Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1972_A1 S33.pdf [ 11.93MB ]
Metadata
JSON: 831-1.0060141.json
JSON-LD: 831-1.0060141-ld.json
RDF/XML (Pretty): 831-1.0060141-rdf.xml
RDF/JSON: 831-1.0060141-rdf.json
Turtle: 831-1.0060141-turtle.txt
N-Triples: 831-1.0060141-rdf-ntriples.txt
Original Record: 831-1.0060141-source.json
Full Text
831-1.0060141-fulltext.txt
Citation
831-1.0060141.ris

Full Text

11157 CHEMICAL APPLICATIONS OF PULSED AND STEADY-STATE NUCLEAR MAGNETIC RESONANCE ' by Keith N. 'Shaw, 'B.Sc. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY "in the Department Chemistry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 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 the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree 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 purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or 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 not be a llowed without my w r i t t e n p e r m i s s i o n . Department o f CH€hA\ S T K - Y The U n i v e r s i t y o f B r i t i s h Columbia Vancouver; 8, Canada . ABSTRACT A general matrix formulation f o r the e f f e c t s of chemical exchange processes i n high r e s o l u t i o n nuclear magnetic resonance (NMR) has been developed which allows a concise d e s c r i p t i o n and e f f i c i e n t numerical c a l c u l a t i o n of exchange modified lineshapes f o r an a r b i t r a r y number of s p i n - s i t e s . A complete d e s c r i p t i o n of a l l rate processes i s contained i n a s i n g l e matrix, and both f i r s t - and second-order spin systems (as described i n terms of a spin density matrix) are accommodated through s p e c i f i c forms f o r a s p i n - s i t e frequency matrix. The hindered r o t a t i o n about the N-C bond i n substituted amides has been studied on a comparative basis using free energies of a c t i v a t i o n derived from f i r s t - o r d e r rate constants obtained by complete analyses of d i g i t a l lineshape data. A v e r s a t i l e FORTRAN computer program has been.used f o r routine i t e r a t i v e lineshape f i t t i n g and i t has been shown that, even with the p r e c i s i o n now attainab l e using t h i s technique, the most r e l i a b l e k i n e t i c parameter i s the free energy of a c t i v a t i o n . Huckel H-MO and semi-empirical SCF-LCAO-MO c a l c u l a t i o n s have been used i n a d e s c r i p t i o n of the e l e c t r o n i c f a ctors determining the b a r r i e r s to hindered r o t a t i o n and the charge d i s t r i b u t i o n s i n the amides studied experimentally. A p p l i c a t i o n of the Fouri e r transform i n high r e s o l u t i o n NMR has also been considered i n d e t a i l , with p a r t i c u l a r emphasis upon quan t i t a t i v e lineshape studies using the data a v a i l a b l e from simple pulsed NMR experiments. The advantages of the pulse method with d i g i t a l data a c q u i s i t i o n have been examined, and the numerical computations and c o r r e c t i v e factors involved have been incorporated into a general and e f f i c i e n t computer program. TABLE OF CONTENTS Abstract Acknowledgements Chapter 1 Introduction Chapter 2 Theory 2.1 Bloch equations 2.2 Modified Bloch equations 2.3 Saturation e f f e c t s 2.4 Zero saturation l i m i t 2.5 Chemical exchange i n f i r s t - o r d e r spin systems 2-6 ^h^^ica^ exch^o^e i n se^ond'-crder spin systems Chapter 3 Instrumentation 3.1 FT-1064 computer-spectrometer i n t e r f a c e unit 3.2 Rf-pulse gate Chapter 4 Experimentation and c a l c u l a t i o n s 4.1 Hindered r o t a t i o n i n substituted amides 4.1.1 N.,N-dimethyl carbamyl ch l o r i d e 4.1.2 N,N-dimethyl carbamyl bromide 4.1.3 Methyl N,N-dimethyl carbamate 4.1.4 NjN-dimethyl carbamyl f l o u r i d e 4.1.5 Formamide 4.2 Hindered r o t a t i o n i n amides., Huckel MO ca l c u l a t i o n s 4.3 Semi-empirical SCF-LCAO-MO c a l c u l a t i o n s Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 Appendix 1 Appendix 2 References Fourier Transform Applications Basic formulation Resonance condition F i n i t e complex f o u r i e r transform Phase corrections Signal zero c o r r e c t i o n High i-esolution NMR Lorentzian Fourier Transform p a i r Gaussian Fou r i e r Transform p a i r 155 163 168 177 189 191 LIST OF TABLES Following page 2.1 Basis (eigen) functions f o r ABX (J = 0) spin system and corresponding energy l e v e l s . 51 2.2 T r a n s i t i o n frequencies f o r ABX (J^g = 0) spin system 51 2.3 Basis functions for general ABX spin system and spin t r a n s i t i o n operator. 67 2.4 Tra n s i t i o n s f o r AB-part of general ABX spectrum 71 4.1 K i n e t i c data for N,N~dimethyl carbamyl c h l o r i d e , neat 100 4.2 K i n e t i c data f o r N,N-dimethyl carbamyl c h l o r i d e , CCli ( s o l u t i o n 103 4.3 N,N-dimethyl carbamyl chloride a c t i v a t i o n parameters 103 4.4 K i n e t i c data f o r N,N-dimethyi carbamyl bromide 107 4.5 K i n e t i c data f o r methyl N,N-dimethyl carbamate 109 4.6 K i n e t i c data f o r N,N-dimethyl carbamyl f l u o r i d e 116 4.7 Spectral parameters f o r N-formamide, acetone s o l u t i o n 121 4.8 K i n e t i c data f o r formamide, acetone s o l u t i o n 125 4.9 Huckel MO data f o r hindered r o t a t i o n i n substituted N,N-dimethyl amides 135 4.10 Parameters f o r CNDO/2 SCF-LCAO-MO c a l c u l a t i o n s 145 4.11 CNDO/2 MO data f o r formamide 146 LIST OF FIGURES Following page 2.1 Motion of an isolated nuclear magnetic dipole 13 2.2 Nuclear spin isochromats in the rotating frame of reference 17 2.3 Two-site exchange absorption mode lineshapes 25 2.4 Saturation effects for two-site chemical exchange system 30 2.5 (a) Modified Lorentzian component spectral lines and resultant absorption mode exchange lineshapes, k < ft 41 2.5 (b) Combined absorption and dispersion functions and resultant absorption mode exchange lineshapes, k < ft 41 2.6 Modified Lorentzian component functions and resultant absorption mode exchange lineshapes, k > ft 45 2.7 AB-part of first order ABX NMR spectrum 52 2.8 Intramolecular exchange lineshapes for AB-part of a first-order ABX spin system 59 2.9 X-part of first-order ABX NMR system 61 2.10 Intramolecular exchange lineshapes for X-part of a first-order ABX spin system 61 2.11 AB-part of general ABX spectrum 72 2.12 X-part of general ABX spectrum 75 3.1 FT-1064 computer sweep control 82 3.2 FT-1064 control sequence 82 3.3 Spectrometer-computer interface unit 84 3.4 Differential amplifier circuit 85 3 . 5 (a) R f - p u l s e gate c i r c u i t 90 (b) C o n t r o l d c - p u l s e g e n e r a t o r c i r c u i t 92 3 .6 R f - p u l s e gate o p e r a t i o n a l c h a r a c t e r i s t i c s 92 4.1 Lineshape f i t s f o r N,N-dimethyl carbamyl c h l o r i d e 100 4.2 Arrhenius p l o t s f o r N,N-dimethyl carbamyl c h l o r i d e , neat l i q u i d and CCli, s o l u t i o n 103 4.3 V a r i a t i o n of a c t i v a t i o n parameters f o r hindered r o t a t i o n i n N,N-dimethyl carbamyl c h l o r i d e 103 4.4 A c t i v a t i o n parameters obtained from complete lineshape analyses f o r N,N-dimethyl carbamyl ch l o r i d e 105 4.5 Lineshape f i t s f o r N,N-dimethyl carbamyl bromide 107 4.6 Arrhenius p l o t f o r N,N-dimethyl carbamyl bromide 107 4.7 Arrhenius p l o t f o r methyl N,N-dimethyl carbamate, CHC13 s o l u t i o n 109 4.8 Temperature dependence of chemical s h i f t s f o r N,N-dimethyl carbamyl f l u o r i d e , CCli» s o l u t i o n 116 4.9 Lineshape f i t s f o r N,N-dimethyl carbamyl f l u o r i d e 116 4.10 Arrhenius p l o t f o r N,N-dimethyl carbamyl f l u o r i d e 117 4.11 Temperature dependence of chemical s h i f t s f o r 1 5N-formamide, acetone s o l u t i o n 122 4.12 Density matrix lineshape f o r 1 5N-formamide ABCX spin system with intramolecular exchange 125 4.13 Arrhenius p l o t f o r 1 5N-formamide, acetone s o l u t i o n 126 4.14 C o r r e l a t i o n of free energy of a c t i v a t i o n f o r hindered r o t a t i o n with Huckel MO d i f f e r e n t i a l TT-energy 136 4.15 E f f e c t s o f v a r i a b l e carbonyl oxygen Huckel theory Coulomb i n t e g r a l f o r methyl N,N-dimethyl carbamate 137 4.16 C o r r e l a t i o n of free energy of a c t i v a t i o n f o r hindered r o t a t i o n with the group e l e c t r o - n e g a t i v i t y of the X-substituent i n N,N-dimethyl amides 139 4.17 Co r r e l a t i o n of free energy of a c t i v a t i o n f o r hindered r o t a t i o n with the N-C ir-bond order obtained from Huckel MO c a l c u l a t i o n s .140 4.18 Huckel MO e l e c t r o n i c charge density maps f o r the (a) unconjugated and (b) conjugated states of formamide 141 4.19 Huckel MO e l e c t r o n i c charge density maps for the (a) unconjugated and (b) conjugated states of carbamyl f l u o r i d e 141 4.20 Structure of formamide used i n CNDO/2 SCF-LCAO-MO ca l c u l a t i o n s • 143 4.21 Total energy differences f o r formamide planar ground state and va r i a b l e geometry hindered r o t a t i o n t r a n s i t i o n state 149 4.22 Structure of carbamyl f l u o r i d e used i n CNDO/2 SCF-LCAO-MO c a l c u l a t i o n s .152 5.1 Pulsed mode NMR resonance conditions 165 5.2 F i n i t e F o u r i e r transform c h a r a c t e r i s t i c s f o r (a) Lorentzian lineshape system; and (b) Gaussian lineshape system 174 5.3 F i l t e r and r f reference phase corrections f o r a Lorentzian lineshape system 182 5.4 Linear frequency dependent phase c o r r e c t i o n f o r a Lorentzian lineshape system 185 5.5 Amplitude function and phase c o r r e c t i o n f o r a Lorentzian lineshape system • 186 5.6 Resolution enhancement, modified Lorentzian l i n e -shapes 187 5.7 Numerical Four i e r transform d i s t o r t i o n due to non-zero average s i g n a l l e v e l 190 5.8 Free induction decay and f i n i t e Fourier transform spectrum f o r dimethyl nitrosamine 192 ACKNOWLEDGEMENTS Sincere thanks are due to Dr. L. W. Reeves under whose supervision t h i s work was accomplished. Thanks are also due to R. Hobson, E. A l l a n and R. C. Shaddick f o r t h e i r cooperation during the course of this work. Assistance from the Chemistzy Department te c h n i c a l s t a f f and the Computing Centre personnel i s also g r a t e f u l l y acknowledged. CHAPTER I. INTRODUCTION In general, the lineshapes and widths of nuclear magnetic reso-nance (NMR) s p e c t r a l l i n e s are p a r t i c u l a r l y s e n s i t i v e to time-dependent processes occurring within a nuclear spin system. Thus, f o r over a de-cade NMR has been applied, to'varying l e v e l s of s o p h i s t i c a t i o n , to the study of chemical rate processes. This experimental method allows the determination of precise rate constants and associated a c t i v a t i o n para-meters f o r molecular processes such as hindered i n t e r n a l r o t a t i o n , molec-ula r isomerism and chemical exchange which may be described by well de-fined f i r s t order rate constants. A quantitative study leads d i r e c t l y to the evaluation of pos s i b l e mechanisms f o r these molecular processes and also may lead i n d i r e c t l y to fundamental information on the e l e c t r o n i c structures of the molecular systems and t r a n s i t i o n states involved. The s i g n i f i c a n c e of the NMR method i n chemical k i n e t i c studies l i e s i n i t s inherent a b i l i t y to provide data f o r r e l a t i v e l y f a s t pro-cesses from measurements on chemical systems at equilibrium, t h i s i n f o r -mation being unattainable through the a p p l i c a t i o n of conventional chemi-ca l methods. Two fundamental c h a r a c t e r i s t i c s of NMR allow such measure-ments : (i) the c h a r a c t e r i s t i c time scales i n magnetic resonance are r e l a t i v e l y slow f o r microscopic processes. It i s well known that molecular motions give r i s e to generalized r e l a x a t i o n e f f e c t s i n a nuclear spin system. Fluctuating magnetic f i e l d s associated with l a t t i c e motions having frequency components corresponding to the resonant spin precessional frequencies may induce t r a n s i t i o n s between the nuclear energy l e v e l s leading to s p i n - l a t t i c e r e l a x a t i o n processes and observable s p e c t r a l e f f e c t s . When molecular systems are studied as non viscous s o l u t i o n s , these random l a t t i c e f i e l d f l u c t u a t i o n s show only as small time-average e f f e c t s and the i n t e r a c t i o n of the nuclear spins with t h e i r l a t t i c e environment may be considered to be correspondingly weak. Under t h i s condition the time-dependent i n t e r a c t i o n of the nuclear spins with f l u c t u a t i n g l o c a l magnetic f i e l d s becomes dominant. These l o c a l f i e l d s may be considered to be associated with the molecular e l e c t r o n i c chemical s h i e l d i n g and i n d i r e c t spin-spin coupling mechanisms, i n the absence of nuclear e l e c t r i c quadrupolar i n t e r a c t i o n s . Again, i f these f l u c t u a t i o n s are s u f f i c i e n t l y rapid, a time-average magnetic f i e l d acts on each nuclear spin r e s u l t i n g i n a s i n g l e resonance sp e c t r a l l i n e . Very slow f l u c t u a t i o n s , however, allow each spin to precess at a frequency c h a r a c t e r i s t i c of i t s chemical envir-onment r e s u l t i n g i n a spectrum of resonance m u l t i p l e t s . In the intermediate region (for a diamagnetic spin system) the l o c a l magnetic f i e l d f l u c t u a t i o n s determine the lineshape and width of the resonance s p e c t r a l l i n e s . Thus i t i s seen that the o v e r a l l form of an observed NMR spectrum i s c r i t i -c a l l y dependent upon the frequencies associated with l o c a l magnetic f i e l d f l u c t u a t i o n s as compared with the difference 3. i n precessional frequencies of nuclear spins i n s p e c i f i c en-vironmental l o c a l magnetic f i e l d s . Since the environmental f i e l d d i fferences are often very small, r e l a t i v e l y slow mol-ecular motions give r i s e to observable s p e c t r a l e f f e c t s i n NMR spectroscopy and hence rate processes having f i r s t - o r d e r rate constants i n the range 10 ^ to 10 ^ sec ^  may be studied, ( i i ) i n the a p p l i c a t i o n of NMR i t i s possible to control the mo-t i o n of the e f f e c t i v e nuclear magnetic dipole moments through the magnitude and form of the impressed r f magnetic f i e l d and, thereby, also the mode of observation f o r a given nuclear spin system. In t h i s manner, both steady-state and pulsed NMR methods may be used to extend the range and accuracy of chemical rate process studies. Moreover, by using a s p e c i f i c type of magnetic resonance detection, a l l phase information r e l a t i n g to the nuclear motion under consideration i s retained. This feature allows d e t a i l e d c o r r e l a t i o n time analysis of the nuclear and molecular motions f o r a spin system and leads, under normal experimental conditions, to a d i r e c t correspondence between the steady-state and pulsed mode responses of a res-onant spin system through Fourier transform a n a l y s i s . The study of chemical rate processes i s concerned with the trans-f e r of resonant nuclear spins between d i f f e r e n t magnetic environments such that a given spin i s under the action of an e f f e c t i v e f l u c t u a t i n g l o c a l magnetic f i e l d . Such a t r a n s f e r process may be described i n terms of a generalized transverse spin r e l a x a t i o n mechanism, the transverse component of the e f f e c t i v e nuclear magnetization giving r i s e to the resonance s i g -4. nal observed under normal NMR experimental conditions. Although the basic concept of transverse spin relaxation in terms of phase relat ion-ships between individual nuclear spins (or spin isochromats) is physi-ca l ly simple, an exact theoretical analysis is often subtle and compli-cated. Analytical formulations for steady-state NMR lineshapes modi-fied by nuclear spin transfer effects are necessarily complicated.even in the simplest cases., and the l i terature in this f i e ld lias become quite extensive. The importance of spin relaxation effects was > recognized in the or iginal NMR work of Bloembergen et a l . * , who considered magnetic moments under the action of random fluctuating magnetic f ie lds ; and 2 concurrently by Bloch , in the introduction of his phenomenological equations of motion for a nuclear spin system incorporating a transverse '" . _ ^ T n o r - ^ o . i , 3 , 4 spin relaxation t ime, x'1'1 -"-333, ouLoivSKy e l a i . * presented the f i r s t formulation for a magnetic resonance lineshape in the presence of transfer processes within a f irst-order (weakly coupled) nuclear spin system. This semi-classical treatment was based upon the Bloch equations, and the only spectral effect, in addition to those due to natural re-laxation processes, i s considered due to random modulation of the nuclear spin precessional frequencies. A similar description for such a nuclear spin system, as suggested by Halm and Maxwell 5, was i n -troduced by McConnell^. This formulation allows a simplified intro-duction of spin transfer probabili t ies direct ly into the phenomenological equations, as proposed by Bloch, through terms analogous to those des-cribing transverse spin relaxation. These terms, involving a charact-e r i s t i c correlation time, are an inherent part of a general stochastic theory of magnetic resonance lineshapes as exemplified by the more s o p h i s t i c a t e d theories of Markoffian random modulation developed by 7 8 9 Anderson and Kubo . The introduction of t r a n s f e r e f f e c t s i n t h i s manner i s also consistent with the basic concept, i m p l i c i t i n the phenomenological Bloch equations, that nuclear magnetic d i p o l a r motion associated with generalized r e l a x a t i o n e f f e c t s may be superimposed on that of an i s o l a t e d nuclear spin i n an applied magnetic f i e l d . For a simple uncoupled nuclear spin system, i t i s possible to avoid an e x p l i -c i t .quantum mechanical d e s c r i p t i o n of spin t r a n s f e r e f f e c t s since the complete spin Hamiltonian, i n the absence of a perturbing r f magnetic f i e l d , contains only secular terms representing nuclear energy l e v e l separations. Furthermore, i n c e r t a i n cases spin-spin coupling may be included through an a d d i t i o n a l secular term i n accordance with normal f i r s t order perturbation theory. For a general c o u p l e d r m r . l o a r s p i r i system, however, magnetic r e l a x a t i o n processes can be r i g o r o u s l y de-scribed only i n terms of non-equilibrium quantum s t a t i s t i c a l mechanics i n the density matrix f o r m a l i s m ^ The basic complication i n a coupled spin system (in a non-viscous l i q u i d ) i s due to the non-secular term describing i n d i r e c t spin-spin coupling. This term may be consid-ered to represent a mixing of basis eigen-states i n the spin system so that f l u c t u a t i n g magnetic f i e l d s associated"with the spin-spin mech-anism may be e f f e c t i v e i n inducing t r a n s i t i o n s between nuclear energy l e v e l s . A general quantum mechanical formulation of spin t r a n s f e r 14 15 e f f e c t s i n NMR has been developed by Kaplan ' and extended by Alexander"^ ^ . Through the d e f i n i t i o n of a spin t r a n s f e r operator, t h i s work shows that the d e s c r i p t i o n of spin t r a n s f e r e f f e c t s i s con-20-2; s i s t e n t with a generalized spin r e l a x a t i o n theory developed by Bloch and takes a form completely analogous to that used i n the s t o c h a s t i c 23 24 25 theories. More recent l y , Johnson ' and Binsch have proposed a l -ternative quantum mechanical formulations f o r intra-molecular t r a n s f e r processes i n which the time dependence of the average density matrix fo r a l l p o s s i b l e molecular spin configurations i s described by a i • . „ . 11,26 L i o u v i l l e operator Pulsed mode NMR d i f f e r s from the steady-state mode i n that a large pulsed r f magnetic f i e l d i s applied to a resonant nuclear spin system to rotate the thermal equilibrium resultant nuclear magnetization into a plane perpendicular to the s t a t i c applied magnetic f i e l d . Following the pulse, nuclear spin isochromats precess i n the transverse plane and dephase due to magnetic f i e l d inhomogeneity and general transverse r e l a x a t i o n processes, the transverse component of the nuclear magnetization giving r i s e to the observed NMR free induction decay s i g n a l . Thus the form of t h i s decay i s also strongly dependent upon 27 nuclear spin t r a n s f e r e f f e c t s . In 1961, Woessner considered the response of the nuclear magnetization during and following a pulsed magnetic f i e l d f o r an uncoupled spin system i n terms of the modified Bloch equations^ previously mentioned. The form of the predicted free induction decay as modified by spin t r a n s f e r e f f e c t s was then 28 q u a l i t a t i v e l y v e r i f i e d experimentally by Reeves and Wells . However, a single pulse experiment can only give information equivalent to that obtained i n steady-state studies of chemical rate processes. On the other hand, pulsed mode NMR becomes a v e r s a t i l e and independent method when multi-pulse sequences are used to produce corresponding spin echo 29 t r a i n s . Such pulse sequences e f f e c t i v e l y remove the influence of magnetic f i e l d inhomogeneity, which normally l i m i t s the range of rates 7. 30 that may be measured by NMR techniques. Carr and P u r c e l l and Meiboom 31 and G i l l have developed s p e c i f i c multi-pulse sequences to reduce syst-ematic errors in the determination of transverse r e l a x a t i o n times with 32 contributions from spin t r a n s f e r processes. In 1963, Luz and Meiboom f i r s t reported the measurement of spin t r a n s f e r rates using such a pulse sequence. A n a l y t i c a l formulations f o r the decay of C a r r - P u r c e l l 33 spin echo amplitudes have been developed by Bloom, Reeves and Wells 34 and Powles and Strange for simple uncoupled spin systems i n accordance with a s t o c h a s t i c theory based upon a c l a s s i c a l s t a t i s t i c a l averaging of the accumulated phases of spin isochromats. S i m i l a r r e s u l t s were 35 36 obtained by Allerhand and Gutowsky ' using the modified Bloch equa-t i o n s . The analysis has been extended to include coupled spin systems 37 38 using Alexander's formalism ' . About the same time, Gutowsky, Void 39 and Wells developed a matrix formulation based upon the Anderson-40 41 Weiss and Banwell-Primas treatments of general spin systems. In the study of chemical rate processes, a fundamental advan-tage of the pulsed NMR method i s an extension to the measurement of rate constants considerably higher than those normally a c c e s s i b l e to q u a n t i t a t i v e steady-state methods. The pulse method, however, shows an inherent lack of s e l e c t i v i t y f o r the various homonuclear spin t r a n s i t i o n s usually observed i n the steady-state h i g h - r e s o l u t i o n NMR spectrum of l i q u i d s . Due to the wide frequency d i s t r i b u t i o n associated with an intense pulsed r f magnetic f i e l d , t h i s f i e l d i n t e r a c t s with a l l n u c l e i of. a given species i n a molecular system. Although i t i s p o s s i b l e to obtain s e l e c t i v i t y i n a very simple spin 42 system using conventional spin echo techniques, general studies of 8. transverse r e l a x a t i o n properties of i n d i v i d u a l nuclear spin t r a n s i t i o n s 43-are most r e a d i l y measured using the phenomenon of rotary spin echoes 45 Comparable studies i n the steady-state mode use double resonance 46-48 techniques . In addition, these double resonance techniques allow quantitative studies of,much slower rate processes than those normally accessible to NMR methods. Hindered i n t e r n a l r o t a t i o n i n molecular systems such as amides, n i t r i t e s and nitrosamines i s a process of considerable chemical i n t e r e s t i n that the magnitude and form of the p o t e n t i a l b a r r i e r to r o t a t i o n i s expected to be d i r e c t l y r e l a t e d to the d e t a i l e d e l e c t r o n i c structure of the system. This r o t a t i o n may be considered i n terms of an i n t r a -molecular nuclear spin t r a n s f e r process, described by a f i r s t - o r d e r rate „„„ ,~ -1- 4- Tn,~ Ml,tn m4-1 , ~ ,1 o n , . t l n k m . n o -v- ^ U n o l 1 „ <- , . n +- ,-, A +• ^ r t i . J i n r of such molecular rate processes, u s u a l l y i n a c c e s s i b l e to chemical kin-e t i c techniques. Of p a r t i c u l a r i n t e r e s t i s the hindered r o t a t i o n about 49 the N-C bond of amides as o r i g i n a l l y postulated by Pauling * , being of importance i n the theories of the structure of p r o t e i n s ^ and many other molecules of b i o l o g i c a l s i g n i f i c a n c e containing the common pep-t i d e f u n c t i o n a l group. Although the IR spectrum of N-methylacetamide had been i n t e r p r e t e d 5 * i n terms of such a r o t a t i o n , the analysis of 52 the NMR spectrum of N,N-dimethylformamide by P h i l l i p s i n 1955 demon-strated unequivocally the existence of hindered r o t a t i o n i n such a 53-57 system. As described i n the reviews a v a i l a b l e , the determination of hindered i n t e r n a l r o t a t i o n rates and the corresponding p o t e n t i a l 5 8 b a r r i e r s , i n accordance with f i r s t - o r d e r absolute rate theory , in substituted amides has become one of the major topics of NMR k i n e t i c 9 . studies. To date, however, very few r e l i a b l e k i n e t i c measurements e x i s t for these systems. Moreover, a comprehensive study of a series of chemically r e l a t e d compounds has not yet been reported. This i s due i n part to the necessity of optimum instrumental operating con-d i t i o n s and i n part to the lack of d e t a i l e d analysis of the r e s u l t s obtained. In addition, the systematic errors inherent in the d i f -ferent NMR methods applied must be c a r e f u l l y considered. Only very r e c e n t l y has s u f f i c i e n t experimental care been exercised i n and soph-i s t i c a t e d analysis been applied to the determination of q u a n t i t a t i v e k i n e t i c data ' " ~.Even though the systematic errors involved have been considered i n some d e t a i l * ^ , the r e s u l t s obtained using steady-state and pulsed mode NMR are seldom consistent. Nonetheless, the v a l i d i t y of such k i n e t i c data i s exemplified by the consistent r e s u l t s obtained (by the f i t t i n g of t o t a l NMR lineshapes) by independent re-search g r o u p s ^ f o r N,N-dimethylformamide. Of necessity, i n the i n i t i a l development of NMR methods f o r k i n e t i c studies, molecular systems were chosen f o r s i m p l i c i t y of analysis rather than chemical s i g n i f i c a n c e . Furthermore, approximate methods have been applied without due consideration of t h e i r v a l i d i t y i n p a r t i c u l a r studies. Chemists are b a s i c a l l y i n t e r e s t e d i n the e l e c t r o n i c s t r u c t -ures of and i n t e r - r e l a t i o n s h i p s between a ser i e s of r e l a t e d molecular systems, f o r example, the substituted amides. For t h i s reason, the purpose of the present thesis i s to develop consistent and v e r s a t i l e t h e o r e t i c a l and experimental techniques f o r e f f e c i e n t analysis of NMR k i n e t i c data i n a systematic study of hindered i n t e r n a l r o t a t i o n i n substituted amides. With the advent of field-frequency locked steady-state spectrometers and d i g i t a l data a c q u i s i t i o n systems, i t i s now possible to obtain r e l i a b l e s p e c t r a l data that may be r a p i d l y pro-cessed to give the parameters of chemical i n t e r e s t . General equations are developed, consistent with a prescribed p h y s i c a l model f o r a given rate process, and the resultant NMR spectra are numerically computed and p l o t t e d using a high speed computer. Experimental lineshapes i n d i g i t a l form can then be compared d i r e c t l y with t h e o r e t i c a l spectra. The well known uncoupled AB spin system, and the r e l a t e d ABX spin system, i s considered i n d e t a i l to allow the development of a more gen e r a l model applicable to steady-state and pulsed mode analyses and a comparison of d i f f e r e n t t h e o r e t i c a l treatments. Under normal experimental conditions, the steady-state and pulsed mode responses of a nuclear spin system are r e l a t e d by a Fourier t r a n s f o r m a t i o n ^ To date, Fourier transform techniques have not been applied to any extent i n high- r e s o l u t i o n NMR spectros-c o p y ^ '^and the emphasis has been upon the enhancement of si g n a l - t o -71 noise r a t i o . In view of increased i n t e r e s t i n lineshape studies, a t h e o r e t i c a l analysis has been made to asce r t a i n the possible advan-tage of such a transformation i n the analysis of pulsed mode NMR data. Towards t h i s end, the necessary computer programs f o r rapid iiumerical computations have been developed. A spectrometer system f o r Fourier transform studies has also been developed and the operational charact-e r i s t i c s of the c r i t i c a l c i r c u i t r y involved has been considered i n de-t a i l . With the a v a i l a b i l i t y of d i g i t a l data a c q u i s i t i o n systems, accurate data may be accumulated very r a p i d l y i n a most convenient form and i t i s shown that t h i s general method i s extremely v e r s a t i l e 1 1 . and does not include many of the inherent disadvantages of the corre-sponding normal steady-state methods. The extension of d i g i t a l tech-niques to multi-pulse NMR procedures also has obvious advantages and may lead to a s i g n i f i c a n t reduction i n the large systematic errors associated with t h i s method at the present time. In t h i s t h e s i s , the b a r r i e r s to i n t e r n a l r o t a t i o n i n a se r i e s of s u b s t i t u t e d amides have been considered using routine t o t a l l i n e -shape f i t t i n g based upon the p h y s i c a l models, t h e o r e t i c a l expressions and computer programs developed. This data has been correlated with that previously a v a i l a b l e . In addition, molecular o r b i t a l c a l c u l a t i o n s 72-74 applying the Huckel and complete neglect of d i f f e r e n t i a l overlap 75-77 (CNDO) approximations have been used to i n t e r p r e t the measured k i n e t i c parameters i n terms of molecular e l e c t r o n i c structure. In p a r t i c u l a r , the p o t e n t i a l b a r r i e r to hindered r o t a t i o n and other (spec-t r a l ) properties of the parent compound formamide have been considered i n d e t a i l f o r comparison, where po s s i b l e , with a l t e r n a t i v e molecular o r b i t a l c a l c u l a t i o n s . The e l e c t r o n i c structure of carbamyl f l u o r i d e and the r e l a t e d compounds ac e t y l f l u o r i d e and acetaldehyde have also been studied to obtain information on the general a p p l i c a b i l i t y of the aforementioned approximate MO methods. 12. CHAPTER 2. THEORY 2.1 Bloch Equations. To allow a consistent development of the theory of nuclear mag-ne t i c resonance e f f e c t s due to nuclear spin t r a n s f e r , a c l a s s i c a l vector model w i l l be considered to describe the dynamics of a general nuclear spin system. As the Bloch equations f o r such a system are fundamental to a l l following discussions, the basis f o r these equations w i l l be b r i e f l y formulated. The motion of an i s o l a t e d nuclear magnetic dipole, u, i n a time independent uniform magnetic f i e l d , II . i s described i n a f i x e d frame r • —o of reference Oxyz by % = yxYH (2.1.1) dt ~ ~° where y i s the nuclear gyromagnetic r a t i o . Assuming the magnetic f i e l d to be directed along the z-axis so that H = kH , (2.1.1) becomes -i/£ = ui0.yu-*k , co0 » Y H 0 (2.1.2) where k i s the un i t vector i n the z - d i r e c t i o n and co i s the Larmor f r e -— o quency (rads. sec *) . This equation of motion i s consistent with quantum 78 mechanical concepts i n that i t i s r e a d i l y shown that u may be replaced by the expectation value <u>. The e f f e c t of a c i r c u l a r l y p o l a r i z e d magnetic f i e l d , H(t), de-fi n e d by 1 3 . H ( t ) = I L c o s o J t . H f t ) = - I - L s i n w t , w > 0 ( 2 . 1 , 3 ) x 1 ' yK J i > o n a n u c l e a r d i p o l e may b e r e p r e s e n t e d i n t e r m s o f a v e c t o r m o d e l a s s h o w n i n F i g . 2 . 1 a . T h a t i s , f o r t h e p a r t i c u l a r g e o m e t r i c a l c o n f i g u r -a t i o n s h o w n , t h e n u c l e a r d i p o l e t e n d s t o p r e c e s s a b o u t t h e m a g n e t i c f i e l d v e c t o r s a n d w i t h a n g u l a r f r e q u e n c i e s <s>^ a n d OJ^ , r e s p e c t -i v e l y . I n a c c o r d a n c e w i t h t h e g e n e r a l e q u a t i o n ( 2 . 