g a sub-  m u l t i p l i c i t y o f e i g h t , however, t h e e f f e c t i v e data p o i n t s e p a r a t i o n i s r e d u c e d t o 0.06 and 0.025 Hz f o r T = 2 and 5 s e c , r e s p e c t i v e l y , and an a c c u r a t e r e p r o d u c t i o n o f t h e u n s a t u r a t e d is obtained.  s t e a d y - s t a t e NMR spectrum  The program LGTRN has been d e v e l o p e d f o r maximum e f f i c i -  ency and t h e c o m p u t a t i o n a l  (CPU) time f o r 1024 i n p u t d a t a p o i n t s and  an 8192 p o i n t F o u r i e r t r a n s f o r m a t i o n , c o r r e s p o n d i n g multiplicity  t o a sub-harmonic •  o f e i g h t , w i t h phase c o r r e c t i o n s and t h e s e t t i n g up o f  f u l l p l o t t i n g data arrays i s t y p i c a l l y  l e s s than 20 s e c . on an IBM  360/67. The  o v e r a l l accuracy  t r a n s f o r m a t i o n i s most c l e a r l y a given off-resonance  o f the numerical  complex f i n i t e  Fourier  shown f o r a s i n g l e s p e c t r a l l i n e w i t h  s h i f t ft, c f . Eq. (5.2.4).  The f r e e i n d u c t i o n  decay as d e f i n e d by s ( t ) i n Eq. (5.1.13) may be c o n s i d e r e d on t h e interval and  0 ^ t 4  = 0.1 sec.  0.64 sec. f o r a L o r e n t z i a n system w i t h ft = 100 Hz The n u m e r i c a l  F o u r i e r t r a n s f o r m a b s o r p t i o n and  d i s p e r s i o n mode d a t a a r e r e p r e s e n t e d by c r o s s e s i n F i g . 5.2a, and the f u l l  l i n e s h a p e s as c a l c u l a t e d i n accordance w i t h Eq. (5.1.13) a r e  shown as f u l l 0.39  lines.  The n u m e r i c a l  d a t a p o i n t s a r e s e p a r a t e d by  Hz, as d e f i n e d by a sub-harmonic m u l t i p l i c i t y o f f o u r , t h e  full-line-width  a t half-maximum b e i n g 3.2 Hz.  I t i s seen t h a t t h e  d e v i a t i o n from t h e e x a c t a b s o r p t i o n mode l i n e s h a p e f o r t h i s p o i n t transformation i s minimal,  t h e d e v i a t i o n b e i n g o n l y 1% and  4% a t f r e q u e n c i e s o f 103.1 and 105.47 Hz, r e s p e c t i v e l y . tional  4096  check on t h e o v e r a l l a c c u r a c y  An a d d i -  o f the F o u r i e r t r a n s f o r m a t i o n  109 is  a f f o r d e d by P a r s e v a l ' s theorem  , i n t h a t f o r an e x a c t t r a n s -  f(x)  Figure  5.2(a)  F i n i t e Fourier transform characteristics for a L o r e n t z i a n l i n e s h a p e system.  h(x)  F i g u r e 5.2(b)  F i n i t e Fourier transform c h a r a c t e r i s t i c s f o r a G a u s s i a n lineshape. system.  175.  2 2 £ Is, I = P E |A I . k n  formation  1  1  1  1  The d i f f e r e n c y between t h e s e two J  f a c t o r s was c a l c u l a t e d t o be l e s s than 2% f o r t h e above l i n e s h a p e function. The  corresponding  f o r a Gaussian  system a r e shown i n F i g . 5.2b. The a b s o r p t i o n and  d i s p e r s i o n mode G a u s s i a n Eqs.  l i n e s h a p e s and n u m e r i c a l d a t a p o i n t s  l i n e s h a p e s , f ( x ) and h ( x ) , a r e g i v e n i n  (5.4.20) and (5.4.21), r e s p e c t i v e l y . The a s s o c i a t e d r e s p o n s e  f u n c t i o n , s ( t ) , i s taken as t h e r e a l p a r t o f t h e f u n c t i o n d e f i n e d i n Eq.  (5.4.22) and i s a l s o shown i n F i g . 5.2b. The G a u s s i a n  absorption  mode l i n e s h a p e f u n c t i o n has c h a r a c t e r i s t i c s d i s t i n c t from those o f the L o r e n t z i a n f u n c t i o n , and a g a i n the d e v i a t i o n from t h e e x a c t  line-  shape i s shown t o be m i n i m a l . The  response s ( t ) f o r a L o r e n t z i a n system may be c o n s i d e r e d  i n t h e form  s(t). = Bexp(-t/T )cosftt, 2  c f . Eq. ( 5 . 1 . 5 ) , and has been assumed above t o be f i n i t e o n l y on t h e i n t e r v a l 0 6 t & T.  I t now remains t o c o n s i d e r t h e e f f e c t o f t h e t r u n -  c a t i o n o f t h i s f u n c t i o n as compared w i t h t h e normal d e f i n i t i o n on t h e interval 0 ^ t  < °°.  The l i n e s h a p e f u n c t i o n f ( x ) i s g i v e n as t h e  F o u r i e r c o s i n e t r a n s f o r m o f s ( t ) , c f . Eq. (5.1.14), and hence T  f ( x ) = B [ e x p ( - t / T ) c o s f t t cos x t d t . 2  o The  f u n c t i o n f (x) c e n t r e d a t x = ft i s then d e f i n e d by T  f ( x ) = | J" e x p ( - t / T ) c o s [ ( f t - x ) t ] d t , +  2  o  (5.3.8)  176.  A 1+  T ^ U L )  2  -  (5.3.9) where A i s a n o r m a l i z a t i o n c o n s t a n t . limit T  T h i s e q u a t i o n shows t h a t i n t h e  0 ° , t h e f u n c t i o n f (x) d e s c r i b e s a normal L o r e n t z i a n a b s o r p t i o n  mode l i n e s h a p e on t h e i n t e r v a l 0 ^ t < °°. F o r a f i n i t e v a l u e o f T, howe v e r , t h i s l i n e s h a p e i s m o d i f i e d by o s c i l l a t o r y terms w i t h an o v e r a l l amplitude f a c t o r  (0, - x) exp (-T/T ) . 2  F o r * - Q, t h e e x p o n e n t i a l i s  dominant and d e f i n e s an a m p l i t u d e o f < 1% o f t h e normal L o r e n t z i a n f u n c t i o n maximum f o r T > 7T^.  I n a d d i t i o n , t h e c o s i n e and s i n e terms  i n Eq. (5.3.9) tend t o c a n c e l f o r a l l x and hence t h e o v e r a l l o f t r u n c a t i o n i s e x p e c t e d t o be n e g l i g i b l e f o r t h e T l i m i t above.  effect  defined  T h i s i s shown t o be t h e case f o r the L o r e n t z i a n a b s o r p t i o n  mode l i n e s h a p e f ( x ) c o n s i d e r e d i n F i g . 5.2a, f o r which T = 6.4T .  