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Electron spin resonance study of the chlorpromazine cation Tapping, Robert L. 1968

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AN ELECTRON SPIN RESONANCE STUDY 0? THE CHLORPROMAZINE CATION BY ROBERT L TAPPING B . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF We accept t h i s t h e s i s as .conforming to the r e q u i r e d standard Master of Science i n the Department of Chemistry THE UNIVERSITY OF BRITISH COLUMBIA October, 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e ' a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a - i i -ABS TRACT The cation r a d i c a l of chlorpromazine i n solut i o n was studied i n d e t a i l by electron spin resonante. The 16 l i n e spectrum was interpreted i n terms of a nitrogen atom, two equivalent protons at the f i r s t side chain carbon atom, and three almost equivalent protons from the r i n g system. The r e l a t i v e magnitudes of the s p l i t t i n g constants require many of the spe c t r a l l i n e s to be coincident, and the res-u l t i s the l 6 l i n e spectrum observed. Analysis of the s p l i t t i n g constants was done using Htickel molecular o r b i t a l ' calc u l a t i o n s , from which i t was deduced that the chlorprom-azine cation structure i s folded about the N-S axis, with an included angle of 104°. The s p e c t r a l asymmetry observed in s u l f u r i c acid s o l u t i o n was interpreted in terms of random molecular motions causing a f l u c t u a t i n g environment•to a r i s e at the nuclear positions. This leads to a modulation e f f e c t on the nuclear magnetic moment, and i s responsible for linewidth v a r i a t i o n . Further broadening due to exchange effects i s discussed q u a l i t a t i v e l y . The asymmetry of the spectra enable the sign of the nitrogen s p l i t t i n g to be estimated -- i t was found to be p o s i t i v e . A b r i e f discussion of the e l e c t r i c a l properties of chlorpromazine, using HMO calculations, was included, and some discussion of the mechanism of chlorpromazine drug action was also considered for completeness. - i i i -TABLE OF CONTENTS page A b s t r a c t i i Table of Contents i i i L i s t of Tables v L i s t of Fig u r e s v i Acknowledgement . v i i i INTRODUCTION 1 EXPERIMENTAL METHODS A: ESR Spectrometers 3 B: Solvents Used • 3 C: Compounds I n v e s t i g a t e d 4-D: V a r i a b l e Temperature Apparatus 4 E: Sample p r e p a r a t i o n by e l e c t r o l y t i c o x i d a t i o n 5 F: Sample preparation, using chemical o x i d a t i o n . 5 THEORETICAL ASPECTS I HYPERFINE SPLITTING AND SPIN DENSITIES A: Proton Hyperfine S p l i t t i n g i n Aromatic R a d i c a l s 7 B: Nitrogen Hyperfine S p l i t t i n g 9 C: Hyperconjugation and Spin P o l a r i s a t i o n C o n t r i b u t i o n s 14 D: C a l c u l a t i o n of Spin D e n s i t i e s ( i ) Hlickel Molecular O r b i t a l Theory 17 ( i i ) McLachlan S e l f - C o n s i s t e n t F i e l d Theory 19 I I PARAMAGNETIC RELAXATION IN LIQUIDS A: The S p i n Hamiltonian . 2 1 B: Random C o r r e l a t i o n s . 2 3 C: Linewidth and the Nuclear Quantum Number 2 5 - i v -page RESULTS AND DISCUSSION A: A Brief Introduction 3 0 B: The Hyperfine structure of CPZ + * 3 0 C: Httckel Calculations 32 D: Comparison of Calculated Spin Densities with Experiment 5 9 E: The "Anomalous" Spectrum 65 F: The Correlation Time 68 G: Line width V a r i a t i o n 71 H: Possible Exchange Ef f e c t s 81 Some B i o l o g i c a l Aspects of Chlorpromazine 85 A: E l e c t r i c a l Properties 85 B: Mechanism of Chlorpromazine Drug Action 93 BIBLIOGRAPHY 9 8 APPENDIX 1 The Fermi Contact Hamiltonian 1 0 3 APPENDIX 2., Proton Hyperfine Interaction Mechanism 104 APPENDIX 3 Derivation of the Spin Hamiltonian 1 0 7 APPENDIX k The Correlation Function and i t s Properties 1 0 9 APPENDIX 5 The Correlation Function for a Simple Case 112 APPENDIX 6 A Random Function Treatment of JC,(t) Ilk APPENDIX 7 The Bloch Equations a Br i e f D e f i n i t i o n 117 APPENDIX 8 V i s c o s i t y of s u l f u r i c acid as a Function of Temperature . 118 LIST OF TABLES TABLE I Spin Densities for Chlorpromazine TABLE II Linewidths Calculated for a = 6.5 1 TABLE III Calculated T^ times for Chlorpromazine i n s u l f u r i c acid solution - v i -LIST OF FIGURES page ---Figure l a CPZ + i n HgSO^ glass at - 9 0°C 3 3 l b CPZ + at 9°G 34 1c CPZ + at 16°C 35 Id CPZ + at 22°G • 36 le CPZ + at 50°G 3 ? If GPZ + at ?5°C 3 8 l g CPZ + at 92°G 39 Ih . CPZ+ at 1 11°C 40 l i GPZ + at 131°C 41 Figure 2 a "changed species" (CS) at 2 l . 8 ° C 42 2 b CS at 59°C 4 3 2 c CS at l 4 l ° C 4 4 2 d low resolution spectrum of "changed" 4 5 chlorpromazine species . Figure 3 a .CPZ i n aluminum chloride/nitromethane 46 3 b ' GPZ i n cone, hydrochloric a c i d 4 ? 3 c GPZ+ i n 3 0 $ phosphoric a c i d 48 3 d CPZ + i n a c e t o n i t r i l e 4 9 Je CPZ reacted with I 2 i n a c e t o n i t r i l e : d i -me thoxy ethane . 5 0 Figure 4 a computed s t i c k diagram for GPZ + 51 4 b computed spectrum for GPZ + -- linewidth L O G 52 4 c . computed spectrum f o r GPZ +--linewidth 1.5G 5 3 4 d computed spectrum for CPZ +--linewidth 2.0G 5 4 v i i -page Figure 4 e computed spectrum for C P Z + — l i n e w i d t h 2.5G 5 5 + Figiire 5 linewidth v a r i a t i o n of CPZ in s u l f u r i c a c i d with temperature 8 7 Figure 6 v a r i a t i o n of energy (of highest-occupied molecular o r b i t a l ) with f o l d i n g angle,9 88 Figure 7 some representative phenothiazine drugs 89 Figure 8 v a r i a t i o n of sidechain bending and nitrogen spin density 9 0 - v i i i -A G KNOW LED GEM EN T I wish to thank my resear c h supervisor, Dr. J.B. Farmer for his encouragement and assistance i n this work. I also wish to thank Dr. R. Srinivasan for some he l p f u l re-marks, and Mr. D. Kennedy for his assistance i n using the-HMO computer program-. Mr. T. Markos of the electronics shop also deserves thanks for answering f a l s e alarms and a s s i s t i n g with the spectrometer, besides maintaining the upkeep of the spectrometers. INTRODUCTION The many and i n t r i c a t e problems involved i n un-derstanding such an apparently simple thing as a l i v i n g c e l l have resulted i n much attention being focussed on the bio-l o g i c a l sciences by students of the physical sciences* The chemical and physical questions are often within the realm of physical science, e s p e c i a l l y when separated from the bio-l o g i c a l structure, and so there i s no b a r r i e r preventing overlap of the d i f f e r e n t d i s c i p l i n e s . The separation of the sciences into various departments has often resulted i n dup-l i c a t i o n of experiments and loss of p o t e n t i a l applications to a subject requiring a fresh approach. Now, however, many f i e l d s described variously as biophysics, neurophysics, bio-chemical physics, quantum biology, etc. are o f f e r i n g a chal-lenge to s c i e n t i s t s who want to apply pure physics, f o r ex-ample, to the problems of c e l l membranes. Electron spin resonance (esr) studies are e s p e c i a l l y adaptable to many b i o l o g i c a l problems, and the systems studied often involve very complicated molecules. Careful approxim-ations are then needed to apply the theories of esr to such molecules, without l o s i n g too much information about the bio-molecule. Simple biomolecules are, of course, being studied most, and. i n t h i s work a r e l a t i v e l y simple drug molecule was examined. Drugs are not usually species native to l i v i n g ( e s p e c i a l l y animal) systems, but do interact i n i n t e r e s t i n g and generally unknown ways (unknown in the sense that the mech--2-anisms of drug a c t i o n are not pre c i s e l y understood at the c e l l u l a r and molecular l e v e l s , i n many cases). Chlorpromazine (CPZ): JL ^ CI/6 (CH 2 ) 3 N(CH 3 ) 2 where the. numbers indicate the numbering convention to be used i n thi s work, i s e a s i l y available to esr study, and is an asymmetric structure. Most esr theory has been estab-li s h e d for symmetrical species, for mathematical convenience, so an esr study of CPZ should enable a comparison to be made between the re s u l t s generally obtained for symmetrical mol-ecules and the int e r p r e t a t i o n of a molecule l i k e chlorprom-azine . Chlorpromazine i s usually used as a hydrochloride, and i s a white p o l y c r y s t a l l i n e s o l i d that i s r e a d i l y oxidis-able by b i o l o g i c a l , chemical, and electrochemical means, and also exhibits both impurity and i n t r i n s i c semi-conductor pro-perties . The simple theories to be applied to this esr study of CPZ are hoped, to provide evidence that the techniques of chemical physics, f o r example, may be of use i n explaining b i o l o g i c a l functions. Most of the work here w i l l concentrate on explanation of the esr spectra and physical properties of CPZ, but some q u a l i t a t i v e discussion w i l l be given to the rol e of chlorpromazine as a drug. - 3 -EXPERIMENTAL METHODS A: ESR SPECTROMETERS Two spectrometers were used. The f i r s t , designated ESR - 3 , i s s i m i l a r to the Var i a n V-4-500 lOOKHz u n i t operating a t X-band frequency (—9200MHz), with microwave power s u p p l i e d by a V a r i a n V - 1 5 3 k l y s t r o n w i t h Kepco t r a n s i s t o r i z e d DC power supply to s t a b i l i s e the f i l a m e n t v o l t a g e . The magnet i s a V a r i a n V-4012 12" model with 2 . 5 " pole gap, modulated a t lOOKHz. The magnetic measured with a proton resonance magnetometer, w i t h the probe connected to an FM-modulated 20Hz o s c i l l a t o r . The o s c i l l a t o r s u p p l i e s a v a r i a b l e frequency of about 14-MHz to the probe, and the frequency i s read o f f a Hewlett Packard 54-25L e l e c t r o n i c counter a f t e r being made to beat with a s i g -n a l generator. Power was monitored w i t h a Hewlett Packard kQOC microwave power meter. The other spectrometer was a Va r i a n E - 3 bench model. This instrument has a 4 " magnet with 1.2" pole gap and i s mod-u l a t e d a t lOOKHz. Microwave frequency and power are read o f f the microwave b r i d g e , and f i e l d measurement i s by means of a F i e l d i a l . Accuracy i s ±3% of the f i e l d reading. T h i s , of course i s c o n s i d e r a b l y l e s s accurate than the proton resonance magnetometer, which i s accurate to about 0.1G. The E - 3 has an X-Y recorder f o r s i g n a l output. B; SOLVENTS USED Nltromethane (CH-^NOg) and a c e t o n i t r i l e (CH^CN) were obtained from Eastman Organic Chemicals (Spectro Grade), and were used without f u r t h e r p u r i f i c a t i o n . When not i n use, the solvents were kept under vacuum. S u l f u r i c a c i d (E^SO^) was obtained from B r i t i s h Drug Houses (Analar, 95>5% min.) and was dried on a vacuum before use. C: COMPOUNDS I WES T l GATED Chlorpromazine-HCl was obtained from Poulenc, and was used without further treatment. Promazine-HCl was a g i f t of J. Wyeth Co., and was also used without further t r e a t -ment. For the e l e c t r o l y t i c oxidation, the a u x i l i a r y com-pound used as supporting e l e c t r o l y t e was LiClO^ (G.F. Smith Chemical company), and was dried over vacuum for several days before use. It was found that the e l e c t r o l y s i s f a i l e d to give a stable r a d i c a l i f a l l compounds were not dried care-f u l l y before use. Chemical oxidation i n nitromethane/aluminum.chloride used aluminum chloride obtained from B r i t i s h Drug Houses. The highly hygroscopic nature of A l C l ^ meant that i t had to be stored over a vacuum a l l the time. D; VARIABLE TEMPERATURE APPARATUS Temperature v a r i a t i o n was effected using a Varian V-45^7 Variable Temperature Accessory. For low temperatures dry nitrogen gas was passed through a sta i n l e s s s t e e l c o i l which was immersed i n a suitable coolant. The cooled gas then passed through a c y l i n d r i c a l dewar equipped with a s t i c k heater, and into a sp e c i a l dewar situated i n the cavity. To control the temperature either the flow rate could be adjusted or the heater could be used to warm the gas. For high temp-- 5 -erature work, the cooling device was omitted and the s t i c k heater used to warm the gas to the desired temperature. Temperature was measured with a copper/constantan thermocouple and a Rhodes potentiometer. During a l l runs d i f f e r i n g from room temperature, the cavity was kept at a reasonably constant temperature by passing dry nitrogen gas through i t . The temperature v a r i a t i o n for a l l runs was never worse than ±2°C, and was often constant to within 0 . 2 degrees for a p a r t i c u l a r run. Et SAMPLE PREPARATION - BY ELECTROLYTIC OXIDATION An e l e c t r o l y s i s c e l l designed by P.H.H. Fischer ( 1 0 5 ) was used, with best r e s u l t s obtained using a c e t o n i t r i l e as the solvent and LiClO^ as the supporting e l e c t r o l y t e . The solut i o n was prepared i n the c e l l by d i s s o l v i n g CPZ and LICIO^ In the a c e t o n i t r i l e and then degassing on a vacuum l i n e . Passage of 2 0 amps at 5 to 10 volts f o r 30 to 60 minutes produced the c h a r a c t e r i s t i c red CPZ cation. The concentration of CPZ i n - 3 the i n i t i a l s o l u t i o n was about 10 M for most s a t i s f a c t o r y r e s u l t s . F: SAMPLE PREPARATION USING CHEMICAL OXIDATION S u l f u r i c a c i d oxidation was used most frequently. This was most e a s i l y carried out by preparing a sample of CPZ i n 9 5 > 5 % HgSO^ beforehand, and tr a n s f e r r i n g a portion.to an esr sample tube equipped with a stopcock. The solut i o n could then be degassed and esr spectra recorded. Optimum concentrations for t h i s method were 10 J to 10 M. Oxidation i n AlCl^/CH^NOg was considerably more precise. -6-A sidearm. was a t t a c h e d t o t h e e s r sample tube below the s t o p -cock, and i n t h i s s i d e a r m were p l a c e d t h e r e q u i s i t e amounts of A l C l - j and CEZ. . .Nitromethane was t h e n d i s t i l l e d i n t o the s i d e a r m , and the r e a c t i o n completed. The r e s u l t a n t r e d s o l -u t i o n of CPZ +'was t h e n t r a n s f e r r e d t o the bottom of the e s r tube. I n t h i s manner, the r e a c t a n t s were k e p t degassed a t a l l t i m e s and the c o n c e n t r a t i o n of the i n i t i a l c o u l d be f a i r l y a c c u r a t e l y known. Optimum c o n c e n t r a t i o n s here were found t o be I O - 3 t o 10"^M. O x i d a t i o n s i n p h o s p h o r i c and h y d r o c h l o r i c (cone.) a c i d s was e f f e c t e d i n the same manner as f o r s u l f u r i c a c i d , y i e l d i n g s i m i l a r r e d s o l u t i o n s . The s p e c t r a of t h e s e s o l u t i o n s were not examined i n g r e a t d e t a i l . Temperature s t u d i e s were made only, on the s u l f u r i c a c i d s o l u t i o n s . A t t e m p t s t o do the same f o r the a c e t o n i t r i l e and n i t r o m e t h a n e s o l u t i o n s r e s u l t e d i n the s o l v e n t s (which were under vacuum) b e i n g t r o u b l e s o m e due t o b o i l i n g and evap-o r a t i o n above 40°G. -7-THEORETICAL ASPECTS I HYPERFINE SPLITTING AND SPIN DENSITIES  A: PROTON HYPERFINE SPLITTING IN AROMATIC RADICALS When aromatic hyperfine s p l i t t i n g s were f i r s t ob-served by Weissman et a l ( 1 - 3 ) i n 1 9 5 3 , the f i r s t explanat-ion was that somehow there was i n t e r a c t i o n of the el e c t r o n i c magnetic moment with proton moments. Such an i n t e r a c t i o n was considered necessary to explain why naphthalene anions ex-h i b i t e d detailed hyperfine structure ( l ) . Clearly the elec-tronic wave function was not zero at some proton positions, a f a c t that contradicted the general ideas of aromaticity at that time. In the conventional planar aromatic molecule the p i - e l e c t r o n charge resides i n the lobes of the carbon p z o r b i t a l s — either above or below'"the aromatic r i n g . There is then a node at the carbon nucleus through which the un-paired electron, i n a r a d i c a l , must become zero. Thus no i n t e r a c t i o n ( i n a conventional sense) between the electron and proton (through the C-H sigma o r b i t a l ) seemed possible. Weissman (l) speculated that zero-point vibrations by the proton may be responsible, since only small admixtures of states are required to produce the observed magnitudes of hyperfine s p l i t t i n g s ( 3 ) . This i s because a pure hydrogen Is o r b i t a l has a s p l i t t i n g of about 5 1 0 G . Subsequently the vibronic i n -t e r a c t i o n approach was shown to be wrong (4), since the s p l i t t -ing constants were not proportional to the square roots of the nuclear masses, as a vibronic i n t e r a c t i o n theory would predict ( 5 ) • -8-In e l i m i n a t i n g bending v i b r a t i o n s of the C-H bond, Venkataraman and F r a e n k e l ( 4 ) suggested that the unpaired e l e c t r o n i s not i n a pure p i s t a t e i n an aromatic r a d i c a l (or any p i r a d i c a l ) , and t r a n s f e r s some s p i n d e n s i t y to the protons by c o n f i g u r a t i o n i n t e r a c t i o n between the sigma and p i s t a t e s . H.M.McConnell ( 6 - 9 ) and s e v e r a l others examined such a mech-anism, and the most e x t e n s i v e work i s McConnell's -- much of what immediately f o l l o w s i s based on McConnell's i n v e s t i g a t i o n s . The d e t a i l s of McConnell's theory are presented q u i t e f u l l y i n Appendix 2 , but a b r i e f d i s c u s s i o n of the use of "McConnell's r e l a t i o n " i s g i v e n here. E s s e n t i a l l y , the s p i n p o l a r i s a t i o n a r i s i n g from the presence of an unpaired e l -e c t r o n i n a p i system can produce an a p p r e c i a b l e s p i n p o l a r -i s a t i o n i n s-atomic o r b i t a l s of the aromatic protons by an atomic exchange c o u p l i n g mechanism. Both Weissman ( 1 0 ) and McConnell have e x p l a i n e d t h i s , and the r e s u l t i s o f t e n w r i t t e n q u a n t i t a t i v e l y as: a H = Qccf>c ( 3 - D TT ir where a " i s the p r o t o n h y p e r f i n e c o u p l i n g c o n s t a n t , f t i s the u n p a i r e d s p i n d e n s i t y a t the carbon atom of the C-H bond, and Q^ c i s the s p