1 . 2 ) , co 1 = y H , . ( 2 . 1 . 4 ) U n d e r n o r m a l e x p e r i m e n t a l c o n d i t i o n s , i s a s s o c i a t e d w i t h a l i n e a r l y p o l a r i z e d m a g n e t i c f i e l d , H ( t ) , d e f i n e d b y : H x ( t ) = 2 H 1 c o s o ) t , H ( t ) = 0 , ( 2 . 1 . 5 ) a s p r o d u c e d i n a s i m p l e c o i l . A f i e l d o f t h i s f o r m , h o w e v e r , m a y b e c o n -79 s i d e r e d i n t e r m s o f t w o c o m p o n e n t c i r c u l a r l y p o l a r i z e d f i e l d s , o n e o f w h i c h i s e q u i v a l e n t t o t h a t d e s c r i b e d b y E q . ( 2 . 1 . 3 ) . T h e c o m p l i c a t e d o v e r a l l m a g n e t i c d i p o l a r m o t i o n d u e t o t h e i n d e p e n d e n t m a g n e t i c f i e l d s , H a n d H i s s i m p l i f i e d b y c o n s i d e r i n g * O "1 80 t h e s y s t e m i n a r o t a t i n g r e f e r e n c e f r a m e , O u v z , t o . e l i m i n a t e e f -f e c t i v e l y t h e t i m e d e p e n d e n c e a s s o c i a t e d w i t h H ^ . I f t h e t i m e d e p e n d e n c e o f y_ i n t h e r o t a t i n g f r a m e o f r e f e r e n c e i s d e f i n e d b y ( d y / d t ) , i t f o l l o w s t h a t = + , u?r= -uyk ( 2 - 1 . 6 ) w h e r e OJ (>0) i s t h e a n g u l a r f r e q u e n c y o f t h e r o t a t i n g f r a m e . T h e r e f o r e , f r o m E q s . ( 2 . 1 . 2 ) a n d ( 2 . 1 . 6 ) i t i s s h o w n t h a t F i g . 2.1 Motion of an i s o l a t e d nuclear magnetic dipole 14. (i^)r = /*: * (w0-u)r)k , (2.1.7) and t h i s equation of motion may be expressed i n the form where H r r i s an e f f e c t i v e magnetic f i e l d i n the z - d i r e c t i o n and - e f f . " ^ e f f = W o " W r ' ^ m a ^ ^ e a s s u m e c ^ t n a t the magnetic f i e l d vector I-I i s along the u-axis of the r o t a t i n g frame and that co^  = co, c f . Eq. (2.1.3) Under these conditions, the nuclear magnetic dipole precesses about a re s u l t a n t time-independent magnetic f i e l d H: H = H,i + H __k, (2.1.9) — 1— e f f — as shown in V i a . 2.1b. with an anuijlar freouenr.v Q ci vp.n hv JL * [ U - u ^ f + l ^ ] * (2.1.10) The nuclear magnetic resonance condition may now be defined as that corresponding to a maximum time-independent perturbation of the nuclear d i p o l a r motion by the applied magnetic f i e l d , H(t) , t h i s con-d i t i o n being defined by co = co = co . (2.1.11) r o v J Thus, (JJq may be r e f e r r e d to as the resonance (Larmor) frequency f o r an i s o l a t e d nuclear spin. Also, i n accordance with Eqs. (2.1.8, 10, 11): and thus the condition on co defi n i n g resonance also determines the 15. steady-state condition f o r the nuclear system i n the r o t a t i n g frame of reference. That i s , under t h i s condition, an i n i t i a l l y in-phase r e l a t i o n s h i p between y_ and H^, corresponding to maximum i n t e r a c t i o n , i s time-independent. At exact resonance, the res u l t a n t magnetic f i e l d i n the r o t a t i n g reference frame i s simply and hence the magnetic dipole precesses about the u-axis at the frequency OJ^. In the f i x e d reference frame, t h i s resonance precessional motion appears as a slow nutation superimposed upon the Larmor precessional motion with a cor-responding change in. dipole o r i e n t a t i o n , as defined by the angle 3 i n Fig . 2.1b. Under steady-state nuclear magnetic resonance conditions, a spe c t r a l l i n e of f i n i t e width i s observed. The e f f e c t s of contri b u t i n g magnetic f i e l d s at. t h e s i t e of a specific, nuclear spin w i t h i n a. given spin system, i n the absence of an applied o s c i l l a t o r y magnetic f i e l d , may be considered i n terms of: (i) s t a t i c f i e l d inhomogeneity, ( i i ) spin-spin r e l a x a t i o n , and ( i i i ) s p i n - l a t t i c e r e l a x a t i o n . A l o c a l s t a t i c magnetic f i e l d f o r a given nuclear spin i s determined by the molecular e l e c t r o n i c environment, giving r i s e to a basic chemical s h i f t e f f e c t , and by the smaller inhomogeneity associated with the ap-p l i e d magnetic f i e l d , H^. These f i e l d s give r i s e to a resonance frequen-cy d i s t r i b u t i o n and an associated s p e c t r a l l i n e broadening. The exact form of the inhomogeneity broadening i s d i f f i c u l t to define e x p l i c i t l y , but a common assumption made i s that the resonance frequency d i s t r i b u -t i o n may be described by a Lorentzian lineshape function, f(w), of the 16. general form (2.1.12) where ca i s the independent frequency v a r i a b l e and oi^ i s the mean reso-nance frequency f o r a given s p e c t r a l l i n e ; A i s a normalization constant and 5 i s a line-width parameter. The c h a r a c t e r i s t i c line-width at h a l f -maximum, A/2, i s then given simply as 2/£. An a l t e r n a t i v e lineshape function i s that corresponding to a Gaussian frequency d i s t r i b u t i o n as described by the lineshape function $Cw) = A e*p [ - ^ ( w - u ) , ) * ' ] (2.1.13) with a corresponding line-width 2(2£n2) /£. The l i f e t i m e within a nuclear Zeeman energy l e v e l may be con-sidered to be l i m i t e d due to simultaneous mutually induced t r a n s i t i o n s between adjacent resonant nuclear spins, these t r a n s i t i o n s being i n -duced through a time-dependent magnetic d i p o l a r i n t e r a c t i o n . In accordance with the quantum mechanical uncertainty p r i n c i p l e , such a process leads to an e f f e c t i v e broadening of the Zeeman energy cor-responding to a f i n i t e width f o r the spe c t r a l l i n e associated with the induced t r a n s i t i o n s , t h i s l i f e t i m e broadening being described by a Lorentzian lineshape function. A c h a r a c t e r i s t i c spin-spin relax-a t i o n time, T20' m a y then be defined to describe t h i s line-width i n accordance with Eq. (2.1.12), with £ = T„ . T_ i s normally r e f e r r e d n 2o 2o to as the natural transverse spin r e l a x a t i o n time. It should be noted that, i n more general terms, a c h a r a c t e r i s t i c time defined i n t h i s manner e f f e c t i v e l y describes the time-dependent phase r e l a t i o n s h i p s 1 7 . between precessing resonant nuclear spins within a given spin system. Thus the concept of a generalized transverse spin r e l a x a t i o n time i s a v e r s a t i l e means of describing many rel a x a t i o n processes giving r i s e to spin dephasing e f f e c t s . In contrast to spin-spin r e l a x a t i o n pro-cesses showing a conservation of energy within a spin system, spin-l a t t i c e r e l a x a t i o n processes involve a net tr a n s f e r of energy between the spin system and i t s l a t t i c e environment to maintain thermal equil-ibrium conditions f o r the complete s p i n - l a t t i c e system. Also, due to the i n t e r a c t i o n between the spin system and i t s l a t t i c e environment, f l u c t u a t i n g environmental magnetic f i e l d s may induce t r a n s i t i o n s be-tween nuclear Zeeman energy l e v e l s leading to an a d d i t i o n a l s p e c t r a l l i n e broadening which may be described by a s p i n - l a t t i c e r e l a x a t i o n time, T,. x Under the assumption that the complete resonance s p e c t r a l lineshape i s described by a Lorentzian function, a t o t a l transverse spin r e l a x a t i o n time, T^, may be defined by A- = J_ + i + J , + J L . (2 .1 .14) •k where T^ and £ are r e l a x a t i o n times describing f i e l d inhomogeneities and a d d i t i o n a l transverse spin r e l a x a t i o n processes, r e s p e c t i v e l y . In the study of l i q u i d systems, motional averaging determines a general condition: i n the absence of s p i n - l a t t i c e r e l a x a t i o n mechanisms due to paramag-n e t i c species and nuclear quadrupclar i n t e r a c t i o n s . 18. To allow a d i r e c t a p p l i c a t i o n of the s e m i - c l a s s i c a l concepts o u t l i n e d above, i t i s now considered that a macroscopic nuclear spin 2 29 isochromat ' , M(co , <{>) , may be defined as the resultant magnetization due to a l l spins having a resonance frequency COq and an associated time-dependent phase angle (j). In t h i s manner, the resonance frequency d i s t r i b u t i o n f o r a. given s p e c t r a l l i n e i s simply r e l a t e d to the prop-e r t i e s of a macroscopic nuclear spin system. The equation of motion f o r such an isochromat i n the r o t a t i n g frame of reference with an angular frequency co (corresponding to that of the applied o s c i l l a t o r y magnetic f i e l d H(t)) i s given i n accordance with Eqs. (2.1.8) and (2.1.9) as = Xl£l*,4>>**]-^i - i V j , - ^ C M x - M o j k (2.1.16) dt 12, \z T i where u, V, have been defined as the Cartesian components of the isochromat M(x, (j)), as shown i n F i g . 2.2, and x i s now the independent variable, defined as x = co - co , while M i s the thermal equilibrium o o value of M . The t o t a l e f f e c t i v e magnetic f i e l d H appearing i n t h i s equation i s 11 = H i + (x/y)k_, c f . Eq. (2.1.9). It should be noted that the v a r i a b l e x has been defined on the i n t e r v a l -°° < x < 0 0 such that x > 0 corresponds to oo > OOq . From Eq. (2.1.16), the component equations of motion are given as AM' + JL u. - >cV • = o - X-U. + i _ V = - U3± NU • dt T*. ^ - u i i ^ = - ^ ( ^ - ^ o ) (2.1.17) 19. where O J^ = yH^, c f . Eq. ( 2 . 1 . 4 ) . These equations form the basis for the formulation of nuclear spin t r a n s f e r e f f e c t s i n the following section. 2.2 Modified Bloch Equations. Nuclear spin transfer e f f e c t s i n an uncoupled AB spin system w i l l be considered i n i t i a l l y i n terms of stoc h a s t i c p r i n c i p l e s to develop a simple p h y s i c a l model for such e f f e c t s . The steady-state NMR spectrum foi' t h i s spin system may be represented as shown below: A B 0 'A l B Site-A, associated with a d i s t i n c t e l e c t r o n i c magnetic environment and chemical s h i f t , may be considered i n terms of a mean spin isochromat, M , with a resonance frequency x^ = -Q r e l a t i v e to the average frequency aj Q: ft, (2.2.1) i n the normal r o t a t i n g frame of reference. At any given time, the f r a c -t i o n a l population of nuclear spins associated with s i t e - A , p defines A. the magnitude of the spin isochromat M^. In the absence of spin t r a n s f e r e f f e c t s , M^ and M^ precess independently in the applied magnetic f i e l d H . Due to a r e v e r s i b l e molecular process (such as a hindered rotation) described by f i r s t - o r d e r rate constants and k , a resonant nuclear spin may be considered to be transferred between the s i t e s A and B i n a random s t a t i s t i c a l manner. The basic assumptions made i n describing t h i s t r a n s f e r process are as follows: (i) a l l spins remain i n s i t e - A with a mean l i f e t i m e x. A. u n t i l an instantaneous trans f e r takes place into s i t e - B . Precessional e f f e c t s i n the t r a n s f e r i n t e r v a l are neglected and as transfer into a s i t e of the same type has no observable e f f e c t on the spin system, only t r a n s f e r i n t o a d i f f e r e n t s i t e i s considered; ( i i ) f o r any spin i n s i t e - A , there i s a constant p r o b a b i l i t y k. per u n i t time for t r a n s f e r into s i t e - B . t h i s prob-A. - -a b i l i t y being i n v e r s e l y proportional to the f r a c t i o n a l population p^; ( i i i ) the l i f e t i m e x^ i s independent of the associated spin r e l a x a t i o n times T,. and T „ A ; IA 2k' (iv) nuclear spin isochromats relax independently except f o r spin t r a n s f e r e f f e c t s . Under these assumptions, the s i t e l i f e t i m e x^ and the corresponding rate constant k^ are simply r e l a t e d by k A = x ~ \ (2.2.2) 81 and i n accordance with the p r i n c i p l e of d e t a i l e d balance P A k A = p B k B . (2.2.3) w i t h P A + P B = ] - • As the observed NMR s i g n a l i s due to the transverse component of the nucl e a r magnetization (and f o r a n a l y t i c s i m p l i c i t y ) a complex transverse magnetization f o r s i t e - A , G^, i s defined i n the r o t a t i n g frame of reference by G A = I U A + i v A l > ( 2 ; 2 - 4 ) as shown i n F i g . 2.2. The Bloch equations f o r the s p i n isochromat M may now be expressed i n complex form i n accordance with Eqs. (2.1.16) and (2.1.17) as cfct 1 (2.2.5) where e. = ioo and io = w - fl. Assuming n e g l i g i b l e s a t u r a t i o n 1 -p A A O and q u a s i - s t e a d y - s t a t e c o n d i t i o n s : w1 ->• 0, M . ->- M and Eq. (2.2.5) J. Z A O A reduces to <U*k + U A + t w ) & A = - U U 3 i P a / ^ 0 (2-2.6) with M q A = P^M > a n c^ the thermal e q u i l i b r i u m z-component magneti-z a t i o n f o r the complete nuclear s p i n system. A s i m i l a r equation a p p l i e s to the s i t e - B magnetization, so that f o r the complete s p i n system: G = G A + G B. (2.2.7) F o l l o w i n g M c C o n n e l l ^ , t h e t i m e - d e p e n d e n c e o f t h e t r a n s v e r s e m a g n e t i z a t i o n due t o n u c l e a r s p i n t r a n s f e r p r o c e s s e s may now be d e s c r i b e d b y : s u c h t h a t - k . G d e f i n e s t h e r a t e o f t r a n s f e r o f t r a n s v e r s e m a g n e t i z a t i o n A A. f r o m s i t e - A and k i s t h e f i r s t - o r d e r r a t e c o n s t a n t f o r t h i s t r a n s f e r . The t i m e - d e p e n d e n c e i s i n c o r p o r a t e d i n t o t h e n o r m a l B l o c h e q u a t i o n s , E q s . ( 2 . 2 . 6 ) , t o g i v e a m o d i f i e d e q u a t i o n o f m o t i o n f o r s i t e - A m a g n e t i z a t i o n : and a c o u p l e d e q u a t i o n f o r s i t e - B m a g n e t i z a t i o n w i t h a . = •—^— + k - ito , c f . E q . ( 2 . 2 . 5 ) . A s s u m i n g t h e t o t a l t r a n s -2A v e r s e r e l a x a t i o n t i m e , T^, as d e f i n e d i n t h e a b s e n c e o f s p i n t r a n s f e r p r o c e s s e s , t o be s i t e i n d e p e n d e n t , i t f o l l o w s t h a t + i u = r^ + i (x + ft) ,• x = to - O J Q ( 2 . 2 . 1 0 ) w i t h r^ = + k^ . These m o d i f i e d B l o c h e q u a t i o n s may now be e x -p r e s s e d i n t h e m a t r i x f o r m _ X Cx s + l u), 1 P ( 2 . 2 . 1 1 ) ..15: ~ ~ 1 °~ The t e r m ioo M Q P may be r e f e r r e d t o as a d r i v i n g t e r m , as i n t h e a b s e n c e o f an a p p l i e d o s c i l l a t o r y m a g n e t i c f i e l d : to^ = 0 . Under s t e a d y - s t a t e c o n d i t i o n s , ( d G / d t ) = 0 , and i t f o l l o w s f r o m E q . ( 2 . 2 . 1 1 ) t h a t R-G = - iw.M P, ( 2 . 2 . 1 2 ) = — 1 o— v 23. or e x p l i c i t l y , PA G c 6 4 The t o t a l complex transverse magnetization may now be derived from Eq. (2.2.12) as a function, of the independent v a r i a b l e x: whe re 6 -C = 6 + lC and (2.2.13) The NMR absorption mode corresponds to the component trans-verse nuclear magnetization i n quadrature phase with the applied o s c i l -l a t o r y magnetic f i e l d vector, H , as shown i n r i g . 2.2. Thus the ab-sorption mode lineshape, V(x), i s given i n accordance with Eq. (2.2.13) as the imaginary part of G (x), v i z . , 6 Z 4 CZ (2.2.14) This lineshape function applies to a general-population two-site un-coupled AB nuclear spin t r a n s f e r system under normal steady-state NMR conditions. The term w^ M expresses the l i n e a r dependence of s i g n a l 2 4 . amplitude upon the magnitude of H , i n the absence of saturation e f f e c t s , and may be considered as a normalization f a c t o r . The two-site equal population system (p^ = p = 0 . 5 ) i s of considerable chemical i n t e r e s t , and i n accordance with Eq. ( 2 . 2 . 1 4 ) , the lineshape function V(x) takes the s i m p l i f i e d form where 6 • ^ ( l + a l c ) + ^ - ^ ( 2 . 2 . 1 5 ) T z v T z and A i s the normalization constant. From Eq. ( 2 . 2 . 1 5 ) , the p o s i t i o n s of the function V(x) maxima are determined as x 1 = ± ( f t 2 - 2 k 2 ) 1 5 rad. sec." ( 2 . 2 . 1 6 ) For k << ft, x' = ± ft, and i n t h i s l i m i t of very slow spin t r a n s f e r , the NMR spectrum consists of two spectral l i n e s separated by the chemical s h i f t 2ft. As the rate of spin t r a n s f e r increases, x' decreases and thus a condition f o r s p e c t r a l l i n e coalescence may be defined as x' = 0 and /2 k - ft. ( 2 . 2 . 1 7 ) It i s now convenient to consider general nuclear spin t r a n s f e r e f f e c t s i n terms of (i) slow spin t r a n s f e r : k < 0 . 2 ft, ( i i ) intermediate spin t r a n s f e r : k - ft, ( i i i ) f a s t spin t r a n s f e r : k > 5 ft. The exchange lineshapes f o r a representative equal population system, - 1 25. as given i n accordance with Eq. (2.2.15), are shown i n Fig. 2.3 for a range of values of the c h a r a c t e r i s t i c parameter k/Q,. For c l a r i t y , these lineshapes have been-normalized, "through the constant A, to an a r b i -t r a r y maximum independent of k. It i s important to note that the chemical s h i f t between s i t e s i n the absence of exchange, 2fi, defines the o v e r a l l range of the rate constant k that may be determined from NMR lineshape analyses. Also, the general population absorption mode lineshape derived f o r the two-site exchange system, Eq. (2.2,14), i s 87 consistent with that given by Gutowsky and Holm ". The v a l i d i t y of the phenomenological Bloch equations has 20 been considered i n d e t a i l and, under the assumptions o u t l i n e d above, modified Bloch equations may be extended to describe nuclear spin t r a n s f e r e f f e c t s i n a general f i r s t - o r d e r NMR exchange system. A matrix formulation f o r an n - s i t e system, based upon the equations of motion derived above f o r the simple two-site exchange system, allows a concise d e s c r i p t i o n of general t r a n s f e r e f f e c t s and also leads to expressions which are r e a d i l y adapted to e f f i c i e n t computer calcu-l a t i o n s . I n i t i a l l y , i t i s assumed that exchange processes modulate a l l frequency d i f f e r e n c e s , |to^  - w_. | , associated with n d i s t i n c t s i t e s of d i f f e r i n g Larmor frequency, OK . Again following McConnell^, a t r a n s f e r of nuclear magnetization may be described f o r the £-mode i n the form (2.2.18) k/fl = 0.032 0.16 0.64 3.2 0.5 P A = 0.65 Two-site exchange absorption mode lineshapes 26. cf. Eq. (2.2.8). The indices refer to site and the double indices refer to f i r s t order transfer rate constants between sites such that k.. £. i j 1 represents the rate of transfer of £-mode magnetization from site i to_ site j . This form of notation is important in defining matrix elements for a multi-site exchange process. The time-dependence of the u-mode magnetization associated with s i t e - i may now be expressed in the Bloch form cit u kl i 1 1 (2.2.19) with w^  = co - . As usual, co is the angular frequency of the o s c i l -latory radio frequency f i e l d defining the normal rotating frame of reference. <o. is defined as the sit.e-i resonance frenuencv. and T_. is ' l 2i the corresponding spin-spin relaxation time including a contribution from magnetic f i e l d inhomogeneity. The total u-mode magnetization for the system is then given by u = Z u^, the site magnetization iso-chromats being independent except for transfer effects. Considering this magnetization as a column vector, u, Eq. (2.2.19) may be expressed in matrix form to include a l l sites, v i z . , d e c -t- fc-.UL - W.V = 0 (2.2.20) oft " = " 7 with = T^ + j£, where is a diagonal n x n matrix with general element T • and K is the transfer rate matrix, n x n, with elements 2i = ' K U = gV^ and ^ = -V-^y , L*L C 2- 2- 2 1) 27. S i m i l a r l y , w i s a diagonal frequency deviation matrix., n x n with elements w^  and V i s the v-mode magnetization vector with elements V.. The off-diagonal elements of the rate matrix K are i n d i v i d u a l f i r s t order rate constants f o r each p a i r of s i t e s , and the diagonal elements are the sums of rate constants from each s i t e - i . Thus, the diagonal elements represent the o v e r a l l rate processes f o r n s i t e s . Also, any column has a zero sum, consistent with d e t a i l e d balance of rate processes, + - 0 S i m i l a r matrix equations can be written f o r the v- and z-mode magne-t i z a t i o n transfers i n v o k e d i n a chemical exchange process: (2.2.22) where = yH^, = + JC with T^ a diagonal s p i n - l a t t i c e relaxa-t i o n matrix, n x n with elements T.., T,. being the s p i n - l a t t i c e relax-l i li b 1 ation time f o r s i t e - i ; P i s a. column vector of the f r a c t i o n a l population fo r each s i t e such that Z p. = 1 and M . = M p. where M i s the thermal . . 1 oi o i o e q u i l i b r i u m value f o r the z-mode magnetization i n s i t e - i . Solving the coupled matrix equations (2.2.20), (2.2.22), the•steady-state V-mode magnetization i s given e x p l i c i t l y by 28. L- § 2 . - i l - T i • (2.2.23) with . _ Z -^ /-Here I i s the un i t n x n matrix and e and .e^ are the inverse matrices corresponding to R^  and R^, r e s p e c t i v e l y . For an a r b i t r a r y reference frequency to , the independent frequency v a r i a b l e x may be defined as x = to - w such that f o r s i t e i w. = x - f t . i I ft. = to. - to a. l o (2.2.24) where ft. i s the chemical s h i f t with respect to to . The vector V l r o — now defines the steady-state NMR absorption mode spectrum as a function of x, v i z . , tt Vtt) = ^ VLfcO = I - i (2.2.25) 1=1 The elements of the matrix w i n Eq. (2.2.23) are now defined by Eq. (2.2.24) and I i s the row vector with each of n elements equal to unity. The corresponding matrix equations f o r the steady-state u-and z-mode magnetizations are now given i n terms of V as u. ~ g z . w . V (2.2.26) 29. This formulation allows a compact and v e r s a t i l e d e s c r i p t i o n of general nuclear magnetization t r a n s f e r e f f e c t s i n a f i r s t - o r d e r NMR spectrum, 46 48 i n c l u d i n g those associated with s p i n - l a t t i c e r e l a x a t i o n ' and nuclear spin t r a n s f e r saturation. 2.3 Saturation E f f e c t s . Under normal steady-state conditions, a saturation f a c t o r S' may be defined for a given NMR s p e c t r a l l i n e i n the form / 1 / S = ± - , 0< S ^ 1 1 + CJ^TIU Now, since i n the presence of chemical exchange saturation e f f e c t s are J . I 1 ^-i _ i - j-i— — r> • „ r?~ r i o o ? •> _i i — are contained i n that part defined as S' = I + Wi , C 2- 3- 1) 2 i t i s convenient to replace co^  by an equivalent parameter 3 defined i n terms of an average saturation f a c t o r S' as where the part defined i n terms of the r e l a x a t i o n times T^ and T^ i s an average value f o r a l l s i t e s . In t h i s way, Eq. (2.3.1) i s written i n the form S ' = I + g £ 2 . (2.3.2) A complete analysis of saturation e f f e c t s would include the i n t e r -83 r e l a t i o n s h i p with the dev i a t i o n from steady-state conditions Under normal experimental conditions, however, l o c k e d - f i e l d spectrom-eters allow a very good approximation to the required slow passage conditions and the above formulation i s adequate. The e f f e c t s of satu r a t i o n on the NMR absorption mode l i n e -shape have been studied f o r a p a r t i c u l a r two-site (A- and B-sites) chemical exchange system described by the s p e c t r a l parameters: p. = 0.5, A 20 = \0. - 0U\ = 10.0 Hz and T„. = T„_ = 0.64 sec. Also, as i s usual 1 A B1 2A 2B f o r l i q u i d systems, i t i s assumed that T.. = T_.. The spin- s p i n r e l a x -X A A a t i o n time, T O A , chosen corresponds to a Lorentzian f u l l width at h a l f -i-r\. maximum of 0.5 Hz and 20, i s the chemical s h i f t between s i t e s i n the absence of exchange, c f . Eq. (2.2.24). By means of an i t e r a t i v e numerical a n a l y s i s , using Eqs. (2.2.23) and (2.2.25), values of the sa t u r a t i o n parameters 3 and S' were determined corresponding to a pre s c r i b e d mean deviation of the lineshape f u n c t i o n , V(x), from that f o r the reference l i m i t of zero s a t u r a t i o n ( 3 = 0 ) . A mean per-centage d e v i a t i o n , AV, was defined f o r N data points as u=i .100 i n which V\ (x) and \\ (x) are the reference and saturated lineshape f u n c t i o n values, r e s p e c t i v e l y , corresponding to the frequency value x^. The r e s u l t s of such an analysis are shown i n F i g . 2.4 f o r AV =2% and 5% as plot s of 3 and S' as a function of l o g j 0 ( k / 2 Q ) . Figure 2.4 Saturation parameters f o r two-site chemical exchange system. It i s seen that saturation e f f e c t s f o r a f i x e d magnitude of the i r -r a d i a t i o n r f magnetic f i e l d , 2H , are most severe i n the l i m i t s of very slow (k << ft) and very f a s t (k >> ft) exchange corresponding to sp e c t r a l l i n e s of minimal width (0.5 Hz). For a mean deviation AV = 5% the values of H^ for k/2ft = 0.01 and 1.0 are computed for 1H NMR ( Y h = 2.7 x 10* rads sec" 1 gauss - 1) to be 0.56 x IO" 2 and 9.4 x 10 2 mgauss, r e s p e c t i v e l y . These f i e l d s correspond to a s p e c t r a l frequency d i s t r i b u t i o n of only 0.002 and 0.04 Hz. However, 84 Grunwald et a l . have shown experimentally that H^ f i e l d s an order of magnitude greater than those c a l c u l a t e d above give n e g l i g i b l e d i s t o r t i o n due to saturation for chemical exchange systems. This implies that, i n actual f a c t a complicated combination of steady-state passage conditions and r f power l e v e l s (H ) determine observed sat u r a t i o n e f f e c t s . As H^ i s increased, saturation gives r i s e to a general broadening of s p e c t r a l l i n e s and leads to an increased apparent f i r s t - o r d e r rate constant, k. A 5% deviation as considered above may lead to an error i n a f i t t e d k value of 5-10%. However, an i r r a d i a t i o n f i e l d i n the intermediate exchange region (k - ft) with a magnitude approximately twenty times greater than that causing d i s t o r t i o n i n the slow exchange region (k << ft) i s s t i l l acceptable (see F i g . 2.4). In the intermediate exchange region, the s p e c t r a l l i n e s are of maximal width (~ ft rad. s e c . - 1 ) and hence minimal inten-s i t y . S t r i c t l y , a s p e c t r a l l i n e i n t e n s i t y i s l i n e a r l y p r oportional to Hj only i n the absence of saturation e f f e c t s , but i t i s seen that the i n t e n s i t y and hence signal-to-noise r a t i o of a recorded lineshape may be increased to l e v e l s acceptable f o r the measurement of r e l i a b l e 32. data f o r lineshape f i t t i n g without adverse d i s t o r t i o n due to sat u r a t i o n , by increasing H following an experimental l i n e a r i t y check f o r n e g l i g i b l e saturation i n the slow exchange l i m i t . Relative H f i e l d strengths corresponding to n e g l i g i b l e saturation d i s t o r t i o n may then be estimated over a complete range of the c h a r a c t e r i s t i c parameter k/2fl from F i g . 2.4, and hence r e l i a b l e spectra f o r complete lineshape f i t t i n g may be ob-tained, i n e f f e c t , i n the l i m i t of zero saturation. 2.4 Zero Saturation Limit.' For a general f i r s t - o r d e r NMR n - s i t e exchange system, the zero saturation l i m i t may be defined by w ->• 0 (to = yti^) and thus the modified Bloch equation describing the steady-state z-mode magnetization i s given in matrix form i n accordance with Eo (2.2.22) as with the inverse of the matrix R . In the absence of chemical ex-change, R = £i» a n c * n e n c e M_z = M P_. That i s , the z-mode magnetization i n each s i t e i s d i r e c t l y proportional to the s i t e f r a c t i o n a l population, p..*, and i s independent of s p i n - l a t t i c e r e l a x a t i o n e f f e c t s . This con-d i t i o n i s shown to be v a l i d also i n the presence of chemical exchange f o r a l l exchange rates and i s considered i n d e t a i l f o r a general two-s i t e system at a l a t e r point. The modified Bloch equations describing chemical exchange i n the l i m i t of zero saturation reduce to two coupled matrix equations given i n accordance with Eqs. (2.2.20) and (2.2.22) : 3ft ' ~ = ~ (2.4.1) •from which the steady-state v-mode magnetization i s given e x p l i c i t l y by V = - ui± M 0 C | P (2.4.2) with ik = +- W. JL2.. w Again, the steady-state NMR absorption mode spectrum i s determined by V(x) as defined i n Eq. (2.2.25). In the l i m i t of zero saturation, Eq. (2,4 = 2) show? that the NMR ahsomtion i n t e n s i t y i s l i n e a r l y r e l a t e d to the e f f e c t i v e magnitude of the i r r a d i a t i n g r f magnetic f i e l d , , and i s independent of s p i n - l a t t i c e r e l a x a t i o n e f f e c t s f o r a l l chemical exchange rates. Within the l i m i t s discussed i n the preceding s e c t i o n , t h i s i s the expression d e f i n i n g the lineshape function V(x) normally used i n the analysis of steady-state NMR data. Although the absorption mode lineshape function, V(x), i s given e x p l i c i t l y by Eq. (2.4.2) i n terms of the r e a l matrix £, the independent v a r i a b l e x i s contained i n the frequency deviation matrix w and hence an evaluation of V(x) over a s p e c i f i e d frequency range requires an inversion of the matrix £ for each value of x. An a l t e r -native formulation i n the l i m i t of zero saturation based upon Eq. (2.4.1) allows a much s i m p l i f i e d c a l c u l a t i o n of V(x). A complex transverse magnetization, G, may be defined f o r an n - s i t e exchange system i n 34. vector form as u + iV , (2.4.3) where the component V describes the v-mode magnetization associated with the j - s i t e . From Eq. (2.4.1), the steady-state transverse magne-t i z a t i o n i s now given by 1^ + L W -1 . P and the corresponding complex lineshape function G(x) follows as -1 ? (2.4.4) where I_ i s as defined previously, c f . Eq. (2.2.25). The absorption mode lineshape function, V(x), i s simply the imaginary part of G(x). The independent v a r i a b l e x appears now only i n the diagonal matrix vv, and thus i t i s possible to transform the matrix [R + iw] in t o a completely diagonal form hence allowing a ready evaluation of G(x) following a sing l e matrix d i a g o n a l i z a t i o n . Now consider the matrix R defined by R = 1 + K - i f i (2.4.5) and r e l a t e d to R by [R + iw] = R + ixj^. In these equations, £ i s a n x n diagonal matrix with elements Q,y the chemical s h i f t f o r the j - s i t e , Eq. (2.2.24), and I, i s the unit n x n matrix. In terms of the diagonalized matrix J\ corresponding to R, Eq. (2.4.4) may be expressed i n the form l - 1 - - i . (k iH) = - t u ^ M.0 I. S 5 (2.4.6) 3 5 . where S is the matrix which diagonalizes R., viz., A = Sf 1 -R-S (2.4.7) Following this transformation, the evaluation of inverse matrix ele-ments is reduced to the determination of reciprocal diagonal elements of the matrix ./V + ixj^ , and the complex function G (x) is defined in component form by Gurt = - c u i ^ g I : (2-4.8) where Xj_ is theJ-th diagonal element of the matrix iV. S_.^  and ( S - 1 ) ^ are the jJ-th elements of the transformation matrices S. and S~1, re-spectively. As the matrix elements involved are a l l complex, the computation of each spectral data point defined by G(x) requires a number of operations in complex arithmetic on a digital computer. These operations are relatively slow and hence the absorption mode lineshape function, V(x), is f i n a l l y given in terms of operations in real arithmetic only by defining a matrix J5 as the real part of the matrix S/j^  A + ix_l] ~ 1 1. .That i s , in accordance with Eq . (2.4.6), V(x) = A I/FvP, in which A is a normalization constant, and the matrix elements B., j k are given by where a..^ and b j j ^ (and Xj and x|) are the real and imaginary parts 3 6 . of S. • (S *) , (and X ), r e s p e c t i v e l y ; d i s defined in. terms of the J K independent v a r i a b l e x as d-j - L^x t + (fj_ + *>) Analogous to Eq. (2.4.8), V(x) i s now given as V(x) = A E I . IB p . (2.4.9) A computer program based upon Eqs. (2.4.7) and (2.4.9) has been de-veloped and has been shown to give r e l i a b l e and e f f i c i e n t i t e r a t i v e t o t a l lineshape analyses f o r m u l t i - s i t e exchange systems. Although s p e c i f i c d e t a i l s w i l l be discussed i n a following chapter dealing with experimental a p p l i c a t i o n s , one aspect of a gen-e r a l i t e r a t i v e lineshape f i t t i n g procedure should be outlined at t h i s j . i . point. In the (m + 1 ) L 1 i t e r a t i o n , the r e f i n e d rate constant f o r any two s i t e s i and j i s determined as k. m + 1 = k. m + Ak.. i l i J i l and thus i n accordance with Eq. (2.4.5), m^+1 = T 2 + (K m + AK) - i£ = T 2 + K m + 1 - i£ where the elements of the matrix Aj( are defined i n terms of the incre-mental rate constants Z\k , c f . Eq. (2.2.21). Now, i n the mt*1 i t e r -a t i o n, transformation matrices have been determined i n accordance with Eq. (2.4.7) such that Am = ( S - 1 ) m - R m ' S m . 