177.  5.4  Phase C o r r e c t i o n s .  In p u l s e d mode NMR  the f r e e i n d u c t i o n decay s i g n a l i s d e t e r -  mined by the t r a n s v e r s e n u c l e a r m a g n e t i z a t i o n o f a complex f u n c t i o n S ( t ) , c f . Eq.  and i s d e s c r i b e d i n terms  (5.1.4), which i s d e f i n e d f o r a  g e n e r a l f i r s t - o r d e r s p i n system by^  S(t) = exp(-t/T ) 2  I B [cosft t + i s i n f l t ] i  (5.4.1)  i  i n the normal r o t a t i n g frame o f r e f e r e n c e .  i s an a m p l i t u d e f a c t o r  f o r the component decay w i t h the c h a r a c t e r i s t i c f r e q u e n c y c f . Eq.  (5.1.2).  The  terms o f isochromat  t r a n s v e r s e m a g n e t i z a t i o n may  £L = ah  - co , o  be c o n s i d e r e d i n  components G_(x, cb) i n the r o t a t i n g frame, as shown  i n F i g . 2.2, w i t h time-dependent phase a n g l e s cb d e f i n e d w i t h r e s p e c t t o t h e r e a l u - a x i s o f the complex t r a n s v e r s e p l a n e .  A l l isochromat  phase i n f o r m a t i o n i s r e t a i n e d e x p e r i m e n t a l l y by u s i n g an r f phase sens i t i v e d e t e c t i o n scheme, i n w h i c h the s i g n a l S ( t ) i s mixed w i t h a r e f erence s i g n a l S  and i n t e g r a t e d o v e r a time i n t e r v a l t > I/Aw,  Aw i s t h e r f a m p l i f i e r b a n d w i d t h . m u l t i p l i c a t i o n w h i c h may  This mixing  corresponds  be r e p r e s e n t e d i n g e n e r a l  S ( t ) = Re S ( t ) - S Q  r  where  to a s i m p l e  as  ,  (5.4.2)  where S ( t ) i s t h e r e a l o u t p u t s i g n a l and the r e f e r e n c e s i g n a l i s def i n e d i n complex form i n a f i x e d frame o f r e f e r e n c e as  S  r  = b-exp -i(u> t + 4>) o  r  That i s , t h i s s i g n a l has a f r e q u e n c y frequency  •  (5.4.3)  e q u a l t o the r f p u l s e  and an a s s o c i a t e d phase angle cb  which may  be  irradiating  adjusted  178.  a r b i t r a r i l y and i s d e f i n e d w i t h r e s p e c t t o the f i e l d H ( t ) d e f i n e d by |HjJexp -i(co t + TT/2), c f . Eq.  ( 2 . 1 . 5 ) , c o n s i s t e n t w i t h the  Q  assumption  t h a t the f i e l d v e c t o r Hj_ l i e s a l o n g the v - a x i s o f the r o t a t i n g as p r e v i o u s l y d i s c u s s e d .  The  phase s e n s i t i v e d e t e c t o r e f f e c t i v e l y  f r e q u e n c i e s t o co = to  references a l l output  0  dent r e f e r e n c e f i e l d v e c t o r shown i n F i g . 2.2  output  may  be  (x = 0) .  The  time-indepen-  r e p r e s e n t e d i n the r o t a t i n g frame  such t h a t S  The  frame,  r  = b exp(i<j> ).  (5.4.4)  r  s i g n a l f o r a g e n e r a l complex response S ( t ) i s g i v e n i n  accordance w i t h Eqs. S ( t ) = Re{b o l  = be  (5.1.4), e  -t/T  -t/T  (5.4.2) as  B.(cosft.t + i sinft.t) i. I I J  B.  cosft.t cost!) l r  coscb r  + i sind) } r  r  - sinfi.t s i n * l r  r  I  F o r a component decay i n Eq. takes the  (5.4.1) and  r  .  (5.4.5)  ( 5 . 4 . 5 ) , the F o u r i e r t r a n s f o r m o f s ( t )  form  g  '  W  =  1 + iT (x+ft) 2  +  1  +  Ccos* -i sin(}) )  iT Cx-0)2  r  r  Ccos* i r +  sin* )' r  (5.4.6) T l i e r e f o r e , on the f r e q u e n c y  i n t e r v a l 0 x < °°, the complex  f u n c t i o n c e n t r e d at x = Q, i s g i v e n i n accordance w i t h Eq.  t  F o r s i m p l i c i t y , i t has been assumed t h a t T ^ 2  lines.  = T2  lineshape (5.1.8) as  for a l l spectral  179.  (5.4.7)  Thus i t i s shown t h a t t h e r e a l p a r t o f t h e complex F o u r i e r f u n c t i o n , f ' ( x ) , i n general  transform  corresponds to a l i n e a r combination o f  L o r e n t z i a n a b s o r p t i o n and d i s p e r s i o n mode l i n e s h a p e f u n c t i o n s f ( x ) and h(x)  as d e f i n e d i n Eqs. (5.1.2) and ( 5 . 1 . 6 ) , r e s p e c t i v e l y .  For the  p a r t i c u l a r case i n w h i c h t h e r e f e r e n c e f i e l d v e c t o r i s i n t h e d i r e c t i o n o f t h e u - a x i s o f t h e r o t a t i n g r e f e r e n c e frame, <t> • = 0 and from Eqs. (5.4.5) and (5.4.7) i t f o l l o w s t h a t  S ( t ) = b e x p ( - t / T _ ) E B.cosft.t O  2  1  ;  1  1  v  and G< (x) = b £ A. [ f . (x) - i h ( x ) ] , I  x  consistent with the expression derived-previously  (Eq. (5.1.11)) under  the assumption t h a t t h e f r e e i n d u c t i o n decay was d e t e r m i n e d by u-component n u c l e a r  magnetization.  The a b s o r p t i o n mode l i n e s h a p e , f ( x ) , i s o b t a i n e d from Eq. (5.4.7) as  f ( x ) = f ' (x)cos(J)  Both f ( x ) and h'(x) 1  + h'(x)sincb  r  .  are a v a i l a b l e from a n u m e r i c a l  (5.4.8)  complex F o u r i e r  t r a n s f o r m a t i o n o f d i g i t i z e d d a t a i n t h e time domain and hence t h e phase  180.  a n g l e tf> may be a d j u s t e d t o d e f i n e f ( x ) t o an a c c u r a c y d e t e r m i n e d by the c r i t e r i o n used t o d e f i n e t h e c o r r e c t a b s o r p t i o n mode l i n e s h a p e . B a s e l i n e e q u a l i z a t i o n over s p e c i f i c frequency to  ranges has been shown  g i v e p a r t i c u l a r l y e f f i c i e n t n u m e r i c a l phase c o r r e c t i o n .  Ernst  69 and Anderson  have used an a r e a r a t i o c r i t e r i o n f o r t h i s r f phase  adj ustment. I t has been n o t e d t h a t t h e h i g h e s t f r e q u e n c y  contained i n  the r e s p o n s e S ( t ) t h a t can be a n a l y z e d u s i n g a s a m p l i n g  technique i s  x = N / 2 T where N i s the number o f samples i n t h e time domain and x i s the r f pulse r e p e t i t i o n p e r i o d .  