i n p o l a r i s a t i o n constant (or sigma-pi f a c t o r , e t c.) f o r the case i n q u e s t i o n . U s u a l l y the r e l a t i o n s h i p j d e s c r i b e d by eq ( 3 - D i s assumed to be l i n e a r , and i n such an assumption there i s a p o t e n t i a l hazard. McConnell's d e r i v a t i o n , g i v e n i n Appendix 2 , d e s c r i b e s the s p l i t t i n g g e n e r a l l y as: a j ^ l 6 4 0 f j f j ( 3 - 2 ) -9-The s p i n a t t e n u a t i o n f a c t o r f , can then be a s s o c i a t e d with Q, but s i n c e f i s , i n g e n e r a l , expected to d i f f e r from p r o t o n to proton, Q must a l s o be s u s c e p t i b l e to the same be-h a v i o u r . I t i s u s u a l l y convenient to ignore such behaviour, however, i n t r e a t i n g most f r e e r a d i c a l s . B: NITROGEN HYPERFINE SPLITTINGS The o b s e r v a t i o n t h a t s p i n p o l a r i s a t i o n was respons-i b l e f o r p r o t o n h y p e r f i n e s p l i t t i n g s can c l e a r l y be a p p l i e d to any other nucleus p o s s e s s i n g a s p i n magnetic moment and a t t -ached t o , or i n , a r i n g system or p i system. Karplus and F r a e n k e l ( 1 3 ) examined C^ 3 i n g r e a t d e t a i l and showed that s p l i t t i n g s are dependent on the p i - e l e c t r o n s p i n d e n s i t i e s a t nearest-neighbour atoms as w e l l as a t the C J nucleus i t s e l f . E x p e r i mental v e r i f i c a t i o n has shown t h i s theory to be essent-i a l l y c o r r e c t ( 1 3 , 1 4 ) . Other atoms c o n s i d e r e d i n the same manner are F^^ ( 1 5 ) and N ^ ( 1 6 - 2 0 ) . these treatments, how-ever, r e q u i r e c o n s i d e r a b l y more d i s c u s s i o n . ; . The f l u o r i n e case i s not of i n t e r e s t here, except to mention t h a t H i n c h c l i f f e and M u r r e l l ( 1 5 ) o b t a i n only order of magnitude agreement and c e r t a i n l y more work i s needed i n t h i s p a r t i c u l a r case. As r e f e r e n c e s 1 6 - 2 0 show, c o n s i d e r a b l e disagreement as to an e x p l a n a t i o n f o r ^ x h y p e r f i n e s p l i t t i n g s was the s i t u a t i o n f o r some time. A thorough and reasonable treatment of the problem, however, has been g i v e n by Henning ( 2 1 ) , and a p p l i c a t i o n of the theory has been q u i t e s u c c e s s f u l . Thus a b r i e f d e s c r i p t i o n of the method i s u s e f u l . C l e a r l y s r e l a t i o n s h i p l i k e eq ( 3 - D c o u l d be used f o r h y p e r f i n e i n t e r a c t i o n s . In f a c t , C a r r i n g t o n and Santos Veiga ( 1 6 ) used such a r e l a t i o n s h i p and claimed f a i r s u c c e s s . However, one important a s p e c t may have been over-looked by some, and t h a t i s t h a t r e s u l t s l i k e those of Ca r r -i n g t o n were f o r a f a m i l y of l i k e compounds. Use of eq ( 3 - 1 ) f o r any atom should s t r i c t l y be l i m i t e d to such f a m i l i e s i n order to have meaningful use of the r e l a t i o n s h i p . Henning a c t u a l l y performed a l e a s t - s q u a r e s f i t and an e r r o r t e s t on: a N =QNf]J ( 3 - 3 ) and a p p l i e d i t to )f , | f - d i p y r i d y l , 1 , 4 - and 1 , 5-diazonaphth-a l e n e , and p y r a z i n e . His r e s u l t s i n d i c a t e d t h a t a r e l a t i o n -s h i p of the form of eq ( 3 - 3 ) does not e x i s t f o r the g e n e r a l case. I f the i s o t r o p i c h y p e r f i n e i n t e r a c t i o n between an e l e c t r o n i c moment ( g e ^ e S (k) )• and a n u c l e a r moment ( g n ^ n . I n ) , d e s c r i b e d by the Fermi c o n t a c t Hamiltonian (see Appendix 1 ) i s r e p r e s e n t e d as: a N = 8 / 3 g n p n # / M s | i ( r n k ) S z ( k ) \ ^ > ^ then can be c o n s i d e r e d to be the s t a t e i n which the f r e e r a d i c a l system e x i s t s , and l|>(0) i s the ground s t a t e w i t h eigenvalue M g s i n c e i t i s an e i g e n s t a t e of Sz=; S z ( k ) . [fyi 0)^ 4o c a n b e r e p r e s e n t e d as where € r e p r e s e n t s anC(.spin andQ a @ s p i n . (The above i s f o r an a n i o n but the r e s u l t can be a p p l i e d e q u a l l y w e l l to c a t i o n s ) . The sigma and p i o r b i t a l s can be w r i t t e n more ex--11-p l l c l t l y (21): TVi = | ' a i r ( p z ) r (3 - 5 ) where the molecular o r b i t a l s are l i n e a r combinations of carbon and nitrogen 2 p z atomic o r b i t a l s with z perpen-d i c u l a r to the molecular plane, and ^ for the bonding and antibonding cases, respectively. The o r b i t a l s making up the j in-plane bonds are l i n e a r 2 i combinations of sp hybrids, ( h x | h y ) , of atoms x and y con-s t i t u t i n g the x-y bond, where 4 j i x\h^ i s the overlap i n t e g r a l . It should be noted that the sigma o r b i t a l s are necessarily symmetric i n the molecular plane, and further that both the nitrogen Is and lone-pair o r b i t a l s are s i m i l -a r l y symmetric, so the sigma system just described can i n -clude the "odd" nitrogen o r b i t a l s . Now i f excited states are designated tjj^, i n t e r e l e c t r o n i c repulsions (represented as ^ e 2 / * ^ ) can mix ty0 with ^ , giving: M t ?L<^l«MO/(E0-Ev)] 4< ( 2 - 7 ) to f i r s t order (see, for example, any text on simple f i r s t order perturbation theory, i . e . ref. 22 p 2 ^ 5 ) . Because the t o t a l electronic Hamiltonian i s a sum of one-electron and two-electron operators, only si n g l y and doubly excited configurations need be considered ( 2 1 ) . Thus, considering only those configurations V^ - which give a f i r s t order contribution to the spin density at the nuclear pos-i t i o n r n , the spin density there can be written: - 1 2 -since p ( r n ) = j ^ < £ 2 S z k f ( r n k ) v j ' d T ( 2 2 ) and as i n eq ( 3-8). In other words, the unpaired spin density at the nucleus, which i s controlled by the delta function of electron-nuclear p o s i t i o n , i s the difference between the average number of spins and p spins (governed by 2 S z k ) — thus i f there are more otspins the spin density i s p o s i t i v e , and vice-versa. For the ground configuration Cj^the Tf o r b i t a l con-ta i n i n g the odd electron has a node i n the molecular plane, so that the f i r s t term i n (3-8) must vanish. Thus, consid-ering the non-vanishing contributions to the spin density i t can be noted that the delta function requires the electron to*jump"between o r b i t a l s which are non-vanishing at the nuc-leus the spin density at the nucleus i s therefore produced by one-electron"jumps^between excited states, r e s u l t i n g i n the p o s s i b i l i t y of two spin doublet states and one spin quartet ( 6 , 2 1 ) . The ground configuration l | 0 i s a doublet, so the quartet can be ignored and two doublets can be derived: pure si n g l y excited: (|&=^ ^J^C, • . -QCTp'. . . /T^ I-JC^--^^*...TT R T or vjf^fapp-p**) ( 3 - 9 ) pseudo singly excited : ^ = 14 -jMc* ™ ^ Q ^ Equations ( 3 - 9 ) are well-known wave functions ,(6, 2 1 , 2 2 ) , and i t can be seen immediately that ^ scannot contribute to the spin density since i t cancels i n t e r n a l l y ( 2 1 ) , so ^ps i s the only configuration needed to f i r s t order. Using the appropriate matrix elements ( 1 2 ) : ' - 1 3 -<^s\g'\^o> = ( 3 / / 6 ) Q(f N ) G pn< N) ( 3 - 1 0 ) where. M ~ ^ ( r n k ) S z ( k ) It Now rf l= <£a i r(p z) r, as before, so <Mf\^= - ( ^ ) ^ a n + 1 | r a n + 1 ( S < Q ( p z ) r | e 2 / r l 2 | ( p z ) s > and the atomic o r b i t a l spin density matrix forVJ/ 0is ( 2 1 ) : 7T = an+l,r an+l,s ( 3 - 1 1 ) Therefore combining gives: a N = UrUpf^Sg&fZ. ( 3 - 1 2 ) and thus the hyperfine coupling matrix Q N can be written: Q ? S = - l 6 / 3 f f g n ? n £ £ ^ ^ ' PX Q ( r N ) ( 7 ^ ( ( r N ) ( 3 - 1 3 ) In eq ( 3 - 1 3 ) "the summation over i and p should, i n p r i n c i p l e , be extended over the excitations of a l l the sigma bonds i n the molecule, but i t i s convenient to r e s t r i c t the summation to sigma o r b i t a l s adjacent to the atom i n question, i . e . ( l s ) N , 6NC» ^ N C a n d nN f o r R I I T R O S E N : Then ^ ( r ^ ) G ^ r ^ ) can be considered n e g l i g i b l e for a l l other o r b i t a l s . Such a r e s t r i c t i o n i s necessary for ease of computation, and i s commonly used i n s i m p l i f i e d molecular orb-i t a l theory. McConnell's r e l a t i o n s h i p (eq ( 3 - 1 ) ) i s based on thi s assumption. In other words, i t i s assumed that the exchange i n t e g r a l for non-nearest-neighbour atoms i s vanish-ingly small. N So a matrix for Q can be set up: -14-0 N 0 N Q N A RTN [ - N N N ' \ . . . yNCl-^NC2 Q•• = \ QcN Q Q C where, by symmetry, N N N N / Q C 1 N = Q C 2 N fa0 QCC/ n N _nN Q C 1 C 1 W C 2 C 2 so, from eq (3-12) & N = QNNfNN + Q C C ( f c i C l + P c 2 C 2 ) + ._ . , . N N IT if + ( Q N C + Q C N ) ( p C l N + f c 2 N } ^ Henning shows i n d e t a i l (21) that the bond e x c i t a t i o n ^ T f * ^ / N N i s antisymmetric, which leads, to the r e s u l t that Q ^ Q = - Q C N ' Thus eq (3-l4) becomes: a * QNNfNN- + Q C C ( f c i C l + pC2G2 } ( 3 ~ 1 5 ) Henning also carried out a similar argument for a C-H fragment, considering the only relevant e x c i t a t i o n to # H be 6" r-*(J , and arr i v e d at a matrix representation for Q : CH CH / H H v _H / QHH QHC \ . . . ' . Q = g H ) ' since the proton does not \ Q C H Q C C / H H H bear a p - o r b i t a l , Q H H = 0 » ^HC=<^CH='0' s i n c e n o e x c i t a t i o n can occur between them. Thus the only non-vanishing matrix element H i s Q Q C . as shown e a r l i e r by McConnell (6). C: HYPERCONJUGATION AND SPIN POLARISATION CONTRIBUTIONS How important i s hyperconjugation i n aromatic sys-tems? The question has been debated by several authors (24-29) i n i n considerations and analyses of esr spectra, with the s i t u a t i o n so far tending to favour hyperconjugation over spin p o l a r i s a t i o n for the i n t e r a c t i o n of methyl and meth-ylene groups with an aromatic r i n g system. These groups are the ones of most i n t e r e s t here, and w i l l be the only ones -15-considered. McConnell's equation ( 3-1) for protons has often been applied to methyl protons ( 2 4 , 2 5 ) , and methylene protons ( 2 7 ) with some success, but the obvious inconsistency i n Q values (varying from 2 0 to 3 0 gauss, with no p a r t i c u l a r corr-e l a t i o n within "f a m i l i e s " of molecules) suggests that a more fundamental r e l a t i o n s h i p holds. Indeed, i n thi s experiment an apparent Q value i n the neighbourhood of 11G i s obtained i f spin p o l a r i s a t i o n conditions are considered. The whole question of hyperconjugation versus spin p o l a r i s a t i o n , which was ra i s e d i n some d e t a i l by Colpa and de Boer ( 2 6 ) , seems better expressed as "how much does spin p o l a r i s a t i o n contrib-ute to hyperconjugation (or vice-versa) when considering the methyl and methylene hyperfine structure i n esr spectra". Levy (28) has considered this point i n quantitative terms, arid his discussion w i l l be applied here. I t should be noted, however, that some modification may be needed to completely explain experimental r e s u l t s , but this w i l l be further elab-orated l a t e r . Levy's formulation i s useful i n the sense that i t s accuracy depends larg e l y on the molecular o r b i t a l treatment used to obtain the c o e f f i c i e n t s of the appropriate wave function, and as such i s very adaptable to s i m p l i f i e d theories. Using HUckel theory Levy obtained reasonably good r e s u l t s -- cer-t a i n l y more consistent than many previous explanations ( 2 5 , 2 6 ) . Two expressions are derived, one for methylene protons and one for methyl protons. The derivation i s straightforward and i s - 1 6 -based on calculations of the spin densities at the protons concerned, and using Slater atomic o r b i t a l s (see, for ex-ample, r e f . 3 0 ) an expression for the t o t a l spin density at the proton can be calculated from: where 4> = £ C o C ( 3 - 1 7 ) i The f i n a l r e s u l t s are: •(cc,+ec,.) ( 3„i8) where the spin p o l a r i s a t i o n contribution i s included i n the l a s t term and i s derived from ( 2 7 ) : * s P * Q<? = Q(Q*+Ci'') = -3-o9p ( 3 - 1 9 ) where C and G" represent the adjacent carbon atoms. For the methyl case, acH3 = 2/9.6+ /3./7CC2 + miCn-i + 3.997<*£' + 0.mCLCc, ( 3 - 2 0 ) where again the l a s t term represents the spin pol-a r i s a t i o n contribution. Care must be taken i n applying these relationships to non-simple systems (either a l i p h a t i c or aromatic) since often many assumptions have been made i n a r r i v i n g at a s i t -uation i n which the use of such relationships as eqs. ( 3-l8) and ( 3 - 2 0 ) i s plausible, and the r e s u l t s may not be e n t i r e l y j u s t i f i a b l e . However, i f the necessary care i s taken, the re s u l t s should indicate the extent to which hyperconjugation -17-and/or spin p o l a r i s a t i o n contribute to the esr spectra. D; CALCULATION -OF SPIN DENSITIES The close r e l a t i o n s h i p between spin densities and the observed hyperfine s p l i t t i n g i n esr spectra leads to the conclusion that a knowledge of the c o e f f i c i e n t s of the r e l -evant molecular o r b i t a l s (and hence a knowledge of the un-paired electron density) leads to an estimation of the hyp-erfine coupling constants. In using any molecular o r b i t a l theory to estimate spin densities a very important point to consider i s the ease of c a l c u l a t i o n of the desired values. Since this usually implies a s i m p l i f i e d treatment, the par-ameters calculated from such c o e f f i c i e n t s should r e f l e c t the same degree of approximation. The two methods to be consid-ered here for molecular o r b i t a l calculations are the Htlckel molecular o r b i t a l method (HMO) and the McLachlan self-con-s i s t e n t f i e l d theory (SCF). A b r i e f discussion of each might be useful: (i) Htlckel molecular o r b i t a l theory ( 3 2 , 3*0; B a s i c a l l y , a l l xvave functions ^ are solutions to ; the Schrbdinger e q u a t i o n . J ^ ^ i r ^ , where J ^ i s the Hamiltonian operator which includes a l l interactions between m electrons and n nuclei. The lack of exact expressions for has usually meant that considerable approximation must be made, however. Further, since the wave function contains far more information than can be obtained from i t at any time, the approximations made must be suitable for the purpose. The f i r s t approxim-a t i o n here i s that lp can be considered the product of a set of (T-bonds, which are further d i v i s i b l e into r e l a t i v e l y non--18-i n t e r a c t i n g l o c a l i s e d bonds, and ff-bonds ; $ = <krfar ( 3 - 2 1 ) Thus i f i t i s assumed that c^ - can be considered as a product of 2 - c e n t e r bonds only ( i . e . the i n d i v i d u a l (T-bonds are considered to determine ), and (j>^ i s approximated i n the LCAO method as a combination of 2 p z o r b i t a l s , each of which shares the same nodal plane (the carbon s k e l e t o n i n planar aromatic r i n g s ) : f o r the LCAO molecular o r b i t a l s . The 'jT'-systern i s considered alone, and the Hamiltonian can be i n i t i a l l y considered a one-electron operator, and to solve f o r the set of c o e f f i c i e n t s g i v i n g the best energy f o r the molec-u l a r o r b i t a l the v a r i a t i o n p r i n c i p l e can be used (see, f o r ex-a m P l e , r e f . 35). _ f ^ ^ In other words, eq ( 3 - 2 2 ) i s the ex p e c t a t i o n value of ^  over the Hamiltonian operator and represents an approximation to the energy of (^  . Any wave f u n c t i o n ( i • e. the approximation to be used) w i l l t h e r e f o r e give a higher energy than the ground-state energy EQ — where ^' r^ ,6 = Eg. Min-i m i s i n g eq ( 3 - 2 2 ) with respect to the c o e f f i c i e n t s leads to ( 3 2 ) : £ C Y ( H Y t - 6 ^ f c ) = 0 . ( 3 - 2 3 ) where ^ = [ ^ H < i ? s i t = H*f = S <QS IT. - %( I f there are n c o e f f i c i e n t s , c l e a r l y there w i l l be n eq-- 1 9 -u a t i o n s of the' form ( 3 - 2 3 ) where t ranges from 1 t o n. Now, i n the H i i c k e l a p p r o x i m a t i o n the Coulomb xn-t e g r a l s , H ^ r i t , r e p r e s e n t a p p r o x i m a t e l y the energy of an e l e c t r o n i n a (carbon) 2 p o r b i t a l ; so f o r the H * - l a t t i c e , con-s i s t i n g of carbons e n t i r e l y , i t can be assumed t h a t a l l R"rr a r e e q u a l and thus Hv*^ The r e s o n a n c e , or bond, i n t e g r a l s , E L , , , r e p r e s e n t an a t o m i c o r b i t a l i n t e r a c t i o n energy, and i f r and s a r e not bonded H r s=0, i f r and s a r e a d j a c e n t and bonded, H r s=^. The o v e r l a p i n t e g r a l s , S r g , can be r e p r e s -e n t e d by the d e l t a f u n c t i o n , s r s=o^ s, s i n c e S r r = l f o r norm-a l i s e d a t o m i c o r b i t a l s , and S„_ i s assumed t o be 0 f o r r / s . r s F i n a l l y , i t can be shown t h a t ( 3 2 ) : £ j a <X -t W j p (3-24) f o r the energy of the j 1 t h m o l e c u l a r o r b i t a l . I f heteroatoms a r e t o be i n c l u d e d , m o d i f i c a t i o n of t h e Coulomb and resonance i n t e g r a l s can. be e f f e c t e d as f olloxtfs : , H r s = * = ( 3 - 2 5 ) 0 ) S ^ f i i where h and k a r e d e t e r m i n e d somewhat s e m i - e m p i r i c a l l y ; f o r each heteroatom i n q u e s t i o n ( 3 2 , 3 6 ) . The s p i n d e n s i t y can now be c a l c u l a t e d a t the r 1 t h atom from the f o l l o w i n g r e l a t i o n -s h i p : .p V ( 3 - 2 6 ) where c^ i s the c o e f f i c i e n t of the h a l f - o c c u p i e d mol-e c u l a r o r b i t a l a t the atom i n q u e s t i o n . T h i s r e l a t i o n s h i p f o l l o w s from (3-8) and the wave f u n c t i o n j u s t d e s c r i b e d . ( i i ) M c L a c h l a n s e l f - c o n s i s t e n t f i e l d t h e o r y ( 3 3 , 3 9 ) : The M c L a c h l a n SCF method' a p p l i e s . b e s t t o a l t e r n a n t - 2 0 -hydrocarbons ( 3 2 , 3 3 , 3 9 ) and some d i f f i c u l t y a r ises when applying this method to non-alternant systems, esp e c i a l l y i f heteroatoms are present. I t does, however, explain negative spin densities and has been of some use i n calculations on n i t r o - and cyano-aromatic r a d i c a l s ( 3 6 , 3 7 ) * The SCF t o t a l ff-electron Hamiltonian can be written: #=(-4V A -£V*)4&geV<« ( 3 - 2 7 ) where V r i s the p o t e n t i a l due to the screened nucleus r. McLachlan ( 3 9 ) showed that electrons of<yand^spin occupy d i f f e r e n t o r b i t a l s without a f f e c t i n g the e l e c t r o n i c wave function. In other words, they occupy s p a t i a l l y d i f f e r e n t o r b i t a l s . Thus the t o t a l I f - e l e c t r o n wave function becomes tyv = ^(( j^cO^O) 4f&)?Ci0.