37. Thus i n the (m+l) t' 1 i t e r a t i o n , i t f o l l o w s that a matrix A' deter mined by A1 = ( S - ' f - l f ^ - S i s approximately d i a g o n a l . The matrix A/ may now be d i a g o n a l i z e d th e x a c t l y w i t h c o n s i d e r a b l e r e d u c t i o n i n computation time. The (m+3.) i t e r a t i o n t r a n s f o r m a t i o n m a t r i c e s , s"1"1 * and ( S - 1 ) ! l 1 + * are r e a d i l y ob-ta i n e d as A m + 1 = T _ 1-A'-T = C r 1) m + 1-R m + 1-s m + 1 w i t h , S m + 1 = S m-T and ( S " 1 ) 1 " * 1 =T1-(S~1)m. Having defined the , . ,m+l „m+l , , n_i.m+l' . . complex matrices A , j | and (S ) m t h i s manner, the compu-t a t i o n of the lineshape f o r the r a t e constants k™+± reduces to an a p p l i c a t i o n of Eq. ( 2 . 4 . 9 ) . The complex matri x f o r m u l a t i o n of chemical exchange processes i s most r e a d i l y i l l u s t r a t e d by an analysis of the simple t w o - s i t e system i n the l i m i t of zero s a t u r a t i o n . Such a system may be consid-ered w i t h exchange s i t e s A and B defined by r e l a t i v e chemical s h i f t s (rads. sec"-') flA = - f l and fl^ = fl i n terms of the independent frequency v a r i a b l e x w i t h COq = (co^ + Wg), c f . Eq. 2 . 2 . 2 4 ) , and f r a c t i o n a l s i t e p o p u l a t i o n s p and p . The r e l a x a t i o n , r a t e and chemical s h i f t matrices are then determined as: 3 8 . ft .OL 0 In the l i m i t of zero s a t u r a t i o n , the z-mode magnetization i s given by Eq. (2.2.22) as K li-Xi-i 5 w i t h and tip" 1 D = 1 * A k, 1 6 k, A A k . k, Ti-6 Using Eq. 2.2.3), i t i s r e a d i l y shown th a t the matrix product J^'T^-P_ reduces to P_ i n t h i s l i m i t , thus showing the v a l i d i t y o f the modified Bloch equations as expressed i n Eq. (2.4.1). For a n a l y t i c s i m p l i c i t y i t w i l l now be assumed that the s p i n - s p i n r e l a x a t i o n times f o r the two s i t e s are the same so th a t the matrix becomes a s c a l a r m a t r i x , . v i z . , _I where i s the 2 x 2 u n i t matrix. The exchange matrix R, defined by 2 Eq. (2.4.5), may now be expressed i n the form (2.4.10) 3 9 . and the diagonal matrix A. corresponding to K - i f l j i s now defined by A = S~ K - i f l •S. If , i n i t i a l l y , an equal population system i s considered, the diagonal elements of the matrix .A above are given by X = k ± a 3.. a = [ k 2 - fl2]^ , (2.4.11) with k = k^ = k^. The associated transformation matrices are given as S = 1 1 , s; (2.4.12) i v j i c r c p l+ = (u ± i f l ) / k . As the matrix K. - i£j, or R., i s rion-Hermitean, the matrix elements X are i n general complex and the column vectors defining the transformation matrix are complex and non-orthogonal. These vectors are normalized i n accordance with the condition S*.S.. + S*S,. = 1 .11 3 3 h 3-3 f o r j ,X = 1, 2 and s!\ the complex conjugate of S_. . It should also be noted that a l l normalization constants have been combined i n the f a c t o r k/2a i n Eq. (2.4.12). The complex lineshape function, G(x), i n accor-dance with Eq. (2.4.6) now takes the form GcLi.) - - C u i 1 N l 0 I . S . A + ( - i + i * ) l -1 _i 5 . P (2.4.13) The parameter a, as defined i n Eq. (2.4.11), i s r e a l or pure imaginary f o r k > fl and k < fl, r e s p e c t i v e l y . Thus, from Eq. (2.4.13), the complex 40. lineshape function for rate constants i n the range 0 £ k ^ ft i s expressed i n terms of a r e a l parameter e as a«> . u U ^ e ) + U^.Ve) ) ( 2 . 4 . 1 4 ) with A = - u nM , r = =• + k and e = ft2 - k 2 . The associated 1 o T 2 lineshape function for rate constants i n the range ft^k < °° i s & w - + _u±kU (2.4.15) Eqs. (2.4.14) and (2.4.15) allow a l u c i d and concise d e s c r i p t i o n of the NMR absorption lineshape function V(x) f o r the equal population two-site chemical exchange system. In general, the diagonal matrix elements determine the p o s i t i o n s and lino-widths of the component, s p e c t r a l l i n e s and the matrix S. determines the i n t e n s i t i e s of these l i n e s . Thus, for O ^ k ^ f t , Eq. (2.4.14) defines absorption mode com-ponents corresponding to modified Lorentzian l i n e s described by the lineshape function Vc*) = A . (r+k)+4>c , CrA)~^ X ( (2.4.16) In the absence of exchange, k = 0 (r = , E = ft) , i t i s shown that 2 the spectrum consists of pure Lorentzian l i n e s centered at x = ±ft with f u l l widths at h a l f maximum of 2/T^, consistent with the normal d e f i n i t i o n s of 2ft as the chemical s h i f t between s i t e s and T^ as the parameter describing s p e c t r a l line-width i n the absence of exchange 8? e f f e c t s In the presence of exchange, the sp e c t r a l components may 41. be considered i n terms of a superposition of pure Lorentzian ab-sorption and associated dispersion functions, the degree of mixing being determined by the factor k/e i n Eq. (2.4.16). For example, Thus i t i s shown that the Lorentzian type component has a f u l l l i n e -width at half-maximum of 2 ( ^ + k), l i n e a r l y increasing with k, and 2 a p o s i t i o n e < Q, such that the component separation, 2e, decreases with increasing k. A s i m i l a r function V_(x) i s obtained for the com-ponent centred at x = -e. These b a s i c lineshape c h a r a c t e r i s t i c s are 2.5a, the modified Lorentzian component s p e c t r a l l i n e s , V (x) and V_(x), are shown as dashed l i n e s ; and the resultant lineshape function, V(x), i s shown as a f u l l l i n e , cf. F i g . 2.3. In F i g . 2.5b, the com-bined absorption and dispersion functions, with contributions from the component l i n e at x = e being given by the f i r s t and second terms, r e s p e c t i v e l y , in Eq. (2.4.17), are shown as the dashed l i n e s . Again, the computed lineshapes have been normalized, through the f a c t o r A, to an a r b i t r a r y maximum independent of k. the observation of an NMR s i g n a l proportional to a s p e c i f i c component of the transverse nuclear magnetization, G(x), as an o s c i l l a t o r y l i n e -a r l y p o l a r i z e d component at the i r r a d i a t i o n frequency to i n the f i x e d laboratory frame of reference. The d i r e c t i o n of a s p e c i f i c component for the component centred at x c, V (x) i s given by (2.4.17) Experimentally, use of an r f phase-sensitive detector allows k/ft = 0 . 6 4 0.80 Figure 2.5(a) Modified Lorentzian component s p e c t r a l l i n e s and resultant absorption mode exchange lineshapes. Figure 2.5(b) Combined absorption and dispersion functions and resultant absorption mode exchange lineshapes. of t r a n s v e r s e magnetization i s defined by the r e l a t i v e phase angle between the detector reference r f f i e l d v e c t o r H and the r f i r r a d i a t i o n f i e l d v e c t o r d e f i n i n g the u-axis o f the normal r o t a t i n g frame of refe r e n c e , c f . F i g . 2.2. Thus i n the absence of exchange e f f f e c t s , by s e t t i n g cj> = 90 the s p e c i f i c component corresponding to. the v-mode magnetization v e c t o r V i s observed. The absorption mode spectrum con-s i s t s of pure L o r e n t z i a n l i n e s (assuming a simple 1/T -type s p i n - s p i n r e l a x a t i o n mechanism) centred at x = ±ft o r , w = w ± ft. The general d i s p e r s i o n mode spectrum i s described by a lineshape f u n c t i o n u(x) given i n accordance w i t h Eq. (2.4.16) as This form o f spectrum i s observed by s e t t i n g cj> = 0. The f i r s t term i n Eq.(2.4.19) describes the d i s p e r s i o n mode l i n e centred at x = +ft i n the l i m i t o f no exchange. In the presence of chemical exchange f o r k < ft, the mixing o f normal absorption and d i s p e r s i o n mode NMR s i g n a l s a s s o c i a t e d w i t h the v- and u-mode components, r e s p e c t i v e l y , of the tra n s v e r s e magnetization i n the r o t a t i n g frame of refer e n c e may now be considered q u a l i t a t i v e l y i n terms of the behaviour of the tran s v e r s e components of the i n d i v i d u a l nuclear s p i n isochromats, M(x, cj)). In the absence of chemical exchange, T^ i s the r e l a x a t i o n time d e s c r i b i n g the isochromat dephasing and consequential s p e c t r a l — r (2.4.18) where u^ ( x ) , analogous to V (x) i n Eq. (2.4.17), i s (2.4.19) line-width due to l o c a l magnetic f i e l d inhomogeneity g i v i n g r i s e to d i s t r i b u t i o n s of Larmor frequencies centred at the A - and B - s i t e resonance frequencies, x = ±ft. This type of isochromat dephasing i s a coherent e f f e c t . In the presence of chemical exchange, however, the i n d i v i d u a l spin isochromats are involved i n a tr a n s f e r between s i t e s corresponding to d i s t i n c t l o c a l magnetic f i e l d s (chemical s h i f t s ) at random times. Conversely, i t may be considered that each spin isochromat experiences a randomly f l u c t u a t i n g l o c a l magnetic f i e l d with a fundamental frequency component i n the associated f r e -quency d i s t r i b u t i o n of k rads. s e c . - 1 . This random process a l t e r s the form of isochromat dephasing, which may now be described i n term 33 85 of a p r o b a b i l i t y function ' f o r the isochromat r e l a t i v e phase d i s t r i b u t i o n , and the time average e f f e c t for a l l isochromats i s observed as an e f f e c t i v e mixing of the normal u- and v-mode trans-verse magnetizations as described by Eqs. (2.4.16) and (2.4.18). It should be emphasized that the function V(x) by d e f i n i t i o n describes the spectrum as observed i n the presence of exchange processes f o r the phase s e n s i t i v e detector r e l a t i v e phase (J> = 90 , and t h i s w i l l always be r e f e r r e d to as the absorption mode s i g n a l . Now, i f the fundamental frequency associated with the f l u c t u a t i n g l o c a l magnetic f i e l d due to a chemical exchange process becomes comparable to the frequency diffe r e n c e between exchange s i t e s , 2ft, the modification of the b a s i c T^ isochromat dephasing i s expected to be most s i g n i f i c a n t This i s ac t u a l l y observed i n the form of maximal broadening and coalescence of the component spectral l i n e s f o r rate constants k - ft. For rate constants k - ft, i t follows from Eq. (2.4.16) that r V(x) = A l + + (2.4.20) Thus i t i s seen that the lineshape c h a r a c t e r i s t i c s are strongly dependent upon the second term, i n addition to the broadening de-scrib e d by the parameter r , due to the fa c t o r k/e 0 0. It i s this f a c t o r that describes the well known s e n s i t i v i t y of the observed absorption mode lineshape to rate constant f o r a p a r t i c u l a r chemical s h i f t d i f f e r e n c e between exchange s i t e s i n the region about component l i n e coalescence. For k = ft (e = 0) the second term i n Eq. (2.4.20) now makes no e f f e c t i v e contribution to the lineshape function V(x) and thus the spectrum consists of a si n g l e Lorentzian l i n e centred at x = 0 with f u l l width at half-maximum of 2r = 2\\ + ft] and inten-s i t y f a c t o r 2Ar. Q u a l i t a t i v e l y , t h i s i s consistent with the simple isochromat. model f o r exchange e f f e c t s previously discussed. As the rate of isochromat t r a n s f e r between exchange s i t e s increases, or conversely, as the fundamental frequency associated with the f l u c t u -ating l o c a l magnetic f i e l d f o r any isochromat increases, the time average contribution from the normal dispersive u-mode magnetization i s maximized and then e f f e c t i v e l y cancelled out f o r the p a r t i c u l a r condition k = ft. This c a n c e l l a t i o n coincides with an exact averaging of the component l i n e separation, 2e, to zero. Also, f o r a l l rate constants k > ft there i s no e f f e c t i v e contribution to the lineshape from a dispersive type function. 2 45. For rate constants k > ft, the absorption mode lineshape function i s given i n accordance with Eq. (2.4.15) as with r = — + k, a = (k 2 - ft2)2. That i s , the spectrum consists of 2 a modified Lorentzian l i n e centred at x = 0 which may be described as the superposition of two Lorentzian components, that g i v i n g a p o s i t i v e contribution to V(x) being defined as with a f u l l width at half-maximum of 2(r - a) and an i n t e n s i t y f a c t o r A ( l + k/a). The parameters r and a determine a linewidth which de-creases with increasing k and a maximum value for the i n t e n s i t y f a c t o r of 2A. The other component defined i n Eq. (2.4.21) gives a negative contribution and i s a Lorentzian function with line-width 2(r + a ) , increasing l i n e a r l y with k, and i n t e n s i t y factor A(l - k/a). Again, f o r k - ft the lineshape i s strongly dependent upon the f a c t o r k/o; (as a -»- 0) and the observed lineshape i s p a r t i c u l a r l y s e n s i t i v e to the rate constant k i n t h i s region. In the l i m i t of very f a s t ex-change (k » ft) the component V (x) becomes dominant and the absorption mode spectrum consists of a s i n g l e Lorentzian l i n e with line-width 2/12 a n < * a n i n t e n s i t y factor 2A, consistent with the f a c t that the i n t e n s i t i e s associated with the component l i n e s at x = ±ft i n the l i m i t of slow exchange have now combined i n the sing l e l i n e centred at the mean Larmor frequency. These lineshape c h a r a c t e r i s t i c s are shown i n Fi g . 2.6 f o r two values of k/ft > 1 where again the computed (2.4.21) (a) (b) k/n = 3 . 2 Figure 2.6 Modified Lorentzian component functions and resultant absorption mode exchange lineshapes. 4 6 . l i n e s h a p e s . h a v e b e e n n o r m a l i z e d t o a n a r b i t r a r y m a x i m u m i n d e p e n d e n t o f k . I n F i g . 2 . 6 a , t h e c o m p o n e n t L o r e n t z i a n f u n c t i o n s V ( x ) a n d V _ ( x ) a r e s h o w n as d a s h e d l i n e s ; a n d t h e r e s u l t a n t l i n e s h a p e f u n c t i o n V ( x ) i s s h o w n as a f u l l l i n e . I n F i g . 2 . 6 b , t h e f u n c t i o n s V ( x ) a n d V ( x ) a r e s h o w n t o b e v e r y n e a r l y e q u i v a l e n t . A s i m i l a r m a t r i x t r e a t m e n t f o r t h e m o r e g e n e r a l u n e q u a l p o p u l a t i o n , p A ^ p . ^ , t w o - s i t e e x c h a n g e s y s t e m u s i n g E q s . ( 2 . 4 . 1 0 ) a n d ( 2 . 4 . 1 3 ) l e a d s t o c o m p l e x l i n e s h a p e f u n c t i o n s f o r 0 J ~ k *S fl: few - j U+4)-MfA)t + U-4> W i - ^ ) f a n d f o r ft ^ k < 0 0 : l i\r+<£ ) + i X Ir-nt) + U . 4 - . 2.3) w h e r e t h e c h a r a c t e r i s t i c p a r a m e t e r s a r e d e f i n e d b y w i t h k = % ( k A + k g ) , 2 f l = w B - o ) A = fl& - , a n d w = (to + to ) . I n t h i s c a s e t h e p a r a m e t e r s e a n d a r e m a i n c o m p l e x , b u t t h e y may b e c o m p a r e d d i r e c t l y w i t h t h e a n a l o g o u s p a r a m e t e r s u s e d i n t h e a n a l y s i s o f t h e e q u a l p o p u l a t i o n s y s t e m . I t i s i n t e r e s t i n g t o n o t e t h a t t h e s e p a r a m e t e r s now d e t e r m i n e b o t h c o m p o n e n t l i n e p o s i t i o n ( i m a g i n a r y p a r t ) and line-width (real part) f o r a l l rate constants k, whereas i n the equal population case the r e a l parameters e and a determined only l i n e p o s i t i o n or line-width, r e s p e c t i v e l y . In the l i m i t of slow ex-change, for a general population two-site exchange system analogous to that already discussed, i t i s shown from Eq. (2.4.22) that the absorption mode lineshape function s i m p l i f i e s to the form \)(^  - A { fft - V - 7 - U - ^ A ) — - J C 1(2.4.24) Hi us, i n the absence of exchange (k = 0), the spectrum consists of pure Lorentzian l i n e s centred at x = -Q and x = +0, with i n t e n s i t y factors p and (1 - p ) = p , r e s p e c t i v e l y , and f u l l widths at h a l f -A A D maximum of Z/'T^' -*-n t n e l i m i t of very f a s t exchange the second term i n Eq. (2.2.23) becomes dominant and the absorption mode spectrum consists of a si n g l e Lorentzian l i n e centred at x = (1 - 2p )ft with a line-width 2/T . That i s , f o r the A /. unequal population exchange system the s p e c t r a l l i n e i n t h i s l i m i t i s p ositioned away from the mean frequency x = %C°^ + ftg) = 0 towards the resonance frequency corresponding to the larger population ex-change s i t e . A reformulation of the imaginary parts of Eqs. (2.4.22) and (2.4.23) shows that these general equations are equivalent to the expression previously derived, cf. Eq. (2.2.14), f o r a two-site ex-change system. 48. 2.5 F i r s t - O r d e r J-Coupling. In previous s e c t i o n s , a - m u l t i - s i t e nuclear spin exchange system has been considered i n terms of d i s t i n c t s i t e s of d i f f e r i n g Larmor frequency U K , an i m p l i c i t assumption being that each s i t e i s associated with a p a r t i c u l a r molecular e l e c t r o n i c environment, and corresponding r e l a t i v e chemical s h i f t fl^. The e f f e c t of chemical exchange on spin-spin mu l t i p l e t s i n NMR spectra was recognized i n the o r i g i n a l work of Gutowsky and c o - w o r k e r s ' \ and various app-roximate a n a l y t i c a l expressions have been developed to derive k i n -e t i c data f o r s p e c i f i c cases^*' ^6,87^ j _ C O U p i j n g j _ n a general f i r s t - o r d e r NMR exchange system, however, i s r e a d i l y included i n a matrix formulation of exchange processes based upon a simple stoch-a s t i c model as already developed. A general spin Hamiltonian for a system of N-spins i n the absence of exchange processes may be considered i n the form (2.5.1) whore i t i s assumed that a l l terms i n t h i s Hamiltonian are expressed i n energy units h = h/2iT, h being Planck's constant. A term de-s c r i b i n g the i n t e r a c t i o n of the observing r f magnetic f i e l d with the nuclear spin system i s not necessary i n the following a n a l y s i s . }-{ i s a generalized Zeeman term yi0 = - W D ^ X^i (2.5.2) l = i with a reference Larmor frequency w rad. secT 1 defined as co = yH - A, A > 0. (2.5.3) o o As usual, I ^ i s the z-component spin angular momentum operator for the i ^ - s p i n and H i s the magnitude of the s t a t i c magnetic f i e l d applied i n the p o s i t i v e z - d i r e c t i o n of the Cartesian r o t a t i n g and f i x e d reference frames. In Eq. (2.5.3), y i s the nuclear gyromag-n e t i c r a t i o for the resonant spins i n a general hetero-nuclear system and A i s a frequency s h i f t parameter, defined to be p o s i t i v e , determining the o r i g i n for the independent frequency v a r i a b l e x, cf. Eq. (2.2.24). This form of frequency v a r i a b l e allows an a r b i -t r a r y choice of frequency o r i g i n and a s i m p l i f i e d analysis of general NMR spectra, while being d i r e c t l y r e l a t e d to the experi-mental i r r a d i a t i n g frequency co. K Q i s the chemical s h i f t term M - ^ l ^ i (2.5.4) in which fi^ i s the chemical s h i f t for the i ^ - s p i n , or s i t e - i , r e l a t i v e to the reference Larmor frequency to , cf. Eq. (2.2.24). In t h i s manner, ft^ may be considered as the s i t e - i Larmor frequency i n terms of the independent v a r i a b l e x. The d e f i n i t i o n of Ji^ i n terms of the observable may be r e l a t e d to the concept of a 88 p o s i t i v e e l e c t r o n i c s h i e l d i n g parameter o^, describing the l o c a l time-independent magnetic f i e l d i n the z - d i r e c t i o n : H. = (1 - a.)H , R B 1 1 0 M th such that }ia = yH Z a . l .. Increased s h i e l d i n g of the i -spin, O j X Z 1 as described by an increased value, thus corresponds to a de-crease i n the Larmor frequency at constant magnetic f i e l d H . In the p a r t i c u l a r case for which A = 0 (to = yH ) , a l l s i t e Larmor r o ' o 50. frequencies are such that to. ^  co and hence a l l r e l a t i v e chemical 1 0 s h i f t s are negative. The p a r t i a l Hamiltonian may be r e f e r r e d to as a f i r s t -order coupling term and i s given as >C -£ I Z l * * L ( 2 - 5 ^ where J._. i s the i n d i r e c t (scalar) spin-spin coupling constant between the i ^ ' 1 - and j ^ - s p i n s . Thus , K ^ i s a second-order coupling term of the form In the ^ - r e p r e s e n t a t i o n , with basis functions as simple products of eigen-functions of the spin operator I , the term }i , corresponds 2 ' J to off-diagonal elements i n the spin Hamiltonian matrix "H_. A f i r s t -order NMR spectrum may be defined by the condition J . . « |Q. - Q.\ R 1 3 1 i 3 1 and, as the matrix ^ i s diagonal i n a l l off-diagonal matrix ele-ments may be neglected i n a f i r s t - o r d e r determination of s p e c t r a l t r a n s i t i o n frequencies and i n t e n s i t i e s . The Hamiltonian term )/j2^ , however, has been shown to give r i s e to a spin-spin r e l a x a t i o n mechanism which may be described by Bloch type equations f o r weakly 89 coupled nuclear spin systems . The e f f e c t of such a r e l a x a t i o n mechanism i n addition to chemical exchange processes i s not included at t h i s point. In the I^-representation, the e f f e c t i v e spin Hamiltonian f o r a weakly coupled ( f i r s t - o r d e r ) spin system, W - Ho + X n + X j ( 1 ) (2.5 .7 ) i s diagonal and a l l terms con-cspond to secular energies d e f i n i n g f i r s t - o r d e r spin t r a n s i t i o n s . The energy E^ associated with the basis function (also an eigen-function o f ^ ' j cf)^  i s now given as E£ = - a o l m U ~lni\i +i i, Jij m£i m£j ' ( 2 - 5 > 8 ) c fJ where I . cj)- = m..cj) , and <*>. = IT ^ .. . I . i s the spin operator cor-Z X Kr X JC Xs ^ = | X/X Z X th responding to the i -part, , of the product function cf)^ . Tlius, f o r spin l-h, = +h> -h f o r = ct, 3 , where a and 3 are the spin eigen-functions defined by I z^|a> = +%|a>, ^ j j ^ = _ ^ | 3 > . Spin systems commonly occurring i n studies of m u l t i - s i t e exchange processes, e s p e c i a l l y hindered i n t e r n a l r o t a t i o n , may be analyzed as f i r s t - o r d e r ABX and ABX„ l-h systems with J n = 0. 6 A B As the simple ABX system includes a l l the s a l i e n t features of a general a n a l y s i s , t h i s p a r t i c u l a r system w i l l now be considered i n some d e t a i l . The f i r s t - o r d e r spin Hamiltonian for an ABX (J^g = 0) spin system may be expressed i n accordance with Eq. (2.5.7) as -H1 = -%[lzK + TzB + : z x ] + ^ z A " T Z B ] + V Z X with chemical s h i f t s defined as : ioA = co - ft, co„ = co + ft and A o B o co = co - fl where co = ^(co. + con) and hence A = YH - %(co. + con) , X o X o A B o A B cf. Eq. (2.5.3). Simple product basis functions, <j>^ , i n the I -representation f o r t h i s 3-spin (I =h) system are given in Table Z.l. , where in general cj)^  = ^^C^gC^^J by d e f i n i t i o n , these functions arc als o eigen-functions of the spin Hamiltonian 7^'. 'Die corresponding energy l e v e l s , as derived from Eq. (2.5.8), are also given i n Table 2 TABLE 2.1 BASIS (EIGEN) FUNCTIONS FOR ABX ( J A B = 0) SPIN SYSTEM, I = y NUCLEI, AND CORRESPONDING ENERGY LEVELS 1 Basis function, (j) acta Energy l e v e l , E^ 3 1 _ 1. T a a3a 3on - ~ OJ 0 + j % + « - j J. 4 wo + 4 % " A + y J-1 1 O 1 -j 1 1 n M _ - — 2 ' A Z a33 3a3 333 y wo - \ fix + n + Y J. 2 U ° I n I T 7 % + 7 J + J ± = 4 C J BX ± JAX) ; J A r J B X > o T A B L E 2.2 TRANSITION F R E Q U E N C I E S F O R A B X (J^g = 0) S P I N S Y S T E M T r a n s i t i o n Number a 1 2 3 4 T r a n s i t i o n Type A A Energy levels (1,3), (2,4) (5,7), (6,8) (1,2), (3,4) (5,6), (7,8) Frequency x 4 J A X 4 J A X n T j B X "4 J B X 1 2 3 4 X X X Y (1,5) (3,7) (2,6) ( 4 , 8 ) -ft^+j_ -,a,+.T. "A i These t r a n s i t i o n numbers correspond to those used i n Figs . 2.7 and 2.9. The bracket (£,m) r e f e r s to energy l e v e l s E£ and E m , and corresponding eigen-functions <f>^ and <j)m, given i n Table 1. J± = 1 ( J B X 1 J A X > ' J B X ' J A X > 0 52. The allowed t r a n s i t i o n frequencies,in terms of the independent v a r i a b l e x, are l i s t e d i n Table 2.2. The eigen functions and <j>m corresponding to the energy l e v e l s determining these spectral t r a n s i t i o n s are i n d i -cated by the brackets (I, m). By d e f i n i t i o n ft > 0 > ft. > ftv, and i t D A A i s assumed i n i t i a l l y that J > J > 0 such that J = ^ ( J D V ± J . v ) > 0. D A A A — D A A A In the absence of exchange, for a molecular system without a preferred conformation, a l l t r a n s i t i o n i n t e n s i t i e s i n the AB-part of the NMR spectrum are equal, as are those i n the X-part. In a more general case, these i n t e n s i t i e s may be determined by the f r a c t i o n a l populations of a number of po s s i b l e conformations. The AB-part of the ABX spectrum consists of the four trans-i t i o n s 1 - 4 (each of which i s doubly degenerate as = 0) as shown \T\ P"i rr o y ^ I i 1 1 c v>o T_i^r->-i nrr duis to tl^s X~ s jp i n clct^oXjTiincc fciXx* e f f e c t i v e Larmor frequencies x = -ft ± % J ^ and x = ft ± %Jg^> which may i n turn be considered to define four d i s t i n c t exchange s i t e s . Such s i t e s are analogous to the two environmental s i t e s with Larmor frequencies x = ± ft i n the absence of J-coupling. In a general d e s c r i p t i o n of ex-change e f f e c t s , a b a s i c environmental (chemical s h i f t ) s i t e with an associated X-spin state may now be r e f e r r e d to as spin s i t e - j with Larmor frequency x_. , cf. Eq. (2.2.24). In t h i s manner x.. may be con-sidered as an e f f e c t i v e chemical s h i f t . Thus f o r the J-couplings defined above, spin s i t e - 1 corresponds to a b a s i c A - s i t e associated with an a X-spiri s t a t e , as shown by the eigen-functions corresponding to the energy l e v e l s defining the Larmor ( t r a n s i t i o n ) frequency x = -ft-%J : 1 A A (aaa, Baa) and (aga, 33a), cf. Tables 2.1 and 2.2. Thus i t i s seen that J-coupling i n a f i r s t - o r d e r NMR spectrum f o r a spin system with chemical 2 J AX a4J 2 J A X a-4J 2 J B X '2 J 8X k » a 2J+ 2 J -f (b) X A a 2 a 8 k ? 23-K •a-i|JAX| - o + | | J A X | n-|JBX Q + i J 2 J B X k -•1-J i J Figure 2.7 AB-part of f i r s t - o r d e r ABX spectrum exchange processes, simply increases the number of possible exchange s i t e s to be considered in a d e s c r i p t i o n of the system by modified Bloch equations. .. Consistent with the s e m i - c l a s s i c a l vector model previously discussed, the nuclear magnetization associated with a general spin s i t e - j may be considered i n terms of a spin isochromat M(x^ , (()_.) with a c h a r a c t e r i s t i c Larmor frequency x.. and a r e l a t i v e phase (with respect to the u-axis of the normal r o t a t i n g frame of reference uvz ) cj)_. , cf. F i g . 2.2. The f r a c t i o n a l population of spin s i t e - j , p.., determines the magnitude of M(x_. , <})_.). In the presence of a chemical exchange process defined i n terms of a f i r s t - o r d e r rate constant k s e c . " 1 , the spin i s o -chromat i s involved i n a random tr a n s f e r between spin s i t e - j and spin s i t e - i described by a rate constant k... c f . Eq. (2.2.18). Conversely, i t may be considered that t h i s isochromat experiences a randomly f l u c -tuating l o c a l magnetic f i e l d with a fundamental frequency component i n the associated frequency d i s t r i b u t i o n of k rad. s e c . " 1 . The b a s i c assumption for a s t o c h a s t i c d e s c r i p t i o n of such an exchange process i n a f i r s t - o r d e r nuclear spin system may be summarized as follows: (i ) the isochromat M(x_. , (}>_.) remains in spin s i t e - j with a mean l i f e t i m e T . u n t i l a random instantaneous t r a n s f e r to a d i f f e r e n t J s i t e takes place, such that precessional e f f e c t s i n the t r a n s f e r i n t e r v a l may be neglected; ( i i ) the s i t e l i f e t i m e x.. i s independent of the associated spin-spin and s p i n - l a t t i c e r e l a x a t i o n times, T^^ and T^ .. , r e s p e c t i v e l y ; ( i i i ) i n d i v i d u a l spin isochromats r e l a x independently except for s i t e t r a n s f e r e f f e c t s ; and (iv) f o r the spin isochromat M(x^ , cb^ ) i n spin s i t e - i there i s a. constant p r o b a b i l i t y per unit time, kj^> f ° r t r a n s f e r into s i t e - i , t h i s p r o b a b i l i t y being i n v e r s e l y p r o p o r t i o n a l to the f r a c t i o n a l s i t e population p_. . Under these assumptions, a s i t e lifetime. T and associated rate constants, k_.^ , a r e simply r e l a t e d as £ k.. = T . _ 1 , (2.5.10) l i 1 where i ( i / j ) includes a l l allowed t r a n s f e r s i t e s connected with 82 s i t e - j . Also in.accordance with the p r i n c i p l e of d e t a i l e d balance fo r the rate processes, s i t e population and rate constants f o r any two spin s i t e s s a t i s f y p.k. . = p.k.'. (2.5.11) These generalized r e l a t i o n s h i p s may be compared with those f o r the simple two-site exchange system i n i t i a l l y discussed, c f . Eqs. (2.2.2) and (2.2.3). Exchange e f f e c t s i n a f i r s t - o r d e r ABX (J = 0) NMR spectrum « AB may now be described i n terms of t h i s s t o c h a s t i c model. Only general intramolecular exchange processes (for example, hindered i n t e r n a l ro-tation) w i l l be discussed e x p l i c i t l y . The b a s i c exchange process i s assumed to be a t r a n s f e r of nuclear magnetization between s p i n - s i t e s d i s t i n g u i s h e d by chemical s h i f t s f o r the A- and B-spins, - ft and + ft, r e s p e c t i v e l y , cf. Eq. (2.5.9). This process i s defined i n terms of a s i n g l e f i r s t - o r d e r rate constant, k s e c . " 1 , which i s n e c e s s a r i l y de-f i n e d as a reduced rate constant f o r a general exchange system. The AB-part of the ABX spectrum i s shown i n F i g . 2.7a f o r the spin system i n the absence of exchange (k = 0) with coupling constants J^^, > J > 0 In F i g . 