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Electron spin resonance of X- and Y- irradiated potassium difluoromalonate and electron paramagnetic… Mustafa, Mohammed Rafi 1969

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ELECTRON SPIN RESONANCE  OF X- AND Y-IRRADIATED POTASSIUM  DIFLUOROMALONATE AND ELECTRON PARAMAGNETIC  RESONANCE OF COPPER  ( I I ) COMPLEX  WITH TRIFLUOROACETATE LIGANDS by  B.Sc.  MOHAMMED RAFI MUSTAFA ( H o n o u r s ) , U n i v e r s i t y o f S i n d , . P a k i s t a n , 1963  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  We a c c e p t  this  THE  t h e Department of Chemistry  t h e s i s as c o n f i r m i n g  to the r e q u i r e d  UNIVERSITY OF BRITISH DECEMBER 1969  COLUMBIA  standard  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  further  for  scholarly  by h i s of  agree  this  written  thesis at  the U n i v e r s i t y  make tha  it  purposes  for  freely  permission may  representatives. thesis  in p a r t i a l  financial  December  8,  1969  Columbia,  British  by  for  gain  Columbia  shal1  the  that  not  requirements  reference copying  t h e Head o f  understood  Chemistry  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, Canada  of  for extensive  permission.  Department o f  Date  is  of  available  be g r a n t e d  It  fulfilment  of  I agree and this  be a l l o w e d  or  that  study. thesis  my D e p a r t m e n t  copying  for  or  publication  without  my  TO MY PARENTS AND  TEACHERS  ABSTRACT  In P a r t I of t h i s t h e s i s an i r r a d i a t e d s i n g l e  crys-  t a l of dipotassiura d i f l u o r o m a l o n a t e monohydrate has been s t u d i e d by E l e c t r o n Spin Resonance. On X - i r r a d i a t i o n the y i e l d s CF(COO~)  2  crystal  r a d i c a l with a h i g h l y a n i s o t r o p i c h y p e r f i n e  t e n s o r , c h a r a c t e r i s t i c of a  1 9  F  nucleus. In a d d i t i o n , some  other l i n e s were a l s o observed which were too weak to be anal y z e d . On Y - i r r a d i a t i o n , the c r y s t a l y i e l d s mainly the same r a d i c a l i n a d d i t i o n to two  types of CF2C00" r a d i c a l s . In one  type, the c a r b o x y l group was the C?2  found to be n e a r l y coplanar with  fragment while i n the o t h e r , i t was  c u l a r . A broad c e n t r a l l i n e i n each spectrum the p o s s i b i l i t y of the presence of C 0  An u n r e s t r i c t e d Hartree-Fock approximation was  2  nearly perpendiindicated  also  radical.  c a l c u l a t i o n , u s i n g the  of Intermediate Neglect of D i f f e r e n t i a l  Overlap  c a r r i e d out on a number of f l u o r i n a t e d r a d i c a l s to c o r r e -  l a t e the  1  1 9  F  h y p e r f i n e t e n s o r s o b t a i n e d i n the present study  with those obtained p r e v i o u s l y . The  theory was  used t o c a l c u -  l a t e the t h e o r e t i c a l h y p e r f i n e t e n s o r s f o r v a r i o u s r a d i c a l s and was  found to g i v e a reasonably good agreement with the  experimental r e s u l t s . On t h i s b a s i s the h y p e r f i n e tensor f o r •  CHFCOO" r a d i c a l was  p r e d i c t e d . The c a l c u l a t i o n a l s o showed  t h a t one of the c a r b o x y l groups i n CF(C00~)2 r a d i c a l s i s cop l a n a r with the r a d i c a l plane while the other makes an of 85°.  angle  In P a r t  I I , a c r y s t a l of  hydrate, containing at  77°K and  i t was  approximately consisting  small  amounts o f  found t h a t  tetragonal  zinc  the  Cu  trifluoroacetate d i -  Cu + +  + +  has  been  ion resides  environment w i t h the  in  ground  2 o r b i t a l which i s coupled  of mainly d  studied an state  z orbital, tion  of  studied  x  perhaps through v i b r o n i c  interactions. Also  a  copper t r i f l u o r o a c e t a t e i n t r i f l u o r o a c e t i c a c i d at  interpreted state  2  to d  room t e m p e r a t u r e and by  at  77°K and  assuming a t e t r a g o n a l  being pure d  0  0  x -y z  . z  the  2 —y  soluwas  r e s u l t s were  symmetry w i t h  the  ground  TABLE OF CONTENTS  PART I :  An E l e c t r o n i S p i n Irradiated malonate  Resonance Study  o f X- and y -  Single C r y s t a l s o f Potassium  Difluoro-  Monohydrate.  Chapter 1  2  Page INTRODUCTION  1  THEORETICAL ^(A) . The D e t e r m i n a t i o n o f t h e g - and t h e AT e n s o r s f r o m t h e O b s e r v e d g - V a l u e s and Hyperfine S p l i t t i n g s . (B). C a l c u l a t i o n o f the M o l e c u l a r E l e c t r o n i c Structure  3  12  EXPERIMENTAL (A) . P r e p a r a t i o n o f t h e Sample  19  (B) . I r r a d i a t i o n  20  (C) . The S p e c t r o m e t e r  21  (D) . Measurements  24  (E) . M o l e c u l a r O r b i t a l C o m p u t a t i o n s (i) . CF(COO-) R a d i c a l ( i i ) . CF COO- and CHFCOO" R a d i c a l s ( i i i ) . CHFCONH and C F C O N H R a d i c a l s  28 29 32 34  2  2  2  4  7  2  2  RESULTS (A) . The C F ( C O O " )  2  (B) . The CF COO~  Radicals  2  Radical  35 46  V  Chapter  Page  (C). M o l e c u l a r O r b i t a l  5  Calculations  58  DISCUSSION (A) . The  1 9  F Hyperfine Coupling Tensors  (B) . A n i s o t r o p i c I n t e r a c t i o n Calculations (C) . F e r m i - C o n t a c t Calculations (D) . O r i e n t a t i o n  and INDO  Interaction  of Radicals  68 72  and INDO  83  i n the C r y s t a l  89  PART I I : An E l e c t r o n  P a r a m a g n e t i c Resonance S t u d y o f Cu  in  and S i n g l e  Trifluoroacetic Acid  Zinc  Trifluoroacetate  Crystals of  Dihydrate.  1  INTRODUCTION  92  2  THEORETICAL  99  3  EXPERIMENTAL (A) . P r e p a r a t i o n o f t h e Sample  106  (B) . Measurements  107  4  RESULTS  108  5  DISCUSSION  121  vi  Page  BIBLIOGRAPHY  126  APPENDICES:-  APPENDIX A: The Method o f S o l v i n g for  CF(COO~)  2  the Hamiltonian  and CF COO~ 2  135  Radicals  APPENDIX B: The R o o t h a a n E q u a t i o n s  14 3  APPENDIX C: C o m p l e t e N e g l e c t o f D i f f e r e n t i a l  152  Overlap APPENDIX D: I n t e r m e d i a t e N e g l e c t o f D i f f r e n t i a l  161  Overlap APPENDIX E : The P r o b l e m o f S p i n C o n t a m i n a t i o n  165  L I S T OF TABLES  Page TABLE I  The P r i n c i p a l  V a l u e s and t h e P r i n c i p a l  Axes o f A- and g - T e n s o r s f o r C F ( C O O ~ )  43 2  Radical TABLE I I  The P r i n c i p a l  V a l u e s and t h e P r i n c i p a l  55  Axes o f A- and g - T e n s o r s f o r C F C O O ~ ( I ) 2  Radical TABLE I I I  The P r i n c i p a l  V a l u e s and t h e P r i n c i p a l  56  Axes o f A- and g - T e n s o r s f o r C F C O O ~ ( I I ) 2  Radical TABLE IV  INDO S p i n  Densities  f o r CF(COO~)  TABLE V  INDO S p i n  Densities  f o r CF COO~(I)  60  Densities  f o r CF COO~(II)  61  Densities  f o r CHFCOO~(I)  62  Densities  f o r CHFCOO~(II)  63  Densities  f o r CHFCONH (I)  64  2  Radical  2  59  Radical TABLE V I  INDO S p i n  2  Radical TABLE V I I  INDO S p i n Radical  TABLE V I I I  INDO S p i n Radical  TABLE IX  INDO S p i n Radical  2  viii Page  TABLE X  INDO Spin D e n s i t i e s f o r CHFCONH (II) 2  65  Radical TABLE XI  INDO Spin D e n s i t i e s f o r CF CONH (I) 2  2  66  Radical TABLE XII  INDO Spin D e n s i t i e s f o r CF CONH (II) 2  2  67  Radical TABLE XIII  Comparison of Some F l u o r i n a t e d R a d i c a l s  69  TABLE XIV  Some F l u o r i n a t e d R a d i c a l s Arranged i n  76  the Order of I n c r e a s i n g D e l o c a l i z a t i o n TABLE XV  T h e o r e t i c a l Hyperfine  Tensors f o r Some  80  F l u o r i n a t e d R a d i c a l s Computed from INDO Spin TABLE XVI  A n a l y s i s o f L i n e a r i t y Between a p  TABLE XVII  Densities  S  N N  N  and  86  as C a r r i e d out by Pople e t a l .  S  Comparison Between the Observed t r o p i c Hyperfine  Coupling  Iso-  88  and the Value  C a l c u l a t e d from Pople's C o r r e l a t i o n TABLE XVIII  The D i a g o n a l i z e d  A- and g-Tensors with  the D i r e c t i o n Cosines f o r the s i t e  115  ix  Page  TABLE XIX  The D i a g o n a l i z e d A- and g - T e n s o r s w i t h the D i r e c t i o n QlP  TABLE XX  P 2  Cosines  f o r the S i t e  3  A Summary o f S p i n H a m i l t o n i a n for  116  Cu  + +  Ion.  Parameters  120  I  L I S T OF FIGURES  Page FIGURE 1  Energy One  FIGURE 2  1 9  Levels f o r a Radical Containing  F  nucleus  Energy Two  1 9  10  Levels f o r a Radical Containing  F  11  nuclei  FIGURE 3  B l o c k D i a g r a m o f 100 k c . ESR S p e c t r o m e t e r  23  FIGURE 4  The C r y s t a l  27  H o l d e r w i t h Dewar Used f o r  Low T e m p e r a t u r e Work FIGURE 5  The C o o r d i n a t e  S y s t e m Used  f o r CF(COO")  31  2  Radical FIGURE 6  The C o o r d i n a t e  Systems Used  f o r CF COO~  33  2  Radicals FIGURE 7  A Typical  ESR S p e c t r u m o f X - I r r a d i a t e d  CF (COOK) .H 0, the Magnetic 2  2  being FIGURE 8  (0.5, 0, -0.8660)  2  H 0  the Magnetic  and  Parallel  Angular ting,  FIGURE 10  i n the D i r e c t i o n  Field  An ESR S p e c t r u m o f X - I r r a d i a t e d C F ( C O O K ) . 3 7 2  FIGURE 9  2  36  Field  in^ab-Plane  t o a-Axis  Variation  the Magnetic  Angular  Being  2  Variation  of the Hyperfine S p l i t Field  Being  39  i n bc-Plane  of the Hyperfine  40  xi  Splitting,  the Magnetic  Field  Being i n  ca-Plane FIGURE 11  Angular V a r i a t i o n Splitting,  of the Hyperfine  the Magnetic  Field  41  Being i n  ab-Plane FIGURE 12  The  P o s i t i o n o f the Magnetic  to  45  ESR S p e c t r u m o f Y - I r r a d i a t e d  47  Locate the Observed FIGURE 13  A Typical  g-Value  CF (COOK) .H 0, the Magnetic 2  in FIGURE 14  2  2  the D i r e c t i o n  2  2  i n bc-Plane  and P a r a l l e l  Angular V a r i a t i o n Splitting Magnetic  Field  Magnetic  of the Hyperfine  2  the  2  Being  i n ca-Plane  Magnetic  f o r CF COO~ R a d i c a l s , 2  Field  53  the  Angular V a r i a t i o n of the Hyperfine Splitting  52  Being i n bc-Plane  f o r CF COO~ R a d i c a l s , Field  Being  to b-Axis  f o r CF COO~ R a d i c a l s , Field  50  Angular V a r i a t i o n of the Hyperfine Splitting  FIGURE 17  Being  An ESR S p e c t r u m o f y - I r r a d i a t e d 2  FIGURE 16  Field  (-0.5, 0.8660, 0)  CF (COOK) .H 0, the Magnetic  FIGURE 15  Field  the  Being i n ab-Plane  54  xi i  FIGURE 18  Splitting and  o f the D Z  State  i n Octahedral  T e t r a g o n a l F i e l d s , t h e Ground  Being  B  l g  State  . 2  FIGURE 19 Splitting  101  o f the  102 D State  i n Octahedral  and T e t r a g o n a l F i e l d s , t h e Ground Be i n g A  State  2  l g (  FIGURE 20  A Typical Single the  EPR S p e c t r u m o f C u  C r y s t a l of Zinc  Magnetic F i e l d  + +  in a  109  Trifluoroacetate  Being  i n the D i r e c t i o n  (0.9848, 0, -0.1736) FIGURE 21  An EPR S p e c t r u m o f C u C r y s t a l of Zinc Magnetic F i e l d and  FIGURE 22  Parallel  in a  Single  Being  i n the ca-Plane .  t o the a-Axis of the Hyperfine  Splitting  i n a Single  Zinc  Field  of C u  + +  Trifluoroacetate,  Being  of  Zinc  Field  the Magnetic  i n the bc-Plane  of C u  + +  i n a Single  Trifluoroacetate,  Being  111  Crystal  Angular V a r i a t i o n o f the Hyperfine Splitting  110  T r i f l u o r o a c e t a t e , the  Angular V a r i a t i o n  of  FIGURE 23  + +  Crystal  the Magnetic  i n the ca-Plane  112  xii i  FIGURE 24  Angular V a r i a t i o n o f the Hyperfine Splitting of  Zinc  Field FIGURE 25  An EPR  of C u  i n a Single  Crystal  T r i f l u o r o a c e t a t e , the Magnetic  Being i n the ab-Plane Spectrum o f C u  acetic Acid FIGURE 26  + +  113  Solution  + +  in Trifluoro-  a t Room  An EPR  Spectrum of C u  acetic  acid Solution at  + +  Temperature  in Trifluoro77°K.  117  118  ACKNOWLEDGEMENT  I w i s h t o t h a n k P r o f e s s o r W. C. L i n s i n c e r e l y f o r his expert  g u i d a n c e and c o n t i n u o u s  the  of this  course  tance,  study.  help  I never  t o D r . F . G. H e r r i n g  and g u i d a n c e , p a r t i c u l a r l y  est  appreciated.  i n the theoretical  f o r h i s help.  my s t a y  and  John T a i t  ening  Kennedy, P u i Wah L a u , James Hebden, N a r e s h D a l a i and D r . M. D. S a s t r y  efforts  keep t h e s p e c t r o m e t e r s  f o r many h o u r s o f e n l i g h t -  o f Mr. J o e S a l l o s and Mr. Tom Markus t o i n e x c e l l e n t c o n d i t i o n a l l the time,  a l s o a c k n o w l e d g e d . I am a l s o t h a n k f u l t o Mr. G l e n n  who a l w a y s welcomed me w i t h vide nitrogen  Dean o f S t u d e n t s '  results after  a smile  and n e v e r f a i l e d  c y l i n d e r s , even a t odd  I am a l s o i n d e b t e d the  t o my f r i e n d s  discussions.  The  are  at the  Columbia.  Acknowledgement must a l s o be e x p r e s s e d Messrs David  work.  C. A. M c D o w e l l a l s o f o r h i s i n t e r -  i n t h i s work and h i s e n c o u r a g e m e n t d u r i n g  University of British  i n assis-  f r o m whom I r e c e i -  f o u n d h i m "busy" whenever I r e q u e s t e d  I wish t o thank P r o f e s s o r  throughout  The many h o u r s he s p e n t  c o u n s e l l i n g and d i s c u s s i o n s a r e g r e a t l y  I am h i g h l y i n d e b t e d ved  encouragement  and f o r g r a n t i n g  to pro-  times.  to President  Affairs)  Lenin  Gage  (at that  time,  f o r h i s a p p r e c i a t i o n o f my  t h e UBC G r a d u a t e F e l l o w s h i p ,  t h e d a t e o f a p p l i c a t i o n had e x p i r e d .  long  L a s t but not the l e a s t , felt  appreciation  during  t o N i g h a t , my  I have t o e x p r e s s my  heart-  w i f e , f o r h e r encouragement  t h e c o u r s e o f s t u d y and f o r h e r h e l p i n p r e p a r i n g t h e  manuscript  of t h i s  thesis.  PART I  ELECTRON SPIN RESONANCE STUDY OF IRRADIATED SINGLE CRYSTALS OF DIPOTASSIUM DIFLUOROMALONATE MONOHYDRATE  CHAPTER  ONE  INTRODUCTION  Most o f t h e work on  electron  on o r i e n t e d f r e e r a d i c a l s o v e r concerned with the The  p r e s e n t work i s one  with  fluorine  rine  and  unpaired the  ling  of those  reason  electron density  fluorine  is entirely  itself due  hyperfine^interaction r  than  fluorine  to the  fluo-  for this  i s the presence  p-orbital  the a n i s o t r o p y i n proton s p i n d e n s i t y on  a  anisotropic  proportional to r electron  c o u p l i n g i s n a t u r a l l y more  coup-  neighbouring  i s attached. Since the  i s inversely  of  and  3  where  the  nuc-  anisotropic  the proton c o u p l i n g .  The ling and  between  i n t h e much l a r g e r a n i s o t r o -  i s t h e d i s t a n c e between t h e u n p a i r e d  l e u s , the  been  exceptions which d e a l  (or s p i n d e n s i t y ) i n a  while  atom t o w h i c h t h e p r o t o n  few  major d i f f e r e n c e  couplings l i e s  f o r m e r . The  t h e p a s t t e n y e a r s has  of the proton h y p e r f i n e c o u p l i n g .  c o u p l i n g . The  proton  py o f t h e  of  study  s p i n r e s o n a n c e [/(esr)  systematic study  in oriented radicals h i s co-workers(1-3),  began w i t h the  malonic  acid  diated  c a r b o x y l i c a c i d s has  ginal  findings.  o f the p r o t o n h y p e r f i n e coupt h e work o f  f i r s t one  S u b s e q u e n t work on mainly  b e i n g on  McConnell X-irradiated  t h e e s r o f X-  or  y^irra-  s u b s t a n t i a t e d these  ori-  2  It in  crystals  c a n be s a i d  that  of carboxylic  h i g h energy r a d i a t i o n  t h e r a d i c a l s which  acids,  after  being  are stable  subjected t o  a r e , i n most c a s e s , t h o s e f o r m e d by  t h e b r e a k a g e o f a C-H bond  a t the a-position  from t h e c a r -  b o x y l g r o u p . Thus i n t h e c a s e o f m a l o n i c a c i d ^ ,  the pre-  dominant  radical  radical  CH COO"", l e s s 2  s p e c i e s was  stable  kage o f a C-C bond  formula acid of  of a radical  group  f o r m e d by t h e b r e a -  a t t h e a p p o s i t i o n was t h e r a d i c a l 2  . In t h i s  was f o r m e d by t h e b r e a -  and t h e removal o f a c a r b o x y l  (C00~) GHCH (COO"!) formed  '  v  2  than the f i r s t ,  A n o t h e r i m p o r t a n t example kage o f a C-H bond  CH(COO"") . A s e c o n d  radical,  i n y-irradiated  not only  o f the  succinic  the hyperfine  coupling  a- p r o t o n was o b s e r v e d , b u t t h o s e o f t h e two 8> p r o t o n s  were a l s o  studied.  I t s h o u l d be m e n t i o n e d  that,  besides the  b r e a k a g e o f t h e C-H bond  a- t o t h e COOH g r o u p , t h a t o f t h e  C-H  i n an amide h a s a l s o been  a t o t h e CONH  2  group  o b s e r v e d e . g . , CHFCONH amide  v  2  found i n X - i r r a d i a t e d m o n o f l u o r o a c e t -  '*  Although the l i t e r a t u r e radicals  a c i d s o r amides  must be b o r n e  n o t a b l e example CH  abounds i n t h e examples o f  formed by t h e b r e a k a g e o f an a C-H bond  carboxylic it  frequently  i n mind  are irradiated  when t h e  w i t h X- o r y r a y s ,  t h a t t h i s d o e s n o t a l w a y s happen.  i s t h a t o f a l a n i n e . The o n l y  CHCOOH, i n y - i r r a d i a t e d  stable  A  radical,  J l - a l a n i n e was formed by t h e b r e a -  3  kage o f a C-N b o n d  . Examples o f o t h e r t y p e s o f b o n d -  v  breakage a r e n o t l a c k i n g will  be c o n c e r n e d  cals,  once t h e y  i n the l i t e r a t u r e .  with the e l e c t r o n i c  While  most r a d i c a l s  electron radicals  contact with  elsewhere  i n which the unpaired  t h e r e were some r a d i c a l s  t r o n was f o u n d  t h e mechanism o f t h e i r '.  i n a a-molecular  e l e c t r o n r e s i d e s mainly  the proton being  i n the nodal  i n which t h e unpaired orbital  such  a p r o t o n m i g h t be p o s s i b l e .  elec-  that the d i r e c t  These l a t t e r  were v e r y o f t e n c h a r a c t e r i z e d by l a r g e p r o t o n fine  f 14) v  s t u d i e d were t h e s o c a l l e d TT-  i n a ir-molecular o r b i t a l , w i t h plane,  work  structures o f the r a d i -  a r e formed, r a t h e r than  f o r m a t i o n which has been f o r m u l a t e d  However t h i s  ones  isotropic  hyper-  interactions.  One  o f t h e b e s t examples o f  the formyl r a d i c a l , of carbon  HCO, w h i c h was o b t a i n e d by t h e r e a c t i o n  monoxide w i t h a t o m i c  t o l y t i c decomposition hyperfine  hydrogen, produced  o f hydrogen i o d i d e ,  c o u p l i n g c o n s t a n t was f o u n d  examples a r e X - i r r a d i a t e d cyanoacetylurea electron radical  a-electron radicals i s  o f the formula  a a-electron radical,  isotropic  t o be 137 g a u s s .  m a l o n a m i d e ' a n d  . The f o r m e r  by t h e p h o -  yielded,  Other  X-irradiated  among o t h e r s , a a -  NH COCH CONH, and t h e l a t t e r , 2  2  CNCH CONHCONH. 2  In c o n t r a s t t o t h e r a d i c a l s  exhibiting  proton  hyper-  4  fine  coupling,  radicals  showing f l u o r i n e  emphasis w i l l of  interaction.  carboxylic  i n this  acids  loss  and t h e i r  case t o o r e s u l t s  a C-F bond, p r e f e r a b l y  the  I n what f o l l o w s , t h e  be o n t h e s p e c i e s o b t a i n e d by t h e i r r a d i a t i o n  fluorinated  irradiation or  t h e r e have been r e l a t i v e l y fewer examples o f  1  9  a t an a p o s i t i o n ,  o f t h e c a r b o x y l g r o u p may a l s o  first  F hyperfine systematic  was  carried  gle  crystals  perature the  analysis  o f the f l u o r i n e  of trifluoroacetamide with  and s t u d i e d  2  Teflon, the  hyperfine  y-rays  structure sin-  a t room tem-  R o g e r s and K i s p e r t ^  the trifluoroacetamide reported  at 77°K  2 1  ^  and,  irrain  by L o n t z and G o r d y , t h e y  found  C F 3 . Some e v i d e n c e was a l s o  given  radicals.  The two f l u o r i n e s  i n the  w h i c h were f o u n d  t o be e q u i v a l e n t  a t room  radicals,  the presence o f  CF^COIS^ r a d i c a l s ,  to resolve  t h e b r e a k i n g o f a C-F bond l e a d i n g t o  t o the r a d i c a l  trifluoromethyl  sometimes  They i r r a d i a t e d  2  2  addition  were a b l e  o u t by L o n t z and G o r d y ^ ^ .  and o b s e r v e d  although  occur.  i n the i r r a d i a t e d  formation o f CF CONH . L a t e r ,  diated  for  structure  The  i n t h e b r e a k a g e o f a C-H  A l t h o u g h R e x r o a d and Gordy the  derivatives.  NH2CO  t e m p e r a t u r e , d i d n o t remain so a t 7 7 ° K .  Cook and c o - w o r k e r s ^ ) 2 2  of  single  crystals  out the X - i r r a d i a t i o n  o f m o n o f l u o r o a c e t a m i d e and f o u n d t h e c l e a -  vage o f a C-H bond, l e a v i n g established  carried  the r e l a t i v e  t h e CHFCONH  2  r a d i c a l . They  signs of the p r i n c i p a l  also  elements o f  5  the  fluorine hyperfine  tensor  e l e m e n t s were n e g a t i v e , positive and  as w i t h o t h e r  Whiffen^-^  rosuccinate  and f o u n d  t h e two  the i s o t r o p i c hyperfine fluorine-containing  smaller  coupling  and obtained  being  r a d i c a l s . Rogers  i r r a d i a t e d s i n g l e c r y s t a l s o f sodium  perfluo-  (C00~)CF CF(C00~).  the r a d i c a l ,  s t u d y was i n t e r e s t i n g i n t h a t ned  that  2  t h e r a d i c a l so o b t a i n e d  contai-  $ f l u o r i n e atoms. The two 3 f l u o r i n e s were f o u n d t o be  n o n e q u i v a l e n t and, as e x p e c t e d , had lower h y p e r f i n e t h a n t h e a f l u o r i n e atom;* T h i s  i s logical  s h o u l d have l o w e r s p i n d e n s i t i e s  The  trifluoromethyl  since  splittings  the $ f l u o r i n e s  than t h e a ones.  r a d i c a l , C F , f i r s t reported 3  by  (21) R o g e r s and K i s p e r t diated by  liquid  was a l s o  v  F e s s e n d e n and S c h u l e r ^ ) .  They a l s o  4  w h i c h gave C F , C H F 3  Another w  2  at liquid  nitrogen  and C H F r a d i c a l s 2  temperature  respectively.  i n t e r e s t i n g s t u d y was c a r r i e d o u t by  A t room t e m p e r a t u r e , 2  peratures,  Srygley  the i r r a d i a t i o n resulted  i n t h e f o r m a t i o n o f CF COONH^ w h i c h e x h i b i t e d  racteristic  radical  the i r r a d i a -  h o i r r a d i a t e d a s i n g l e c r y s t a l o f ammonium t r i -  fluoroacetate. mainly  studied  studied  d i f l u o r o - and m o n o f l u o r o m e t h a n e s i n t h e  k r y p t o n and xenon m a t r i c e s  Gordy(^)  in irra-  h e x a f l u o r o e t h a n e and t e t r a f l u o r o m e t h a n e  tion of t r i f l u o r o - ,  and  f o u n d t o be p r e s e n t  fluorine hyperfine  structure.  the i r r a d i a t i o n r e s u l t e d 4  However a t l o w tem-  i n the formation o f the  i o n CF COONH "*. The u n u s u a l l y 3  the cha-  large  hyperfine  split-  6  tings  showed  that the f l u o r i n e s possessed  a very  large  spin  density.  In t h e p r e s e n t ssium and  salt  study,  a single  of difluoromalonic acid,  s t u d i e d a t room t e m p e r a t u r e .  ted mainly  CF (COOK) 2  The X - r a y  i n the formation of CFfCOO"^  C-F bond. A weak c e n t r a l of CO2""  crystal  radical.  a l s o observed  line  In a d d i t i o n ,  indicated  of the o r i e n t a t i o n s , The sample was t h e n  irradiated  with  the p o s s i b l e  2  of a  presence  weak i i n e s  were i n most  t o o weak o r u n r e s o l v e d .  yrays  c i e s o b t a i n e d was a g a i n t h e C F ( C O O ~ )  resul-  by t h e c l e a v a g e  c o u l d be made s i n c e ,  t h e y were e i t h e r irradiated  was  irradiation  some e x t r e m e l y  b u t no measurements  2  of the pota-  and t h e m a i n s p e -  radical.  The c e n t r a l  peak became s t r o n g e r b u t was q u i t e b r o a d ,  partly  overlap with other  c o u l d n o t be c a r r i e d  out'ton t h i s identified stronger  line.  lines However  i n the case  i n one t y p e  C F 2 C 0 0 "  s a m p l e , were now much  l e d to the i d e n t i f i c a t i o n  radicals.  planar while  -COO g r o u p was a l m o s t  since  so t h a t t h e  i n the other, the plane of  p e r p e n d i c u l a r t o t h e FCF p l a n e . I t  was f o u n d  t h a t t h e p l a n a r r a d i c a l was formed  ties  the non-planar  than  o f two  T h e s e were d i s t i n g u i s h a b l e  t h e -COO g r o u p was i n t h e FCF p l a n e  r a d i c a l was c o m p l e t e l y the  t h e weak l i n e s w h i c h c o u l d n o t be  of X-irradiated  and measurements  types o f the  and measurements  due t o t h e  one.  i n larger  quanti-  CHAPTER  TWO  THEORETICAL  A proper understanding of the concept H a m i l t o n i a n and tial  be  and  found  out of the experiments  d i s c u s s i o n of the r e s u l t s .  