Higher s i g n a l frequencies are  down-converted and g i v e r i s e t o s p u r i o u s s p e c t r a l l i n e s as d e t e r m i n e d by a F o u r i e r t r a n s f o r m a t i o n , and h i g h f r e q u e n c y  n o i s e s i m i l a r l y down-  converted  leads t o a degradation o f s i g n a l - t o - n o i s e r a t i o i n the  frequency  range o f i n t e r e s t , 0 < x < N / 2 T .  the output  T O minimize  these  from t h e r f phase s e n s i t i v e d e t e c t o r , s ( t ) , i s f o l l o w e d b y  a low pass f i l t e r t o a t t e n u a t e t h e f r e q u e n c i e s x > N/2x. output  effects,  s i g n a l s ^ ( t ) i s then f e d t o a s a m p l i n g  d e v i c e and an A-D con-  v e r t e r t o o b t a i n t h e f i n a l d a t a i n d i g i t a l form. ever, introduces a frequency  The f i l t e r  T h i s f i l t e r , how-  dependent phase s h i f t which  t h e d e t e r m i n a t i o n o f a c o r r e c t a b s o r p t i o n mode l i n e s h a p e .  complicates The e f f e c t  o f t h e f i l t e r i n t h e time domain i s d e s c r i b e d by a c o n v o l u t i o n r e l a 102 tionship  and hence i t i s more c o n v e n i e n t  to define the Fourier  t r a n s f o r m f u n c t i o n s s f x ) and s ^ f x ) and t o c o n s i d e r t h e e f f e c t o f t h e o f f i l t e r i n t h e f r e q u e n c y domain. I n t h i s manner  — 00  181.  where T(x) i s the complex t r a n s f e r f u n c t i o n d e f i n i n g the f i l t e r a c t e r i s t i c s , such t h a t s^Cx)  T(x)s  =  (x).  char-  This t r a n s f e r f u n c t i o n  may  be c o n s i d e r e d i n the form  T(x) = D(x)exp i 9 ( x )  so t h a t D(x)  determines an a t t e n u a t i o n and  i s t i c frequency sampling  (5.4.9)  dependent phase s h i f t .  8(x) d e f i n e s a c h a r a c t e r -  I f i t i s assumed t h a t  d e v i c e does not i n t r o d u c e a f u r t h e r f r e q u e n c y  dependent  a t t e n u a t i o n o r phase s h i f t , the F o u r i e r t r a n s f o r m o f the s i g n a l may  be  c o n s i d e r e d e q u i v a l e n t t o g j (x) .  dance w i t h Eqs.  (5.4.7) and Eq.  s (x) f  = f»(x)  + i h  1  the  filtered  Therefore, i n accor-  (5.4.9), s^(x) i s given  by  (x)  = D(x)[f(x) - i h(x)][cos(6(x)'+  <f>) r  + i sin(6(x) +  c^)]  (5.4.10) and  the a b s o r p t i o n mode f u n c t i o n f ( x ) i s now f(x)  g i v e n i n g e n e r a l form as  = D ( x ) { [ f ' ( x ) c o s 6 ( x ) + h' ( x ) s i n 0 ( x ) ] cos<J> _ 1  - [ f • ( x ) s i n O ( x ) - h' ( x ) c o s 6 ( x ) ] s i n c f ^ } .  (5.4.11) 102  The  t r a n s f e r f u n c t i o n f o r a s i n g l e s e c t i o n low-pass RC f i l t e r  is  H U t L l  (5.4.12)  and hence D(x)  = [ l + (xRC)  ]  , c o s 8 ( x ) = D(x) and s i n 0 ( x ) = -xRCD(x).  For t h i s p a r t i c u l a r f i l t e r , f ( x ) i s independent o f D(x)  and  from  182.  Eq.  (5.4.11) i t f o l l o w s t h a t  f(x)  = [f»(x)cos<j>  + h'(x)sin<J> ] + xRC [ f (x)sin<|> 1  - h (x) c o s c f j . 1  ' (5.4.13) This expression mode l i n e s h a p e  allows a simple  evaluation o f the corrected  i n terms o f t h e complex F o u r i e r t r a n s f o r m  and h ' ( x ) , t h e r e f e r e n c e  absorption  functions f ( x ) 1  r f phase a n g l e cj> and t h e f i l t e r parameter RC.  T h i s l a t t e r p a r a m e t e r may be e m p i r i c a l l y v a r i e d o v e r a s m a l l range t o o b t a i n an improved l i n e s h a p e  from n u m e r i c a l  t a i n e d experimentally using a standard low-pass f i l t e r .  Fourier transform  data ob-  sampling device i n c o r p o r a t i n g a  The e f f e c t o f f i l t e r and r f r e f e r e n c e p h a s i n g i s i l -  l u s t r a t e d i n F i g . 5.3 f o r a r e p r e s e n t a t i v e L o r e n t z i a n l i n e s h a p e by ft = 100 Hz and T The o f f - r e s o n a n c e  = 0.1 s e c . (3.2 Hz f u l l - w i d t h a t half-maximum).  s h i f t o f 100 Hz ensures t h a t t r a n s f o r m o v e r l a p  t o r t i o n , as d i s c u s s e d i n s e c t i o n 5.2 i s m i n i m a l . time constant 160 Hz.  defined  i s 0.001 s e c , c o r r e s p o n d i n g  The r e a l and i m a g i n a r y  dis-  The s p e c i f i c RC f i l t e r  to a c u t - o f f f r e q u e n c y o f  parts o f the F o u r i e r transform s^(x)  are shown i n F i g . 5.3a, t h e f r e e i n d u c t i o n decay f u n c t i o n s ( t ) b e i n g d e f i n e d by 1024 d a t a p o i n t s o v e r a t i m e i n t e r v a l o f 0.64 s e c . a computational  sub-harmonic m u l t i p l i c i t y o f f o u r t h e d a t a  With  points  shown i n t h e f r e q u e n c y domain have a s e p a r a t i o n o f 0.39 Hz, t h e c o r r e s p o n d i n g fundamental f r e q u e n c y b e i n g  1.56 Hz.  The f i l t e r  frequency  dependent phase c o r r e c t i o n determines t h e r e a l f u n c t i o n f ' ( x ) shown i n F i g . 5.3b, and an a d d i t i o n a l f r e q u e n c y independent r f r e f e r e n c e . phase c o r r e c t i o n o f - 2 0  determines t h e f i n a l L o r e n t z i a n  absorption  mode l i n e s h a p e shown i n F i g . 5.3c. The s i n g l e s e c t i o n RC f i l t e r an a t t e n u a t i o n o f -3dB a t t h e c u t o f f f r e q u e n c y x  gives  = (2TrRC)~~ Hz and  F i g u r e 5 .3  F i l t e r and r f r e f e r e n c e phase c o r r e c t i o n s f o r a L o r e n t z i a n l i n e s h a p e system.  183.  -40 dB at x  £ n  = lOOx . c  An i n c r e a s e d a t t e n u a t i o n o f -80 dB a t t h e f r e -  quency x^ i s o b t a i n e d by u s i n g a t w o - s e c t i o n RC f i l t e r , each s e c t i o n b e i n g matched and i s o l a t e d , w i t h a t r a n s f e r f u n c t i o n  2-1 2 where w = xRC and D(x) = (1 + w ) , c o s 0 ( x ) = (1 - w ) D ( x ) and s i n O ( x ) = -2wD(x). in  this  Again  f (x)  i s shown t o be independent o f D(x) and  case  f(x)  = (1 - w ) [ f ' ( x ) c o s c t + h'(x)sin<J> 2  r  +• 2w[f' (x)sin<J>  ]  - h» (x)cos<J>  (5.4.15)  In g e n e r a l , e x p e r i m e n t a l l y , i t i s not p o s s i b l e to o b t a i n the r e s p o n s e s (t)  f o r t -»- 0 because o f t h e f i n i t e r e c e i v e r r e c o v e r y  time  f o l l o w i n g a Tr/2-pulse d u r i n g which t h e f r e e i n d u c t i o n decay s i g n a l i s not o b s e r v a b l e ,  and hence t h e f u n c t i o n s ( t ) i s t o be c o n s i d e r e d on t h e 103  interval t  Q  ^ t < °°.  s h i f t i n t h e frequency  The t i m e s h i f t  t  > 0 c o r r e s p o n d s t o a phase  domain which may be c o n s i d e r e d i n terms o f t h e  F o u r i e r t r a n s f o r m o f a m o d i f i e d r e s p o n s e p ( t ' ) d e f i n e d , on t h e i n t e r v a l 0  t  1  < °°, t o be e q u i v a l e n t t o s (t) s (x) = [ m  for t  p(V) e^,(-i*i') i f '  ^ t < °°, t h a t i s , ,  i'-i-t  0  0  c>Cj>(ut ) I s(f) t*j> [ - U t ) 0  2  ev-p {Ui ) 0  SU)  dt (5.4.16)  where s ( x ) i s t h e F o u r i e r t r a n s f o r m o f s ( t ) as d e f i n e d on the normal i n t e r v a l 0 ^ t < °°.  By a g a i n c o n s i d e r i n g t h e F o u r i e r t r a n s f o r m f u n c t i o n  184.  s  ( ) as e q u i v a l e n t t o g j ( x ) i n Eq. ( 5 . 4 . 7 ) , i t f o l l o w s t h a t f ( x ) i s x  m  g i v e n i n terms o f t h e f r e q u e n c y dependent phase s h i f t x t  f(x) = f'(x)cosxt  Q  by  + h'(x)sinxt .  0  (5.4.17)  0  I t i s t o be n o t e d t h a t a l l phase s h i f t s i n v o l v e d i n t h e e v a l u a t i o n o f phase c o r r e c t e d l i n e s h a p e s may now be combined as ^  = $  r  + S(x) + x t  so t h a t  g'(x) = f • ( x ) + i ' h ' ( x )  f (x)  = f (x)cosci)  + h' (x)sin<t>  h(x)  = f (x)sin<J>  - h'(x)cos<j>  and  The  d e f i n i t i o n  I  o f  th  ma.v  be  e x t e n d e d  t o  .  i n c l u d e  (5.4.18)  a n v  a d d i t i o n a l  nha.se  factors. In l i q u i d s under h i g h r e s o l u t i o n c o n d i t i o n s t h e r e l a x a t i o n times T^ and T^ a r e o f t h e o r d e r o f seconds and t h e normal Tr/2-pulse w i d t h s used a r e 10 - 100 usee. . An average r e c e i v e r r e c o v e r y time i s 10 usee., and hence t h e phase f a c t o r x t 1 kHz f o r *H NMR i s n e g l i g i b l e .  over a frequency  range o f  F o r C NMR i n v o l v i n g c h e m i c a l 1 3  shifts  (and hence s p e c t r a l w i d t h s ) o f the o r d e r o f 10 kHz, however, t h i s f a c t o r may n o t be n e g l e c t e d .  I n s o l i d s w i t h much s h o r t e r r e l a x a t i o n  t i m e s , t h i s phase f a c t o r becomes v e r y s i g n i f i c a n t as t h e s p e c t r a l f r e q u e n c y w i d t h i s extended t h e above approximate when t  Q  by up t o t h r e e o r d e r s o f magnitude.  l i n e a r phase c o r r e c t i o n may not be adequate  becomes s i g n i f i c a n t w i t h r e s p e c t t o  t o t a l s i g n a l sampling  Also,  time.  which d e f i n e s the  I n t h e l a t t e r case i t i s p o s s i b l e t o use  Q  185.  a Tr/2-Trpulse sequence w i t h a p u l s e s e p a r a t i o n T < the s p i n echo c e n t r e d a t t = 2x.  and t o sample  The F o u r i e r t r a n s f o r m o f s ( t ' ) i s  o b t a i n e d f o r O ^ t ' < , as t ' = 0 now c o r r e s p o n d s t o t = 2 T , and i t OT  i s w e l l known t h a t t h e s p i n echo s i g n a l on t h i s i n t e r v a l i s e q u i v a l e n t to t h e normal f r e e i n d u c t i o n decay. The phase e f f e c t c o r r e s p o n d i n g undefined  t o the response s ( t ) being  f o r 0 4 t < t has been t r e a t e d n u m e r i c a l l y , and i s i l l u s o J  t r a t e d i n F i g . 5.4 f o r t h e r e p r e s e n t a t i v e L o r e n t z i a n l i n e s h a p e by ft = 100 Hz and T  2  = 0.1 s e c , c f . F i g . 5.3.  decay, s ( t ) , a s s o c i a t e d w i t h a Tr/2-pulse w i d t h  defined  The f r e e i n d u c t i o n o f - 1 msec, was de-  f i n e d by 1024 d a t a p o i n t s o v e r a time i n t e r v a l o f 640 msec, w i t h t  = 1 msec.  F i g . 5.4a shows t h e F o u r i e r t r a n s f o r m  f u n c t i o n f ' (x) ,  c f . Eq. (5.4.17), w i t h t h e f r e q u e n c y domain sub-harmonic d a t a f points separated  by 0.39 Hz.  A l i n e a r f r e q u e n c y dependent phase c o r -  r e c t i o n as g i v e n by Eq. (5.4.17) determines t h e a b s o r p t i o n mode l i n e shown i n F i g . 5.4b, where t h e phase f a c t o r x t a t a f r e q u e n c y o f 100 Hz.  The s m a l l n e g a t i v e  t a k e s t h e v a l u e 36° d e v i a t i o n , as compared  w i t h t h e c o r r e c t l i n e s h a p e shown i n F i g . 5.4d, i s p r e s u m a b l y due t o the i n t e g r a l a p p r o x i m a t i o n  i n v o l v e d i n Eq. (5.4.16).  I t i s now i n t e r -  e s t i n g t o compare t h e above a b s o r p t i o n mode l i n e s h a p e w i t h  that  shown i n F i g . 5.4c o b t a i n e d b y u s i n g a f r e q u e n c y independent phase c o r r e c t i o n , c f . Eq. (5.4.8).  