2.7a, the b a s i c A and B environmental s i t e s are in d i c a t e d along with the X-spin'state corresponding to each AB-exchange s p i n - s i t e . A l l po s s i b l e transfers between the four AB s p i n - s i t e s may be represented by: (2.5.12) the forward transfers only, of the p a i r s defined by Eq. (2.5.11) being shown for c l a r i t y . Consistent with assumption ( i ) above, i t i s to be assumed that the X-spin state i s unchanged i n a s i t e t r a n s f e r i n t e r v a l and hence the allowed transfers are 1 - 3 and 2 - 4 as defined by the rate constants k ^ ^ 3 1 ) and k^^ ^ 4 2 - ' ' This i - s r e a d i l y seen from a comparison of the spin eigen-f unctions, c f. Table2.1 associated with the doubly degenerate t r a n s i t i o n s 1 and 3: E E E ^A^B^X aaa 3aa ^A^B^X aaa a3a (2.5.13) a 3a 63a 3aa 33a where 1 i s a f i r s t - o r d e r A-spin t r a n s i t i o n as defined by the usual spi n t r a n s i t i o n operator I*: I*|3eta> = |aaa> and I*|63a> = |a3a>. S i m i l a r l y , 3 i s a B-spin t r a n s i t i o n and thus the t r a n s f e r as defined i s between b a s i c A and B environmental s i t e s as required. A l s o , a l l product eigen-functions concerned have an a X-spin part. This type of s i t e t r a n s f e r i s to be compared with the t r a n s f e r s 1 - 2 and 2 - 4 i n v o l v i n g a change i n X-spin state. The simple s t o c h a s t i c model considered above i s consistent with a quantum mechanical treatment of exchange processes. I t may be assumed that the molecular system following a nuclear s p i n - s i t e trans-f e r has a s p i n Hamiltonian of the same form as i t had i n i t i a l l y , and t h a t - t h i s Hamiltonian d i f f e r s only i n that the non-equivalent A- and B-spins have interchanged magnetic p r o p e r t i e s . An operator may now be d e f i n e d * 5 ' * ^ to describe the change of t o t a l nuclear spi n s t a t e , under exchange, as determined by spin eigen-functions within the l i m i t s o f s e p a r a b i l i t y of a wave-function f o r the complete molecular-spin 20 system . For an intramolecular exchange process, the nuclear spin state of an i n d i v i d u a l molecule following' a s p i n - s i t e t r a n s f e r i s determined completely by the o r i g i n a l s tate. This i s a c t u a l l y the fundamental d i f f e r e n c e between such a process and an intermolecular chemical exchange process i n which the state of an i n d i v i d u a l molecule following s i t e t r a n s f e r depends upon the o r i g i n a l state of t h i s mol-17 ecule and also upon the state of the i n t e r a c t i n g molecule involved For an intramolecular AB s p i n - s i t e t r a n s f e r i n the f i r s t - o r d e r ABX system, an exchange operator, P, may be defined by P |aa^ > = |aa£ > and P |a3£ > = |3a£ >. Thus, i t i s seen that t h i s operator defines exactly the t r a n s i t i o n s involved i n the allowed s i t e t r a n s f e r , as defined in terms of the s t o c h a s t i c model above and represented i n Eq. (2.5.13) f o r the eigcn-functions having an a X-part. S p i n - s i t e s 1 and 2 d i f f e r only i n the X-spin s t a t e , the eigen functions associated with t r a n s i t i o n 2 being (aa3, 3a3) and (a33, 333), Table I. Thus the t r a n s f e r 1 - 2 does not contribute to NMR s p e c t r a l c h a r a c t e r i s t i c s s p e c i f i c a l l y associated with the s t o c h a s t i c exchange b a s i c A- and B-spin s i t e s . Such s i t e t r a n s f e r s , however, correspond to t r a n s i t i o n s induced by the i n t e r a c t i o n of a randomly f l u c t u a t i n g magnetic f i e l d with the X-spin of the ABX spin system leading to X-spin 90 l i f e t i m e l i m i t i n g as described by a c o r r e l a t i o n time . In ad d i t i o n i f the X-spin has a quadrupole moment (1>1), the i n t e r a c t i o n of a randomly f l u c t u a t i n g e l e c t r i c , f i e l d gradient may lead to a s i m i l a r r e l a x a t i o n mechanism with a d i f f e r e n t c o r r e l a t i o n time, x^. For the s i m i l a r t r a n s f e r s 1 - 2. and 2 - 3 , the c o r r e l a t i o n time X £ may be considered to define rate constants: k,„ = k„, = x - 1 . A l s o , such 12 34 c a mechanism may lead to e f f e c t i v e t r a n s f e r s 1 - 4 and 2 -3-, and i n t h i s case the rate constants k, . and k„_ would be defined as a sum « 14 23 of contributions from.exchange and X-spin t r a n s i t i o n processes. The o v e r a l l e f f e c t of these a d d i t i o n a l spin r e l a x a t i o n processes on NMR s p e c t r a l c h a r a c t e r i s t i c s would be expected to depend upon the param-eters ( T ^ ) ' 1 and ( T ^ ) " 1 . In accordance with Eq. (2.4.6), a complex lineshape function G(x) determining the steady-state NMR spectrum associated with n spin-s i t e s i s expressed i n the form G(x) = A I _ - S - [ A + (1/T 2 + i x U ] " 5-S-^P (2.5.14) 58. i n which A. i s the diagonal matrix corresponding to an n x n matrix J< - iQ , the diagonal matrix Q being defined by the s p i n - s i t e Larmor frequencies x_. in terms of the independent frequency v a r i a b l e x. It has been i m p l i c i t l y assumed that the spin-spin r e l a x a t i o n time i s the same f o r a l l s p i n - s i t e s i n defining the r e l a x a t i o n matrix as a s c a l a r matrix ( l / T ^ ) ^ , with 1^  the n x n unit matrix. This s i m p l i f i c a t i o n i s allowed by assuming that the spin-spin r e l a x a t i o n time i s independent 9] of s p i n - s i t e and that a l l non-secular r e l a x a t i o n processes ', as de-s c r i b e d by off-diagonal elements i n T^, are n e g l i g i b l e . Although ad-d i t i o n a l r e l a x a t i o n processes associated with the J-coup led X-spin and considered above as contributing to s p i n - s i t e t r a n s f e r are not fur t h e r considered at t h i s point, these processes are equivalently described bv off-diagonal elements i n the r e l a x a t i o n matrix T.. Lineshaue char-a c t e r i s t i e s f o r the AB-part of the ABX ( J A G = 0) spectrum i n the presence of exchange are concisely described through a 4 x 4 rate matrix K defined by Eq. (2.2.21) with k^ ^  = 0 f o r any t r a n s f e r not allowed i n terms of the s t o c h a s t i c exchange model. It i s to be noted that the diagonal matrix element k „ corresponds to the inverse s i t e -l i f e t i m e T T 1 , cf. Eq. (2.5.10). Thus the K and Q matrices for an equal population exchange system with coupling constants J > J > 0, D A A A and s p i n - s i t e s as ordered i n F i g . 2.7a are given e x p l i c i t l y as IC = k 0 -k 0 0 k 0 -I 0 k 0 0 -k 0 k 59. and 0 0 0 0 0 0 0 0 0 0 0 • (-O^^^yJzt (2.5.15) The vectors P_ and I_ i n Eq. (2.5.14) are defined by four equal elements 0.25 and 1.0, r e s p e c t i v e l y . 'The extension to a more general unequal population system only requires the evaluation of K. matrix elements i n accordance with Eqs. (2.2.21) and (2.5.11), consistent with the s i t e -population vector, P_, elements. Equation (2.5.14) allows a very e f f i c i e n t numerical analysis of the NMR absorption mode lineshape function, V(x), given as the r e a l part of G(x), and the development of computer programs f o r r a p i d i t e r a t i v e f i t t i n g s of t h i s t h e o r e t i c a l function to experimental data. Absorption mode lineshapes have been c a l c u l a t e d f o r the AB-part of an ABX spectrum defined by the a r b i t r a r y parameters Q •- 4.0 Hz, J._ = 0, J . v = +2.0, J D V = +5.0 Hz, T_ = 0.64 sec. A D A A D A Z (0.5 Hz f u l l - w i d t h at half-maximum) and are shown i n F i g . 2.8a f o r rate constants i n the range 0 v< k 4 200 s e c . - 1 . The lineshape functions have been normalized to a maximum i n t e n s i t y independent of k. I t i s seen that i n the l i m i t of fa s t exchange (k >> ft, t h i s part of the spectrum reduces to a doublet. This feature i s c h a r a c t e r i s t i c of an A 2X spin system and i s consistent with the expectation that i n t h i s exchange l i m i t b a s i c A and B environmental s i t e s become equivalent. As shown i n Fi g . 2.7a, f o r J.,, and J n v of the same si g n , the resultant b ' AX BX ^ ~ lin e s are centred at x = ±%J with J = %(J„„ + J.„). These p o s i t i o n s + + BX AX r Figure. 2.8 Intramolecular exchange lineshapes f o r the AB-part of a f i r s t - o r d e r ABX spin system. 6 0 . may be modified, however, by temperature dependent chemical s h i f t s , oi and u n . A B . Now f o r an ABX (J = 0) spin system with J and J„ of A D A A D A d i f f e r e n t sign, the f i r s t - o r d e r spin Hamiltonian may be expressed i n accordance with Eq. (2.5.9) as i . + i _ + i v zA zB zX J j I . — I [ zA zB X zX ^ A x ' V z X + JBX IzB IzX (2.5.16) where i t i s assumed that J,. < 0 and J„,r > 0. Thus the corresponding AX BX r b energy l e v e l s , as determined using Eq. (2.5.8) are given by s u b s t i t u t i n g ~^AX^ "*r°r ^AX ^ n ^able X J the allowed t r a n s i t i o n s are s i m i l a r l y derived from those l i s t e d i n Table 2. The AB-part of the spectrum f o r t h i s spin system i s shown i n F i g . 2.7b, i n which the ordered t r a n s i t i o n s 1 and 2 now correspond to 3 and a X-spin sta t e s , r e s p e c t i v e l y . Therefore, i n t h i s case, the intramolecular AB exchange process i s described by the rate constants designated as and k^^j moreover, the ra t e matrix JC f o r an equal population system i s given e x p l i c i t l y as k o o - k o V -IL o o -k k o - j i o o k/ (2.5.17) This matrix defines the absorption mode lineshapes shown i n F i g . 2.8b for J - - 2.0 Hz and J n v = + 5.0 Hz. In the region of coalescence A A D A (k - fl) and in the f a s t exchange l i m i t , for a given rate constant k 61. describing the intramolecular exchange process, these lineshapes are d i s t i n c t l y d i f f e r e n t from those previously discussed, c f . F i g . 2.8a. Hence the r e l a t i v e signs of the coupling constants J and may be determined d i r e c t l y from lineshape f i t t i n g of an exchange modified spectrum; i n general, t h i s a d d i t i o n a l information i s r e a d i l y a v a i l a b l e through a simple study of the temperature dependence of a f i r s t - o r d e r NMR spectrum (or part of) f o r a spin system undergoing chemical exchange. Of course, such information i s equivalent (or complementary) to that obtained using the well-known double resonance techniques. This l i n e -shape method, however, may allow the determination of r e l a t i v e signs of coupling constants f o r systems to which double resonance techniques are i n a p p l i c a b l e or only applied with great d i f f i c u l t y . A furt h e r con-s e n n e n c . f i o f t h e d i f f e r e n t r e l a t i v e <;ion<; o f J a n d J i s t h a t t h e r e -' 1 " ' ----- - AX * BX " sultant l i n e s i n the f a s t exchange l i m i t spectrum are now centred at x = ± hJ , where J = k(Jnv - \j,v\) , a s shown i n F i g . 2.7b. D A A X Exchange e f f e c t s i n the X-part of an ABX (J^g = 0) spectrum are also described by 4 x 4 matrices K and Q , completely analogous to those given i n Eqs. (2.5.15) and (2.5.17). A general X-part spectrum f o r J g ^ > > 0 may be represented as shown i n F i g . 2.9a i n the absence of exchange (k = 0) and i n the f a s t exchange l i m i t (k » J _ ) , with J = % ( J - J ); the corresponding absorption mode lineshapes — D A AX. f o r the a r b i t r a r y ABX s p e c t r a l parameters defined above ( Q - 4.0 Hz, =2.0 Hz) are shown in Fi g . 2.10a. For an equal population system, i t i s to be noted that the exchange process i s defined i n terms of the s i n g l e matrix element ~ = -k, as shown i n Fig..2.9a with the s p i n - s i t e s involved corresponding to the AB spin (a) AB aa 1 pa 2 ap k PP 4 23 -a X k »J_ AB pa aa 1 2 PP 3 ap n-J . - n - J . - a -n+J. k » J . Figure 2.9 X-part of f i r s t - o r d e r ABX spectrum 62. states a3 and 3a. S i m i l a r l y , F i g . 2.9b shows the general X-part spectrum f o r J < 0 and J > 0. In t h i s case the f a s t exchange l i m i t may be defined by k >> J , with J = h(Jnv + The ! J +' + ^ BX 1 AX1 corresponding (fl = 4.0 Hz, J = - 2.0 Hz) absorption mode lineshapes A. A are shown i n F i g . 2.10b, and again, the r e l a t i v e signs of the coupling constants and Jg give r i s e to marked differences i n these l i n e -shapes allowing a simple sign determination from experimental spectra. A lineshape analysis of both the AB- and X-parts of an ex-perimental ABX spectrum allows a check on the i n t e r n a l consistency of the f i t t i n g procedure used to obtain rate constants. Also, the coa-lescence condition f o r the AB-part of the spectrum i s defined by the chemical s h i f t d i f f e r e n c e 29., whereas the corresponding condition for the X-oart i s defined bv the narameters J = J*>f.T_.„ - J...1 and ' " " BX A X ' J+ = ^ ( J g x + I ^ A X ^ ^ o r ^ e s a m e a n c * d i f f e r e n t r e l a t i v e signs of and Jgy, r e s p e c t i v e l y . Thus, i f J_ (or J + ) and 0, d i f f e r s i g n i f i c a n t l y , a lineshape analysis of both parts of the spectrum allows an accurate determination of rate constants over an extended temperature range leading to more r e l i a b l e a c t i v a t i o n parameters f o r the intramolecular exchange process involved. 2.6 Second-Order J-Coupling. In a general nuclear spin system, the non-secular part of the spin Hamiltonian describing i n d i r e c t spin-spin coupling may be considered to represent a mixing of b a s i c spin eigen-states. As mixed states are 11 92 most conveniently described by a density matrix ' , spin t r a n s f e r e f f e c t s i n a second-order (tightly-coupled) spin system can be r i g o r -ously described only i n terms of quantum-statistical mechanics''^ using 63. the spin density matrix formalism"''4' ^ . An intramolecular chemical exchange process w i l l be considered i n terms of a density matrix f o r an ABX spin system to develop a more general model f o r exchange and to show a c o r r e l a t i o n with the s e m i - c l a s s i c a l modified Bloch equations already presented f o r a f i r s t - o r d e r (J^B = ^ sPxn system. The spin Hamiltonian f o r a general ABX spin system i n a f i x e d frame of reference may be expressed i n the form X -to C D . tx) o + X ^ + HT + X T I . + I _ + I zA zB zX + ft I . - I zA zB + V z X + JI .1 + J A V I .1 v + J n v I n I v + % J ( l t l " + I.I*) (2.6.1) zA zB AX zA zX BX zB zX v A B A B 1 where J i s the AB i n d i r e c t spin-spin coupling constant and a l l other parameters have been previously defined, cf. Eqs. (2.5.1) and (2.5.9). For s i m p l i c i t y , the r e l a t i v e chemical s h i f t s f o r the A- and B-spins have been defined as ft^ = -ft and ftg = +ft, c f . Eq. (2.5.4). In a more general formulation, the Zeeman and chemical s h i f t terms i n Eq. (2.6.1) would be expressed as • o ' ' ^-flL = " (WQ - ft) I . + I D + I V zA zB zX ftAI . + ft„I _ + ft I A zA B zB X zX (2.6.2) with ft = %(ft. + ftn) . A B In the Schrodinger representation with basis functions cj)^, an N-spin system i s described by the wave function i K t ) = S ^ ( t ) ^ , (2.6.3) 64. and the time-dependent expansion c o e f f i c i e n t s define an ensemble-average spin density m a t r i x ^ , £, with elements P j £ ( t ) = «J>j [p U £>= C j ( t ) c * ( t ) . (2.6.4) 92 A s i m p l i f i e d Schrodinger equation of motion f o r th i s density matrix i s now obtained as 1 - ^ ^ - A ^ (2'6" 5) 21 by adding phenomenological terms to define spin-spin and s p i n - l a t t i c e r e l a x a t i o n , such that £ ° i s the ensemble thermal equilibrium density matrix. The Hamiltonian i s given i n Eq. (2.6.5) i n matrix form f o r the chosen ba s i s {<J>^ }- A random intramolecular spin t r a n s f e r , as de-fin e d by a c o r r e l a t i o n time T . m a y h o described by an a d d i t i o n a l t e^m i n the density matrix equation of m o t i o n ^ ^ X (2.6.6) where E i s a spin operator defined under the assumptions that following spin t r a n s f e r the spin Hamiltonian has the same form and that non-equivalent spins have simply interchanged magnetic p r o p e r t i e s . In Eq. (2.6.6), x may be considered as the average time between spin trans-f e r s and hence a f i r s t - o r d e r rate constant f o r the t r a n s f e r process may be defined as k = x~ 1. Under the action of an observing r f f i e l d , H_ , the spin system i s most conveniently considered i n an i n t e r a c t i o n representation corresponding to the c l a s s i c a l r o t a t i n g reference frame p r e v i o u s l y disqussed. The equation of motion f o r the density matrix i n such a 6 5 . representation, and i n the l i m i t system shows a minimal deviation i s given i n accordance with Eqs. ponent form as of zero saturation such that the spin from thermal equilibrium conditions, ( 2 . 6.1), ( 2 . 6 . 5 ) and ( 2 . 6 . 6 ) i n com-- k (2.6.7) where = yH^ and x i s the independent frequency v a r i a b l e (rad. sec." 1) defined as x = co - U q , cf. Eq. (2.2.24). In the high temperature approximation''"^, an evaluation of the matrix element describing the i n t e r a c t i o n of the spin system with the i r r a d i a t i n g r f f i e l d i n Eq. (2.6.7) i s s i m p l i f i e d by defining a r e a l constant C such that t u » 1 < < | > ^ [ / ' , X ^ t I x 6 + I 3c/ ] l ^ > " -cCCfA^tAj (2-6.8) where C = caoco^/2kT, k i n t h i s instance being the Boltzmann constant, The component quantum number M0 i s given by M. = E mp., cf. Eq. (2.5.8); .-v XV ^ ~ | A/X that i s , for the basis function cb^: I + 1^ + 1 ^ = ^^p-The expectation value of the complex transverse nuclear mag-n e t i z a t i o n , G_, as defined i n Eq. (2.4.3) and shown i n F i g . 2.2, i s now + given d i r e c t l y i n terms of the spin system t r a n s i t i o n operator I and the frequency v a r i a b l e x as G(x) = A<I+> = Tri&'l} (2.6.9) where A may be treated as a normalization constant and Tr{...}•denotes + the trace of the matrix product j ^ - ^ over the spin states. The matrix elements p.~ are i n general complex''""'" and show an i m p l i c i t x'dependence 1 + through Eq. (2.6.7). The t r a n s i t i o n operator I defines the non-zero elements of the matrix I i n the basis *{4>^1 and hence e f f e c t i v e l y de-termines the density matrix elements defining G(x) i n Eq. (2.6.9). Consistent with the normal s e l e c t i o n r u l e f o r magnetic dipolar trans-i t i o n s , the t o t a l ABX spin system (I = h spins) operator may be de-f i n e d as I*" = I* + 1^ + I*, such that l V = &M M +1<j>£, cf. Eq. (2.6.8) J £' j The basis functions connected by t h i s operator d i f f e r i n only one part £ ^ of the product function cj)^, cf. Eq. (2.5.8), corresponding to the component operator i t . In t h i s manner, i t i s p o s s i b l e to assign t r a n s i t i o n s to a s p e c i f i c spin within a given spin system i n the f i r s t -order J-coupling l i m i t . It should be noted at t h i s p o i n t , however, that the operator defined above does not determine the combination t r a n s i t i o n s of f i n i t e i n t e n s i t y normally associated with a t i g h t l y coupled nuclear spin system. These combination t r a n s i t i o n s may be determined only by g e n e r a l i z i n g the operator 1^  to connect a l l 6 7 . basis function <JK and cjj^  with - M_. = + 1 . The product basis functions <|>£ f o r an ABX (I = h) spin system and the corresponding upper diagonal I_+ matrix are shown i n Table 2.3., the elements determining combination t r a n s i t i o n s being i n parentheses. For.the basis functions ordered as shown i n Table 2 . 3 the matrix elements I*^ and I*^ define A-spin trans-i t i o n s i n the f i r s t - o r d e r l i m i t as already described for an ABX (J^g = 0 ) spin system, c f . Table 2 . 2 . The corresponding B-spin t r a n s i t i o n s are de-+ + fined by the matrix elements 1 ^ and I j 4 > ^ n a c c o r d a n c e with Eq. ( 2 . 6 . 9 ) , the density matrix elements P'21' P 3 1 ' P 4 2 A N C * / ° 4 3 describe an AB-part of + a general ABX spectrum. Thus the I_ matrix i n the factored form i n d i -cated i n Table 2 . 3 shows that the ABX t r a n s i t i o n s may be considered i n terms of sets of density matrix elements (*? ti*) = A| I/7,* +^|3 + ^ 4 +yC?3 4 ) + l^+//ST+^Y/78) * 1 1 where p.. = p.. for the Hermitean density matrix p. The density matrix elements within each of these sets are now defined by coupled equations of motion of the form given i n Eq. ( 2 . 6 . 7 ) . A basic intramolecular exchange process may now be considered to correspond to a t r a n s f e r of nuclear magnetization between s p i n - s i t e s distinguished by chemical s h i f t s for the coupled inequivalent A- and B-spins, -ft and ft, r e s p e c t i v e l y . An example of such a process i s the hindered r o t a t i o n i n a molecular system having two possible conformations giving r i s e to the AB-spin inequivalence, t h i s r o t a t i o n being defined i n terms of a s i n g l e f i r s t - o r d e r rate constant, k sec. 1 . For the chosen product basis functions <$>^, the operator E given i n Eq. ( 2 . 6 . 6 ) and Table 2.3 Basis Functions f o r ABX Spin Systems I = h Nuclear Spins <j>£ aaa aBa Baa BBa aaB aBB BaB BBB 3 1 1 1 1 1 1 3 2 2 2 2 2 2 Spin T r a n s i t i o n Operator Matrix i n Basis ( ^ J " -0 0 1 o i i o o l CD o o .CD (1) o o o 8 0 68. describing the above exchange process for an equal population system may be defined by E|aa£x> = |aa£x>, E|a3£ > = |Ba£> .A. A. (2.6.11) App l i c a t i o n of t h i s exchange operator i n Eq. (2.6.7) shown that under steady-state conditions (-^ - £ = 0) the density matrix elements d e s c r i -bing an AB-part of the ABX spectrum i n the presence of exchange are given by sets of coupled equations which may be expressed i n the general matrix form: R* -G' iCP (2.6.12) such that V 4 l[*+/iZ.+£(3"+T A}c)] U - k X X r+i, [ * - ^ + ^ ( T - K J 6 ) C ) J 1 1 and X u _ k A-- - i d 1 1 (2.6.13) where r = ^ + k. I t i s assumed that spin-spin r e l a x a t i o n e f f e c t s i n 2 the absence of exchange are defined by the s i n g l e r e l a x a t i o n time T , c f . Eq. (2.6.5). The above matrix equation i s equivalent to those pre-v i o u s l y derived i n terms o f s e m i - c l a s s i c a l modified Bloch equations f o r the simple two-site exchange system, cf. Eq. (2.2.12), and a m u l t i - s i t e exchange system i n the f i r s t order J-coupling l i m i t , (cf. Eq. 2.4.4). In the t i g h t l y coupled (J ± 0) ABX spin system, l i n e a r combinations of the vector elements Gj may be considered to define transverse s p i n -s i t e magnetizations analogous to those associated with the isochromats M ( x j , ^ j ) i n t n e f i r s t - o r d e r l i m i t , c f . F i g . 2.2. That i s , a s p i n - s i t e magnetization vector G_ i s defined by D-G = -iCQ (2.6.14) where J) i s the diagonal matrix corresponding to R/ . In t h i s manner, the elements of the matrix J) determine the p o s i t i o n s and general lineshape c h a r a c t e r i s t i c s f o r s p e c i f i c s p i n - s i t e s and the elements of the vector Q determine the o v e r a l l s p e c t r a l i n t e n s i t i e s associated with these s i t e s . I t should be noted that, through the d e f i n i t i o n of the general spin density matrix element i n Eq. (2.6.4), t h i s diagonal matrix i s equivalent to that obtained i n a basis {^ }^ c o n s i s t i n g of eigen-functions of the spin Hamiltonian defined i n Eq. (2.6.1). The above diagon-a l i z a t i o n procedure i s also d i r e c t l y r e l a t e d to that already used i n the s i m p l i f i e d f i r s t - o r d e r l i m i t , cf. Eq. (2.4.6). The d e t a i l e d form of the AB-part spectrum associated with the vector elements designated as Gj = and G^ = ma>' n o w be determined i n terms of the vector elements G^ and G^ (defining s p i n - s i t e s 1 and 3) corresponding to the 7 0 . diagonal, matrix elements and given from Eq. (2.6.13) as D = r + i x + % ( J + J + ) + D , D„ = r + i x + J s ( J + J ) - D 3 + with D_ = h 4(ft2 - k 2) + J(J - 4ik) + J_(J_ - 4ft) ^  and J + = %GJg X ± J ^ ) In accordance with Eq. (2.6.14), a partial lineshape function may now be expressed in the form 3> + V (2.6.15) where the normalization constant A includes C and the complex Q_ vector elements corresponding to and G^  have been defined as = a + id and = b + ie. These intensity vector elements are now determined by equating Eqs. (2.6.13) and (2.6.15), that is (r + k) + i x + J + hJ+ = (a + id){r + i[x + h(J + J +) - DJ } + + (b + ie){r + i[x + % ( J + J +) + DJ }. Thus the par t i a l lineshape corresponding to spin-sites 1 and 3 is given in terms of G^(x) and G^(x), where 1 X with 3_ = (%J - ik)/D . As usual, the absorption mode spectrum is de-scribed by the imaginary parts of G^(x) and G^(x). In the absence of 71. exchange (k = 0 ) , Eq. (2.6.16) shows that G (x) defines a L o r e n t z i a n a b s o r p t i o n mode l i n e w i t h a f u l l - w i d t h at half-maximum o f 2/T rad. s e c . " 1 centred at x^ = - % ( J + J + ) - D_ with a r e l a t i v e i n t e n s i t y of 1 - J/2D . This component l i n e i s represented as s p i n s i t e - 1 i n F i g . 2.11 showing the complete AB-part of the ABX spectrum, where i t has been assumed that > > J > 0. As the independent frequency v a r i a b l e x and the common r e l a x a t i o n time form p a r t of the diagonal elements of the mat r i x R' i n Eq. (2.6.12), t h i s m a t r i x may be reformulated to give R + (i + i x U l G ' = - i C P 1 , (2.6.17) 2 where _I. i s the u n i t m a t r i x . S p e c t r a l c h a r a c t e r i s t i c s a s s o c i a t e d w i t h exchange are now defined i n terms o f a diagonal matrix A corresponding to R, where i n accordance w i t h Eq. (2.6.14) [A + d + i x ) i ] G = - iCQ. (2.6.18) L 2 The elements o f the matrix yV, A.., and the corresponding i n t e n s i t y f a c -t o r s Q^. are given i n Table 2.4 for a l l s p i n s i t e s - j i n the AB-part o f an ABX spectrum. Thus i t i s seen that i n the absence of exchange, the de n s i t y m a t r i x elements P^' ^24 a n c* ^34 determine t n e s p e c t r a l c h a r a c t e r i s t i c s f o r the s p i n s i t e s 1 - 4 , t h i s p a r t of the spectrum c o n s i s t i n g of a t y p i c a l AB q u a r t e t . I t i s i n t e r e s t i n g to note the c o r r e l a t i o n between these p a r t i c u l a r d e n s i t y m a t r i x elements and the energy l e v e l s d e f i n i n g the degenerate AB s p i n s i t e s f o r a f i r s t - o r d e r ( J A B = 0 ) ABX s p i n system as l i s t e d i n Table 2.2. For the type of i n t r a m o l e c u l a r exchange process under con-s i d e r a t i o n f o r an ABX s p i n system, the exchange operator E determines Table 2.4 AB-Part o f an ABX Spectrum M a t r i x Element X. 3 % ( J + J ) + D + -I n t e n s i t y Factor Q 1 - 3 -h(J - J + ) + D + 1 + 3 h(J + J + ) - D_ 1 + 3 i(J - J J - D 1 - 3, =2(J - d + J 1- Jt_ i - y - % ( J + J J + Ej 1 + j % ( J - J + ) - E_ 1 + Y h(J + J + ) - E + 1 " Y, 4 (ft 2 - k 2 ) + J ( J ±. 4ik) + J _ ( J _ - 4ft) %[^4(ft 2 - k 2 ) + J ( J ± 4ik) + J _ ( J _ + 4ft) \hJ ± i k ] / D ± , Y ± = [hJ ± i k ] / E ± ^ J B X * JAX) the independence of the two q u a r t e t s forming the AB-part o f the spectrum. Thus the form o f the matrices i n Eq. (2.6.13), as d e f i n e d by the s p i n t r a n s i t i o n operator I + (and E) a l l o w s a r e l a t i v e l y simple a n a l y t i c a l f o r m u l a t i o n of the ab s o r p t i o n mode lineshape f o r a com-ponent AB q u a r t e t . From Eq. (2.6.16) and s i m i l a r expressions f o r G^(x) and G^(x), t h i s lineshape i s obtained f o r an equal p o p u l a t i o n exchange system as where B ± . J W / ( ^ + k ) ± ifj The independent frequency v a r i a b l e w has been defined,to s i m p l i f y the above e x p r e s s i o n , as w = x + hJ f o r the AB qu a r t e t centred on x = - % J + J c f . Table 2.4. A l s o , i n t h i s manner Eq. (2.6.19) may be a p p l i e d i n gen-era'l f o r a s i m i l a r i n t r a m o l e c u l a r exchange process i n an a r b i t r a r y AB s p i n system. Such an a n a l y t i c a l expression f o r the absor p t i o n mode lineshape allows a very e f f i c i e n t i t e r a t i v e comparison w i t h experimental data. A l s o , i t i s r e a d i l y seen that Eq. (2.6.19) reduces to the l i n e -shape equation p r e v i o u s l y d e r i v e d f o r the simple two s i t e uncoupled AB s p i n system, c f . Eq. (2.2.15) w i t h x =. w, from s e m i - c l a s s i c a l modified Bloch equations. Using Eq. (2.6.19) and a s i m i l a r expression f o r the AB q u a r t e t centred at x = hJ+, lineshapes have been c a l c u l a t e d f o r a range o f the parameter k/ft and are shown i n F i g . 2.11. These abso r p t i o n (2.6.19) 73. mode lineshapes are defined by the parameters 9 = 4.0 Hz, J = 4.2 Hz, J A Y = 10.0 Hz, J = 15.0 Hz and T = 0.64 sec. (0.5 Hz f u l l - w i d t h at AA 13 A 2 half-maximum), and may be compared w i t h those f o r the f i r s t - o r d e r ABX s p i n system shown i n F i g . 2.8a. A b a s i c i n t r a m o l e c u l a r exchange process has been considered i n terms of a t r a n s f e r of nucl e a r magnetization between s p i n - s i t e s d i s t i n g u i s h e d by d i s t i n c t chemical s h i f t s f o r the coupled i n e q u i v a l e n t A- and B-spins. The operator d e s c r i b i n g t h i s process, c f . Eq. (2.6.6), may be g e n e r a l i z e d to i n c l u d e an unequal p o p u l a t i o n exchange system. An example,of t h i s more general exchange process i s the hindered ro-t a t i o n i n a molecular system having two p o s s i b l e conformations, one of which i s p r e f e r r e d . The exchange terms i n the component equation of motion f o r the densitv matrix. F,n . f2.6.71 . mav be considered i n the form k^ <<j)_. |EpE|(J)^> - k^ p ^ , where k^ i s the p r o b a b i l i t y f o r a t r a n s f e r from a b a s i c A-spin s i t e w i t h a f r a c t i o n a l p o p u l a t i o n p . In terms of a matrix equation of the general form given i n Eq. (2.6.12), the den-s i t y matrix elements p ^ and are now def i n e d by PA = -IC - 1\-^K r & + i [ x - ^ . ( w & J ( ) ] fe (2.6.20) where r = — + k wi t h p. + p D = 1 and p.k. = p n k n . This matrix A T 2 A r A r B r A A rB B fo r m u l a t i o n f o r an unequal p o p u l a t i o n system allows a much simp l e r c a l c u l a t i o n o f lineshapes, f o r the p a r t i c u l a r exchange process under 23 c o n s i d e r a t i o n , than an a l t e r n a t i v e procedure proposed by Johnson The p a r t i a l lineshape a s s o c i a t e d w i t h the above de n s i t y m a t r i x elements i s given by G^(x) and G (x) i n Eq. (2.6.16) where the parameters D and $ are now defined as and 3 . J. - iV - A T - A-p J&> w i t h AJ = J + - ( P A J B X + P B J A X ) > 2 k = k A + k B a n d A P = P A - P B- The complete AB-part spectrum may now be defined i n terms o f the matrix elements X^ and corresponding i n t e n s i t y f a c t o r s Q.. , c f . Eq. (2.6.18), as given i n Table 2.4 wi t h modified parameters: 3>. 7. L E ± = l [ 4 - ( ^ - ^ ) + T(T^4-:k) 4- T_(T_ + 4vro.) + 4-1 k (LT-X, Lljh)j % J ±. C.L — AT - A.pa2v V ± .- [ J j f c l k , +- AT ^ A.p <_OvJ / (2.6.21) I t has been assumed t h a t a l l J - c o u p l i n g constants i n the ABX sp i n system have the same s i g n . For the p a r t i c u l a r exchange process considered, the form of the AB-part spectrum may be c r i t i c a l l y dependent upon the r e l a t i v e signs o f these coupling constants due to terms such as ±4ikJ and ±4J ft i n the c h a r a c t e r i s t i c parameters D + and E . Thus the r e l a t i v e signs of J and J D V w i t h respect to the AB coupling A X B X constant J determine the form o f the AB-part spectrum i n a manner s i m i l a r to t h a t f o r a f i r s t - o r d e r ABX system as i l l u s t r a t e d i n F i g . 2.8, and a complete lineshape f i t to experimental data may allow a. simple determination of the r e l a t i v e signs of a l l coupling constants f o r a general ABX s p i n system. The X-part of an ABX spectrum has been shown to be d e t e r -mined by the s i x s p i n d e n s i t y m a t r i x elements P^^> $26' ^27' ^36' ^37 and p^g3 c f . Eq. (2.6.10). The elements p^j and p ^ are a s s o c i a t e d w i t h combination t r a n s i t i o n s , and comparison w i t h the energy l e v e l s determining the X-part of a f i r s t - o r d e r ABX spectrum, c f . Table 2 shows that (2, 7) and (3,6) are not i n c l u d e d i n t h i s l i m i t . The chosen b a s i s f u n c t i o n s (J). , cj>r, (f>. and <))_ are i n v a r i a n t under the ex-1 o 4 o change operator E as defined i n Eqs. (2.6.6) and (2.6.11), and hence p and p are d i r e c t l y r e l a t e d , through Eq. (2.6.14), to the t r a n s -i t 4o verse magnetization associated with s p i n - s i t e s 1 and 6 as shown i n F i g . 2.12. In the presence of exchange, these s p i n - s i t e s correspond to simple L o r e n t z i a n l i n e s determined from Eqs. (2.6.7) and (2.6.14) as GcAiL) = LA Ik &c La) - ik \ + ^ U + * f i * - T + ) ( 2 . 6 < 2 2 ) The remaining s p i n - s i t e magnetizations are determined by a 4 x 4 matrix R i n Eq. (2.6.16), that i s k = 0.1. sec 7 6 . iZ IZ 7~ X iZ z - i + XiJh 0 (2.6.23) where [_R + + i (x + ft )}JJG_' = -iCPJ . The elements o£ the v e c t o r P_* corresponding to combination t r a n s i t i o n s (2 and 5 i n F i g . 2.12) are set equal to zero to allow a c o n s i s t e n t a p p l i c a t i o n of the g e n e r a l i z e d I + operator i n the e v a l u a t i o n of the general s p i n d e n s i t y m a t r i x equation of motion term given i n Eq. (2.6.8). The lineshape fun r*+- i f~\ T\ n f^r~\ A, s c r i b i n g t h i s p o r t i o n of the ABX spectrum i s most r e a d i l y evaluated by co n s i d e r i n g the above equation i n the general forms given i n Eqs.(2.6.16) and (2.6.17). That i s , a diagonal m a t r i x A_ i s d e r i v e d from the matrix R through a s i m i l a r i t y t r a n s f o r m a t i o n as described i n d e t a i l f o r the f i r s t - o r d e r ABX s p i n system, c f . Eq. (2.5.14). For a t i g h t l y coupled s p i n system, however, the vec t o r I_ i n Eq. (2.5.14) must take the same form as P_ to allow f o r the e f f e c t o f combination t r a n s i t i o n s on the o v e r a l l lineshape. Computed lineshapes f o r the X-part of an equal p o p u l a t i o n ABX s p i n system are shown i n F i g . 2.12 f o r the parameters chosen to d e f i n e the AB-part spectrum, c f . F i g . 2.11. Again, these lineshapes may be compared with those f o r the f i r s t - o r d e r ABX system, c f . F i g . 2.10a. I t i s to be noted t h a t f o r p a r t i c u l a r cases, f o r example, when , » J i n a heteronuclear system, i t i s p o s s i b l e 77. to t r e a t the X-part spectrum as that i n the f i r s t - o r d e r l i m i t . In t h i s approximation, the o v e r a l l lineshape i s simply obtained i n terms of G^(x) and (x) given i n Eq. (2.6.22) and a lineshape f o r an e f f e c t i v e two-s i t e exchange system w i t h s p i n - s i t e frequencies defined i n Table 2 as x , = - f l - J and x = - f l + J , c f . Eq. (2.2.15) w i t h fl = J . The 3 x - 4 x - n <-b a s i c c o n d i t i o n f o r t h i s approximation, of course, corresponds to a complete neglect of the o f f - d i a g o n a l elements i n J i n the m a t r i x R. In g e n e r a l , the e f f e c t o f d i f f e r e n t r e l a t i v e signs of J . v and J i s AA DA very s i m i l a r to that i l l u s t r a t e d f o r the f i r s t - o r d e r case i n F i g . 