But  f o r convenience  and  details  that  i n c a r r y i n g out v a r i o u s c a l c u l a t i o n s  be g i v e n i n t h e f o l l o w i n g p a g e s . mulas and pression  the  other d e t a i l s  of these and  can  oria  is directly i n -  i n t h i s work,  Some d e r i v a t i o n s  o f the  are g i v e n i n the appendices.  f o r t h e h y p e r f i n e c o u p l i n g was  t h e d e r i v a t i o n has  presen-  self-sufficiency,  summary o f t h e p a r t o f t h e t h e o r i e s  volved  but  The  and  i s essen-  i n a g r e a t number o f b o o k s , r e v i e w a r t i c l e s  g i n a l papers. brief  spin  of the theory of m o l e c u l a r o r b i t a l s  f o r the c a r r y i n g  tation  of  d e r i v e d by  will for-  The  ex-  Dr. W.  C. L i n  n e v e r t h e l e s s n o t been p u b l i s h e d e l s e -  where .  (A).  The  observed  d e t e r m i n a t i o n o f t h e g - and g-values  It  was  and  hyperfine  netic  system  whose d e g e n e r a c y  from  the  splittings:-  shown by P r y c e ^ G )  t h a t the e n e r g i e s o f the ground  the A-tensors  a n c  j  Abragam and  Pryce  s t a t e o r s t a t e s o f a paramag-  o r near degeneracy  i s removed  by  8  an e x t e r n a l m a g n e t i c n i a n which ning  field,  involved only  t o the o r b i t a l s  form. T h i s type  energy o p e r a t o r  f o r the  study  spin  A clear  division  and  given  ged  of  labour  the v a r i o u s i n t e r a c t i o n s  i n equation  3H.g.S  =  Here the  first  effective field  .(.JL) i n w h i c h t h e  +  ^ S . A  electronic  a symmetric  3x3  The  The  coupling  arising  from the  The  lear  spins  1  third  (  constant  effect.  A^ ).  by  parameters  the  from  1  the  i  )  t h r e e terms are  . l  (  i  )  the  2^g  -  ( i ) N  interaction  8 i s the  the  pre-  into  interaction  8 H.I  the  between  the  e x t e r n a l magnetic  account the  the  g is  spin-orbit  hyperfine  between S and  interactions  the e x t e r n a l magnetic  (1)  ( i )  N  the  hyperfine coupling tensors  term r e p r e s e n t s  and  arran-  Bohr magneton and  second term r e p r e s e n t s  I ^ ) , the  with  i n v o l v e d . For  terms  magnitude.  t e n s o r which takes  spin vectors  these  s p i n v e c t o r S and  coupling  lear  i s achieved  o f t h r e e terms i s s u f f i c i e n t  term r e p r e s e n t s  v e c t o r H.  has  convenient  u s u a l l y c o n s i s t s o f many  i n the d e c r e a s i n g order of  H  which  interpretation.  sent purpose, a Hamiltonian is  pertai-  o f p a r a m a g n e t i s m by e l e c t r o n  A spin Hamiltonian representing  Hamilto-  of Hamiltonian  i s t h e most  s e p e r a t i o n o f t h e work o f d e t e r m i n i n g t h e work q f t h e i r  a  i n c l u d e d i n the parameters  'Spin Hamiltonian'  resonance.  d e s c r i b e d by  spin operators. A l l effects  c o u l d be  must assume a t e n s o r i a l b e e n known as  c o u l d be  field,  of the g  M  ^  nucbeing nucbeing  9  the g - f a c t o r s f o r 1 ^ , i n t h i s case, the F n u c l e i and 3 1 9  N  i s the n u c l e a r magneton. In the present case, the summation i n the second and the t h i r d terms i s over the two f l u o r i n e n u c l e i f o r CF COO~ r a d i c a l s and there w i l l be no summation 2  i n the case o f CF(COO~) . 2  I t i s shown i n Appendix A t h a t the Hamiltonian in  equation  (1) y i e l d s the energy l e v e l s given by  E(Mg,M < ,M.,. ) 1J  (2)  T  C  (A)  S  z  and I  v z  = g£HM  + %|AMM ,H) | (Mj +M-r (1)  s  l)  (  2  )  S  and M - i =<i_  where M =<S >=±h ~  given  ( l )  )  >=+%  (2) (3)  .  *  ' being the z-components o f the operators S and  l ( i ) respectively,  g i s the observed  g - f a c t o r , H i s the mag-  n i t u d e o f the a p p l i e d f i e l d and A' i s given by A'(Me,H)  lt-2  =  :V mk - ^ N / S> mk>|/ A  e  H  M  (4)  5  N  T.m  where k,m=l,2,3 r e p r e s e n t i n g the three c o o r d i n a t e axes, A ^ i s the element o f the h y p e r f i n e t e n s o r i n the m*"* row and the 1  k  t n  column, 6;«& i s the Kronecker d e l t a and 1 (m=l,2,3) a r e ' mk m  the d i r e c t i o n c o s i n e s o f the magnetic f i e l d . The energy l e v e l s f o r CF(COO~) cating  2  are shown i n f i g u r e  1 with the s o l i d l i n e s  indi-  the allowed o r s t r o n g t r a n s i t i o n s and the broaken l i n e s  showing the f o r b i d d e n o r weak t r a n s i t i o n s . S i m i l a r l y f i g u r e 2 shows the energy l e v e l s f o r CF COO~ r a d i c a l with two f l u o r i n e 2  nuclei.  10  i 1_  2  2  1  1_ 2  2  J L  /  8 H  + lA'(l.H)  H-^A'(^,H)  | g j H +4"A'(-1,H)  2  -^g/3H -  2  FIGURE 1  Energy L e v e l s f o r a R a d i c a l One  1^  G  t  l 9  l ^ I . H )  Containing  F Nucleus. The F u l l Lines Show  t h e Allowed T r a n s i t i o n s While the Dotted Lines I n d i c a t e the Forbidden T r a n s i t i o n s .  11  FIGURE 2  Energy L e v e l s Two  1 9  F  for a Radical  N u c l e i . The  FT.11  Containing  Lines  A l l o w e d T r a n s i t i o n s V/hile t h e Lines  I n d i c a t e the  Forbidden  Show  the  Dotted Transitions.  It the observed  i s shown  i n Appendix A t h a t  hyperfine s p l i t t i n g ,  from e q u a t i o n ( 4 ) ,  HFS, c a n be e x p r e s s e d  k m  k *  as  m  where  G = 2g e H/g e N  g  e  . \  e  N  being the f r e e - e l e c t r o n g - f a c t o r . g j-> t  Appendix A t h a t  g  obs  =  0  the observed  s  V X ) S £  l  m  l  i m "n where g ^ a r e ' t h e e l e m e n t s m  With  equations  n  g  m  i  g  n  g-value  i s g i v e n by  •  (5) and  (7) t h e e l e m e n t s  and  the h y p e r f i n e s p l i t t i n g s  a r e fme^svfred f o r v a r i o u s o r i e n -  structure:-  a l l modern c a l c u l a t i o n s o f t h e m o l e c u l a r  structure  a r e based  on what i s known as t h e  The a p p r o x i m a t i o n s  involved i n  these equations are the s e l f - c o n s i s t e n t f i e l d ment o f t h e e l e c t r o n  tals  g value  magnetic'field.  R o o t h a a n \ Xequations  of atomic  o f t h e g-  once t h e o b s e r v e d  (B). C a l c u l a t i o n o f the m o l e c u l a r e l e c t r o n i c  electronic  ( 7 )  o f the g-tensor.  t h e A - t e n s o r s c a n be d e t e r m i n e d  Almost  (6)  shown i n  i  and  t a t i o n s of the  I t i s also  ~  orbitals  repulsions  (SCF) t r e a t -  and t h e l i n e a r  combination  (LCAO) d e s c r i p t i o n o f t h e m o l e c u l a r o r b i -  (MO). I n p r a c t i c e , a l i m i t e d s e t o f a t o m i c  orbitals is  13  used  and t h e y  ment i s a l s o  are g e n e r a l l y of the S l a t e r known a s t h e H a r t r e e - F o c k  t y p e . The SCF  (H-F) SCF i n t h e s e n s e  that  a d e t e r m i n a n t a l ^formgof t h e t o t a l m o l e c u l a r  tion  i s used  original  i n s t e a d o f the simple  Hartree  equations  treatment.  product  form  used  F  c... C= uv v i  S l^j  .e. uv v i i  <j)v, i n t h e  LCAO  r  (i|^=£<J> c J ,  description  of  i s the o r b i t a l  v  between t h e two a t o m i c  of the  energy  orbitals  (8) the  atomic  and S V  i s the overlap  y v  cj>^ and <J>. F ^  potential  . which d e s c r i b e the e l e c t r o n i c  consistency for  solved to find  till,  of  i s achieved  the c a l c u l a t i o n  then  put  energies, part of the l a t t e r  v  i s the matrix of kinetic  being  dependent  structure.  guessing  equations are  a new s e t o f c ' s . The p r o c e s s the output  i s repeated  s e t o f c's equals  s e t o f c ' s c a n be u s e d  Self-  a s e t o f the c's  o f t h e F * s . The e i g e n v a l u e  i n one i t e r a t i o n ,  s e t . The f i n a l  by f i r s t  orbitals  o r b i t a l \l>.  molecular  element o f t h e Roothaan o p e r a t o r which c o n s i s t s  the c  i n the <  c  . are the c o e f f i c i e n t s vi  on  func-  form, t h e Roothaan x7  I n component  where Vsxo^c  and  wave  may be w r i t t e n a s f o l l o w s .  E v  treat-  the i n -  £or t h e c a l c u l a t i o n  the molecular p r o p e r t i e s .  At still  this  level  quite intractable  o f approximation,  the c a l c u l a t i o n i s  due t o t h e d i f f i c u l t i e s  the e v a l u a t i o n o f m u l t i c e n t e r i n t e g r a l s operator. This i s specially  present  involved i n i n the energy  true f o r l a r g e r molecules  which  14  a r e , however, o f most c h e m i c a l i n t e r e s t s ? to s i m p l i f y  Many  approximations  t h e R o o t h a a n Q ) e q u a t i o n s have t h e r e f o r e b e e n  devi-  (29 — 33) s e d among w h i c h t h o s e o f P o p l e  and  h i s co-workers  t o be most c o n v e n i e n t . T h e s e a p p r o x i m a t i o n s lect  of c e r t a i n  empirical in  integrals  parameters.  and  There  O v e r l a p ) (29-32)  a  n  d  t  h  a r e two  e  involve  the  neg-  the s u b s t i t u t i o n of o t h e r s  t h e P o p l e method: t h e CNDO  tial  seem  v  levels  (Complete  I N D  o  of  by  approximation  Neglect of  Differen-  (Intermediate Neglect of  (3 3 \  Differential of the type <J>^ and  Overlap) <j> <j>  .  The  former  to d i f f e r e n t  neglects a l l products  n e g l e c t s only those  atoms. T h e r e  i s one  p r e s e r v e d as  i n which  important  i n t h e P o p l e method w h i c h needs m e n t i o n i n g .  r i a n c e of Roothaan^J is  A  while the l a t t e r  belong  aspect  v  The  inva-  e q u a t i o n s under u n i t a r y t r a n s f o r m a t i o n s  i t s h o u l d be  but  this,  then, r e q u i r e s  further  approximations. I n what f o l l o w s ,  a s h o r t summary o f t h e method w h i c h  i n c l u d e s o n l y t h e most i m p o r t a n t e q u a t i o n s u s e d lations will dices  B,  C,  be  given. Greater d e t a i l s  D and  paramagnetic  E . A l s o we  systems,  case of o p e n - s h e l l  a spin  3 spin.  be c o n c e r n e d  i s s i m p l e r and  i n appen-  only with  the  is different  c a n be  case  c o n s i d e r e d as  systems.  I n an o p e n - s h e l l s y s t e m with  found  calcu-  i . e . , t h e o p e n - s h e l l m o l e c u l e s . The  of c l o s e d - s h e l l molecules afj s p e c i a l  will  c a n be  i n the  f r o m q,  ( C o n v e n t i o n a l l y p £ q ) . We  p,  t h e number o f  electrons  t h e number o f e l e c t r o n s s t a r t w i t h f o r m i n g two  with sets  15  of LCAO MO's, one f o r a l l e l e c t r o n s with a s p i n and the other f o r those with 6 s p i n . Thus we have  y  -"-M  and  ib. 1  B  = V^c  3  An u n r e s t r i c t e d  < f >  (10)  ( i n the sense t h a t the r e s t r i c t i o n of- the  assignment o f one a s p i n and one 6 s p i n t o the same space orbital  i s relaxed) Hartree-Fock m o l e c u l a r wave f u n c t i o n  has the form V = |i|» (l)a-(l)y» (2)o(2) a  if (p)(x(p)i (^+l)&i(|+l) p y 1 •- • -•" '••  0  1  a  2  e  ...  4/ (n)B(n)| q e  •. (11)  where n i s the t o t a l number o f v a l e n c e e l e c t r o n s (n = p + q ) 1 The p a r t i a l charge d e n s i t y and bond order m a t r i c e s c o r r e s ponding  t o equations  P  a  yv  (9) and (10) can be w r i t t e n as  »'Vc? c  a  Z—r iy i v  (12)  i  and P  3  yv  =  r'cf c? y *v 1  where- the summation  (13)  i s c a r r i e d over the occupied v a l e n c e -/orbi-  t a l s o n l y . A l s o we d e f i n e a s p i n d e n s i t y matrix Q  yv  =  P« - pf yv yv  (14)  16  The v a r i a t i o n a l  treatment o f the o r b i t a l  l e a d s t o the Roothaan's simplification convenient  equations  c's  ( A p p e n d i x B) w h i c h ,  by CNDO a p p r o x i m a t i o n s  form f o r the Roothaan  coefficients  after  (Appendix C ) , g i v e  operators F  and  a  a  .  yv  yv  Thus  ' W - ^ V V + ^ A A - V - ' C - »yy, T J L A - S <. P B B - V T A B +  < 1 5 )  and  F and  a  yv  similar  (16) we  = B£r,S - P AB yv yv'-AB  tt  e x p r e s s i o n s f o r F^ and F^ . I n e q u a t i o n s yy yv  AB  *<  =  e  A  +  S  where g£ i s an e m p i r i c a l t h e atom A and  the t o t a l  g^  B  '  (15)  (  parameter  i s termed  B  / J y  1  and  )  d e p e n d i n g on t h e n a t u r e is  by  yy  where  (18) P  I ^ and A^  7  a bonding parameter.  c h a r g e on atom A d e f i n e d  AA  affinity  (16) '  have  B  of  v  yv  =  pa yv  +  P  B  y  V  are the atomic i o n i z a t i o n p o t e n t i a l respectively  f o r the o r b i t a l  s i o n o f the t h e o r y , only  and  <J>^. i n an e a r l i e r  the i o n i z a t i o n p o t e n t i a l  p l a c e o f t h e a v e r a g e ^(^y+Ay)?  ^  u  t  electron  later  on,  was  i t was  ver-  used i n  found  that  17  the r e s u l t s improved by u s i n g the average o f the i o n i z a t i o n p o t e n t i a l and the e l e c t r o n a f f i n i t y . and Y ^ B *  S  Z  A  i s the core charge  ^he 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 c a l c u l a t e d as the  two-center coulomb i n t e g r a l i n v o l v i n g valence s f u n c t i o n s centered on atoms A and B. Thus  Y  AB = i T A S  The  y  ( 1 ) S  B  i  ( 2 ) d T  d T  2  (  3  B  u v  A B  S  Y V  -^P^Y  -P„„V  A B  } {z Z R^-P >  B  R A  +  A  +P .P  BB BA  A  )  'y v  J  + -Z r E E { 2 P A<B V y v  where R  9  t o t a l energy o f the molecule can be w r i t t e n as  L  A  1  B  y  A A  V  A B  l]  AA BB ABjJ A  (20)  i s the d i s t a n c e between the c e n t e r s o f atoms A and  B, while V,  =  AB D  Z  By ' A B  (21)  When we i n c l u d e the products  o f two atomic wave ,  f u n c t i o n s a t the same center as i n the INDO (Appendix D), equations  F  yy  =  u  p y  +  i f  {  p  X X  (  ^ l  x  x  approximation  (15) and (16) become  )  -  p  x x  (  ^  x  l y M  }+  ]T<PBB-ZB> Y A B  B T A  x  (22)  where y i s on c e n t e r A, and Fa = (2P - P ) ( y v | y v ) - p (yylvv) yv yv yv' yv a  a  1  1  (23)  18  where y^v  , but both  a r e c e n t e r e d on A.  integrals  (yv|Xa) a r e d e f i n e d  In e q u a t i o n  Using  slater  are reduced  U  type t o the  S  s  = -Js(I  U  p  = -h ( I  from  + Ap)-(Z  A  i n Appendix  5  , but  tes with  total  contaminating similar  +i(Z  single  was  used.  u  2 5 )  - fjG  (26)  1  and  F  2  A  - f)F  •  2  are S l a t e r ' s  interaction  (27)  notations  integrals  as  out-  the m o l e c u l a r  (11)  i s not  wave f u n c t i o n o f  an e i g e n f u n c t i o n o f  the the  c o n t a i n s t h e components o f s e v e r a l s p i n  spin  S>%.  I t i s necessary  components by  annihilation,  greater d e t a i l  U^  D.  t h e use  3 4  The  1  electron  to that of Lowdin^ ).  of  A  A A  Here G  type g i v e n i n equation 2  A A  - J )Y +iG +(2/25) ( Z  Unfortunately,  S  integrals  <  I  A  for various one-center  operator  the core  (24)  2  " ^HH  - %)Y  boron t o f l u o r i n e .  lined  1  and  S  p  a  p orbitals,  V  +  + A )-(Z  s  v  form  " -^S  f o r hydrogen,  U  s and  the  as  (yv|Xa) = j ^ i j ( l ) * J ( 2 ) { l / r > < | > ( l ) < | > ( 2 ) d T d T i 2  (23)  as  of a projection  E.  contamination  these  operator,  I n t h e p r e s e n t work, t h e  i n t r o d u c e d by Amos and  problem of s p i n i n Appendix  to a n n i h i l a t e  sta-  method  Snyder^  i s treated in  3 5  ),  CHAPTER THREE  EXPERIMENTAL  (A).  P r e p a r a t i o n of the  The in  this  of  the  study  Sample;-  potassium was  salt  of d i f l u o r o m a l o n i c acid  k i n d l y s u p p l i e d by  P r o f e s s o r A.  used  E.  U n i v e r s i t y o f B i r m i n g h a m . I n h i s method o f  Pedlar  prepara-  (36) tion  35  dized  by  acetone  gm.  6.38  of octafluorocyclohexa-1,4-diene  gm.  of potassium  (the mole r a t i o  d u r e A o f B u r d o n and decolorized  and  dissolved  i n 400  charcoal,  and  evaporated. ml.  n e u t r a l i z e d with  evaporated  dipotassium  by  salt  was  ml.  1:1.2) u s i n g  of  proce-  ethereal extract of  dried  with  solid  residue  the  a n h y d r o u s magne(30 gm.)  was  water, d e c o l o r i z e d with  0.2N  aqueous p o t a s s i u m  external indicator.  under reduced  single  crystals  slow e v a p o r a t i o n . t h a t the  would p r e s e n t were f o u n d  an  oxi-  pressure  The  to give  hydrosolution  22  gm.  of  difluoromalonate.  The tion  The  of d i s t i l l e d  x i d e u s i n g m e t h y l r e d as then  '. The  v  aqueous s o l u t i o n was  sium s u l f a t e  was  p e r m a n g a n a t e i n 100  diene:KMnO^ was (37)  Tatlow  was  acid  The  itself  were grown f r o m aqueous reason was  handling problemsi  t o be  f o r u s i n g the  highly hygroscopic The  crystals  so  solu-  potassium and  obtained,  the monohydrate from the m i c r o a n a l y s i s which  20  gave 15.36% c a r b o n lated was  values  carried  of out  The acid  itself  coated  by  in  or a f t e r  approximately  different  s p e c t r a was of  not  t o be  analysis  as h y g r o s c o p i c  as  crystals  the  were  (a f a s t - d r y i n g m a t e r i a l made by U.  Ohio, which d i d not  3><2xl mm.  The  calcu-  department.  added p r e c a u t i o n , t h e  irradiation).  The  in  g i v e an e s r  S.  signal  c r y s t a l s were q u i t e r e q u l a r another  and  were  size.  X - i r r a d i a t i o n was  placed  rance  I t was  low  a t the  f o r dry immersed  other hole  temperature box  out with  intensity  that after not  the  precooled  through  sample  a T e f l o n b o t t o m and  by  passing  i n a Dewar c o n t a i n i n g l i q u i d  t h e box  of the  source esr  a b o u t two  of the h o l e s ^ s e r v e d  as t h e e x i t  a 40kV  hours  critical.  irradiation,  with  s i d e s . One  n i t r o g e n gas  served  rays entered  found  f u r t h e r d o s a g e was  in a cylindrical  holes  carried  p e r i o d s of time'and the  observed.  For  coil  Borda of t h i s  f a c e s p e r p e n d i c u l a r t o one  irradiation,  two  9,  16.24% r e s p e c t i v e l y .  found  as an  as a g a i n s t t h e  Irradiation:-  The for  P.  Tygon p a i n t  shape w i t h  (B).  Mr.  s a l t was  Stoneware, A r k r o n before  15.90% f l u o r i n e  15.38% and  and,  with  and  as  was having  the  ent-  through  nitrogen.  f o r the n i t r o g e n gas. t h e T e f l o n b o t t o m and  a  The The  the  Xirra-  21  d i a t i o n was  carried  The cobalt  f o r 2-3 h o u r s .  y - i r r a d i a t i o n was done i n GAMMACELL  60 i r r a d i a t i o n  unit  manufactured  E n e r g y o f Canada L t d . , C o m m e r c i a l Ottawa.^Increasing effect  after  The  the  Products,  the  a  Atomic  P. 0. Box 93,  d i d n o t seem t o h a v e any  a minimum e x p o s u r e o f 20 m i n u t e s w i t h a s o u r c e  s t r e n g t h o f about  after  the dosage  by  220,  250,000  Rads/hour.  c r y s t a l s were s e e n t o a c q u i r e a b r o w n i s h  irradiation.  tint  The c o l o r was more e a s i l y j o b s e r v a b l e i n  case o f X - i r r a d i a t i o n .  (C).  The S p e c t r o m e t e r : -  The measurements were c a r r i e d  o u t on an X-band e s r  s p e c t r o m e t e r h a v i n g a 12" V a r i a n V-3900 magnet mounted on a V-3921 r o t a t i n g s c a l e which vertical  allowed the  ing  to  R e g u l a t o r which p r o v i d e d d i r e c t  kilogauss. unit  be r o t a t e d  by a V a r i a n V-2501 F i e l d i a l  t i o n by d i g i t a l in  magnet  l o c k and an a z i m u t h a l about  a x i s w i t h a p r e c i s i o n o f ± 0 . 2 5 ° . The m a g n e t i c  was c o n t r o l l e d Field  b a s e w i t h a cam a c t i o n  control dials  T h i s r e g u l a t o r was  by w h i c h  the f i e l d  field  Mark I I M a g n e t i c  magnetic  indicating  the  field  the f i e l d  selec-  intensity  a l s o provided with a scann-  c o u l d be v a r i e d  uniformly  in a  22  preselected  time  interval.  The m a g n e t i c tions tem  c a u s e d by l i n e  voltage  n o i s e s and m a g n e t i c  controlled  withithe vered  fluctuations,  system changes,  u s e d was  f r e q u e n c y range  the f i e l d  modulation c o i l s  The low t e m p e r a t u r e  hooked o n t o t h i s  through a  cavity  magnet r o t a t i o n s .  is  temperature  klystron  temperature  cavity  the microwave b r i d g e . T h i s freely  around fea-  on t h e v e r t i c a l  so t h a t ,  f o r such purposes, a V a r i a n could not allow  The r e s o n a n t f r e q u e n c y o f t h e c a v i t y l o a d e d w i t h a Dewar  was  f o r keeping  a t 77°K. A b l o c k d i a g r a m o f t h e s p e c t r o m e t e r  shown i n F i g u r e  field  3.  was m o d u l a t e d  ted  by t h e e l e c t r o n i c s  tic  Frequency  by a 100 k c . u n i t c o n s t r u c -  group o f t h i s department.  C o n t r o l U n i t used  f r o m t h e d e t e c t o r was  The Automa-  to lock the k l y s t r o n  t h a t o f t h e r e s o n a n t c a v i t y , was  signal  deli-  a c c e s s o r i e s , however, c o u l d n o t be  t h a n 9.5 Gc. when i t was  The  to  sys-  were f r e e t o r o t a t e  V-4531 M u l t i - p u r p o s e c a v i t y was u s e d , w h i c h  the  varia%  electronic  a V a r i a n V-153/6315  a l l o w e d t h e magnet t o be r o t a t e d  axis.  less  against  o f 8.5-10.0 Gc. The k l y s t r o n  cavity without e f f e c t i n g  ture  stabilized  t h e m i c r o w a v e t o a V a r i a n V-4533 c y l i n d r i c a l  which  the  was  s e n s o r p r o b e , mounted on one magnet p o l e c a p . The  klystron oscillator  in  field  frequency  a l s o made l o c a l l y .  fed into  a Lock-in  The  amplifier,  KLYSTRON POWER  SUPPLY POWER M E T E R  KLYSTRON  THERMISTOR ISOLATOR  MOUNT  VARIABLE  I  ATTENUATORS 20 dB  L  r  DIRECTIONA  DIRECTIONAL  COUPLER 20  dB  20  dB  20  dB  LOCK-IN  FREQUENCY COUNTER  AMPLIFIER  PHASE  ||  DETECTOR 3  FIELDIAL P O W E R SUPPLY  R E C O R D E R S  T  IOO K c . IMODUL A T O R  FIGURE 3  B l o c k D i a g r a m o f t h e 100 k c  B  ESR  lOOKc. (OSCILLATOR  Spectrometer  P H A S E  (j  SHIFTER  i|  24  M o d e l RJB cations  4 87 s u p p l i e d  I n c . and  was  by E l e c t r o n i c s ,  r e c o r d e d on  Missiles  a Hewlett-Packard  d e r M o d e l M o s e l e y ,700'5B.'vA H e w l e t t - P a c k a r d with  a 5256A P l u g - I n u n i t  f r e q u e n c y . The Models  i2i-3L  taining of  Fieldial through  samples  was  c o u n t e r . The  from the observed version  factor.  changed by  was  '  directions  calibrated  u s i n g . NMR  The  varied  by  accurate f i e l d  probes., con-  a variable'RF oscibe observed- on the.  could  a shunt of s u i t a b l e  .  .  a polarizing  691, U n i v e r s i t y - P a r k , could  i n s i d e , the c a v i t y . .  New  be J  '  Mexico  rotated '  ' .'  .'  were  c.' The  crystal  by * Magna D e v i c e s , ' •  8 8 0 0 1 , by  • \  found  throughout  the.use  in a vertical ,;.,,'•'  *  these;, d i r e c t i o n s . •  d e s i g n a t e d as a, b and  Box  be-  microscope,' the  t h e o r t h o g o n a l r e f e r e n c e axes  mounted i n a sample h o l d e r s u p p l i e d  could  ',  of the c r y s t a l  was  the c r y s t a l  by. a.' c o n -  length.  . \  of, maximum e x t i n c t i o n  t o be  t h e n be o b t a i n e d  f r e q u e n c y range' o f t h e " p r o b e s  e x a m i n a t i o n under  and were a r b i t r a r i l y  which  counter  by M a g n i o h I n c . ,  t o be .parallel.-'.to t h e :edges.;' C o n s e q u e n t l y were c h o s e n  frequency  f r e q u e n c y s i m p l y by m u l t i p l y i n g  applying  On  X-Y.recor  to"measure the microwave  frequency could  (D) ,< M e a s u r e m e n t s : -  ,  Communi-  o f a' p r o t o n - l i t h i u m m i x t u r e . The' p o s i t i o n  and. t h e r e s o n a n c e  frequency  used  124-3L' s u p p l i e d  the p r o t o n resonance  llator  was  and  .\  of  plane / '/.-'.  25  Although esr  the c r y s t a l  observations indicated  and,  s t r u c t u r e was n o t known, t h e  that  t h e c r y s t a l was  i n general, four nonequivalent s i t e s  When t h e m a g n e t i c only  two s i t e s  parallel  field  was  could  and when t h e m a g n e t i c  t o any o f t h e t h r e e a x e s , one s i t e were u s e d  be o b s e r v e d .  i n any o f t h e ab, be o r c a p l a n e s  were o b s e r v e d  These c r i t e r i a  orthorhombic  to align  was  the c r y s t a l  field  was  observed.  i n the magnetic  field.  The  c r y s t a l was  means o f f a s t - d r y i n g Good Y e a r  Tire  attached to the c r y s t a l  P l i o b o n d Cement m a n u f a c t u r e d  Company and i n t r o d u c e d i n t o  with  the a-axis i n a v e r t i c a l  llel  t o the magnetic  ted  as f o l l o w s :  orientations and  i f any  several  by r o t a t i n g  on e i t h e r  ded  showed f o u r  crystals tals.  a desired  sites,  rotation until  s i d e o f the i n i t i a l o r i e n t a t i o n  a t every  The c r y s t a l  field.  