-'•) (3-28) McLachlan used Huckel molecular o r b i t a l s as s t a r t i n g o r b i t a l s and calculated the e f f e c t s of electronic interactions (elec-tron-electron perturbations) on the HMO c o e f f i c i e n t s . To do this he defined the modified Coulomb i n t e g r a l cxj =s o(r + 3>C* j m . ( 6 ( 3 - 2 9 ) where c r are the c o e f f i c i e n t s of the r'th atomic o r b i t a l of the h a l f - f i l l e d HMO, a n d t h e Coulomb i n t e g r a l i n the Huckel approximation. If c - ^ i s the MO c o e f f i c i e n t , then the electron-electron perturbation becomes ( 3 9 ) : ( 3 - 3 0 ) Thus the spin density becomes, i n McLachlan's method (3 3 ) • n ?r = C*,n*i + Z ( C V ~ C v ) ( 3 - 3 D and i n calculations A is usually taken to b e ~ 1 . 2 . This theory works we 13. with alternant hydrocarbons, as mentioned -21-e a r l i e r , and a l s o with some n o n - a l t e r n a n t hydrocarbons ( f l u o r a n t h e n e , acenaphthylene), u s i n g Q =-24.2G. As w i l l be shown l a t e r , t h i s v alue of Q i s most s u i t a b l e f o r t h i s work, a l t h o u g h a p p l i c a t i o n of any of the t h e o r i e s j u s t out-l i n e d to chlorpromazine r e s u l t s i n d e v i a t i o n s from experiment t h a t suggest c a r e f u l i n t e r p r e t a t i o n of any q u a n t i t a t i v e v a l u e s . I I PARAMAGNETIC RELAXATION IN LIQUIDS  A: THE SPIN HAMILTONIAN Very o f t e n the h y p e r f i n e s t r u c t u r e of s p i n resonance s p e c t r a i n l i q u i d s e x h i b i t s a d e f i n i t e asymmetry or a l i n e -width a l t e r n a t i o n e f f e c t . Such phenomena have been shown to be due to n u c l e a r o r i e n t a t i o n , s o l v e n t v i s c o s i t y , e t c . , and most have been w e l l - d i s c u s s e d (40-57)« The r e l a x a t i o n e f f e c t s t o be d i s c u s s e d here are those e x p l a i n i n g s p e c t r a l asymmetry and l i n e w i d t h narrowing, s i n c e the more complicated l i n e w i d t h v a r i a t i o n s are probably not a p p l i c a b l e to t h i s experiment. The q u i t e complicated r e l a x a t i o n and d e n s i t y matrix treatments of such as K i v e l s o n and Freed and F r a e n k e l (43,44,49-51,54) w i l l not be co n s i d e r e d i n any d e t a i l , t h e r e f o r e . Before approaching any t h e o r e t i c a l d e r i v a t i o n s , some p o s t u l a t e s on the nature of the system to be s t u d i e d have t o be made, i n order to choose the a p p r o p r i a t e Hamiltonian to d e s c r i b e the p e r t i n e n t i n t e r a c t i o n s . McConnell (40), i n h i s o r i g i n a l treatment assumed t h a t i n s o l u t i o n an ordered s t a t e e x i s t s . That i s , the paramagnetic s p e c i e s being considered -22-behaves as i f i t were i n a regular c r y s t a l f i e l d , but un-l i k e s o l i d - s t a t e c r y s t a l behaviour i s able to undergo Brovmian motion. Ideally, then, the paramagnetic ion acts as a tumbling microcrystal. Such motion causes a f l u c t u a t i n g force to act randomly on anisotropic g-tensor and hyper-finej i n t e r a c t i o n s . Thus the greater t h i s anisotropy, the great-er t;he r e l a x a t i o n e f f e c t . If the c r y s t a l f i e l d surrounding the ion i s then assumed to have a ce r t a i n symmetry, the app-ropriate spin Hamiltonian can be applied with respect to the random f l u c t u a t i n g forces. The treatment outlined by Pake (58) i s s u f f i c i e n t for this treatment, e s p e c i a l l y since only an order of magnitude agreement i s expected. S l i c h t e r (59) and Abragam (60) also give thorough discussions, as do the other references l i s t e d at the beginning of this section. A major approximation to be made i s that the micro-c r y s t a l exerts an a x i a l l y symmetric force on the paramagnetic ion (this i s somewhat j u s t i f i a b l e simply by noting that the majority of the unpaired spin density resides on the nitrogen atom, and primarily there i n the p z o r b i t a l . Thus i n r e l a t i o n to the nitrogen part of the molecule a x i a l symmetry i s a f a i r approximation), and the appropriate spin Hamiltonian i s (see Appendix 3) '• transforming to laboratory co-ordinates, and noting that fo r spin \ systems D = 0 , and defining:-d =--/3/\i] +% Ai , b = / V A i (3_33) -23-the Hamiltonian can be written (see r e f . 58 f o r a l l terms): $ = 5pH 0S^a |. 5 + / 3 (A 9 j i H 0 + [>I?X3cO619-i)Sife + V ^ s M 9 c ^ e ( l + e ^ + L^)S i i -. ! K 2 ( 3 o » l 0 - O ( l + S . + I.S + ) (3-34) which was the way McConnell f i r s t wrote the Hamiltonian. In eq (3-34) the time-dependence of the terms inGand^has been l e f t implied, rather than writing e(fc),#fc)etc.. Terms i n -volving S + and S_ w i l l give r i s e to electron s p i n - l a t t i c e r e laxation, wheras terms i n I + and I_ w i l l give nuclear re-laxation e f f e c t s . B: RANDOM CORRELATIONS The next problem i s to determine the behaviour o f $ i n a randomly f l u c t u a t i n g environment (which w i l l cause the spins to experience a modulated magnetic f i e l d ) , and to do thi s a knowledge of random functions i s necessary. This i s more f u l l y discussed i n Appendix 4. The most formidable, or perhaps uncertain, problem involves the d e f i n i t i o n of the c o r r e l a t i o n functions Gft) and cflz)(appendix 4.) where: &CX) * TJCmTl2 g(t) - Qr(o)O)tX) (3-35) and f ( x ( t ) ) i s a random function of time since x(t) i s by d e f i n i t i o n . Any convenient d e f i n i t i o n of cj(t), then, w i l l define (kt), since G(o) i s usually available i n the problem being considered — i t i s c e r t a i n l y evaluable i n this case. For mathematical ease, then, the most commonly used form of g(t) i s as follows: -24-9C0) = I , o£Z-~oa}=o (3_36) and cjitc) -where rCc i s the c o r r e l a t i o n time for the system being considered. There i s c l e a r l y a lack of p r e c i s i o n i n the d e f i n i t i o n of , and the only r e a l l i m i t i s that be small forlTl»\. In other words, the c o r r e l a t i o n must be poor (a small value of the c o r r e l a t i o n function) when the i n t e r v a l being considered deviates considerably from the true one. Appendix 5 shows how the c o r r e l a t i o n function can be derived, and Appendix 6 gives the relaxation rate 1/T^ i n terms of the c o r r e l a t i o n function already derived. S l i c h t e r (59) and Pake (58) give very complete derivations of the s p i n - l a t t i c e re-laxation, and Abragam (ref. 6 0 , p 283) also gives a good treatment. Following these references, and Appendix 6 , the general r e s u l t can be derived that: = Wfc* . t f f G C t ) it (3-37) Now the time-dependent perturbation can be written as the product of a spin operator, A Qp, and a time-dependent part, F ( t ) , that describes the v a r i a t i o n of the l a t t i c e co-ordinates with time:-$,Ct) = A^ F(t) (3-38) or, more completely, = 2, ^ Fit) (3-39) f o r each nucleus. Clearly, i f the Hamiltonian considered applicable f o r the problem is separable into a spin and l a t t i c e product, the r e l a x a t i o n effects of the perturbation can be treated. Con--25-sider, then, that: GrCtV - l|(cos9tb))|* ^p(-in/rc) (3-4o) where ^(005 9Cb)) = <w|^tb)|l?> and <g(c) = «yp(-ir|/rc) and j l t o ) =• ZZc/(\ + td^f^-) ~- see Appendix 4 so = k i < kIVIm > | £ I (b)|22V6 +^ 2J ( 3 - 4 D For the case of an aromatic I T - r a d i c a l , S=^ and i n the esr case the transitions occur between ms=+-| and m s = - | r , with Amj^O. The s p a t i a l part of the operator w i l l be a funct-ion of angle, as already described. Thus, s e l e c t i n g from eq ( 3 - 3 4 ) the term: since i t i s the most important s p i n - l a t t i c e term — the others become n e g l i g i b l e i n comparison for t h i s case. Then: G-ct) - A U ^ + K)* \ l ^ 3 > \ 2 \ ^ e ^ 9 ^ ^ ~ <30 (A^^Wo -f U R ) 2 * ^  ( 3 - 4 3 ) therefore £ T = *f%0(/^ pHo bmLf W<./(\\LOK1) ( 3 - 4 4 ) Which i s the equation to be used i n this case. C: LINEWIDTH AND THE NUCLEAR QUANTUM NUMBER Asymmetric spectra i n electron spin resonance have been studied and explained for many cases ( 4 0 , 4 l , 4 3 , 4 5 , 4 6 , 6 5 ) and f o r the purposes of this work a b r i e f d e s c r i p t i o n of one ef f e c t w i l l be given. Again, the part of the Hamiltonian to be considered dominant w i l l be the part that i s diagonal i n nuclear quantum number, as th i s term w i l l be most affected by variation s i n the magnetic environment at the nucleus (for this s i m p l i f i e d case). Complete d e t a i l s w i l l be omitted p r i n c i p --26-a l l y because the results obtained are concerned primarily with an explanation of the observed r e s u l t s , and only second-a r i l y with quantitative agreement. If a s i m p l i f i e d f i r s t - o r d e r perturbation i s consid-ered, and i t i s assumed that b4*cgpHQ (this i s not a strong assumption, since experimental work indicates that these two terms are nearly equal),and i f the diagonal term i n eq ( 3 - 3 4 ) : i s considered, then i f the microcrystal i s fixed i n space, the diagonal element of t h i s term f o r some state Iwi .tw^ corr-esponds to a l o c a l hyperfine magnetic f i e l d d i splacing the e l -ectron resonance frequency by S>v =r SOO/M Furthermore (since C\V>^ 3 [3 H 0 ) : kSto - W3 (3co$26>-l) m £ (3-45) and, as i s well known, under rapid tumbling the angular term tends to an average value of zero and the resonance i s undisplaced. Therefore, knowledge of the e f f e c t o f ^ on eq (3-45) should indicate the manner i n which the dipole-dipole coupling w i l l a f f e c t the resonance (since ^  {^>coiG-\) occurs i n the d i -OQ /j.s Xi y)C S r) polar coupling Hamiltonian, tftJi - "5^3d^n \ "" )• Narrowing arises here because of the f l u c t u a t i n g magnetic en-vironment of the ion and i f such an environment i s considered to have two values, s h i f t i n g the center of the resonance, o) 0 , by either + £ or -$ , then the l i f e t i m e of existence i n either state can be defined to be \ . The spin can therefore precess at u)0i$ with the magnetic f i e l d varying randomly at either l o c a l e . I f , however, no f l u c t u a t i o n between the environments occurred, then the decay time (T2, the Bloch transverse rela x a t i o n time, see Appendix 7) w i l l be T 2=l/£. The f l u c t u a t i o n of each precessing spin, i s , i n e f f e c t , a random walk about the spin's phase angle with respect to the r o t a t i n g frame. Then a mean-square phase difference, b ^ , can be defined ( 6 6 ) : A/* S r tOu) 2 ( 3 ^ 6 ) where n i s the number of changes of environment, and %S then measures the mean phase accumulation per second. Since Tg i s e s s e n t i a l l y the time i n which in-phase precessing spins become 1/e out of phase ( 5 8 ) , and ) must be of the order of one radian, and i f the elapsed time i s T^; So I (Ti/tc^cS)2- ( 3 - 4 7 ) and the linewidth i s equal to 1/T2 f or */S or i f Tc>/^ , 1/T 2=1/T^^. In other words, i f /<$ the environment has fluctuated before the transverse spin component has decayed i n conseq-uence of the T 2~/£ decay. Therefore ZitYfi i s the c r i t e r i o n for averaging away broadening effects (S i s the frequency measure of dipolar coupling). General theories by Kubo and Tomita ( 6 7 ) and other considerations of density matrix and rela x a t i o n treatments show that <S can be taken to be the amplitude of any randomly f l u c t u a t i n g perturbation diagonal in. m ( 5 8 ) » which averages to zero over long times and has a c o r r e l a t i o n time . This -28-cannot be a p p l i e d to l o n g Tu (a r i g i d l a t t i c e ) where the l i n e -width i s S ( T C > / & ) . O f f - d i a g o n a l random p e r t u r b a t i o n s can c o n t r i b u t e to s p i n - l a t t i c e r e l a x a t i o n , as was shown i n the previous s e c t i o n , and under c o n d i t i o n s of extreme narrowing , 1/T^ can con- . t r i b u t e a term comparable to I/T2 ( 5 8 ) to the t o t a l l i n e w i d t h because the f i n i t e s p i n s t a t e l i f e t i m e , , broadens energy-l e v e l s i n accordance wi t h T.jAE'Ma, Even i n c l u d i n g t h i s e f f e c t , however, l/T^^h i s s t i l l q u i t e v a l i d as an order of magnit-ude e x p r e s s i o n . In o r g a n i c r a d i c a l s the a v e r a g i n g away of a n i s o t r o p i c e f f e c t s i s u s u a l l y n e a r l y complete i n l o w - v i s c o s i t y s o l u t i o n s . T h i s Is because T^is s h o r t and g- and A-tensor a n i s o t r o p y i s u s u a l l y not l a r g e . Hence the i s o t r o p i c Hamiltonian: iC - SpWoSi + (3-48) i s o f t e n a p p r o x i m a t e l y v a l i d . I f i t can be shown t h a t the molecule obeys the a x i a l H a m i l t o n i a n and i f there are reasons f o r c o n s i d e r i n g g- and A-tensor a n i s o t r o p y , then the d i a g o n a l term of the a x i a l Ham-i l t o n i a n : ^(A3fo!4o-vbl2:)(3c^20-i)52: ( 3 - 4 9 ) and f o r s u f f i c i e n t l y r a p i d motion, u s i n g eq ( 3 - 4 5 ) ; S = Yzi&^Wo + bm^ftx ( 3 - 5 0 ) so \ * rt (AjpHo -+ b™x)7V- ( 3 - 5 1 ) r e p r e s e n t s the m o t i o n a l ( r e s i d u a l ) l i n e w i d t h to be ex-pected. C l e a r l y , the n u c l e a r quantum number, m-j-, w i l l de-termine the l i n e w i d t h , and q u i t e d i f f e r e n t l i n e w i d t h s are to be expected for d i f f e r e n t values of nij. In general, 1/T 2 w i l l be less for negative m-j- than p o s i t i v e . Such an observ-ation, or the opposite, should enable the manner of nuclear s p l i t t i n g to be determined for esr spectra. F i n a l l y , a more general r e s u l t (which- i s e s s e n t i a l l y the same as eq ( 3 - 5 1 ) . but can include as many terms or effects as wanted) was developed by Kivelson ( 4 5 ) : 1/T 2 = Km| + Lmj + C ( 3 - 5 2 ) where the l i n e a r term i s the intramolecular dipole i n -t e r a c t i o n just considered and i s responsible for spectral a-symmetry. For the nitrogen atom L i s negative'(68), and i n general eq ( 3 - 5 2 ) enables the sign of the s p l i t t i n g constant to be determined. Spectral narrowing or broadening due to exchange effects w i l l not be considered here, but w i l l be discussed very b r i e f l y i n the next section when the s p e c t r a l lineshapes are considered. -30-RESULTS AND DISCUSSION A: A BRIEF INTRODUCTION The spectra observed i n s u l f u r i c acid, shown i n f i g s , l a - i , were assigned to the cation of chlorpromazine (CPZ +), and the spectra shown i n f i g s . 2a-d were assigned to another oxidation product, to be discussed more f u l l y l a t e r . The t r a n s i t i o n to the spectrum exhibited i n f i g s . 2 was not expected, but adds usefu l information to the expected behav-iour of chlorpromazine -- th i s w i l l be more f u l l y discussed when the e l e c t r i c a l properties of chlorpromazine are consider-ed. This w i l l also be correlated with the biophysical and biomedical aspects of CPZ. For the majority of the discuss-ion the properties of CPZ w i l l be the main point of interest. Spectra obtained under d i f f e r e n t circumstances are shown i n f i g s . 3a-e and w i l l also be discussed l a t e r -- no discussion of the spectra of promazine hydrochloride (for the structure of t h i s , and other common phenothiazine drugs, see Appendix 9) i s given since attempts to obtain resolved and useful spectra i n various media were unsuccessful. It was hoped that some information on the esr spectrum of a symmetrical, but very s i m i l a r , " r e l a t i v e " of CPZ would add to the knowledge of CPZ i n p a r t i c u l a r , and a l l phenothiazine drugs i n general. B; THE HYPERFINE STRUCTURE OF CPZ + Analysis of the spectrum of CPZ proved d i f f i c u l t . The sixteen l i n e spectrum observed c l e a r l y consisted of many co-incident (accidentally) l i n e s , so no easy separation of - 3 1 -components was possible. Assignments were effected by exam-ina t i o n of a) the spectra of phenothiazine and some of i t s simpler derivatives ( 6 9 - 7 9 )» and b) by examination of the Htlckel spin densities at various r i n g positions. Neither a) nor b) was p a r t i c u l a r l y successful, but closer examination of the electrolysed CPZ samples ( f i g . 3 a ) combined with a com- • puter-simulated spectrum f i n a l l y r esulted i n an excellent agreement between observed and calculated spectra ( a l l esr simulations were made using a program written by C.S-. Johnson of Yale University and P.J.Black of t h i s u n i v e r s i t y ) . The s p l i t t i n g constants obtained were not e n t i r e l y expected from considerations of either the other phenothiazines, or the lim-i t e d work done on chlorpromazine already. Hox^ever, the " f i t " , suggested the following basic s p l i t t i n g constants: a^ = 6 . 7 0 0 G aj = 3.400 G (^-1) a~ = 1.620 G a^ = 1 . 