2.10b. A d e t a i l e d a n a l y s i s of a general ABX s p i n system has allowed a c o n s i s t e n t development of the p h y s i c a l model and m a t r i x f o r m u l a t i o n r e q u i r e d f o r a d e s c r i p t i o n o f i n t r a m o l e c u l a r exchange e f f e c t s i n a t i g h t l y coupled s p i n system. Although a n a l y t i c a l lineshape equations may be d e r i v e d i n p a r t f o r more complicated s p i n systems, a formu-l a t i o n of exchange e f f e c t s a p p l i c a b l e to the numerical computation of the lineshape f u n c t i o n s G(x) and V(x) f o r a general t i g h t l y coupled N-spin system may now be described i n terms of the p r o p e r t i e s of the d e n s i t y matrix model considered above. In g e n e r a l , having determined the d e n s i t y matrix elements r e q u i r e d i n the d e s c r i p t i o n of an NMR spectrum (or p a r t of one) through the g e n e r a l i z e d t r a n s i t i o n operator I i n Eq. (2.6.9), these elements must be evaluated i n accordance w i t h the component equation of motion, Eq. (2.6.7). In the chosen b a s i s {(j)^}, the equation of motion term i n v o l v i n g the independent frequency v a r i a b l e i n an i n t e r a c t i o n r e p r e s e n t a t i o n i s simply Kf i x « ( . j |[p Z I z i l l * a > = - i x p j £ (2.6.24) 78. The e f f e c t i v e s p i n Hamiltonian i n t h i s r e p r e s e n t a t i o n may now be con-s i d e r e d i n the form }(^ + Xj"^ + > analogous to that defined i n Eq. (2.6.1). The p a r t X^ + X j ^ i s diagonal i n the b a s i s i4>£} and hence i t f o l l o w s that w i t h the Hamiltonian m a t r i x element }i ^  d e fined by ti M Hji. = - dl*-J*r + T^^JLl^J-i (2.6.26) where = m £ i ^ ' c : | r. ^ c l - (2.5.8). I t i s to be noted t h a t the term 7iji£ - X'jj defines a f i r s t - o r d e r t r a n s i t i o n energy and hence the s p i n d e n s i t y matrix element defines d i r e c t l y a s p i n - s i t e i n t h i s l i m i t , as p r e v i o u s l y considered f o r p a r t i c u l a r s p i n systems. Such a r e l a t i o n s h i p 92 i s d e f i n e d i n general through the d e f i n i t i o n of a L i o u v i l l e operator ', such that the elements o f the d e n s i t y matrix jo may be considered as components of a v e c t o r i n a s u i t a b l y d e f i n e d vector space. The d e r i -41 v a t i o n super-operator used by Banwell and Primas i n a d i r e c t c a l c u -l a t i o n " of NMR s p e c t r a i s i n f a c t i d e n t i c a l to the L i o u v i l l e operator 92 (2) defi n e d by Fano . As the second-order coupling Hamiltonian t e r m , X j mixes f u n c t i o n s i n the b a s i s ify^} and corresponds to the o f f - d i a g o n a l p a r t of the complete s p i n Hamiltonian, t h i s term leads to a mixing of s p i n d e n s i t y m a t r i x elements. Thus i t f o l l o w s that <*.|Cp,WJ 2 ) ]|V = x £ r p j r - X ^ , , (2-6.27) where X j <j>£ = <J>^, and X j <f>j = <j>j , • S i m i l a r l y , the term d e f i n i n g exchange e f f e c t s i s evaluated as k«j). |EpE|(j>£> = kp.„ £ 1 I, (2.6.28) where E<J)£ - <j)^M. C o n s t r u c t i o n of a v e c t o r G_' with elements as determined by the t r a n s i t i o n operator I and a l s o i n d e x i n g v e c t o r s a s s o c i a t e d w i t h the e f f e c t i v e d e n s i t y m a t r i x s u b s c r i p t c o n t r a c t i o n , s p e c i f i c s p i n Hamiltonian m a t r i x elements and the b a s i s f u n c t i o n s connected by the exchange operator E allows the matrix R i n Eq.(2.6.14) to be e f f i c i e n t l y assembled on a computer. A l l elements o f t h i s m a t r i x are n u m e r i c a l l y evaluated i n terms o f the normal s p i n Hamil-t o n i a n elements d e f i n e d f o r a s p e c i f i c s p i n system i n the b a s i s ify^), by making use of Eqs. (2.6.24) - (2.6.28). The computation of the ab s o r p t i o n mode lineshape f u n c t i o n V(x) then reduces to a p p l i c a t i o n s of the b a s i c equations given i n Eq. (2.4.6) and (2.4.9). The over-a l l d i m e n s i o n a l i t y of the m a t r i x R f o r an i n t r a m o l e c u l a r exchange process i s determined by the number of d e n s i t y m a t r i x elements de-f i n e d by I + i n Eq. (2.6.9), t h a t i s , the number o f allowed and com-b i n a t i o n t r a n s i t i o n s f o r a given s p i n system. As the d i a g o n a l i z a t i o n of the complex m a t r i x R and the i n v e r s i o n of the a s s o c i a t e d complex t r a n s f o r m a t i o n m a t r i x J5 are the time determining computational pro-cedures, a l l f a c t o r i z a t i o n s of these matrices as d e f i n e d by the form of the Hamiltonian f o r a s p e c i f i c s p i n system should be taken i n t o account. A computer program GENLIN has been developed on the b a s i s of the s i m p l i f i e d d e n s i t y m a t r i x f o r m u l a t i o n o u t l i n e d above, and w i l l be described i n more d e t a i l as r e l a t e d to the a n a l y s i s o f a p a r t i c u l a r 4-spin system i n an experimental s e c t i o n of t h i s t h e s i s . I t i s to be noted t h a t , d u r ing the course of the work 9 3 described here, Binsch independently proposed a s i m i l a r method numerical a n a l y s i s f o r t i g h t l y coupled s p i n systems based upon a more formal theory i n the L i o u v i l l e r e p r e s e n t a t i o n . 81. CHAPTER 3 INSTRUMENTATION .-3.1 FT-1064 Computer I n t e r f a c e Complete lineshape analyses of chemical exchange processes u s i n g NMR i n v o l v e sets of data p o i n t s d e f i n i n g the ex p e r i m e n t a l l y recorded s t e a d y - s t a t e s p e c t r a , a set o f d i s c r e t e data p o i n t s being f i t t e d to a t h e o r e t i c a l lineshape equation to o b t a i n a s p e c i f i c r a t e constant. The tedious manual conversion of the recorded data to d i g i t a l frequency and corresponding amplitude values may be e l i m i n a t e d by using an e l e c t r o n i c s i g n a l sampling device and an a n a l o g - t o - d i g i t a l (A-D) converter l i n k e d to a small computer w i t h a magnetic memory core to st o r e the d i g i t a l i n f o r m a t i o n . Of course, the computer must scan i t s memory l o c a t i o n s (channels) i n s y n c h r o n i z a t i o n w i t h the spectrometer frequency (or f i e l d ) sweep so that the s i g n a l amplitude data p o i n t i n any given memory l o c a t i o n may be a c c u r a t e l y assigned a frequency value d e r i v e d from two c a l i b r a t i o n frequencies. The d i g i t a l i n f o r m a t i o n may be t r a n s f e r r e d to a f u l l - s c a l e d i g i t a l computer, to allow an e f f i c i e n t i t e r a t i v e lineshape f i t , v i a an incremental magnetic tape o r a d i r e c t l i n e i f t h i s i s a v a i l a b l e . The FABRITEK FT-1064 computer has been used, "in c onjunction w i t h the spectrometer-computer i n t e r f a c e u n i t d e s c r i b e d below, to give an automatic d i g i t i z a t i o n and f i t t i n g o f NMR lineshape data. The i n t e r f a c e u n i t has been designed to allow a general a p p l i -82. c a t i o n o f the FT-1064 computer with only simple m o d i f i c a t i o n s o f spectrometer c i r c u i t r y . I n i t i a l l y i t w i l l be assumed that the NMR s i g n a l - t o - n o i s e r a t i o i s such that a s i n g l e scan of the stea d y - s t a t e spectrum i s s u f f i c i e n t t o o b t a i n data o f a q u a l i t y warranting a complete lineshape a n a l y s i s . A block diagram showing the b a s i c u n i t s o f the i n t e r f a c e i s given i n F i g . 3.1, and the as s o c i a t e d t i m i n g c o n t r o l sequence may be represented as shown i n F i g . 3.2. In the s i n g l e scan mode, the spectrometer sweep .mechanism i s used under normal o p e r a t i n g c o n d i t i o n s and the NMR s i g n a l at the output of the spectrometer recorder a m p l i f i e r i s fed continuously to a high impedance input d i f f e r e n t i a l a m p l i f i e r , A, and then to the sampling A-D converter, SW/1, of the FT-1064 computer. This d i f f e r e n t i a l a m p l i f i e r acts as a b u f f e r between the sampling device and the spectrometer and a l s o allows a v e r s a t i l e gain and dc l e v e l c o n t r o l independent of the normal spectro-meter c o n t r o l s . At time t Q the c o n t r o l b i s t a b l e m u l t i v i b r a t o r , B, changes to i t s a c t i v e s t a t e under the a c t i o n of the manual sweep t r i g g e r , T. This m u l t i v i b r a t o r i s used as a master c o n t r o l and as a t r i g g e r b u f f e r , the m u l t i v i b r a t o r s t a t e at any time being shown by the i n d i c a t o r I . At t h i s time, t Q , a t r i g g e r pulse from the c o n t r o l b i s t a b l e i n i t i a t e s the FT-1064 channel sweep and simultaneously a p o s i t i v e pulse i s ac-coupled to the input of the spectrometer recorder a m p l i f i e r . The e m i t t e r f o l l o w e r F l acts as a b u f f e r between the c o n t r o l b i s t a b l e and the FT-1064 sweep u n i t which has a r e l a t i v e l y low input impedance. The monostable m u l t i v i b r a t o r M s u p p l i e s a 5 msec pulse of an amplitude s u f f i c i e n t to give a w e l l - d e f i n e d v e r t i c a l marker on the recorder c h a r t , t h i s marker a c c u r a t e l y determining the p o s i t i o n o f the i n i t i a l S V V E E R R E S E T CD CD S W E E P D I G I T A L O U T P U T R E A D O U T — ^ F2 R E C O R D E R I N P U T A R E C O R D E R O U T P U T F i g . 3.1 FT-1064 computer sweep c o n t r o l I Sweep trigger Reset trigger Bistable control voltage i i i i I 1 FT-1064 aate -FT-1064 sweep voltage i i i i i i Marker pulse i i i i i — , , 1 1 1 1 1 F i g . 3.2 FT-1064 c o n t r o l sequence 83. d i g i t i z e d data p o i n t as the sweep times used correspond to 50 msec -2 sec per channel. A 5 msec pulse has been found to give a f a s t r i s i n g marker w i t h i n the normal response time l i m i t of the r e c o r d e r system. In the time i n t e r v a l t 0 to t i the c o n t r o l b i s t a b l e remains i n the a c t i v e s t a t e , the FT-1064 gate voltage i s p o s i t i v e and the l i n e a r channel sweep voltage changes as shown i n F i g . 3.2. The analog NMR s i g n a l i s sampled and d i g i t i z e d and a l s o recorded on the spectrometer chart over t h i s time i n t e r v a l , the d i g i t a l s i g n a l amplitude data being continuously s t o r e d i n the computer magnetic core memory. At time t i , the FT-1064 channel sweep i s complete and a p o s i t i v e p u l s e d e r i v e d from the t r a i l i n g edge of the gate, i n the i n v e r s i o n and d i f f e r e n t i a t i o n u n i t F2, t r i g g e r s the monostable m u l t i v i b r a t o r , M, to give a second marker which determines the p o s i t i o n o f the f i n a l d i g i t i z e d data p o i n t . The c a l i b r a t i o n frequencies corresponding to the i n i t i a l and f i n a l d i g i t i z e d data p o i n t s are now obtained by d i r e c t measurement of the observing f r e -quency r e l a t i v e to the NMR f i e l d l o c k i n g frequency by matching the r e -corder pen p o s i t i o n w i t h the chart markers. In t h i s manner these frequencies are r e a d i l y determined to ah accuracy o f ± 0.1 Hz, which r e q u i r e s a counter gate time of at l e a s t 10 sees and i n t u r n prevents an accurate frequency measurement during a spectrum scan, and are al s o shown t o be r e p r o d u c i b l e w i t h i n these l i m i t s . For an N-channel sweep, the frequency a s s o c i a t e d w i t h channel n and s i g n a l amplitude V n i s simply given by = w 0 + n . Aw,with the frequency increment Aw de f i n e d as Aw = (wi - wo)/(N-l) where w Q and Wi are the c a l i b r a t i o n frequencies cor-responding t o the markers set at times t 0 and t j , r e s p e c t i v e l y . The c o n t r o l b i s t a b l e i n i t s a c t i v e s t a t e prevents an ina d v e r t a n t computer 84. sweep, and preceding a new sweep t h i s b i s t a b l e s t a t e must be changed under the a c t i o n of the manual reset t r i g g e r T at a time t 2 , as shown i n Fig.3.2. The spectrometer recorder may now be r e p o s i t i o n e d f o r another s i n g l e scan and the FT-1064 channel sweep i s i n i t i a t e d again by the manual sweep t r i g g e r , T, at time t$. In t h i s way the spectro-meter may be swept i n an a r b i t r a r y d i r e c t i o n and d i g i t i z e d data may be s t o r e d i n independent s e c t i o n s of the computer memory core. The d e t a i l e d c i r c u i t diagram f o r the i n t e r f a c e u n i t i s given i n F i g . 3.3, and the p e r t i n e n t c o n t r o l waveforms and volta g e s are inc l u d e d on t h i s diagram. The manual t r i g g e r c i r c u i t s centred on the 2N3646 high speed NPN s i l i c o n s w i t c h i n g t r a n s i s t o r s TI and T2 are p a r t i c u l a r l y simple and have been shown to be completely r e l i a b l e i n ope r a t i o n . With switch SI open, TI i s b i a s e d i n t o conduction and the c o l l e c t o r v o l tage i s h e l d at +0.4V wh i l e the base c a p a c i t o r Cx i s charged to +4.5V. When the switch SI i s c l o s e d , the c a p a c i t o r C i i s discharged and TI i s c u t - o f f as the base v o l t a g e f a l l s t o near OV. The c o l l e c t o r voltage r i s e s and the p o s i t i v e going output t r i g g e r pulse f o l l o w i n g d i f f e r e n t i a t i o n has an amplitude o f ^  2V w i t h a r i s e time o f 0.1 usee. When SI i s reopened, Ci recharges w i t h a time constant R i C i and a spurious t r i g g e r waveform (normally a s s o c i a t e d w i t h switch bounce i s e l i m i n a t e d . The emitter-coupled c o n t r o l b i s t a b l e m u l t i v i b r a t o r , T3 and T4, i s of conventional design and uses a s i n g l e MC-715 MRTL dual t r i p l e - i n p u t gate w i t h a s w i t c h i n g t h r e s h o l d o f +0.7V and s w i t c h i n g times of the order o f 0.1 usee. The quiescent and a c t i v e s t a t e s of the c o n t r o l b i s t a b l e correspond to T4 c o l l e c t o r v o l t a g e s o f +0.2 amd +4.2V, r e s p e c t i v e l y . A s i m i l a r MRTL c i r c u i t , T7-T9, i s used f o r the monostabl 3900 Rl 1 0 0 0 : •0.4-640 0.001 4 7 Tl SWEEP s , H | J •20 CI 560 560 T3 ^6800 2 2 0 Ll 1000 S 3900 0 1 i . r 560 r : 0.001 560 2 7 0 0 ; T4 T 2 6800; 2 0 -r — o +6V 4.2 • 0.2 - 1 > T10 RESET | S2 47 2N3646 T MC 715 B 2N3646 T 2N3566 I T5 BISTABLE 0 001 , CONTROL ^t-gXJQ 1 0 K < ? 2 2 0 "2.4 0.2-1 0 0 0 : 2 2 0 0 T FT-1064 T 6 j GATE ^ . W - ^ NPUT Y ^00V 1 Q K 1 0 0 0 1 f^P\ FT-1064 SWEEP TRIGGER 6 4 0 O -o +6V R2 470 K 640 560 10K 560 0.01 C2 560 560 T7 T8 0.1 T9 MARKER PULSE 2N3646 2N3646 1N914 F l F2 MC-715 F i g . 3.3 Spectrometer-computer i n t e r f a c e u n i t 85. m u l t i v i b r a t o r , t h e o u t p u t p u l s e w i d t h b e i n g d e t e r m i n e d b y t h e t i m e c o n s t a n t R2C2. The i n d i c a t o r c i r c u i t c o n s i s t s o f a s e r i e s c o n n e c t e d lamp, L I , and s i n g l e t r a n s i s t o r , T10, s u c h t h a t t h i s t r a n s i s t o r i s b i a s e d i n t o c o n d u c t i o n and t h e lamp i s on when t h e c o n t r o l b i s t a b l e i s i n i t s a c t i v e s t a t e . The c i r c u i t r y f o r t h e d i f f e r e n t i a l a m p l i f i e r , A i n F i g . 3.1, i s shown i n F i g . 3.4 and i s b a s e d upon P h i l b r i c k o p e r a t i o n a l a m p l i f i e r s , t y p e s 1009 and 1301. The 1009 i s a g e n e r a l p u r p o s e l o w - n o i s e u n i t w h i c h f e a t u r e s an FET i n p u t s t a g e t o p r o v i d e an i n p u t impedance o f 20M f o r t h e u n i t y g a i n c o n f i g u r a t i o n u s e d . T h i s a l l o w s e s s e n t i a l l y c o m p l e t e i s o l a t i o n o f t h e computer s a m p l i n g u n i t f r o m t h e s p e c t r o m e t e r s y s t e m , even though d c - c o u p l i n g i s r e t a i n e d t o p r o v i d e an u n d i s t o r t e d low f r e q u e n c y s i g n a l r e s p o n s e . The h i g h i n p u t impedance a l s o a l l o w s a s i m p l e dc z e r o l e v e l c o n t r o l t h r o u g h a v o l t a g e summing a t one i n p u t o f th e d i f f e r e n t i a l a m p l i f i e r . The 1301 o p e r a t i o n a l a m p l i f i e r i s u s e d as a v a r i a b l e g a i n i n v e r t i n g dc a m p l i f i e r w i t h a maximum g a i n o f 10 w i t h s w i t c h S I i n p o s i t i o n 1. The o v e r a l l l i n e a r i t y o f t h e a m p l i f i e r was ch e c k e d o v e r t h e o u t p u t v o l t a g e range -5 t o +5V and was f o u n d t o be b e t t e r t h a n 0.4%. The o u t p u t o f t h i s a m p l i f i e r may be f e d d i r e c t l y t o t h e FT-1064 SD/1 s a m p l i n g A-D c o n v e r t e r w i t h an i n p u t impedance o f 5K. An o s c i l l o s c o p e d i s p l a y o f the computer memory contents over 1024 channels i s a v a i l a b l e and the d i g i t a l data p e r t a i n i n g to a p a r t i c u l a r memory l o c a t i o n i s i n d i c a t e d on the analog s i g n a l d i s -p l a y by an i n t e n s i f i e d s p o t . T h i s d i g i t a l d i s p l a y a l l o w s a most convenient method o f s e l e c t i n g a s m a l l e r number of data p o i n t s to be f i t t e d to a t h e o r e t i c a l l i n e s h a p e , the a s s o c i a t e d frequency 86. being d i r e c t l y r e l a t e d to the channel (memory l o c a t i o n ) number as p r e v i o u s l y d e s c r i b e d . In g e n e r a l , the d i g i t a l data i s t r a n s f e r e d from the memory core to an incrementa l magnetic tape us ing a FABRITEK FT-282 c o n t r o l u n i t w i t h a KENNEDY Model 1400 tape r e c o r d e r . A d i g i t i z e d s i g n a l ampl i tude va lue i s s t o r e d and read out as a word c o n s i s t i n g o f two b y t e s , each byte being s i x b i n a r y b i t s . T h i s d i g i t a l word f o r m , however, i s o n l y compat ib le w i t h IBM 360 a r i t h m e t i c f o l l o w i n g a c o n v e r s i o n to the s tandard 18 b i t s f o r t h i s system. Thus a mixed FORTRAN-IV/SYMBOLIC program GET was developed t o a l l o w the e f f i c i e n t reading o f the magnetic tape i n the form o f independent o r d e r e d b y t e s , the unpacking o f s e l e c t e d two byte words and the s e t t i n g up o f an a r r a y o f e q u i v a l e n t three byte words f o r the i t e r a t i v e l i n e s h a p e f i t t i n g . A l t h o u g h 1024 data p o i n t s were a v a i l a b l e from the FT-1064 computer, an optimum minimal number o f data p o i n t s were a c t u a l l y used f o r the l i n e s h a p e f i t t i n g . For two- and f o u r -s i t e exchange s p e c t r a , i t was found t h a t the accuracy o f the f i t t i n g was not i n c r e a s e d above 64 o r 128 data p o i n t s . Thus these data p o i n t s were s e l e c t e d a t o p t i m i z e d i n t e r v a l s over the frequency range o f the d i g i t i z e d data by the program GET and the t o t a l CPU t ime r e q u i r e d f o r a c o m p l e t e l y automat ic l i n e s h a p e f i t to the taped d i g i t a l data was o f the o r d e r o f 10 sees per spectrum u s i n g the IBM 360/67 system. The complete i n t e r f a c e u n i t d e s c r i b e d above has been used w i t h most s a t i s f a c t o r y r e s u l t s w i t h both a JE0LC0 C-60 and a VARIAN HA-100 spec t rometer , o n l y the a d d i t i o n o f the two r e c o r d e r i n t e r - c o n n e c t i o n s shown i n F i g . 3.1 being r e q u i r e d to l i n k the spectrometer to the FT-1064 computer. 87. I f the s i g n a l - t o - n o i s e r a t i o f o r a s i n g l e scan spectrum i s i n s u f f i c i e n t to a l l o w a r e l i a b l e l i n e s h a p e a n a l y s i s , i t i s necessary to operate the computer-spectrometer system i n a m u l t i - s c a n accumula-t i v e mode. In t h i s case the i n t e r n a l sweep frequency o s c i l l a t o r o f the NMR spectrometer i s r e p l a c e d by an e x t e r n a l v o l t a g e c o n t r o l l e d o s c i l l a t o r d r i v e n i n s y n c h r o n i z a t i o n w i t h the FT-1064 channel sweep by the dc v o l t a g e a v a i l a b l e , as shown i n F i g . 3 . 2 . Thus the normal r e c o r d e r sweep mechanism i s d i s a b l e d , but the i n t e r n a l f i e l d l o c k i s r e t a i n e d . Once the r e q u i r e d s i g n a l - t o - n o i s e r a t i o i s a t t a i n e d the d i g i t a l data i s used as p r e v i o u s l y d e s c r i b e d , and a permanent c h a r t r e c o r d i n g i s o b t a i n e d by f e e d i n g the computer memory analog output to the spectrometer r e c o r d e r o p e r a t i n g i n i t s normal l i n e a r sweep mode us ing the above i n t e r f a c e u n i t to t r i g g e r the analog r e a d - o u t and to p l a c e the f requency c a l i b r a t i o n markers . 88. 3.2 Rf-pulse gate An i d e a l pulsed NMR spectrometer would c o n s i s t o f a t r a n s -m i t t e r s u p plying a high i n t e n s i t y r f pulse over a very s h o r t time i n t e r v a l , as compared with the n u c l e a r r e l a x a t i o n times i n v o l v e d , to r o t a t e the n u c l e a r magnetization through a w e l l - d e f i n e d angle 3, c f . F i g . 2,1; and a r e c e i v e r that would i n s t a n t l y record the f r e e induc-t i o n decay s i g n a l f o l l o w i n g the pulse without d i s t o r t i o n or the a d d i t i o n of n o i s e . In p r a c t i c e , i t i s d i f f i c u l t to produce a large constant amplitude r f magnetic f i e l d a s s o c i a t e d w i t h an a c c u r a t e l y timed pulse having a c l o s e l y c o n t r o l l e d width and minimal r i s e and f a l l times. A l s o , i t i s only p o s s i b l e to have a low noise (so that o v e r a l l s e n s i t i v i t y i s l i m i t e d by the thermal noise i n the sample c o i l ) l i n e a r r e c e i v e r w i t h a minimal time f o r recovery from overload f o l l o w i n g the r f p u l s e . In a d d i t i o n , an r f phase coherent system i s 29 r e q u i r e d to r e t a i n the n u c l e a r s p i n isochromat phase i n f o r m a t i o n inherent i n the detected f r e e i n d u c t i o n decay or s p i n echo, and to be able to a c c u r a t e l y determine resonance c o n d i t i o n s and the d e t a i l e d form o f the s p i n isochromat motion under the a c t i o n of s p e c i f i c types of r f pulses i n a general m u l t i - p u l s e sequence. An r f phase s e n s i t i v e d e t e c t i o n scheme has also been shown t o provide a c c u r a t e l y l i n e a r d e t e c t i o n o f weak s i g n a l s and to allow f u l l usage o f p o s t - d e t e c t i o n 120 121 i n t e g r a t i o n ' . In high r e s o l u t i o n NMR a p p l i c a t i o n s , the o v e r a l l s t a b i l i t y o f the spectrometer system i s c r i t i c a l . F o l l o w i n g the 29 30 o r i g i n a l pulsed NMR experiments o f Hahn and Carr and P u r c e l l , a number of spectrometer systems have been de s c r i b e d i n the l i t e r a t u r e 110 However, F o u r i e r transform a p p l i c a t i o n s and the study of 89. chemical exchange us i n g m u l t i - p u l s e sequences r e q u i r e p a r t i c u l a r l y s o p h i s t i c a t e d i n s t r u m e n t a t i o n . . Indeed, the l a r g e systematic e r r o r s apparent i n the k i n e t i c parameters d e r i v e d from pulsed mode NMR data may be l a r g e l y due to the lack o f such apparatus. One of the most c r i t i c a l component u n i t s i n a pulse spectro-meter i s the r f - p u l s e gate. This gate i s r e q u i r e d to provide pulses o f constant width and amplitude w i t h a very high r f suppression i n the time i n t e r v a l f o l l o w i n g each' p u l s e . The r a t i o o f the amplitude of an r f pulse (on-period) to the steady r f feedthrough ( o f f - p e r i o d ) may be r e f e r r e d to as the gate r e j e c t i o n r a t i o , a. The r f feedthrough may be considered to be n e g l i g i b l e as long as the resonance s a t u r a t i o n f a c t o r S1 , c f . Eq. (2.3.1), i s i n the range 0.99 < S' < 1.0. The f a c t o r 0.99 corresponds to a s a t u r a t i o n e f f e c t d i s t o r t i o n of the free induc-t i o n decay o f about 1% over the e n t i r e measurement time i n t e r v a l . I f the iT/2-pulse width i s assumed to be 0.01 T2, where T2 i s the t o t a l t r a nsverse r e l a x a t i o n time, the r e s t r i c t i o n on the s p i n - l a t t i c e r e l a x a -t i o n time T i s a t i s f y i n g the above s a t u r a t i o n c o n d i t i o n i s T i < 4 x IO" 6 a T2 ^* . In l i q u i d s Ti - T2, and t h i s i m p l i e s t h a t the minimum r e j e c t i o n . r a t i o r e q u i r e d i s a = 2 x 10 5, or 110 dB. For normal high r e s o l u t i o n NMR i n l i q u i d s T 2 ^ 0.6 sec, and hence a pulse width of 0.01 T2 corresponds to 6 msecs. In studying chemical exchange pro-cesses, however, the e f f e c t i v e transverse r e l a x a t i o n time may be s i g n i f i c a n t l y decreased and hence a iT/2-pulse width o f the order of micro-seconds i s d e s i r a b l e . In F o u r i e r transform a p p l i c a t i o n s l a r g e phase e r r o r s that are d i f f i c u l t to q u a n t i t a t i v e l y c o r r e c t may a r i s e through the general i n a b i l i t y t o measure the f r e e i n d u c t i o n decay 90. s i g n a l f o r t imes t -+ o . To m i n i m i s e these e r r o r s i t i s necessary t o use r e l a t i v e l y s h o r t i T / 2 - p u l s e s and a r e c e i v e r w i t h a f a s t r e c o v e r y from o v e r l o a d . Thus general s p e c i f i c a t i o n s f o r an r f - p u l s e gate i n a p u l s e spectrometer used f o r chemical exchange s t u d i e s and h igh r e s o l u -t i o n F o u r i e r t r a n s f o r m a p p l i c a t i o n s may be g i v e n a s : i T / 2 - p u l s e w i d t h = 10-20 y s e c , r i s e and f a l l t ime = 0.1 ysec and r e j e c t i o n r a t i o a = 10 6 (120 dB) . The o n l y gate systems i n the l i t e r a t u r e t h a t appear to meet the above s p e c i f i c a t i o n s are those o f Blume ^ and Lowe and T a r r •j I C 1 p p The Blume gate has been adapted by C l a r k and Bloom and c o -w o r k e r s . T h i s c i r c u i t uses an expensive 7077 p l a n a r t r i o d e as a g r o u n d e d - g r i d r f a m p l i f i e r so t h a t the very low c a t h o d e - t o - p l a t e c a p a c i t a n c e o f 0.01 r f s e v e r e l y r e s t r i c t s r f f e e d t h r o u g h , w i t h c a r e -f u l e l e c t r o n i c c o n s t r u c t i o n . Lowe and T a r r use a t r a n s i s t o r i s e d r f gate w i t h a r e l a t i v e l y low r e j e c t i o n r a t i o f o l l o w e d by a m u l t i - s t a g e gated h igh power r f a m p l i f i e r . The gate d e s c r i b e d here has been developed to a v o i d the use o f s p e c i a l components, an e l a b o r a t e con-s t r u c t i o n or a gated high power r f a m p l i f i e r . In a d d i t i o n , i t has been des igned to be a s imple and adaptable u n i t t h a t may be r e a d i l y l i n k e d w i t h s tandard NMR and r f i n s t r u m e n t a t i o n w h i l e meeting the s p e c i f i c a t i o n s o u t l i n e d above. The d e t a i l e d c i r c u i t diagram f o r the r f p u l s e gate o p e r a t i n g a t 10MHZ i s g i v e n i n F i g . 3 . 5 ( a ) , and the p e r t i n e n t v o l t a g e l e v e l s and waveforms are i n c l u d e d on t h i s d iagram. The dc supply v o l t a g e s were chosen to match those a v a i l a b l e from the s t a b i l i z e d power supply f o r 125 the TEKTRONIX 160 s e r i e s pulse generators used to set up general m u l t i p l e F i g . 3.6 (a) Rf-pulse gate 91. and/or r e p e t i t i v e pulse sequences. Of course, a l l power leads are h e a v i l y f i l t e r e d to prevent r f leakage between u n i t s o f the complete pulse spectrometer. The 6GY6, VI, i s operated as a low gain c l a s s A gated r f a m p l i f i e r w i t h a tuned input and an untuned output. This vacuum tube i s a sharp c u t - o f f pentode with dual c o n t r o l g r i d s , g i and g 3 , and low i n t e r - e l e c t r o d e capacitances. The tuned g r i d c i r c u i t r y i s c a r e f u l l y s h i e l d e d through the vacuum tube socket, but t h i s i s a c t u a l l y the only mechanical r f s h i e l d i n g used. VI i s b i a s e d i n t o c u t - o f f i n the gate quiescent s t a t e ( o f f - p e r i o d ) by a dc b i a s Vg3 = -12V, and the r f feedthrough f o r t h i s stage f o r the maximum inp u t voltage o f 0.8V p-p i s l e s s than 15 mV p-p. This feedthrough i s due mainly to the g r i d - t o -p l a t e capacitance which has a minimum value of 0.026 pF. When the gate system i s d r i v e n i n t o the a c t i v e s t a t e (on-period) by a p o s i t i v e c o n t r o l pulse a p p l i e d to the second c o n t r o l g r i d , g 3 , the voltage Vgs i s h e l d at near the cathode p o t e n t i a l by the l i m i t i n g diode Dl and the 6GY6 acts as an a m p l i f i e r w i t h Vgj = -0.4V. This a m p l i f i e r gives a maximum output of 12V p-p and hence the r e j e c t i o n r a t i o f o r t h i s f i r s t stage, of the gate system i s of the order o f 10 3 (60 dB). I t should be noted here that as the capacitance between the two c o n t r o l g r i d s i s l e s s than 1 pF, e x c e l l e n t r f i s o l a t i o n of the c o n t r o l pulse c i r c u i t r y i s very simply a t t a i n e d . The low p l a t e load r e s i s t a n c e , R l , and p l a t e -to-cathode capacitance of 6pF determine the r i s e and f a l l times f o r the r f pulse and a s s o c i a t e d p e d e s t a l i n the stage output c i r c u i t as l e s s than 0.2 usee. The 6CY5, V2, acts as a pulse i n v e r t e r 92. to g i v e an u n d i s t o r t e d g a t i n g waveform f o r the 6JC6, V 3 , output a m p l i f i e r . T h i s t e t r o d e has a low g r i d - t o - p l a t e c a p a c i t a n c e (0.03 pF) and r f feedthrough i s minimised i n t h i s s tage by us ing the p a r t i c u l a r c o n f i g u r a t i o n shown i n F i g . 3 . 5 ( a ) . As t h i s s tage i s untuned, the r f g a i n i s l e s s than u n i t y which f u r t h e r reduces feedthrough i n the q u i e s c e n t stage and a l s o determines the 6JC6 i n p u t v o l t a g e o f 4V p-p (maximum) i n the a c t i v e s t a t e . The 6JC6 i s operated as a h igh g a i n c l a s s C a m p l i f i e r and the c o n t r o l g r i d b ias i s a d j u s t e d to g i v e a maximum r e j e c t i o n r a t i o and minimum p u l s e r i s e and f a l l t i m e s , which are f a c i l i t a t e d by the low-Q p l a t e c i r c u i t . Aga in a low g r i d - t o - p l a t e 3 c a p a c i t a n c e o f 0.02 pF leads to a stage r e j e c t i o n r a t i o o f 4 x 10 (72 d B ) . The r f gate output i s o b t a i n e d through an i n d u c t i v e l i n k , the o u t p u t impedance being approx. 100 fi. The output p u l s e c h a r a c t e r i s t i c s are i l l u s t r a t e d i n F i g . 3.6 and may be summarised as f o l l o w s : output v o l t a g e <: 20V p - p , r f feedthrough < 15 yV p - p , r i s e - t i m e < 0.4 usee and f a l l t ime < 0.1 y s e c . The feedthrough v o l t a g e was measured u s i n g a h igh g a i n l i n e a r r f a m p l i f i e r w i t h an i n p u t impedance o f 75 n a t 10MHz and a d e t e c t i o n t h r e s h o l d o f 2yV p - p , and hence the o v e r a l l r e j e c t i o n r a t i o f o r the gate c i r c u i t may be c o n s i d e r e d to be a c c u r a t e l y determined as g r e a t e r than 120 dB. The o p e r a t i n g frequency o f the r f gate d e s c r i b e d above i s r e a d i l y changed to any frequency up to 20 MHz by v a r y i n g o n l y Cl and C 2 , and the output p u l s e c h a r a c t e r i s t i c s are shown to be w i t h i n the 10MHz l i m i t s g i v e n above. The r f gate c o n t r o l monostable m u l t i v i b r a t o r c i r c u i t i s g i v e n i n F i g . 3 . 5 ( b ) . T h i s p u l s e g e n e r a t o r was developed as an i n t e g r a l F i g . 3.6 (b) C o n t r o l dc-pulse generator Gate c o n t r o l waveform and output r f p u l s e . (0 .5 ysec/cm) Output r f p u l s e (0 .5 ysec / cm, 5V/cm) F i g . 3.6 R f - p u l s e gate o p e r a t i o n a l c h a r a c t e r i s t i c s . part o f the r f gate system to a t t a i n the r f pulse r i s e and f a l l times p r e v i o u s l y s p e c i f i e d , a TEKTRONIX 161 pulse generator being inadequate i n t h i s r e s p e c t , and to s i m p l i f y the r f i s o l a t i o n of t h i s c o n t r o l c i r c u i t r y . The cathode-coupled m u l t i v i b r a t o r using the 6DJ8 double 123 124 t r i o d e , VI and V2, was designed f o l l o w i n g w e l l known p r i n c i p l e s ' the c r i t i c a l g a t i n g pulse f a l l time being minimised through a c a r e f u l study o f the m u l t i v i b r a t o r c h a r a c t e r i s t i c s , the use o f the p l a t e -catching diode Dl and i s o l a t i o n o f the m u l t i v i b r a t o r c i r c u i t r y by a cathode f o l l o w e r , V3. This f o l l o w e r c o n f i g u r a t i o n reduces the e f f e c -t i v e capacitance at the output p l a t e (V2) o f the m u l t i v i b r a t o r , t h i s capacitance being the major f a c t o r d e f i n i n g the output pulse f a l l time. The p u l s e - w i d t h , t w , i s governed by the time constant R i C i and the g r i d b i a s f o r VI, and hence a v e r s a t i l e width c o n t r o l i s obtained through the v a r i a t i o n of both of these parameters by Rl and R2. For the com-ponent values shown, t w may be v a r i e d from 5 to 100 usees and may be r e a d i l y adjusted to w i t h i n 1% of a s p e c i f i e d value. The inherent pulse width j i t t e r i s measured as l e s s than 0.05% of the pulse-width. The output pulse amplitude may be v a r i e d , through R3, between 1 and 30V. The gate c o n t r o l pulse i s shown i n F i g . 