until  slightly  one  site  t a k e n t o be 0 ° w i t h b  The s p e c t r a were t h e n  then, r o t a t e d  angle  each o f the spec-  5° i n case o f t h o s e on y - i was  correc-  the c r y s t a l  1 0 ° i n t h e c a s e o f measurements on  and e v e r y  were  N e x t t h e magnet was r o t a t e d  t o the magnetic  first  and t h e b - a x i s p a r a -  S l i g h t misalignments  was o b s e r v e d . T h i s o r i e n t a t i o n was parallel  the  the c a v i t y ,  t h e magnet t h r o u g h  vertical  showed two s i t e s .  by  s p e c t r a were t a k e n a t d i f f e r e n t  of the spectra  was g i v e n a s l i g h t tra  field.  direction  h o l d e r by  vertically  r r ;  recor-  X-irradiated  adiated  through  crys90°  26  to bring  the b-axis  t a t i o n with  to the v e r t i c a l  the c-axis  parallel  direction  and t h e o r i e n -  t o the magnetic f i e l d  was  o t a k e n t o be 0 . The s p e c t r a in  xn t h e c a - p l a n e were t h e n  a s i m i l a r manner. F i n a l l y  tached  t h e t c r y s t a l was  and mounted.back o n t h e h o l d e r  pendicular t i o n with  t o the magnetic f i e l d . the a-axis  parallel  with  For a rectangular  the c-axis  t o the magnetic f i e l d  c a v i t y was u s e d holder  n o t be u s e d . I n s t e a d ,  per-  was  i n the ab-plane.  accuracy  taken This  o f ±0.25°.  low t e m p e r a t u r e work, as p r e v i o u s l y  r o t a t e d . The c r y s t a l could  taken o u t , de-  In t h a t case the o r i e n t a -  t o b e y O ° and t h e s p e c t r a were r e c o r d e d p r o c e d u r e e n s u r e d an e s t i m a t e d  taken  mentioned,  and t h e magnet c o u l d  n o t be  s u p p l i e d by Magna D e v i c e s  also  t h e c r y s t a l was mounted on t h e  edge o f a P e r s p e x r o d c o n n e c t e d t o a c y l i n d r i c a l  piece of  T e f l o n which c o u l d  be r o t a t e d  be r o t a t e d . The c r y s t a l  in  a h o r i z o n t a l plane with  of  a p r o t r a c t o r . Below t h e p r o t r a c t o r , t h e r e  cylindrical  piece  mouth o f a s m a l l placed passing cooled uid  an a c c u r a c y  could  o f T e f l o n which c o u l d Dewar  dry nitrogen  was  cooled  the help  a hollow  be f i t t e d  ( F i g u r e 4 ) . The Dewar was  i n t h e c a v i t y and t h e c r y s t a l was  a t the i n turn  down by %  g a s t h r o u g h t h e Dewar. The gas was p r e -  by t h e p a s s a g e t h r o u g h a c o p p e r c o i l  nitrogen.  immersed i n l i q -  The wave g u i d e s n e a r t h e c a v i t y were k e p t  f r o m c o n d e n s e d m o i s t u r e by p a s s i n g them.  o f ±2° with  dry nitrogen  free  gas t h r o u g h  27  PROTRACTOR  POINTER  NITROGEN  O U T L E T S T E F L O N TO  HOLD  PERSPEX  li  CYLINDER CRYSTAL  ROD  1  DEWAR  NITROGEN  FIGURE 4  The C r y s t a l  H o l d e r W i t h Dewar Used  Temperature  Worko  INLET  f o r Low  28  •••,  ' ;it,h'-this ..set-up w  be c o r r e c t e d  orilyby  warmed s l o w l y  0  s i n c e the ;method l  -  was used o n l y  t o s e e whether  r a d i c a l s was i m p r o v e d ; i n • t h e  X-  sample,. ';the a l i g n m e n t problems, were; n o t c r i t i c a l .  (E) . M o l e c u l a r O r b i t a l  T  Computations:  .  D u r i n g t h e c o u r s e . o f t h i s ; work, t h e . c o m p u t i n g l i t i e s , a t the University IBM  of British  7044 t o an IBM 360/67 c b m p u t e r .  INDO c a l c u l a t i o n s ' were c a r r i e d using-programs F. G. H e r r i n g limited  •  :  s i g n a l due t o C ^ C O O  irradiated  e  by c u t t i n g o f f " t h e g a s supply.^ t k k e n , o u t arid  remounted . B u t the  t r i a l and" - e r r o E . The c r y s t a l V h a d " t .'>j  written  Columbia  changed  Initially  faci-  f r o m an  t h e CNDO. and  '  o u t o n .the IBM ^7 04 4 -computer  i n , t h e m a i n by D r s , P. J.' " B l a c k and  t o g e t h e r w i t h Mr. D a v i d Kennedy and m y s e l f . , The  c o r e s p a c e p f t h e . 7044 n e c e s s i t a t e d  t h e ;;use; of' two  programs.. T h e f i r s t p r o g r a m ' c o n s t r u c t e d heces'safyvd.ata: f o r CNDO. o r INDO . c a l c u l a t i o n ; " by . f e e d i n g , t h e • c o o r d i T i a t e W o f • t h e [atoms,  their  n u c l e a r . c h a r g e s , . number o f b a s i s o r b i t a l s  at  '  e a c h . atom . e t c and. t h e MO.parameters" s u c h as-, &&°f'"*y'- ,'G , F 1  \etc was  as; d i s c u s s e d the actual  earlier  i n C h a p t e r Two. The; s e c o n d  2  program  SCF calcul.at.i6n>,,With' the'^adven-t- of-'..the new  computer-;.these  two programs;' were j o i n t ; t o g e t h e r ^ to ';f6rm • t h e  •program'' c a l l e d  UBCMOL. • T h i s  :  1  work;.was. c a r r i e d by -Dr>' F.G. ;  ?  Herring  29  who as  also included  a d d i t i o n a l forms o f p a r a m e t r i z a t i o n  theFischer-Kolmar  and  v  (3 R)  a n d t h e Klopmann-Ohno  ;  t h e whole p r o g r a m was c o n v e r t e d  Further,  thea v a i l a b i l i t y  makes l a r g e c o m p u t a t i o n s less  of filing like  wprkers'  (i).  3 1  s y s t e m i n IBM 360 s e r i e s  thepresent  molecules previously published '  3 3  '  CF(COO~)  4 0  2  (39) ' methods,  v  t o double p r e c i s i o n .  one s i m p l e r a n d .  t e d i o u s . The p r o g r a m was c h e c k e d a g a i n s t  for various  such  the results  by P o p l e a n d c o -  ).  Radical;-  T h i s r a d i c a l was t r e a t e d a s an 8 - c e n t e r p r o b l e m . I t s s t r u c t u r e i s n o t known s o t h a t some  45-electron reasonable  g e o m e t r y h a d t o be assumed. Thus i t was assumed t h a t t h e f r a g ment  F—•-.—-6 -1 r  was  planar.  carboxyl t h e r (41).  I tissalso  it  i n m a l o n i c a c i d t h e two  groups a r e approximately i f t h e same h o l d s  potassium s a l t , should  known t h a t  was d e c i d e d  angles  angles  g r o u p s i n CF(COO~)2  t o r o t a t e theplane  basis  o f one o f t h e c a r b o x y l  therest  doing  were t a k e n t o be 1 2 0 °  a l l bond a n g l e s  radical  t o one a n o t h e r . On t h i s  groups w h i l s t m a i n t a i n i n g so,  t o one ano-  f o r d i f l u o r o m a l o n i c a c i d and i t s  t h e two c a r b o x y l  a l s o be a t r i g h t  at right  ofvhthe  radical  planar. In  and t h e b o n d -  30  l e n g t h s u s e d were a s f o l l o w s : -  -V  0  0  C - C = 1.53A, C - 0 = 1.265A, C - F = 1.38A These b o n d - l e n g t h s were u s e d o n t h e b a s i s o f t h e c r y s t a l structure o f malonic acid o f P o p l e and c o - w o r k e r s  . L a t e r o n , when t h e r e s u l t s  on i s o t r o p i c  hyperfine  splitting  (42) appeared ous  , i t was f o u n d t h a t  pairs  the bond-lengths o f v a r i -  o f atoms w h i c h t h e y h a d d e d u c e d  from t h e e q u i l i -  brium geometries o f v a r i o u s molecules, determined method, were n o t v e r y d i f f e r e n t  by INDO  from t h e ones used  i n this  s t u d y . Hence i t was n o t f o u n d t o be n e c e s s a r y t o change t h e b o n d - l e n g t h s employed system used  f o r the calculation  The iterations  i n t h e p r e s e n t c a s e . The c o o r d i n a t e  INDO c a l c u a l t i o n was f o u n d t o t a k e many more  t h a n t h e CNDO c a l c u a l t i o n .  CNDO c a l c u l a t i o n was p e r f o r m e d and  this  energy  to the r a d i c a l  was r o t a t e d  against the  and t h e minimum e n e r g y was f o u n d t o o c c u r  8 5 ° . The c a l c u l a t i o n was a l s o p e r f o r m e d  a n g l e s between 8 0 ° and 9 0 ° t o l o c a t e i t was s e e n t h a t  at intermediate  t h e minimum  accurately  t h e minimum was n o t d i f f e r e n t  N e x t , a few o r i e n t a t i o n s calcualtions  till  p l a n e . The m o l e -  f o r e a c h c o n f i g u r a t i o n was p l o t t e d  angle o f r o t a t i o n  only  on t h e p l a n a r c o n f i g u r a t i o n  1 0 ° i n s t e p s and t h e c a l c u l a t i o n was r e p e a t e d  g r o u p was p e r p e n d i c u l a r  cular  and  Hence, a t f i r s t ,  t h e n t h e p l a n e o f one o f t h e c a r b o x y l g r o u p s  through  at  i s shown i n F i g u r e 5.  around  8 5 ° were c h o s e n  b u t t h e minimum was n o t s h i f t e d .  from 8 5 ° .  f o r INDO  31  FIGURE 5  The C o o r d i n a t e CF(C00~)  9  System Used  Radical,  f o r the  32  ( i i ) . CF_COO" and CHFCOO"" R a d i c a l s ; -.".  •  As w i l l ' b e d i s c u s s e d s l a t e r . ,\CWe •'. have: two t y p e s o f  CE^COO" r a d i c a l s . group  I n one t y p e ^ c a l l e d  radical  i s i n t h e same p l a n e , as. the." FCF,/fragment l  other type,, c a l l e d  radical  II, d t- is  I, the carboxyl. w h i l e i n ,the  almost .perpendicular  :  .  t o t h a t p l a n e . F i g u r e ; 6 shows theV^pordih'ate.vsystem'^used. .for ;the'.'MO,.calculations. The b o n d - l e n g t h s " a n d "bond-angle's^, were, .', 'the same' as ^ t h o s e u s e d :f or- t h e .CF"|CO.Q  , '. : r  ;  • '••  ;' "•„'•' As m e n t i o n e d  earlier,  d'if l u o r o m a l b h a t e i o n , . CF, (C00~) ,V 2  2  ';}.'.. '.  i t was: assumed t h a t , i n ' t h e ' the~ two j c a r b o x y 1 \groups ;  1  . •" ' 1  were a l m o s t p e r p e n d i c u l a r : tp/"each: other'.'^Hence r t h e l o s s / o f - \ one  group  o n i r r a d i a t i o n y i e l d s - t h e , p l a n a r , C F ^ C O O " - r a d i c a l ';  (I) , w h i l e the. l o s s o f ' t h e other, <J.eads t o t h e f o r m a t i o n o f 'v 7  the noriplanar r a d i c a l ion a n a l o g o u s ^CHF  ( I I ) .. C a l c u l a t i o n s were;^also  performed  CHFCOO" '.radical'-; With.., 'the-.; c a r b o x y l - g r o u p  plane,, c a l l i n g  i n the -  i t - CHFCOO" (I) ,V and . p e r p e n d i c u l a r "to t h e  \ CHF p l a n e , , c a l l i n g ; i t CHFCOO" ( I I ) . T h e C o o r d i n a t e system" was .the same a s t h a t ' u s e d ;f dr. . t h e - ' c d r r e s p o h d i n g CF, COO": r a d i c a l s 2  'with' t h e d i f f e r e n c e  that  a hydrogen  atom-was ' s u b s t i t u t e d f o r .  ' t h e f l u o r i n e . i n t h e yz-pl'ane , ( F i g u r e , }6) ,.- rising C-H=1.'09AL.\', '• . A l t h o u g h . the. CHFCOO - 'radical---has ^ n o ^ .tally,,.., t h e -\calculMtip'ns w e r e ' p e r f o r m e d . in- order'} to; see^.the\, ; trend, o f s p i n ' d e n s i t y - v a r i a t i o n ' inVfche.!C--F;, regipn' ;in d i f f e r :  ent; •(ae^pcalization\*ehy£rpnme.nts'.^ *  V'/:..'':•  33  Pi y  •  C;  •cx  ( a ) . CF COO (I) 2  Ql  y  \  •Cr  (b).  FIGURE 6  CF COO~(II) 2  The C o o r d i n a t e S y s t e m f o r t h e CF COO~ 2  Radicals,  34  -J ( i i i ) .' C H F C O N H ^  '  arid-CFnCONHo  ^Radicals': -•' :  i n the - case o f each.of t h e s e - r a d i c a l s again', two ' "' p o s s i b i l i t i e s ' w e r e studied,' i.e.' the' CONH^;group>;'beihg. co- ,  •  J  planar -with /the FCF .or •' HCF fragment' ( r a d i c a l ,,10 o r ^being;^'' :'•;;,:  r  1  ;  ••'.'.'-, perpend i c u l a r t o ^ the" plane .of, ;th'is fragment.', (radical!, 'II):.,,. -„...- '  The bond-lengths  >'".,.'  . '/''':''.-'- "'  C - N = -1-.37A and.:..N - H =' 1.00A 7  . •were used.  • •'i-V-  •f":  '•.-"''"• -•' '•.'•.-,.' >';•/'C  CHAPTER FOUR  •  (A) . The CF (COO")  . RESULTS  .  Radical:-  F i g u r e 7 shows a t y p i c a l s t u d y . The f o u r valent sites direction  \ •  spectrum  obtained i n this  s t r o n g l i n e s were a t t r i b u t e d  o f C F ( C O O ~ ) , When t h e m a g n e t i c 2  t o two n o n e q u i field  was i n a  n o t i n any o f t h e p l a n e s a b , be o r c a , t h e s p e c t r u m  showed e i g h t  lines  due t o f o u r n o n e q u i & a l e n t  s i t e s . Further  when t h e m a g n e t i c . f i e l d was p a r a l l e l t o any o f t h e symmetry a x e s a , b o r c , t h e s p e c t r u m " s h o w e d two l i n e s a spectrum  due t o one s i t e  as shown  i n F i g u r e 8. F i n a l l y  entation  9 was same as t h a t o b t a i n e d a t 180-8. A l l t h e s e §  features  showed t h a t  the c r y s t a l  belonged  o b t a i n e d a t an o r i -  t o the orthorhombic  system. •  The obscurred  the p o s i t i o n s  Cjtbe" r e l a t i v e fine The  presence o f extra  coupling  radicals  of the forbidden transitions  signs of the p r i n c i p a l tensor could  allowed t r a n s i t i o n s  resolved  l i n e s due t o o t h e r  values of the  1  9  so t h a t F hyper-  n o t be d e t e r m i n e d , e x p e r i m e n t a l l y .  were f o u n d t o be s h a r p and w e l l -  a t a l l o r i e n t a t i o n s o f the magnetic  For. e a s e o f d e s c r i p t i o n ,  field.  t h e two s i t e s  each o f t h e t h r e e p l a n e s a r e l a b e l l e d  •  observed i n  P and Q i n s u c h a way  FIGURE 7 A T y p i c a l ESR S p e c t r u m o f X - I r r a d i a t e d D i f l u o r o m a l o n a t e Monohydrate, (0,5,  Single  C r y s t a l of Potassium  the Magnetic F i e l d 0, -0.8660)  Being  i n the D i r e c t i o n  38  /V/v:-: ' v,.'  A-A./^tha^At^  from; 109 to J80° . but, l a r g e r , from. 100° to ^170°, A t 0 ° ; 90° \  •'/.r^;-;;;  .•>  •' /  y  smafrie'rNth^  r - ; -  ,-;--v.  and: ;180 ,-/;of :,couV^  -..\,.  ^ ' V ; "  angur <X\  :  :  (  'jar: v a r i a t i o h , p f t  h  ;  e  '  '  >  t  w  o  .v,:.-..:''is-, shown.•in!'Figur:e's.^9:/:VjO^  : siite's:'^''';/'•• f i e I'd being  . ;.in bc> ca and' vab planes ^re\spectiyel^ the experimental ,-W';,;" ;-><y,;  >• 'C j  points.  x'XX'X'^^X^^^  The A-tenspr was ;obtained-,;f ro'm ^eg-uatiori'v"(4)', by:': • 'k V,' ..-'' ^' Xi A-A\A ''•>''^ A'*;-vAV A;A:':'--A^'^'F-'';--Ai^-*; • A .•employing a ,leas:t^squaresAfitting,vtetihn-ique'i" •vin\-4file^pke'S'e'rft'"'^"-'-'H'V. . '*'fX' ;•.:' ! \y ',,,, ^ " ; 7 A ' K . A X , A .• ' ' ' " A'"" ;,-• ' ' ^'' 5'''.^''>''i- ^^•^^ ^,V 1  '•*\v'-'  (  y:  :  ;  ,  ,  ;  f  t  ;:;  f  • ' ' :case, we have to -assbciate 'one; particular set of data ff rom, •/ s  ;  •-,s\one:;plan^'vwith''''either "of, th,e' two setsof -data • ' .from; each .of : ,;  uy  -u^-'  .••;>, < : \ > r , -b.>-  t  the. second and third planes ...For instance the site. P.. in be. ;' plane, could be associated with either P orin^.ca< plane): •.,:• »'?s:^<-S?,•••>••;< f-'*vV'• I' v ' r ' ' v . - . " ' • i^'.^i'-^%•','••>• M'V^V7:^.^r-'':.:r^*-.^?s,V,:.-. ;,*v. givnig, two; p o s s i b i l i t i e s j . i.;e-"P.^P^Opr: P j'Q . '-Each^f;..^tKese':^v ''-., * two ^combinations could: then be associated with 'either 'P. or jc-'- :.-.V-\,-.^.-. .; V  F  i..,  , .  '.•.•p-'*-;-^^%V' v>V.jc:8.^.*.- -' v  t  (  1  ;  1  r  v  v  v  2  ^••'/:.Q;in;^he;"'ia'b^;p^  A.-'  PiP  p 2  3f - i 2Q3' P  P  jaj|j|o i.e.;,-;..., p  i  Q  2  p  3 '  p  i  Q  2  Q  where 'the ' i h d i c e s ' i i ; ^ ^aaiid^^^.'-"-  3  stand for. be,,,' ca .and;.ab p l a n e s . i e s p e c t i v e l - y ; ? S i m i j ^ l y V t h e ;-i v  , • . s i te|iQ' i n the' be plane a l s o. gave .four :comi?i-n&tiph's^-! -Thus;,we "-'«'•;.1had, e i g h t p6.ssabilities'; i n .(.all.  ;  i  . ', f o r the; l^ast-squares; ;  v  v  were' t r i e d  :  Using'a 'eomputer^prbgram./.Cj'"' '.- -, ;  :<  They ,;were; P. ^ P;P - ^ .P^ Q Q y / Q ^ Qy/and^Q ^Q;.^ . T h e ^ h y p e r f ine 2  !  fitt  but o n l y four- of them gave convergent r e s u l t s . : 2  " '  3  , t e n s o r s /.obtaii^ed/.f pr.'/the.^ four''convergent  (  •  cpmbinatibn  ~£X a'ppr^ximatejy^ithe same ;?in',magni£ude''but :diiff-^^ sJLgns'' •' '*>*' . ''Vr~- ", • -\v'•• v'«-5'-'.. '.".V"'A ;,'P' '''v?''" •:;•'. '^'-^2>~ -'•''• : ; , ,,- .•, _ " - v v • V , - ; ; , ..•H..-r ,:^ '•> v , y - . . • , - . y .-•.*'A-v:.'; '/; . v / - o f the >f o ff-fd-i a g o n a l * elements .i.::Thu^;..fpr;'P^P:P-g, all' . 'th'e»tof f:  ;  .  /  -  •  >  f  r  :v  4  ;  2  ;  ,;  FIGURE 9  Angular V a r i a t i o n of the Hyperfine S p l i t t i n g the Magnetic  Field  Being  i n the  bc-P^ane,  f o r CF ( C 0 0 ~ )  2  Radical,  FIGURE 10  Angular V a r i a t i o n the Magnetic  Field  of the H y p e r f i n e S p l i t t i n g Being  i n the  ca-Plane.  f o r CF(COO~)  2  Radipal  FIGURE 11  Angular V a r i a t i o n o f the H y p e r f i n e S p l i t t i n g the Magnetic  Field  Being i n the  ab-Plane.  f o r CF(COO")  2  Radical  ^;Jy;f.Qr . ' . ^  "•;  • <•': •/•"•,"'•:-v-  --..Ci.  - ' V  v  -  v "  •'- .  v  b  -  v  •- '':,-v  .  •, :  Vv •  S  •' - ' f 4 ^ ' ; d i a g o n a l c ^ , e l e m e n t s . ; b e i m g ^ n e g a t i v e V - ' T h e / e i e m e n t i s V f b r ^ t h e f o u r . ' •  •' ;  ;  v t >:'. -'•, >  t e n s o r s ; w e r j e ••>av.er.aged. ' w , i t h b u t < ? r e g a r d ' :  t o : - s i g n s Vand , £ h e ^av.eryy< ; :  ^•ffK.'^ raged• t e n s o r , was  ;•'  •\-'}',:*-:- >-\-,^-f  ^'-^i  ?  • ' '  ;V''  A.--">V-^:> "v-'-:---•'.' -,,y- • • .yvv>:'•;><:•' • ;;•/' >v;;V'  •<-  94:.?7^ v  'V-'-'A-*' r  •"''  i••  . 10.6  •' Th'e.. t e n s o r s  'for  -' V % ' : \ : V . v  ; .  ;  V •  1 1 2 ,.4-  -Theva'fcbv.e.Vtensor^-'±-s-vobvl'.oualy  :  ;A,y:"7  Mother, •cbmb.inat'ibns  , f o r V \ t h e i P' P-P;, '.c6mB.iha\£i6nt..^- -*f*> :  c o u l d ''be''  6b t ' a i  ,:  Ae1l-ts i m p :  :  ly':-'-'^  .^0 A — X :r '"Kvi;^*v!''S'V'.'>i!L3i;^^'•-•••'^^'• ' ' v ' . ^ -v.-':''.;.';^ [ ; b y . , c h ' ^ n g i h g ' . . . t h ' 6 ' ^ « i g n - s > o.f • t h e < f a p j ( r o p r i a t e ' o f ' f ^ d l a g o n a l e l e m e n t s . ' :  •\''« •. * : :  ;  ;  ;  .The . . t e n s o r s 'so  -.  o b t a'ine^d . w e r e d i a g o n a l l z e d " y i e  lding .the ; p r ; i h r  r  -  ' ' y ' ' Q " \ ^ J ' / - K ^ v ' - ' 't^fV^-'-'-'-'OfTt^:' 'v'-; v ^ ' V ^ .;"'^•v..•: --V.; ' V v V^''"'-C'i'p.sfi-Rvalues' 'and. t h e ; d i r e b t l q n : c o s i n e s q'f • t h ' e ; ^ ^ r l n c i p a ^ l ' ^ • a x e ^ *•„.•: ;  ?  :  ;  ;  * ' v r'yv ^^;;''^ 1  • f'\ '••' :  :  • ;  V^V;.;T^ obtained  iv,^;,'iV'.'  by changing  a l l . t h e s i g n s . f i n t w o . c q i u m i n s • a t - a. [tirne'.'•  ,  {• vy '\«/SSi /•;^\'i''.^''/"^h'®^ L  ;  ,  ;  < .'; > ' • ; ^> • s q u a r e s ;  f i t t i n g ,were  :  "•<{•"' .* •' ,  :  :  used'  t o - ' r e ' c a i c u i a t e .-'the ' . h y p e ' r f .InW-'^spi'it'T-.• - •  ''fcMg's^'JraridirvthexxJ^au^lts'ydr^'^o^y ih;^Figures 9l.%Q* /;  ast";'V?' '  andvllff  \V;^;.;- l . ' s p M d ; ; i i r t  •;,  v r t ' ^ > v . ( v ^ '.-•'.«•.-•.•:;'.-;-<r;emarkabl'y^go6d4\Mvishould-'/.be*' •.me.h'tione.d*;. Her"e'.'>ith''|it^fee; -;iie:l a:r!!*v> -' ?  KV.-':^ ,, 7-' t l y e - sd^hsi:o^fV- the,' ' p r a h b i j j a l } i v a ] ^  * - ''<•"*• % v . ' v ; l ,V-'\'; UK  ; ;  - ' , ; ; . i - -1'--."0A'V^'-'-^'ft^ :  ]"'<\';.yX -^ ---sor^ A "was - c o n s t r u c t e d ^ .-• -'.-^ A:'iv ;^\;^v * ^y^iU/^ ;  ;  >  :  ,. ;  :  r/^I.Vjsh'o^n••  ::  ,-  l  :  ,  by^th'e'yfqll^  - •'  "byv^  •  '  H-  43  TABLE'I  The  Principal  V a l u e s and t h e P r i n c i p a l Tensors  f o r CF(C00~)  Direction Principal  Values  Axes  of  A- and g  Radical  Cosines of P r i n c i p a l  Axes  ^ • V  1  1 9  F  ...  m  0  0  Hyperfine.  Coupling  Tensor*-' •  '•'C\ \  . 208 ± 2  gauss  0.705  0.075  0.cZ06  27 ± 2  gauss  -0.160  0.986  0.055 V.  17 ± 2  gauss  0.691  0.151  -0.707  -' '• '•'•••0 g-Tensor:-  2.0009  • f! Ji  ±^oom  0.545  0.101  2.0044 ± 0.0003  0.118 .  0.973  2.0039 ± 0 . P 0 0 3  0.830  -0.205  •  \\  1 ,'  0.832 ' -0.196 ,fj -0.519  44  A = T.J^.T"  1  where A^ i s a d i a g o n a l t e n s o r w i t h  the p r i n c i p a l  values  t o 208, -27, -17 and T i s t h e t r a n s f o r m a t i o n m a t r i x by  the d i r e c t i o n  to r e c a l c u l a t e  c o s i n e s . The r e s u l t i n g  equal  formed  t e n s o r A, when u s e d  the hyperfine s p l i t t i n g s ,  gave p r a c t i c a l l y t h e  same c u r v e s . The c a l c u l a t e d p o s i t i o n s o f t h e ' f o r b i d d e n ' t r a n s i t i o n s were, however, d i f f e r e n t . the  I t i s unfortunate  f o r b i d d e n t r a n s i t i o n s were n o t r e s o l v e d , o t h e r w i s e  relative  s i g n s o f the p r i n c i p a l values  The g-values  g-tensor  to equation  was o b t a i n e d  c o u l d be  by f i t t i n g  (7) . The v a l u e s o f g ^ g  that the  determined.  the  were  observed calculated  from the resonance c o n d i t i o n  g  o  b  =  s  where h is,R<lanck's and  H i s the f i e l d  first  constant,  (28) v i s t h e microwave  corresponding  t o an u n s p l i t  frequency  line.  To  order H = H  where field  hv/8H  and H lines  present  Q  =  ( H  x  2  )/2  (29)  a r e t h e p o s i t i o n s o f t h e l o w - and t h e h i g h -  2  respectively  case,  + H  as shown i n F i g u r e  the h y p e r f i n e s p l i t t i n g  a second o r d e r  12(a).  In the  was n o t t o o s m a l l and  c o r r e c t i o n had t o be a p p l i e d . The l i n e  H  was 2  closer equal  t o H than to H . Q  Instead  as shown  i n F i g u r e 12(b) and H was n o t  the r e l a t i o n  g i v e n by t h e f o l l o w i n g  H  H  H,  ( a ) H<  H, H  H,  1  H  IH.  H,  H  H,  H  H 2  H  H  C  FIGURE 12  2  H,3  3  c)  (b)  H,  i  •  (d)  The P o s i t i o n o f t h e M a g n e t i c F i e l d t o L o c a t e t h e O b s e r v e d g - V a l u e (a) One N u c l e u s , F i r s t O r d e r (b) One N u c l e u s , Second O r d e r (c) Two N u c l e i , F i r s t O r d e r (d) Two N u c l e i , Second O r d e r .  H 4  E  46  e q u a t i o n ,holds;~ /  "  "' ( H H = H^; l ;  -H  2 f t  Hrw,  )  X  , ^ } : • ' '  ; • ... ' ' .' v. (30)  . '.,«-' '  .. E q u a t i o n  :  • '„' .'"'v-  v  (30)  was s u b s t i t u t e d  c a l c u l a t i o n of. g  , .'  o  b  g  ,•  .  ;  - • s b r . .The  r e s u l t s a r e 'shown '.In.'TableV^  final  being  -  "\. t h a t  t  i'-:'; .v'--'  .employed; f o r ; t h e  obtained  changing  asyy ;  1  ; pbmp'arison^iof ;-±he r e s u l t s , g i v e n ' i n ' T.able^I •. shows • r  t h e A..and - the• g V t e n s o r s  1  normal t o the  radical  :  are" d i a g d h a l ' i n  approximately  i f one, assumes t h a t t h p : ' l a r g e s t y ^ ' y • 1  r  ' ^ p r i n c i p a l 'Va'iu'e^.ofthe-\.A';:tenbbr:Vcorre  :  ' to. •a^p^-i'-rectaon';^' '..  p l a n e i t t i e n t h e ' s m a l l e s t v a l u e ,,of • t i l e . g .  i . ••';:•?'•' te-h''sor-via' -pa ral-lel-.;tQ\th'is;\'dire.Gtioni .a s i t u a t i o n w h i c h ^  is';^.>-  . ';  ;  i  ,  1 ;  f a m i i ' i a r r to't-hat,yob  B;)V. The . C F . r  y  T  N'", ••; : .^'^V&'^V'''  .- y/'-'v  :  v  • .  A' ten-: . . ;  by • s i g n  ?'/.-•<.''•' •: y . ' * ;  }\ t h e •.same.; axis;, system,' A l s o  "v'''  obtained-  a g a i n ' f o r t h e V s i t e P,^  v -....signs•for.'other . s i t e s c a n .be  "'\;- \; ',•  one  • : ' /  /• '' ••" ''-V \<y. "•'•'•••'''A::  analogous t o the  '/".'^  :  ',  ;  d e s c r i .bed" b e d a b o v e ' - • { -/'  ,  ;  , ;", by. a p r o c e d u r e  :  1  .'' " '  ' (28) . f o r t h e ^  T h e - p r l n c i p a l ^'values 'of ; t h e y g tensor", were  •>•[''' the- d i r e c t i o n , c o s i n e s  r  i n t o , equation  r t A  thej.CH (gOO )^' r a d i c a i ^ ^ . , -  " .-RadxcalsJir ^ - — ' - " b  :  i  l  J  ^^y^v"  v .>;---v-<. 'j  ^;^^;- ••'.". ^  /•. --\.^.- : t  '' '  .v,, ' 'j;^..',"". ; .. 1  >,'\(,*• . ,Flgui"e';' 1-3' 'sh'o^ws. a '.typ.ic^lV.e'xanipi.e- pfthe-r^spectra,.''; ,'. \ ;  '"\. ''.••<;:•-,  - A ; - - ' * .  »  : •'..«••.•;'- ; ' t • c., \0 v !.—'">7-  •'• ,r'.V,'' c  v  of'••"Y'-ir'radiated', d i p o t a s s l u m  ;••'')',".' - f j - ' ; ' « . ' . ? " V v /tV^:*:*' • -. ' * v ' > " i -  f  > ' ' ^  -''^  v  ,,:  '', , i ' ' ; ^ ' - - - ' - i , , , ' i '•''' •  d i f l u o r b i h a l o n a t e . The f o u r  •I"- I-' ' • . '  5  r  -strong  48  l i n e s near-/the. c e n t e r s p e c i e s , CF (C00"")  (marked A)/are' due, to' .the'- main; rraaddiiccaal .  in- twc>. 'symmetry r e l a t e d s i t e s ./ There.;aire -  2  r  good t h e o r e t i c a l ; a r i d experimental' t h e two  that  ., ;  f l u o r i n e c o u p l i n g : tensors i n a r a d i c a l , o f the f o r - ;•>*•  :  mula CF.COO" should be n e a r l y ax'ially symmetric? and, should ;  •>']:;  2  '....M"-.i  :  •  '  - - v .  -  vA'  1  . ; . - .  V ? f & Y ' ^  have: t h e i r unique p r i n c i p a i axes,-parallel. tOveachT'other-i , ;* 5^ ". 2  0  2  For t h i s reason one would expect an e s r .jiatter'n 'pf .(three ^equar 1  I l l y spaced  r  l i n e s with, t h e ' ' i n t e n s i t y ' r a t i o ; •- >j : . 2 , fl '-/and' wi^brji-, t  ,'• -.> 'i-*:;, •  •at the' most,/a .small s p l i t t i n g of, the c e n t r a l iine/fprHsome• ';- , 1  • o r i e n t a t i o n s ; ' In, any.'case,'.Half o f the; s p l i t t i n g .between tHe ..