The c o r r e c t i o n a n g l e , as d e f i n e d by a  baseline e q u a l i z a t i o n i n a determination  of the c o r r e c t  lineshape,  o  i s +36 .  I n g e n e r a l , however, f o r a s e r i e s o f s p e c t r a l l i n e s o v e r  an extended f r e q u e n c y r a n g e , t h e l i n e a r phase c o r r e c t i o n must be applied.  f  f(x)  (x)  (a)  (b)  £(x)  f(x)  MKI'.IMWJSBS  F i g u r e 5.4  L i n e a r f r e q u e n c y dependent phase correction f o ra Lorentzian l i n e shape system.  186.  In g e n e r a l , a phase c o r r e c t i o n as g i v e n i n terms o f Eqs. (5.4.8) and (5.4.17) i n v o l v e s an i t e r a t i v e n u m e r i c a l  f i t t i n g o f the  a b s o r p t i o n mode l i n e s h a p e t o s e l f - c o n s i s t e n c y as d e f i n e d by a specific  c r i t e r i o n for a correct lineshape.  The phase independent p a r t  o f t h e complex l i n e s h a p e f u n c t i o n g ( x ) , c f . Eq. ( 5 . 1 . 8 ) , may be cons i d e r e d t o be d e f i n e d by an a m p l i t u d e  f u n c t i o n * ^ A(x) , where  g(x) = A(x) e x p ( 1 0 ( x ) ) w i t h A(x) = [ f ( x ) + h ( x ) ] 2  2  a general Fourier transform  J s  (5.4.18)  and 0(x) = t a n [ - h ( x ) / f (x)] .  l i n e s h a p e f u n c t i o n g'(x) w i t h an a r b i t r a r y 2  phase f a c t o r , c f . Eq. (5.4.18), f  | 2  (x) + h  , 2  2  i t i s shown t h a t f (x) + h (x) =  ( x ) ; t h a t i s , t h e amplitude  from t h e n u m e r i c a l  f u n c t i o n i s simply  derived  F o u r i e r t r a n s f o r m f u n c t i o n s f ' ( x ) and h ' ( x ) .  F u r t h e r m o r e , f o r a L o r e n t z i a n l i n e s h a p e as d e f i n e d i n Eqs. and  Now, f o r  _ 1  (5.1.2)  ( 5 . 1 . 6 ) , i t i s seen t h a t t h e a b s o r p t i o n mode f u n c t i o n i s g i v e n 2  d i r e c t l y as f ( x ) = A ( x ) . e f f e c t i v e phasing  I n t h i s p a r t i c u l a r case, a v e r y  c o r r e c t i o n i s o b t a i n e d by e v a l u a t i n g f ( x ) d i r e c t l y ,  as i l l u s t r a t e d i n F i g . 5.5 f o r t h e s t a n d a r d  Lorentzian  lineshape  p r e v i o u s l y d e s c r i b e d w i t h an a s s o c i a t e d phase angle o f 20 quency o f 100 Hz.  error involved.  domain d a t a p o i n t s e p a r a t i o n i s 0.39 Hz. 2  and 5.5c show t h e a m p l i t u d e  As u s u a l , F i g s . 5.5b  f u n c t i o n s A(x) and A ( x ) , r e s p e c t i v e l y ,  where t h e l a t t e r has been r e - n o r m a l i z e d by A ( x ) .  at a fre-  N u m e r i c a l F o u r i e r t r a n s f o r m f u n c t i o n s a r e shown  i n F i g . 5.5a, i l l u s t r a t i n g t h e p h a s i n g the frequency  simple  I t i s seen t h a t A(x)  t o t h e maximum v a l u e  takes t h e form o f a m o d i f i e d  defined Lorentzian  l i n e s h a p e w i t h an i n c r e a s e d f u l l - w i d t h a t half-maximum o f 2/3 2 rad. s e c . . The n u m e r i c a l l y d e r i v e d f u n c t i o n A (x) i s a v e r y good - 1  Figure  5.5.  A m p l i t u d e f u n c t i o n and phase correction f o r a Lorentzian l i n e shape system.  \  187.  approximation  to an exact  L o r e n t z i a n a b s o r p t i o n mode l i n e s h a p e i n  t h a t t h e d e v i a t i o n a t x = 93.6 Hz (two l i n e w i d t h s from the c e n t r e f r e q u e n c y ) i s l e s s than 2% and t h i s f r e q u e n c y i n t e n s i t y approximately  4% o f the maximum.  c o r r e s p o n d s to an  T h i s l i n e s h a p e may be  compared w i t h t h a t shown i n F i g . 5.5d as o b t a i n e d u s i n g the f u l l phase c o r r e c t i o n s p r e v i o u s l y d e s c r i b e d . With r e g a r d t o e f f e c t i v e l i n e - w i d t h r e d u c t i o n and i n t e n s i t y enhancement l e a d i n g to i n c r e a s e d s p e c t r a l r e s o l u t i o n through numerical  methods, m o d i f i e d L o r e n t z i a n a b s o r p t i o n mode f u n c t i o n s  may be d e f i n e d i n g e n e r a l by (5.4.19)  w i t h n = 1, 2, ••• and A a n o r m a l i z a t i o n c o n s t a n t shape f u n c t i o n (n = 1 ) .  f o r the normal  line-  These m o d i f i e d f u n c t i o n s have i d e a l s p e c t r a l  l i n e s h a p e c h a r a c t e r i s t i c s i n t h a t they a r e symmetric about x = ft, they do n o t have z e r o s , t h e e f f e c t i v e l i n e - w i d t h d e c r e a s e s w i t h i n c r e a s i n g n and t h e s i g n a l - t o - n o i s e r a t i o i s improved when t h i s i s g r e a t e r than u n i t y . g i v e n as l ^ S T , ,  1  F o r n = 2 and 4, t h e f u l l  ratio  l i n e - w i d t h s are  and 0.84T" , r e s p e c t i v e l y , as compared w i t h the 1  normal L o r e n t z i a n w i d t h o f 21^  rad. s e c . . - 1  The n u m e r i c a l  reso-  l u t i o n enhancement as d e t e r m i n e d by the f u n c t i o n s f 2 ( ) 3 f ^ M x  a n <  1 S  i l l u s t r a t e d i n F i g . 5.6 f o r two L o r e n t z i a n l i n e s c e n t r e d a t x = -ft and x = ft and f o r v a r y i n g v a l u e s o f t h e p a r a m e t e r OT^in  t h e above t w o - l i n e spectrum may be d e f i n e d i n terms o f t h e r a t i o  o f maximum i n t e n s i t y t o t h e i n t e n s i t y at x = 0. responding for  Resolution  to a l i n e s e p a r a t i o n o f a f u l l  F o r ftT^ = 0.16, c o r -  l i n e - w i d t h , t h i s r a t i o i s 1.3  n = 1 and 2.1 f o r n = 4, g i v i n g a f r a c t i o n a l r a t i o i n c r e a s e o f 1.6  0.32  Figure  5.6  R e s o l u t i o n enhancement, m o d i f i e d L o r e n t z i a n l i n e s h a p e system.  