3 . 6 , and the dc pulse r i s e and f a l l times are measured under normal r f . gate o p e r a t i n g c o n d i t i o n s as 0.1 and 0.05 usee, r e s p e c t i v e l y . These s w i t c h i n g times allow an r f pulse to be defined i n terms o f an i n t e g r a l number of r f c y c l e s , and als o allow the development of a completely coherent pulse system. In such a system, the pulse gate waveform i s locked to the r f waveform and hence a sequence o f r f pulses w i t h an i d e n t i c a l phase may be generated. The c o n t r o l pulse generator i s t r i g g e r e d by a 6V p o s i t i v e 9 4 . pulse at the c o n t r o l g r i d of VI. A l l a d d i t i o n a l timing t r i g g e r s are derived from the low impedance pulse outputs A and B i n F i g . 3.6(b). In the complete pulse spectrometer, the pulse from the r f gate i s used as the input to a conventional high power r f a m p l i f i e r u s i n g push-push frequency doublers (to give a f i n a l o p e rating frequency of 10, 20 or 40 MHz) and p u s h - p u l l c l a s s C a m p l i f i e r s . Rf pulse shape d i s t o r t i o n i s minimised by u s i n g only low-Q tuned c i r c u i t s . With a s t a b i l i z e d high v o l t a g e power supply f o r the f i n a l r f a m p l i f i e r , up to 1.6 kW (400 V p-p i n t o a 100 Q, r e s i s t i v e load) i s a v a i l a b l e i n a pulse sequence with a minimum pulse i n t e r v a l o f 60 usees. -A t y p i c a l TT/2-pulse width f o r 'H NMR (y^ = 2.7 x 10 4 rad. sec 1 gauss *) i s 10 usees, and t h i s corresponds to an Hi magnetic f i e l d o f 5.8 gauss over the sample volume. I t should be noted t h a t , f o r the Varian V-4300 crossed-c o i l probe used,, the t r a n s m i t t e r c o i l i s 25 mm i n diameter w h i l e the sample tube i s normally 5 mm w i t h a corresponding 8 mm r e c e i v e r c o i l . Although the c r o s s e d - c o i l system attenuates the a v a i l a b l e r f power i n a p u l s e , i t does give a high homogeneity Hi f i e l d over the sample volume which i s u s u a l l y more important i n high r e s o l u t i o n a p p l i c a t i o n s . In a d d i t i o n , when t h i s c o i l system i s used i n c o n j u n c t i o n w i t h a 125 m o d i f i e d LEL l i n e a r r f a m p l i f i e r , the time f o r recovery from over-load f o l l o w i n g an r f pulse i s t y p i c a l l y l e s s than 5 usees when the probe i s c o r r e c t l y balanced. An improved r f p h a s e - s e n s i t i v e d e t e c t i o n scheme uses a HP 10514B broad band mixer, w i t h inputs from the LEL 125 s i g n a l a m p l i f i e r and a s i m p l i f i e d reference a m p l i f i e r , f o l l o w e d by a low output impedance dc a m p l i f i e r which d r i v e s a l l r e c o r d i n g devices \ 95. i n c l u d i n g the FABRITEK FT-1064 computer. The o v e r a l l l i n e a r i t y o f the r e c e i v e r system has been checked over the input range o f 10 uV to 10 mV p-p at a gain o f 10 k to be b e t t e r than 2%. 9 6 . CHAPTER 4 EXPERIMENTATION AND CALCULATIONS 4.1 Hindered r o t a t i o n i n amides. 4.1.1 N,N-dimethyl carbamyl c h l o r i d e . This compound was chosen f o r an i n i t i a l study of the hindered r o t a t i o n about the N-C bond i n amides, due t o the e l e c t r o n d e l o c a l i s a -t i o n i n the NCO bond system, us i n g the steady-state NMR methods des-c r i b e d i n d e t a i l i n Chapter 2 of t h i s t h e s i s . Although a number of s u b s t i t u t e d amides have been s t u d i e d , the parameters d e r i v e d from the NMR data show a large variance and a s e m i - q u a n t i t a t i v e c o r r e l a t i o n o f the data a v a i l a b l e i s not p o s s i b l e . The only example o f an N-C r o t a -t i o n a l b a r r i e r that has been s t u d i e d using complete NMR lineshape analyses i n independent l a b o r a t o r i e s , and f o r which the k i n e t i c para-meters are apparently c o n s i s t e n t w i t h i n the c a l c u l a t e d experimental e r r o r s , i s that f o r N,N-dimethyl formamide ^ , 6 1 ^ However, t h i s r e s u l t i s f o r t u i t o u s i n view of the f a c t t h a t one of the models used ^ f o r t h i s f o u r - s i t e exchange process i s i n c o r r e c t . Thus the k i n e t i c data obtained here f o r N,N-dimethyl carbamyl c h l o r i d e (DMCC) i s intended f o r comparison with t h a t a v a i l a b l e from a recent independent study usi n g a complete lineshape a n a l y s i s ^ and s p i n echo s t u d i e s to s u b s t a n t i a t e the general v a l i d i t y and r e l i a b i l i t y o f the a p p l i c a t i o n of NMR methods t o the study o f hindered r o t a t i o n . DMCC was obtained from K § K L a b o r a t o r i e s Co. L t d . , and was p u r i f i e d by a double vacuum d i s t i l l a t i o n and s t o r e d over molecular s i e v e s . Experimental and l i t e r a t u r e p h y s i c a l constants agreed f o r t h i s compound. As DMCC i s a l i q u i d (b.p. = 165°C at 760 mm) t h i s amide was stu d i e d neat and i n the non-polar s o l v e n t carbon t e t r a c h l o r i d e (6 mole % ) . In p reparing the NMR samples, 2 mole % o f te t r a m e t h y l s i l a n e (TMS) was added to provide a s t a b l e f i e l d - f r e q u e n c y lock s i g n a l and 2 mole % of dioxane was a l s o added as a reference peak as i t i s co n v e n i e n t l y placed with r e s p e c t to the DMCC spectrum and has a temperature indepen-dent l i n e w i d t h . The samples .were thoroughly degassed by the usual freeze-pump-thaw c y c l e and were sealed i n vacuo i n t h i n - w a l l e d NMR tubes o f 5 mm o.d.. The NMR s p e c t r a were obtained on a JEOLCO JNM-C-60H spectr o -meter used i n the i n t e r n a l lock mode and equipped w i t h the JES-VT-2 v a r i a b l e temperature c o n t r o l l e r . The temperature i s normally monitored by a thermocouple p l a c e d near the sample i n the temperature c o n t r o l gas stream. As a temperature measurement i s c r i t i c a l i n q u a n t i t a t i v e k i n e t i c s t u d i e s , e s p e c i a l l y when i t i s only p o s s i b l e to apply complete lineshape analyses over a r e l a t i v e l y s m a ll temperature range, the tem-perature was c a l i b r a t e d w i t h a second thermocouple immersed i n the same volume o f sample contained i n an open non-spinning NMR tube and a l s o by r e p l a c i n g the sample with the ethylene g l y c o l sample s u p p l i e d by Va r i a n . In a d d i t i o n , the temperature was checked before and a f t e r a number o f sp e c t r a were recorded. Two conclusions were immediately evident. While the use of the non-spinning sample does not give a r e l i a b l e measurement of the sp i n n i n g sample temperature, the instrument thermo-couple temperature reading can be a c c u r a t e l y c o r r e l a t e d w i t h that given from the ethylene g l y c o l chemical s h i f t - t e m p e r a t u r e equation reported by Van Geet . As the absolute e r r o r i n the sample temperature measure-ment was estimated to be ± 0.3°C, i t was a l s o shown th a t the sample temperature was constant w i t h i n these l i m i t s over the r e c o r d i n g time f o r a number o f s p e c t r a at a given c o n t r o l l e r temperature s e t t i n g . Thus, i n g e n e r a l , i t i s p o s s i b l e to use the instrument thermocouple measurements c a l i b r a t e d at four temperatures with the g l y c o l sample to reduce the time i n v o l v e d w i t h c o n t i n u a l sample replacement. Sweep rates as low as 0.05 Hz sec 1 were used to ensure a good approximation t o slow passage c o n d i t i o n s and s p e c t r a were checked f o r p o s s i b l e d i s -t o r t i o n due to s a t u r a t i o n , c f . s e c t i o n 2.3. At each temperature, f i e l d curvature and homogeneity and r f phase adjustments were made to ensure rep r o d u c t i o n of the true s p e c t r a l lineshape. The r e s u l t a n t l i n e w i d t h at half-maximum o f the reference peak was h e l d w i t h i n the range 0.3 -0.5 Hz. M u l t i p l e s p e c t r a were obtained at each temperature and any s p e c t r a that were not p r e c i s e l y r e p r o d u c i b l e were discard e d . The hindered r o t a t i o n i n DMCC may be considered as an equal p o p u l a t i o n t w o - s i t e exchange process and hence the s t e a d y - s t a t e NMR absorption mode lineshape f u n c t i o n , V ( x ) , i s given by Eq. (2.2.15) i n accordance w i t h the s t o c h a s t i c model described i n d e t a i l i n s e c t i o n 2.2. The experimental s p e c t r a l data i s converted to a s e r i e s o f N d i g i t a l amplitude v a l u e s , V (XJ[) , and associated, f r e q u e n c i e s , x^ . These f r e -quencies are d e r i v e d from two c a l i b r a t i o n frequencies measured d i r e c t l y as the d i f f e r e n c e between the sweep (observing) frequency and the f i x e d f i e l d lock frequency, to an accuracy of ± 0.1 Hz. A FORTRAN-IV computer program PLONK has been developed to allow an e f f i c i e n t i t e r a t i v e l i n e -shape a n a l y s i s o f the d i g i t i z e d data. The i t e r a t i v e f i t t i n g i s based upon 99. a simple but effective search routine which rapidly converges to a best-fit parameter value through a series of optimised parameter increments, the best-fit value corresponding to a minimum in the sum of 127 squares of deviations , S, where N S = Z [ V C x i ) - V C x J ] 2 i=l and V(xi) is the theoretical lineshape amplitude for x=x^. The line-shape function is normalised to the experimental data, V ( x ^ ) , through the constant A in Eq. (2.2.15). The parameters to be considered in a general lineshape analysis are the chemical shift between exchange sites 2ft, cf. Eq. (2.2.1), the transverse relaxation time T 2 describing the residual linewidth in the absence of exchange and the f i r s t order rate constant k defining the exchange process. T?_ is defined in terms of the linewidth of the reference peak, where i t is implicitly assumed that the magnetic f i e l d homogeneity is described by a Lorentzian frequency dis-tribution, and is considered to be a fixed parameter. A standard non-128 linear least squares regression analysis using ft and k as variable parameters was shown to be unreliable, especially for data obtained above the coalescence temperature, and this is probably due to the unusual forms of the partial derivatives required, [ — j and ( ) X i X i Therefore, the program PLONK takes k as the variable parameter for a specific ft value, with the option of being able to vary ft in a well-defined manner to determine the coupled parameter values corresponding to a minimum S value. This appears to be the most satisfactory form of analysis to determine a temperature dependence of the chemical s h i f t , 100. 29. For N = 20-60, the CPU time i n v o l v e d f o r each k i t e r a t i v e f i t i s of the order o f 4 sees on an IBM 360/67 system, and the s e t t i n g up o f a l l a rrays f o r c o n t r o l of a CALCOMP p l o t t e r takes l e s s than 1 sec. T y p i c a l lineshape f i t s f o r DMCC (neat) u s i n g the computer program d e s c r i b e d above and a constant chemical s h i f t 29, = 6.80 Hz at 60 MHz are shown i n F i g . 4.1, and the r a t e constants obtained by averaging the k values from the f i t t i n g o f m u l t i p l e s p e c t r a at a l l temperatures are summarised i n Table 4.1. I t should be noted that the mean d e v i a t i o n , c a l c u l a t e d as 1 N i £ [V 'CxiO - V ( x i ) ] / V ( x i ) , i = l f o r the s p e c t r a shown i n F i g . 4.1 i s 1.5%. The Arrhenius p l o t f o r DMCC (neat) i s shown i n F i g . 4.2. The a c t i v a t i o n energy, E a , f o r the hindered r o t a t i o n i s determined as 17.64 ± 0.52 k c a l s . mole 1 from a l e a s t squares f i t to the usual 129 equation k = A exp(-E a/RT), (4.1.1) where T i s the absolute temperature, A i s the frequency f a c t o r f o r the rat e process and R i s the u n i v e r s a l gas constant (1.986 c a l . deg 1 mole Both E a and A are assumed to be temperature independent, and the a c t i v a -t i o n energy i s def i n e d as the d i f f e r e n c e between the average energies of the ground s t a t e and the a c t i v a t e d t r a n s i t i o n s t a t e i n v o l v e d i n the 130 hindered r o t a t i o n . A thermodynamical f o r m u l a t i o n of r e a c t i o n rates shows t h a t TABLE 4.1 K i n e t i c data f o r N,N-dimethyl carbamyl c h l o r i d e (neat) Temp.°C k sec 28.7 4.32 29.2 4.52 33.0 6.35 33.2 6.35 35.8 8.16 70 A i i n A 39.4 11.46 40.5 10.55 41.5 13.59 46.8 23.00 48.4 25.99 50.8 32.02 E a = 17.64 ± 0.52 k c a l s . mole" 1 AH # = 17.05 k c a l s . mole" 1 at 25°C AS # = 0.8 ± 1.6 c a l deg" 1 mole" 1 AG # = 16.82 k c a l s . mole" 1 F i g . 4.1 Lineshape f i t s f o r N,N~dimethyl carbamyl c h l o r i d e , two-s i t e equal p o p u l a t i o n exchange system. \ 101. k = -j- exp (-AG#/RT) 4 k# ^4-1-2^ where fe i s the Boltzmann constant, n i s Planck's constant and A G i s the free energy of a c t i v a t i o n . K may be i n t e r p r e t e d as an e q u i l i b r i u m constant f o r the interchange between ground and a c t i v a t e d s t a t e s , and may t h e r e f o r e be de f i n e d i n terms of the increase i n energy i n forming the t r a n s i t i o n s t a t e , AE*, by the general r e l a t i o n s h i p d , „# AE , A T ^ Ink = . (4.1.3) Thus i t fo l l o w s that E a = AE* + RT and the enthalpy of a c t i v a t i o n , AH , may now be de f i n e d as AH* = AE* - PAV # , where AV i s the increase i n volume on going i n t o the t r a n s i t i o n s t a t e . For the unimolecular hindered r o t a t i o n process, AV = 0 and i n t h i s p a r t i c u l a r case AH* may be c a l c u l a t e d d i r e c t l y from the experimental a c t i v a t i o n energy at a given temperature as AH* = E & - RT . (4.1.4) S u b s t i t u t i o n f o r E a i n Eq. (4.1.1) now allows a determination o f the entropy o f a c t i v a t i o n , AS , i n accordance w i t h Eq. (4.1.2) as 102. AS # = R [ l n ^ - 1 ] , (4.1.5) w i t h AG # = AH # - TAS # (4.1.6) The a c t i v a t i o n parameters c a l c u l a t e d f o r DMCC (neat) at 25°C are in c l u d e d i n Table 4.1 and are i n very good agreement w i t h those obtained by Neuman et a l . ^  ( E g = 16.9 ±0.5 k c a l s . mole l, AG^ = 16.8 k c a l s mole 1 , AS = -1.6 c a l s . deg 1 mole l ) . I t has p r e v i o u s l y , . 59, 60, 64 . . . • * i • u + • . been noted ' that as i n c r e a s i n g care i s taken m o b t a i n i n g k i n e t i c data u s i n g NMR methods, the values of E and A tend t o i n c r e a s e b ' a whi l e the f r e e energy of a c t i v a t i o n remains n e a r l y constant. The data presented here e x e m p l i f y ; the tre n d s . For hindered r o t a t i o n s i n N.N-dimethvl amides the ent.ronv of a c t i v a t i o n s i s exnected to be 47 59 r e l a t i v e l y small ' . As t h i s entropy change i n l i q u i d s i s most probably a s s o c i a t e d w i t h a d i f f e r e n c e i n the s o l v a t i o n s t r u c t u r e s f o r the ground and t r a n s i t i o n s t a t e s i t i s p r e d i c t e d t h a t there w i l l be a small i n c r e a s e i n entropy on formation of the t r a n s i t i o n s t a t e , i n t h a t the d i p o l e moment f o r t h i s s t a t e i s normally s m a l l e r than t h a t f o r the p l a n a r ground s t a t e . W i t h i n the e r r o r l i m i t s given above, and c a l c u l a t e d as the s t a t i s t i c a l 95% confidence l i m i t s , t h i s i s shown to be the case f o r DMCC st u d i e d as a neat l i q u i d . In comparison, the entropy of a c t i v a t i o n d e r i v e d from spin-echo data i s ~ -10.5 c a l s . deg. 1 mole 1 and the corresponding a c t i v a t i o n energy f o r DMCC (neat) i s 14.0 ± 0.9 k c a l s . mole 1 I t i s to be noted, however, t h a t under the assumption that AS =0. the a c t i v a t i o n energy may be c a l c u l a t e d from the spin-echo data at the coalescence t e m p e r a t u r e ^ 103. to be 17.5 k c a l s mole , i n e x c e l l e n t agreement with the s t e a d y - s t a t e value presented above. In a d d i t i o n , the f r e e energy o f a c t i v a t i o n i s determined as AG* = 16.6 k c a l s m o l e - 1 , c f . 16.8 k c a l s mole" 1 Table 4.1. So l u t e - s o l v e n t i n t e r a c t i o n s f o r the s t r o n g l y p o l a r DMCC molecule (p - 4.08D may be expected to i n f l u e n c e the magnitude o f the p o t e n t i a l b a r r i e r f o r hindered r o t a t i o n i n t h i s amide. Thus DMCC was a l s o s t u d i e d as a d i l u t e (6 mole %) s o l u t i o n i n the non-polar s o l -vent CCl4 to minimise such i n t e r a c t i o n s . The k i n e t i c data obtained from t o t a l l ineshape analyses, u s i n g a constant chemical s h i f t 29. = 7.1 Hz at 60 MHz are summarised i n Table 4.2. The corresponding Arrhenius p l o t i s shown i n F i g . 4.2 and the a c t i v a t i o n energy i s determined as 17.05 ± 0.47 k c a l s mole 1. Thus t h i s parameter shows only a small s o l v e n t dependence. The entropy o f a c t i v a t i o n i s again very small but f o r m a l l y n e g a t i v e , AS = -0.6 ±1.5 c a l s deg 1 mole Although the hindered r o t a t i o n i n t h i s p a r t i c u l a r amide i s shown to be n e a r l y independent o f s o l u t e c o n c e n t r a t i o n , the form o f the magnetic ' 132 133 an i s o t r o p y of the carbonyl group ' , which i s assumed to be the dominant mechanism g i v i n g r i s e to the chemical s h i f t between the N-methyl groups c i s and trans to t h i s f u n c t i o n a l group, i s changed s i g n i f i c a n t l y from t h a t i n the neat l i q u i d system. This change i s presumably due to the break-down o f a s p e c i f i c s o l u t e - s o l v e n t i n t e r -a c t i o n . In order to con s i d e r the i n t e r - r e l a t i o n s h i p s between d e r i v e d a c t i v a t i o n parameters, a l l data a v a i l a b l e f o r the hindered r o t a t i o n i n DMCC have been c o r r e l a t e d i n Table 4.3. The AG*, AH* and AS* values have been c a l c u l a t e d f o r a f i x e d temperature o f 25°C. The o v e r a l l TABLE 4.2 K i n e t i c data f o r N,N-dimethyl carbamyl c h l o r i d e (6% C C l ^ s o l u t i o n ) 0.5 . 0.44 5.9 0.52 15.2 1.25 30.5 6.30 34.5 9.45 39.3 13.54 44.0 18.83 44.6 18.85 48.8 31.58 49.5 33.62 54.5 51.05 54.8 55.55 58.0 93.42 63.2 137.10 E a = 17.05 ± 0.47 k c a l s . mole" 1 AH # = 16.46 k c a l s . mole" 1 at 25°C AS* = -0.6 ± 1.5 c a l . deg" 1 mole" 1 AG* = 16.64 k c a l s . mole" 1 TABLE 4.3 N,N-dimethyl carbamyl c h l o r i d e a c t i v a t i o n parameters r e f . E AG # AH* AS''! a 17, .64 + 0, ,52 16, .82 17, ,05 + 0, ,52 0, ,8 + 1, ,6 a 17, .05 + 0. ,47 16. ,64 16. ,46 + 0. ,47 -0. ,6 1, ,5 b 16, .9 + 0. ,5 16, ,8 16, .3 + 0, ,5 -1. ,6 + 2, .0 b 17, .7 + 0, .9 16, .3 17, .1 + 0, ,9 2, ,6 + 3, ,2 c 14, .0 + 0. ,9 16, ,6 13, ,4 + 0, .9 -10, ,6 + 2. ,7 c 9, .7 + 0. ,5 16, .6 9, ,1 + 0, ,5 -25, ,3 + 2. .8 c 8, .6 + 1. ,7 16. ,6 8, ,0 + 1. ,7 -28. .9 + 5, ,5 d 7. ,3 + 0. ,5 16, .4 6, ,7 + 0. .5 -27, , 1 + 2, ,4 e 6, ,8 + 0. ,2 16, .2 6, .2 + 0, ,2 -33, ,6 + 1, ,0 f 8. .65 + 0. ,88 16, ,3 8, ,06 + 0, ,88 -27, ,5 + 3, ,0 a This work b R.C. Neuman, D.N. Roark and V. Jonas JACS 89^ , 3412, 1967 c A. A l l e r h a r d and H.S. Gutowsky J.Chem.Phys. 41_, 2115, 1964 d M.T. Rogers and J.C. Woodbrey J.Phys.Chem. 66, 540, 1962 e J.C. Woodbrey and M.T. Rogers JACS 84_, 13, 1962 f "E. Krakower, Ph.D. Th e s i s , UBC, 1967. \ s \ F i g . 4.2 Arrhenius p l o t s f o r N,N-dimethyl carbamyl c h l o r i d e . o 6% C C l ^ s o l u t i o n © neat l i q u i d 20 «> 16 o E. rd u * 12 < 8 4 o a,b a . b 30 -20 10 0 c,f d. 5 AS" cals.deg. 1mole~ 1 F i g . 4.3 V a r i a t i o n o f a c t i v a t i o n parameters f o r hindered r o t a t i o n i n N,N-dimet.hyl carbamyl c h l o r i d e . 104. v a r i a t i o n i n AH* and AS*, as obtained u s i n g d i f f e r e n t methods of a n a l y s i s and a l s o d i f f e r e n t s o l v e n t s and s o l u t e c o n c e n t r a t i o n s , i s shown g r a p h i c a l l y i n F i g . 4.4. The l a r g e v a r i a t i o n i n these parameters f o r i d e n t i c a l chemical systems i s due to r e l a t i v e l y s m a ll e r r o r s i n the experimental determination of s p e c i f i c r a t e constants and/or tempera-t u r e s . Although these e r r o r s may give r i s e to only a small change i n the slope of the corresponding Arrhenius p l o t over the temperature range f o r a p a r t i c u l a r study, t h i s temperature range i s u s u a l l y very small (20-80°C) and hence r e l a t i v e l y l a r g e v a r i a t i o n s i n the E and A cL values determined from Eq. (4.1.1) soon become evident. This of course leads to corresponding v a r i a t i o n s i n AH and AS i n accordance w i t h Eqs. (4.1.4) and (4.1.5), r e s p e c t i v e l y . In g e n e r a l , as shown i n F i g . 4. 47, 64 4 4, higher E (All ) values are a s s o c i a t e d w i t h higher A values cl , and v i c e v e r s a . Except f o r a s i n g l e p o i n t , however, a l i n e a r V # • # c o r r e l a t i o n between AH and AS i s obtained and t h i s i m p l i e s that the f r e e energy of a c t i v a t i o n , AG , i s approximately i n v a r i a n t to small systematic e r r o r s inherent i n the d e t e r m i n a t i o n of E and A, c f . Eq. (4.1*6). Thus i t becomes evident t h a t the only s i g n i f i c a n t parameter 64 obtained from NMR data may indeed be the f r e e energy of a c t i v a t i o n The a c t i v a t i o n parameters f o r DMCC determined by complete lineshape analyses are shown i n d e t a i l i n F i g . 4.5, t h i s method of a n a l y s i s g i v i n g the most r e l i a b l e and h i g h e s t p r e c i s i o n estimates of these parameters. The data obtained i n independent s t u d i e s are con-s i s t e n t w e l l w i t h i n the c a l c u l a t e d e r r o r l i m i t s , but i t i s s t i l l not p o s s i b l e to r e l i a b l y compare the e n t h a l p i e s of a c t i v a t i o n determined f o r the neat amide and the amide i n a CCH s o l u t i o n . Again the f r e e 105. energy of a c t i v a t i o n becomes the most s i g n i f i c a n t parameter, and i s determined f o r AS = 0, c f . Eq. (4.1.6) and F i g . 4.4, as AG = 16.65 k c a l s mole 1. In t h i s p a r t i c u l a r case i t i s i n t e r e s t i n g to note that the slope of the l i n e a r p l o t shown i n F i g . 4.5 defines a c h a r a c t e r i s t i c temperature o f 258°K, which i s to be.compared w i t h that of 298°K at which AH and AS have been c a l c u l a t e d . This temperature d i f f e r e n c e may be considered to i n d i c a t e a small dependence o f AG* on so l v e n t and thus may be used to s e m i - q u a n t i t a t i v e l y describe s o l u t e - s o l v e n t i n t e r -a c t i o n s f o r a s e r i e s of s u b s t i t u t e d amides. For the hindered r o t a t i o n i n an amide, a f r e e energy change i s expected to be defined p r i m a r i l y by i n t r a m o l e c u l a r e f f e c t s and hence i s much simp l e r to i n t e r p r e t than the energy of a c t i v a t i o n which i s s e n s i t i v e to i n t e r m o l e c u l a r e f f e c t s . A l s o , t h e r e i s a general tendency i n r a t e processes i n s o l u t i o n f o r 134' 135 enthalpy and entropy changes to compensate each other ' so that, the e f f e c t i v e change i n f r e e energy i s reduced and becomes l e s s s e n s i -t i v e to e x t e r n a l e f f e c t s . This compensation e f f e c t i s simply i n t e r -p r e t ed i n terms o f a s o l u t e - s o l v e n t i n t e r a c t i o n . Any i n t e r a c t i o n that leads to a stronger b i n d i n g of s o l v e n t molecules to a s o l u t e molecule lowers the enthalpy of the system, and a l s o , by r e s t r i c t i n g the f r e e -dom of motion o f both s o l u t e and sol v e n t molecules t h i s i n t e r a c t i o n lowers the entropy. This c h a r a c t e r i s t i c i s shown by the parameters presented i n F i g . 4.4. In t h i s p a r t i c u l a r case, i t may be considered t h a t a decrease i n entropy f o r the pl a n a r ground s t a t e ( r e l a t i v e to tha t f o r the hindered r o t a t i o n t r a n s i t i o n s t a t e ) o f DMCC as a neat l i q u i d , due t o a d i p o l a r s o l u t e - s o l u t e (solvent) i n t e r a c t i o n , leads t o a I 1 1 : 1 1 1 1 1 1 1 - 4 -2 0 2 A A S cals.deg" mole F i g . 4.4 A c t i v a t i o n parameters obtained by complete lineshape a n a l y s i s f o r N,N-dimethyl carbamylchloride e 6% CCl t, s o l u t i o n o neat l i q u i d 106. s m a l l p o s i t i v e AS value as compared w i t h a commensurate negative As" value f o r a d i l u t e CCli, s o l u t i o n i n which such an i n t e r a c t i o n i s presumably minimised. In g e n e r a l , although d i f f e r e n t s o l v e n t s and s u b s t i t u e n t s i n f l u e n c e AH* and AS* i n a complex manner, the p a r t i a l compensation e f f e c t i s of such a form t h a t t h e i r i n f l u e n c e If on AG i s much s i m p l e r i n form. U n t i l , the present s t e a d y - s t a t e NMR lineshape data * supplemented by spin-echo and double resonance are data o f comparable p r e c i s i o n over an extended temperature range, i t appears t h a t the f r e e energy of a c t i v a t i o n i s the only parameter t h a t w i l l a l l o w q u a n t i t a t i v e c o r r e l a t i o n s f o r a s e r i e s of s u b s t i t u t e d N,N-dimethyl amides. 4.1.2 N,N-dimethyl carbamyl bromide N,N-dimethyl carbamyl bromide (DMCB) was chosen as a h a l o - s u b s t i t u t e d N,N-dimethyl amide i n a s e r i e s t o al l o w a c o n s i s t e n t study o f s u b s t i t u e n t e f f e c t s on the hindered r o t a t i o n about the amide N-C bond. DMCB was prepared by s a t u r a t i n g ~10 gms of N,N-dimethyl 136 carbamyl c h l o r i d e (DMCC) by bu b b l i n g a mixture of.HBr and N 2 gases through the neat l i q u i d c h l o r i d e kept at 0°C. The N 2 stream removed any HC1 or C l 2 formed and s i g n i f i c a n t l y i n c r e a s e d the y i e l d o f the carbamyl bromide. The bromide was p u r i f i e d by m u l t i p l e d i s t i l l a t i o n s at ~10 mm, the b o i l i n g p o i n t of the f i n a l product being 63°C at t h i s p ressure. The product was i d e n t i f i e d by NMR and was shown to c o n t a i n ~2% o f the carbamyl c h l o r i d e , t h i s being v e r i f i e d by the elemental a n a l y s i s : 107. C C a l c u l a t e d 23.68 Found 24.54 This l i q u i d amide was s t u d i e d neat as s o l u t e - s o l v e n t i n t e r a c t i o n s were shown to have only a small e f f e c t on the hindered r o t a t i o n i n the s i m i l a r DMCC system, and on the a c t i v a t i o n parameter of i n t e r e s t at t h i s p o i n t ( A G ), as discussed i n the preceding s e c t i o n . The NMR sample was prepared as p r e v i o u s l y o u t l i n e d . Complete l i n e -shape analyses f o r t h i s t w o - s i t e equal p o p u l a t i o n exchange system, usi n g the i t e r a t i v e f i t t i n g program PLONK, gave the r e s u l t s summarised i n Table 4.4. T y p i c a l lineshape f i t s are shown i n Fig.. 4.5. These f i t s were not improved ( w i t h i n the r a t e constant e r r o r bounds) by co n s i d e r i n g a temperature dependent chemical s h i f t 2ft, c f . Eq. (2.2.1), and thus the c i s - t r a n s N-methyl chemical s h i f t may be assumed to be 5.4 + 0.2 Hz at 60 MHz over the temperature range -15°C t o 70°C. As t h i s chemical s h i f t i s reduced by 20% and 32% from those f o r DMCC and the parent amide DMF (N,N-dimethyl formamide, 2ft = 8.1 Hz), r e s p e c t i v e l y , c o n s i d e r a b l e magnetic anisotropy a s s o c i a t e d with the C-Br bonding system i s i n d i c a t e d . This anisotropy i s of a form that p a r t i a l l y compensates that due to the C = 0 system, and t h i s compensation e f f e c t i s a l s o shown by the C - C l bonding system i n t h a t t h i s chemical s h i f t f o r DMCC i s reduced by 16% from that f o r DMF. However, i t i s to be noted t h a t these chemical s h i f t v a r i a t i o n s are due i n p a r t to i n t e r -133 137 molecular i n t e r a c t i o n s and e l e c t r i c f i e l d e f f e c t s ' . The a c t i v a t i o n parameters c a l c u l a t e d f o r DMCB at 25°C are given i n Table 4.4, H Br Br + C l 3.95 52.63 4.30 - 54.11 TABLE 4.4 K i n e t i c data f o r N,N-dimethyl carbamyl bromide (neat) Temp. °C k sec" -10.2 0.62 - 2.0 2.06 - 1.0 2.14 7.2 3.60 7.3 3.52 11.5 5.52 15.7 7.35 16.2 7.28 16.4 8.67 19.0 11.2 19.4 14.0 19.7 15.0 24.0 20.9 24.0 21.5 32.5 33.7 36.6 60.2 48.8 125.5 55.0 161.9 58.7 300.3 61.6 464.1 continued/... TABLE 4.4 continued E a = 15.25 ± 0.36 k c a l s . mole 1 AH* = 14.66 k c a l s . mole" 1 at 25°C AS* = -3.3 ± 1.2 c a l s . deg" 1 mole AG* = 15.66 k c a l s . mole" 1 T = 24.0°C 36.6 k = 20.9 s e c " 1 60.2 • F i g . 4.5 Lineshape f i t s for"N,N-dime.thyl carbamyl bromide, t w o - s i t e equal p o p u l a t i o n exchange system. 2.8 -2.0 o C D g 1.2 0.4 0.4 -H 3 C ^ 2 . H 3 C 2.8 3.2 3.6 1/T°K x 1 0 3 4.0 F i g . 4.6 Arrhenius p l o t f o r N,N-dimethyl carbamyl bromide 108. the energy of a c t i v a t i o n f o r the hindered r o t a t i o n being determined from the Arrhenius p l o t shown i n F i g . 4.6.as 15.25 ± 0.36 k c a l s . mole 1. Again the entropy of a c t i v a t i o n AS* = -5.3 + 1.2 c a l s . deg. 1 mole 1 i s determined to be r e l a t i v e l y s m a l l , as expected f o r the type of r a t e process under c o n s i d e r a t i o n . 4.1.3 Methyl N,N-dimethyl carbamate The methyl e s t e r of N,N-dimethyl carbamic a c i d , methyl N,N-dimethyl carbamate (DMCO), was s t u d i e d to allow a comparison of the 0-methyl group w i t h halogens and pseudo-halogens as s u b s t i t u e n t s i n an amide s e r i e s . I f hindered r o t a t i o n about the N-C bond i s present i n t h i s carbamate, the r a t e process may be analysed i n terms of a simple two-s i t e equal p o p u l a t i o n exchange system as t h i s compound i s not expected to have a p r e f e r r e d conformation and the s p i n - s p i n c o u p l i n g between the N-methyl and 0-methyl protons w i l l be n e g l i g i b l e . However, Drago 138 et a l have reported the N-methyl groups i n t h i s carbamate to be eq u i v a l e n t i n the neat l i q u i d and i n a number of s o l u t i o n s at a l l temperatures down to -46°C, showing that the b a r r i e r to r o t a t i o n about the N-C bond i s i n h e r e n t l y low, that i s l e s s than about 8 k c a l s . mole 1 139 A l a t e r study by L u s t i g et a l showed that t h i s b a r r i e r to r o t a t i o n i s i n c reased i n chloroform s o l u t i o n s and t h a t a c i s - t r a n s N-methyl ine q u i v a l e n c e i s observed at ~-25°C, the a s s o c i a t e d chemical s h i f t being dependent upon s o l u t e c o n c e n t r a t i o n . The increased n i t r o g e n lone p a i r d e l o c a l i s a t i o n i n a CHCI3 s o l u t i o n , l e a d i n g to an N-C r o t a t i o n 109 . observable by NMR, i s presumably due to a hydrogen-bonding i n t e r a c t i o n between the carbonyl oxygen lone p a i r e l e c t r o n s and the s o l v e n t molecules which would tend to p r e f e r e n t i a l l y s t a b i l i z e the amide + resonance form represented as N = C = 0. Thus DMCO i s s t u d i e d here as a 10 mole % C H C I 3 s o l u t i o n to o b t a i n an estimate o f the e f f e c t of such a s o l v e n t i n t e r a c t i o n on the N - C = 0 bond system and to allow a c o r r e l a t i o n w i t h a simple molecular o r b i t a l model developed i n a l a t e r s e c t i o n . A 10% s o l u t i o n corresponds to a near maximal N-methyl chemical s h i f t (2ft) o f only 1.8 ± 0.1 Hz at 60 MHz and at -30°C. DMCO was prepared through the r e a c t i o n o f dimethylamine w i t h methyl chloro-formate as de s c r i b e d by Hartman and Brethen^'''. The l i q u i d product was p u r i f i e d by d i s t i l l a t i o n s at ~40 mm and s t o r e d over molecular s i e v e s , the f i n a l product having a b o i l i n g p o i n t of 58°C at t h i s p r e s s u r e , c f . l i t . value 56.5 - 57°C 1 4 1. The NMR sample was prepared as p r e v i o u s l y d e s c r i b e d f o r DMCC, with Spectrograde deuterochloroform which had been thoroughly d r i e d over molecular s i e v e s . Owing t o the very small chemical s h i f t 2ft, c f . Eq. (2.2.1), the temperature range f o r the k i n e t i c study i s l i m i t e d to 25°C. There-f o r e , to o b t a i n r e l i a b l e r a t e constants from complete lineshape analyses, a l a r g e number o f s p e c t r a were f i t t e d at. each temperature u s i n g the program PLONK, as desc r i b e d i n s e c t i o n 4.1.1. The s h i f t 2ft was shown t o be temperature independent (± 0.1 Hz) w i t h i n the lineshape f i t t i n g e r r o r l i m i t s a t t a i n a b l e . The averaged r a t e c o n s t a n t s , k s e c " 1 , are given i n Table 4.5 and the corresponding Arrhenius p l o t i s shown i n F i g . 