6 0 0 G where a^ i s the nitrogen-l4 s p l i t t i n g constant, a)- the s p l i t t i n g due to the- CHg group attached to the nitrogen atom 2 ? (carbon number 15, or C - 1 5 ), and a and a J the s p l i t t i n g s due to r i n g protons. Some confusion exists with these constants — i n i t i a l l y , from Huckel c a l c u l a t i o n s , i t was expected that there would be four almost equivalent r i n g protons (this equiv^ alence i s not equivalence i n the symmetrical sense, but i n the sense that the spin densities are almost equal -- this means that the s p l i t t i n g constants w i l l be almost equal, i f the proton spin p o l a r i s a t i o n , or Q, constants are the same at: a l l these positions. In other words, "accidental" equiv-alence i s involved) , but the r e s u l t s indicate that only-three r i n g protons are contributing to the structure obs-erved, and these are nearly equivalent. Further, the spect-r a l f i t requires two protons to be more equivalent than the other (although for a more r e a l i s t i c f i t a l l protons should have s l i g h t l y d i f f e r e n t s p l i t t i n g c o n s t a n t s — this i s not warranted i n view of the approximate nature of much of the c a l -culation, and the term "equivalent" w i l l continue to be used to describe protons with s i m i l a r spin d e n s i t i e s , even i f there i s no s t r u c t u r a l symmetry). If two protons have a=1.62G, and one has a=1.60G, then eq. (4-1) holds. However, i f the opposite i s true then a* should be.v3.37 G. The f i n a l ass-ignment of this w i l l be made i n the next sections, a f t e r a discussion of the Huckel calculations has been made. G; HUCKEL CALCULATIONS Huckel calculations (done using a computer program written by D.Kennedy, and modified s l i g h t l y to do a McLachlan c a l c u l a t i o n also by P.Nakano, both of this university) were carri e d out on a planar structure i n i t i a l l y , but i t was l a t e r considered better to modify these calculations to include f o l d i n g about the N-S a x i s . The exact angle of f o l d i n g was determined by carrying out calculations for a series of angles and f i t t i n g the r e s u l t s to the s p l i t t i n g constants considered i n the previous section. This method was r e s t r i c t e d primarily to a®, and somewhat less to aX.t since most of the e f f e c t of s t r u c t u r a l f o l d i n g about the N-S axis should be evidenced at, or near, the nitrogen atom. Folding, or flapping, of the side--33-F ig 1a C P Z + In H 2 S C ^ g lass a t~-90°C A l l s p e c t r a t h a t f o l l o w a re a t room temperature, u n l e s s otherwise s p e c i f i e d . Spectrometer c o n d i t i o n s f o r a l l s p e c t r a were about: 3250G f i e l d w i t h 0.1 to 0 . 7 5 G mod-u l a t i o n ; 9.10 GHz microwave frequency a t a 5 m i l l i w a t t pow-er l e v e l . -34-F i g l b C P Z 4 at9°C (all spectra shown in figs 1a-i a re for l-LSQ solution) Fig 1c CPZ+ at 16°C Fig 1e CPZ+ at 50°C -38-Fig 1f CPZ+ at 75°C - 3 9 -Fig 1g CP2+ at 92°C -40-Fig 1h CPZ+ at 111 °C -41-7 5 2 G 1; CPZ+at131°C - note the change becoming apparent on the right where the temperature was drifting up towards 136°C -42-Fig2a "changed species" (CS) at 21-8°C -43-Fig 2b CS at 59°C -44-Fig 2c CS at 141°C Fig 2d low resolution spectrum of "changed" CPZ species shown in figs 2a-c Fig 3b CPZ* in cone HCI Fig 3e CPZ reacted with L, spectrum in. 50:50 acetonitriie:dimethoxyethane Q CD" - 5 1 -Q to' i — i O LU cr •• a. o o ~ r r 6.0 12.0 18,0 FIELD (&RJ5SI 24. Q ~ i 30. Fig 4a computed stick diagram for CPZ*-Fig 4b computed spectrum for CPZ+ in KSO with linewidth 1-0 G -54--55-Fig 4e computed spectrum linewidth 2-5G -56-chain was considered but should not be applicable here since the -CH hyperfine coupling constant i s known for whatever p o s i t i o n the side-chain takes, even i f the exact orientation of the sidechain remains unknown. This aspect of the structure w i l l be considered i n more d e t a i l subsequently. Thus, the 2 o f i n a l assignment of a and.a J could be made once the best set of s p l i t t i n g constants was known. As w i l l be given l a t e r i n more d e t a i l , the amount of f o l d i n g was determined to be app-roximately 3 8 degrees from planar, f o r both aromatic rings. Table I summarises the Huckel c a l c u l a t i o n r e s u l t s . The spin densities at carbons 1., 3 . 9 , and 11 are the largest and ar'e considered here to be those responsible f o r the r i n g proton hyperfine structure, and since the spin density at carbon 1 i s consistently the lowest of these four for both Huckel and McLachlan ca l c u l a t i o n s , the observed hyperfine s p l i t t i n g i s ' assigned to carbons 3 , 9 , and 11. This i s somewhat a r b i t r a r y but necessary i n order to account for the observed spectrum properly..- The spin density calculations could well be i n error on the amount of influence the chlorine atom at p o s i t i o n 2 has on either of the neighbouring protons, since no estimates of such an influence are a v a i l a b l e , and none were estimated be-cause of the lack of l i t e r a t u r e on the subject. Thus the three protons were considered responsible for the f i n a l struc-ture. It i s possible that correct Interpretation of the C l r o l e would give one of carbons- 1 and 3 a larger spin density-: than has been calculated, and therefore diminish the other so that the i n t e r p r e t a t i o n given i s e s s e n t i a l l y correct. In any - 5 7 -T A B L E i SPIN DENSITIES ATOM: N C-15 C-12 C-1'4 C-l e - 3 C-9 - c - 1 1 S 0° HMO .232 .000 . 044 . 044 . 0 5 2 . 0 6 6 . 0 6 2 . 0 5 6 .137 McL. . 3 l 4 - . 0 0 6 . 0 1 7 . 0 1 8 . 0 6 6 . 0 7 6 .071 . 0 7 1 .141 10° HMO . 2 3 9 .000 .046 .046 .046 . 0 7 1 . 062 . 0 5 5 .144 McL. • 3 3 0 -.000 . 0 1 7 .018 . 0 5 9 .083 .070 . 0 7 0 .149 HMO . 2 3 7 .000 .044 .044 .045 . 0 6 9 . 0 6 0 . 0 5 3 .141 McL. . 3 2 3 - . 0 0 6 . 0 1 6 .018 .057 .080 . 0 6 8 . 0 6 8 .146 3 0 ° . HMO . 2 7 2 .000 . 0 4 5 .046 . 040 . 0 6 7 .058 .048 .154 McL. • 395 -.000 .013 .014 . 0 5 3 .077 . 0 6 6 . 064 . 1 5 5 3 8 ° HMO .307 .000 .043 .04 3 . 0 3 6 . 0 5 6 . 0 5 3 . 0 4 3 . 1 6 0 McL. .469 -.008 . 0 0 5 . 0 0 6 . 0 5 1 . 0 6 1 . 0 5 8 . 0 5 5 . 1 5 2 40° HMO . 3 1 0 .000 . 0 4 3 .044 . 0 3 5 . 0 6 3 . 0 5 4 .042 . 1 6 2 McL. .481. - . 0 . 0 0 .005 .006 . 0 5 0 . 0 7 1 . 062 . 0 6 0 .154 45° HMO • 338 . 0 0 0 .041 .041 • 0 3 3 . 0 6 0 . 052 . 0 3 9 . 1 6 7 McL. .5^8 -.000 -.008 - . 0 0 3 . 049 . 0 6 6 . 0 6 0 . 0 5 9 .148 HMO means HULckel calculated values, McL. means McLachlan se l f - c o n s i s t e n t fieltfl c a l c u l a t i o n s . Notice the occurrence of negative spin densities i n the McLachlan calculations where the Httckel calculations indicate a zero spin density. - 5 8 -c a s e , the C l atom i s e x p e c t e d t o have c o n s i d e r a b l e i n f l u e n c e on the u n p a i r e d s p i n d e n s i t y a t the n e i g h b o u r i n g atoms. The s p i n 3 / 2 s t a t e n e c e s s a r y t o o b t a i n the f i n a l s p e c t r a l agreement cannot be a s s i g n e d t o the C l atom, s i n c e i n most a r o m a t i c s y s -tems q u a d r u p o l a r r e l a x a t i o n e f f e c t s i n s o l u t i o n a r e e x p e c t e d t o be s u f f i c i e n t t o remove any C l h y p e r f i n e s p l i t t i n g . F u r -t h e r , the p r o t o n s ( o r , r a t h e r , the carbons t h e y a r e bonded t o ) must have a s i g n i f i c a n t s p i n p o l a r i s a t i o n i n t e r a c t i o n t o a c c o u n t f o r o b s e r v a b l e s p l i t t i n g s . B e f o r e any f u r t h e r comparisons w i t h e x p e r i m e n t a l v a l u e s a r e a t t e m p t e d , a b r i e f d i s c u s s i o n on the H u c k e l c a l -c u l a t i o n s w i l l be g i v e n . The heteroatom t e c h n i q u e ( r e f . 3 2 , chaps. 4 and 5 ) was used t h r o u g h o u t t o a c c o u n t f o r a l l heteroatoms and the s i d e c h a i n . If. °<a and 0,are the Coulomb and resonance i n t e g r a l s . , f o r the benzene r i n g , then the r e l a t i o n s h i p s : can be used, where X r e p r e s e n t s the heteroatom i n q u e s t i o n . Most, i f not a l l , of the v a l u e s f o r . . h x and k c x a r e ap p r o x i m -a t e , and few a t t e m p t s have been made t o e s t a b l i s h whether unique v a l u e s of these parameters e x i s t . I t i s p o s s i b l e t o use a t e c h n i q u e ( t h e a u x i l i a r y i n d u c t i v e p a r ameter, or AIP) f o r h e t e r o a t o m s , whereby the i n d u c t i v e e f f e c t s of the h e t e r o * atoms i s a c c o u n t e d f o r . I n o t h e r words, the more e l e c t r o -n e g a t i v e heteroatom p o l a r i s e s the C-X bond, thus i n c r e a s i n g the e f f e c t i v e e l e c t r o n e g a t i v i t y a t the c a r b o n atom. F or a g i v e n bond, C^-X, i t i s p o s s i b l e t o w r i t e : - 5 9 -where cf i s the a u x i l i a r y inductive parameter, and values varying from 1 / 1 0 to 1 / 8 for & are commonly used. Since ex-periment suggests that such an inductive e f f e c t i s n e g l i g i b l e a f t e r one cT-bond ( 3 2 ) , and since much discussion on the actual effectiveness of c o e x i s t s , l i t t l e weight w i l l be assigned to i t here since computations suggest that the same r e s u l t s are achieved by modifying h i t s e l f , and including any inductive e f f e c t s i n the o r i g i n a l Huckel assumption empirically. The appropriate parameters used, therefore, were: h N = 0 . 9 5 ; h s = 1 . 5 0 ; h c l = 3.0'0» h H 3 = 0 kC-N = 1 , 0 ; kC-S = 1 ' 0 s kC-Gl = °' 6' kG-H 3 = 3 , 0 I n i t i a l l y these values were taken from the l i t e r a t u r e ( 3 2 , 7 9 ) , but h^ and k G _ C ] L were l a t e r modified to the above values. D: COMPARISON OF CALCULATED SPIN DENSITIES WITH E X P E R I M E N T . The calculated spin densities are for a model that has an angle of 104 degrees between the aromatic rings. This w i l l be further discussed i n the next section. The a c t u a l ex-perimental/computational comparison w i l l not be d i r e c t l y made on the spin densities, b\it on the s p l i t t i n g constants. The f i n a l assignment of s p l i t t i n g constants i s as follows: a^ = 6 . 7 0 G at = 3 . 4 0 G ( 4 - 4 ) a% = 1 . 6 2 G a 3 = 1 . 6 0 G 1 where a describes the -CR2 s p l i t t i n g a t the C-15 p o s i t i o n , 2 a the s p l i t t i n g due to the nearly equivalent protons a t C - 3 - 6 0 -and C - 9 , and a-^  d e s c r i b e s the s p l i t t i n g due to the pro t o n a t C-11. Splittings.-:from other protons must be assumed to be too s m a l l to be observed, or completely hidden i n the spectrum. H y p e r f i n e c o u p l i n g from the n a t u r a l abundance of S-*3 ( 0 . 7 4 $ ) i s p r o b a b l y not observable here, even i n the AlCl^/CH^NOg system, a l t h o u g h i t has been observed i n the s l i g h t l y s i m i l a r c a t i o n s of t h i a n t h r e n e and v a r i o u s s u b s t i t u t e d t h i a n t h r e n e s w i t h a S about 8 or 9 G ( 8 0 ) . Htickel c a l c u l a t i o n s made f o r the t h i a n t h r e n e s d i d not i n c l u d e the s u l f u r d - o r b i t a l s , as s i g n i f -i c a n t agreement was obtained u s i n g only the p - o r b i t a l s — f u r -t h er d i s c u s s i o n of t h i s p o i n t w i l l be g i v e n when the energy l e v e l s of CPZ a r e co n s i d e r e d . The computation of the n i t r o g e n h f s constant was made u s i n g eq ( 3 - 1 5 ) ' C o n s i d e r a t i o n of the Q values to be used H was determined mainly by Q Q Q , the pro t o n sigma-pi parameter. The many l i t e r a t u r e v a l u e s of t h i s parameter have a l l been a s s i g n e d to gi v e good experimental agreement with c a l c u l a t i o n s . Consequently, v a l u e s of Q r a n g i n g from - 2 2 . 5 ( f o r the benzene anion) to - 3 0 . 0 have been used. R e c e n t l y , C a r t e r and Vincow (81), working on the benzene c a t i o n , have suggested that f o r c a t i o n s Q Q C should be higher than f o r a n i o n s . Using the MO and McLachlan s p i n d e n s i t i e s showed t h a t i n t h i s experiment Q c c = - 2 9 . 2 G . T h i s i s a good r e s u l t f o r a t y p i c a l aromatic c a t i o n , and i s c l o s e to the c a l c u l a t e d value of -28 G ( f o r a n a n i o n ) , but s l i g h t l y h igher as p r e d i c t e d by C a r t e r and V i n -cow. U s i n g t h i s v alue to f i n d from the data presented by Henning ( 2 1 ) suggested t h a t Q J J N = 2 0 . 0 G and Q Q C = 7..0 G (by - 6 1 -a polynomial f i t routine, MPOLY, available from^the U.B.C. computing center). Thus: or 6 . 7 0 = ( 2 0 . 0 ) ((j?) + ( 7 . 0 ) ( 0 . 0 8 6 ) ( 4 - 6 ) therefore @J = 0 . 3 0 4 8 This value i s i n excellent agreement' with the HIv30 value of O . 3 0 6 8 , but i n poor agreement with the McLachlan value of 0 . 4 6 9 . Further, throughout the v a r i a t i o n i n ©, the values of Qp( and ^  remain almost constant, ranging from 0.041 to 0.046 (see Table I ) . Thus a deviation from any molecular shape that amounts to only 1 0 . 0 0 2 0 i n the unpaired spin density i n d i c -ates that the bridging carbons C-12 and C - l 4 are not p a r t i c -u l a r l y s e n s i t i v e to f o l d i n g about the N-S axis. For the sake of comparison the values of the r i n g proton s p l i t t i n g constants are: HMO: a 2 = 1 . 6 3 5 G & J = 1 . 1 7 G ( 4 - 7 ) McLachlan: a 2 = 1 . 7 5 2 G a^ = 1 . 6 0 6 G ( 4 - 8 ) Clearly a combination of the two theories gives almost perfect agreement with experiment -- no attempt to further examine these constants w i l l be made, however. For consist-ency with the r e s t of the calculations the HMO spin densities w i l l be used. A detailed study on a series of rela t e d cations would give better information on the nature of the various spin p o l a r i s a t i o n constants , and a lea s t squares analysis s i m i l a r to that made by Henning f o r a series of N-heterocyclic aromatic anions could then be used to give more reproducible values. The values used here, Q Q Q = -29 . 2 G , Q ^ J J = 2 0 . 0 G , and Q Q C = ? . 0 G are very consistent for cations, i f allowance i s made for the f a c t that the nitrogen constants perhaps do not vary as much' for a cation as for an anion, with respect to the proton spin p o l a r i s a t i o n constants. No precedent exists for this hypothesis, but the computer c a l c u l a t i o n mentioned e a r l i e r suggests that the value of Q = - 2 4 . 2 G that gives the used values of for anions ( 2 1 ) would be between - 2 8 and -30 G for the cation, assuming that the Q N stay the same. This, then, was the assumption used and the amount of f o l d i n g about the N-S axis was determined from the best f i t of the experimental s p l i t t i n g constants to the appropriate set of spin d e n s i t i e s . Using such a ca l c u l a t i o n a value of 1 0 4 deg-rees f o r the angle between the aromatic rings was established. Other values of 9 cannot give agreement of calculated s p l i t t i n g constants, using the spin p o l a r i s a t i o n constants just described, with both a N and the proton s p l i t t i n g constants. i C a l c u l a t i o n of a , the s p l i t t i n g due to the protons QX carbon 15» the carbon attached to the nitrogen atom, was based on some s p e c i a l assumptions. In c a l c u l a t i n g the HMO par-ameters the -CH2CH2CH2N(CH3)2 sidechain attached to the n i t r o -gen atom was assumed to be e f f e c t i v e l y a -CH^ group, and the part was treated as a heteroatom with the appropriate par-ameters (as already given). Of course, when the spin density at this heteroatom i s considered, i t must be remembered that-the r e a l s i t u a t i o n i s a -CH2 group. Using these approximations an HMO c a l c u l a t i o n can be used to investigate the importance of h y p e r c o n j u g a t i o n over s p i n p o l a r i s a t i o n , or v i c e - v e r s a . Use of a simple s p i n p o l a r i s a t i o n treatment i s not warranted i n t h i s case, s i n c e attempts to f i n d a reasonable Q f o r the methyl s p i n p o l a r i s a t i o n f a i l e d , and use of e x i s t i n g a c c e p t e d v a l u e s f o r methyl and methylene groups (Q=-3.09G, r e f . 2 7 ) f a i l e d to g i v e reasonable r e s u l t s . I t i s h i g h l y prob-a b l e , t h e r e f o r e , t h a t h y p e r c o n j u g a t i o n i s much more important i n t h i s case than i s s p i n p o l a r i s a t i o n . In f a c t , i t seems l i k e l y t h a t there i s no s p i n p o l a r i s a t i o n c o n t r i b u t i o n . T h i s w i l l be d i s c u s s e d now. The h y p e r c o n j u g a t i o n i n CPZ + should a r i s e from i n t e r -a c t i o n of the TT-system about the n i t r o g e n atom wi t h the G-H sigma bonds -- f o r e x c i t e d s t a t e s and carbonium ions hyper-c o n j u g a t i o n has been shown to be q u i t e important ( 3 2 ) , a l t h o u g h i t may not be f o r n e u t r a l molecules. Thus, i n c o n s i d e r i n g CPZ +, Levy's equation, ( 3-20), f o r the s p l i t t i n g constant due to h y p e r c o n j u g a t i o n w i t h a methyl group was.used, and H M O . v a l u e s of the c o e f f i c i e n t s were employed. So: a C M 3 = 319.8C„ -+tt.nCt +101.1 CttCt * 3.997C MC t/ * 0.973 C c ^' - 3. 0<K<? ( 4 - 9 ) and . d c H 3 = ?>.Zll - (4-10) p In eq (4-10) the s p i n p o l a r i s a t i o n c o n t r i b u t i o n , - 3.09 Cr/ i s d e l i b e r a t e l y l e f t u n i n c l u d e d . T h i s term, when evaluated, i s -0.948G. Thus a C H ^ i s reduced to 2.264 i f s p i n p o l a r i s -a t i o n i s i n c l u d e d . S i n c e a,-,jj i s to be equated with a to o b t a i n an approximate measure of the v a l i d i t y of eq ( 4 - 9 ) here, the v a l u e a ^ j j ^ = 3.212 G g i v e s very good agreement, con--64-si d e r i n g the approximations used, wheras the value SLQRJ = 2.264 i s u n r e a l i s t i c a l l y low. Thus, by neglecting the effects of spin p o l a r i s a t i o n i n t e r a c t i o n at the methyl and nitrogen pos-i t i o n s much better agreement beti^een experiment and c a l c u l a t -ion i s obtained. Condidering that the molecule under con-s i d e r a t i o n i s a cation and not a neutral molecule, hypercon-jugation may well be a dominant e f f e c t because of the tendency of a l i p h a t i c groups (especially the methyl group) to donate electrons — the posit i v e nature of the nitrogen atom should enhance this e f f e c t and thus increase the hyperconjugative i n t e r a c t i o n . This i s most c e r t a i n l y the experimental r e s u l t here, and for cations i n general a similar mechanism and res-u l t may be expected. Levy's calculations and comparisons for a series of anions indicated that the effects of hyperconjug-a t i o n had been over-emphasised consistently. This i s i n ex-c e l l e n t agreement with the mechanism just postulated -- i n the case of the anion the negative charge would tend to opp-ose the flow of electrons from the methyl group (or methylene group). Thus Levy's calculations might be expected to be modified s l i g h t l y for a p p l i c a t i o n to esr spectra, with the treatment he gives (eq_s 3-18 and 3-20) being more applicable to neutral species i n the sense that the hyperconjugation/ (lot spin p o l a r i s a t i o n balance isyjthe same for the r a d i c a l as for the neutral analogue. In any event, the experimental f a c t here i s that CPZ + interacts with the aromatic r i n g almost ex-c l u s i v e l y by means of hyperconjugation mechanisms which are enhanced by the cati o n i c nature of the aromatic system. - 6 5 -E:- THE "ANOMALOUS" SPECTRUM The spectra shown i n figure 2 are termed "anomal-ous" because they were t o t a l l y unexpected. When the temper-ature study was being made on CPZ +, a l i n e a r r e l a t i o n s h i p be-tween linewidth and temperature was looked-for. However, at a temperature of 125°C the spectrum began to change, and by l J 6 0'C the change was complete. Subsequently i t was found that at room pressure (the esr sample was evacuated to about 1 0~^torr) prolonged heating at 100°C produced the same r e s u l t . This heating had to be greater than 1 2 0 hours, since less heating resulted simply i n a s l i g h t l y broadened, but normal spectrum — heating to around 1 3 0 ° i n the esr samples produced a change i n 10 minutes. The change observed was not r e v e r s i b l e upon cooling, although d i l u t i o n with water sometimes restored the red colour (from a c h a r a c t e r i s t i c tan of the changed specie) but destroyed the spectrum. The red colour generally d i s -appeared quite quickly. Analysis of this spectrum proved d i f f -i c u l t , but some approximate values of s p l i t t i n g constants were obtained that generated calculated spectra of si m i l a r shape. Lack of r e s o l u t i o n prevented a more serious assignment. The 2 3 1 constants obtained were a and a. remaining e s s e n t i a l l y un-changed, varying from 6 . 