^ 1  two' outer l i n e s should be a f a i r l y a good measure' of t  h  e  .  h y p e r f i n e ' s p l i t t i n g I f . there are-two syitimetry^*rel,aj:ed sites,/:.: ;  \  ' • ' ? * ' ; • ' •  one  "  " ' . • ' • "  •  '  '•'••''  :  '  ' - ' . • ' • '  : ' ' • • ' ' . ' : - ' ' - „ " . ' ' ; . ' " "  .-  s  '-:  ^  ''. »  v  £ .  -  .  f  . . ' A * "  .-, •  should obtain-an e s r p a t t e r n o f f i v e l i n e s with .intensity  r a t i o s 1:1:4 :1:1. I n F i g u r e , 13 the f o u r l i n e s /-.marked' B /are' the f o u r outer, l i n e s o f t h i s p a t t e r n and aire due t o a " r a d i c a l , ;,''  ' • ''  :  .',' .\ ' . -  •  .• . -  y '•'.__•:,.• ..  •;•'. ^",'.'; "," " ;'V"' ' .(  of the formula C F C O 0 ~ . A n o t h e r s e t o f f o u r l i n e s , marked" C, ;  2  of weaker, i n t e n s i t y has been i n t e r p r e t e d ' as due/ to .another, '•>.••'•. r a d i c a l o f the same chemical. formula, i.e.,, ,CF COO"". >"'/' :  2  i.'-  -As both of - the either  1  w i l l ,be s h o w n - l a t e r , " t h e s e  ,•' :  ;  ',  ' -  t- •  two r a d i c a l s . ,  same 'formula^ ;>we£e' a c t u a l l  although  i n the';->jos'sf. of.'"  one o r /'the' pther> c a r b o x y l - g r o u p o f C F (C0O~) ' and, i n / 2  2  /fact,; - had V s l i g h ^ t i y d i f f e r e n t ,cpnf i g u r a t i o n s . ;The/. f i r s t ; < s u c h .;•/ 1  :  radical,  whose/esr  s i g n a l 'is  as. the/second,/•..wi^jl/be  1  approximately,:four  '• c a l l e d / r a d i c a l /  w i t h ' weaker- s i g n a l / w i l l  :  be l a b e l l e d  'times as' s t r o n g  while'.''th'e/'.other  a s r a d i c a 1 I I i "The l  one  rather '  49  broad  line,(marked  D) a t t h e c e n t e r o f t h e s p e c t r u m  superposition of several lines, i.  • the c e n t r a l  ii. of  lines  eg.,  f o r both,  i s the  radicals  f o r some o r i e n t a t i o n s , a t l e a s t ,  I  R  and I I ,  the forbidden  lines  t h e C F ( C 0 0 ~ ) - r a d i c a l , and 2  iii. C02~  possibly, .^4)  central  t  D  U e  line  unresolved  due t o t h e r a d i c a l  t o i t s b r o a d n e s s and u n r e s o l v e d c o u l d n o t be a n a l y z e d .  (marked E) w h i c h - w e r e s e e n n o t be  lines  nature,  this  The two v e r y weak  o n l y i n some o r i e n t a t i o n s  •  lines could  identified.  When t h e m a g n e t i c f i e l d axes a> b o r c , t h e s i t e s  was p a r a l l e l  t o any o f fche  o f e a c h o f t h e two r a d i c a l s  coales-  ced  a s c a n be s e e n ' i n F i g u r e 14. The method f o r d e t e r m i n i n g ,  the  c o u p l i n g t e n s o r s from  as t h a t u s e d  f o r the CF(COO~)  d a t a was a v a i l a b l e but  the experimental  forradical  radical.  I, the overlapping with  for a l l orientations  b e , c a and a b ,  lines  from  t h e main  i n t h e c a p l a n e . Hence f o r  radical,  ling  t e n s o r h a d t o be s u p p l e m e n t e d w i t h  t h e p r o c e d u r e - f o r t h e d e t e r m i n a t i o n o f t h e coup  cess. This consisted of guessing ca plane, doing  guesses u n t i l  II the  measurements o f i t s h y p e r f i n e  this  the  For radical  f o r a l l the three planes  s p e c i e s CF(C00~)2 prevented splitting  2  d a t a was t h e same  a trial:and  two e x p e r i m e n t a l  error pro  points i n  t h e l e a s t - s q u a r e s f i t and c h a n g i n g t h e  t h e s t a n d a r d < e d e v i a t i o n . was minimum. The  set  o f h y p e r f i n e tensor elements f i t t e d  for  the remaining  final  the experimental  two p l a n e s v e r y w e l l w i t h  a standard  data  devia-  51  tion  comparable- i n m a g n i t u d e t o ,that" o b t a i n e d  ;A11'; e x p e r i m e n t a l l y  splittings  shown as c l o s e d c i r c l e s resolvable* lines  The  a r e shown •• as o p e n c i r c l e s c u l a t e d v a l u e s , u s i n g the solid  lines, for r a d i c a l  Figures  15-17*  experimental  are  shown i n T a b l e s  ,' ' The  broaken-lines / "I '  diagonalized A tensors  P.J-.PJJP'^. The  metry-irelated  '^••:X-  -  The manner as  \  v  ) _ radical.  f o r t h e CF {C00~-)^'radical.  for. equation  .  I and  crystal cosines :•  II • "\  .three .sym-'^ c a n -be-, o b t a i n e d  i n 'two  - A Y - A A ,  i n the  column's ;•' '••'•X v  similar  :.  -<The measurements -of • g ^ ^ ,  i s t h e , same as t h e  Q  order  the-ppsition pf  central  :  l i n e H' ,Hg! as -/ 2  shown i n F i g u r e 12.(c) and; t h e u s u a l p r o c e d u r e ,,is'to d e t e r m i n e the p o s i t i o n of t h i s central and  line.  l i n e . c o u l d n o t be >  c o u l d be  J  In the p r e s e n t  Cr;  tables/-'correspond  ''  a p r o b l e m . To f i r s t  (28)  as ,  V'V Av, •''.': l/T  f o r the remaining  g t e n s o r s were a l s o o b t a i n e d  'however-,: p r e s e n t e d H  tensors  "•'  cal-  f o r radical II i n  i n these  1  f o r CF (COO  -The  I I I r e s p e c t i v e l y . The . d i r e c t i o n  s i g n s o f ,/the d i r e c t i o n  •>"•  a t a t i m e as  17.  for radicals  s i t e s ; i n . an o r t h o r h o m b i c  by-changing the  and  • • ' /'•'  c o s i n e s o f the p r i n c i p a l - axes g i v e n t o the s i t e  16  final: tensors, are:presented  I -and  I I and  p o i n t s for, radical/; II  i n F i g u r e s .15,  :  '('•  j  I !with -the, f i e l d - i n .the ca, p l a n e  ••• v-':.•  ;. A  f o r \radi'caj^,3[' ,are\'' \\  i n F i g u r e s . 15^and 17. t h e r e being, no  for; r a d i c a l  as shown i n F i g u r e 16;.  f o r r a d i c a l II.-/  c a s e , however, the^.:  s e e n and. hence- t h e mid'-poiritv of, H , /  u s e d t o l o c a t e t h e p o s i t i o n o f H . I f we  in-  52  FIGURE 16  Angular V a r i a t i o n o f the Hyperfine S p l i t t i n g w i t h t h e M a g n e t i c F i e l d i n t h e ca-Plane«  f o r t h e CF^COO™  Radicals  FIGURE  17  Angular V a r i a t i o n of the Hyperfine S p l i t t i n g with the Magnetic F i e l d i n the ab-Plane  f o r t h e CF COO"" R a d i c a l s 2  55  TABLE I I The  Principal  Values Tensors  and  the  F  m  n  Tensor:-  181v± 2 gauss  2  Axes  Hyperfine  Coupling  1  Cosines of P r i n c i p a l  Values 1  1 9  Axes o f A- and g -  f o r CF.COO"(I) R a d i c a l  Direction Principal  Principal  0.091  0.940  0.328  13  +: 2 g a u s s  0.757  0.149  -0.637  11  ± 2 gauss  0.647  -0.306  0.698  2.0077 ± 0.0008  0.598  -0.299  0.743  2.0057 ± 0.0008  0.801  0.239  -0.548  2.0014 ± 0.0008  -0.014  0.924  0.383  2.0077 ± 0.0008  0.588  -0.308  0.748  2.0057 + 0.0008  0.808  0.186  -0.559  1.9986 ± 0.0008  0.033  0.933  0.358  s t  nd  Order  order  g-Tensor:-  g-Tensor:-  56 TABLE I I I  The P r i n c i p a l  V a l u e s and  g-Tensors  the  Principal  f o r CF COO"(II)  Direction Principal  2  A-  and  Radical  Cosines of P r i n c i p a l  Axes  m  n  F Hyperfine  Coupling  1  of  Values 1  1 9  Axes  Tensor:-  188  ± 2 gauss  0.078  0.678  0.730  19  ± 2 gauss  0.965  -0.234  0.114  7 ± 2 gauss  0.249  0.696  -0.673  2.0079 + 0.0008  0.568  0.577  -0.587  2.0069 + 0.0008  0.817  -0.480  0.319  2.0012 ± 0.0008  0.098  0.661  0.744  2.0078 + 0.0008  0.572  0.576  -0.584  2.0069 ± 0.0008  0.816  -0.470  0.335  1.9981 ± 0.0008  0.081  0.669  0.739  s t  nd  Order  order  g-Tensor:-  g-Tensor:-  57  elude  the  second  the p o s i t i o n line. that H  b  order e f f e c t s  of H i s not  T h i s d i f f i c u l t y was a single  through  the  line  due  the  first  o r d e r , as  second  overcome as  H = Js(H  H  a  as  to  the  into  "  I b H  second H  a  i f we  4 H  while  include then  H ) ) } 2  a  b  H2  and  then  shown i n F i g u r e 1 2 ( d ) ,  + H ) <1 +  and  a  (31)  a b  (32)  J  H  when t h e e l e c t r o n i n t e r a c t s w i t h a s e c o n d splits  assumes  )  h  (  a  central  into H  a spin-*s n u c l e u s  I Now  splits  shown i n F i g u r e 1 2 ( c ) , b u t  order e f f e c t ,  radical,  f o l l o w s . I f one  to a free e l e c t r o n  + H  a  i n the CF(C00~)2  same as t h a t o f t h e  i n t e r a c t i o n with  H = %( H to  the  as  splits  spin-*s n u c l e u s ,  i n t o H-j and  H4.  Thus  order (  = Jj(H, +  (2 H  +  4 H  -  H ) ) \ 2  X  (33)  1 2 H  and (H. - H ) ^ 4H H 3  H  K  H ){1  = Js(H, +  A  +  3  >  (34)  4  where  H I f we we  2  = H  substitute  can determine and  3  = J (H 5  1  equations  + H)  (35)  2  (33),  the p o s i t i o n s  (34)  and  o f H from  (35)  into  (32),  the p o s i t i o n s  of  .  I t was  found  that, with  second  order  correction,  58  the p r i n c i p a l  g value  corresponding  v a l u e o f t h e A t e n s o r was responding  uncorrected  to the  largest ^principal  c o n s i d e r a b l y s m a l l e r than  v a l u e . The  remaining  two  a p p r e c i a b l y a f f e c t e d . The  for  the  and  are  shown i n T a b l e s  r a d i c a l s with  (S). Molecular  IV shows t h e  t h e CF ( C 0 0 ~ ) radical  INDO s p i n d e n s i t i e s  radical,  2  plane.  The  with  to the  the  s p i n d e n s i t y b e f o r e p r o j e c t i o n and  one  u p p e r number i n e v e r y the  lower  a f t e r p r o j e c t i o n . Analogous r e s u l t s  CF CONH (II) r a d i c a l s  2  2  2  row  one  at is  i s that  f o r CF COO""(I), 2  2  C F C O N H ( I ) and  C-F  c a r b o x y l group  2  XII  correction  i n the  C F C O O " ( I I ) , CHFCOO"(I), C H F C O O " ( I I ) , C H F C O N H ( I ) , 2  tensors  III.  85°  obtained  second order  g  g  Orbital Calculations:-  Table region of  I I and  without  cor-  principal  v a l u e s were, however, n o t two  the  are given  CHFCONH (II), 2  i n Tables  V-  respectively.  In t h e t h a t the  COO  above m e n t i o n e d r a d i c a l s , i t h e ..label  o r CONH  fragment w h i l e  the  2  group i s c o p l a n a r w i t h  label  the  CHF  (I) means or  (II) i n d i c a t e s t h a t the p l a n e  group i s p e r p e n d i c u l a r to the plane  o f CHF  o r CF-  CF of  2  the  fragment.  TABLE IV INDO  C  2Px  Spin Densities  °2p  C y  2p  f o r CF(COO" )2  F z  2s  Radical  F  2Px  F  2p  y  2p  :  0.0410 0.0139 0.0017 0.0018  0.8134 0.8054  0.0023 0.0008  -0.0021 -0.0021  0.0244 0.0082  0.0034 0.0011  -0.0006 -0.0006  -0.0014 -0.0005  0.0199 0.0067  0.0048 0.0016  -0.0006 -0.0007  0.0001 0.0000  0.0013 0.0005  0.0006 0.0002  0.0007 0.0005  -0.1960 -0.1953  0.0005 0.0005  0.0005 0.0003  0.0003 0.0002  0.0476 0.0475  0.0004 0.0001  0.0003 0.0003  -0.0031 -0.0011  0.0002 0.0001  0.0000 0.0000  -0.0001 -0.0001  0.0004 0.0001  0.0029 0.0010  -0.0030 -0.0030  0.0003 0.0001  -0.0097 -0.0032  -0.0025 -0.0008  0.0011 0.0009  0.0000 0.0000  •0.0231 •0.0076  TABLE V INDO S p i n D e n s i t i e s  C  C  C  F  F  2  P  C  2Px  2  Py  C  2p  2  F z  2s  2px  2p  y  2p;  0.0461 0.0157  2s  °2Px C  2s  f o r CF COO~(I) R a d i c a l  y  2p  z  2s  2P  X  2p  y  2p  z  0.0000 0.0000  0.8006 0.7881  0.0045 0.0016  0.0000 0.0000  0,0153 0.0051  0.0056 0.0019  0.0000 0.0000  0.0035 0.0012  0.0212 0.0071  -0.0055 -0.0019  0.0000 0.0000  -0.0003 -0.0001  0.0011 0.0004  0.0006 0.0002  0.0000 0.0000  -0.1888 -0.1868  0.0000 0.0000  0.0000 0.0000  0.0000 0.0000  0.0447 0.0444  -0.0021 -0.0007  0.0000 0.0000  -0.0005 -0.0002  0.0005 0.0002  0.0005 0.0002  0.0000 0,0000  -0.0003 •0.0001  -0.0040 -0.0014  0.0000 0.0000  -0.0047 -0.0016  -0.0097 -0.0032  -0.0024 -0.0008  0.0000 0.0000  0.0027 0.0009  O  •0.0240 •0.0079  TABLE V I  INDO S p i n D e n s i t i e s  C  2s  C  2P  X  °2p  y  F  2Px  C  2p  y  °2p  F z  2s  F  2Px  0.0000 0.0000  0.8497 0.8482  0.0023 0.0008  0.0000 0.0000  0.0155 0.0052  0.0031 0.0011  0.0000 0.0000  0.0018 0.0006  0.0204 0.0069  -0.0055 -0.0019  0.0000 0.0000  -0.0002 -0.0001  0.0013 0.0004  0.0006 0.0002  0.0000 0.0000  -0.1925 -0.1923  0.0000 0.0000  0.0000 0.0000  0.0000 0.0000  0.0437 0.0436  2Py  -0.0020 -0.0007  0.0000 0.0000  -0.0006 -0.0002  0.0007 0.0002  0.0005 0.0002  0.0000 0.0000  2  -0.0031 -0.0011  0.0000 0.0000  -0.0045 -0.0015  -0.0095 -0.0032  -0.0027 -0.0009  2p  z  2s  F 2  F  C  Radical  2  P x  P  F 2Py  2p  5  0.0453 0.0154  C  C  2s  f o r CF COO~(II)  z  0.00005 0.0000  -0.0003 -0.0001 0.0028 0.0009  -0.0259 -0.0085  TABLE V I I  INDO S p i n  Densities  C 2s  C  2P  C X  f o r CHFCOO"(I) R a d i c a l  2P,  F r  F 2s  F  2P  X  o  2Py  2P:  0.0553 0.0189 C  ^Px  0.8314 0.8155  2Py  0.0042 0.0015  0.0000 0.0000  0.0262 0.0088  2p  0.0026 0.0009  0.0000 0.0000  -0.0026 -0.0009  0.0151 0.0051  -0.0079 -0.0027  0.0000 0.000  -0.0013 -0.0005  0.0019 0.0006  0.0012 0.0004  0.0000 0.0000  -0.2397 -0.2367  0.0000 0.0000  0.00005 0.0000  0.0000 0.0000  0.0695 0.0689  -0.0015 -0.0005  0.0000 0.0000  -0.0034 -0.0012  0.0022 0.0007  0.0009 0.0003  0.0000 0.0000  0.0003 0.0001  -0.0057 -0.0019  0.0000 0.0000  -0.0043 -0.0014  -0.0113 -0.0037  -0.0034 -0.0011  0.0000 0.0000  0.0023 0.0008  C  F  z  2s  2P  X  2p  y  F  F  cn  0.0000 0.0000  F 2P  Z  •0.0184 •0.0061  TABLE  INDO S p i n  C  2s  2  Px  Densities  c~ 2p  2 y  P  VIII  f o r CHFCOO"(II)  F Z  2S  Radical  2Px  F  2  P  y  2p  5  0.0538 0.0183 0.0000 0.0000  0.8841 0.8822  0.0007 0.0002  0.0000 0.0000  0.0254 0.0086  0.0014 0.0005  0.0000 0.0000  -0.0029 -0.0010  0.0168 0.0055  0.0076 0.0026  0.0000 0.0000  -0.0008 -0.0003  0.0022 0.0007  0.0011 0.0004  0.0000 0.0000  -0.2408 -0.2406  0.0000 0.0000  0.0000 0.0000  0.0000 0.0000  0.0657 0.0657  0.0009 0.0003  0.0000 0.0000  -0.0033 -0.0011  0.0022 0.0007  0.0008 0.0003  0.0000 0.0000  0.0003 0.0001  0.0036 0.0012  0.0000 0.0000  -0.0035 -0.0012  -0.0118 -0.0039  -0.0042 -0.0014  0.0000 0.0000  0.0022 0.0007  •0.0217 •0.0071  TABLE IX  INDO S p i n  C  2s  C  2p  Densities  y  2p  f o r CHFCONH  F z  2s  (I) R a d i c a l  F2  Px  F 2p  y  2P  S  0.0294 0.0099 0.0000 0.0000  0.5021 0.3456  0.0000 0.0000  0.0000 0.0000  0.0201 0.0068  0.0012 0.0004  0.0000 0.0000  0.0004 0.0001  0.0135 0.0045  -0.0041 -0.0014  0.0000 0.0000  0.0000 0.0000  0.0022 0.0007  0.0008 0.0003  0.0000 0.0000  -0.1474 -0.1038  0.0000 0.0000  0.0000 0.0000  0.0000 0.0000  0.0421 0.0309  -0.0004 -0.0001  0.0000 0.0000  -0.0036 -0.0012  0.0003 0.0001  0.0002 0.0001  0.0000 0.0000  0.0006 0.0002  0.0003 0.0001  0.0000 0.0000  -0.0007 -0.0002  -0.0071 -0.0024  -0.0031 -0.0010  0.0000 0.0000  0.0006 0.0002  -0.0168 -0.0056  TABLE X  INDO S p i n  C  C  C  F  f o r CHFCONH^II)  F 2  Px  2 p  y  2s  Radical  2P  F X  2p  y  -0.0129 -0.0132  0.8465 0.8401  -0.0058 -0.0019  -0.0102 -0.0103  0.0269 0.0091  *.  -0.0012 -0.0004  -0.0018 -0.0017  -0.0015 -0.0005  0.0236 0.0080  2s  -0.0064 -0.0022  0.0021 0.0022  0.0007 0.0002  0.0042 0.0014  0.0012 0.0004  0.0047 0.0042  -0.2476 -0.2466  0.0032 0.0031  -0.0008 -0.0001  -0.0012 -0.0008  0.0728 0.0725  0.0013 0.0004  0.0031 0.0032  -0.0046 -0.0016  0.0004 0.0001  -0.0002 0.0000  -0.0010 -0.0009  0.0007 0.0003  0.0041 0.0013  0.0069 0.0069  0.0004 0.0001  -0.0125 -0.0042  -0.0067 -0.0022  -0.0032 -0.0024  0.0000 0.0000  2  P  *P  2P;  0.0450 0.0154  2s  y  C  F  2s  Densities  Z  •0.0345 -0.0113  TABLE XI  INDO S p i n  C  C  C  C  2Px  C  2p  y  2P  f o r CF CONH (I) 2  Z  2  2s  Radical  F 2P  F  2  P  y  X  <7\  2Px  0.0000 0.0000  0.5042 0.3558  Py  0.0017 0.0006  0.0000 0.0000  0.0160 0.0054  0.0010 0.0003  0.0000 0.0000  0.0017 0.0006  0.0123 0.0041  -0.0046 -0.0015  0.0000 0.0000  0.0000 0.0000  0.0019 0.0006  0.0008 0.0003  0.0000 0.0000  -0.1417 -0.1020  0.0000 0.0000  0.0000 0.0000  0.0000 0.0000  0.0389 0.0290  -0.0020 -0.0007  0.0000 0.0000  -0.0019 -0.0006  0.0007 0.0002  0.0006 0.0002  0.0000 0.0000  -0.0001 0.0000  -0.0002 -0.0001  0.0000 0.0000  -0.0024 -0.0008  -0.0067 -0.0021  -0.0030 -0.0010  0.0000 0.0000  0.0023 0.0007  2s  2p  2P,  0.0312 6-0.0105  2s  2  F  2s  Densities  z  •0.0163 •0.0054  TABLE X I I  INDO S p i n D e n s i t i e s f o r C F C O N H ( I I ) R a d i c a l 2  C  2s  C  2p  y  C  2P,  2  F 2s  F„ 2Px  F„ 2Py  2P:  0.0455 0.0156 2  C  Px  2p  2s  F  2p  x  F 2Py  -0.0139 -0.0145  0.8058 0.7989  -0.0032 -0.0010  -0.0049 -0.0048  0.0210 0.0071  -0.0015 -0.0005  -0.0017 -0.0017  0.0008 0.0003  0.0205 0.0069  -0.0068 -0.0023  0.0024 0.0025  0.0008 0.0002  0.0034 0.0011  0.0010 0.0004  0.0047 0.0043  -0.2209 -0.2197  0.0014 0.0013  -0.0007 -0.0001  -0.0012 -0.0009  0.0608 0.0605  -0.0010 -0.0004  0.0011 0.0011  -0.0022 -0.0008  0.0006 0.0002  0.0004 0.0001  -0.0003 -0.0003  -0.0002 -0.0001  0.0031 0.0010  0.0069 0.0069  -0.0014 -0.0005  -0.0117 -0.0039  -0.0064 -0.0021  -0.0030 -0.0022  0.0019 0.0007  •0.0321 •0.0105  CHAPTER F I V E  DISCUSSION  ( )•  The  A  1 9  F  Hyperfine  The hyperfine  prigin  Coupling  of the  large anisotropy of a - f l u o r i n e  c o u p l i n g t e n s o r has  appreciable  spin density  Tensors:-  been shown t o be  i n the f l u o r i n e  to the  free radical  characteristic  a b o u t 0.1  symmetry.  to the  radical  fluorine (45) coupling .  2p^  orbital  some f l u o r i n a t e d  to give a f l u o r i n e  and  large  the  a small  tensor with  radicals  near of  positive  have been shown  perfect cylindrical  '(2025)  .  ''" ', more a c c u r a t e  l  and  a  v  Although  ry  . Thus  shows a  plane  paral-  23) '  tensor  an  It implies a large spin population  i n the  isotropic  orbital  a-fluorine hyperfine  coupling perpendicular cylindrical  c a r b o n 2p  to  2p o r b i t a l (22  lei  due  J  departure  appreciable  .  measurements have i n d i c a t e d a  from i t ^ 2 , 2 3 , 4 6 ) ^  s e n c e o f an  symmet^;  This  i s mainly  spin density  (about  due  to the  0.8)  i n the  slight pre2p IT  orbital Table  of  XIII  the  c a r b o n atom t o w h i c h t h e  contains  a collection  radicals  p r e v i o u s l y observed  radicals  observed  is  doubt about the  relative  numbers i n p a r e n t h e s e s  are  of r e s u l t s  together  i n the p r e s e n t  fluorine  with  study.  for the  is  attached.  fluorinated results  In cases  where  for  there  s i g n s of the p r i n c i p a l v a l u e s , those  c a l c u l a t e d by  using  the  the  opposite  69  TABLE  XIII  C o m p a r i s o n o f Some F l u o r i n a t e d R a d i c a l s  Radical  S.No:  Tensor+  References  Hyperfine  Anisotropic A o  A  A  zz  XX 1  COO~CF CFCOO" 71  S80  -12  -67  23  2  CF^CONH 2 2  75  103  -51  -51  20  3  CF^CFCONH^ 3 2  74(60)  127(141) -62 (-86) -64 (-82) 46  4  CHFCONH  56®  133  5  CF COONH  6  CF COO"(I)  68 (52) 113 (129) -55 (-39) -57 (-41)  7  CF COO~(II)  71(54)  8  CF ( C 0 0 ~ )  84 (55) 124 (153) -57 (-82) -67 (-72)  2  4  2  2  2  -60  -72  22  72 (53) 116(135) -58 (-86) -58 (-86) 25  117 (134) -52 (-35) -64 (-47)  present work  2  2  -The v a l u e s o f t h e t e n s o r e l e m e n t s a r e i n g a u s s . The R a d i c a l CF COONH i s s i m i l a r t o t h e CF COO 2 4 2 o b t a i n e d i n the p r e s e n t study. The NH  radicals  p a r t need n o t be c o n s i d e r e d .  70  relative  s i g n s between t h e  values. Also of the  C-F  bond  and  i s perpendicular  t h e most n e g a t i v e  ( z - d i r e c t i o n ) as  In a l l p r e v i o u s ined  is  i n Table  d o u b t has will  be  evidence  b e e n c a s t on  the  radicals,  corresponding ment t o be  the  usual  radical  plane to  practice.  the r a d i c a l s  contrary;  assumption^ )  f  conta-  fragment  although th  2  hyperfine  i n Tables  for radical  e  some  above model  i n the  p l a n e ) , we  2p  I I and  of the  may  easily  I I . Considering  come t o t h e  fluorine  region i s less  I I s h o w i n g t h a t i t has orbital  as  compared  than  this to  conclusion  from a x i a l  two  I I I shows t h a t  x-direction ( i . e . perpendicular  f o r I I . A l s o the d e p a r t u r e  carbon  tensors  I i s somewhat s m a l l e r  element f o r r a d i c a l  more i n r a d i c a l i n the  t o the  component  element i s p a r a l l e l  i s the  to the  this  as g i v e n  s p i n d e n s i t y i n the  I than  largest  followed.  l a r g e element  radical  principal  been assumed t h a t t h e  A comparison o f the CF^COO  small  discussions of  X I I I , i t has  planar, without  the  been assumed t h a t t h e  coupling tensor  (x-direction) the  i t has  l a r g e and  for  the  ele-  the that radical  symmetry i s  a'larger spin density to r a d i c a l  I . These  71  f e a t u r e s can spin the  densities  seen i n T a b l e s V f o r p l a n a r and  density  due  to  i n the  an  fluorine  uncertainty The  w h i c h show t h e  p l a n a r one. that  We  radical  the  signal  and  radical.with  due  to  the  COO"  has to  less  the  the to  spin  easier  the  of  CF  that  in  C00~  assuming  the  CF ~ 2  a  in a less  that  carbon  2p^  the  non-  group i n  fact  of  the  r a d i c a l with  resulting  2  i n the  r e s u l t of  the  de-  than the  justified  I I has  delocalization  seems r e a s o n a b l e  CF (COO"") , the 2  cally  hybridized  follows: 2  one  of  CCC  p l a n e . The  C-F  bonds i n most o f  the  indicated  a more s t a b l e  density  nate i o n ,  t i o n of  be  after  group c o p l a n a r w i t h the  that  i s as  the  have a l s o  radical  irradiation  J  figure  partly  odd  the  a sta-  planar  orbital is electron  into  INDO c a l c u l a t i o n w i l l  be  i n more d e t a i l l a t e r .  It  sp  fourth  a weaker s i g n a l . The  group. T h i s  discussed  C00~  r i s e to  plane perpendicular  radical  i n the  of  same  o r b i t a l w h i c h may  7r  i s s l i g h t l y more s t a b l e  fragment thus g i v i n g stronger  2p  should, therefore,  I has  INQO  non-planar configurations  INDO r e s u l t s  planar configuration  of  VI  2  cimal point.  ble  and  CF COO~ r a d i c a l . T h e s e numbers show a l m o s t t h e  spin be  be  that  In  the  central  resulting  what h a p p e n s d u r i n g undamaged  in tetrahedral  i r r a d i a t i o n leads to  CFCCOO")^ and  the  CF  loss  COO".  bonds.  perpendicular (i) the  molecules r e s u l t i n g  ( i i ) the  molecules y i e l d i n g  difluoromalo-  carbon o r b i t a l s are  c a r b o x y l groups i s n e a r l y  of In  a COO" case  the  basiAlso  to  the  breakage  of  i n the  forma-  g r o u p i n some  ( i ) , after  the  72  l o s s o f a f l u o r i n e atom, the f l u o r i n e r e t a i n e d i n the r a d i c a l moves i n t o the plane o f the three carbon  atoms to g i v e  r i s e t o a p l a n a r sp^ h y b r i d i z a t i o n on the c e n t r a l atom. In case  (ii),  carbon  a C00~ group i s removed, but s i n c e the  two c a r b o x y l groups are nonequivalent,  there should be two  kinds o f the CF COO~ r a d i c a l s so formed. In one, the C00~ 2  group r e t a i n e d i n the r a d i c a l w i l l be coplanar with the CF^ fragment while i n the o t h e r , i t w i l l n o t . F u r t h e r d i s c u s s i o n o f t h i s p o i n t w i l l be taken up l a t e r .  (B). A n i s o t r o p i c I n t e r a c t i o n and INDO C a l c u l a t i o n s ; -  I f we designate the s p i n d e n s i t y i n a p a r t i c u l a r o r b i t a l by p^, where X denotes the center  ( f l u o r i n e and/or  carbon) and.Y the type o f the o r b i t a l r e l a t i v e t o the F-C bond, then a p r i n c i p a l value o f the a n i s o t r o p i c p a r t o f the f l u o r i n e c o u p l i n g tensor  F (A ) may be r e p r e s e n t e d app">» \jL Ut  r o x i m a t e l y as f o l l o w s .  (36) where o(.=x, y o r z and 0  F aa  =  (r  2  - 3aa)/r  5  (37)  where r i s the v e c t o r from the f l u o r i n e atom t o the e l e c t r o n .  73  The  first  three  terms i n e q u a t i o n  (36) a r e t h o s e  (22) considered assumed <2p |0 F  by Cook e t a l .  . For the f i r s t  two t e r m s ,  they  that  F  |2p > = <2p |0 F  x  F  F z  |2p  >-=  F  1082 g a u s s  (38)  and  <2p |6 F  F  |2p > = F  yy  IT  <2p |8 -|2p F  F  ZZ  TT  H'  = <2p |0 F  These v a l u e s an  F  |2p  SCF 2p a t o m i c o r b i t a l  and  <2p |3 F  > =  F  F  *a XX  N  TT  were o b t a i n e d  i n t e g r a l was e v a l u a t e d  by  y  F  1  > =  |2p > F  1  * 0 '  -541 g a u s s  from the v a l u e s  (39)  of ( " " ) r  f o r neutral fluorine ^  by them u s i n g  3  a v e  for  •* . The t h i r d  8  t h e method o f M c C o n n e l l  S t r a t h d e e ' f ) . The f o u r t h t e r m w h i c h was n o t c o n s i d e r e d 2  Cook e t a l , , r e p r e s e n t s  hyperfine  coupling  the o v e r l a p fluorine  region  tensor  the c o n t r i b u t i o n to the f l u o r i n e from t h e s p i n d e n s i t y  between t h e 2 p  atoms. An e s t i m a t e  orbitals  u  o f the order  residing i n  on c a r b o n and  o f magnitude o f (49)  this  integral  c a n be made u s i n g  a Mulliken  approximation  v  '  viz.,  <2 ^ |2p£>= P  [< p ^  a  where S_  2  i s the overlap  | 2p^> integral  +  <2pC |  | p C >] 2  between t h e 2p  ,40,  orbitals  o f c a r b o n and f l u o r i n e .  