188.  f o r the modified f u n c t i o n f .fx). 6 f o r QT  2  This f r a c t i o n a l r a t i o increase i s  = 0 . 2 4 and 18 f o r OT^ = 0.32. To d e t e r m i n e the g e n e r a l v a l i d i t y o f an e f f e c t i v e phase  c o r r e c t i o n through Gaussian  the e v a l u a t i o n of the amplitude  f u n c t i o n A ( x ) , the  f u n c t i o n may be considered, as an example o f a l i n e s h a p e f u n c -  t i o n w i t h d i f f e r e n t c h a r a c t e r i s t i c s as compared w i t h t h e L o r e n t z i a n function.  The a b s o r p t i o n mode G a u s s i a n  l i n e s h a p e f u n c t i o n , c f . Eq.  ( 2 . 1 . 1 3 ) , may be c o n s i d e r e d i n t h e form  f ( x ) = A exp(-%T (x  -  2  2  ft) ),  w i t h a l l p a r a m e t e r s d e f i n e d as p r e v i o u s l y . maximum i s 2(2£n2) T  rad. s e c . .  2  - 1  2  (5.4.20)  2  The f u l l - w i d t h a t h a l f -  The c o r r e s p o n d i n g  f u n c t i o n may he e v a l u a t e d i n terms o f t h e jrwevse  d i s p e r s i o n mode  Fourier  t r a n s f o r m  f ( t ) b y a c o n t o u r i n t e g r a t i o n , g i v e n i n A p p e n d i x 2, as  T (x-ft) 2  r h(x)  =  A  exp (-JgT (x-ft) ) 2  2  /2 e-*.p(q ) d r j 2  T (x-fi) •2  = A " VT  *  I  1  o  i n a c c o r d a n c e w i t h t h e g e n e r a l complex l i n e s h a p e f u n c t i o n d e f i n e d i n Eq.  (5.1.8).  These G a u s s i a n  f u n c t i o n s a r e shown i n F i g . 5.2b, as d e t e r -  mined by a n u m e r i c a l F o u r i e r t r a n s f o r m a t i o n o f t h e r e s p o n s e s ( t ) , where  s ( t ) = B exp(-t /2T )[cosftt + i sinftt] 2  2  c f . Eq. ( 5 . 1 . 4 ) .  From Eqs.  (5.4.22)  (5.4.20) and (5.4.21), i t f o l l o w s t h a t  189.  f(x) =  A(x)\  1 +  1  o and hence t h e a b s o r p t i o n mode f u n c t i o n i s n o t g i v e n s i m p l y i n terms o f A(x).  As t h e i n t e g r a l t e r m i s symmetric i n x about x = 0, A(x)  a m o d i f i e d G a u s s i a n l i n e s h a p e w i t h an i n c r e a s e d l i n e - w i d t h . shown n u m e r i c a l l y f o r T  defines  This i s  = 0.1 s e c . i n t h a t t h e normal f u l l - w i d t h a t  half-maximum i s 3.7 Hz ( c f . 3.2 Hz f o r a L o r e n t z i a n f u n c t i o n ) and t h e f u l l - w i d t h f o r t h e l i n e s h a p e d e s c r i b e d by A(x)  i s 6.8 Hz.  Thus, i n  2 g e n e r a l , i t i s p o s s i b l e t o use t h e f u n c t i o n A (x) as an a p p r o x i m a t i o n t o an a b s o r p t i o n mode l i n e s h a p e o n l y f o r s p e c t r a l l i n e s c l o s e l y desc r i b e d by a L o r e n t z i a n l i n e s h a p e f u n c t i o n , as i s t h e case f o r h i g h r e s o l u t i o n NMR i n l i q u i d s . 5.5  S i g n a l Zero C o r r e c t i o n . I n p u l s e d mode NMR, a s i g n a l v o l t a g e f o l l o w i n g p h a s e - s e n s i t i v e  d e t e c t i o n , f i l t e r i n g and.dc a m p l i f i c a t i o n may be r e p r e s e n t e d by s' ( t ) = C + s ( t ) ,  (5.5.1)  where s ( t ) i s t h e f r e e i n d u c t i o n decay s i g n a l and C i s a c o n s t a n t t a g e , i n g e n e r a l d i f f e r e n t from z e r o .  A s u f f i c i e n t condition for the  e x i s t e n c e o f a F o u r i e r t r a n s f o r m o f s'(t) i s a b s o l u t e +  vol-  integrability,  C O  ^ | s ' ( t ) | d t < , and t h e c o n s t a n t C does n o t s a t i s f y t h i s c o n d i t i o n . 00  —  CO  I t i s r e a d i l y shown, however, t h a t t h e i n v e r s e F o u r i e r t r a n s f o r m o f 2TTC6  (x) ( w i t h 6 (x) t h e D i r a c d e l t a f u n c t i o n ) i s s i m p l y t h e c o n s t a n t C.  T h i s i m p l i e s t h a t C i s a s s o c i a t e d w i t h an a n a l y t i c a p p r o x i m a t i o n * * ^ t o 6(x)  i n t h e f r e q u e n c y domain.  That i s , t h e F o u r i e r t r a n s f o r m o f C,  190.  c ( x ) , may be c o n s i d e r e d i n t h e form +00  c(x) = C ] e x p ( - i t x ) d t —  00  (5.5.2)  The p a r a m e t e r t  d e f i n e s t h e l i m i t s o f i n t e g r a t i o n f o r a f i n i t e com-  p l e x F o u r i e r t r a n s f o r m a t i o n as c o n s i d e r e d i n s e c t i o n 5.3.  The above  e q u a t i o n d e f i n e s an o s c i l l a t o r y f u n c t i o n w i t h a maximum v a l u e 2 C t x = 0 and zeroes a t x = mr/t r a d . s e c . n m interval 0 ^  - 1  m  at  f o r n = 1, 2, • • - o n t h e .  x ^ t . As c ( x ) i s r e a l , t h e g e n e r a l F o u r i e r t r a n s f o r m  o f s' ( t ) may be e x p r e s s e d  i n t h e form  g'(x) = [ c ( x ) + f ' ( x ) ] + i h ' f x ) ,  (5.5.3)  c f . Eq. (5.4.18). I f the o f f - r e s o n a n c e s h i f t s ft. > 0 a r e m i n i m i z e d  as d e s c r i b e d  1  i n s e c t i o n 5.2, t h e f u n c t i o n c ( x ) may g i v e r i s e t o an o s c i l l a t i o n w i t h a f r e q u e n c y dependent a m p l i t u d e superimposed on a s p e c t r a l of i n t e r e s t .  lineshape  T h i s e f f e c t i s i l l u s t r a t e d i n F i g . 5.7 f o r a r e p r e s e n t a t i v e  L o r e n t z i a n l i n e s h a p e d e f i n e d by ft = 40 Hz and T s ' ( t ) as d e t e r m i n e d  = 0.1 sec.  The s i g n a l  by 1024 d a t a p o i n t s o v e r an i n t e r v a l o f 0.64 s e c .  i s shown i n F i g . 5.7a, where s' ( t ) = 2.0 + 10.Oexp(-t/T ) c o s f t t .  The  a b s o r p t i o n mode n u m e r i c a l F o u r i e r t r a n s f o r m f u n c t i o n , c ( x ) + f ( x ) , i s 1  d e f i n e d i n F i g . 