6 to 6 ,8G, which brackets the value obtained for CPZ ( assuming this "new" species i s not CPZ ). 1 The change i n a , the s p l i t t i n g due to the group attached to the nitrogen atom, i s quite i n t e r e s t i n g , however. Analysis indicated that a =6.8G, and the s p l i t t i n g i s described by a doublet of equal I n t e n s i t i e s . This, of course, i s character-- 6 6 -i s t i c of a spin ^ system -- i n this case undoubtedly a proton. To account for these r e s u l t s s t r u c t u r a l l y i s not s t r a i g h t -forward, however. It i s immediately obvious that the s p l i t t i n g of the + group attached to the nitrogen atom i n CPZ i s one half that of the same, or comparable, s p l i t t i n g f o r the new species. This could be coincidence, or could r e f l e c t a s i t u a t i o n i n which a proton has been l o s t from the -CHg group i n CPZ . If the spin density remains e s s e n t i a l l y unchanged at a l l other positions then a s p l i t t i n g constant for the remaining proton would be twice as large as for the previous two protons, but only two esr l e v e l s would r e s u l t from the single proton, changing the spectrum r a d i c a l l y . In this p a r t i c u l a r case, also, there i s the large complication that the new s p l i t t i n g i s numerically the same as f o r the nitrogen atom, so an i n -t e r a c t i o n of an unpaired spin with these two centers would r e s u l t i n a quartet of i n t e n s i t y r a t i o 1 : 2 : 2 : 1 , and sep-a r a t i o n of peaks of 6.8G. Further i n t e r a c t i o n with the spin density at the r i n g protons would add structure around and between these peaks, as observed. Further, taking into account the broadening of the h i g h - f i e l d l i n e s , and checking with a low-resolution spectrum ( f i g . 2 d ) , the experimental i n t e n s i t y r a t i o s are 1 : 2 : 2 . 3 : 0 . 9 , which is i n excellent agree-ment with the calculated values for the case of a l o s t proton. A c a l c u l a t i o n using the same equation as before, eq ( 4 - 9 ) , was made (modifying the c o e f f i c i e n t s to account for the loss of one proton), y i e l d i n g : -67-aCH = 7-24 G And a spin p o l a r i s a t i o n treatment for a single proton attached to the nitrogen atom d i r e c t l y gives: aNH = 7»53 G The l a t t e r c a l c u l a t i o n was made to include the p o s s i b i l i t y that the temperature change was, i n e f f e c t , loss of the side-chain and protonation of the nitrogen atom, presumably from the solvent. This consideration was made because of the re-semblance of the spectrum to an unresolved phenothiazine spect-rum, which also has a 1 : 2 : 2 : 1 i n t e n s i t y d i s t r i b u t i o n . The two calculations offer no d e f i n i t i v e answer to the s t r u c t u r a l change problem, although the hyperconjugation c a l c u l a t i o n i s s l i g h t l y better. Energetically, i t might be expected that loss of a proton to the solvent at temperatures greater than 125°G ( i n concentrated s u l f u r i c acid) i s the more l i k e l y answer. The report ( 9 3 ) that a second electron may be re-leased from the side-chain nitrogen, giving an intense blue d i - c a t i o n could not be substantiated either experimentally or t h e o r e t i c a l l y (by simulating a spectrum to f i t the observed and predicted r e s u l t s ) . Further, i t was not possible to i n -clude the side-chain nitrogen atom i n the spectral structure analysis for either mono- or d i - c a t i o n i c structures. No blue s o l u t i o n could be obtained i n solut i o n (although i t i s r e a d i l y obtained i n the s o l i d (93))» and the excellent agreement of k the given r e s u l t s with calculations suggests that the side-chain i s not involved i n the esr considerations beyond the f i r s t carbon atom -- this i s the usual aromatic s i t u a t i o n . -68-F: THE CORRELATION TIME The asymmetry exhibited i n the spectra can be read*-i l y interpreted i n terms of anisotropy i n the g- and A-ten-sors, which leads to a l i n e a r dependence of linewidth on the nuclear quantum number, mj. Aromatic molecules generally do not e x h i b i t large anisotropics (except for s p e c i a l cases l i k e nitrenes, where there can be appreciable s p i n - o r b i t coupling i n c e r t a i n states; i . e . i n the t r i p l e t s tate), and i n chlor-promazine the p r i n c i p a l anisotropy i s expected to a r i s e from the nitrogen atom. I t was found impractical to try to estim-ate the anisotropy of the other contributing atoms and Inter-actions, so only the parameters of the nitrogen atom are to be considered. F i r s t , however, a-discussion of the c o r r e l -a t i o n time i s appropriate. I t can be seen from f i g s . 1 that, at higher tempera-tures, the spectra become more symmetrical and better r e s o l v -ed. This is due to the lowering of the c o r r e l a t i o n time, Te , as the v i s c o s i t y of the solvent decreases. The determination of the c o r r e l a t i o n time i s quite a d i f f i c u l t problem, and i s not usually attempted exactly for magnetic resonance experi-ments. The more elaborate theories of r e l a x a t i o n i n l i q u i d s require only an estimate of the c o r r e l a t i o n time, often using precedents calculated i n s e l f - d i f f u s i o n work, and thus only order of magnitude agreement with whatever the true c o r r e l -a t i o n time i s required. An exact c a l c u l a t i o n would, of course, be preferable. Most of those using c o r r e l a t i o n times approximate -69-them using the Stokes expression for the viscous force on a sphere, which, combined with the theory of Brownian motion i n l i q u i d s (55,58,82) gives: where a i s the Stokes' radius of the r o t a t i n g p a r t i c l e , and y the s o l u t i o n v i s c o s i t y . E d e l s t e i n et a l (53) found that values of a calculated from eq (4-11) were considerably small-er than the molecular dimensions would suggest. Further, O'Reilly (82) found, using quadrupolar r e l a x a t i o n times obtain-ed from the nmr spectra of a serise of Cl-containing molecules, that % calculated from eq (4-11) were an order of magnitude too" long. However, O'Reilly used the molecular volume for i n eq (4-11) and E d e l s t e i n found that this was not always v a l i d . Any comparison of c o r r e l a t i o n times that compare some experi-mentally-determined value of ^ with that determined from eq (4-11) must specify what a i s . This i s not straightforward, and i s the reason many authors use eq (4-11), and i f unexpected dev-iat i o n s occur, modify their parameters (53,58). Considering the physical basis for eq (4-11), however, leads to the concl-usion that the Debye r e l a t i o n s h i p was derived for a spherical macroscopic object, and incorporated into the Stokes express-ion, and i s not d i r e c t l y applicable to the question of how a molecule or miorocrystal moves, subject to Brownian motion, through a s o l u t i o n of known v i s c o s i t y . Attempts to calculate a more precise value of for the CPZ case were made using an expression for the free volume of a rigid-sphere molecule r o t a t i n g randomly i n a solution. Buehler et a l (85) give: - 7 0 -V; = [ 2&/3 <? - Vs c*<r - if c^-c*)& + 8/3 S i n " 1 u t + % - Sin~'uJ - si/f' b) (4-12) where c=a//2, a=i/2r 0 , r Q being the c o l l i s i o n sphere r a d i u s , and: rvi = Y - Y t Of ~CZ)/*fl U s i n g a p p r o p r i a t e values of r Q (assumed here to be about o the same as the dimensions of chlorpromazine, or about 10A), v a l u e s of v ^ r a n g e d from 5 to 7 angstroms. Such val u e s were s u r p r i s i n g l y c l o s e to those expected (4 to 6 2.), i n c o n s i d e r -a t i o n of the remarks of E d e l s t e i n e t a l ( 5 3 ) * Thus, s u b s t i t -u t i o n of v f f o r a i n eq (4-11) was c o n s i d e r e d a p p r o p r i a t e , and v a l u e s of T£ obtained ranged from 2 . 6 5 x 1 0 to 1.42x10*" seconds, a t room temperatures (2l-22°C). T h i s seems to be a reasonable r e s u l t , e s p e c i a l l y when compared to the value -11 f o r H 20 of 3 x 1 0 sees. ( r e f . 5 8 , p l l O ) , which has a s m a l l e r r a d i u s and v i s c o s i t y than chlorpromazine -- s m a l l enough to account f o r the 1/1000 to 1/100 d i f f e r e n c e i n c o r r e l a t i o n time. T h i s r a i s e s the f a c t t h a t i t should be p o s s i b l e to determine values of Xt q u i t e r e a s o n a b l y i f a standard, such as water, was a c c u r a t e l y known. Comparisons of dimensions and v i s c o s i t y then should enable c o n s i s t e n t values of ^ to be obtained. A thermodynamically obtained value of the c o r r e l -- 7 1 -a t i o n time i s a v a i l a b l e , and Watts et a l ( 8 6 ) have derived the u s e f u l r e l a t i o n s h i p : P'j/kT = constant ( 4 - 1 3 ) where D i s the d i f f u s i o n c o e f f i c i e n t for the l i q u i d . Then using ( 8 3 ) : (f _ j£ i * / ^ « p ( - ^ ) . x p ( - « E / i e T ) . (4-1*) where/\E i s the b a r r i e r height for d i f f u s i o n , w i s the 2 work of formation of a hole, and d the mean-square distahbe between l a t t i c e s i t e s (w a r i s e s from considering that a mol-ecule p a r t i c i p a t i n g i n the random motion of a l i q u i d can be considered trapped i n a hole and subject to Brownian c o l l i s -ions with other holes and p a r t i c l e s ) . The use of eq ( 4 - l 4 ) i s not warranted here, since an estimation for w i s not av-a i l a b l e f o r chlorpromazine. However, i f w i s known, an ex-pression for t^. can be formulated: = (^fD) 4 * f ( -w/ teT ) ( 4 - 1 5 ) which shows that exact values of ^ are by no means un-a v a i l a b l e for the r i g h t systems. A d e t a i l e d study on l i n e -width variations would then not be subject to discrepancies at the solute/solvent l e v e l where approximations are often made, and where, perhaps, fewest approximations are desirable. G: LINEWIDTH VARIATION To apply any knowledge of the c o r r e l a t i o n function and c o r r e l a t i o n time to a r a d i c a l tumbling randomly i n a -vis-cous solvent, some.knowledge of the anisotropy i n the g- and A-tensors i s necessary. Such information i s a v a i l a b l e often from glass spectra or spectra of randomly-oriented s o l i d s ( 8 5 ) . - 7 2 -The spectrum of CPZ + i n a glassy matrix of H2S0i,, a t - 9 0°G gave some i n d i c a t i o n of A "by measurement of the s p l i t t i n g of the outermost peaks ( f i g l a ) . Further analysis of this spectrum was accomplished by analysis of other nitrogen-containing aromatics, as outlined by McConnell ( 7 D « + McConnell studied the i n t e r c a l a t i o n of CPZ i n D N A helices ( 7 1 ) where the heli x axis i s held p a r a l l e l and per-pendicular to the applied f i e l d . CPZ intercalates such that the aromatic plane (probably the expected axis i f the mole-cule were planar) i s perpendicular to the h e l i c a l axis. Thus an esr study of the orientation i s r e l a t i v e l y simple. The spectra obtained (very similar to the one obtained here) were not considered good enough to measure the A and g tensor e l -ements, but i^re used as a check of estimated parameters. The estimated parameters are obtained by considering a similar molecule whose A and g parameters are known. Then a r e l a t i o n -ship can be proposed: A ^ / a = constant ( 4 - l 6 ) for a p a r t i c u l a r nitrogen-containing similar molecule, and where a i s the i s o t r o p i c s p l i t t i n g observed i n solution, and AJL i s the A component i n the required d i r e c t i o n i (with respect to the symmetry a x i s ) . Using this leads to a value for the constant of 2 . 3 5 and 0 . 3 3 for the perpendicular and p a r a l l e l d i r e c t i o n s , respectively, of d i - t - b u t y l nitroxide. Thus, for chlorpromazine: 2 . 3 5 = A , / 6 . 7 0 , A , . = 1 5 . 8 G ( 4 - 1 7 ) 0 . 3 3 = A f t / 6 . 7 0 , A J J = 2 . 2 G - 7 3 -The measured i s o t r o p i c g-value for several spectra was remarkably constant, and checking with a DPPH i n t e r n a l stand-ard (contained i n a c a p i l l a r y mounted inside the sample tube) gave the same r e s u l t of g= 2 . 0 0 6 2 6 . McConnell's g | | = 2 . 0 0 3 , and the g|| estimated from the glass spectrum obtained, here, g|j=2.002 agree well, so the r e l a t i o n s h i p : g = l / 3(g u + 2 g j ) can be solved for gj^. This yields gj_ = 2 . 0 0 7 9 l which i s s l i g h t l y higher than McConnell's estimate of 2 . 0 0 6 . Thus i t should be f e a s i b l e , using correct values of 4 g=| g^-gj, %>. and temperature to apply eq ( 3 - 5 D and obtain an estimate of the linewidth v a r i a t i o n to be expected. The calculations made e s s e n t i a l l y ignored any anisotropy i n the proton hyperfine and Zeeman interactions, and thus considered only the nitrogen atom as the contributing factor. Further, c o n s i d e r i n g . a l l the approximations made, only order of magnitude agreement was ex-pected, and so the neglect of proton hyperfine anisotropy fo r this particular,:. case, i s probably more than within the o v e r a l l experimental uncertainty. Table II shows the r e s u l t s of the ca l c u l a t i o n s , and as can be seen, the asymmetric spectra of CPZ + i n H 2 SO^ at temperatures where the v i s c o s i t y i s s t i l l high (see Appendix 8 for the v i s c o s i t i e s of s u l f u r i c acid) i s well explained. As the v i s c o s i t y decreases, the shorter corr-e l a t i o n time shows that, under the experimental conditions used, the h i g h - f i e l d broadening should vanish, leaving an e s s e n t i a l l y symmetrical spectrum (of course, under conditions such that the measurable linewidth i s of the order of m i l l i -- 7 4 -o TABLE II LINEWIDTHS CALCULATED FOR a=6.5A temperature nitrogen spin state +1 0 -1 2 8 2 °K . 1 6 2 G 1 . 4 7 G 4 . 1 G 2 8 9 . 1 2 3 1 . 1 2 3 . 1 2 9 5 . 0 9 . 80 2 . 2 3 3 0 4 . 0 6 .64 1 . 5 0 3 2 3 .04 . 3 2 . 8 3 5 3 3 3 . 0 2 2 . 2 5 . 5 8 3 5 0 . 0 2 . 1 7 5 .48 3 6 0 . 0 1 6 .142 • . 3 9 3 8 5 . 0 1 2 . 1 0 . 2 9 3 9 5 . 0 1 0 • .09 . 2 5 5 3 9 6 . 0 1 0 . 0 8 8 . 2 5 0 4 0 9 . 0 0 8 .08 . 2 2 These numbers are for only a diagonal contribution from the spin Hamiltonian, a T-j term may contribute as much as the above T 2 values ' i f the c o r r e l a t i o n time i s of the r i g h t s i z e . Also neglected are any exchange effects on hyperfine broadening, and any other terms i n the Hamiltonian that might give r i s e to broadening. -75-gauss, broadening effects would be more pronounced, and s l i g h t differences more e a s i l y seen). Attainment of the 16 l i n e spectrum shown i n f i g . 1 allowed better s p l i t t i n g constant measurements to be made, also, which further showed that using lOOKHz modulation, where the modulation-broadening amounts to about ? 0 milligauss, even i n the absense of any other broadening e f f e c t , very l i t t l e further r e s o l u t i o n i s l i k e l y since the l 6 l i n e s are coincident to the extent that only 1 0 0 milligauss orpso separarate the coincident l i n e s . Examination of the calculated linewidths shows that best r e s u l t s seem to occur for a, the Stokes' radius, of the order 6 . 5 angstroms. Thus i t i s tempting to see i f the app-ropriate c o r r e l a t i o n time for this value of a, at some set temperature, would be useful i n c a l c u l a t i n g a value for the s p i n - l a t t i c e r e l a x a t i o n time, T-j_. Kivelson ( 4 5 ) derived an expression for T^ which showed that T^ i s dependent on the same mj and m| terms as was T 2. At the l i m i t s of high temp-erature and low concentration, where T^* T2 ( 4 5,8?), a per-f e c t l y symmetrical spectrum should a r i s e . In this experiment i t appears that this condition may have been approached. If the dipole-dipole i n t e r a c t i o n i s dominant (for which an ex-pression for T^ was calculated i n section I I I ) , then eq ( 3 - 4 4 ) i s applicable. This r e s u l t s i n the values of T^ shown i n Table I I I . Stephen and Fraenkel (88) determined l/T-^ for H 2 0 - 4 - 1 to be of the order of 2 x 1 0 sec", where u^T^te \ i s assumed. — 8 Q In this experiment 7 ^ 1 0 " and C 0 ~ 3 x l 0 ^ , so w t c i s c e r t a i n l y not applicable. At room temperature, inhere the v i s c o s i t y of s u l --76-TABLS III SOME REPRESENTATIVE T. TIMES POR VARIOUS VALUES OF a, AND NITROGEN SPIN STATE M =0 a 6,00 8. 6 . 50 R 7 . 0 0 A5 temperature T^ (sees) T^(seos) T 1(sees) 2 82 °X ,171xl0" 2 . 2 l 7 x l 0 " 2 . 2 7 1 x l 0 ~ 2 289 . 1 3 0 " . 1 6 5 » . 2 0 6 " 2 9 5 • 9 3 5 x l 0 " 3 .119 " .148 » 3 0 4 . 