Using the  overlap  t h e C-F bond l e n g t h o f  integral  S  1.388,  the value o f  was f o u n d t o be 0.120 9. A l s o t h e N  74  expression  o f M c C o n n e l l and S t r a t h d e e  )c-F f r a g m e n t ,  /2p |6 C  F  , modified  gave  |2p^>= 5 . 5 ( 1 - 3 c o s 0 ) + 2 . 2 ( c o s 9 2  the f i e l d  x a x i s . Using equations the value  -33.2  (41)  d i r e c t i o n and t h e  and < j > i s t h e a n g l e between t h e p r o j e c t i o n o f t h e v e c -  tor representing  that  - l)cos2<|>  2  where 0 i s t h e a n g l e between t h e f i e l d z-axis  for a  d i r e c t i o n o n t h e xy p l a n e , and  (38) ,  (39) and ( 4 1 ) ,  of the i n t e g r a l i n equation  i t was f o u n d  (40) i s 6 5 . 2 g a u s s , ;  g a u s s and -32.0 g a u s s i n x, y and z d i r e c t i o n s  tively.  respec-  A study o f the s p i n d e n s i t i e s f o r f l u o r i n a t e d  radi-  CF cals,  given  i n Tables  (rrO.l t o -0,2) so t h a t spin density sity  will  I V - X I I shows t h a t  orbital  A consideration possible  as t h a t  c a u s e d by t h e s p i n  i . e . , the t h i r d  o f t h e terms i n e q u a t i o n  t o understand  to  some e x t e n t ,  the  r a d i c a l s given  and  OB^CONP^, f o r i n s t a n c e , signs  coupling  i n Table XIII. there  was no a m b i g u i t y  2  =+0.8 and a d e q u a t e l y  assumed  F  distribution of  =-0.016, p =+0.119 and a fr accounted f o r the l a r g e anisotropy 2  p  p  about t h e  o f the f l u o r i n e hyper-  F o r CHFCONH , Cook e t a l . F  a spin density  tensor f o r  I n t h e c a s e o f CHFCONB^  o f the p r i n c i p a l values tensor.  (36) makes  the v a r i a t i o n i n  a n i s o t r o p i c part o f the f l u o r i n e hyperfine  fine  den-  term i n equar^  the  relative  off-diagonal  TT  t i o n (36).  it  i s quite . large  t h e c o n t r i b u t i o n from t h i s  be as l a r g e  on c a r b o n 2p  p„  75  due  to the  ation CF  large  f l u o r i n e 2p^  spin density  from a x i a l  symmetry due  to the  bond. As  dicular  and  assume, as w i l l  CF CONH 2  to the  T a b l e s X and  same i n t h e  ributions  are  anisotropy  plane,  show t h a t two  p  radicals  relatively  i n CF CONH 2  requires  that  radical.  The  there  CONH  then the and  2 p  2  fourth  2  are  i s not  i n the  can  d e l o c a l i z e d over the  of  fact  p~  f l u o r i n e 2p.  JT  second  orbital  f l u o r i n e and  i n the  bond i s e x p e c t e d  negative  will  increasing d e l o c a l i z a t i o n . This  remarkably true  and  I V - X I I shows t h a t we  can  study of  arrange the  increase  i s found  to  fluorinated radicals in  i n c r e a s i n g d e l o c a l i z a t i o n , as  The  r a d i c a l s w h i c h have been l a b e l l e d as (or CONH ) g r o u p c o p l a n a r 2  given  i n Table  (I) i n t h i s with  the  CF  CHF)  fragment, w h i l e t h o s e which have been l a b l e l l e d  this  group p e r p e n d i c u l a r  to the  CF  spin d e n s i t i e s i n Tables  order  COO"  CF  i n the  presumably  assumption  the  have t h e i r  of  the  this  density  TT s p i n d e n s i t y  and  and  2  shows  TT s p i n  2  former  off-diagonal  t o be  the  i n CHFCONH i n the  the  cont-  that  0.0725 f o r CHFCONH  unreasonable since  2  in  approximately  bonds i n C F C O N H . The 2  that  e v e n t t h e s e two  than that F  i s a reduction  2  later,  spin densities i n  p£  ( i n any  i s smaller  This  be  term  group i s perpen-  2  s m a l l ) . Hence t h e  trend.  with  the  TT  INDO s p i n d e n s i t y - o f  0.0605 f o r C F C O N H  be  the  discussed  r a d i c a l s , the  2  radical  XII  be  a the  devi-  (36).  I f we 2  slight  p o l a r i z a t i o n of  mentioned above, they n e g l e c t e d  of equation  CHFCONH  and  plane of  the  XIV.  table (or  2  ( I I ) have  fragment.  76  TABLE XIV  Some F l u o r i n a t e d R a d i c a l s A r r a n g e d  i n the Order o f Increa-  sing Delocalization  Radical  S.No:  P p 2  (projected)  12  CHFCONH (II)  -0.2466  2  CHFCOO"(II)  -0.2406  3  CHFCOO"(I)  -0.2367  4  CF CONH (II)  -0.2197  5  CF(COO )  -0.1953  6  CF COO~(II)  -0.1923  7  CF COO"(I)  -0.1868  8  CHFCONH (I)  -0.1038  9  CF CONH (I)  -0.1020  2  2  2  _  2  2  2  2  2  2  77  the spin density J  Since effect rease  o f the f o u r t h term A  and i n c r e a s e A  x x  CF2CONH2/  A xxv  and A  y v  F >  '> ' A y y  F  C  xx  H  F  z z  < A yy  us now i n c l u d e o t h e r  scheme. The r a d i c a l radicals  i n equation .  obtained  C  F  2  4  CF  cipal  2  fragment.  values  CT^COO"" (II) for  i s similar study.  the r e l a t i v e  v  e  r  t  as w i l l  the r a d i c a l s ,  F  scheme, r e l a t i n g  XIII  i n this CF COO~ 2  I t i s hard i s coplanar  i n each then  XIII  2  of CFfCOO"^  perpendicular  to the r a d i c a l  be m e n t i o n e d  the i n t e g r a l s  by  u s i n g p u r e SCF 2p o r b i t a l s  spin densities  arises  i s almost  p  plane.  of  i n equations  hold  find  the d e l o c a l i z a -  t h a t one o f t h e c a r b o x y l g r o u p s  necessarily  case  and XIV t h a t  from the f a c t  i t should  we  assume  f i t i n t h e above-  the hyperfine tensor with  Finally  with  f o r C ^ C O O " (I) ,  COO~CF CFCOO~,  i n case  t o say  signs of the p r i n -  i n Tables  The s m a l l e r a n i s o t r o p y  e  (42)  F  to the  be d i s c u s s e d l a t e r ,  except  n  be t o dec-  and CF (COQ~) 2 a r e n o t known, b u t i f we  from a comparison o f r e s u l t s  not  i  t  from T a b l e  t h e moment t h a t a l l t h e t h r e e e l e m e n t s  tion.  a  hold  o f the hyperfine tensors  are p o s i t i v e ,  all  ?  C  i n the present  Further  <  (36) w i l l  w h e t h e r t h e -COONH4 g r o u p i n t h e f o r m e r the  e  ;' A $z f z < A zz  F  radicals  CF COONH  n  s  Thus f o r CHFCONH2 and  the f o l l o w i n g i n e q u a l i t i e s  X  Let  ^  that the values  (38) and (39) were and h e n c e t h e s e  i n the present  have been o b t a i n e d  obtained  values  may  d i s c u s s i o n since the  by u s i n g  the b a s i s s e t  78  of  Slater  hyperfine sors. be  type  atomic  tensor using these  sent study  account  rise  l e d t o much s m a l l e r  t h a t these  and t h e known e x p e r i m e n t a l 2  and C F C O N H 2  of the i n t e g r a l s  2  different  ten-  should  i n the preFor this pur-  r a d i c a l s were f i r s t  chosen.  could not adequately  tensors of other  be due t o t h e f a c t  t o an e n t i r e l y  radicals.  t h a t t h e amide g r o u p  This  gives  environment f o r the e l e c t r o n  compared t o a c a r b o x y l g r o u p . T h i s w i l l study  integrals  results.  so o b t a i n e d  f o r the experimental  may p a r t l y  the a n a l y s i s of the  using the spin d e n s i t i e s obtained  p o s e , t h e CHFCONH values  In f a c t  values  I t was t h e r e f o r e t h o u g h t  evaluated  The  orbitals.  be c l e a r  from the  o f T a b l e s V and V I o r V I I and V I I I on one hand  and  IX and X o r XI and X I I on t h e o t h e r , w h i c h shows t h a t t h e spin densities  a r e n o t so s e n s i t i v e  c a r b o x y l g r o u p as t h e y  to the r o t a t i o n  are to the r o t a t i o n  of the  o f t h e amide  g r o u p . S i n c e we do n o t know t h e g e o m e t r i e s  o f t h e CHFCONH  and  e r r o r i n the  CF CONH 2  2  radicals decisively,  a slight  g e o m e t r y c o u l d l e a d t o a l a r g e change i n t h e s p i n Because o f t h i s ,  the data  r a d i c a l s were n e x t integrals;  be  noted,  in  t h e two r a d i c a l s  2  chosen t o e v a l u a t e  2  t h e above-mentioned  i s o u r l a c k o f knowledge  signs o f the tensor elements.  I t should  however, t h a t t h e e n v i r o n m e n t o f t h e f l u o r i n e i s almost  the  c l o s e resemblance o f t h e i r  and  I I I ) , Thus t h e r e l a t i v e  radicals.  densities.  f o r C F C O O ~ ( I ) and C F C O O ~ ( I I )  but the problem here  about the r e l a t i v e  2  t h e same as c a n be s e e n hyperfine tensors  signs should  atom  from  (Tables I I  be t h e same i n b o t h  C a l c u l a t i o n s were p e r f o r m e d on t h e a s s u m p t i o n  79  that but  t h e two s m a l l e r e l e m e n t s o f t h e t e n s o r s were the r e s u l t s  radicals.  failed  t o reproduce the tensors f o r other  Hence i t was n e x t assumed t h a t  ments f o r b o t h r a d i c a l s calculation culation  negative,  were p o s i t i v e .  a l l the three  The r e s u l t s  ele-  of this  a r e summarized i n T a b l e XV. The method o f c a l -  i s as f o l l o w s :  Substituting  equation  (40) i n t o (36)  we g e t  §5a"  <2  <P5  >  P  TT  11  <I  °aa ' P * P 2 < 2 p 2  > +  r  P a  I 6£  F .AF a  0  | 2p< > (43)  ^P^p/^Trrr^p^^pSl^al^ Now i f t h e o n e - c e n t e r i n t e g r a l s  arCconstrained  t o be a x i a l l y  s y m m e t r i c , we h a v e  < ^  1 6 5 x I P ? > = < P a 1 L I Pa>= " 2  5  2  2  2 < 2  P a I °xx I  =-2<2pF|6 |2p >=-2<2p |6 |2p > F  F  F  F  y  A ~xx F  (44) i n t o  P ? > = "  a=x,  (43) and p u t t i n g  < & * >  y , z, we g e t  = (P-hQ)A + R<2p^|6 |2pJ> F  K  = -^(P+Q)A + R<2pJ|6 y |2p^> F  ^y  2  (44)  = A (say)  F  z  Substituting  2  F  '(•4 5)  - ( ^ P - Q ) A + R<2p^|0 z |2p^>J F  where P =/  + 2 p  ^  F  CF  ' R  TT  and  : A = <2p .|6 |2p> > TT TT XX F  F  FF  1  O  =P2p  CF +  IT  ^ T T T r ^ p  TT  (46) (47)  80  TABLE XV  Theoretical  Hyperfine Tensors Computed  from  f o r Some F l u o r i n a t e d  INDO S p i n  Anisotropic S.No:  Densities  Hyperfine (gauss)  Tensor  Radical yy  A„„ zz  118  -54  -64  A  xx  A  1  CF(COO )  2  CF COO~(I)  113  -51  -62  3  CF COO~(II)  117  -52  -64  4  CHFCOO"(I)  142  -68  -74  5  CHFCOO" ( I I )  143  -68  -76  6  CHFCONH (I)  64  -28  -36  7  CHFCONH (II)  151  -67  -84  8  CF CONH (I)  62  -27  -35  9  CF  134  -59  -74  -  2  2  2  2  2  2  2  CONH (II) 2  Radica  81  Using the a n i s o t r o p i c p a r t s o f the h y p e r f i n e t e n s o r s f o r the CF COO"*(I) and CF COO~(II) r a d i c a l s 2  2  ( c o n s i d e r i n g the  three elements t o be p o s i t i v e , as d i s c u s s e d above), the i n t e g r a l s i n equations  (46) and (4 7)  s o l v i n g the simultaneous  were e v a l u a t e d by  equations  (45)  and  u s i n g the  p r o j e c t e d s p i n d e n s i t i e s . The r e s u l t s were as f o l l o w s : -  PS| XXI PS>  84.0  gauss  <2 J|6 |2 ;>  -45.0  gauss  -38.7  gauss  1290.0  gauss  < 2  6  2  F  P  y  ^ f|:5  F  P  z  P  |2 ;> P  A The  =  (48)  values o f the i n t e g r a l s i n equation  (4 8)  were then  used i n c o n j u n c t i o n with the s p i n d e n s i t i e s t o compute the t h e o r e t i c a l h y p e r f i n e t e n s o r s f o r other r a d i c a l s as g i v e n i n Table XV.  The  r e s u l t s i n Table XV show some i n t e r e s t i n g  f e a t u r e s . F i r s t o f a l l we g e t a t e n s o r f o r the C F ( C 0 0 ~ ) r a d i c a l which c l o s e l y resembles the experimental if  2  tensor  we take a l l the three elements o f the t e n s o r t o be ^>n  p o s i t i v e . Secondly, and CF CONH 2  2  although  the agreement f o r CHFC0NH  r a d i c a l s i s not very good, i t would get even  worse i f the p l a n a r structure-were c a l s . F o r t h i s reason observed dicular  the  CHF  assumed f o r these  i t may be concluded  experimentally to  2  or  have CF^  the  CONH  fragment.  radi  t h a t the r a d i c a 2  group  perpen  I t may be men-  82  tioned  t h a t i n t h e above c a l c u l a t i o n  t h e NH  2  part of the  CONH2 g r o u p was c o n s i d e r e d t o be c o p l a n a r w i t h f r a g m e n t . The p o s s i b i l i t y  II,  b u t t h e a g r e e m e n t became e v e n w o r s e . A n o t h e r  for  t h e nonagreement may be t h a t t h e e x a c t g e o m e t r y and C F C O N H 2  metry used correct for  radicals  The  of  the corresponding  c l o s e agreement o f t h e e x p e r i m e n t a l  tensors f o r the CF(COO~)  2  radical  the i r r a d i a t i o n  CHF(COOK) , o r potassium 2  i s achieved  and  ( w i t h i n +8  of the tensors f o r other r a d -  a c a r b o x y l g r o u p , e . g . , t h e CHFCOO"  Whenever t h i s  tensor  a c a r b o x y l group.  which has n o t been o b s e r v e d med f r o m  reason  c o n t a i n i n g an amide g r o u p i s more s e n -  g a u s s ) l e a d s t o an e s t i m a t e i c a l s with  I and  the hyperfine  t o t h e changes i n geometry than  calculated  radicals  i s n o t known and t h e g e o -  and, as mentioned e a r l i e r ,  r a d i c a l s with  tation  f o r both  i n INDO c a l c u l a t i o n s may n o t n e c e s s a r i l y be  the r a d i c a l s  sitive  2  CON  f r a g m e n t t o be t w i s -  through  2  study  2  ted  CHFCONH  9 0 ° was a l s o  o f t h e NH  the  radical  y e t , b u t w h i c h c o u l d be  of potassium  monofluoromalonate,  monofluoroacetate, i t will  CH FC00K. 2  be a c h e c k on t h e o r i e n -  o f t h e c a r b o x y l g r o u p i n CHFCOO" r a d i c a l  a l s o show how f a r t h e t h e o r y h a s s u c c e e d e d the h y p e r f i n e t e n s o r f o r t h i s  for-  radical.  and i t w i l l  i n predicting  83  (C)• Fermi-Contact I n t e r a c t i o n  The  and INDO  Calculations:-  theory o f i s o t r o p i c h y p e r f i n e s p l i t t i n g ,  ari-  s i n g from the Fermi-contact i n t e r a c t i o n , has been d e c r i b e d in detail  elsewhere ^ ® *• . The f l u o r i n a t e d r a d i c a l s e x h i b i t  a small and u s u s a l l y p o s i t i v e i s o t r o p i c h y p e r f i n e due  t o a small s p i n d e n s i t y  From the value of I ^2 s ting for a 17110 the  1 9  i n the f l u o r i n e 2s o r b i t a l .  I '  0  coupling  t  n  e  i  s o t r  °pi  c  F nucleus has been c a l c u l a t e d  gauss i f the odd e l e c t r o n  hyperfine  split-  t o be equal  to  i s localized exclusively i n  f l u o r i n e 2s o r b i t a l ( 5 0 ) . However we cannot use t h i s SCF  value i n the p r e s e n t case, due t o the reasons mentioned i n the p r e v i o u s s e c t i o n , w i t h 2s S l a t e r  INDO  spin density  i n the f l u o r i n e  orbital.  The  i s o t r o p i c hyperfine coupling  constant a , f o r N  a nucleus N i s g i v e n by a  = (4 r/3)g[3g B <S >- <*|p(r ) | »> 1  N  N  T  N  z  N  where ¥ i s the wave f u n c t i o n  of the system, which i n the  p r e s e n t case, i s the u n r e s t r i c t e d defined  i n equation  (11). r  nucleus N and the q u a n t i t y  (49)  N  Hartree-Fock wave  i s the p o s i t i o n v e c t o r  p(r ) N  i s the s p i n d e n s i t y  function of the opera-  t o r evaluated a t the nucleus N and d e f i n e d as  P<-HN>  =  £ ' z k * < * k - r„) k 2 s  (50)  84  where r ^ i s t h e p o s i t i o n is  t h e z-component  the  Dirac delta  vector of the k  of the spin of k  t  n  t  n  electron,  e l e c t r o n and 6 ( r )  is  f u n c t i o n . W i t h ¥ d e f i n e d as i n e q u a t i o n ( 1 1 ) ,  we g e t  <f|p(r J|*>  5^P %(r )« (r )  =  N  yv  N  v  :(51)  N  yv where that  p  y  i s same as Q y  v  a l l c o n t r i b u t i o n s t o t h e summation  negligible  except  the nucleus only  those  when ^  the s o r b i t a l s  N  p S  f o r other o r b i t a l s  =  p  o n atom N,  a t the nucleus  a r e z e r o . Thus e q u a -  |<J>  (r ) |  (52)  2  N  N s orbital  (r ) |  i s the probability  2  N  evaluated  a t the nucleus.  density of  S u b s t i t u t i n g equa-  (52) i n ( 4 9 ) , we g e t a  N  ^ V3)g8g B <s >-lU  =  4  N  N  The q u a n t i t y i n t h e b r a c e s  N  =  i s constant as Q . N  (4TT/3) c 3 g e < S > J  (53)  z  atom and may be a b b r e v i a t e d  Q  so  are  S  that o r b i t a l ,  of  (51)  centered a t  i s the spin population i n the valence  N N  t h e atom N and | <j>  tion  centered  have n o n - z e r o d e n s i t i e s  N N  of  i n equation  assume  (51) becomes < ¥ | p ( r ) |V>  where  ( 1 4 ) . Now  and c(>^ a r e b o t h  N. A l s o among t h e o r b i t a l s  whereas t h e d e n s i t i e s tion  defined i n equation  V  N  N  2  1  fora particular  type  Thus  | <t> (^l S N  2  (54)  that  a  N  =  (55)  85  Pople,  Beveridge  in  fact,  at  specific  a c o r r e l a t i o n between t h e computed  isotropic a linear  and Dobosh(42) d e m o n s t r a t e d t h a t t h e r e  atoms  i n a number o f m o l e c u l e s  hyperfine coupling constant relationship  correlated  i s i m p l i e d by e q u a t i o n  linearity,  they  obtained  and t h e c a l c u l a t e d  Pople such  values  we  should  ( i ) . Among t h e n u c l e i sufficient  considered  two  important  f o r *H;  for  1 9  F  deviation  as 22 g a u s s . The v a l u e  c a r r y i n g o u t any c o r r e l a t i o n , we  spin densities.  For the molecules  of Q  and Dobosh, t h e y  the "contaminated"  The p r e s e n t  were  for  obtained  spin  study  of contaminating  significantly.  In f a c t  change i n by  Pople,  change and h e n c e  they  densities.  i n v o l v e s l a r g e s y s t e m s and t h e and p r o j e c t i o n show t h a t t h e a n n i -  s p i n s changes t h e s p i n  the s p i n d e n s i t i e s  i n a b o n d i n g were  decide  came t o t h e c o n c l u s i o n t h a t t h e  r e s u l t s o f INDO c a l c u l a t i o n  volved  F  should  considered  a n n i h i l a t i o n d i d n o t make a s i g n i f i c a n t  hilation  there  t h e r e were n o t  w h e t h e r t h e s p i n a n n i h i l a t i o n makes s i g n i f i c a n t  used  using  gauss.  ( i i ) . Before  Beveridge  of  points:  for correlation,  was  the  obser-  XVI. B e f o r e  this  44829  several  between t h e  n i n e d a t a p o i n t s and t h e s t a n d a r d  was  ( 5 5 ) . From t h e N  i n Table  recall  data points only  as h i g h  atoms. Such  o f Q ;/for  more t h a n case  observed  c o u p l i n g c o n s t a n t s . The r e s u l t s  e t a_l. have been summarized correlation,  densities  and t h e  a t those  atoms w h i c h gave t h e b e s t c o r r e s p o n d e n c e ved  spin  was,  found  densities  i n the o r b i t a l s i n -  t o be a f f e c t e d  to a larger  ex-  86  TABLE  Analysis  o f L i n e a r i t y Between a  XVI  and  N  f  as C a r r i e d  out  N N By; P o p l e , B e v e r i d g e and Dobosh  Nucleus  1 3  No: o f D a t a P o i n t s Q , N  Gauss S t d :  Dev:,  »H  141  540  7.3  C  26  820  23.8  29  379  2.3  5  889  2.7  9  44829  22.2  1 9p  Gauss  87  t e n t than those i n the o r b i t a l s i n v o l v e d i n IT bonding.  The above arguments seem t o suggest  t h a t the c o r -  r e l a t i o n c a r i i e d out by Pople et: al_. i s not very  satisfacto-  r y . Table XVII c o n t a i n s the v a l u e s of a^, c a l c u l a t e d by u s i n g the contaminated  as w e l l as pure s p i n d e n s i t i e s , f o r some  f l u o r i n a t e d r a d i c a l s together with the experimental In these c a l c u l a t i o n s the value of o b t a i n e d by Pople e t a l . i . e . , 44829  was  results.  the same as t h a t  gauss.  Table XVII shows t h a t the c o r r e l a t i o n i s poor and the computed v a l u e s are, i n g e n e r a l , much s m a l l e r than the observed  values.  88  TABLE  XVII  C o m p a r i s o n Between t h e O b s e r v e d ling  Isotropic  and t h e V a l u e C a l c u l a t e d  a , N  Radical  from P o p l e ' s  calculated a , N  contami- using  nated  spin  project-  ed s p i n ties  Correlation  calculated  using  densities  H y p e r f i n e Coup-  d e n s i - a^,  (gauss)  Exptl:  (gauss)  (gauss)  CHFCONH (I)  36  14  CHFCONH (II)  54  18  CF CONH (I)  36  14  CF CONH (II)  45  18  CF(COO )  27  10  84 (55)  CF COO"(I)  27  10  68 (52)  CF COO"(II)  27  10  71 (54)  CHFCOO-(I)  54  18  CHFCOO"(II)  49  18  2  2  2  2  -  2  2  2  56  75  89  (D).  O r i e n t a t i o n of Radicals i n the C r y s t a l ; -  A comparison o f r e s u l t s shows t h a t t h e g t e n s o r sent is  study  the  largest  normally  A value  CF(C00"*)2 r a d i c a l  expected^  (known as " u n i q u e  0.706). T h i s d i r e c t i o n plane  The  unique  angles  c a n be t a k e n  and t o t h e C-C-C p l a n e principal  CF COO-(II) 2  by s i g n  the  carboxyl is  almost  radicals the  being  the  (0.705. 0.075,  J  t o be n o r m a l t o t h e r a d i -  o f t h e undamaged  and  molecule.  CF2C00~(I)  and  (0.078, 0.678,  are the p r i n c i p a l  axes  0.730)  f o r o n l y one obtai-  c h a n g e s as d e s c r i b e d e a r l i e r .  i t was  crystal  axis") of the  s i t e o f each r a d i c a l , the o t h e r s b e i n g  Although n o t known,  axis for  a - and c - a x e s ,  axes f o r t h e r a d i c a l s  Again<j;these  crystallographic  with  3  a r e (0.091, 0.940, 0.328)  respectively.  principal  This  i s nearly perpendicular to the c r y s t a l l o -  1 2  ned  . The p r i n c i p a l  c o s i n e s f o r t h e s i t e P.P P  cal  i n the pre-  p a r a l l e l to the A tensor.  g r a p h i c b - a x i s and makes e q u a l direction  I , I I and I I I  f o r each o f t h e r a d i c a l s  i s approximately  the s i t u a t i o n  i n Tables  the c r y s t a l  inferred  structure  from  of malonic  group i s c o p l a n a r w i t h perpendicular are completely  t h r e e unique  axes,  mation about the angles  structure  2  CNDO c a l c u l a t i o n s acid^l)  Now  p l a n a r , a study  and  is  from  t h a t one o f t h e  t h e CCC p l a n e  (85°) t o i t .  mrntioned  o f CF2ICCOOK)  while  i f both  the other  CF COO~ 2  of the angles  above, s h o u l d y i e l d  between t h e CCC p l a n e  between infor-  and t h e two  90  COO p l a n e s v From t h e above c o n s i d e r a t i o n s one w o u l d that  the unique  a x i s o f one CF COO~ r a d i c a l  llel  to that of the CF(COO )  2  -  2  radical  ted  sign  s h o u l d be p a r a -  and t h a t o f t h e o t h e r  s h o u l d be p e r p e n d i c u l a r t o i t . C a l c u l a t i o n s none o f t h e p o s s i b l e  combinations  showed  yielded  of Y - i r r a d i a t e d malonic  could  acid  t h e COO 2  group  -  fragment.  the c r y s t a l  they  found  that  They i n t e r p r e t e d  result  this  forces preventing the rotation  also,  -  the plane  perpendicular to  2  planar configuration. case  t h a t one CH^COO  i n CH COO~ was n e a r l y  bond w h i c h w o u l d be n e c e s s a r y  present  and f o u n d  studied the  and t h e o t h e r , a l t h o u g h p r e s e n t ,  n o t be a n a l y z e d . I n f a c t  the CH to  expec-  result.  r a d i c a l was n o n p l a n a r  of  that  this  H o r s f i e l d , M o r t o n and W h i f f e n ^ esr  expect  f o rthe r a d i c a l  I f t h e same s i t u a t i o n  tb^.be due about  t h e C-C  t o adopt t h e  prevails  t h e a n g l e between t h e u n i q u e  i n the  principal  a x e s o f t h e two CF COO~ r a d i c a l s w o u l d be a measure o f t h e 2  a n g l e between t h e two C-C bonds o f t h e undamaged It  was f o u n d  113° which  t h a t one c h o i c e o f s i g n s y i e l d e d  i s n o t f a r from  tent with t h i s get  the tetrahedral  c h o i c e , i t was a l s o  found  an a n g l e o f 8 4 ° between t h e u n i q u e  CF(COO ) _  and C F C O O ( I ) r a d i c a l s _  2  2  ween t h o s e o f C F ( C O O ~ )  2  an a n g l e o f  angle. Consis-  t h a t one c o u l d  principal  i s strictly  2  c a r b o x y l group  tly  p e r p e n d i c u l a r t o t h e CCC p l a n e  axes  of  and an a n g l e o f 5 9 ° b e t -  and CF COO""(II) r a d i c a l s .  one  molecule.  parallel  But i f  and t h e o t h e r  i n t h e undamaged  stric-  molecule,  91  both  o f these  angles  s h o u l d be 9 0 ° . T h e s e d i s c r e p a n c i e s  c o u l d v e r y w e l l be due t o e i t h e r tion of  n o t b e i n g e x a c t l y as assumed o r t h e p a r t i a l  t h e C-C bond a t t h e t i m e  If cipal is  the o r i g i n a l configura-  A values  truely  tioned  of r a d i c a l  the difference  f o r the r a d i c a l s  indicative  earlier,  be  t h e one  nonplanar  with  the r a d i c a l less  prin-  C F C O O ~ ( I ) and C F C O O ( I I ) -  2  2  geometry, then,  as men-  (I) w i t h A||=181 g a u s s s h o u l d be  t h e one w i t h more s p i n d e r e a l i z a t i o n structure while  formation.  between t h e l a r g e s t  of the r a d i c a l  the r a d i c a l  rotation  suggesting  a planar  ( I I ) w i t h A,|=188 g a u s s  spin d e r e a l i z a t i o n  should  suggesting the  structure.  