5.7b by d a t a p o i n t s s e p a r a t e d by t h e sub-harmonic f r e quency 0.39 Hz.  The s u p e r p o s i t i o n o f c ( x ) i s c l e a r l y shown over t h e  f r e q u e n c y range 0 t o 65 Hz, t h e c o n s t a n t C b e i n g 20% o f t h e maximum v a l u e o f t h e f r e e i n d u c t i o n decay s i g n a l . The  constant C i s r e a d i l y determined  as t h e mean s i g n a l  level  f(x)  0 F i g u r e 5.7  40  x(Hz)  Numerical Fourier transform d i s t o r t i o n t o non-zero average s i g n a l l e v e l .  due  191.  f o r t •+ T, T b e i n g t h e s a m p l i n g time i n t e r v a l .  A numerical  correction  of -C may then be a p p l i e d t o t h e sampled s i g n a l d a t a t o o b t a i n a good approximation to the s i g n a l f u n c t i o n s ( t ) .  A numerical F o u r i e r trans-  f o r m a t i o n o f s ( t ) then g i v e s f ' ( x ) and h ' ( x ) as r e q u i r e d f o r the g e n e r a l phase c o r r e c t i o n s d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n .  T h i s method f o r  a s i g n a l z e r o c o r r e c t i o n has been shown t o be both r e l i a b l e  and e f f i c -  i e n t by u s i n g a s u b r o u t i n e DIRCON i n t h e FORTRAN program LGTRN p r e v i ously outlined.  192.  5.6  High R e s o l u t i o n F o u r i e r T r a n s f o r m NMR  The b a s i c c o n c e p t s  and c o m p u t a t i o n a l methods i n v o l v e d i n a  F o u r i e r t r a n s f o r m NMR e x p e r i m e n t have been c o n s i d e r e d i n d e t a i l i n the preceding sections  o f t h i s t h e s i s , w i t h p a r t i c u l a r emphasis upon the  p o s s i b i l i t y o f u s i n g t h i s t e c h n i q u e f o r q u a n t i t a t i v e 1ineshape: s t u d i e s . As an example h i g h r e s o l u t i o n a p p l i c a t i o n , the spectrum o f dimethyl  n i t r o s a m i n e has been o b t a i n e d a t a r e l a t i v e l y low H  c o r r e s p o n d i n g to an o p e r a t i n g f r e q u e n c y o f 1 0 . 0 MHz.  field  Q  This p a r t i c u l a r  s p i n system was chosen because o f the s i m p l i c i t y o f the c h e m i c a l l y s h i f t e d equal i n t e n s i t y d o u b l e t spectrum and the magnitude o f chemical s h i f t , :  45 Hz a t 60 MHz and 30°C.  As the p u l s e  the  spectrometer  13 d e s c r i b e d i n s e c t i o n 3 . 2 was d e s i g n e d s p e c i f i c a l l y f o r a f i e l d o f 9.4 frequency f o r  C studies  k g a u s s , i t was a l s o c o n v e n i e n t t o use the above o p e r a t i n g NMR.  The sample was p r e p a r e d as an 8% C C l ^ s o l u t i o n ,  the s o l v e n t carbon t e t r a c h l o r i d e b e i n g used to e l i m i n a t e the background 1 s o l v e n t NMR e x p e r i m e n t .  in  large  ' ' H s i g n a l as o b t a i n e d i n the n o n - s e l e c t i v e  pulsed  T h i s sample i n a 5mm od tube was degassed u s i n g the  usual freeze-pump thaw c y c l e s . The f r e e i n d u c t i o n decay s i g n a l f o l l o w i n g phase  sensitive  d e t e c t i o n was sampled u s i n g a FABRITEK FT-1064 A-D c o n v e r t e r w i t h a f i l t e r i n g time c o n s t a n t  of 50 y s e c s ,  1 2 - b i t amptitude r e s o l u t i o n .  the d i g i t a l  The o f f - r e s o n a n c e  d a t a c o r r e s p o n d i n g to s h i f t , which i s  s a r y t o p r e v e n t the l i n e s h a p e d i s t o r t i o n s d e s c r i b e d i n s e c t i o n  neces5.2,  was a d j u s t e d to be 27 Hz u s i n g a f r e q u e n c y s y n t h e s i s e r w h i l e the Ho f i e l d s t a b i l i t y was m a i n t a i n e d by a VARIAN f l u x  super-stabiliser.  Fig.  5.8  Free i n d u c t i o n decay and f i n i t e  Fourier  t r a n s f o r m spectrum f o r d i m e t h y l  nitrosamine.  193.  The r e s u l t a n t  f r e e i n d u c t i o n decay s i g n a l s i t ,  z e r o c o r r e c t i o n as d i s c u s s e d i n s e c t i o n 5 . 5 , o v e r a time i n t e r v a l o f 1.5 the r e a l  sees..  following a function  i s plotted i n F i g . 5.12(a)  U s i n g the computer program hGTRN,  and i m a g i n a r y p a r t s o f the f i n i t e complex F o u r i e r  f u n c t i o n a r e shown i n F i g . 5 . 1 2 ( b ) and ( c ) ,  respectively.  s u b - h a r m o n i c a n a l y s i s as d e s c r i b e d i n s e c t i o n 5 . 3 ,  the  s p a c i n g o f the o u t p u t d a t a p o i n t s i s reduced to 0.08 m u l t i p l i c i t y o f 8 and 8192  t r a n s f o r m data p o i n t s .  r e l a t i v e l y narrow l i n e s a r e w e l l  transform Through  frequency  Hz u s i n g a s u b -  In t h i s manner  d e f i n e d by a t l e a s t twenty d a t a  For t h e o p t i m a l f i l t e r time c o n s t a n t used a f r e q u e n c y dependent correction,  of s e c t i o n 5.4,  o f phase c o r r e c t i o n  was not r e q u i r e d .  However, the  phase  reference  T h i s spectrum  o f 288 d a t a p o i n t s on the f r e q u e n c y i n t e r v a l 19.9 shift  points.  used to produce the f i n a l a b s o r p t i o n mode spectrum  shown i n F i g . 5 . 1 2 ( d ) was computed as + 2 0 . 5 ° .  the chemical  the  i s d e t e r m i n e d as 7.57  l i n e w i d t h a t half-maximum i s 0.7  ± 0.08  < x < 39.9  Hz.  consists H z , and  The c o r r e s p o n d i n g  H z , the s m a l l d i s t o r t i o n o f the  line-  shape b e i n g due t o the form o f the Ho f i e l d inhomogeneity which i s d i f f i c u l t to a d j u s t  very  by o b s e r v i n g the f r e e i n d u c t i o n decay i n the h i g h  r e s o l u t i o n NMR l i m i t . Thus i t has been shown t h a t a spectrum o f a q u a l i t y w h i c h i s a t l e a s t comparable w i t h t h a t o b t a i n e d from the t e d i o u s slow passage s t e a d y - s t a t e t e c h n i q u e may be v e r y r e a d i l y o b t a i n e d u s i n g the rapid Fourier transform technique.  