6 3 0 » .800x10-3 . 9 9 9 x 1 0 ~ 3 3 2 3 . 1 2 5 » . 440 11 .549 " 3 3 3 . 2 3 7 " . 3 0 1 " .375 " 3 5 0 .196 " .248 " . 3 0 8 »• 3 6 0 . 1 6 9 1 1 .213 . 2 6 5 " 3 8 5 .117 " .147 " .182 " 395 . 1 0 6 " . 1 3 3 " .164 » 3 9 6 . 1 0 3 " .128 " .159 " 409 . 088 » . 1 0 9 1 1 .135 " These values are calculated somewhat approximately from the s p i n , l a t t i c e term of the a x i a l spin Ham-i l t o n i a n that i s considered most important to mod-ul a t i o n of the nuclear and spin moments by l a t t i c e f luctuations or random molecular motions. f u r i c a c i d i s about 40cp., Hyde and Brown (87) f i n d about 1.4x10"^. I t would appear, then, t h a t t e t r a c e n e (the com-pound s t u d i e d by Hyde and Brown) and chlorpromazine, which have a s i m i l a r s t r u c t u r e , e x h i b i t a s i m i l a r s p i n - l a t t i c e r e -l a x a t i o n p r o c e s s , s i n c e the experimental T^'s here are around 10 r sees.. For t e t r a c e n e , i t was found e x p e r i m e n t a l l y t h a t T i o ^ f c e O r ^ ' and s i n c e l / T j * T^/^^j—see eq (3-44) — u>Xc»I f o r the ex p e r i m e n t a l c o n d i t i o n s used. T h i s i s the same ex-p e r i m e n t a l s i t u a t i o n here, but experimental v a l u e s of T^ were not measured so no check on the dependence of the s p i n - l a t t i c e r e l a x a t i o n time on the c o r r e l a t i o n time was p o s s i b l e . The theory used to d e r i v e T^, and thus eq (3-44), p r e d i c t a l i n e a r dependence, and t h i s i s what i s found by Hyde and Brown. The importance of s p i n - l a t t i c e r e l a x a t i o n processes i n dete r m i n i n g the observed l i n e w i d t h cannot be q u a n t i t a t i v e l y determined without an experimental measurement of T^, but i n t h i s case the l i n e w i d t h may be more n e a r l y : 1/T| =. 1/T 2 + 1/TJL' as suggested by McConnell (40). I f t h i s Is the case, then the c a l c u l a t e d l i n e w i d t h s , T2^", may be as much as one-h a l f times too s m a l l f o r the case to 2-^ 2^ | , Such i s the s i t -u a t i o n f o r t e t r a c e n e i n s u l f u r i c a c i d (87). Thus the r a t h e r s m a l l v a l u e s of 1/T 2 shown i n Table I I are probably too s m a l l because of the n e g l e c t of the s p i n - l a t t i c e r e l a x a t i o n . The T^ e f f e c t s can be added i n , but s i n c e the exact amount of T^ c o n t r i b u t i o n to the r e l a x a t i o n process i s not known, and s i n c e exchange broadening i s a l s o l i k e l y p r e s e n t ( t h i s w i l l be d i s --78-cussed s h o r t l y ) , only approximate values could be given, and these would therefore have diminishing significance with r e-spect to the approximations made to estimate the T 2 values. To examine the linewidths a b i t further; i t i s poss-i b l e to express the o v e r a l l linewidth as: 1/T2 = Km2 + Lmj + C (4-18) Now, an asymmetric spectrum allows an estimation of the sign of the s p l i t t i n g constant to be made. In this case, as-suming the nitrogen to be responsible for the asymmetric shape observed, the sign of the nitrogen hyperfine coupling constant, a N , can be determined i f L i s known (since only the li n e a r term i n eq (4-18) w i l l give an asymmetric contribution). 1 3 3 3 The sign of the s p l i t t i n g constants for C J and S (89 , 9 0 ) have been determined i n this fashion. In this instance, and L can be written i n Kivelson's notation ( 4 5,89): where <£l * & < 0 I ( 3co^9: - l ) / f * I 0 > ( 4 - 2 0 ) u - 4 ^ <ol s»w 2 0:^p(^(/t ) / r 3lo> ( 4 - 2 1 ) and where ^ 0 | . . l c ^ i s the expectation value for the op-erator i n question for the molecular ground state. Since, + for CPZ , g x = g y f s i m p l i f i c a t i o n i s possible, and since for n i t r o g e n - l 4 : %sps3a^  V'cil - 38.1 HWt from reference 89, a^ must be posit i v e and thus L must be negative since g z i s greater than both g x and g . Since experimentally the broadest lines appear on the h i g h - f i e l d - 7 9 -side of the spectrum, i t i s expected that l i n e s with p o s i t i v e mj w i l l be narrowest, l i n e s with negative mj broadest. The h i g h - f i e l d l i n e s therefore are for negative mj and thus the s p l i t t i n g constant a^ i s p o s i t i v e . This i s the accepted re-s u l t ( 2 0 , 2 1 ) . If i t were possible to assign a linex\ridth and pos-i t i o n to every l i n e i n the spectrum, i t would be plausible to attempt an analysis of the r e l a x a t i o n effects of a l l the par-t i c i p a t i n g atoms, and thus determine i f there i s any anisotropy due to the proton hyperfine i n t e r a c t i o n . In other words, the resultant l i n e s would have measurable linewidths and the contributions from various methylene protons' spin states and r i n g protons' spin states to the linewidth could be sub-tracted out of the resultant linewidth to give the linewidth due to the nitrogen interactions with the unpaired spin. The r e s u l t should be comparable to the linewidths calculated for this spectrum (GPZ + being considered). This i s a q u a l i t a t i v e analysis, but such an analysis could be useful. A modification of this treatment to a set of coincident l i n e s i s possible, whereby the s t i c k diagram f o r the spectrum ( f i g 4 a here) could be used to determine the i n d i v i d u a l contributions to each observed l i n e . This was done i n this case, giving a set of 16 equations that could be solved for the expected l i n e -width contributions for the various nitrogen spin states. In other words, equations l i k e the following can be obtained: - ( 5 t £ * + < O l ( 4 - 2 2 ) -80-and L i s the l i n e w i d t h of the p a r t i c u l a r r e s u l t a n t l i n e , b the c o n t r i b u t i o n to the l i n e w i d t h from the I st a t e of the'methylene proton s p l i t t i n g , and <^ ol or<i| the appropriate c o n t r i b u t i o n from the n i t r o g e n s p l i t t i n g . The p a r t i c u l a r value measured i n eq (4-22) gives an average r e s u l t of 0 . 5 G . The s i g n i f i c a n c e here i s that the number i s always p o s i t i v e f o r a l l the sp e c t r a t h a t were a c c u r a t e l y c a l i b r a t e d . The o r i g i n a l assumption that the n i t r o g e n nucleus c o n t r i b u t e s most to the an i s o t r o p y broadening i s s u b s t a n t i a l l y c o r r e c t . Attempts to f i n d s i m i l a r c o r r e l a t i o n s f o r the proton s p l i t t i n g s r e s u l t e d i n negative r e s u l t s -- there was no c o n s i s t e n t value f o r any s p l i t t i n g c o n t r i b u t i o n to the broadening f o r the spectra con-s i d e r e d , and the numbers themselves were s m a l l . Thus the e f f -e c t i v e c o n t r i b u t i o n s are p r a c t i c a l l y zero, as expected. Only the signs of these numbers can be considered u s e f u l , the mag-nitude c a l c u l a t e d i s not n e c e s s a r i l y r e p r e s e n t a t i v e of true l i n e w i d t h d i f f e r e n c e s , as w i l l be explained s h o r t l y . I t i s p o s s i b l e , then, u s i n g t h i s r e l a t i v e l y e m p i r i c a l method to ex-amine c e r t a i n l i n e s , to a s s i g n the s i g n of the s p l i t t i n g con-st a n t s i f the parameters of eq (4-18) are known. U s u a l l y , how-ever, only a v e r i f i c a t i o n of the s i g n can be made, since the p a r t i c u l a r s t a t e c o n t r i b u t i n g to the s p l i t t i n g must be known before the d i f f e r e n c e s expressed i n equations l i k e eq (4-22) can be i n t e r p r e t e d p r o p e r l y . The q u a n t i t a t i v e aspects of eq (4-22) cannot be em-phasised because the r e s u l t a n t l i n e s do not n e c e s s a r i l y have L o r e n t z i a n shape. Theories used to d e r i v e the r e l a x a t i o n times--81-have assumed t h i s shape f o r the s p e c t r a i n s o l u t i o n , but the r e s u l t a n t of a s e r i e s of n e a r l y c o i n c i d e n t l i n e s , w h i c h may be i n d i v i d u a l l y L o r e n t z i a n i n shape, i s not n e c e s s a r i l y L o r e n t z -i a n . The B l o c h f o r m u l a t i o n of r e l a x a t i o n t i m e s p r e d i c t s a L o r e n t z i a n l i n e s h a p e f o r u n s a t u r a t e d resonance c o n d i t i o n s , but o n l y f o r w e l l - s e p a r a t e d l i n e s . Thus a l t h o u g h the f i n a l l i n e -shape may appear t o be L o r e n t z i a n , the a c t u a l l i n e w i d t h i s not the sum of t h e i n d i v i d u a l L o r e n t z i a n components, or any s e r i e s of s i m p l e f r a c t i o n s of them (91)> Thus, i n t h i s c a s e , i t would be unwise t o a t t a c h too much emphasis t o the measur-ed l i n e w i d t h s i n the f i n a l spectrum -- i n f a c t , as shown <in f>BT f i g . 5. the r e s u l t a n t l i n e w i d t h s do not a l l e x h i b i t the same b e h a v i o u r w i t h t e m p e r a t u r e . S i m u l a t e d s p e c t r a a l s o had d i f f -e r e n t l i n e w i d t h s t o the a c t u a l s p e c t r a , and i n g e n e r a l i t can be assumed t h a t l i n e w i d t h a n a l y s i s of the more c o m p l i c a t e d e s r s p e c t r a w i l l n ot show the t emperature dependence e x p e c t -ed w i t h o u t some knowledge of the b r o a d e n i n g due t o each nuc-l e u s p r e s e n t -- t h i s i n f o r m a t i o n i s o f t e n not a v a i l a b l e and thus many of the more c o m p l i c a t e d e s r s p e c t r a a r e l e f t un-a n a l y s e d . A d e t a i l e d c o m p u t a t i o n of the a c t u a l l i n e s h a p e s f o r each l i n e h ere was not a t t e m p t e d , so q u a n t i t a t i v e a s p e c t s of eq (4-21) must be l e f t a t the e m p i r i c a l l e v e l a l r e a d y de-s c r i b e d . H: POSSIBLE EXCHANGE EFFECTS + F i g u r e 3 shows some s p e c t r a of CPZ o b t a i n e d i n media o t h e r t h a n s u l f u r i c a c i d . I t can be seen t h a t the s p e c t r a i n p h o s p h o r i c and h y d r o c h l o r i c a c i d s a r e v e r y s i m --82-i l a r to those i n s u l f u r i c a c i d . T h i s was expected s i n c e the v i s c o s i t y of these s o l v e n t s i s h i g h enough to cause m o t i o n a l - 2 - 1 broadening ( v i s c o s i t i e s g r e a t e r than ~10 - 10 p o i s e are i n the range necessary to broaden the spectrum of C P Z + ) . The lower v i s c o s i t i e s of a c e t o n i t r i l e and nitromethane (around .4 to .8 c p ) , however, were expected to show symmetrical s p e c t r a of approximately the same shape as those f o r s u l f u r i c a c i d a t higher temperatures. In nitromethane ( f i g 3&) a w e l l - r e s o l v e d but s-shaped spectrum was observed, and l o w e r i n g the concen-t r a t i o n f a i l e d to remove the s-shape and s t i l l g i v e a r e s o l v e d spectrum. Thus the expected r e s o l u t i o n , but not the shape, was o b t a i n e d . The a c e t o n i t r i l e s p e c t r a were a l l s i m i l a r to t h a t shown i n f i g 3d» and here the unusual f e a t u r e i s t h a t not o n l y i s the s-shape r e t a i n e d , but the l i n e w i d t h broadening has caused enough o v e r l a p to obscure f o u r l i n e s . Attempts to r e s o l v e these s p e c t r a f a i l e d , and l o w e r i n g the c o n c e n t r a t i o n below about 10~-Vi r e s u l t e d i n l o s s of r e s o l u t i o n . Thus i t seems t h a t i n these media there i s an exchange broadening e f f -e c t . The apparent removal of t h i s e f f e c t as the temperature i s i n c r e a s e d f o r the s u l p h u r i c a c i d system c o u l d not be dup-l i c a t e d w i t h the l o w e r - v i s c o s i t y s o l v e n t s , s i n c e they began to b o i l b e f o r e any change was observed. For the s u l f u r i c a c i d examples, the " f l a t t e n i n g " of the o v e r a l l l i n e s h a p e i s prob-a b l y due to a volume i n c r e a s e i n the s o l v e n t a t higher temp-e r a t u r e s thus e f f e c t i v e l y i n c r e a s i n g the d i s t a n c e between the. m i c r o c r y s t a l s or paramagnetic c e n t e r s . T h i s would d i m i n i s h the exchange i n t e r a c t i o n between these c e n t e r s . . -83-The exchange i n t e r a c t i o n was not considered quant-i t a t i v e l y here, but a b r i e f description w i l l be given i n most-l y q u a l i t a t i v e terms. Exchange interactions generally occur In non-dilute systems, and a many-particle model of a system containing paramagnetic centers shows that Coulomb interactions between the electrons of neighbouring ions can lead (consider-ing a l s o the Pauli p r i n c i p l e ) to an i n t e r a c t i o n of the form: = Sc-Sj (4-23) where T;j = j f^C*)^ (yjUa/*;) W2jHft(*MtU^ ^~ 2 ^ and $ pi fa) describes the r e s u l t of electronic over-lap. Because large overlap leads to the general r e s u l t that U/fl and L^ Q are not orthogonal i f A and B r e l a t e to d i f f e r e n t centers, i t i s customary to consider J ^ j as a small perturbat-ion on the e l e c t r o n i c energies within the ion, rather than attempt computations on a new 2-center system where each cen-ter i s on a d i f f e r e n t molecule. Thus large overlap w i l l be neglected. The exchange i n t e g r a l , J^y f a l l s off r a p i d l y for large i n t e r - i o n i c distance, so the exchange i n t e r a c t i o n can be l o c a l i s e d to neighbouring pa i r s : Me = - £<2 Jcj^ -Sj (4-25) where the sum i s over a l l nearest-neighbour pairs, and usually a l l nearest-neighbour pairs are assumed to have iden-t i c a l l o c a t i o n within the system, thus J^^ = J. Calculations of the e f f e c t of exchange in t e r a c t i o n w i l l not be given, but r e s u l t s for solution study show (58) that the e f f e c t of exchange i n t e r a c t i o n is e s s e n t i a l l y • a narr-owing, since exchange tends to average out electronic dipolar -84-i n t e r a c t i o n s which normally would g i v e a very broad s p e c t r a l u t i o n s p e c t r a are narrowed o v e r a l l , but i n d i v i d u a l h y p e r f i n e l i n e s become broadened. T h i s a r i s e s when the c o n c e n t r a t i o n of the sample i s such t h a t exchange of the e l e c t r o n s on neigh-b o u r i n g n u c l e i occurs, and the h y p e r f i n e l i n e s broaden by an amount 1/tOe, where i s the exchange frequency ( r e f 60, p 5 0 2 ) , r e s u l t i n g i n l/<)g being the e f f e c t i v e l i f e t i m e of an e l e c t r o n a t the n e i g h b o u r i n g nucleus. As the exchange frequency becomes comparable w i t h the energy of the h y p e r f i n e i n t e r a c t i o n , the h y p e r f i n e s t r u c t u r e g r a d u a l l y broadens and c o a l e s c e s i n t o a s i n g l e l i n e . In other words, the h y p e r f i n e i n t e r a c t i o n i s no l o n g e r c e n t e r e d a t one nucleus, but becomes averaged over p a i r s of n e a r e s t - n e i g h b o u r s ' n u c l e i . Thus, f o r the case of CPZ +, i t appears t h a t a concen-t r a t i o n of c a t i o n s s u f f i c i e n t to observe the spectrum by esr techniques i s h i g h enough to cause some exchange i n t e r a c t i o n . Removal of t h i s i n t e r a c t i o n was expected through the use of the nitromethane/aluminum c h l o r i d e o x i d a t i o n technique, s i n c e i t has been r e p o r t e d t h a t t h i s technique y i e l d s c l o s e to 100^ of the r a d i c a l s p e c i e s (80). Thus, s i n c e the same source quoted r e s u l t s t h a t i n d i c a t e t h a t s u l f u r i c a c i d o x i d a t i o n y i e l d s only about 1% r a d i c a l , i t should be p o s s i b l e to o b t a i n r e s o l v e d esr -5 -4 s p e c t r a f o r around 10 - 10 M s o l u t i o n s of CPZ i n AlCl^/CH^NOg. Many attempts to do t h i s f a i l e d , with the optimum c o n c e n t r a t i o n f o r good s p e c t r a i n both cases being about 10 M^. Thus, i f i t i s indeed exchange e f f e c t s t h a t g i v e the s-shaped s p e c t r a , and i t p r o b a b l y i s , then the r a d i c a l - g e n e r a t i n g a b i l i t y of both l i ( r e f 22 , pp 205-208). S o l i d s show t h i s c l e a r l y , but s o l -- 8 5 -systems must be about the same for chlorpromazine. ^ Some B i o l o g i c a l Aspects of Chlorpromazine A; ELECTRICAL PROPERTIES The report (9*0 that chlorpromazine behaves as an impurity semi-conductor below 32°C, and as an i n t r i n s i c semi-conductor above that temperature indicates the extremely low i o n i z a t i o n p o t e n t i a l of CPZ. The 32°C t r a n s i t i o n point has been associated with a crystallographic change which a f f e c t s the sidechain as well as the r i n g system (9**-) — a very im-portant feature of this i s that this change occurs i n the re-gion just below body temperature. Interestingly, the change occurs only i n the p o l y c r y s t a l l i n e material, and not in. s i n -gle c r y s t a l s ( 9 5 )• Careful study of the esr spectra from room temperature to 5 0 degrees revealed no unusual change i n the molecular structure of the r i n g and f i r s t part of the side-chain. The change at 125°C might be the same change reported fo r the conductivity studies, or a d i f f e r e n t e f f e c t e n t i r e l y , but no change at 32°C i n solu t i o n occurs. However, the cond-u c t i v i t y studies were made on compressed p e l l e t s of the poly-c r y s t a l l i n e state, and this would have the e f f e c t of lowering the i o n i s a t i o n energy, or, equivalently, reducing the size of the energy gap. This i s e s s e n t i a l l y what happens when carbon condenses into the diamond l a t t i c e , which i s an i n t r i n s i c semi-conductor at the appropriate temperature. Thus the s i t u a t i o n at 32°C might be equivalent to that at 125°C, i f i t i s assumed that the change to i n t r i n s i c semi-conductivity i s equivalent - 8 6 -to an increase i n the number of c a r r i e r s a v a i l a b l e . This means t h a i something l i k e a d i - c a t i o n i s formed, as described e a r l i e r when considering the changed spectrum above 125°C. The change then was not described e x p l i c i t l y as a change from a mono- to a d i - c a t i o n , but this could be the r e s u l t of losing a proton from the f i r s t sidechain methylene group