Hence i t seems  likely  that CF COO~(I), the 2  more d o m i n a n t r a d i c a l ,  was f o r m e d by t h e l o s s o f t h e c a r -  b o x y l group o r i g i n a l l y  parallel  CF COO~(II), the less 2  t o t h e CCC p l a n e  dominant r a d i c a l ,  l o s s o f t h e c a r b o x y l group o r i g i n a l l y CCC  plane  was f o r m e d by t h e  perpendicular to the  i n t h e undamaged m o l e c u l e . The INDO c a l c u l a t i o n s  t h a t t h e energy the non-planar one  while  i s stabler.  showed  o f t h e p l a n a r CF^COO (I) r a d i c a l was lower than t h a t o f CF^COO (II) r a d i c a l by 0.56 eV, showing t h a t t h e former  PART I I  ELECTRON PARAMAGNETIC  RESONANCE  STUDY OF COPPER  ( I I ) IN SINGLE  CRYSTALS OF ZINC TRIFLUOROACETATE DIHYDRATE  CHAPTER ONE  INTRODUCTION  The main d i f f e r e n c e between the e l e c t r o n p a r a magnetic resonance  (epr) o f t r a n s i t i o n metal complexes and  the e l e c t r o n s p i n resonance lies  (esr) o f o r g a n i c f r e e  i n the o r i g i n o f the magnetogyric r a t i o  factor  radicals  (Y>) o r the g-  (gS=Y * where 8 i s the Bohr magneton) o f the system. N  While i n the f r e e r a d i c a l s , g i s very l i t t l e d i f f e r e n t  from  the f r e e s p i n g-value of 2 (2,0023 t o be e x a c t ) , i n the t r a n s i t i o n metal complexes  i t usually d i f f e r s  considerably  from 2. T h i s i s mainly due t o the f a c t t h a t i n the former, the o r b i t a l angular momentum i s quenched  t o a very l a r g e  extent w h i l e i n the l a t t e r , due t o th^^^^f^%!0eJfefei^.-.ogbi-i? ••coupling",-, t h i s  i s not so. This accounts f o r the somewhat  d i f f e r e n t nomenclature being c u s t o m a r i l y used i n two cases f o r e s s e n t i a l l y the same phenomenon, i . e . , e s r f o r o r g a n i c f r e e r a d i c a l s and epr f o r t r a n s i t i o n metal complexes. Apart from t h i s , t h e r e i s l e s s obvious d i f f e r e n c e between the two c a s e s . With o r g a n i c molecules, ground t r i p l e t s a r e comparat i v e l y r a r e , not t o mention the h i g h e r s p i n m u l t i p l e t s . With t r a n s i t i o n metal complexes,  on the o t h e r hand, d o u b l e t s ,  t r i p l e t s and up t o q u i n t e t s are o f common o c c u r r e n c e . T h i s , i n f a c t , i s due t o the u n u s u a l i s t a b i l i t y o f p a r t i a l l y ,  filled  i n n e r s h e l l s , the d - s h e l l s i n t h i s case. When the m u l t i p l i -  93  city  i s g r e a t e r than  spectra in  the  2,  showing f i n e spin  there i s the p o s s i b i l i t y  of the  s t r u c t u r e w h i c h r e q u i r e s an e x t r a  epr term  Hamiltonian.  The p l e x e s c a n be  theory of the epr of t r a n s i t i o n metal  found  i n many books (65-71)  Reviews o f e x p e r i m e n t a l  work may  a n c  a l s o be  j  r  e  v  found  i  e  w  com-  (72-82) ^  s  i n these  sour-  (82) ces. That  o f Kuska and  literature  up  t o 1966.  the e l e c t r o n i c c a n be  one  structure  F o r complexes w i t h  c o n f i g u r a t i o n c a n be  looks a t i t , the  respect,  the epr  Cu  later).  same as t h o s e  and,  2 reflects and,  i n fact,  spin  mat-  no  fine  + +  ions  In  this  of free  radi-  Hamiltonian.  the e l e c t r o n i c  from  i t i s r e p r e s e n t e d by  a n g u l a r momen-  s t r u c t u r e of the  I n a t r u e l y o c t a h e d r a l complex five-fold  degeneracy of the d o r b i t a l s  resulting  i n a higher doubly-degenerate  complex.  (0 >;group)  the  h  is partially e  a  This deviation  the c o n t r i b u t i o n of the o r b i t a l  indirectly,  ion,  + +  No  However, t h e g - f a c t o r i s c o n s i d e r a b l y d i f f e r e n t  t e n s o r i a l q u a n t i t y i n the  tum  and  w i t h two  + +  of  , or i t  i n the d - s h e l l .  be m e n t i o n e d  s p e c t r a are the  2 as d i s c u s s e d e a r l i e r  from  a single C u Q  system i s a d o u b l e t  state w i l l  survey  d e s c r i b e d as d  i s e x p e c t e d , (complexes o f C u  hence a t r i p l e t  cals.  ' gives a detailed  v  c o n s i d e r e d as a p o s i t i v e h o l e  t e r how  and  Rogers  level  and  removed a  lower  g  triply-degenerate  t g 2  level.  The  resulting  i n a Eg  that  a system i s n a t u r a l l y  such  2  h o l e r e s i d e s i n t h e eg  s t a t e . J a h n and  T e l l e r ' h a v e  u n s t a b l e and  will  levels  shown  distort  94  ( i f not already d i s t o r t e d move t h e o r b i t a l ting  the  b  The  ig(  d x  e  L  will  be  a l o n g a C4  o  t r u e and + +  l i g a n d s may  n o t be  be  shown l a t e r ,  of  the o r b i t a l  f  t  n  orbital  orbital  resul-  the a ^ g f d ^ )  distortion  e  t h e d^2  in this  and  is a  i s less  resulting  stable i n an  i s an e l o n g a t i o n , t h e o p p o s i t e will  be  in a  state.  belongs  ground  stable o r b i t a l s i s  However  e.g., be  involved  to a  state.  the  lower In  such  d _ or d o o 2 x^-y^ z  to p r e d i c t .  T h i s i s where t h e  determina-  be o f t h e g r e a t e s t h e l p . As  the g-tensor w h i c h t h e odd  i s very s e n s i t i v e electron  sometimes t h a t due  (or t h e h o l e )  to the l i g a n d s w i l l the e l e c t r o n i c  will  to the  type  occu-  s t u d y o f t h e h y p e r f i n e c o u p l i n g t e n s o r due  great value i n understanding  to  a l s o be  structure  of  of  complex.  The tive  t o be  e p r work on  reviewed  here  studies of only a certain be  I  t o form  then i t n a t u r a l l y  t i o n of the g-tensor w i l l  the  the  w i t h a non-degenerate  i s more d i f f i c u l t  and  symmetry, t h a t o f  a l l t h e same o r c h e l a t e s may  a c a s e , whether the l e a s t '  + +  4h*  axis, be  The  complex i s n o t a t r u e o c t a h e d r o n ,  symmetry g r o u p  Gu  D  the system  the c o o r d i n a t i o n ,  The  r  to r e -  state.  split  I f the d i s t o r t i o n  when t h e C u  pies.  o f the ground  orbitals will orbitals  state.  t o some o t h e r c a u s e s )  have a l o w e r  thus the h o l e w i l l  ig  in  g  2  compression and  degeneracy  complex w i l l  group.  due  copper  in detail.  complexes The  results  c l a s s of the C u  c o n s i d e r e d , namely t h e c o p p e r  i s too  + +  exhaus-  o f the  complexes  s a l t s of a l k y l  and  epr will  aryl  95  carboxylic  acids,  plex studied  which.is, of course, related  t o t h e com-  i n t h e p r e s e n t work.  (85)  Shimada and resonance rate,  co-workers  i n the s i n g l e  c r y s t a l s of copper  C u ( H C O O ) . 4 H 0 and 2  2  room t e m p e r a t u r e s c r y s t a l s was  ' studied  formate  2  assuming  that  the C u  s u r r o u n d e d by  four  o x y g e n atoms and  nal  d i s t o r t i o n . Later,  out  a detailed  + +  i n each of the  Wagner and c o - w o r k e r s  low t e m p e r a t u r e  two  of the  water  tetrago-  study o f the s i n g l e  a Cu:Zn r a t i o - ^ o f 0.2-3.4:100. A t t h a t  two  carried  Cu(HCOO)2.2H2O d i l u t e d w i t h i s o s t r u c t u r a l z i n c  ing  tetrahyd-  i t s d i h y d r a t e , Cu(HCOO) . 2 H 0 a t  molecules, the l a t t e r l y i n g along the a x i s  of  paramagnetic  crystals salt  hav-  time the c r y s t a l  ) structure clinic tain  d a t a was  c r y s t a l of zinc  four  t y p e was  and  results  t y p e when s u b s t i t u t e d able to c a l c u l a t e  known t o c o n environment site  o f the second  o c t a h e d r o n b u t formed  Cu  o f t h e mono-  o f the  o c t a h e d r o n o f o x y g e n atoms f r o m  o x y g e n s f r o m two show t h a t  cell  s i t e s . Each  formate groups w h i l e t h a t  a distorted two  the u n i t  types of l i g a n d  symmetry-related  a distorted  different  and  f o r m a t e d i h y d r a t e was  m e t a l i o n s w i t h two  e a c h h a v i n g two  also  available  + +  by  four  "?  type  water  was  molecules  d i f f e r e n t f o r m a t e g r o u p s . The  ions prefer  first  epr  the s i t e s of t h e ^ f i r s t  i n s m a l l amounts. Wagner e_t a l . were  a l l the parameters  of the s p i n H a m i l t o n i a n  qi un ic tl eu d id ni gf f et rh e n q t u af d rr ou m p otlheo s ei n ot fe r aShimada e t at lh e. i r r e s u l t s c t i o n and (85)  were  96  Reddy and study  of  the  salt.  unit c e l l of  four  the of  odd the  single crystals  The  so  triclinic  t h a t the  lines  '  v  of  Ba2Cu(HCOO)g.4H 2 0,  tetrahydrate, zince  Srinivasan  was  d i b a r i u m copper  c r y s t a l s contained  spectra  were q u i t e The  orbital  spin Hamiltonian could  be  explained the  d  but  9  temperature. the  This  case of N i  Copper a c e t a t e first  studied  as  a  t h e y were s u r p r i s e d gave more t h a n  s i t i o n of  the  ions w i t h the  that, unlike  t o be  due  to  electron spin vectors absorption  assuming and  one  of  and  with  found  in  c r y s t a l s .  H 2  °  w  Gordy^ ^  a  s  and  9 <  copper formate, i t They e x p l a i n e d  "|;hes?s)imultaneous of  the  increasing  NaF  Lancaster  anisotropic line.  "anamolous r e s o n a n c e "  by  2  to f i n d  one  parameters  monohydrate, C u ( C H F C O O ) «  powder by  that  z  s i t u a t i o n i s s i m i l a r to the  i o n i n i r r a d i a t e d L i F and  +  the  orbital  0  x -y interaction  the  z  consisting  only  z  neighbouring d ? o r b i t a l ,  site&per  r e s u l t s showed  ^2  between  formate  one  simple,  i n d^2  a vibronic interaction  a detailed  d i l u t e d with i s o s t r u c t u r a l  at a l l o r i e n t a t i o n s .  electron  reported  two  this tran-  neighbouring  Cu  a s i n g l e quantum". S i m i l a r work (91)  was  c a r r i e d out  the  same r e s u l t s  be  due  The + 1  to  the  by  B l e a n e y and but  The  m e t a l 10ns  explained  binuclear  r e s u l t s were f i t t e d  Bowers the  v  '  and  in binuclear  found  anamolous r e s o n a n c e  configuration^" of  t o the  they  conventional configuration  copper spin  to  acetate.  Hamiltonian  existain pairs.  97  for  a triplet  lished  their  of copper  state. Later r e s u l t s on  Kumagai and  i n t e r p r e t a t i o n was  by L a n c a s t e r and  Gordy^^;  t h e o r y was  (921  g i v e n by  Bleaney  most r e c e n t e p r s t u d y o f c o p p e r  and  pub-  crystals  similar  evidently  to they  (91) Bowers '. The d e t a (93) Bowers ' l a t e r . The  were unaware o f t h e work o f B l e a n e y and lied  v  the epr study of the s i n g l e  a c e t a t e but t h e i r  the s p e c u l a t i o n s  co-workers  v  v  a c e t a t e was  done by  Kokoszka  (94) and  co-workers  . They u s e d  t a t e monohydrate c o n t a i n i n g onal  impurity  Cu-Zn  single  about  and d e t e r m i n e d  crystals  0.5%  zinc  o f copper  as a  ace-  substituti-  t h e g - and A - v a l u e s f o r t h e  units. (95,96) I t was  f o u n d by Abe  that  monohydrate, Cu(CH CH COO) ,H 0, l i k e 3  nuclear  2  s t r u c t u r e . He  mined t h e p a r a m e t e r s  2  2  carried  copper p r o p i o n a t e  t h e a c e t a t e , had  out a d e t a i l e d  s t u d y and  a bideter-  o f t h e s p i n H a m i l t o n i a n . L a t e r Abe  and  (97) Shirai  studied  single  c r y s t a l s o f c o p p e r - n - b u t y r a t e mono-  h y d r a t e , C u ( C H 2 C H C H C O O ) 2 . H 0 , c o p p e r m o n o c h l o r o a c e t a t e mono2  2  2  h y d r a t e , C u ( C H C 1 C 0 0 ) H 0 , and 2  2 <  2  copper  h y d r a t e , Cu (CClgCOO) .H <0, among w h i c h 2  2  f o u n d t o have t h e b i n u c l e a r "normal  resonance"  t r i c h l o r o a c e t a t e monothe f i r s t  two  were  s t r u c t u r e while the l a s t  ( c f "anamolous r e s o n a n c e "  one  showed  above).  (98) Herring resonance  and  co-workers  in a polycrystalline  Cu (CgH,-COO)  2  and  confirmed that  studied  sample o f c o p p e r  the  paramagnetic  benzoate,  t h e sample c o n t a i n e d n o t  only  98  the  salt  i n the  the  c o p p e r was  in binuclear  binuclear present  exist  in a concentration  configuration  The xylates,  c o n f i g u r a t i o n , but  in binuclear the  have a b n o r m a l l y  with  form. T h i s  alkyllow  the  and  exception  presence of 1^3) .  the that  acetate  binuclear  and  other  c o n f i g u r a t i o n . For  w o u l d be  and  carboxylates, this  a paramagnetic resonance study of  roacetate  of  of i n t e r e s t .  carbo-  formate, the  fact  copper(II) temperature  antiferromagnetic Thompson  magnetic s u s c e p t i b i l i t y  c o p p e r t r i f l u o r o a c e t a t e , Cu(CF3COO)2> the  copper  i s a l s o r e f l e c t e d by  i n t h e s e compounds  4  unlike  of  copper  m a g n e t i c moments s h o w i n g t h e  Yawney(i^ ) c a r r i e d out  that  .  aryl-carboxylates  v a r i a t i o n which i n d i c a t e s the interactions  weaker t h a n  above e x a m p l e s i n d i c a t e t h a t  i n general,  t h a t most o f  '  a l s o some o f  and  studies  concluded  that,  i t d i d not  r e a s o n i t was copper (II)  on  have  thought trifluo-  CHAPTER  TWO  THEORETICAL  Cu tomary  has a d  configuration.  and c o n v e n i e n t , t o t r e a t  positive hole.  In t h i s  I t i s both, cus-  i t as a s y s t e m w i t h one  f o r m a l i s m , one need n o t d i s t i n g u i s h  between t h e g r o u p t h e o r e t i c a l and t h o s e o f t h e o r b i t a l s .  designations of the states  Thus one c o u l d c o n s i d e r  the free  2 ion  D t e r m as b e i n g  field  into  s p l i t by a s t r i c t l y 2  a lower l y i n g  E  spanned  q  octahedral  by d 2 _ 2 x  a n <  v  crystal  ^ & 2  ^  an<  2 an u p p e r tal  T  spanned  2 g  degeneracy  by d  x y  i s not s p l i t  , d  y  z  and d  x z  . The d o u b l e t  by s p i n - o r b i t  expect a J a h n - T e l l e r d i s t o r t i o n which w i l l eracy  d o e s . The c r y s t a l original  ding  to the f o l l o w i n g  2 T  The g r o u n d s t a t e or  B,„(d «  B „(d„„) Zg xy 0  O  lower than  degeneracies are s p l i t  accor-  scheme: —  2g  distortion  h a s now a symmetry o f D ^ ,  0^, and t h e o r b i t a l  %  distortion  degeneracy but a t e t r a g o n a l  field  and we  remove t h e d e g e n -  i n order t o lower the energy. A t r i g o n a l  does n o t remove t h i s  the  coupling  orbi-  ~  ' '  2 B  A  2g  c a n be e i t h e r  and t h e u p p e r  l g  +  +  2 f i  2  £  lg  g  the non-degenerate  states  ' - ( d ^ )  are the non-degenerate  and t h e d o u b l y - d e g e n e r a t e E „ ( d „ .d ) . E a c h o f t h e s e •* g y z ' xz  100  is,  of course,  the  lowest of the  ligand is  A  ig(  doubly  lower  octahedron  lowest  state  still  z  2)  i  lowest  s  Eg  i s A and  and  A  l a  i s concerned,  along  a C. a x i s , ^  (Figure 19). and  g  T  In t h e s e  B  The  l g  f a r as  i f the B _(d ~) -y x^—y^ n  0  J  of D  are  4 h  A  the  figures,  A  j f  2  and  and u  B ,  and  2 g  respectively.  above s i m p l e wave f u n c t i o n s a r e t h e  f u n c t i o n s o f the c r y s t a l coupling operator  XL.S  o p e r a t o r . The  ( A  t h e n be  to construct matrices  l g  be  considered  wave f u n c t i o n s f o r t h e g r o u n d  spin doublet  2  2  B  l g  ),  correct  t o second  f o r the  order,  Zeeman  P  the h y p e r f i n e  interaction  S are the o r b i t a l  a n g u l a r momentum and  l a r momentum o p e r a t o r s r e s p e c t i v e l y , a n g u l a r momentum, x i by e q u a t i o n  (69)  can  (56)  [(L.I)+£{ML-+1)-X> ( S . I ) - f S ( L . S ) ( L . I ) - | 5 ( L . I ) (L.S)]  where L and  state  interaction  .j8H. (L+2S) and  eigen-  spin-orbit  b e i n g much s m a l l e r , can  as a p e r t u r b a t i o n . The or  field  the  l e v e l s o f the o c t a h e d r a l  2 g  t h e s e p a r a t i o n s between  and  used  i n s p i n . As  whereas i f i t i s c o m p r e s s e d ,  s e p a r a t i o n between t h e E field  levels  i s elongated  ( F i g u r e 18) d  two  degenerate  and  sa  numerical  £ i s given  (57) spin  L i s the t o t a l  angu-  orbital  parameter, P i s d e f i n e d  by  2U+D-4S 5  where 1=2.  S(2£-l) (2JI+3) (2L-1)  Comparison w i t h  the m a t r i x of the  ( 5 8 >  spin  Hamiltonian  101  102  FIGURE  19  S p l i t t i n g of the Tetragonal F i e l d s ,  S t a t e i n O c t a h e d r a l and t h e G r o u n d S t a t e B e i n g A^g,  103.  = %mzSz  s  H  % » ( ^  +  H  +  V + A S , I  y  gives, the f o l l o w i n g e x p r e s s i o n s p a r a m e t e r s . These r e s u l t s but  ?  +  B (  ig  B  g  g  x -y2  ( d  2  have been o b t a i n e d '•previously',': ,  ) :  -  ' 2  2  f o r Ground. S t a t e - ' b e i n g  ;  '  '; •  ^ \  2 :  : ' ' ' • ' y ' \ '  :  '  2  J-  =." P { - ( l - q C ) -7-7.q.C-8pC-|q .C -' ~ ^ q C } ' ' 2  2  2  2  2;  i  g  (  P{-x(l.^q C -2p' c ) 2  ;  2  The S p i n H a m i l t o n i a n 2 ) ^  d z  gj.  =  2  ' :  being  2  . (64)  '  . :=•'• ' P { - x d - 3 5 ) + 7 +  + (15/7) t; } 2  --'  (65)  AV'' :  (66)  f s ) - 4 - ( 4 5 / 7 ) 5 - (5.7/14).? }. -  P{-Xd-  2  ;  •• , •„  X = X/(A +y).,. PC = \/A 2  ..(67) ,  2  (60) to, (67) ,' we have  • '• '. , .  )  (.63).  2  P a r a m e t e r s f o r Ground S t a t e  -  2  2 - 6? - 6 C  Cr  2  0  v  = .2 - 3 C  In e q u a t i o n s  2  6  V'\.;'.'''.-; .:v:' ^  g„  =  2  (  (62)  -TT'q' « -7P C: }",  2  2  :  (61) '  ?  X  ;  B.  '  2  = 2 - 2qC - 4 p f  x  (ii).  ••  .  2 - Spr. - 3q X - •* 4 p q C  =  A  y  ?  A.  A  x  t h e ' d e r i v a t i o n s have n o t b e e n g i v e n ' g k p l i 6 i i t l y .  B  B  x  f o r the' spin-'Hamiltoniari  (i)..The-Spin Hamiltonian-Parameters  ;  S I + S I y ) (59)  • '  lf  ,• ,  -' X/A  2  .  .  •' •• V : "' . (68). ./'  104  and P = 2$g 3 <r N  where g  N  >  (69)  i s t h e w e i g h t e d mean n u c l e a r g - f a c t o r f o r t h e two  copper i s o t o p e s ,  6 3  Cu  magneton and < r 3 > ground s t a t e r a d i a l related  and  6 5  Cu  (=1.513), 8  function, x i s a numerical  to.the Fermi-contact  For  the determination  meters, B l e a n e y h a s  o f the experimental  given a formula  i n strong magnetic f i e l d s ,  Hamiltonian  g i v e n by e q u a t i o n  f o r the allowed  b a s e d on t h e s p i n  to h i s formula,  2  I  2  2  2  e  o  2  2  2  2  2  2  J  the p o s i t i o n  XKI+D-mj }  2  2  e  -{{{A -B )gn g _ /(2H g 6 K g >}sin ecos 0m 2  order  i s g i v e n by  H = H -(Km /g 6)-{B (A +K )/(4H g e K g ) o  para-  (59) and u s i n g t h e s e c o n d  perturbation theory. According line  parameter  interaction.  transitions  H o f a resonance  i s the nuclear  N  i s t h e mean i n v e r s e cube r a d i u s f o r t h e  -  closely  _ 3  N  o  2  2  2  2  2  e  I  (70)  where  :  g  H  2  2  0 being  ? 2 2 c o s 6 + g^ s i n 9  = hv/gB  Q  K g  2  =  2  the angle  =  A g 2  2 H  cos 9 + 2  B g 2  2 x  sin 0 2  between t h e z - a x i s and t h e d i r e c t i o n o f  the magnetic f i e l d ,  itij=<I > where I i s t h e n u c l e a r z  s p i n and  105  v i s the microwave cular can  be  tively  transitions  f r e q u e n c y , The obtained  i n the  parallel  (70) .  perpendi-  frozen solution  e v a l u a t e d by p u t t i n g 0 e q u a l i n equation  and  t o 0° and  90°  spectra respec-  CHAPTER THREE  EXPERIMENTAL  (A).  P r e p a r a t i o n o f t h e Sample:-  Copper t r i f l u o r o a c e t a t e ving  copper(II)  diluted  2:1  solution of  was  by d i s s o l -  carbonate i n excess t r i f l u o r o a c e t i c a c i d ,  (v/v) w i t h d i s t i l l e d left  to evaporate  copper t r i f l u o r o a c e t a t e  were s t o r e d  was p r e p a r e d  w a t e r . The r e s u l t i n g  s l o w l y . The deep b l u e  blue crystals  so o b t a i n e d were h y g r o s c o p i c and  i n a dessiccator  containing  silica  gel.  The  m i c r o a n a l y s i s gave 14.41% c a r b o n and 1 , 2 7 % h y d r o g e n w h i c h showed t h e c r y s t a l s H 1.23%  f o r dihydrate;  C 15.63%, H 0.65%  Single  of zinc  ning varying ving  crystals  mixtures of zinc  described  f o r monohydrate.  trifluoroacetate  c a r b o n a t e and c o p p e r c a r b o n a t e  and c a r r y i n g  a b o v e . The c r y s t a l s  on t h e b a s i s  intensity  C 14.70%,  contai-  c o n c e n t r a t i o n s o f c o p p e r were grown by d i s s o l -  fluoroaceticsacid  sen  t o be t h e d i h y d r a t e . C a l c u l a t e d :  in t r i -  o u t t h e c r y s t a l l i z a t i o n as f o r t h e p r e s e n t work were  o f minimum l i n e w i d t h and optimum  cho-  signal  and were f o u n d t o have a Cu:Zn r a t i o o f 0.0075:1.  M i c r o a n a l y s i s gave 14.50% c a r b o n and 1.28% h y d r o g e n w h e r e a s the to  zinc  c o n t e n t , d e t e r m i n e d by a t o m i c a b s o r p t i o n , was  be 19.71% s h o w i n g t h a t  the c r y s t a l s  of zinc  found  trifluoroace-  107  t a t e were t h e d i h y d r a t e . by Mr.  (B).  P.  Borda o f t h i s  The m i c r o a n a l y s i s  was  carried  department.  Measurements:-  Since three  reference  the c r y s t a l  s t r u c t u r e was  n o t known, t h e  axes a, b, c were c h o s e n t o be p a r a l l e l  the  e d g e s o f t h e c r y s t a l s a n d t h e measurements were  out  a t every  tal  details  10° o f a x i a l  r o t a t i o n a t 77°K. The  are described  Observations  i n Part  were a l s o c a r r i e d  spectrometer. the  The m a g n e t i c  DPPH r e s o n a n c e .  field  experimen-  was  o u t on a  0.5%  in trifluoroacetic  a t room t e m p e r a t u r e and a t 77°K u s i n g  to  carried  I.  s o l u t i o n o f copper t r i f l u o r o a c e t a t e acid  out  a Varian  E-3  calibrated against  CHAPTER  FOUR  RESULTS  Figure  20 shows a t y p i c a l  the p r e s e n t  study.  The l i n e s  while  marked B a r e due t o a n o t h e r .  those  was  taken  g-value ved  respectively.  tions.  lines  as t h e o b s e r -  F i g u r e s 22, 23 and 24 show t h e c a l c u l a t e d hyperfine s p l i t t i n g s The c a l c u l a t e d  splittings  experimental  lines  was t a k e n  splittings  P, w i t h  f o r the other  p o i n t s shown as f u l l  a t some o r i e n t a t i o n s  f o r t h e two s i t e s  a r e shown i n f u l l  corresponding  site  lines  called  experimental  show t h e c a l -  Q, w i t h t h e  c i r c l e s . The h a l f - o p e n  i n these  figures  show t h a t t h e  were n o t r e s o l v e d a t t h e s e  I t was e v i d e n t f r o m  and  i n b e , c a and ab p l a n e s  shown as open c i r c l e s . The d o t t e d  circles  spec-  k i n d . The s p l i t t i n g b e t w e e n two s u c c e s s i v e  f o r one s i t e , c a l l e d  culated  orienta-  and F i g u r e 21 shows a  a t the c e n t e r o f the spectrum  experimental  points  A t some  site  as a m e a s u r e o f t h e h y p e r f i n e s p l i t t i n g w h i l e t h e  g-value.  lines  obtained i n  marked A a r e due t o one  t i o n s o n l y one s i t e was o b s e r v e d trum o f such  spectrum  orienta-  F i g u r e s 22, 23 and 24 t h a t t h e r e  were o n l y two p o s s i b l e c o m b i n a t i o n s  i . e . , P!Q Q3 2  the nomenclature being  t h e same as t h a t u s e d  method f o r d e t e r m i n i n g  t h e A- and t h e g - t e n s o r s  same as d e s c r i b e d b e f o r e . The r e s u l t s  a n t  3  Qi 2 3' P  P  i n P a r t I . The was  also the  o f t h e d i a g o n a l i z e d A-  IOO An EPR at  Spectrum o f C u  + +  Gauss i n a Single  77°K, t h e M a g n e t i c F i e l d  a-Axis.  C r y s t a l of Zinc  Trifluoroacetate  B e i n g i n t h e c a - P l a n e and P a r a l l e l  t o the  H//b  (/)  D <  O  o z hk _J CL CO  180  ANGLE,DEGREES FIGURE 22  Angular V a r i a t i o n o f the Hyperfine S p l i t t i n g of  Zinc  Trifluoroacetate  f o r Cu ++ i n a S i n g l e  a t 77 K, t h e M a g n e t i c  Field  Being  Crystal  i n the bc-Plane,  H//c  0-| O  H//o  1  1 20  1  |  40  1  |  60  1  1  1  80  H//c  |  1  IOO  |  1 140  1  120  1  I  1  160  180  ANGLE,DEGREES FIGURE 23  Angular V a r i a t i o n  of the Hyperfine S p l i t t i n g  Zinc T r i f l u o r o a c e t a t e  a t 77°K, THe  Magnetic  of C u Field  + +  i n a Single  Being  i n the  Crystal  ca-Plane.  of  H//o  H//b  H//Q  ANGLE,DEGREES FIGURE 24  A n g u l a r V a r i a t i o n o f t h e H y p e r f i n e S p l i t t i n g o f Cu i n a Single Crystal Z i n c T r i f l u o r o a c e t a t e a t 77°K t h e M a g n e t i c F i e l d B e i n g i n t h e ab-Pla,ne. (?  .114 .-••''•••'•.'"'•:''V';'  ,,  ' '•• "' •' J - ' v " - ' '''r  V " 'S-  •,;.  •''.A'i  :  VVV-rJ- •  ;,:thajt\^  :  :  X  ^'—•'•''''''"^^  ..- vof- the \ r e s u l t s ; " ^ ^ ^  •'  / V a l u e s ' and .;alsp; the,' v a l u e s .for the.two .s4&^.s..