4.8. The energy o f a c t i v a t i o n , E ; , i s determined as 15.18 ± 0.58 k c a l s mole" 1 TABLE 4.5 K i n e t i c data f o r N,N-dimethyl carbamate (10% C H C I 3 s o l u t i o n ) Temp. °C k sec -23.9 0.25 -22.5 0.41 -16.0 0.76 -14.6 1.05 -14.4 1.15 - 9.3 1.36 - 9.3 1.44 - 8.9 1.57 - 8.3 1.68 - 5.4 2.23 - 5.2 2.46 - 2.7 3.18 - 2.2 3.42 - 1.1 4. 18 - 1.0 4.43 1.5 5.36 E = 15.18 ± 0.58 k c a l s . mole 1 cl AH # = 14.54 k c a l s . mole" 1 at 25°C AS* = -2.0 ± 2.2 c a l s . deg" 1 mole AG* = 15.19 k c a l s . mole" 1 F i g . 4.7 Arrhenius p l o t f o r methyl N 3N-dimethyl carbamate (10% CHC13 s o l u t i o n ) 110. and thus i t i s shown that the carbonyl-CHCI3 i n t e r a c t i o n i n c r e a s e s the b a r r i e r to hindered r o t a t i o n f o r DMCO by about 8 k c a l s . mole This i m p l i e s t h a t , i n ge n e r a l , the magnitude of E g(or AG ) f o r the N - C r o t a t i o n may be a s e n s i t i v e measure o f i n t e r r a o l e c u l a r i n t e r a c t i o n s w i t h an amide C = 0 group, when these parameters are d e r i v e d through complete NMR lineshape analyses. The small entropy o f a c t i v a t i o n determined from the above k i n e t i c data f o r DMCO, AS =-2.0 ± 2.2 c a l s deg 1 mole i m p l i e s t h a t the carbonyl - CHCI3 i n t e r a c t i o n and hence s o l u t e - s o l v e n t s t r u c t u r e i s very n e a r l y i n v a r i a n t to the change i n o v e r a l l d i p o l a r c h a r a c t e r o f the amide system i n forming the t r a n s i t i o n s t a t e f o r the r a t e process under c o n s i d e r a t i o n . The magnitude o f the chemical s h i f t 2ft i s d i f f i c u l t to i n t e r p r e t as i t w i l l depend upon a m o d i f i c a t i o n o f the form o f the C = 0 magnetic a n i s o t r o p y due to the C H C I 3 hydrogen-bonding and a l s o the magnetic a n i s o t r o p y a s s o c i a t e d w i t h 137 the C - O C H 3 group, i n a d d i t i o n to s o l v e n t and e l e c t r i c f i e l d e f f e c t s 4.1.4 N,N-dimethyl carbamyl f l u o r i d e N,N-dimethyl carbamyl f l u o r i d e (DMCF) was s t u d i e d as a f u r t h e r member o f a s e r i e s of h a l o - s u b s t i t u t e d N,N-dimethyl amides showing hindered r o t a t i o n about the N - C bond. Although t h i s compound i s w e l l known as a s p e c i f i c enzyme i n h i b i t o r " ' ' 4 2 , very l i t t l e i n f o r -mation i s a v a i l a b l e on i t s e l e c t r o n i c s t r u c t u r e as compared with other 144 amides of importance i n biochemical systems . A study of t h i s com-pound allows a d i r e c t comparison of the data a v a i l a b l e from molecular o r b i t a l c a l c u l a t i o n s f o r N,N-dimethyl formamide (DMF) and DMCF and f o r 111. the parent compounds formamide and carbamyl f l u o r i d e . Carbamyl f l u o r i d e 145 i s s t a b l e only at r e l a t i v e l y low temperatures and thus i t i s not p o s s i b l e to exp e r i m e n t a l l y determine the b a r r i e r to hindered r o t a t i o n f o r t h i s p a r t i c u l a r amide. DMCF was prepared by a simple exchange r e a c t i o n between N,N-dimethyl carbamyl bromide (DMCB) and AgF, usin g C H 3 C N as a s o l v e n t . The y i e l d was low (~ 10%), however, and an improved method o f p r e p a r a t i o n 146 u t i l i z e s SbF3 as a f l u o r i n a . t i n g agent . The product was p u r i f i e d by vacuum d i s t i l l a t i o n s at ~10mm (bp = 35°C), and a l s o by a d i s t i l l a t i o n at 760 mm (bp = 129°C) usin g a spinning-band column, and gave an elemental a n a l y s i s : C H F C a l c u l a t e d 39.58 6.59 20.88 % Found 39.81 6.58 20.68 % DMCF was s t u d i e d as a 16 mole % CCli* s o l u t i o n to minimise 147 s o l u t e - s o l v e n t i n t e r a c t i o n s f o r t h i s h i g h l y p o l a r compound (u = 4.02D ). The NMR sample was prepared i n the manner p r e v i o u s l y described and as usual dioxane was used as the reference peak d e f i n i n g the tran s v e r s e r e l a x a t i o n time, T2, i n the absence of chemical exchange. DMCF gives an A 3 B 3 X spectrum, the methyl group A 3 being trans to the carbonyl oxygen atom and res o n a t i n g at a lower f i e l d than the B3 group. This s p e c t r a l assignment i s c o n s i s t e n t w i t h the general NMR chemical s h i f t c h a r a c t e r i s t i c s a s s o c i a t e d w i t h the magnetic anisotropy o f the carbonyl 132,133,148 _ i i • ^ . ^1 -, l , r 1 9 „ group . I n a d d i t i o n , the unequal H - F spm-spm c o u p l i n g constants are determined i n the absence o f exchange - 20°C) 112 as | j f I = 0.3 + 0.05 and | - J , , V | = 0.8 + 0.05 Hz, c o n s i s t e n t w i t h a 1 AX 1 - 1 BX1 ' 152 normal t r a n s c o u p l i n g ( J g ^ ) being g r e a t e r than a c i s c o u p l i n g . Th i s assignment i s s i m i l a r to that made f o r N,N-dimethyl f o r m a m i d e ^ , i n which case =0.35 and J g ^ = 0.60 Hz. These couplings may be com-pared w i t h those f o r the r e l a t e d a c e t y l compounds CH 3CF0 (Jj_,p = 7.6 Hz) and CH3CH0 ( J j ^ = 2.85 Hz), showing the e f f e c t o f the i n t e r p o s e d N atom and the N-C double bond c h a r a c t e r on these i n d i r e c t s p i n - s p i n couplings The chemical s h i f t between the A and B methyl groups (20) i n DMCF i s very s m a l l , s o l v e n t and c o n c e n t r a t i o n dependent, and a l s o temperature dependent. For example, the values o f 20 f o r the neat amide and a 16 mole % CCl l ( s o l u t i o n at ^ -20°C are 1.2 + 0.1 and 2.20 + 0.05 Hz, r e s p e c t i v e l y , at 100 MHz. The corresponding s h i f t at 30.5°C f o r the CC1,, s o l u t i o n i s 1.75 + 0.05 Hz. This p a r t i c u l a r s h i f t was a l s o measured at 220 MHz and was shown to be a c c u r a t e l y c o n s i s t e n t w i t h the o v e r a l l assignment made f o r the A B - t r a n s i t i o n s . The methyl proton chemical s h i f t i s predominantly due to the combined e f f e c t o f the magnetic s u s c e p t i b i l i t y a n i s o t r o p i e s o f the C-F and C=0 bond systems. The V e l a t i v e l y small 20 value thus i n d i c a t e s t h a t the forms o f the s u s c e p t i b i l i t y tensors a s s o c i a t e d w i t h the e l e c t r o n i c charge d i s t r i b u -t i o n s l o c a l i s e d i n these two bonds are such t h a t the two methyl s h i e l d ! regions have very s i m i l a r c h a r a c t e r i s t i c s . This i n t u r n may imply that d e l o c a l i s a t i o n o f f l u o r i n e 2 ^ e l e c t r o n s , as represented by the resonance form N - C = F , i s s i g n i f i c a n t i n the ground s t a t e o f DMCF. 1 0 -Such a charge d i s t r i b u t i o n would lea d to an an i s o t r o p y o f the C-F bond which i s not a x i a l l y symmetric and i s comparable to t h a t noi-mally 113. a s s o c i a t e d with the C=0 system. The temperature dependence o f 29 may be a s c r i b e d to a temperature dependent i n t e r m o l e c u l a r d i p o l e - d i p o l e a s s o c i a t i o n which a f f e c t s the magnetic a n i s o t r o p i e s of the C=0 and/or C-F groups, and hence i t becomes apparent that the r e d u c t i o n o f s o l u t e - s o l v e n t i n t e r a c t i o n s i n a CCli+ s o l u t i o n may inc r e a s e the p r o b a b i l i t y of s o l u t e - s o l u t e i n t e r a c t i o n s between the p o l a r DMCF mole-c u l e s . Although chemical s h i f t s are g e n e r a l l y dependent upon d e t a i l e d s o l v e n t i n t e r a c t i o n s , i n d i r e c t s p i n - s p i n couplings are u s u a l l y very n e a r l y independent of these i n t e r m o l e c u l a r e f f e c t s . In the case of an amide such as DMCF, however, such i n t e r a c t i o n s may s t a b i l i z e the + ground-state resonance form N = C - F l e a d i n g to enhanced couplings 0. -between the methyl protons and the f l u o r i i i e n u c l e a r s p i n due to the incr e a s e d N-C double-bond c h a r a c t e r . This i s a c t u a l l y observed e x p e r i -m e n t a l l y , i n tha t | j | = 1.1 ± 0.1 Hz f o r neat DMCF. This i s an increase of 0.3Hz as compared with the value f o r the DMCF/CCli, s o l u t i o n , showing that the CCli* s o l v e n t does break down s p e c i f i c s o l u t e - s o l u t e i n t e r a c t i o n s to some extent. As the four-bond s p i n - s p i n c o u p l i n g between the A 3 and B 3 methyl groups i n N,N-dimethyl amides i s n e g l i g i b l e the 1H NMR spectrum f o r the DMCF s p i n system may be analysed i n terms of an AB-part o f a f i r s t - o r d e r ABX (J^g = 0) spectrum as described i n d e t a i l i n s e c t i o n 2.5 of t h i s t h e s i s and i l l u s t r a t e d i n F i g . 2.7. The couplings and J g ^ are shown to be temperature independent ( w i t h i n experimental e r r o r ) , as the averaged c o u p l i n g i n the l i m i t o f very f a s t 114. exchange (k»ft) i s determined as 0.6 + 0.1 Hz corresponding to a c a l c u l a t e d c o u p l i n g J + = ^ ^ A X * ^BX^ " - 0.10 Hz. The observed lineshapes i n the presence of exchange unambiguously lea d to the c o n c l u s i o n t h a t the r e l a t i v e signs of these couplings are the same, as i m p l i c i t l y assumed i n the above d e f i n i t i o n o f the parameter J . The c h a r a c t e r i s t i c AB- p a r t of a f i r s t - o r d e r ABX spectrum i s shown i n F i g . 2.7 (a) f o r J K V > J Y > 0, where f o r k»ft the spectrum reduces DA A X to an A 2X form w i t h an e f f e c t i v e c o u p l i n g J . In such an a n a l y s i s f o r DMCF i t has been assumed th a t the two-bond s p i n - s p i n couplings between the methyl protons and the amide 1 4 N n u c l e a r s p i n are n e g l i g i b l e . Although a 1li - { 1 4N} double resonance study* *^ of N - s u b s t i t u t e d amides i n d i c a t e d that these couplings were indeed n e g l i g i b l y s m a l l , the 151 152 data a v a i l a b l e from f u r t h e r s t u d i e s u s i n g 1 5N- amides ' leads to c a l c u l a t e d 1 4 N c o u p l i n g constants J X 1 . and J. I T, of 0.7 and 0.8 Hz, r e s -r • NA NB p e c t i v e l y . N e v e r t h e l e s s , i t may be assumed that the methyl protons are completely decoupled by quadrupolar l l ,N n u c l e a r s p i n - l a t t i c e 153 r e l a x a t i o n . I t should be noted, however, t h a t the X-part o f the complete A 3B 3X spectrum may show e f f e c t s due to the c o u p l i n g t h i s three-bond c o u p l i n g being enhanced by the N-C p a r t i a l double bond ch a r a c t e r . The '"'N - XH c o u p l i n g J f o r N,N-dimethyl formamide i s 152 c a l c u l a t e d as 10.4 Hz , and hence the corresponding 1 4 N - 1 9 F c o u p l i n g may be o f the order o f 15 Hz. For a s c a l a r c o u p l i n g of t h i s magnitude, the lkH quadrupolar r e l a x a t i o n mechanism may give only a p a r t i a l s p i n -s p i n decoupling due to the temperature dependence of the molecular r e o r i e n t a t i o n g i v i n g r i s e to the time-dependent e l e c t r i c f i e l d g radient 115. at the 1 "*N nucleus. The f i r s t - o r d e r r a t e constants, k, d e s c r i b i n g the hindered r o t a t i o n i n DMCF have been determined us i n g t o t a l lineshape analyses i n terms of the s t o c h a s t i c model f o r such a m u l t i - s i t e exchange process developed i n s e c t i o n 2.5. The absor p t i o n mode lineshape f u n c t i o n , V ( x ) , i s the r e a l p a r t o f the complex f u n c t i o n G(x) given i n Eq. (2.5.14). The diagonal m a t r i x A and a s s o c i a t e d t r a n s f o r m a t i o n matrices and S_ 1 appearing i n t h i s expression are der i v e d from a 4 x 4 matrix [K_ - i f l j d e f i n i n g the s p e c i f i c r a t e process f o r the equal p o p u l a t i o n DMCF exchange system and the Larmor frequencies f o r the four s p i n - s i t e s to be considered ( c f . F i g . 2.7). The matrices r e q u i r e d f o r > > 0 have been given e x p l i c i t l y i n Eq. (2.5.15). A FORTRAN-IV computer program GPLONK was developed to al l o w a r a p i d numerical c a l c u l a t i o n of general m u l t i - s i t e exchange lineshapes based upon the component expressions given as Eqs. (2.4.8) and (2.4.9). Representative i n t r a -molecular exchange lineshapes f o r the AB-part o f a f i r s t - o r d e r ABX s p i n system, as c a l c u l a t e d u s i n g GPLONK, are shown i n Fig.2.8. The CPU time i n v o l v e d on an IBM 360/67 system f o r each 600 p o i n t spectrum, i n c l u d i n g the s e t t i n g - u p o f c o n t r o l arrays f o r a CALCOMP p l o t t e r , i s l e s s than 4 sees which e x e m p l i f i e s the advantages o f a p p l y i n g a matrix f o r m u l a t i o n , and the s p e c i f i c component form p r e v i o u s l y described i n d e t a i l , to lineshape c a l c u l a t i o n s . The program GPLONK, i n co n j u n c t i o n w i t h a subroutine GFITT, a l s o allows an e f f i c i e n t automatic i t e r a t i v e f i t t i n g o f a t h e o r e t i c a l lineshape to d i g i t i z e d experimental data. The sub-r o u t i n e GFITT i s based upon the simple search through optimised parameter increments p r e v i o u s l y d e s c r i b e d , c f . s e c t i o n 4.1.1, where the 1.16. m a t r i x K_ i s changed i t e r a t i v e l y i n accordance w i t h the general procedure o u t l i n e d i n s e c t i o n 2.4 f o l l o w i n g Eq. (2.4.9). M u l t i p l e s p e c t r a were obtained f o r the DMCF C C l ^ s o l u t i o n at each o f nine temperatures over the range -15.2 to 78.2°C usin g a FABRITEK FT-1064 computer with the spectrometer-computer i n t e r f a c e described i n s e c t i o n 3.1. A t y p i c a l sampled and d i g i t i z e d spectrum f o r DMCF i s shown i n F i g . 3.5. Because o f the very small methyl chemical s h i f t 20 f o r DMCF, and hence NMR lineshape frequency i n t e r v a l , i t i s d i f f i c u l t to o b t a i n a c c u r a t e l y r e p r o d u c i b l e s t e a d y - s t a t e s p e c t r a even with the o v e r a l l s t a b i l i t y a v a i l a b l e from a f i e l d - f r e q u e n c y locked spectrometer. Some t y p i c a l lineshape f i t s are shown i n F i g . 4.9 and the average r a t e constants obtained from m u l t i p l e f i t s at each temperature are summarised i n Table 4.6. The temperature dependences o f the chemical s h i f t s w and O J^, c f . Eq. (2.2.24), d e r i v e d from these l i n e -shape f i t s are shown i n F i g . 4.8 along with the mean frequency d e f i n i n g the reference zero f o r the independent v a r i a b l e x. A l l frequencies are measured with respect to that f o r TMS at constant f i e l d . The v a r i a t i o n i n methyl chemical s h i f t s i s presumably due to a ste r e o -s p e c i f i c i n t e r m o l e c u l a r i n t e r a c t i o n which may lead to a small change i n the form o f the a n i s o t r o p i c s h i e l d i n g a s s o c i a t e d w i t h the C=0 and C-F groups i n DMCF. Thus the chemical s h i f t 20 v a r i e s from 2.2 to 1.4 Hz over the temperature range -15.2 to 78.2°C. The cou p l i n g constants J . Y and J g ^ were assumed to be temperature independent, and the chemical s h i f t s corresponding to minimum e r r o r m u l t i p l e lineshape f i t s were found to be c o n s i s t e n t to w i t h i n + 0.05 Hz. Thus t h i s lineshape f i t t i n g procedure may be used to determine small chemical s h i f t s and/or TABLE 4.6 K i n e t i c data f o r N,N-dimethyl carbamyl f l u o r i d e (16 mole % i n CCK) temp., °C k, sec 1 e r r o r 12.2 0.076 4.8 15.6 0.159 4.2 30.2 ' 0 . 5 5 0 3.7 32.5 0.692 4.1 37.6 1.99 2.2 44.9 2.29 1.2 46.5 2.57 1.3 53.8 5.11 1.3 62.7 9.55 3.9 78.2- 26.3 4.6 E = 1 8 . 3 + 0 . 6 k c a l s . m o l e - 1 a l o g 1 ( ) A = 12.9 AH* = 17.7 + 0.6 k c a l s . mole" 1 AS* = -1.4 + 2.1 c a l . d e g " 1 . mole" 1 AG* = 18.2 ± 0.6 k c a l s . mole" 1 Hz -292 - 293-294H 295 -296 / / / / / / /a . / / / / ' ' t / 9 • / . o/ / a / ,0 / / / / / /o / / / 0 2C1 2.2 1.75. 1.4 Hz T - 4 0 0 40 80 T°C F i g . 4.8 Temperature dependence of chemical s h i f t s f o r N,N-dimethyl carbamyl f l u o r i d e 62.7 9.55 F i g . 4.9 T o t a l lineshape f i t s f o r N,N-dimethyl carbamyl f l u o r i d e 117. c o u p l i n g constants to t h i s p r e c i s i o n ' , which could not be obtained under normal NMR s p e c t r a l r e s o l u t i o n . I t i s to be noted t h a t the e r r o r s ( c f . s e c t i o n 4.1.1) f o r the DMCF f i t t e d lineshapes are maximal i n the l i m i t s o f very slow and very f a s t exchange. This may be due to the d i f f i c u l t y i n r e c o r d i n g the ste a d y - s t a t e lineshapes over a frequency range o f only 5Kz, but i n a c t u a l f a c t these lineshapes were a c c u r a t e l y r e p r o d u c i b l e w i t h i n the e r r o r l i m i t s i n v o l v e d . A l s o , i n the very slow exchange l i m i t , the r e s i d u a l l i n e w i d t h s were g r e a t e r than the reference widths and hence a d d i t i o n a l t r a n s v e r s e r e l a x a t i o n e f f e c t s may have been observed i n t h i s p a r t i c u l a r amide. Such e f f e c t s may a r i s e from anis o -t r o p i c nuclear d i p o l e - d i p o l e , quadrupolar and chemical s h i f t i n t e r a c t i o n s through incomplete averaging by molecular r e o r i e n t a t i o n * ^ ' . In a d d i t i o n , enhanced c r o s s - r e l a x a t i o n s a s s o c i a t e d w i t h these a n i s o t r o p i c , tensor i n t e r a c t i o n s may give r i s e to d i f f e r e n t i a l l i n e w i d t h s i n the absence o f hindered r o t a t i o n i n DMCF. Such r e l a x a t i o n processes have not been i n c l u d e d i n the a n a l y s i s r e p o r t e d here. The Arrhenius p l o t f o r DMCF i s most s a t i s f a c t o r y , however, and i s shown i n F i g . 4.10. The a c t i v a t i o n energy i s determined as 18.3 ± 0.6 k c a l s . mole l. The f r e e energy o f a c t i v a t i o n i s then c a l c u l a t e d at 25°C as AG =18.2 +0.6 k c a l s . mole 1, with an entropy o f a c t i v a t i o n AS = -1.4 + 2.1 c a l . deg 1mole 1which i s o f a magnitude t y p i c a l o f the hindered r o t a t i o n s i n a l l o f the s u b s t i t u t e d N,N-dimethyl amides s t u d i e d here. -1.6 1 2.3 3.0 3 2 2 A ~ ~ 3J6~ 1/T°Kx103 F i g . 4.10 Arrhenius p l o t f o r N,N-dimethyl carbamyl f l u o r i d e 118. 4.1.5. Formamide Although some s e m i - q u a n t i t a t i v e k i n e t i c data were" a v a i l a b l e for the hindered r o t a t i o n i n formamide*^^ ,*^^ ), a t o t a l l i n e shape a n a l y s i s had not been r e p o r t e d and hence t h i s parent amide was s t u d i e d to o b t a i n r e l i a b l e a c t i v a t i o n parameters f o r comparison w i t h those f o r the s u b s t i t u t e d N,N-dimethyl amide s e r i e s already d i s c u s s e d . I t i s a l s o o f i n t e r e s t to compare the e x p e r i m e n t a l l y determined r o t a t i o n b a r r i e r s i n formamide and N,N-dimethyl formamide (DMF) to a l l o w a f u r t h e r c o r r e l a t i o n w i t h the corresponding s e r i e s of parent amides, which are u s u a l l y e x p e r i m e n t a l l y i n a c c e s s i b l e but are c e r t a i n l y more convenient f o r molecular o r b i t a l c a l c u l a t i o n s being the s i m p l e s t molecular c o n t a i n i n g the N-OO bond system. Of course, the b a r r i e r i n formamide i s o f fundamental importance as a r e f e r e n c e p o i n t f o r a l l d e s c r i p t i o n s o f s u b s t i t u e n t e f f e c t s on the hindered r o t a t i o n about the N-C bonds i n amides and o f the bonding c h a r a c t e r i s t i c s i n these compounds. In the course o f the work d e s c r i b e d here, an independent NMR lineshape study was r e p o r t e d * ^ and the a d d i t i o n a l data now a v a i l a b l e may be used t o a s c e r t a i n the a p p l i c a b i l i t y o f the general lineshape f i t t i n g procedure t o a more complicated t i g h t l y - c o u p l e d n u c l e a r s p i n system. As already d e s c r i b e d i n d e t a i l i n s e c t i o n 2.6, the a n a l y s i s o f chemical exchange i n a t i g h t l y - c o u p l e d (second-order) s p i n system r e q u i r e s a complete s p i n d e n s i t y m a t r i x treatment. In c o n t r a s t to DMF, there are-two c o m p l i c a t i n g f a c t o r s i n an NMR study o f the hindered r o t a t i o n i n formamide. F i r s t l y , the quadru-p o l a r l l |N nucleus leads t o very broad resonances f o r the d i r e c t l y 119. bonded amino protons e l i m i n a t i n g a l l chemical s h i f t i n f o r m a t i o n f o r these protons*^*, and hence i t i s necessary to use an i s o t o p i c a l l y r 158 159 e n r i c h e d 1 5N-formamide sample or double r f - i r r a d i a t i o n The former s i m p l i f i c a t i o n i s p r e f e r a b l e f o r a q u a n t i t a t i v e l i n e s h a p e study as the d i s t o r t i o n i nherent i n s p i n - s p i n decoupling i s d i f f i c u l t to e v a l u a t e . Secondly, the i n t e r m o l e c u l a r exchange o f the b a s i c amino protons o f formamide gives a d d i t i o n a l s p i n t r a n s f e r e f f e c t s i n the s t e a d y - s t a t e spectrum and the i n c l u s i o n of such e f f e c t s i n a lineshape a n a l y s i s leads to a l a r g e i n c r e a s e i n the computer core and time r e q u i r e d f o r numerical c a l c u l a t i o n s . In p r i n c i p l e i t i s p o s s i b l e to determine the r a t e s of the i n t r a m o l e c u l a r (hindered r o t a t i o n ) and i n t e r m o l e c u l a r exchange processes s i m u l t a n e o u s l y from lineshape f i t s , but as the l a t t e r process i s not o f c u r r e n t i n t e r e s t i t may be supp-ressed over the temperature range f o r an i n t r a m o l e c u l a r exchange study u • « ' 158 by u s i n g acetone as a s o l v e n t 1 5 N - e n r i c h e d (98 atom %) formamide was obtained from Merck Sharp and Dohme and was s t u d i e d as a 10 mole % acetone (spectro-grade) s o l u t i o n . The commercial formamide was used without f u r t h e r p u r i f i c a -t i o n and was d r i e d w i t h the s o l v e n t over molecular s i e v e s . Hexamethyl-d i s i l o x a n e (^2%) was added to the NMR sample f o r f i e l d - f r e q u e n c y l o c k i n g . A separate s i n g l e l i n e was not r e q u i r e d f o r a r e f e r e n c e l i n e -shape standard as the acetone peak i s a v a i l a b l e and a convenient standard i s a l s o present i n the C-part o f the ABCX 1 5N-formamide 1l\ spectrum. That i s , i n the absence o f i n t e r m o l e c u l a r exchange there are f o u r l i n e s that, have widths t h a t are i n v a r i a n t t o exchange e f f e c t s , 120. c f . s e c t i o n 2.5, and hence these l i n e s may be used to estimate the l i n e w i d t h a s s o c i a t e d w i t h magnetic f i e l d inhomogeneity and slow i n t e r -molecular exchange which may be present at higher temperatures. The NMR sample was thoroughly degassed by the usual freeze-pump-thaw c y c l e and was sealed i n vacuo i n t h i n w a l l e d tubes o f 5 mm o.d.. The Ml NMR s p e c t r a were obtained at 100 MHz with a V a r i a n HA-100 spectrometer equipped w i t h a V-6031 variable-temperature probe and temperature c o n t r o l l e r . The V-6031 c o n t r o l l e r maintains a preset temp-erature f o r the N 2 gas he a t i n g or c o o l i n g the sample, and the gas temperature i s sensed at about 5 cm from the sample volume i n the spectrometer r e c e i v e r c o i l . Thus the sample temperature was measured before and a f t e r r e c o r d i n g a spectrum using a standard methanol f o r g l y c o l ) sample and i n t e r p o l a t i o n o f the -OH chemical s h i f t s obtained i n 126 HA-mode using the data o f Van Geet . The temperature thus determined i s estimated to be accurate to ± 0.3°C and was shown to be s t a b l e , w i t h i n these l i m i t s , over a p e r i o d o f about 20 mins. The s p e c t r a were recorded at sweep r a t e s o f 0.02 or 0.05 Hz sec 1 to best approximate st e a d y - s t a t e c o n d i t i o n s , and w i t h a low amplitude r f f i e l d (y 0.02 mgauss) to minimise lineshape d i s t o r t i o n due to s a t u r a t i o n e f f e c t s . At each temperature, to ensure r e p r o d u c i b i l i t y , at l e a s t f o u r s p e c t r a were d i g i t i s e d and st o r e d using a F a b r i t e k FT-1064 computer i n t e r f a c e d w i t h the HA-100 as described i n s e c t i o n 3.1. A l l frequencies were measured, usin g the V-4315 counter, w i t h reference to HMDS as the i n t e r n a l standard. An optimum H 0 f i e l d homogeneity was maintained over the temperature range -15 to 83°C corresponding to a 0.4 Hz r e s o l u t i o n ( T 2 = 0.8 sec.) and a 121. reference lineshape that best approximated a symmetrical L o r e n t z i a n shape. Wilmad PS-S05 5 mm t h i n - w a l l e d sample tubes were used to a t t a i n r e p r o d u c i b l e s p i n n i n g c h a r a c t e r i s t i c s and f i e l d homogeneity c o n t r o l . The hindered r o t a t i o n i n 15N-.formamide may be considered i n terms o f a mutual s p i n t r a n s f e r of the A- and B- spins i n an ABCX nu c l e a r s p i n system, f o r which the b a s i c equal p o p u l a t i o n c h e m i c a l l y s h i f t e d s i t e s are predominantly due to the diamagnetic s u s c e p t i b i l i t y a n i s o t r o p y o f the carbonyl group. Consistent w i t h p r i o r s t u d i e s w i t h i n t h i s s e c t i o n , the amino B- proton which i s c i s to the carbonyl oxygen i s assigned to h i g h - f i e l d o f the A-proton, and thus the C- s p i n i s the formyl proton which i s to l o w - f i e l d r e l a t i v e to both the A- and B-protons. Owing to the s p e c i f i c i n t e r a c t i o n o f the acetone s o l v e n t molecule w i t h the s o l u t e molecule, i t i s t o be expected that the amino protons w i l l have chemical s h i f t s showing s i g n i f i c a n t temperature dependences, w h i l e s p i n - s p i n couplings are u s u a l l y l e s s s e n s i t i v e to i n t e r m o l e c u l a r i n t e r a c t i o n s . Therefore, i n i t i a l l y , i t i s necessary t o c a r e f u l l y analyse the ABCX spectrum i n the absence o f exchange e f f e c t s to determine the form o f the temperature dependences, i f any, f o r the s p e c t r a l parameters. The s p e c t r a were analysed u s i n g a computer program NMRFIT, which i s a much modif i e d v e r s i o n o f 162 LAOCOON g i v i n g a very e f f i c i e n t i t e r a t i v e f i t o f s p e c t r a l parameters to t r a n s i t i o n frequences, by a s s i g n i n g a l l 24 l i n e s over the tempera-t u r e range -12.4 t o 30.2°C. The r e s u l t a n t chemical s h i f t s and couplings are given i n Table 4.7, along w i t h those obtained from the 158 e a r l i e r study o f an acetone s o l u t i o n o f the same c o n c e n t r a t i o n Table 4.7 S p e c t r a l parameters f o r 1 5N-formamide \ JAB JAC JBC JAX JBX Jcx r e f . 12.4 -4.9 4.9 -84.6 2.8 1.8 13, .5 90, .1 87, .5 15, .1 t h i s wo -5.0 -5.7 5.7 -90.3 2.8 1.7 13. .5 90. ,2 88. .3 15. .2 t! 3.0 :7.0 7.0 -97.4 2.8 •1.7 13. .5 90. .0 87, .6 15, .3 1 ? 30.2 -10.3 10.3 -118.8 2.8 1.6 13. .5 90. .0 87. .6 15. .5 II -10.7 10.7 -118.8 2.9 1.7 13. ,5 91. .0 88. ,0 16. .4 (158) -12.1 12.1 -97.0 2.8 1.7 13. .4 89. ,7 86. ,4 16. ,4 (152) -18.5 18.5 -69.8 2.6 1.6 13. ,5 89. . 3 87, i 15. .6 (160) -9.9 9.9 -102.5 2.7 1.7 13. .5 -- -- (159) 10 mole % acetone s o l u t i o n , -12.4 to 30.2°C 35 mole % acetone s o l u t i o n , 30°C 14.1 % diglyme s o l u t i o n , 25°C 22.2% acetone s o l u t i o n A l l chemical s h i f t s r e f e r to 100 MHz NMR. 122. , +. . . . ,. . . . • 152,159,160 and those from more recent s t u d i e s u s i n g s i m i l a r s o l v e n t s Only r e l a t i v e l y small v a r i a t i o n s i n the values o f the c o u p l i n g constants are evident f o r d i f f e r e n t c oncentrations o f formamide i n acetone and diglyme as s o l v e n t , and are even s m a l l e r over the tempera-t u r e i n t e r v a l o f 42°C f o r the 10 mole% acetone s o l u t i o n . . Thus a l l c o u p l i n g constants are assumed to be temperature independent i n the f o l l o w i n g lineshape a n a l y s i s . As i s normal, the trans c o u p l i n g J g ^ i s much grea t e r than the c i s c o u p l i n g J^Q> both being enhanced due to the p a r t i a l double bond c h a r a c t e r o f the in t e r p o s e d N-C bond. The chemical s h i f t s ft., ftn and ft , however, show r e l a t i v e l y l a r g e v a r i a t i o n s A D L w i t h s o l u t e c o n c e n t r a t i o n and temperature. In Table 4.7, the reference frequency d e f i n i n g the independent frequency v a r i a b l e x f o r a lineshape a n a l y s i s i s taken as the average of the s h i f t s co. and U L , A b cf-. Eq. (2.2.24), and hence ft„ < ft. < 0 < ftD. The s h i f t s co. , coD and L A D A D are measured r e l a t i v e to hexamethyl-disiloxane and UL, i s shown to be co approximately constant w h i l e co and co (ft = co -to ) have r e l a t i v e l y A. Lt A. A. U strong temperature dependences as i l l u s t r a t e d i n F i g . 4.11. As the chemical s h i f t s d e r i v e d u s i n g NMRFIT were r e p r o d u c i b l e to w i t h i n + 0.2 Hz and as the v a r i a t i o n over the temperature i n t e r v a l -12.4 to 30.2°C was l i n e a r , as depi c t e d by the open p o i n t s i n F i g . 4.11, i t was assumed that a l i n e a r e x t r a p o l a t i o n could be used t o determine the chemical s h i f t s used i n a lineshape f i t at any temperature up to 85°C. The a c t u a l lineshape f i t s showed that the v a r i a t i o n s of a l l chemical s h i f t s over t h i s temperature range are indeed l i n e a r , i n that the chemical s h i f t s corresponding to the minimum e r r o r f i t s are shown as the closed - 8 2 0 - 2 0 0 2 0 ' 60 100 TEMP °C F i g . 4.11 Temperature dependence of the chemical s h i f t s f o r 5N-formamide, 10% acetone s o l u t i o n 123. p o i n t s i n F i g . 4.11. The a b s o r p t i o n mode ste a d y - s t a t e NMR lineshapes f o r the ABC-p a r t o f the ABCX 1 5N-formamide system as modified by mutual t r a n s f e r of the A- and B-spins have been c a l c u l a t e d u s i n g a FORTRAN-IV computer program GENLIN. The theory f o r these c a l c u l a t i o n s has been discussed i n d e t a i l i n s e c t i o n 2.6 and the computer program i s based upon the matrix equation Eq. (2.6.17), which i s d e r i v e d from the general com-ponent equation of motion f o r the s p i n d e n s i t y m a t r i x , c f . Eq. (2.6.7). The s p i n t r a n s i t i o n operator m a t r i x , I + , i n the b a s i s of simple product f u n c t i o n s f o r the f o u r s p i n system i s very s i m i l a r to that given e x p l i c i t l y f o r the ABX (or ABC) s p i n system i n Table 2.3. In f a c t the m a t r i x given i n Table 2.3 forms a block diagonal p a r t of the complete ABCX t r a n s i t i o n m a t r i x which defines 30 t r a n s i t i o n s , i n c l u d i n g allowed and "combination" t r a n s i t i o n s , i n the ABC-part of the ABCX spectrum and hence the fundamental matr i x R 1, c f . Eq. (2.6.12), i s a 30 x 30 m a t r i x w i t h complex elements. To s i m p l i f y the c a l c u l a t i o n f o r the l a r g e number o f numerical s p e c t r a l data p o i n t s (M000) r e q u i r e d i n an i t e r a t i v e lineshape f i t t i n g procedure f o r such a complicated spectrum, the m a t r i x R' i s transformed to diagonal form and the frequency dependent pa r t i s r e t a i n e d as the s c a l a r m a t r i x x l _ , I_ being the 30 x 30 u n i t matrix. In comparison w i t h Eq. (2.4.6) g i v i n g the complex lineshape f u n c t i o n G(x) f o r a f i r s t - o r d e r s p i n system, the diagonal m a t r i x Q i s d e r i v e d from the m a t r i x R given i n Eq. (2.4.5) where Q i s now an o f f - d i a g o n a l m a t r i x . The elements of the m a t r i x ft 92 93 are f o r m a l l y defined by the L i o u v i l l e Hamiltonian operator , but 124; are very simply determined using a simple a l g o r i t h m based upon Eq. (2.6.27). On d i a g o n a l i s a t i o n , i n the absence of exchange e f f e c t s , t h i s m a t r i x d i r e c t l y determines the second-order t r a n s i t i o n frequencies and a s s o c i a t e d complex i n t e n s i t i e s . I t i s to be noted that the diago- . n a l elements of the matrix ft are f i r s t - o r d e r approximations to the t r a n s i t i o n f r e q u e n c i e s , and indeed the only change i n going to a second-order a n a l y s i s of exchange e f f e c t s u s i n g t h i s m a t r i x f o r m u l a t i o n i s i n the d e f i n i t i o n , o f the s p i n - s i t e frequency m a t r i x , ft, i n Eq, (2.4.5). F o l l o w i n g the s i m i l a r i t y t r a n s f o r m a t i o n as given i n Eq. (2.4.7) , the numerical lineshape computation i s reduced to simple m u l t i p l i c a t i o n s i n r e a l a r i t h m e t i c as p r e v i o u s l y described, c f . Eq. (2.4.8) . The computer program GENLIN i s complicated by i n v o l v e d i n -dexing and s o r t i n g t o maximise the computational e f f i c i e n c y and such d e t a i l s need not be discussed at t h i s p o i n t . The advantage of using the m a t r i x f o r m u l a t i o n described as p a r t of t h i s work, however, i s e x e m p l i f i e d by comparison w i t h the computational methods used i n the independent study of 15N-formamide"1"^^. The CPU time quoted f o r the computation of a s i n g l e s p e c t r a l data p o i n t f o r a 4-spin system u s i n g an IBM 360/75 computer i s 0.3 second* 1^, and hence the time i n v o l v e d f o r a 1000 p o i n t spectrum as r e q u i r e d f o r a r e l i a b l e v i s u a l or numerical comparison with experimental data i s 5 minutes. The CPU time f o r the c a l c u l a t i o n o f the same spectrum using the program GENLIN on an IBM 360/75 computer i s l e s s than 10 seconds, r e p r e s e n t i n g a r e d u c t i o n i n time by a f a c t o r of about t h i r t y . That i s , i n approximately 2 minutes i t i s p o s s i b l e to o b t a i n a completely automatic numerical 125. i t e r a t i v e lineshape f i t over any p r e s c r i b e d frequency range and to a p r e c i s i o n o f the order of 1% f o r the f i r s t - o r d e r r a t e constant d e f i n i n g the i n t r a m o l e c u l a r exchange. The subroutine GFITT used i n c o n j u n c t i o n w i t h GENLIN f o r the i t e r a t i v e f i t t i n g has been p r e v i o u s l y descibed, c f . s e c t i o n 4.1.1. The average r a t e constants obtained from m u l t i p l e lineshape f i t s to both the AB- and C- p a r t s o f the 1 ^ -formamide spectrum over the temperature range 43.3 to 83.0°C are l i s t e d i n Table 4.8 and the complete lineshapes corresponding to these p a r t i c u l a r r a t e constants are shown i n F i g . 