to, say, the solvent. Much discussion on the a b i l i t y of the phenothiazines (see f i g 7 f o r some representative examples) to form a cat i o n i c state has been centered about the energy of the h i g h e s t - f i l l e d molecular o r b i t a l . Most studies have been made using HMO, and Karreman et a l (96) report m^ , i n the expression: fci = a AVIC 3^ (4-26) to be negative. This would indicate that the energy of the h i g h e s t - f i l l e d molecular o r b i t a l i s representative of an a n t i -bonding o r b i t a l , since and are negative? wheras Or l o f f and F i t t s ( 9 7 ) f e e l that consideration of the sulfu r d-orbitals should be taken into account, thus making m^  p o s i t i v e (for chlorpromazine, i t i s very small and p o s i t i v e ) . This i s i n -t u i t i v e l y more reasonable, since most phenothiazines are stable i n the absence of l i g h t , but are non-explosive. They are, i n other words, e a s i l y ionized but not overly unstable i n their ground state. This would indicate that the h i g h e s t - f i l l e d HMO i s close to the energy of an i s o l a t e d carbon 2p o r b i t a l . HUckel calculations performed indicate that CPZ i s bonding i n i t s h i g h e s t - f i l l e d HMO, and the v a r i a t i o n of the energy of thi s o r b i t a l with the angle deviation from a planar structure i s given i n figure 6. I t can be noted that the energy theta (deviation of each ring from a planar structure) Fig 6 variation of energy (HMO) with theta -89-phenoth i az ine (CH2)3N(CH3)2 p r o m a z i n e ( C H 2 ) 3 N ( C H 3 ) 2 c h l o r p r o m a z i n e (CH2)2-th io r idaz ine CH2CHN(CH3)2 CH3 p r o m e t h a z i n e (ay. Cr^ Ch^ OH perphenaz ine Fig 7 s o m e r e p r e s e n t a t i v e phenothiazine drugs -90 •305 300 295 -O—O-.290 0 10 20 30 0 40 50 8 variation of 0 (angle of siclechain and nitrogen spin density ;bending) - 9 1 -appears to become more bonding as theta increases towards 90 degrees (the p l o t i s not of energy i t s e l f , versus t h e t a , but mean that t h i s i s the s i t u a t i o n : a l l that i t r e a l l y i n d i c a t e s i s that as cos8 decreases to 0, the resonance i n t e g r a l f o r the C-N bond approaches the value f o r a C-C bond; t h i s i s because the angular v a r i a t i o n was included i n t o the Huckel c a l c u l a t i o n s as f o l l o w s : where 0 Is the f o l d i n g tingle. The same r e l a t i o n s h i p was used to f i n d the v a r i a t i o n of n i t r o g e n s p i n d e n s i t y w i t h the p o s i t i o n of the a l k y l s i d e c h a i n , as shown i n f i g u r e 8. For the case of the f o l d i n g angle, l i t t l e can be deduced from the mj_ values except that m^  i s p o s i t i v e , and hence the Huckel approximation i n d i c a t e s that the energy of the highest-occu-p i e d o r b i t a l i s lower than that of an. i s o l a t e d carbon 2p orb-i t a l and thus bonding with respect to t h i s carbon o r b i t a l . Even f o r the planar s t r u c t u r e , where i s s m a l l , but p o s i t i v e , a bonding s i t u a t i o n i s apparent, and i t can be concluded t.h* a more s t a b l e c o n f i g u r a t i o n than t h i s (more bonding) w i l l be most favourable, but t h i s w i l l not be the extreme approached a t a f o l d i n g angle of 9 0 ° (when both r i n g s are touching) as f i g u r e 6 would suggest. Thus a compromise i s probable; and e a r l i e r s p i n d e n s i t y c a l c u l a t i o n s showed that an angle of 104° between the r i n g s was c o n s i s t e n t with the esr spectrum. This r e s u l t would appear reasonable despite the i n d i c a t i o n s of f i g -ure 6. The best p o s i t i o n of the s i d e c h a i n would appear to ee of m^  i n eq (4-26) versus t h e t a ) . T h i s , however, does not (4-2.7) - 9 2 -the l o g i c a l one, that is,attached to the nitrogen atom para-l l e l to the N-S ''bond"and. perpendicular to a plane bisecting the rings through the C - 3 , C-Z, C - 9 , and C - 1 0 bonds. The following diagram i l l u s t r a t e s this point: /Ss> This point i s included for completeness, since i t has been postulated ( 9 6 , 9 7 ) that the sidechain i s not i n this pos-i t i o n , but folded under the nitrogen i n a pos i t i o n that can be best described as "tucked under" between the folded rings. Gutmann and Keyzer ( 9 5 ) support this sort of structure, but esr shows only that the nitrogen spin density, calculated by MO methods, should, be about . 3 0 0 - . 3 0 8 . Figure 8 shows that th i s i s the value over a wide range of angles ( 1 0 ° to 6 0 ° ) , and varying the p o s i t i o n does not a l t e r the m^  value of the highest-occupied o r b i t a l appreciably enough to attach any im-portance to t h i s parameter. The charge transfer mechanism has been investigated by Gutmann and Keyzer ( 9 5 ) , with the r e s u l t that chlorpromazine forms a complex with iodine of the form: I^sCPZ :I 2:CPZ : l 2 « where 1°; i s a neiitral molecule. Above 5 0°C, i n a c e t o n i t r i l e s o l ution, this 2 : 3 r a t i o changes to a 1 : 2 complex. Since a 1 :1 complex i s also observed, but not discussed, i t may be postulated that the 2 : 3 complex breaks at the neutral specie, leaving the 1 : 1 and 2 : 1 complexes. This would require weak bonds, which i s t y p i c a l for many b i o l o g i c a l charge-transfer complexes. Since the chlorpromazine probably uses i t s XT-sys-tem to s t a b i l i s e the complex, the donation of charge to the acceptor species may involve the inclu s i o n of the acceptor be-tween the rings , where the negative charge on the acceptor may serve to s t a b i l i s e the close proximity of the two. p a r t i a l l y p o s i t i v e aromatic rings. Interaction at either the nitrogen or s u l f u r atoms i s possible, and both carry appreciable un-paired spin density i n CPZ . The sulfur d -orbitals are nat* u r a l candidates for p a r t i a l bonding, as i s the nitrogen lone-pair system. Sharing between the two centers could, be poss-i b l e f or larger acceptors: no further discussion of this w i l l be attempted, however, since a detailed study of charge trans-f e r complexes of chlorpromazine was not made. A spectrum of the CPZsIg complex i n a c e t o n i t r i l e i s given on page 5 0 , and as can be seen, l i t t l e information i s available from this spec-trum. A s i m i l a r , but broader, spectral shape was obtained for the s o l i d complex. B: MECHANISM OF CHLORPROMAZINE ACTION The t r a n q u i l i s i n g aspects of a r e l a t i v e l y s t r a i g h t -forward molecule l i k e chlorpromazine implicate the complex-i t i e s of the human nervous system. I t has been suggested that the phenothiazines act as charge transfer donors i n b i o l o g i c a l reactions, and i t has further been suggested that most bio-l o g i c a l complexes have the majority of the electronic charge i n the acceptor state (96). Chlorpromazine, l i k e a l l the phenothiazines, donates electrons r e a d i l y from a highest-_g4-f i l l e d o r b i t a l t h a t i s c l o s e to the lowest-unoccupied o r b i t -a l . A t r a n q u i l i s i n g r e a c t i o n can occur by p o l a r i s a t i o n of the e l e c t r i c dotible l a y e r of a c e l l membrane. The drug donates an e l e c t r o n to the acc e p t i n g i n s i d e l a y e r of the membrane, which then becomes negative w i t h respect to the outer l a y e r , and thus p o l a r i s e d . This tends to oppose the normal mode of nerve e x c i t a t i o n , or any membrane c r o s s i n g , by preventing the "sodium pump" a c t i o n . A complete d e s c r i p t i o n of t h i s , and the a c t i o n of nerve c e l l s i s given i n most t e x t s on neur-ology, and a good, but b r i e f , d i s c u s s i o n may be obtained i n re f e r e n c e 1 0 6 . B r i e f l y , the- membrane becomes a c t i v a t e d by a r a p i d i n f l u x of Na ions and a corresponding l o s s of K ions. Since the co n c e n t r a t i o n of Na + over long times i s constant outside the membrane, a "sodium pump" has been p o s t u l a t e d to account f o r the subsequent removal of the Na ions that enter d u r i n g a nerve p u l s e , r e s t o r i n g the membrane p o t e n t i a l and readying i t f o r f u r t h e r a c t i v a t i o n . By b l o c k i n g e i t h e r the sodium pump, or the entrance of Na i n t o the membrane by p o l -a r i s a t i o n of the membrane l a y e r s , nerve a c t i o n i s damped and t r a n q u i l i s a t i o n occurs. This i s probably the mode of t r a n -q u i l i s a t i o n of drugs l i k e chlorpromazine. Chlorpromazine i s c e r t a i n l y a strong enough donor to a c t u n s e l e c t i v e l y , and t h i s would e x p l a i n i t s tendency to not only t r a n q u i l i s e , but a l s o i n t e r f e r e with normal v i s i o n , induce anemic-like c o n d i t i o n s , and cause extrapyramidal c o n d i t i o n s . This w i l l be tr e a t e d b r i e f l y - a f t e r a d i s c u s s i o n of some biochemical aspects. The b i n d i n g mechanisms and metabolism of chlorprom-- 9 5 -a z i n e are l i k e l y a l l based on the donating a b i l i t y of the dru^. P i e t t e and Sandberg ( 9 9 ) used esr s t u d i e s on s p i n -l a b e l l e d e r y t h r o c y t e ghosts (which are o f t e n considered to be good membrane models — they are simply erythrocytes with" a l l hemoglobin and c e l l f u n c t i o n removed) to show that CPZ can b i n d a t q u i t e r e s t r i c t e d ( s p a t i a l l y ) membrane p o s i t i o n s , and i n f a c t can do so s t r o n g l y . I f a s p i n l a b e l i s weakly attached to such a p o s i t i o n , CPZ tends to strengthen the b i n d i n g and immobilise the s p i n l a b e l . Such a p r o t e c t i v e mechanism i s probably presentjythe p r o t e c t i o n of c e l l s a g a i n s t hemolysis; which chlorpromazine does a t low c o n c e n t r a t i o n . At high c o n c e n t r a t i o n chlorpromazine i t s e l f w i l l hemolyse c e l l s . Such p r o t e c t i v e mechanisms r e q u i r e a stronger surface s t r u c t u r e to a r i s e when the p r o t e c t o r i n t e r a c t s a t the sur-face membrane. This i s apparently common to a l l phenothiazine drugs, w i t h p r o t e c t i v e a b i l i t y decreasing i n the order: perphenazine, chlorpromazine, promazine. This order i s the same order f o r decreasing potency as a psychotropic agent. The a c t u a l b i n d i n g mechanism i s l i k e l y e i t h e r by i n t e r c a l a t i o n , i n t o a polymeric s t r u c t u r e or by bonding at the s u l f h y d r y l group of a p r o t e i n . P i e t t e ' s r e s u l t s i n d i c a t e that CPZ can bond a t the s u l f h y d r y l group, and i t has been shown that CPZ i n t e r c a l a t e s (probably i n the c a t i o n i c form) i n t o the DNA h e l i x s t r u c t u r e ( 7 1 ) . I t i s not g e n e r a l l y known, f o r an i n d i s c r i m -i n a t e r e a c t i o n , whether chlorpromazine i s o x i d i s e d f i r s t , or whether, i t becomes o x i d i s e d a f t e r b inding by p a r t i c i p a t i n g i n a b i o l o g i c a l charge t r a n s f e r r e a c t i o n . S p i r t e s ( 1 0 0 ) suggests - 9 6 -t h a t the l a t t e r i s not the case, but h i s r e s u l t s seem some-i s e a s i l y r e v e r s i b l e [95% of the CPZ can be removed by washing w i t h a s a l i n e s o l u t i o n ) and i n d i s c r i m i n a t e i n b i n d i n g s i t e i n a molecule c o n t a i n i n g only p r o t e i n . In systems c o n t a i n i n g l i p o p r o t e i n s the b i n d i n g i s not e a s i l y r e v e r s i b l e ( 1 0 0 , 9 9 ) and suggests t h e r e f o r e a more s p e c i f i c b i n d i n g s i t e . as the phenothiazine s e r i e s i s a l o g i c a l r e s u l t of the non-s p e c i f i c i t y of i n t e r a c t i o n a t many membrane s i t e s . The many met a b o l i t e s of CPZ that are present i n any human body that has been treated, w i t h chlorpromazine (these metabolites are a l l v a r y i n g a t the side c h a i n p o s i t i o n s , where the t e r m i n a l n i t -rogen, i n p a r t i c u l a r , loses e i t h e r of the methj^l groups, or both; a l s o i n v o l v e d i n the metabolism i s the r i n g s u l f u r atom, which can be e a s i l y protonated) can a l l r e a c t a t s i t e s u s i n g the b a s i c r i n g s t r u c t u r e to e f f e c t a bond, but d i f f e r e n c e s can occur i n the side chain e f f e c t s . The d i f f e r e n t phenothiazine drugs 3,11 d i f f e r most markedly i n the side chain, so perhaps t h i s p a r t of the molecule i s the most important i n determining drug a c t i o n d i f f e r e n c e s . Thus i n d i s c r i m i n a t e a c t i o n by one of these drugs a t s e v e r a l s i t e s could r e s u l t i n p o l a r i s a t i o n or h y p e r p o l a r i s a t i o n of some membranes, d e p o l a r i s a t i o n of other b i n d i n g w i t h t r a c e metals (72) that are needed f o r other met-a b o l i c processes, r e d u c t i o n of hemoglobin l e v e l s by hemolysis of the red blood c e l l s when used i n high doses, and e t c . . Chlorpromazine i s given i n l a r g e doses to some of the more I t does appear, however, that the binding Thus the occurrence of side e f f e c t s w i t h such drugs - 9 7 -severe cases of s c h i z o p h r e n i a , i n doses up to 2g per day. Such doses could e a s i l y r e s u l t i n the many side e f f e c t s ob-served (anemia, o c u l a t o r y t r o u b l e , extra-pyramidal symptoms) by i n t e r f e r i n g w i t h c e l l f u n c t i o n and nerve conduction. Of much i n t e r e s t i s the occurrence of extra-pyramidal symptoms, which are w e l l manifested i n Parkinson's disease. I f the area of i n t e r a c t i o n of chlorpromazlne that causes t h i s pro-bable nerve i n t e r f e r e n c e could be i s o l a t e d , much more inform-a t i o n on the mechanism of nerve output would be a v a i l a b l e a t the chemical l e v e l . Knowledge of the spectrum of chlorprom-a z l n e i t s e l f , then, might perhaps enable use of p h y s i c a l meth-ods to be more r e l e v a n t than they have been i n the past. More work i s being done i n t h i s area by p h y s i c a l s c i e n t i s t s , and the f u n c t i o n and s t r u c t u r e of hemoglobin, f o r inst a n c e , has been w e l l examined by esr and d i f f r a c t i o n techniques, w i t h the r e s u l t that much of the behaviour of t h i s molecule i n the human body i s now Ttfell understood. This i s a r e l a t i v e l y sim-p l e example, b i o l o g i c a l l y , and of much more i n t e r e s t and com-p l e x i t y i s the f u n c t i o n of the nervous system and b r a i n , and the manner of i n t e r a c t i o n of outside agents (e.g. drugs) a t s i t e s i n the nervous system i s enabling some in f o r m a t i o n to be obtained a t the molecular and c e l l u l a r l e v e l s . H o p e f u l l y , p h y s i c a l techniques l i k e e l e c t r o n s p i n resonance w i l l be able to be used to solve some of these i n t e r e s t i n g problems. The f u n c t i o n of the p h y s i c a l s c i e n t i s t i n these a p p l i c a t i o n s w i l l be to guide the b i o l o g i s t i n h i s t h e o r i e s , and to open new f i e l d s to study by p h y s i c a l means and t h e o r e t i c a l i n t e r p r e t a t i o n . - 9 8 -3IBLI0GRAPHY • 1 S. I .Weissman, J.Townsend, D.E.Paul, and G.E.Pake, J.Chem Phys 2 1 , 2 2 2 ? ( 1 9 5 3 ) 2 S. I .Weissman, J.Chem Phys 2 2 , 1 1 3 5 ( 1 9 5 4 ) 3 T.LChu, G.E.Pake,D.E.Paul, J.Townsend and S.I.Weissman, J.Phys Chem i 2 » 5 0 4 ( 1 9 5 3 ) 4 B.Venkataraman and G . K . F r a e n k e l , J.Chem Phys 24 , 7 3 7 ( 1 9 5 6 ) 5 C. J. 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P i e t t e , Agressologie £,59 (1968) 100 M.A.Splrtes, A g r e s s o l o g i e 2,189 (1968) 101 A.G.Bolt and I . F o r r e s t , Agressologie 2- 2 0 1 (1968) 102 A.Loewy and P . S i e k e v i t z , C e l l S t r u c t u r e and Function, ( H o l t , Rinehart and Winston, New Yorkj 196JT chapter 15 103 J.Fernandez-Alonso, Advances i n Chemical P h y s i c s , V o l VII ( I n t e r s c i e n c e , London"", 1964) p44ff 104 B.J.Gudzinowicz, J o u r n a l of Gas Chromatography 4,110 (1966) 105 P.H.H.Fischer, Ph.D. Thesis, U.B.C. (1963) 106 R.F.Thompson, Foundations of P h y s i o l o g i c a l Psychology, (Harper and Row, New York, 1967) chapters 5-11 -103-APPENDIX 1 THE FERMI CONTACT HAMILTONIAN Probably the most important i n t e r a c t i o n i n aromatic free r a d i c a l s i s the hyperfine i n t e r a c t i o n a r i s i n g from i n t e r a c t i o n of electronic and nuclear moments and described by the so-called Fermi contact Hamiltonian. D i f f e r e n t notations are often used, and d i f f e r e n t ones are to be found i n this thesis, but a l l are equivalent. We write, then, the Hamiltonian (22,23) i t -- S'Yi s^a^x ini so*,) where - ^  j applying this to some polyelectronic wave function {J> , and w r i t i n g Jl = ^ofS-I = ^ SJt *N--<mff>/3?