Furthermore, a l l data i s a  form f o r f u r t h e r p r o c e s s i n g on a d i g i t a l  computer.  relatively convenient  APPENDIX 1  Lorentzian F o u r i e r Transform P a i r . C o n s i d e r the L o r e n t z i a n l i n e s h a p e the i n t e r v a l -°° < x < the  00  f u n c t i o n f ( x ) , d e f i n e d on  i n terms o f t h e r e a l v a r i a b l e x, e x p r e s s e d i n  form  If  f(x) where k = l / T ^ .  , k > 0  The c o r r e s p o n d i n g complex f u n c t i o n , f ( z ) , g i v e n i n  terms o f the complex v a r i a b l e z = x + i y , has s i m p l e p o l e s  a t z = fl i i k .  These are shown, f o r 0. > 0, i n t h e z-plane as  n  The i n v e r s e F o u r i e r t r a n s f o r m o f the r e a l v a r i a b l e t ^ 0.  o f f ( z ) , f ( t ) , may be c o n s i d e r e d  i n terms  The f u n c t i o n f ( t ) i s t h e n d e f i n e d by a  contour i n t e g r a t i o n f(t) =  where C  1  i s the c o n t o u r shown as p a r t o f t h e c l o s e d c o n t o u r C i n t h e  upper-half-plane.  I n t h e l i m i t R ->• , i t can be shown t h a t 00  4  f(z)ex Citz)dz = P  J  k  c The  00  K  2  +  ( x  _  f l )  2 .  - 0 0  r e s i d u e f o r f ( z ) e x p ( i t z ) a t the s i m p l e p o l e z = ft + i k i s g i v e n as R(ft+ik) = l i m \ [z-(ft+ik)]k explitz) I z-»ft+ik ik 1 [(z-ft)+ik][(z-ft)-ik] j 2  ^  k  exp [ - ( k - i f t ) t ]  and i n accordance w i t h Cauchy's i n t e g r a l  j  1 f (z)exp(itz)dz =  formula  2iR(ft + i k ) k y  r  1  expL-(k  - ift)tj  Thus i t ' f o l l o w s t h a t t h e i n v e r s e F o u r i e r transfoi-m i s g i v e n as f ( t ) = T ^ ~ exp [-dr Z 1  corresponding  -ift)t] 2  2  to  f(x) =  \ j 1 + T (x-ft) Z  2  Now c o n s i d e r t h e L o r e n t z i a n l i n e s h a p e f u n c t i o n h ( x ) , d e f i n e d on t h e i n t e r v a l -co < x <  0 0  , e x p r e s s e d i n the form  k(x-ft) h ( x ) = -y~ 5— k + (x-ft) 1  The  , k > 0.  i n v e r s e F o u r i e r t r a n s f o r m h ( t ) i s d e f i n e d by a contour i n t e g r a t i o n  s i m i l a r t o t h a t d e s c r i b e d above, where the r e s i d u e f o r h ( z ) e x p ( i t z ) a t the s i m p l e p o l e z = ft + i k i s d e t e r m i n e d as  R(^.+ik)  =  h k exp  [-(k-ift)t].  Thus i t f o l l o w s t h a t the i n v e r s e F o u r i e r t r a n s f o r m  h(t)  =  i %-  exp [-(£• 2  l  corresponding to  h  (x)  =  T  1 +  ^  ?  -  T (x-Q) 2  Z  2  ityt]  i s g i v e n as  APPENDIX 2  Gaussian  F o u r i e r Transform  Pair.  C o n s i d e r t h e f u n c t i o n s ( t ) , d e f i n e d on the i n t e r v a l 0  t < °°  i n terms o f the r e a l v a r i a b l e t, i n t h e form  s(t) = where k = l / T ^ .  [-  C  i^-t]  The F o u r i e r t r a n s f o r m o f s ( t ) , g ( x ) , i s g i v e n i n terms  o f t h e r e a l v a r i a b l e x on t h e i n t e r v a l - 0 0 < ?( < °° by 00  o OO  T h i s i n t e g r a l may be e v a l u a t e d through a c o n t o u r ii n t e g r a t i o n j" exp(-z ) d z , where t h e complex v a r i a b l e may be d e f i n e d as z = e + in. and the c l o s e d c o n t o u r C i n t h e z-plane i s c o n s i d e r e d as shown b e l o w :  JUJ.\>  —6-  c  -t-  6  e  \  For t h e above c o n t o u r , Cauchy's i n t e g r a l theorem g i v e s  e x p ( - z " ) d z = 0,  and hence ft C-tfC-t -)  di.  1  +  [-((Ulvj) \| 7  I\  <LYJ  +  e^cp  [-(e + d) -] de 3  ft  Thus i n t h e l i m i t R •> oo, i t i s shown t h a t CO  -°*>  The f i r s t i n t e g r a l on t h e r i g h t hand s i d e i s a gamma f u n c t i o n , and i n the g e n e r a l  case i t f o l l o w s  The r e q u i r e d i n t e g r a n d P  -  1  Vr  that  i s now g i v e n b y t h e s u b s t i t u t i o n h -  (  x  "  a  )  and hence  The G a u s s i a n l i n e s h a p e  f u n c t i o n s f ( x ) and h ( x ) a r e now g i v e n , i n  accordance w i t h the d e f i n i t i o n o f the g e n e r a l complex l i n e s h a p e f u n c t i o n g(x) = f ( x ) - i h ( x ) ,  f(x)  as  ft) ]  = Aexp[-^T (x -  2  2  and  r h(x) = A exp[-l T (x 2  2  \%  ft) ].^lexpCn )dr 2  2  1)  o  where A i s a n o r m a l i z a t i o n c o n s t a n t . 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Chem., 49,  (in press).  NMR S t u d i e s  o f M u l t i - s i t e C h e m i c a l Exchange.  Rotation i n Dimethyl Acetamide, Dimethyl  III.  Hindered  Trifluoro-acetamide  and Benzamide. 4.  E.A. Allan,  R . F . Hobson, L . W . Reeves and K . N . Shaw, J . Am.  Chem. S o c . , 1971  (in press).  H i n d e r e d R o t a t i o n i n N , N - d i m e t h y l Amides w i t h Halogen and Pseudo-halogen 5.  L . W . Reeves, 1971  Substituents. R . C . Shaddick and K . N . Shaw, J . P h y s . Chem.,  (in press).  A Determination  o f the H i n d e r e d R o t a t i o n B a r r i e r i n Unsym-  dimethyl Selenourea 6.  and Comparison w i t h s i m i l a r  K . N . Shaw and L . W . Reeves,  Chem. P h y s . L e t t e r s ,  Compounds. 19 7 1 .  A . S e m i - e m p i r i c a l SCF-LCAO-MO Study o f the H i n d e r e d  Internal  R o t a t i o n i n Formamide. 7.  K . N . Shaw, Rev. S c i . I n s t r . , A Simple S o l i d - s t a t e  rf-pulse  1971  (submitted for  publication)  gate f o r NMR S p e c t r o m e t e r s .  

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