«^ -ge'd^'With :th'e/ following''. f i n a l /results^--',/ 'r':^SVP'Kl  \WXAS-<}y'  /''-'////"V^//^ ' ; ^ ! v « - / ' ///: 'V('vf "-.-''4 * ^/'H,A//=^3;1/;^ :  /,/  :  :  . <vv.  - " . - H / / V •'•///'  j ' v / ^ ^ ' y ; : ^ ) .  $V^**'i?^  >->  •*%%W'C'-^'V, -"h • • • J  r  M*|''fLV^/^V^*>  •  . • \  , / / ^ i / v 7 / / ' /'^V/>.  :  acetic., a'did i s o l u t i o n , ;of copper •;•'trif l u b r b a c e t a t e U a t Ifbom Vt'eitir-'* p e f a t u r e a^id\^at;-77/ K" r j e s p ^ c i t i v e l y ^ •  7  :  .  i' f,  :  • v '"''fy  g e t t h e 'averaged . - h y p e r f i n e ; . s p l i t t i n g : ' A - a 0  •  "  \y'AA  t h e j.mean, g.rvalue '.  c r • = - • ( g . +2gJ)) • '^,K.*":'":-.v - ^ ^ ' ' r — " ' ^ ^^^>1( 7^1V ^ 'A: ;  &X& - \^ :  i l A ' A ^ ^ -  }  ^ S-<X-  :  :  fv'f« , '  ' '•••'„,.',••'. A.'; ..The. spe c t r urri shown i n . F i g u r e ' 25 .1 s. a ; good; .example: of- t h e \ r. • • \...... :  ">;A';>.'•?-^^vaf'iation ofv<lin'ev widthy^'ff.bm^l.ih  ' .afe'r'-.ho'tV b6npefned\ <liefe.jf; v *  '>,th6' m b l ; e e u i a f 4 t u ^ f  • .';:r-  ;  r  :  :  •' /1. v T ( H ^ - ^ i t h ^ t ^ i e ' J s t V ^ 1  :  •<:' ,->"  •• ;• dening> ;arising^;'from ;• t h e m b l e c u l a f • ;  ")••••<" A-:~: yyy--:yyyy-;A\X<?  •'<'-*'~yyKAX*iX)&XX-'~ 1  \ ^ •:• v.:; l i n e d : i n d e t a i l - ' e l s e w h e r e . *V ' ^4-?v  ;-mo.t>i^n;^.^ha's, ^b'eren out-H ^"•' e  v  f  'y^y.: • Ayy^Ay X'yAsyf'- ' " - H '•• ' ^AAdAX'A >y:':y yM^ V-'-v... •;. 1  : > :  115  .TABLE X V I I I  The  D i a g o n a l i z e d A- and g - T e n s o r s w i t h t h e D i r e c t i o n for  the S i t e P Q Q i  2  3  '  :  1  Tensor  Direction  Elements  Hyperfine  •  Cosines  •m  1  Cosines  .' n "  Tensor:-  • 64 ± 1.3  gauss  0.154  0.703  59  ±1.3  gauss  0.884  -0.412  30 ± 1.3  gauss  0.442 .  .0.580  '  p.  -0.694 v  -0.222  : A  , 0.685  ••.IS •g-Tensor:-  2.295  +0.002  -0.633  2.266  ± 0.002  2.045 ± 0.002  ,  0.698 .  -0.336 ;•  0.709  0,347  -0.614  0.312  0.627  0.714 .  .  " Li  116(§  TABLE XIX  The D i a g o n a l i z e d A- and g - T e n s o r s w i t h t h e D i r e c t i o n for  the S i t e  Q P P X  2  Cosines  3  i  Direction Tensor Elements  1  Hyperfine  Cosines  ^ m  n  (I  tensor:-  63 ± 1.3  gauss  -0.151  0.846  -0.511  59 ± 1.3  gauss  0.866  -0.136  -0.481  32 ± 1.3  gauss  0.477  0.516  0.712  2.297 ± 0.002  -0.617  0.735  -0.283 v  2.261 ± 0.002  0.730  0.400  -0.554  2.048 ± 0.002  0.294  0.548  0.783  g-Tensor:-  w  IOO  FIGURE 25  An EPR  Spectrum o f C u  Gauss  + +  in Trifluoroacetic  Acid  Solution  a t Room T e m p e r a t u r e ,  119  The talline  frozen  spectrum  solution  (Figure  l i n e s marked - f , - | , \,  gave  26) s h o w i n g  a typical axial  polycrys-  symmetry. The  \ are the p a r a l l e l  transitions  wiith t h e irij v a l u e s g i v e n by t h e s e numbers and c a n g i v e a c c u r a t e v a l u e s o f A and g the strong sitions  line  and an e s t i m a t e o f B, w h i l e  u  i s an e n v e l o p e o f t h e p e r p e n d i c u l a r  tran-  and c a n be u s e d t o d e t e r m i n e g ^ . From t h e room  temperature  s p e c t r u m , we g e t A = 5 0 ± 2 Q  0.005. The a n a l y s i s o f t h e f r o z e n Bleaney's e q u a t i o n which  Q  spectrum  uses  f o r 9=0 becomes  2  I  and g = 2 . 1 6 9 ±  solution  H = H -Am -(B /2H )(15/4 0  gauss  o  - m-,.)  (73)  2  where ~H  D  = hv/g 6 H  U s i n g m-j. and H v a l u e s f o r t h e f o u r p a r a l l e l  transitions  we  g e t t h e a v e r a g e v a l u e s o f A=141 g a u s s , B=53 g a u s s and g =2.371 whereas t h e p e r p e n d i c u l a r (l  T a b l e XX g i v e s tonian  forC u 2  gave  g =2.072. x  a comparison o f the parameters o f s p i n  + +  i n CF^COOH s o l u t i o n  Zn(CF COO) .2H 0. 3  transition  2  and i n doped  Hamil-  crystals of  120  TABLE XX  A Summary  of Spin Hamiltonian  (Room Temp:)  Parameters  (77°K)  for Cu  (77°K)  IA|  0.01316 c m  |B|  0.00498 cm"  - 1  1  0.00289  cm"  0,00569  cm"  g  u  2.371  2.046  g  x  2.072  2.279  |A | O  g  rt  o  0.004 67  2.169  cm"  1  + +  ion  CHAPTER F I V E  DISCUSSION  The Thompson and acetate,  present  Y a w n e y i n  unlike  in  smaller  the  configuration ugh  the  Cu  ions  + +  l e s s s t a b l e . The  to the  Hence t h e  large  r e s i d u a l charge  left  instability  of  the  binuclear  Baillie carried  out  ions  + +  et a l .  spectroscopic  the  hedral  spectra  could  configuration  be  make t h e  in a  binuclear  on  the  altho-  f l u o r i n e atoms,  copper ions  i s greater  acetate  the  and  therefore  , and on  i n the  larger  former  Agambar and  the  (a02)  Orrell  (109)  a number o f t r a n s i t i o n  i n t e r p r e t e d by metal  for  c o n t r i b u t e ss ttoo  r e l a t e d compounds and  around the  poly-  i s a much weaker  e f f e c t of  studies  m e t a l t r i f l u o r o a c e t a t e s and  partners  t r i f l u o r o a c e t a t e group,  structure  (108)  copper  contributesetowards  should  t r i f l u o r o a c e t a t e than f o r the Cu  the  Cu-Cu d i s t a n c e  of  binuclear  between t h e  acetatefgroup,  inductive  r e p u l s i o n between t h e  that  separation  that  trifluoro-  adopt a  configuration,  f a c t o r which  s i m i l a r i n s i z e to the  b a s e due  the  of  copper  does n o t  than the  m e r i c c o n f i g u r a t i o n . Any separation  showing t h a t  a binuclear  e x i s t i n p a i r s with  a p a i r being  the  i s i n agreement w i t h  copper a c e t a t e ,  configuration.,In ions  study  inferred  assuming the  ion.  octa-  122  Returning (61),  (64)  B-^g,  and  (65)  to the p r e s e n t  situation  x  i n copper  ground s t a t e  equations  show t h a t i f t h e g r o u n d s t a t e  t h e n gn>g >2. T h i s i s t h e  encountered  study,  c o m p l e x e s . On  is strictly  A  jg>  then  (60),  is  strictly  t h a t i s commonly  t h e o t h e r hand, i f t h e  gn< 2<g _. T h i s  situation,  r  J  (89) although The  not  results  common, has i n Table  XX  n e v e r t h e l e s s been o b s e r v e d show t h a t C u  t o - t h a / f o r m e r _ s d . t u a t i o n and parameters p£, using be  q£,  equations  calculated  (60)  from  c o u p l i n g parameter -828  cm  (HO)  - 1  t  X i s s m a l l e r and tive  P and  I  n  to  p£  ( 6 3 ) . The  X. F o r copper  q£  by  calculated constant  1 1 1  + +  t h i s value r a t h e r than  Aj and  ion, X i s equal  from  the  3s and  to the nucleus  cm"  f o r the  1  It will the  free  by  M a k i and  approach,  McGarvey (  The  1 1 ;  ^  A  could  2  spin-orbit to  on  the  the  surroun-  be more r e a s o n a b l e  arising  effec-  3p  spin-orbit  ion value  of  the  g-values,  elec-  Owen^" ^ 1  coupling to  use  in this discussion. from  be worked o u t by more e l a b o r a t e c a l c u l a t i o n s , field  by  c o m p l e x e s , however, t h e v a l u e  However, t h e a c c u r a t e c o r r e c t i o n  ligand  case  c o n s i d e r e d as a change i n t h e  a v a l u e o f -695 + +  The  g  a knowledge o f the  ) , From t h e e x p e r i m e n t a l  i n Cu(H20)g .  i s B^ .  for this  splittings  free C u  screening constant a r i s i n g  ligands f  corresponds  3  the ground s t a t e  t r o n s which are pushed n e a r e r ding  i n CF COOH  x were e v a l u a t e d  and  c a n be  + +  before  same p a t t e r n s as  covalency based  on  could the  suggested  by  .  v a l u e s o f p£  and  q£  calculated  from  equations  123  (60)  and  (61) were -0.04797 and -0.04060 r e s p e c t i v e l y  gave  Aj=14,488 c m  trifluoroacetate series^  12,120 c m  - 1  group i s c l o s e t o  (  1 1 6  and t h e v a l u e s o f A  )  and we  (63) t o d e t e r m i n e  the  s i g n s o f A and B was sign  combinations.  positive, the  fact  P=158xl0  - 4  cm  with  - 1  t h a t i f A was  poss-  assumed  B positive.  ^  Horowitz*  1 1 4  >  n  for Cu  solution + +  and  6 5  Cu  (X=0.32 and P = 3 0 0 x l 0  results  for Cu  and  + +  4  i s i n closer  i o n g i v e n by Abragam cm )  - 4  so t h a t i t i s  - 1  t o assume t h a t A i s n e g a t i v e  The t h a t <3± g >2  Cu  and X=0.2088 and P = 3 5 2 x l 0 ~  The s e c o n d  and  reasonable  6 3  (63) were X=0.6685  B negative  the values  states  found  (62) and  agreement w i t h  and B i s p o s i t i v e .  i n Zn(CF3COO) .2H 0 2  2  show  w h i c h d o e s n o t f i t i n t h e schemes m e n t i o n e d  i . e . , the ground  g-values  t h e r e c o u l d be f o u r  T h e r e f o r e A must be c o n s i d e r e d n e g a t i v e . The  to equations  with  earlier  (62)  the i n f o r m a t i o n about  t h a t t h e m a g n e t i c moments f o r b o t h  solutions  - 1  i n equations  P t u r n e d o u t t o be n e g a t i v e w h i c h i s c o n t r a r y t o  are p o s i t i v e .  cm  I t was  here  (Figure 18).  2  P and X. S i n c e lacking,  obtained  2  A >Aj>A  see t h a t  comes t o  2  and A  x  v a l u e s o f p £ and q£ were s u b s t i t u t e d  and  ible  i n the nephelauxetic 2  thtis seem r e a s o n a b l e The  Assuming t h a t the  1  2  the value o f A f o r Cu(CF3COO) .2H 0  ),  1 0 8  and A =17,118 cm" .  - 1  which  state  i s neither A^  n o r B-^g. The  g  c a n be e x p l a i n e d by a s s u m i n g t h a t t h e A^g and B^g  are coupled  the ground  state  through  the v i b r o n i c  c a n be w r i t t e n •nA  l g  +/(l-Ti)B  l g  as  (  8  7  '  interactions 8  8  ' '  1  0  8  '  1  1  5  so t h a t  * (74)  124  and  the experimental  <3\\  n  =  (  9 n )  p a r a m e t e r s c a n be w r i t t e n as  _  ll l g  A  gj. = n ( g j . )  A  =  +  + d(- l -) n (9"n ) ( g) „ ) _ n  =  B i g  (  7 5  7  6  )  +  d - n )( g ^ s ^  n (A), ig  +  (l-n) ( A ) i g  (77)  n(B). lg  +  ( l - n ) (B)  (78)  A ] L g  A  B  B  (  R  B  A  B  l g  where t h e s u b s c r i p t e d p a r a m e t e r s a r e f o r p u r e  s t a t e s . In  order  equations  to evaluate  )  the v a r i o u s parameters  from  ( 7 5 ) - ( 7 8 ) , a k n o w l e d g e o f p and q i s n e c e s s a r y .  In t h i s  case,  by P i l b r o w  and  the values  Spaeth  o f p = l . l and q=1.4, c a l c u l a t e d f r o m a p o i n t c h a r g e m o d e l , were  Substitution  of equations  (60)-(67)  used.  i n (75)-(78)  gave  £ = - 0 . 0 5 2 7 and n=0.8761 w h i c h means t h a t t h e g r o u n d contains dered so  a b o u t 93% A^g and 7% B ^ . F u r t h e r  cm  the value  - 1  4  cm"  1  f o r A being p o s i t i v e  f o r A being  negative. Again  more r e a s o n a b l e . get  A +y=13,188 2  h e n c e y=3,768  Also cm" ,  the f i r s t  cm  - 1  X=0,3348,  and X=0.8503, P = 7 5 x l 0 ~  from t h e v a l u e s Aj=ll,989  showed  4  s e t of values i s  o f £, p £ , a n d qc, and A =9,420 2  cm  - 1  we ,  and  cm" .  Since acts  1  consi-  o f P came o u t t o be n e g a t i v e  t h a t B h a d t o be n e g a t i v e . C a l c u l a t i o n  P=231xl0~  i f B was  g  t o be p o s i t i v e ,  state  1  the t r i f l u o r o a c e t a t e  as a b i d e n t a t e  group i n t h e c r y s t a l  l i g a n d (-1-08) ^ ^he p o i n t g r o u p  symmetry o f  125  Cu(CF COO)  .2H^0  3  than D .  In t h i s  4 h  i n which d  could  and  0  d  more a p p r o p r i a t e l y d e s c r i b e d  case,  ~  the  g r o u n d s t a t e w o u l d be  orbitals  could  interactions  alone  without having  f o r the  coupling.  remembered t h a t t h e A are  not  and  s u p p o s e d t o be  case the  , g_.  not  parallel  can  associate  value A  y  x  tensors the  g  are  symmetry  as  an  on  a  y  the  i s no  (Tables  nearly  and  z  £>2h  and  f  c  a  n  The  tensors  a  e  c  o  fact  n  s  i  that  as w e l l as  we  d  e  r  e  d  from  A , x  the although  as  above c a l c u l a t i o n was  from n e c e s s i t y  are  corresponding  suggests that ^  we  associate  for assigning  XIX).  symmetric  a p p r o x i m a t i o n . Hence t h e symmetry  to A  It  ®2h  z - a x i s . 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Consider  presents  the  the  us  Our  the  will  be  principal  values  o f the  the p r i n c i p a l  y  axes r e s p e c t i v e l y ,  magnitude o f t h i s  simply  can  be  which  written  a x e s x,  y,  re-  as  z fixed  the p r i n c i p a l  so t h a t t h e y w i l l out  the  (Al)  t h e g and  i n the  directions  not  be  the A  diagonal.  tensors  d i a g o n a l i z e them t o d e t e r m i n e t e n s o r s and  , y ,  =  y  2  SL  l  f  the  the r e s p e c t i v e d i r e c t i o n s  a r e u n i t v e c t o r s a l o n g x,  3  SL ,  the e x t e r n a l f i e l d w i t h  of  in  radi-  axes.  If  H  (1)  S  are not  to f i n d  f r o m t h e e s r s p e c t r a and  of  and  j8 H . g .  =  z  the A t e n s o r s  procedure  equation  f i r s t p a r t of the H a m i l t o n i a n ,  In g e n e r a l , these  g and  which d e s c r i b e s the  i s g i v e n by  choose a s e t o f o r t h o g o n a l  crystal. of  study  Zeeman i n t e r a c t i o n H  Let  spin Hamiltonian  I  2  are the d i r e c t i o n  r e s p e c t to these  field,  H(*iHl  y,  cosines  axes and  z of  H i s the  then  + A  2.H2  + J l  3H3  )  (  A  2  )  136 .3. H  =  H  Jl.u.  V  1=1  1  ~  (A3)  1  S i m i l a r l y the second rank tensor g can be w r i t t e n as 3 2 The  =  3  2-2 mnHmyn m=l n=l g  <  A 4 )  e l e c t r o n s p i n v e c t o r S i s supposed t o be quantized i n  the d i r e c t i o n o f the v e c t o r H.g.  5-1 "  H  Z)X)S l  m  n  £  iWi-HmHn  = II > / A g y / J / J m^mn~n m n =  H  ^  k  n H  (vy..y = 6. ) ~ i ~m im  <  n  n  A5)  where kn  =  n  Y\irrmn g  (A6)  m  Now s i n c e S precesses S  =  about H.g, we may w r i t e  S ^ f k / / ? k?}yi , l z  i  (A7)  N  1  where ^ J E k T  'j  i s the r e n o r m a l i z a t i o n f a c t o r . The Zeeman term  3  of the Hamiltonian  K  =  B H S  then becomes  zX) nyn-E iAK yi n i k  { k  }  1  = ^ ZZ { n i / ^ j } y - y i HS  Zn  -  k  n  l  BHs £k^/S; 2  k  (•, . =6 ) Hn  Hl  nl  137  "  9obs  e H S  (A8)  z  where  (A9) which i s i d e n t i c a l  to equation  (7) i n t h e t e x t .  Next, c o n s i d e r the r e s t o f t h e H a m i l t o n i a n w h i c h r e p r e s e n t s t h e h y p e r f i n e and n u c l e a r Zeeman a c t i o n s . For a CF COO  radical,  _  2  H  =  (S.A  ( 1 )  .I  - g g H.I  ( 1 )  N  g  i t can  ( 1 )  )  ( 2 )  )  N  8 H.I N N~ ~  +  inter-  be w r i t t e n as  (S.A( ).l( ) 2  2  (A10)  where  y  N  y  N  (1) y  f o r the present case and  (2)  t h e s u p e r s r i p t s 1 and 2 i n e q u a t i o n  9  (A10)  N  s t a n d f o r t h e two f l u o r i n e n u c l e i . F o r t h e c a s e o f t h e  • CF(COO ) -  can  2  radical,  the second  term  c a n be d r o p p e d o u t and so  t h e s u p e r s c r i p t s and hence t h e H a m i l t o n i a n **SI  =  S'z'l  " N N5*J g  B  L e t us c o n s i d e r t h e l a t t e r t h e n be e x t e n d e d  case  t o the former  simply (  first case.  and t h e r e s u l t s  becomes A  1  1  )  will  138  The  * Also  Z)S  =  m  ^nHm'Hn  n  the equation  si  fi  Using the  s e c o n d - r a n k t e n s o r A c a n be w r i t t e n as  case  A12  >  ( A l l ) c a n be r e w r i t t e n a s  (  =  equations  <  ?'  " V N ? * ^  A  (  A  1  3  )  ( A 3 ) , (A7) and (A12) we c a n t h e n w r i t e ,  for  o f n e a r l y i s o t r o p i c g,  i  =  =S  m  j  n  Sz'EE'E^i^Hi-HmHn i  m  m  n  "  n  z Zn XmN N  9 N Z*jHj B  H  N  j  j n  - N N (g  B  H/S  z mn}Hn )6  = E nHn z B  (A14)  S  n where  n  B  with M  e  = 2~-Kn m  "  (  9N¥/  M  S»U  (  A  1  5  )  = <S„> = ±3s.  Now I i s q u a n t i z e d effective  field  (S.A-g^p^H)  i n the direction  and we may w r i t e  of the  139  S{y>/j= k}Hm  ( A 1 6 )  B  Hence  [S.A-g B H).I = E B y S N  N  n  E{y>K}l!  I  n  B  m  n  K  SS{B B /Vp^}s i y .u  -  m  n  z  = m fomA^Vz v  7  B  =  z  n  V^mnm> = S  Vz  n  A'S I z  (A17)  z  where  A  '(M ,H) R  * 'n 2.  ^ [ E V ^ k - <9 W N  w h i c h i s same a s e q u a t i o n  The  «mk}]  (A18)  (4) i n t h e t e x t .  Hamiltonian  f o r the case o f CF(COO~)  2  now  becomes  H  =  g8HS  z  +  where g i s t h e o b s e r v e d  JJAVIg value  (A19) and A' i s g i v e n by e q u a t i o n  (A18). The e n e r g y E i s g i v e n by E(M  ,M|) =  g$HMg  +  JsA'Mj  where Mg='<S' >=±^ and M.j±<T >=±%. i  z  g i v e n as f o l l o w s : -  z  (A20) The f o u r e n e r g y  levels are  140  E  1  = E(++) = +%gBH + %A* (+h,E)  E  2  = E(+-) = +hq$K ~ kA* (+%,H) (A21)  E  3  = E(-+) = -Jsg3H + % A ' (-h,n)  E  4  = E( — )  where E (++)  = -%g6H - hA' (-Js H) f  means t h a t K =+H and Mj=+H e t c . The four energy s  l e v e l s a r e shown i n F i g u r e 1 .  T-^, T , T 3 , T  Let  2  4  be the e s r t r a n s i t i o n s  bet-  ween the energy l e v e l s , o c c u r r i n g a t the f i e l d s H-^, II2 #• H 3 , H  T  T T  T  4  l  2 3  respectively,  then  -* hv =  E -E  4  =  gBHi +  •* hv = E - E  4  =  g8H  X  2  2  +  + ki-A*  (J5,H ) 2  (A22) (A23) (A24) (A25)  A '(-^HTJ }  + A'{-h,H )> 2  ->• hv =  Ei-E  3  =  g3H  3  +  4  +  ^ { + A ' ( J S , H ) - A ' (-JS,H ) } 3  ->- hv =  4  E ~E 2  3  =  gBH  3  % { - A ' ( J S , H ) - A ' (-JS,H ) } 4  Subtracting  H  4  -  H-L  (A22) from  = (l/4g3)  4  (A25) and r e a r r a n g i n g , we g e t  {A'  +A'  (%,H!) +  (-Js,Hl)  +  A'  A ' (-JJ,H ) }  4  (A26)  4  S i m i l a r l y s u b t r a c t i n g (A23) from  (%,H )  (A24) and r e a r r a n g i n g , we  get H  3  - H  =  2  (l/4g0)(%,H ) 2  + A ' (-%,H ) - A ' (Jj,!^) 2  + A ' (-JS,H ) } 3  Note t h a t A ' i s a f u n c t i o n o f the f i e l d of  N  can use H , the f i e l d Q  of  due t o the i n c l u s i o n  the term g 8 H . H6wever a t high f i e l d s N  H-^, H , H 2  3  and H  4  (A27)  (^3000 gauss) we  a t the c e n t e r o f the spectrum i n s t e a d i n the e x p r e s s i o n s f o r A ' so t h a t i t  141  does n o t r e m a i n a f u n c t i o n o f t h e f i e l d .  A' (J ,H ) = A' (%,H ) = A' (Js,H ) 5  1  2  Thus  = A« Cs,H ) = A  3  +  4  (say)  (A28)  (say)  (A29)  and A' ( - 3 5 , 1 ^ ) =  A' (-is H )= f  2  A  1  (-Js,H )= A' (-%,H )= A" 3  4  where A  +  and  = Al 7 A 7 , V  A"  A  + m  2  k  %Vo mk 5  ^  )  3  1  )  Hence A  o  =  (l/2gB){A  =  (l/2g3){A  +  + A"}  (A32)  - A"}  (A33)  and h  L  +  where A =H -H-^ and Aj_=H ~H . E q u a t i o n 0  4  3  2  (A32) i s same as  equation  (5) i n t h e t e x t . When S.A.I>>g 3 H.I as i s t h e ~ ~ ~ N N~ ~ c a s e h e r e , t h e n A ~A~=A and h e n c e +  A A.  = A/gB) > = 0 ) Let  nuclei yield  (A34)  us now t u r n t o t h e c a s e  o f two  fluorine  as i n CF COO~. F i r s t o f a l l , t h e two f l u o r i n e 2  nuclei  t h e same h y p e r f i n e t e n s o r so t h a t c o r r e s p o n d i n g t o  equation  (A19) we c a n w r i t e  H = gBHS  + J5A'l  ( 1 ) 7  + JsA'l  <) 2  (A35)  142  so  that E (MJ^MJ  (  and  1  ,1^.  )  (  2  )  hence t h e energy  ) = gjSHMg+JsA'Mj^+JsA'M^ ) 2  l e v e l s , a s shown i n F i g u r e 2, a r e  +5sgBH*+ HA'  =  E(++-)  = tJsgBH  E(+-+)  = +^ggH  E (+—)  = t^gBH - HA' (+>S,H )  E(-++)  = -hgW  E(-+-)  =  E (~+)  = -JsggH  E(  = -HqW  0  -HqW  ~ JJA* ( - S , H ) !  0  ner  a s f o r t h e c a s e o f one  t h e two o u t e r  c a n be worked o u t i n a s i m i l a r  A i s equal lines.  (A37)  + JSA' ( - J S , H )  various transitions  splitting  O  0  The  fine  (+JS,H )  E(+++)  )  (A36)  1  9  F n u c l e u s . The o b s e r v e d  to half  man-  hyper-  o f t h e s p l i t t i n g between  APPENDIX B  The  Consider consisting table.  Each o f these  atomic  geometry,  row o f t h e p e r i o d i c  atoms c a n be c o n s i d e r e d t o be compo-  i n the valence s h e l l s  and an  screened  unpolarizaby two I s  The e x c e p t i o n i s a h y d r o g e n atom f o r w h i c h t h e  core i s the bare  z  o f an a r b i t r a r y  core which c o n t a i n s the nucleus  electrons.  2p  a molecule  Equations  o f atomsefrom the f i r s t  sed o f e l e c t r o n s ble  Roothaan  nucleus.  I f <J> a r e t h e n o r m a l i z e d v  valence  o r b i t a l s i . e . I s f o r h y d r o g e n and 2 s , 2 p , 2p^ and x  for lithium  t o f l u o r i n e , then  c a n be a s s i g n e d t o LCAO MO's  the valence  electrons  g i v e n by  (Bl) V  where t h e summation i s c a r r i e d orbitals first  one r e f e r s  electron  product  i s now b u i l t  of a l l molecular  spin.  o r b i t a l while the  t o t h e m o l e c u l a r o r b i t a l . The t o t a l n -  wave f u n c t i o n  electronic tem,  a l l the valence  and t h e c ' s a r e t h e o r b i t a l c o e f f i c i e n t s . The  index o f c r e p r e s e n t s the atomic  second  rized  out over  up as an a n t i s y m m e t -  o r b i t a l s i n c l u d i n g the  Thus f o r a c l o s e d - s h e l l  t h e t o t a l wave f u n c t i o n  ground  state  sys-  ¥ has t h e form  (B2)  144  where p=n/2. E a c h o r b i t a l  ^  i s doubly occupied,  8 electrons are restricted  and tals  spatial  orbi-  and ¥ i s t h e r e f o r e known a s a ' r e s t r i c t e d LCAO-SCF  molecular ned  to identical  so t h a t a  wave f u n c t i o n ' . When an e l e c t r o n i c s t a t e i s d e f i -  by n o r m a l i z e d  V, i t s e l e c t r o n i c e n e r g y , E , i s g i v e n by  E = yV*H¥dT  (B3)  where V' i s t h e complex c o n j u g a t e o f Y and t h e i n t e g r a t i o n is  carried  operator  o v e r a l l s p a c e and s p i n c o o r d i n a t e s .  H i s d e f i n e d as  £ H , + he J2 j j^k  H =  z  D  where Hj i s a l i n e a r the Hamiltonian of  inter-electron j  t  n  operator  t  n  2 £ i  representing  e l e c t r o n moving i n t h e f i e l d  and t h e s e c o n d t e r m r e p r e s e n t s t h e the distance  between  electrons.  Substituting  =  n  (B4)  1  J  r e p u l s i o n , r j ^ being  and t h e k  E  (rik)"  and H e r m i t i a n  f o rthe j * -  the n u c l e i alone  the  Also the  (B2) and (B4) i n ( B 3 ) , we g e t  H.. +  £ if J  (2J  i j  -  K i j  )  (B5)  where  H  ii  =  f^i^idt  Jij-=  K  ij  =  (B6)  e y^J(s)^*(t){r 2  e  ^ A i (  s  ^ j  (  t  )  s t  ^ s t ^  }  1  - 1  ^ ( s H j(t)dT  *j'(s)*i(t)dT  s t  g t  (B7)  (B8)  145 and J • • 11  =  K. •  (B9)  11  The l a b e l s  s and t i n e q u a t i o n s  electrons.  H^j a r e t h e n u c l e a r - f i e l d o r b i t a l  J^j  and K ^ j a r e t h e coulomb  tively.  L e t us d e f i n e  (B7) and  (B8) r e f e r t o t h e energies  and e x c h a n g e i n t e g r a l s  t h e coulomb and e x c h a n g e  while  respec-  operators  J ' and K. as il Jfi|> (s) = e | / i j j * ( t ) { r 2  k  }" ^ (t)dT |ij; (s) 1  s t  i  t  k  (BIO) K ? ^ ( s ) = e j j > * (t) { r } - i | ; ( t ) d T | i | ; ( s ) 2  1  k  s t  where and  J  k  t  i s one o f t h e s e t {^}.  i  Substituting  (BIO) i n (B7)  ( B 8 ) , we g e t  i j  =  K  V  i  ^  ij  =  / * * V i  =  / • • J  i  *  j  d T  (BID K  Let  us now  d  T  define  =  / * j V j  d  T  the matrices  <f>, c ^ and C  (where  underscoring  indicates  a one-dimensional array  underscoring  shows a t w o - d i m e n s i o n a l a r r a y ) , <j>.  5i  =  :  2i  :  3i  and  single double  as  .}  (B12)  (B13)  146 and  c  l l  c  12  •  c  21  c  22  *  •  Hence we may  •  (B14)  •  w r i t e equation *fc  When we  *^  $ •  =  c  (Bl) as i  <  b u i l d an antisymmetrized  product from LCAO  B 1 5 )  MO's,  our problem i s to f i n d the c o e f f i c i e n t s c ^ f o r which the v  corresponding antisymmetrized  product w i l l have minimum  energy. In order to, do so, l e t us f i r s t of a l l , d e f i n e a few m a t r i c e s , e.g., o p e r a t o r M,  corresponding to every o n e - e l e c t r o n  l e t us d e f i n e M  yv = J A*M<b y v dr Y  T  and  (B16) M  M  =  l l  <M  M  12 M  2 1  I f M i s H e r m i t i a n then M  22  = M where the a s t e r i s k  refers  to  the Hermitian complex i n case of a m a t r i x . Thus c o r r e s p o n ding to the Hermitian o p e r a t o r s , H, j ^ , Kj_ e t c , we  can  Hermitian m a t r i c e s H, J ^ ,  proved,  u s i n g equation  (B15),  y^M^dx  e t c . I t can be e a s i l y  that =  (B17)  C*MCJ  I f we d e f i n e an Hermitian o v e r l a p matrix S, such  S  yv  use  = / M v  d  T  that  < > Bl8  147 and assume -Cip> t o be an orthonormal  Jil^JlijdT  Similarly  Let  from  =  c* S  s e t then  Cj  =  6 .  (B19)  i;  (B6), ( B l l ) and (B17), we g e t  I  H  i i  =  Si  Si  »  J  ij  -  s i ii  si  -  SJ i±  K  ij  =  Si Jj Si  =  Sj J i Sj '  Sj  <  B 2 0 )  the v e c t o r s c^.vary by an i n f i n i t e s i m a l amountafic^ so  t h a t the corresponding v a r i a t i o n Then, from equation 6E =  2 ^ 2 8 R  L  ]£. < ir j  i  Substituting  equation  6E = 2 ^ ( 6 c ) H x  + +  E i s 6E.  (B5), we g e t  +  i  i n the energy  2 6 j  ij  "  6 K  ij)  (  B 2 1  )  (B20) i n (B21) we g e t  C i  +  {( i) (2J -K )c 6 c  j  j  i  (6CJ) ( 2 J - K ) c | + 2 ^ C i H ( 6 c i ) i  S  {Si  ( 2  i  j  £j"V  which on s i m p l i f i c a t i o n  (  6  Si  }+  c* ( 2 J - K ) (<SCj ) | i  i  (B22)  gives  <5E = 2^2 (fici)|'H+'^ ( 2 J - K ) | c j  j  i  +  i or  6E = 2 ^ ( 6 c i ) F c . + 2 y ^ ( 6 c ^ ) F * c ' i  ~  s~  x  ~  ~  ( B 2 3 )  where the s u p e r s c i p t t r e p r e s e n t s the transpose m a t r i x . In  148 equation  (B23),  t h e R o o t h a a n o p e r a t o r F i s d e f i n e d as F  =  H  +  G  "  (B24)  where  ?  =  Z  (  2  ~ j  "  )  5J  3 In t h e language o f s i m p l e i n t e g r a l  (B25)  calculus,  the operators  H, F , G c a n be r e p l a c e d by t h e m a t r i x e l e m e n t s H , F , ~ « ~ yv yv G  respectively  and e q u a t i o n F  = H  yv  (B24) c a n be w r i t t e n as  + G yv  yv  (B26)  where H  yv  / * y h  =  v  "  2  I > A  (  r > ] < l >  v  * T  A and  (B27) G  yv  X )  =  where  X  '  P  Ao { ^l s  (  X  o  " *<ya|vX>}  )  a  (yv|Xo) = j ^ c ) , * ( l ) 4, (l) { r } " 12  v  1 (J)  *(2)  (  j  ) o  (2)d  T i  dT  (B28)  2  and  *  p A  a  «  2  I > U  c  i a  (B29)  i the  summation i n e q u a t i o n  (B29) b e i n g  carried  out over the  o c c u p i e d MO's o n l y . Now s i n c e t h e MO's must r e m a i n gonal, equation  ortho-  (B19) must h o l d and h e n c e 6 ( c * S c.) = 0  i.e., (6c*)  or'  S  C j  +  c*  S  (fiCj)  =  0  I  (B30)  149  since S i s Hermitian.  We have a l r e a d y seen t h a t the best antisymmetr i z e d product w i l l be t h a t one f o r which the energy  E is  minimum so t h a t i t i s necessary t h a t 6E=0 f o r any c h o i c e of  6c^'s i n equation  restrictions  (B23) which i s compatible with the  (B30). M u l t i p l y i n g each o f the equations (B30)  by a f a c t o r ~2£jj_ and adding them t o g e t h e r , we g e t  -  ~  I] i/j  2  < Si>sSj ji 6  e  "  2  Z) ( 2j>i*Sx ji = i»j 6  e  0  or "  2  S  Adding  ^SiJSSj^i  "  2  2  ( 6  Si>S*Sj ij = e  0  (B31)  (B31) t o (B23), we get  6E i  •  j  i  ~Z)f^ ij} E  (b32)  j In f a c t what has been done t o approach  to equation  the a p p l i c a t i o n o f the method o f Lagrangian  (B3 2) i s  Multipliers^ ^ 4  to s o l v e the problem of determining the c o n d i t i o n s of m i n i ma. Our problem i s now reduced  t o the d e t e r m i n a t i o n o f the  c o n d i t i o n s f o r 6E'=0 f o r any c h o i c e o f fc^'s without any r e t r i c t i o n . These c o n d i t i o n s are  150  F  et  ->S  c. e.• = 0  j  (B34)  . .  Taking the complex conjugate of equation t i n g from equation  (B33), we  (B34)  and subtrac-  get  IjSj j i " 4j (e  ) =  0  i.e. £ji  =  eij  (B35)  which means t h a t e^j are the elements of an H e r m i t i a n matr i x e so t h a t equations we  (B33)  and  (B34)  are e q u i v a l e n t and  can w r i t e f o r the whole s e t FC = SCe  We  (B36)  can assume here t h a t e i s a d i a g o n a l matrix  with r e a l d i a g o n a l elements e^, f o r i f i t were not,  we  c o u l d d i a g o n a l i z e u i t by a u n i t a r y t r a n s f o r m a t i o n which does not a f f e c t the s e t of the b e s t MO's Equation  (B36)  have d e t e r m i n e d ^ 8 )  can, t h e r e f o r e , be reduced Fc  L  or  we  =  to (B37)  e Sc i  m  i  i n the language of simple a l g e b r a  L S p v ^ i v where  -  E?yvSvi i v £  i s c a l l e d the o r b i t a l energy  <  B38  >  f o r the m o l e c u l a r o r -  b i t a l ty^.  The  s e t of equations  (B38)  i s known as  'Roothaan  151 Equations' present operator  the  and  our  object  simplest  F by  using  but  i n A p p e n d i c e s C and most e f f e c t i v e  CNDO and  INDO  D will  form of  the  approximations.  be  to  Roothaan  APPENDIX C  Complete N e g l e c t  Before Neglect  laying  of Differential  mentioned e a r l i e r , the  of Differential  Overlap  down t h e a p p r o x i m a t i o n s Overlap,  of the  i t s h o u l d be s t a t e d , a s  that our b a s i c o b j e c t i v e i s to s i m p l i f y  R o o t h a a n LCAOSCF e q u a t i o n s  t o overcome t h e c o m p u t a t i o n a l  (B38) i n o r d e r difficulties.  m e n t i o n e d t h a t a t t e m p t s have b e e n made  t o be a b l e  I t s h o u l d be  previously(51-57)  b u t most o f them have been s u c c e s s f u l o n l y u p t o t h e i n c l u s i o n o f TT e l e c t r o n s o r were l i m i t e d cluding  the treatment  m a t i o n s do n o t l i m i t far  t o p l a n a r systems i n -  o f a e l e c t r o n s . The p r e s e n t one t o t h e n a t u r e  a s t h e g e o m e t r y and t h e s i z e  approxi-  o f t h e p r o b l e m as  o f the molecule  are con-  cerned.  Since unitary  (or orthogonal)  t a n t t o bear destroyed fy For  are invariant  transformations(28)  i n mind t h a t t h i s  the atomic  tions  not destroy  orbital  invariant  i t i s import  which can s i m p l i -  the s o l u t i o n s to the equations.  instance the r o t a t i o n of the coordinate should  f  under  i n v a r i a n c e s h o u l d n o t be  by i n t r o d u c i n g any a p p r o x i m a t i o n  the task o f determining  molecule in  t h e Roothaan e q u a t i o n s  system i n the  t h e i n v a r i a n c e . A l s o t h e change  basis s e t should  l e a v e t h e SCF e q u a -  e.g., i n s t e a d o f a b a s i s s e t c o n t a i n i n g 2s,  153 2p /  2  x  Py' P 2  Z  orbitals  to use t h e h y b r i d i z e d  o n v a r i o u s atoms, one s h o u l d be a b l e o r b i t a l s without  any change i n t h e  results.  The  following  o f CNDO c a l c u l a t i o n s , ance r e q u i r e m e n t the  fullfilment  found  s  yv ^yv* =  rix  a r e c o m p a t i b l e w i t h t h e above  H  e  n  c  e  o f these r e s t r i c t i o n s .  The p r o o f s c a n be  work^ ^). 2  s e t <|>y i s t a k e n t o be o r t h o n o r m a l t  invari-  and a r e g i v e n h e r e w i t h o u t t h e p r o o f s o f  i n the o r i g i n a l  (1) . The b a s i s  approximations, which a r e t h e b a s i s  n  coefficients  e  and t h e c o n d i t i o n  E  c  so t h a t  Cj_y f o r m an o r t h o g o n a l mat-  for orthogonality  for  becomes  , c . = 6. . y i V3 ij  (Cl)  y  (2) . The i n t e g r a l s o f t h e t y p e and  X=a, The n o n - z e r o Y  (3) . The i n t e g r a l s the o r b i t a l s orbitals.  values w i l l  y X  =  be d e n o t e d  by Yy^ i « * » e  (yy|XX)  (C2)  Yyv depend o n l y o n t h e atoms t o w h i c h  <J>^ and <$> b e l o n g a n d n o t o n t h e t y p e o f t h e v  T h i s means t h a t  for a particular  A and B, a l l t h e i n t e r a c t i o n s t o be same. Hence i t i s b e t t e r rather  (B28) a r e z e r o u n l e s s y=v  pair  o f the type Y y  V  o f atoms  a r e assumed  t o use t h e n o t a t i o n Y  A  B  than Y y . V  (4) . f-'fhe i n t e g r a l s i n equation  o f the type  (<|>y |V | <j>) , o c c u r r i n g fi  (B27) , where <j>y and 4> b e l o n g  and V_ i s t h e p o t e n t i a l  v  energy  v  i n H^  t o t h e atom A  o p e r a t o r a t t h e atom B, a r e  v  154  put equal to zero unless of  (cj)^ | V  thejtype  y=v.  | <j>^) a r e t a k e n  B  r e g a r d l e s s o f the nature V  A B  .  t o be  the  a l l integrals  same f o r e v e r y  o f t h e o r b i t a l and  represented  by  Thus V  (5) . The  A  =  B  (^IVgl^),  (C3)  o f f - d i a g o n a l elements H  different  atoms a r e e s t i m a t e d  "yv where 8° and  F u r t h e r , i f y=v,  "  3  i s a bonding  between t h e o r b i t a l s  by  the  on  formula  AB yv S  ( C 4 )  p a r a m e t e r d e p e n d i n g on  t h e atoms A  B.  Using  approximations  operator F given i n equation  F  =  yy  H  +  yy  %P  (1) and  (B26)  W  J  Y  takes  +  W  (2), the  S /  the  form  Y  p  *  Roothaan  (C5)  aa \io J  a(^y) and  F where P  =  yv  H„„ yv  -  hP  y  yv'yv  i s d e f i n e d by e q u a t i o n  roximation  (3), the e q u a t i o n  F  yy  =  H  yy  ' ^yy^AA  (C5)  +  P  (y^v)  (C6)  H  (B2 9) . I n t r o d u c i n g c a n be  AA^AA  +  app-  r e w r i t t e n as  PBBYAB  (C?)  B^A  where  P  the  BB  = EV w Bp  summation b e i n g c a r r i e d  ( C 8 )  over  a l l the o r b i t a l s  of  the  y  155  atom B. A l s o t h e core; m a t r i x tion  elements H  , d e f i n e d by equa-  yy  (B27) c a n be w r i t t e n a s  V  - < * v J - ^ - - v | V - Z, 2  (  A  *yl  IV  v B  B^A =  where U  u  yy " S  represents  Hamiltonian  (  *V BI V  ( C 9 )  ,V  the diagonal matrix  elementiof the  c o n t a i n i n g t h e c o r e o f o n l y t h a t atom t o w h i c h  (J)^ b e l o n g s .  According e l e m e n t H^ and  to approximation  i s z e r o when  v  (4) t h e o f f - d i a g o n a l  and <j> b e l o n g  t o t h e same atom  v  the d i a g o n a l element H  takes  t h e form  yy H  yy  =  u  yy  V  a  b  (  *y °  n  a t o m  A )  ( C 1 0 )  BpA Substituting  F  yy  "  U  yy  equations(CIO)  > ^  ^ A A ^ y  +  i n ( C 7 ) , we g e t  + XI  (  P  BBY  - AB>  (  V  A  B  C  1  1  >  B^A where < J >  i s on atom A. I n t r o d u c i n g a p p r o x i m a t i o n  off-diagonal  (5) , t h e  e l e m e n t s become  F  yv  - AB yv^ yvY B 6  S  P  A  (  ^  V  )  (  C  1  2  )  where <j> i s on atom B. v  While doing  t h e c a l c u l a t i o n s , we s t a r t w i t h  basis  set of Slater  i.e.,  I s f o r h y d r o g e n and 2 s , 2p , 2 p , 2p„ f o r l i t h i u m t o  fluorine.  type  orbitals  f o r the valence  a  v  These o r b i t a l s  a r e g i v e n by  shells,  156  Is  =  (ZVTT) * e x p ( - Z r )  2s  =  (Z /96TT) %r  2p  =  (Z /96TT) 3  1  5  t o show t h e  tive The  nuclear values  r  appropriate  nature  o f the p o r b i t a l  Z for various  atoms  using  Li  Be  B  C  N  1.2  1.3  1.95  2.6  3.25  3.9  overlap  integrals  f o r m u l a s (59) . The center  y y s  =  A B  and  the parameter V  tal  s ^ and  B  A  AB  the  (C14)  and  and  h i s formulas  The  paramaters  Abbreviated  S  core  lithium  } -  1  s  B  ( 2 ) d T  B as  V lB  ( 1 ) {  r  } d T  and  t h e r e f o r e , be  U p 2  5.2  using  1  d T  the  by  Thu  (C14)  2  the v a l e n c e  orbi  charge.  (C15) r-^  i s the  B  The  distance  integrals by  used  form U  X),  two-  s functions.  for this a 2 s  can  be  of  the  R o o t h a a n (*>0)  2  n  d  s  r e s p e c t i v e l y ) f o r the  (denoted  Mulliken'  l  t h e p o i n t B.  w h i c h have t h e  to f l u o r i n e  by  have been e v a l u a t e d  can,  2 s  4.55  a point  c h a r g e o f B and  (C15)  as U  2  core  between t h e e l e c t r o n and type  F  i s c a l c u l a t e d as  Ac  effec  S l a t e r ' s rules(58)  i s calculated using  - / A  i s the  1  Z i s the  O  evaluated  y.__  and  i n v o l v i n g valence  ( l ) { r  A B  treating  V  are  v  integral  Coulomb i n t e g r a l  Y  where Z  S^  s p h e r i c a l harmo  are  H  The  J  normalized  c h a r g e c a l c u l a t e d by  of  (C13)  e x p ( - Z r / 2 ) Y(0,<j>)  J  where Y(8,<f>) i s t h e nic  exp(-Zr/2)  5  U  purpose. 2p  2p  elements  estimated  from  157  the  i o n i z a t i o n p o t e n t i a l s associated with  using  the  2s 2p m  n  formula  u  2s  =  -I„(X,2s 2p )-(m+n-l)Y m  n  5  U Finally  the s t a t e s  =  2 p  X  -I (X,2s 2p )-(m+n-l)Y m  y Y  (C16)  X  n  p  o  the bonding parameters  x x  are determined  as t h e  AB a v e r a g e s g i v e n by SAB  -  *<*  + A  B  B  )  where B ^ and B f i a r e d e t e r m i n e d culations  g i v e as c l o s e r e s u l t s  LCAO-SCF c a l c u l a t i o n s Ba  so f o u n d  functions  H  -0.33076  Li  -0.33076  Be  -0.47776  B  -0.62477  C  -0.77177  N  -0.91878  0  -1.13929  F  -1.43330  L  7  )  so t h a t t h e c a l -  as o b t a i n e d  ( i n atomic  C  from  accurate  The v a l u e s o f  units):-  above t h e o r y was u s e d t o compute t h e wave  and t h e e n e r g i e s w h i c h p r e d i c t e d t h e m o l e c u l a r  geometries, fairly  empirically  on d i a t o m i c m o l e c u l e s .  a r e as f o l l o w s  The  (  bending  successfully  force constants  and r o t a t i o n a l  barriers  f o r s m a l l m o l e c u l e s ^ ^ , b u t t h e main 3 (  l i m i t a t i o n was t h a t t h e bond l e n g t h s o b t a i n e d  by e n e r g y  mini-  158  mization large. tron  were t o o s h o r t  This  difficulty  i n an o r b i t a l  other,  leading  was c o r r e c t e d tion  F  U  M  + ( P  energies  a r i s e s from t h e f a c t  o f one atom p e n e t r a t e s  to net a t t r a c t i o n .  This  that  2  AA- « yy)Y + ,  P  A A  where t h e l a s t  (  P B  B  -  Z  B >  t e r m may  contributing to F  the e l e c o f an-  "penetration"  2  W  B^A  M  were t o o  the s h e l l  by e l i m i n a t i n g t h e " p e n e t r a t i o n  ( C l l ) w h i c h c a n be r e w r i t t e n as  py" uu  ral  and t h e b i n d i n g  term"  effect i n equa-  < BYA -V B>  <C18>  Z  B  A  B^A  be d e s c r i b e d  as p e n e t r a t i o n  . I n the corrected  integ-  version of the  yy theory,  known as CNDO/2, t h e s e t e r m s a r e n e g l e c t e d  by p u t -  ting V  instead  of using  estimated the  =  BYAB  equation  <  >  by an a v e r a g e o f t h e i o n i z a t i o n p o t e n t i a l I  and  yy  = -*  of 5 ( y J  I  + A  Li  (C15). A l s o  C19  is  electron affinity  values  Z  the parameter U  U  The  AB  A^, i . e . ,  ( I  y)f  y  V  +  i-  Be  n  "  < A"*>YAA  <C20)  Z  a t o m i c u n i t s a r e as f o l l o w s : B  C  N  0  F  Jj(Ig+Ag) 0.11415 0.21852 0.35259 0.51639 0.70989 0.93311 1.18603 Jsdp+Ap) 0.04623 0.09419 0.14704 Also  the value  cing  the m o d i f i c a t i o n s  0.20478 0.26736 0.33484  o f % ( I + A ) f o r h y d r o g e n i s 0.26373.  of F take t h e form  g  s  (C19) and  (C20), t h e m a t r i x  0.40720  Introduelements  159  ' v M - ^ W t ^ - V - ' X ' y p - ^ ^ + X )  ( P  BB- B>r Z  A B  Bf&A  and  F  = 3  yv  Using of the molecule  A B  S  " %P  y v  y v  Y  (C22)  A B  t h e above a p p r o x i m a t i o n s ,  the t o t a l  energy  may be w r i t t e n as  E = ]PE + X  C 2 3  AB  A  A  <>  E  B^A  where '  A  A  Z-/  A  A  P U + hV yy uvi y  E, = >  y  (P P -HP* ) Y yy w yv' A A  >  (G24)  T  y  and  E  AB=  E  A  y  E < v B  2 P  yv l B  S B  yv-  J s P  -  where Z  A  and Z  B  P  JvY  B B  V  >  +  A B  B A  +  P  { (  A A  P  Z  A Z  B B Y  a r e the core charges  A  B  /  R  A  B  ) -  P  A A  V  A B  B >  <  and R  A B  E 2 5  >  i s the i n t e r -  nuclear distance.  Before while  t o note  c o n c l u d i n g t h i s Appendix,  t h a t a chemist  Resonance i s m a i n l y  concerned  i t may be w o r t h -  working with E l e c t r o n with paramagnetic  Spin  systems so  t h a t he h a s t o d e a l w i t h o p e n - s h e l l c o n f i g u r a t i o n s i n w h i c h t h e number o f e l e c t r o n s w i t h number o f e l e c t r o n s w i t h q respectively.  a spin  $ spin.  i sdifferent  L e t these  C o n v e n t i o n a l l y p i s taken  q and s i n c e we a r e d e a l i n g w i t h d o u b l e t s  from t h e  numbers be p and t o be l a r g e r  i n the present  than case,  160 we  have one  i.e.,  more e l e c t r o n  p = q + 1.  LCAO MO's define  the p a r t i a l (12)  The  s t a r t with  charge and  w  V  F  z-r  1  and  equations  (15) The  set tant  included  row  and  v  F^  as  v  extended  o f the p e r i o d i c #  since  row  of the  )  .  Roothaan  yields  shown i n t h e t e x t  to include  the  by  table  working  he  and  the  elements  the  basis  However i t i s n o t  impor-  extension here.  Fur-  w i t h o r g a n i c r a d i c a l s does  come a c r o s s t h e p r o b l e m  the f i r s t  6  (16).  t h e r m o r e an e s r c h e m i s t  from  2  J  the d o r b i t a l s ^ 2 )  i n t o the p i c t u r e  c  e  t o summarize t h e d e t a i l s o f t h a t  not u s u a l l y  as  3  t h e o r y was  the second  j  o f CNDO/2 a p p r o x i m a t i o n s  for F^ and  and  coeffi-  yv v ! x c  Ot form  o f the o r b i t a l  V  operator. Application  convenient  (10)  bond o r d e r m a t r i c e s  are the m a t r i x elements  v  (9) and  of  X/yv vi i  c  V  3 F^  d i f f e r e n t sets  equations  ]£ yv vi  v  and  spin,  (13).  g i v e s the Roothaan  ex where F ^  two  equations  density  3  than those w i t h  forming  v a r i a t i o n a l treatment  L,  from  a spin  as shown i n t h e t e x t by  i n equations  cients  We  with  i s mostly  of the p e r i o d i c  of bringing  d orbitals  l i m i t e d to the table.  elements  APPENDIX D  Intermediate  Neglect  of Differential  Among t h e l i m i t a t i o n s that  o f t h e CNDO method, one i s  i t does n o t l e a d t o any s p i n d e n s i t y i n a o r b i t a l s  when a p p l i e d t o p l a n a r a r o m a t i c  compounds. T h i s i s due t o  the e x c l u s i o n o f e l e c t r o n i n t e r a c t i o n s which a r e completely also  neglected  leads t o the f a i l u r e  This extent  limitation  orthogonal  o f the type  when <j>^ and <j> b e l o n g v  $ <|>  a  r  v  e  approximations  f o r the  large  Overlap'  retained  wave f u n c t i o n u s e d  #  I  t  procedure the  u n d e r any u n i t a r y o r  t r a n s f o r m a t i o n s h o u l d n o t be d e s t r o y e d .  Hartree-Fock  This  t o t h e same c e n t e r (33)  o f t h e Roothaan's e q u a t i o n s  same as g i v e n by e q u a t i o n  with  of Differential  however, be remembered t h a t d u r i n g t h i s  invariance  tricted  t o account  #  f r o m t h e same c o n f i g u r a t i o n .  Neglect  (INDO) i n w h i c h t h e p r o d u c t s  should,  i n t h e CNDO method (*>1)  type  has been c o r r e c t e d t o a v e r y  i n the 'Intermediate  the cases  o f t h e exchange  o f the theory  separation of states arising  in  Overlap  The u n r e s -  f o r the molecule i s  (11) i n t h e t e x t .  The f o l l o w i n g  d e s c r i b e t h e INDO method a n d a r e c o m p a t i b l e  the r e s t r i c t i o n  of invariance.  ( 1 ) . The o v e r l a p i n t e g r a l s  Sy  which case  reduce t o  equations^(C26)  V  are neglected unless  y=v,  in  162  y v  c  v i  /  V  / y v  v i  c  i  e  (Dl)  /  V  and  v i  c  V  /  (2) .  v  > v i  i  £  V  The i n t e g r a l s  o f the type  (yX|va) a r e z e r o u n l e s s \x-\  v=a. F u r t h e r (yy|vv)  = Y  A  ( V on A, v on B)  B  where y g i s a p p r o x i m a t e d e l a s  a coulomb  A  (3) . The d i a g o n a l c o r e m a t r i x  H  core  yy  _  ,  u  _  yy  Z-^  Z YAB  ^ g .  =  =  u  +  $  o  )  s  ^  (  y\>  =  yy  +  X T  {  P  x x  s  as  3  )  , p  j  D  4  )  orbitals, z  approximations  (  y  y  1 x  x  )  "  P  x x  (  ^  }  +  ^  ( 2 P  yv  <  P  B  B  -  Z  B > Y  A  B  <DS>  B^A  " Pj )(yv|yv) v  where y ^ v and b o t h sions  B  to  X F  s  then  the elements o f t h e F m a t r i c e s , under these  yy  l  orbitals.  X  F  A  a r e c e n t e r e d on atom A, o n l y i f  I f we u s e a b a s i s s e t o f s , p , p  reduce  s A  BpA  I f y i s c e n t e r e d on A and v on B, core  s  ( °  R  we u s e a b a s i s s e t o f p u r e s and p  H  (  interaction  elements a r e c a l c u l a t e d  \  A l s o U ,=0 when y^X and b o t h yA  (4) .  (D2)  - PJ} <UP|VV) V  a r e c e n t e r e d on atom A . S i m i l a r l y  c a n be w r i t t e n f o r F  8 yy  ft and F ^ . yv  (D6) expres-  B  )  163  The  one-center  S l a t e r ' s parameters  (ss|ss) (sp |sp ) x  x  G  1  i n t e g r a l s are written  and F  =  (ss|p p )  =  (1/3)G  x  (P P |P P ) = x  y  x  (p p  (D8)  (D9)  2  F°  +  (4/25)F  2  (D10)  P°  -  (2/25)F  2  (DID  and F  2  the semi-empirical values of G  are  as  1  , i n atomic  units  follows:Atom  zL  2i  Li  0.092012  0.049865  Be  0.1407  0.089125  B  0.199265  0.13041  e  0.267708  0.17372  N  0.346029  0.219055  0  0.43423  0.266415  F  0.532305  0.31580  the core m a t r i x elements  following  hydrogen,  are determined  from  formulas:U  for  (D7)  1  and  the  =  X X  (P P |P P ) = x x y*y  Finally  follows:-  x  (3/25)F  y  Ip p ) =  X X  'as  2  i n terms o f  s  =  -%(I + A ) g  s  - %Y  R H  (D12)  164  U.  •JS(I +A ) S  S  %F° 013)  -?5(I +A ) p  for  p  - hF°  +  (1/12) G  1  lithium, U_  =  -J5(I +A ) g  (3/2)F° +  hG  1  (P14) U  =  p  for beryllium,  -JsUp+Ap) -  (3/2) F° +  hG  1  and •%d +A ) -  U,  s  s  (Z -Js)F° + A  (1/6) ( Z - | ) G  l  A  (D15)  \3  r  -Jjdp+Ap)  -  (Z -3 )F°+-iG +(2/25) ( Z - f ) F 1  A  5  A  f o r boron t o f l u o r i n e .  The method o f I n t e r m e d i a t e O v e r l a p has been u s e d s u c c e s s f u l l y spin densities splittings aromatic  and h e n c e t o a c c o u n t  Neglect of  t o improve the  4  calculated  f o r the observed  i n cases of small molecules  s y s t e m s ( "7) .  Differential  and f o r l a r g e  symmetry.  uniform  p r e s e n t s t u d y h a s , however,  one o f t h e few i n w h i c h t h e t h e o r y has been a p p l i e d s y s t e m o f low  hyperfine  been  to a  large  APPENDIX E  The P r o b l e m o f S p i n  Contamination  The u n r e s t r i c t e d H a r t r e e - F o c k wave f u n c t i o n (11)  by  equation  i s n o t a pure s p i n m u l t i p l e t  it  i s an e i g e n f u n c t i o n z  fact  i t contains  *  "  =  multiplicity  and g i v e n by  h(p - q)  (E2)  v ,, i s t h e e i g e n f u n c t i o n s +m  cients C , g  + m  ases so t h a t  2(s*+m)+l.  as w e l l  E1  of S , with ' 2  3  the  states  < >  where s ' i s t h e l o w e s t s p i n component  The wave f u n c t i o n  spin  Cs'+m^s'+m  s'  opera-  o f the operator S . In  t h e components o f h i g h e r  ^  i . e . , although  o f t h e z component o f t h e s p i n  t o r S , i t i s n o t an e i g e n f u n c t i o n  given  In equation  ( E l ) , the c o e f f i -  d e c r e a s e v e r y r a p i d l y as t h e v a l u e o f m i n c r e the contamination of higher  goes on d e c r e a s i n g  and t h a t  from h i g h e s t  spin  components  spin  state  Mp+q)  is negligible.  According of  t o Lowdin's s c h e m e ^ ^ ,  s p i n s may be s e l e c t e d  g i v e n by  3 4  t h e component  from f by a p r o j e c t i o n  operator  0  C  166 —|—r k II s(s+l)-k(k+l) k=s+l a  °s  "  a  =  (  E  3  )  where k  S  2  - k(k+l)  Each a n n i h i l a t i n g o p e r a t o r a nent o f the m u l t i p l i c i t y  (E4) a n n i h i l a t e s the s p i n compo-  k  2k+l from  In p r a c t i c e , i t i s  very d i f f i c u l t t o apply the p r o j e c t i o n o p e r a t o r t o e l i m i nate the components of s e v e r a l s p i n states(*>3) ^  As mentioned above, the contamination  decreases  very r a p i d l y as we go to higher s p i n s t a t e s so t h a t , i n p r a c t i c e , i t should be s u f f i c i e n t to a n n i h i l a t e the most important o f the unwanted components, namely, the one with s p i n s'+l, the o t h e r s being r e l a t i v e l y unimportant T h i s can be accomplished a  s'+l a  g  .  v  by a s i n g l e a n n i h i l a t i o n  operator  :  I  +  1  =  S  2  - (s'+l)(s'+2)  (E5)  The new wave f u n c t i o n a g i ^ y i s g i v e n by  q  (E6)  m=0 Let P  and  a  be the p a r t i a l charge d e n s i t y and bond order  m a t r i c e s f o r a and 6 s p i n s r e s p e c t i v e l y and y P  = P V  y  + P V  y  V  ^  e  the t o t a l charge d e n s i t y and bond order matrix. A l s o l e t p =P -P be the s p i n d e n s i t y m a t r i x . L e t us d e f i n e the yv yv yv * H  a  notation  B  x  167 P = P * y  a  (E7)  V  A l s o l e t J and K be the p a r t i a l charge d e n s i t y and bond order m a t r i c e s based on the p r o j e c t e d wave f u n c t i o n  then  ag,.,.-^,  the m a t r i c e s J and K can be w r i t t e n i n terms o f P and Q as follows: J = M~ {A -2A.TrPQ+pq-q-(p+q)TrPQ+3TrPQ+2(TrPQ) 2  2  2  2TrPQPQ}P+{p-TrPQ}Q+QPQ+{(p+q)-4TrPQ-3+2A}PQP+ (PQ+QP){2TrPQ-p+l-A}-2(PQPQ+QPQP)+4PQPQP where Tr stands  (E8)  f o r the t r a c e o f a matrix and A and M are  d e f i n e d as A  =  q - 2{J5(p-q)+l}  (E9)  and M=A -2A.TrPQ+pq-(p+q)TrPQ+2TrPQ+2(TrPQ) -2TrPQPQ (E10) as as ^ • a; a; a; as 2: «* as as a; 2  2  A s i m i l a r e x p r e s s i o n f o r K can be obtained by simply changing  P and Q, and p and q i n equations  (E8) and  The elements o f the p r o j e c t e d s p i n d e n s i t y m a t r i x  inter(E10).  p'  and  yv  charge d e n s i t y and bond order m a t r i x P^  'yv  J  v  are simply given by  yv ~ yv K  (Ell) P' = J + K yv yv yv  D u r i n g t h e p r e s e n t s t u d y , t h e p r o j e c t i o n scheme  168  suggested that  by H a r r i m a n  v  ' was  the c o n t a m i n a t i o n from  also  tried  the s p i n  components  s ' + l i s n e g l i g i b l e and t h e above method tion,  therefore,  gives  fairly  and i t was  larger  of single  accurate spin  found than  annihila-  densities.  

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