<Si><h> where ^ - ^  ^ . - 1 0 4 -APFENDIX 2 PROTON HYPERFINE INTERACTION MECHANISM Much of t h i s appendix i s based on reference 6, and i s presented here i n some d e t a i l i n order to provide completion of the d i s c u s s i o n on proton s p l i t t i n g s . B a s i c a l l y , the problem i s one of c o n s t r u c t i n g s u i t a b l e o r b i t a l s or wave f u n c t i o n s , f o r the aromatic C-H fragment, which f o l l o w c e r t a i n assumptions: a) each normalised atomic o r b i t a l holds only one e l e c t r o n b) the wave f u n c t i o n s d e s c r i b i n g the e l e c t r o n s are to be eigenfunctions of S_ and S^ c) one valence bond f u n c t i o n must desc r i b e the ground s t a t e of the C-H fragment. There are three normalised o r b i t a l s , the P z, S and 6" o r b i t a l s , w i t h the f r e e e l e c t r o n r e s i d i n g predominantly i n the Pz o r b i t a l on the carbon. A t h r e e - e l e c t r o n system i s thus the one under d i s c u s s i o n , and the three s p i n - o r b i t a l c o n f i g u r a t i o n s can be w r i t t e n : where A i s an a n t i - s y m m e t r i z a t i o n and r e n o r m a l i z a t i o n o p e r a t i o n (11) ,(g) (1,2 ,3) = p z (1)CT(2) s(3) i s the s p a t i a l p a r t of the wave f u n c t i o n and <*<&f^etc. are s p i n product b a s i s f u n c t i o n s s a t i s f y i n g the f i r s t p a r t of assumption (b). Now, to o b t a i n e i g e n f u n c t i o n s of s2 i t i s necessary to take l i n e a r combinations of <£t , (f^ and (j)^ g i v i n g doublet -(S=§)~ ~~ e l e c t r o n i c s t a t e wave f u n c t i o n s : - 1 0 5 -and now a l l assumptions are s a t i s f i e d , Cpj approximates a ground c o n f i g u r a t i o n , and a n e x c i t e d c o n f i g u r a t i o n , so c a l c u l a t i o n s of the energy d i f f e r e n c e between cf{ andC^.J V/, shows tha t the two l e v e l s d i f f e r by exchange i n t e g r a l s and the t o t a l one-electron nuclear a t t r a c t i o n exchange energy of the C-H bond ( 6 ) . Since cannot give hyperfine c o u p l i n g w i t h the proton by i t s e l f , a sm a l l admixture of e x c i t e d s t a t e cf^L i s necessary, g i v i n g the mixed ground s t a t e wave f u n c t i o n : ' tft' = <ff *f£f>4 (1) . where f i s a s p i n a t t e n u a t i o n f a c t o r d e s c r i b i n g the amount of admixture, or c o n f i g u r a t i o n i n t e r a c t i o n , and hence the amount of coup l i n g w i t h the nucleus. Thus i f H i s the complete 3 - e l e c t r o n Hamiltonian, a n d f f / ^ / , then f=-(H 2 1/AW), where H 2 1= Hl^}. McConnell (6) expands H 2^ as f o l l o w s : where J(t>) fCj) (efajJfccflJGtiUkj s j p ^ two-e l e c t r o n exchange i n t e g r a l , and s i m i l a r l y f o r Jp<: . I f the c o n d i t i o n Jffr^Jps holds, then Using S l a t e r ' s ( 1 2 ) method to evaluate.the exchange i n t e g r a l , and i f i\ W i s i n the range 5 - 1 5 e v , then f ~ 0 . 0 7 - 0 . 2 . Thus the e f f e c t i v e p e n e t r a t i o n of the odd e l e c t r o n i n t o the <$ system i s gi v e n by f, which a r i s e s from c o n s i d e r a t i o n of an atomic exchange mechanism. -106-So f a r only an i d e a l i s e d , i s o l a t e d C-H fragment has been considered. I n the r e a l case the t o t a l unpaired s p i n d e n s i t y i s d i s t r i b u t e d over the whole TT system—so f o r any proton p o s i t i o n j where, i n g e n e r a l , v|j w i l l not be the same f o r a l l C-H bonds i n most aromatic systems. Now, to d i s c u s s the a c t u a l hyperfine s p l i t t i n g s a r i s i n g from the t h e o r i s e d s p i n p o l a r i s a t i o n , i t can be u s u a l l y assumed ( e s p e c i a l l y i n aromatic, organic f r e e r a d i c a l s ) t h a t Zeeman i n t e r a c t i o n s are much stronger than the hyperfine i n t e r a c t i o n s a t the f i e l d s commonly used. Therefore the energy of the contact (or Fermi) hyperfine i n t e r a c t i o n given by the Hamiltonian (10): where OC '^H) ^he u s u a l d e l t a f u n c t i o n f o r the d i s t a n c e between the e l e c t r o n and the proton. The f i r s t order energy f o r S=|- can be c a l c u l a t e d (6): = ( & f j lfl)(lLTIf/3h) ( M l / x J / ^ j / 2 ^ (3) where Iz=±4, I z=0 f o r allowed e l e c t r o n i c t r a n s i t i o n s , s(0) i s the value of the wave f u n c t i o n a t the nucleus and E j t - D ^ E j C + i ) . " Thus f o r a s p l i t t i n g Hj) i n an e l e c t r o n s p i n resonance spectrum, equation (3) can be w r i t t e n : -107-APPENDIX g DERIVATION OF THE SPIN HAMILTONIAN References 5 and 5 8 can be consulted f o r complete d e t a i l s . The s p i n - o r b i t Hamiltonian i s : -U-S .+ (3tfft+-2.l ) (!) which gives a second-order p e r t u r b a t i o n energy of: £C2) ^ 2Q(Stj-\A<j)S;Us-X1A;sSl-Si- " ( 2 ) where *2 1 -iQ<*h» |tj( Q -f ft y l t j i s an o r b i t i n t e r a c t i o n term between the g r o u n d ' and e x c i t e d s t a t e s , and i f i t i s noted that ^ - A y ^ j w i l l give r i s e to a uniform displacement of a l l l e v e l s ( 5 ) i t can be ignored. W r i t i n g : <j.j - 2(fij-\Aij) (3) as the g-facto r ( d e s c r i b i n g , e s s e n t i a l l y , the amount of s p i n - o r b i t i n t e r a c t i o n ) , and l e t t i n g E n-EQ»^, a t t e n t i o n must focus on the second term. This term w i l l be r e s p o n s i b l e f o r z e r o - f i e l d s p l i t t i n g of the s p i n m u l t i p l e t , and i n an ax-i a l l y symmetric c r y s t a l f i e l d (-^y^ytj ,vtitrvt^) JL i s diagonal ( 6 l ) , so that: - xu» ?c i - - tfttj. ( # + t y • j.,, s; (4) but + Sy - S(S+ I) - S £ Z ( 5 ) Now d e f i n i n g D= (-t^ -r_A )^A• and s u b s t i t u t i n g i n t o (2); Or, completely g e n e r a l l y , -108-where D and E are z e r o - f i e l d s p l i t t i n g parameters and g the " s p i n - o r b i t " f a c t o r . Now, f o r S = - | , D=0, so eqn, ( 7 ) can be transformed i n t o l a b o r a t o r y co-ordinates as f o l l o w s ( i n c l u d i n g the hyper-f i n e i n t e r a c t i o n ) : Take * ^ifafSp •+ ] ? % ) + " " (9) as the Zeeman and hyperfine Hamiltonian. Then i f HQ i s the f i e l d i n the z - d i r e c t i o n , H r = H Q C O s 0 where 19 i s the angle between ;;the symmetry a x i s of the molecule, r , and the l a b o r a t o r y z - a x i s . Then H ° S = H Q S z , and f u r t h e r : S r=S zcos 9 + S x s i n 0 c o s ^ + S y S i n 0 s i n j / ( 1 0 ) = S z c o s 0 + |(S +e"¥- S_e¥)sin£ where Si=(S x±iS y) and j> i s the angle between the xy plane and the pq molecular plane, perpendicular to z and r , r e s p e c t -i v e l y . A f u r t h e r p o i n t i s that can be formulated as the sum of an i s o t r o p i c (and therefore time-independent) and a n i s o t r o p i c (time dependent) term: where < j % = J f3 WoS* + * S• J and jBf, = f ^ « 0 + b l j f c ^ 4&l[si«6> co S ©J[I + £^+ I^/V] ^ where g = l/3(g,| + 2 g i) , a =|(Ay+ 2Aj_) b = A J j-A i -109-APPENDIX 4 THE CORRELATION FUNCTION AMD ITS PROPERTIES (22, 58-60) In the Hamiltonian given i n Appendix 3 and S e c t i o n EX, the po l a r angle 0 can be considered to determine the c o r r e l a t i o n of the molecule. The molecular symmetry a x i s w i l l then determine the c o r r e l a t i o n , or l a c k of i t , f o r a c e r t a i n time p e r i o d , and i f the m i c r o c r y s t a l i s considered to be a u n i t sphere i n shape, then J0 2ft Sill 0d© when c o n s i d e r i n g one of the terms i n (2-34). I f the time average i s not considered over a long p e r i o d , but a s u i t a b l y s h o r t e r one, then c l e a r l y *C3COS^ © —|^ need not be 0 a t any time t i f 0(fe) i s a random f u n c t i o n of time. There i s thus always some time i n t e r v a l a v a i l a b l e a t the end of which the f u n c t i o n w i l l ' not: be zero, and i n f a c t there i s a l s o an i n t e r v a l ending a t X where the s i g n of the random f u n c t i o n , and the' magnitude,. i s approximately the same as a t the s t a r t . When t h i s i s true f o r any s t a r t -i n g p o i n t then the. process i s termed a " s t a t i o n a r y random process" and the time i n t e r v a l i s the c o r r e l a t i o n time. Such a process i s to be considered here. Consider a random f u n c t i o n J(x(ti) and a c o r r e l a t i o n f u n c t i o n defined as: G-Ct, ,b a )- J R ^ ' O a ^ T ( 2 ) where the bar denotes an average. -110-Slnce a s t a t i o n a r y random process i s being considered, | t 2 - t i 1 =t. Then: G(T) = f ( x ( t ) ) f * ( x ( t+ T ) ) (3) and i f G(T) = G(0)g(D.= f f T x T t i n^sCt) and g(0) = 1. (4) •Equations (3) and (4) can be shown more e x p l i c i t l y by consid-e r i n g the app r o p r i a t e p r o b a b i l i t y d i s t r i b u t i o n s . I f the def-i n i t i o n p(xj_, t]_ ;x 2, t2) means that x=x-|_ a t t-^ and x 2 a t t 2 , a:nd i f P(x]_, tj_ jx2, t2) means x=x2 a t t 2 when i t i s known that x=x^ _ at tj_, then: p ( x 1 , t 1 j X 2 f t 2 ) = P ( x 1 , t 1 ; x 2 , t 2 ) p ( x i , t i ) (5) . or p i s simply the product of P and the p r o b a b i l i t y that x^x^ at t-|_. Thus the c o r r e l a t i o n f u n c t i o n can be w r i t t e n : G ( t 1 ( t 2 ) = f l x l " t 1 ) ) f " ' r ( x ( t 2 ) ) ' = j[ p ( x 1 t 1 ; x 2 t 2 ) f ( x 1 ) f ' ' r ( x 2 ) d x 1 d x 2 = j J p ( x 1 t 1 ) P ( x 1 , t 1 ; x 2 , t 2 ) f ( x _ ) f *(x2) ;dx 1dx 2 and s i n c e the d e f i n i t i o n of the. average value of a random f u n c t i o n x ( t ) a t some time t , subject to the p r o b a b i l i t y d i s t r i b u t i o n p(x,t) i s defined by (60,p270): xTtJ = J x p(x,t)dx (?) and f o r f ( x ) , a f u n c t i o n of x and hence a random f u n c t i o n of t i f x i s , __________ ( fTxTtTT =J p(x,t) f ( x ) dx (8) For the s t a t i o n a r y random process G(T) = J | p ( x 1 , x 2 , t ) f ( x 1 ) f ' : : " ( x 2 ) d x 1 d X 2 = j | p ( x 1 ) P ( x 1 , x 2 , r ) f ( x x ) f * ( x 2 ) d x 1 d x 2 (9) and i f G(£) i s de f i n e d (somewhat l o o s e l y ) to be sm a l l when i r l » r , , where T_ i s the c o r r e l a t i o n time, the c o r r e l a t i o n time achieves a mathematical d e f i n i t i o n . F u r t her, since p ( x i , x 2 , T ) = p ( x 2 , x i , T ) so G(--T) = G , r ( T ) = G(T) since p ( x j , X 2 , - D = p(xi,X2,t:) and the l a s t statement i m p l i e s that past p r o b a b i l i t y i s the same as f u t u r e p r o b a b i l i t y . I t i s assumed here that there i s always a time T such that t h i s holds. I t i s now p o s s i b l e to define a s e r i e s of s p e c t r a l d e n s i t i e s , which are F o u r i e r transforms of G,(6o): J ( W ) = G ( t ) e i u r d X ^ ( 1 0 ) J(u)) = 2 G(r)cos(u)£)dr = G('C)e d£ ( 1 1 ) k(o>) = G(r)sinu)tdt ( 1 2 ) Jo and t h e r e f o r e j(u)) = |j(w))-ik ( " i ) ( 1 3 ) The reduced c o r r e l a t i o n f u n c t i o n has already been defined i n equation (4), so i f i t i s represented-as ( 6 0 ) : g('C) = e x p ( - l t l / r c ) (14) then j(w) = 2 V ( 1 + ^ 2 ) (15). This i s o f t e n r e f e r r e d to as- the s p e c t r a l d e n s i t y , and c l e a r l y j(~)) a t a given frequency Co i s a maximum f or TJ. decreasing to 0 f o r e i t h e r very short or very long Z_ . This shows the usual r e l a x a t i o n e f f e c t that can be found when working w i t h l i q u i d s o l u t i o n s i n nuclear magnetic or e l e c t r o n s p i n resonance. -112-APPEND1X 5 THE CORRELATION FUNCTION FOR A SIMPLE CASE ( 5 9 ) I f a random f u n c t i o n i s l i m i t e d i n i t s randomness to an i n t e r v a l about some f i e l d value H Q, and i f the d e v i a t -i o n from HQ i s ±<T, and d e f i n i n g : H 2 = H0-& (1) 111 ~ Ho+ci then f o r the c o r r e l a t i o n f u n c t i o n G(T) = f " T t)f(t+fj (2) i t i s p o s s i b l e to w r i t e f(0.) = H-j_ as a s t a r t i n g assumption ( i t could a l s o have been f ( 0 ) = H 2, l e a d i n g to the same r e s -u l t ) . Now, i f a p r o b a b i l i t y i s defined as- f o l l o w s : P 1 ( T ) = 0) f o r f (t) = (H-L 1) (K 2 (3) and s i m i l a r l y f o r P 2 ( T ) , then: f (0)f (T) . = H 1 [ p 1 ( r ) I I 1 + P2(T)H 2] (4) b y simple s u b s t i t u t i o n . The average of (4) leads to the averages of p^CC) and p 2 ( T ) which p r e d i c t t h a t i f a t T=0 the f i e l d i s % , i t w i l l b e e i t h e r H^ or H 2 a t t i m e T ( / 0). fToirro = H 1 [ H i P i ( T ) + H 2 P 2(r)] (5) a n d a s t - ^ O , p^DW and p 2 ( t ) - - 0 i f f i r ) ^ a t t = 0 . Therefore the p r o b a b i l i t y t h a t the f i e l d w i l l jump from Hj_ to H 2, W, obeys the r a t e equations: dpi. = w ( p 2~ P l) d T " (6) d £ 2 = W(p 1-p 2) d r must hold. S o l v i n g gives ( r e f , 5 9 . p231)s (7) vx(X) + P 2 C O =1 px(r) - P2(T) = Cexp(-2Wt) but G = p 1 ( 0 ) - p 2 ( 0 ) =.p 1(0) sin c e p 2(0)=0, ther e f o r e i t f o l l o w s that s i n c e p ^ ( 0 ) = l f C=l. so fToTfTtT = H X [ H l P l ( t ) + H2P2(r)] = h\!>(e2vJt+ p 2(t>) -fp 2 ( t ) ] - i f f c ^ + J p 2 ( T ) -Jp ?(T)] = c.VW'C (8) t h e r e f o r e fTo)f (f) = ciV a W r= G(t) where f (0)f (tr7 represents the' complete ensemble average ( i . e . t a k i n g i n t o account both p o s s i b l e s t a r t i n g p o i n t s f o r the f i e l d v a l u e , and no t i n g that the f i e l d i s e q u a l l y l i k e l y to be e i t h e r value i n i t i a l l y , so the r e s u l t f o r one s t a r t i n g p o i n t i s the same f o r the ot h e r ) . F i n a l l y , s ince 1/tc = 2W (9) 2 itl Ac f ( t ) f (t+r) = ' e " c (10) where (9) f o l l o w s from c o n s i d e r a t i o n s of s a t u r a t i o n and r a t e equations f o r a s p i n \ p a r t i c l e (see r e f . 58« PP34-38), and Xc i s s u b s t i t u t e d f o r , the Bloch l o n g i t u d i n a l r e l a x a t i o n time. -114-APPENDIX 6 A RANDOM FUNCTION TREATMENT OEJ^t) To apply the c o r r e l a t i o n f u n c t i o n treatment to the Hamiltonian already described (3-34), i t i s convenient to w r i t e : where Jf0 i s the i s o t r o p i c p a r t a n d ^ j ( t ) the perturb-i n g s p i n i n t e r a c t i o n s which depend on l a t t i c e co-ordinates. The spins can then be considered a subsystem of the l a t t i c e ( m i c r o c r y s t a l ) which contains atomic and molecular degrees of freedom--thus a d i f f i c u l t quantum-mechanical c a l c u l a t i o n ' f o r the e n t i r e s p i n and l a t t i c e system can be avoided by t r e a t i n g $!j(t) as a random f u n c t i o n of time. To solve the time-dependent Schrodinger equation (the eigenf unctions and eigenvalues ofjj^are s o l u t i o n s of the time-independent Schrodingerequation, but introduces a s m a l l time-dependent perturbation'and thus s t a t i o n a r y s o l -u t i o n s of the time-independent equation do not e x i s t ) : =, (2 ) and i f zero-order eigenfunctions are u n such t h a t : $d0un = E n u n ( 3 ) then ( 6 2 ) : f ^ Y = ^ a n ( t ) u n e x p ( - i E n t / h ) ( 4 ) S u b s t i t u t i n g eq ( 4 ) i n t o i n t o eq ( 2 ) y i e l d s : Z, rh d a n u n e ' + Z_ a n E n u n e " = >»n (5) Now, m u l t i p l y i n g through by u£ and r e p l a c i n g J | 0 U . a - 1 1 5 -by E n u n , and i n t e g r a t i n g over a l l space, remembering that ( 6 2 ) i U j u n d T = ( 6 ) { 2 j r n d a n u n e . ' +Za nE ) 1u r i e gi v e s : 2 j l n d a n un e ' + ^ a n E n u n e = Za n(E n u n e '+^f,un e ™ ) (7) . \ JEj i-h d a n un e l ^ ^ = ^ a n < ^ ( u n e ^ w ^ ( 8 ) O d t f - S» f - ftf - i G v W , x so /U i-fi dak unuk e dt a n ug^,undr e (9) ' 2rh ^ &&=2^%LM\*J (10) d t Now, i f a n =fa a t t=0, then f o r k/m, and the system i n s t a t e m a t t--0, ak< J t ) = - i A i | < k i | ^ ( f ) l m>e^ dt' (11) where d a k = - i / * 2 a n < u g | j ^ | u n S e ( 1 2 ) dt" f o l l o w s from eq (10), andiO k n= E k-E n/n defines the angular frequency ( 5 8 ) . So the behaviour of the c o e f f i c i e n t a^. w i t h time i s d e f i n e d . Now,, i f the p r o b a b i l i t y that a. system i n a s t a t e m w i l l be i n a s t a t e k a f t e r some time i n t e r v a l t i s ( 6 3 ) : pknr a k a k ( 1 3 ) then the t r a n s i t i o n r a t e between these two st a t e s i s : dPkrn = w k m = ak ^Lk + c- c« (1^) d t d t Therefore, s u b s t i t u t i n g eq ( l l ) ^ i n t o eq (14): Wkm=^- 2<ml^(t)| k> e ^ J < , l f t ( f ) l m > J ^ d t ' + c. :j<kl|^(t\)| i ^ f a [ | f | ( t ) | ^ e i ^ ( t - t ' ) d t " + c c . = ? T 2 1 ( 1 5 ) Now i f J - L ^ t ) i s a random f u n c t i o n , t h i s randomness w i l l , be evidenced i n the matrix elements. Therefore a t r a n s i t i o n r a t e average f o r the m i c r o c r y s t a l i s a measureable q u a n t i t y d e f i n e d by: I k m = h " 2 f* ;(t~r)fCtJ eL"'^dt+ C.C. ( 1 6 ) where T=t-f and f ( t ) = < m | ^ t)\ k^ > -116-Therefore, Wkm= ft-2 G.(T) e ^ d t (1?) Now s i n c e G(f) "becomes very s m a l l i f T^is s h o r t , or long, compared to t ( l o s s of c o r r e l a t i o n ) , the integrand must d i s -appear before t deviates f a r from T^, so the l i m i t s i n eq (17) can be r e p l a c e d by ±03 . F u r t h e r , i n the high-temperature approximation, d e f i n e d by (58-60): tm0 = g(2H 0« kT (18) ( i n t h i s work g p H 0 ~ 6 x l 0 ~ 2 0 , kT~4xl0*" 1^) , and thus 1/T± = W(l+exp(-6/kT)) (20) from c o n s i d e r a t i o n s of the E i n s t e i n c o e f f i c i e n t s of ab-s o r b t i o n and emission (see, i n p a r t i c u l a r , r e f . 64). Thus i f £ « kT 1/T-, = 2W (21) so ( 1 / 2 ^ ) ^ = V = r 2 J G ( t ) e ^ d t (22) - C D And from Appendix .4 eq (22) i s simply the F o u r i e r t r a n s -form of the c o r r e l a t i o n f u n c t i o n , d e f i n e d to be the s p e c t r a l d e n s i t y j (to). APPENDIX 7 THE B L O C H EQUATIONS--A BRIEF DEFINITION The Bloch equations provided a c o r r e c t d e s c r i p -t i o n ( f o r l i q u i d s ) of the magnetic p r o p e r t i e s of ensembles of n u c l e i i n e x t e r n a l magnetic f i e l d s . In an a r b i t r a r y magnetic f i e l d (homogeneous) the equation of motion of the nuclear magnetization f o r an ensemble of f r e e spins i s : dM/dt = ^ M x H (1) In a s t a t i c f i e l d , H Z = H Q say, the trend of the mag-n e t i z a t i o n to an e q u i l i b r i u m value Mz=Mo= XoH0 can be f a i r l y a c c u r a t e l y described by: dM z/dt = -[( M z - M 0 ) / T i j (2) Now, i f the nuclear magnetization i s given a component perpendicular to H Q (by, f o r example, an r f p u l s e ) , the various l o c a l f i e l d s cause the transverse magnetization to decay (be-cause the spins are not a c t u a l l y f r e e , but i n t e r a c t w i t h each other and t h e i r surroundings): dM x/dt.= -Mx/T2, dK y/dt = -K v/T 2 (3). In the presence of an a p p l i e d f i e l d (the sum of a DC f i e l d and a much smaller r f f i e l d ) the motion due to r e l a x -a t i o n can be superimposed on the f r e e s p i n motion, y i e l d i n g : dM/dt = f M x H - M Y i ' + M v j ' - Mz - ' . M 0 k« (4) T2^— T T ~ i ' . J S a n (3- k' being u n i t vectors of the l a b o r a t o r y frame of re f e r e n c e , and T^ the l o n g i t u d i n a l r e l a x a t i o n time, T 0 the transverse r e l a x a t i o n time. - 1 1 8 -APPSNDIX 8 F o l l o w i n g are some values of v i s c o s i t y f o r - - - s u l f u r i c a c i d as a f u n c t i o n of temperature. As can be seen, no simple c o r r e l a t i o n seems to e x i s t . temperature v i s c o s i t y -0 °G 48.4 cp 1 5 3 2 . 8 2 0 2 5 . 4 3 0 1 5 . 7 40 1 1 . 5 5 0 8.82 6o 7,22 7 0 6 . 0 9 80 5 . 1 9 A c e t o n i t r i l e has a v i s c o s i t y of .46 cp a t 0°G Nitromethane has a v i s c o s i t y of " .85 cp a t 0°G, and . 6 2 cp a t 25°G 

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