4.12. These lineshapes may be compared with those obtained at 60 MHz 160^ ^ g e f £ e c t Qf t ] i e temperature dependent chemical s h i f t s and i s c l e a r l y shown i n F i g . 4.12. A l s o , from the form of the minimum e r r o r lineshape f i t s , i t i s r e a d i l y deter-mined that the r e l a t i v e signs of the c o u p l i n g constants J ^ , and are the same. The lineshapes i n F i g . 4.12 may be compared w i t h those c a l c u l a t e d f o r the c l o s e l y r e l a t e d ABX s p i n system, F i g s . 2.8 and 2.10 showing the e f f e c t s of the r e l a t i v e signs o f the c o u p l i n g constants and Jg^- The e f f e c t o f the couplings w i t h the same s i g n i s e s p e c i a l l y obvious i n the C-part o f the spectrum. This s i g n deter-152 mmation i s c o n s i s t e n t w i t h a double resonance study . The upper temperature l i m i t f o r the study of i n t r a m o l e c u l a r exchange i s deter-mined as that at which the a d d i t i o n a l i n t e r m o l e c u l a r exchange e f f e c t s become s i g n i f i c a n t . That i s , the l i n e w i d t h a s s o c i a t e d w i t h the l a t t e r process i n c r e a s e s t o 0.8 Hz at 83°C and t h i s was considered to be the maximum al l o w a b l e c o n t r i b u t i o n f o r the o v e r a l l lineshape f i t s not to be TABLE 4.8 K i n e t i c data f o r formamide (10% acetone s o l u t i o n ) Temp°C k sec * 43.3 3.63 45.5 4.68 47.5 . 6.46 49.0 6.76 52.2 9.55 54.6 10.50 55.0 10.00 ^y.4 15.90 60.0 19.10 67.0 33.90 77.4 69.10 77.6 75.90 78.0 85.10 83.0 120.00 E = 19.27 + 0.37 k c a l s . m o l e " 1 a A H * = 18.68 kc a l s . m o l e " 1 at 25°C A S 1 = 3.0+1.2 c a l s . deg" 1 mole 1 A G * = 17.77 kc a l s . m o l e " 1 126. a f f e c t e d by t h i s approximation w i t h i n the o v e r a l l f i t t i n g e r r o r l i m i t s a t t a i n a b l e . The four narrow l i n e s i n the C-part of the spectrum which may be used to monitor t h i s l i n e w i d t h c o n t r i b u t i o n are c l e a r l y shown i n F i g . 4.12. The Arrhenius p l o t i s shown i n F i g . 4.13 and the a c t i v a -t i o n energy obtained i s 19.3 + 0.4 k c a l s . mole 1 which i s c o n s i s t e n t w i t h the e a r l i e s t estimate f o r t h i s parameter**^, namely 18 + 3 k c a l s . mole J. At 25°C, the f r e e energy o f a c t i v a t i o n i s determined as AG* = 17.8 ± 0.4 k c a l s . mole' 1 w i t h a corresponding entropy of a c t i v a -t i o n AS =3.0 + 1.2 c a l s . deg. •'mole 1. These a c t i v a t i o n parameters may now be compared with those obtained f o r a 1^N-formamide acetone s o l u t i o n ( 2 2 . 2 % ) 1 5 9 : A G # = 17.0 + 1.9 k c a l s . m o l e " 1 and A S # = -5.4 + 2.4 cal.deg. 1 mole 1 , and f o r a 1 5N-formamide diglyme s o l u t i o n (14.1%) 1 6 ° : AG* = 17.8 ± 0.2 k c a l s . m o l e " 1 and AS* = 4.0 + 0.7 c a l s . d e g " 1 . mole 1. Although the entropies of a c t i v a t i o n are p o s i t i v e and negative f o r the lineshape and double resonance s t u d i e s , r e s p e c t i v e l y , they are r e l a t i v e l y small as i s t o be expected f o r an i n t e r n a l r o t a t i o n process. A p o s i t i v e entropy corresponds to a more ordered ground s t a t e , f o r the formamide-acetone complex, and as the p l a n a r ground s t a t e may be expected to have the l a r g e s t d i p o l e moment t h i s i s c o n s i s t e n t w i t h an increased s o l u t e - s o l v e n t e l e c t r o s t a t i c i n t e r a c t i o n l e a d i n g to a more s t a b l e complex i n s o l u t i o n w i t h reduced degrees of freedom. However, there are two p o s s i b l e pyramidal t r a n s i t i o n s t a t e s f o r the formamide hindered r o t a t i o n , and w h i l e these s t a t e s have very d i f f e r e n t d i p o l a r c h a r a c t e r they have n e a r l y i d e n t i c a l energies, as c a l c u l a t e d f o r an i s o l a t e d molecule. Thus i f s o l v a t i o n p l a y s an important r o l e , i t may w e l l be that the more d i p o l a r t r a n s i t i o n s t a t e of the two F i g . 4.13 Arrhenius p l o t f o r 1 5N-formamide, 10% acetone s o l u t i o n 127. p o s s i b l e i s s t a b i l i s e d by s o l v a t i o n l e a d i n g to a decreased entropy and a negative entropy o f a c t i v a t i o n . The d i f f e r e n t t r a n s i t i o n s t a t e s w i l l be f u r t h e r discussed i n a. f o l l o w i n g s e c t i o n , In view o f the l a r g e v a r i a t i o n s i n a c t i v a t i o n parameters normally a s s o c i a t e d w i t h independent NMR de t e r m i n a t i o n s , c f . Table 4.3, i t i s g r a t i f y i n g to see the agreement on both the a c t i v a t i o n energy and f r e e energy o f a c t i v a t i o n f o r formamide as determined u s i n g t o t a l lineshape analyses. Of course t h i s i s the f i r s t comparison o f r e s u l t s from complete d e n s i t y m a t r i x lineshape s t u d i e s and the agreement may be somewhat f o r t u i t o u s i n th a t d i f f e r e n t s o l v e n t s and s l i g h t l y d i f f e r e n t s o l u t e c o n c e n t r a t i o n s were used, although the s o l v e n t s do have s i m i l a r chara-c t e r i s t i c s . N e v e r t h e l e s s , i n g e n e r a l , i t may be expected t h a t more r e l i a b l e r a t e constant determinations are p o s s i b l e f o r more com-p l i c a t e d t i g h t l y - c o u p l e d s p i n systems. I t i s w e l l recognised that the most accurate k i n e t i c data are obtained f o r a simple t w o - s i t e exchange system i n the r e g i o n o f coalescence. As there are a number o f analogous r e g i o n s f o r a more complicated s p i n system spanning a l a r g e r range o f r a t e c o n s t a n t s , i t i s to be expected t h a t more r e l i a b l e primary data may be obtained by lineshape a n a l y s i s over an i n c r e a s e d temperature range. Thus w i t h the advent o f e f f i c i e n t computer programs, such as GENLIN d e s c r i b e d i n t h i s work, to handle more complicated s p i n systems i n an i t e r a t i v e NMR lineshape f i t t i n g procedure, a wider v a r i e t y o f chemical systems of current i n t e r e s t showing k i n e t i c e f f e c t s may be p a r t i c u l a r l y amenable to study u s i n g the NMR technique. 128. 4.2 Hindered r o t a t i o n i n amides, Huckel MO c a l c u l a t i o n s The systematic experimental study of hindered r o t a t i o n about the N-C bond i n a s e r i e s of N,N-dimethyl amides o f the form N-C-X w i t h 6 d i f f e r e n t s u b s t i t u e n t s X, as described i n the preceding s e c t i o n s , has .shown th a t the f r e e energy of a c t i v a t i o n f o r the r o t a t i o n process i s n e a r l y independent of i n t e r - m o l e c u l a r i n t e r a c t i o n s . Thus t h i s f r e e energy may be considered as the a c t i v a t i o n parameter c h a r a c t e r i s i n g the i n t r a m o l e c u l a r r a t e process. In a d d i t i o n , i t i s apparent that systematic experimental e r r o r s inherent i n the determination of s p e c i f i c r a t e constants u s i n g s t e a d y - s t a t e NMR lead to minimal e r r o r i n the e s t i m a t i o n of t h i s p a r t i c u l a r a c t i v a t i o n parameter. In terms of the usual absolute r a t e theory, the e f f e c t o f molecular s t r u c t u r a l or s u b s t i t u e n t changes on a r a t e process i s r e f l e c t e d by the f r e e energy 129 of a c t i v a t i o n and hence t h i s thermodynamic parameter may be c o r r e l a t e d w i t h the changes i n the t o t a l molecular energy, as c a l c u -l a t e d u s i n g molecular o r b i t a l (MO) theory, corresponding to the s t r u c t u r a l or s u b s t i t u e n t changes. The e a r l y work of Woodbrey and 87 Rogers shows that the b a r r i e r to hindered r o t a t i o n i n amides i s s t r o n g l y i n f l u e n c e d by the s u b s t i t u e n t X. In t h i s work the b a r r i e r s were determined us i n g only approximate analyses of steady-state NMR data, however, and the a c t i v a t i o n energies were discussed q u a l i t a t i v e l y i n terms of the e f f e c t o f a given s u b s t i t u e n t on the bond-order of the N-C bond. Now that r e l i a b l e values o f the f r e e energies of a c t i v a t i o n are a v a i l a b l e , a s e m i - q u a n t i t a t i v e c o r r e l a t i o n w i t h data a v a i l a b l e from MO c a l c u l a t i o n s i s f e a s i b l e . Thus a simple model may be developed, 129 w i t h i n the framework o f a l i n e a r combination of atomic o r b i t a l s (LCAO) 72 MO theory , to d e s c r i b e the i n t e r a c t i o n s between the carbonyl group, the n i t r o g e n lone p a i r e l e c t r o n s and the s u b s t i t u e n t X i n a s u b s t i t u t e d amide. A c a l c u l a t i o n of the energy change ( e l e c t r o n i c only i n the s i m p l e s t model) on r o t a t i o n about the N-C bond and d i r e c t comparison w i t h the experimental data then allows an e v a l u a t i o n of the general v a l i d i t y o f the model and p o s s i b l y a simple s e m i - q u a n t i t a t i v e p h y s i c a l d e s c r i p t i o n of the hindered r o t a t i o n and the e f f e c t s o f d i f f e r e n t s u b s t i t u e n t s on the b a r r i e r to r o t a t i o n . Although a number of c a l -c u l a t i o n s based upon simple Huckel molecular o r b i t a l (HMO) theory have been reported f o r amides*^ 4 168^ ^ e only c a l c u l a t i o n of t h i s type f o r 169 a s e r i e s o f r e l a t e d compounds i s t h a t reported by Sand Strom f o r a s e r i e s o f s u b s t i t u t e d N,N-dimethylthioamides. ; In the l a t t e r study, the f r e e energies of a c t i v a t i o n obtained from NMR s p e c t r a show only a crude c o r r e l a t i o n w i t h N-C ir-bond orders c a l c u l a t e d u s i n g a modified 170 171 w-Huckel method ' , while a s l i g h t l y improved c o r r e l a t i o n i s obtained w i t h the corresponding l o s s i n u - e l e c t r o n energy which occurs when the dimethylamino group i s r o t a t e d w i t h respect to the t h i o c a r b o n y l group about the N-C bond. In t h i s s e c t i o n , a simple i r - e l e c t r o n HMO model i s used to c a l c u l a t e both bond orders and e l e c t r o n d e l o c a l i s a t i o n energies f o r a s e r i e s o f s u b s t i t u t e d N,N-dimethyl amides. The a p p l i c a t i o n o f quantum-mechanical methods to the c a l -c u l a t i o n o f molecular p r o p e r t i e s f o r systems of chemical i n t e r e s t at t h i s time s t i l l i n v o l v e s a number of approximations, and consequently the emphasis has been placed upon c o r r e c t numerical agreement w i t h experimental data. In the MO approximation, e l e c t r o n s are assigned to 130. molecular o r b i t a l s which are most co n v e n i e n t l y chosen to be one e l e c t r o n wavefunctions expressed as l i n e a r combinations of a f i n i t e number of atomic o r b i t a l s , c|> . That i s , the LCA0-M0 i s given as TII. = 1 C. <$> (4.2.1) V3 u IV u 172 where the c o e f f i c i e n t s C j U are determined by a v a r i a t i o n a l c a l c u l a t i o n Through neglect o f a l l two-electron i n t e r a c t i o n s , the t o t a l molecular wave-function may be considered as a simple product of these one-electron molecular o r b i t a l s so that the t o t a l e l e c t r o n i c energy, f o r a s i n g l e t ground s t a t e , i s given as the sum of the energies f o r e l e c t r o n s i n the occupied molecular o r b i t a l s . In the simpl e s t LCA0-M0 theory, an e l e c t r o n may be considered t c occupy a one-electron MO i n a c e n t r a l .. f i e l d due to a l l other e l e c t r o n s and n u c l e i i n a given molecule. In t h i s manner the energy a s s o c i a t e d w i t h the MO , E•, i s determined by an e f f e c t i v e H amiltonian, H. In accordance w i t h the v a r i a t i o n a l p r i n c i p l e , f o r t h i s H a m i l t o n i a n , the ei g e n - f u n c t i o n s i>. and eigen-values Ej are given i n 172 terms of a set of simultaneous equations : I [H - 6 E.] C. = 0 v yv yv 3 J j y (4.2.2) where H^ v i s the Hamiltonian matrix element f o r the p a r t i c u l a r b a s i s AO's <f>y and <J)V. I t may be assumed that the b a s i s f u n c t i o n s form an orthonormal set and that a l l e f f e c t s o f AO overlap are des c r i b e d by the o f f - d i a g o n a l matrix elements H ^ v ( y * v ) , and hence 6 U V i s the Kronecker 184 d e l t a f u n c t i o n . I n c l u s i o n o f overlap leads to a d i f f e r e n t charge d i s t r i b u t i o n i n a molecule but has only a small e f f e c t on d e l o c a l i s a t i o n 131. energies and t o t a l i r-encrgies. Thus although i t i s more r i g o r o u s t o i n c l u d e o v e r l a p , i n general no s i g n i f i c a n t improvement occurs i n the r e s u l t s from simple Huckel MO c a l c u l a t i o n s . I n i t i a l l y a l l of the eigenvalues are c a l c u l a t e d u s i n g the s e c u l a r determinantal equation: III - 6 El = 0 , (4 . 2.3) and then the expansion c o e f f i c i e n t s C j V are determined as s o l u t i o n s of the simultaneous equations Eq..(4 . 2 . 2 ) w i t h constant c o e f f i c i e n t s d e f i n e d by H ^ and the s p e c i f i c eigenvalue Ej . Given an e x p l i c i t form f o r the o n e - e l e c t r o n e f f e c t i v e Hamiltonian o p e r a t o r , the m a t r i x elements H u v may be c a l c u l a t e d i n the chosen b a s i s of AO's, {<J>U), and s i m i l a r l y the overlap i n t e g r a l s may be evaluated. However, i n such a s i m p l i f i e d form o f LCAO-MO theory there i s l i t t l e p o i n t i n d e f i n i n g the Hamiltonian operator e x p l i c i t l y and hence the m a t r i x elements are d e f i n e d only i n terms o f e m p i r i c a l parameters. The i n c l u s i o n o f o r b i t a l o v e r l a p as d e s c r i b e d by the i n t e g r a l s S i s not c o n s i s t e n t w i t h the complete n e g l e c t of a l l e l e c t r o n - e l e c t r o n and e l e c t r o n - n u c l e u s i n t e r a c t i o n s , and hence i t must be assumed t h a t such i n t e r a c t i o n s are i n c l u d e d i n the e s t i m a t i o n of the e m p i r i c a l parameters, which are obtained from comparison of the r e s u l t a n t MO data w i t h s p e c i f i c experimental q u a n t i t i e s or by d i r e c t comparison w i t h w e l l d e f i n e d molecular p r o p e r t i e s . A f u r t h e r s i m p l i f i c a t i o n may be considered i n the study of ir-conjugated molecules such as amides, i n that to the l e v e l o f approximation under c o n s i d e r a t i o n the i r - e l e c t r o n s may be assumed to be independent of the O-electrons. That i s , the a - e l e c t r o n s are considered to form a l o c a l i s e d 132. bonding system which does not vary s i g n i f i c a n t l y i n form w i t h s t r u c t u r a l or s u b s t i t u e n t changes. This i s probably a good approximation f o r the s u b s t i t u t e d N,N-dimethyl amides with a s i n g l e v a r i a b l e s u b s t i t u e n t X and a s t r u c t u r a l change corresponding t o a simple r o t a t i o n o f the dimethylamino group about the N-C bond. The d i f f e r e n t i a l energy, AF., corresponding to s t r u c t u r a l or s u b s t i t u e n t changes may now be considered i n the p a r t i t i o n e d form j AE = ( A E 0 + AE n) + AE U , (4.2.4) where AE = £ E , the summation being over the occupied TT- molecular j 3 o r b i t a l e n e r g i e s , and AE n represents any change i n the non-bonding i n t e r a c t i o n energy. Thus i n conjugated molecules, f o r s i m p l i c i t y , the bonding c h a r a c t e r i s t i c s and energy v a r i a t i o n s due t o s t r u c t u r a l and s u b s t i t u e n t changes may now be de s c r i b e d i n terms o f a f r - e l e c t r o n o n l y LCAO-MO model. From c a l c u l a t i o n s based upon t h i s model, the bonding c h a r a c t e r i s t i c s are de s c r i b e d i n terms o f d e l o c a l i s e d molecular o r b i t a l s d e f i n e d as l i n e a r combinations o f 2P - AO's (Tr-type) , and the TT n d i f f e r e n t i a l energy becomes AE + AE i n accordance w i t h Eq. (4.2.4) The f o l l o w i n g simple Huckel c a l c u l a t i o n s are based upon the much s i m p l i f i e d Tr-only LCAO-MO model w i t h the aim of checking the general a p p l i c a b i l i t y o f such a model to a d e s c r i p t i o n of the hindered r o t a t i o n i n s u b s t i t u t e d amides. Attempts have been made to i n c o r p o r a t e 173 ]74 e l e c t r o n r e p u l s i o n i n t o simple Huckel theory ' , but at t h i s l e v e l o f approximation such a procedure i s d i f f i c u l t to j u s t i f y i n t h a t a r e d e f i n i t i o n o f the p u r e l y e m p i r i c a l parameters i s i n v o l v e d . Therefore 133. a n o n - i t e r a t i v e Huckel c a l c u l a t i o n i s used i n t h i s study w i t h the e f f e c t i v e Hamiltonian matrix elements f o r s u b s t i t u t e d amides being defined 72 by e m p i r i c a l Coulomb i n t e g r a l s , a , and resonance i n t e g r a l s , 3 R Y , A L» — A where a x = a° + h x 3 ° and (4.2.5) ec-x = kC-X 3° As only r e l a t i v e energies are s i g n i f i c a n t , the standard Coulomb i n t e g r a l a° may be taken as the energy reference zero and then the matrix elements a v and 3 R Y are defined i n terms o f the standard resonance X C—A i n t e g r a l , 3°<o, which i s u s u a l l y taken as th a t f o r the C-C bond i n b e n z e n e 1 0 . I t i s to be noted t h a t the parameter 3 R Y r e f e r s to N-C, C — A C=0 and C-X Tr-type bonds i n the s u b s t i t u t e d amides >N-C-X, a l l other 0 o f f - d i a g o n a l Hamiltonian matrix elements being defined to be zero. In d e s c r i b i n g the energy v a r i a t i o n f o r hindered r o t a t i o n about the N-C bond, due to d e l o c a l i s a t i o n of the formal lone p a i r e l e c t r o n s on the sp - h y b r i d i s e d N atom, the energy f o r the p l a n a r ground s t a t e i s given as a sum o f T T -MO energies. In the r o t a t i o n t r a n s i t i o n s t a t e , the d i -methylamino group i s r o t a t e d i n t o a plane p e r p e n d i c u l a r to th a t o f the carbonyl group and hence the conjugation between the amino and :C-X 0 groups i s broken down as the symmetry r e l a t i o n s h i p necessary f o r ir-bonding between these two groups no longer e x i s t s . In t h i s case the molecular energy must be considered as a sum of I T -MO energies f o r the C-X group and a c o n t r i b u t i o n from non-bonding e l e c t r o n s on the N atom, 6 E n , c f . Eq. (4.2.4). Thus the d i f f e r e n t i a l 7T-energy a s s o c i a t e d w i t h the 134. hindered r o t a t i o n i s given as AE = E^N-C-X) - {E^CC-X) + 2 h M ] , (4.2.6) TT || 1/ IN 0 o where the energies are given i n terms o f the Hamiltonian m a t r i x : kC-N 0 0 kC-N | k c . k c = o k c - x " 0 1 k c = o k o 0 0 1 k c - x 0 k x (4.2.7) i n u n i t s of 3°, c f . Eq. (4.2.5). The only non-bonding energy i s t h a t f o r the s u b s t i t u e n t s under c o n s i d e r a t i o n . In c o n t r a s t to an ab i n i t i o c a l c u l a t i o n , the values chosen f o r the e m p i r i c a l parameters h^ , and k^_^ are o f primary importance s i n c e they alone may determine the r e s u l t s o f a simple Huckel MO c a l c u l a t i o n and^therefore represent the e s s e n t i a l c h a r a c t e r i s t i c s o f the s i m p l i f i e d model being used here. A Coulomb resonance i n t e g r a l i s expected to be d i r e c t l y r e l a t e d to both the i o n i s a t i o n p o t e n t i a l and e l e c t r o n a f f i n i t y f o r a given atom i n a molecule, and hence to the atomic e l e c t r o -17 6 -17 S n e g a t i v i t y . The concept of v a r i a b l e e l e c t r o - n e g a t i v i t y has been 179 180 introd u c e d by J a f f e and coworkers ' , and t h i s a l l o w s the r i g o r o u s d e f i n i t i o n of an o r b i t a l e l e c t r o - n e g a t i v i t y which may be considered as a measure o f the a t t r a c t i n g power o f an atom, as i t e x i s t s i n a molecule, toward an e l e c t r o n i n a s p e c i f i c type of AO, that i s f o r a s p e c i f i c 135. atomic valence s t a t e . Thus the i T - o r b i t a l e l e c t r o n e g a t i v i t i e s f o r sp 2 - h y b r i d i s e d N,C and 0 atoms have been used to f i x the h^, parameters as 0.7, 0.0 and 1.4, r e s p e c t i v e l y . These parameters may be compared 181 w i t h "standard" values . The parameter h^ has a l s o been modified to allow f o r the hyperconjugative e f f e c t o f the N-methyl groups f o r the amides under c o n s i d e r a t i o n . For the general s u b s t i t u e n t X i n the N,N-dimethyl amides, i t has been assumed t h a t a l l multi-atomic sub-s t i t u e n t groups may adequately be represented as pseudo-atoms*^ 2'* 8^ and are t h e r e f o r e c h a r a c t e r i s e d by a s i n g l e h^ parameter de r i v e d through a l i n e a r r e l a t i o n s h i p w i t h the corresponding group e l e c t r o - n e g a t i v i t i e s . The set of h Y parameters used i n the Huckel MO c a l c u l a t i o n s i s given i n Table 4.9 along w i t h references to the method f o r c a l c u l a t i o n of the group e l e c t r o n e g a t i v i t i e s , namely the use of a v a r i a b l e e l e c t r o -n e g a t i v i t y f o r the c e n t r a l atom i n a group and e q u a l i s a t i o n o f e l e c t r o -179 n e g a t i v i t y i n a l l bonds. The resonance i n t e g r a l , and hence the parameter k^ , c h a r a c t e r i s e s the 7T-bonding between the C atom and the atom or pseudo-atom X and i s expected to be d i r e c t l y r e l a t e d to the bond length y{C-X) 1 8 5 j 1 ^ 6 o r the overlap i n t e g r a l between 2P Z-A0's* 8 7. The k(-,_x values d e r i v e d through a l i n e a r c o r r e l a t i o n w i t h bond lengths 188 and corresponding overlap i n t e g r a l s f o r the carbamyl f l u o r i d e s t r u c t u r e shown i n F i g . 4.22 are 0.90(0.90), 0.80(0.86) and 0.60(0.52) f o r the C=0, N-C and C-F bonds, r e s p e c t i v e l y . As i t i s d i f f i c u l t to estimate overlap i n t e g r a l s f o r pseudo-atoms, the s i m i l a r i t y o f the above sets of parameters i n d i c a t e s that t h i s d i f f i c u l t y may be avoided by usi n g the sim p l e r c o r r e l a t i o n w i t h bond le n g t h s , and hence the set of k„ v para-meters given i n Table 4.9 has been determined i n t h i s way from the Table 4.9 HuckelMO data f o r hindered r o t a t i o n i n s u b s t i t u t e d N,N-dimethyl amides £X \ ' kC-X A E + A G # *Vc PC-0 Pj-O % H 0, .759 21.0 0. ,766 0, .516 0. .753 1 .58 a) CN 4 .17 4. .6 0 .55 0. .737 20.6 0, .752 0, .511 0 .731 1 .59 a) F 3. .95 4 .0 0. .60 °. .728 18.2 0 .746 0 .508 0 .723 1 .60 a) C l 3 .00 1 .4 0 .40 0 .715 16.8 0, .737 0 .502 0 .704 1 .58 Br aj 2. .80 0, .8 0 .32 0 .710 15.7 0, ,735 0, .500 0 .694 1, .61 b) OCH3 2. .68 0. 65 0. 50 0. 639 (5.8) 0. 685 0. 473 0. 613 1. 65 c) SCN 3 .91 3. .85 0 .39 0 .745 (21.4) 0, .757 0 .512 0, .739 1 .59 c) NCS 4, .15 4 .50 0 .54 0 . 737 (20.2) 0, .752 0, .511 0, .731 1 .59 c) N 3 4, .42 4. ,90 0, .54 0, .739 (20.4) 0. ,753 0, .511 0, .734 1, .59 a) NCO 3, .05 1, ,55 0, .56 0, .686 (12.6) 0. ,717 0, .492 0. .674 1 .63 b) NH 2 2 .61 0. .80 0, .60 0, .621 (3.0) 0. ,670 0, .465 0, ,597 1 .66 b) N M e 2 2, .40 0. ,70 0, .60 0, ,609 (1.0) 0, ,661 0, .460 0, .583 1 .67 K, = 0.7 0. 8 N C-•N k^ = 0.0 - 0. 9 C C-•0 k 0 = 1.4 ^ i n u n i t s o f 3 <o, see text. a) W. Gordy and W.J.O. Thomas J . Chem. Phys. 24_, 439, 1956 b) J.E. Huheey J . Phys. Chem. 69_, 3284, 1965 c) J.E. Huheey J . Phys. Chem. 70, 2086, 1966 136. s t r u c t u r e s a v a i l a b l e f o r amides or the r e l a t e d a c e t y l compounds H3C-C-X and formyl compounds H-C-X. n 11 0 C The d i f f e r e n t i a l e n ergies. AE , f o r hindered r o t a t i o n i n sub-s t i t u t e d N,N-dimethyl amides are given i n Table 4.9 along w i t h the N-C TT-bond orders f o r the p l a n a r ground s t a t e s and the C-0 TP-bond orders G T f o r the ground and r o t a t i o n t r a n s i t i o n s t a t e s , P Q - Q a n <^ ^C -0' R E S P E C T ^ _ v e l y . These bond orders are def i n e d i n terms of the LCAO-MO expansion c o e f f i c i e n t s as : p = 2 I C. C. , (4.2.8) where <f> and cj) are 2P -AO's on bonded atoms and the summation i s over \i v Zi the occupied rr molecular o r b i t a l s . A corresponding TT-charge d e n s i t y i s d e f i n e d as: p = 2 T, C. C. , (4.2.9) l W j 1U JU J and these d e n s i t i e s f o r the N atom i n the amide ground s t a t e s , q^, are al s o l i s t e d i n Table 4.9. As the NMR spectrum f o r N,N-dimethyl urea shows a s i n g l e sharp peak f o r the N-methyl protons down to -118°C, the f r e e energy of a c t i v a t i o n f o r hindered r o t a t i o n i n t h i s amide may be estimated as 3 k c a l . mole 1 at 298°K. The c o r r e l a t i o n diagram f o r f r e e energy of a c t i v a t i o n , # AG , and Huckel MO d i f f e r e n t i a l fr-energy, AE^, given as F i g . 4.14 i n c l u d e s the estimated energy f o r X=NH2 as an experimental p o i n t and a l s o a p r e l i m i n a r y value of AG = 20.6 + 0.8 kca l . m o l e " 1 f o r N,N-dimethyl carbamyl cyanide (X=CN). The c o r r e l a t i o n obtained i s e x c e l l e n t , which 22 18 o E i u < m H 3 C v \ / N - - C H 3 C X0 X SCN CN N 3 NC5 F Cl Br OCHc NCO 6 o' OCH< NH-N(CH 3) : 0.60 0.65 0/70 AE 0.75 F i g . 4.14 C o r r e l a t i o n of f r e e energy of a c t i v a t i o n f o r hindered r o t a t i o n w i t h Huckel M0 d i f f e r e n t i a l u-energy 137. shows th a t the method of p a r a m e t e r i s a t i o n i s c o n s i s t e n t f o r such a s e r i e s of compounds and a l s o i n d i c a t e s t h a t the change i n Tr-energy on b r e a k i n g down the conjugation through the N-C bond may be a dominant f a c t o r i n determining the b a r r i e r t o hindered r o t a t i o n about t h i s bond. I t i s of some i n t e r e s t to consider i n more d e t a i l the c a l c u l a t e d d i f f e r e n t i a l Tr-energy f o r methyl N,N-dimethyl carbamate (X=0CHg), as e x p e r i m e n t a l l y i t was found that the b a r r i e r to hindered r o t a t i o n i n c r e a s e d by at l e a s t 8 k c a l s . mole 1 w i t h chloroform as a s o l v e n t , c f . s e c t i o n 4.1.3. With the h Y and k values l i s t e d i n Table 4.9, AE ' i s c a l c u l a t e d as 0.639 3° which corresponds to a AG value o f 5.8 k c a l . mole l, as shown by the open p o i n t i n F i g . 4.14, which i s c o n s i s t e n t w i t h the experimental r e s u l t s . Now i f i t i s p o s t u l a t e d t h a t the chloroform forms a s t e r e o - s p e c i f i c hydrogen-bond w i t h the carbonyl lone p a i r a - e l e c t r o n s , t h i s i n t e r a c t i o n would modify the Coulomb i n t e g r a l f o r the carbonyl oxygen atom and hence the Huckel parameter h^. Such a s o l u t e - s o l v e n t i n t e r a c t i o n would be expected to i n c r e a s e the c o n t r i b u t i o n from the resonance form I I shown below i n the carbamate ground s t a t e , and thus t h i s i n t e r a c t i o n i s d e s c r i b e d by an increase i n the magnitude of h . N-C-X <—>- N=C-X ( i ) ( I D The e f f e c t of a v a r i a b l e h^, w i t h a l l other parameters as l i s t e d i n Table 4.9, i s shown i n F i g . 4.15; and i t i s seen t h a t the h. value 0.721 0.70 LU < H 3 C x N / 0 C H 3 N - l - C H 3C / \ Ox w 0 I 0\ o JO 'O \ o' / o'o' .O OO / / ^ / / 0.66-P o h0 / / 0.64 / o/ o ' / / / / / R N-C c=o 0.62-1.4 0.68 0.24 1.8 0.70 0.32 2.2 0.72 0.40 2.6 0.74 0.48 h R 0 R N-C OO F i g . 4.15 E f f e c t s o f v a r i a b l e carbonyl oxygen Huckel theory Coulomb i n t e g r a l , h , f o r methyl N,N-dimethyl carbamate c a l c u l a t i o n 138. # corresponding to AE = 0.705 3 ° and hence AG =15.2 kcal.mole , as TT shown by the c l o s e d p o i n t f o r X=0CH3 i n F i g . 4.14, must be increased to 2.6. Thus although the simple Huckel TT-MO model p r e d i c t s an i n -creased b a r r i e r to r o t a t i o n f o r the s p e c i f i c i n t e r a c t i o n at the carbonyl oxygen, the change i n the h^ parameter r e q u i r e d to reproduce the mag-nitud e of the increase i n AG*, as determined e x p e r i m e n t a l l y i n s e c t i o n 4.1.3, i s r e l a t i v e l y l a r g e . I t i s a l s o p o s s i b l e that the b a r r i e r to r o t a t i o n about the N-C bond may be a f f e c t e d by the chloroform forming a hydrogen bond w i t h the methoxy oxygen lone p a i r o - e l e c t r o n s . Such an i n t e r a c t i o n may lead to an i n c r e a s e d c o n t r i b u t i o n from the resonance form I I I shown below, and i s d e s c r i b e d i n terms of the simple Huckel TT-MO "1 /> + • N - G - X -*—*. N=G=0 X . . . II (.4.2.11) 0 ( I I I ) model by an i n c r e a s e i n the magnitude of h., f o r X=OCH3. Thus AE i s J • X 5 TT c a l c u l a t e d as 0.702 3 ° f o r h = 1.95, as compared w i t h 0.65 i n Table 4.9, A corresponding to AG = 15 k c a l s . mole. 1. An increase i n the b a r r i e r to r o t a t i o n i s p r e d i c t e d as a consequence of the s p e c i f i c s o l u t e - s o l v e n t i n t e r a c t i o n at the methoxy oxygen atom, but again the change i n the h Y parameter r e q u i r e d i s r e l a t i v e l y l a r g e . In so f a r t h a t the increment i n h Q r e q u i r e d to give a c a l c u l a t e d AE^ ^ 0.71 3 ° i s much l e s s than the corresponding increment i n h Y , a s p e c i f i c i n t e r a c t i o n at the carbonyl atom i s p r e d i c t e d to have a more s i g n i f i c a n t e f f e c t on the b a r r i e r to r o t a t i o n . This i s c o n s i s t e n t w i t h the i n t e r a c t i o n e f f e c t being t r a n s -f e r r e d through the carbonyl Tr-bond, t h i s bond being described by 139. k _ = 0.9 as compared w i t h k „„ =0.5. Of course, i f the i n t e r a c t i o n between s o l u t e molecule and 0 atom i s only described by a r e l a t i v e l y l a r ge change i n h^, the value of k^_^ should a l s o be v a r i e d and thus the commensurate change i n h^ w i l l be reduced. From the c o r r e l a t i o n diagram given i n F i g . 4.14 the f r e e energies of a c t i v a t i o n f o r hindered r o t a t i o n i n N,N-dimethyl amides w i t h pseudo-halogen s u b s t i t u e n t s X = SCN, NCS, N3 and NCO, and a l s o i n the symme-t r i c a l t e t r a m e t h y l urea (X = N(CH 3 ) 2 ) may be p r e d i c t e d from the c a l -c u l a t e d AE va l u e s . These AG values are i n c l u d e d i n Table 4.9 and TT are shown as open p o i n t s i n F i g . 4.14, The AG - AE c o r r e l a t i o n i n d i c a t e s that as the e l e c t r o - n e g a t i v i t y o f the s u b s t i t u e n t i n c r e a s e s , the b a r r i e r to hindered r o t a t i o n a l s o i n c r e a s e s . This i s c o n s i s t e n t w i t h an inc r e a s e d c o n t r i b u t i o n from the resonance form IV shown below corresponding to a decreased b a r r i e r , as a d e l o c a l i s a t i o n of u-electrons — NT* (4.2.12) 0^ 0 (IV) on the atom or pseudo-atom s u b s t i t u e n t group X i s p o s s i b l e f o r a l l o f the s u b s t i t u e n t s considered here. As shown i n F i g . 4.16, however, there i s no simple c o r r e l a t i o n between the free energies o f a c t i v a t i o n AG , as measured or p r e d i c t e d i n accordance w i t h the Huckel TT-MO model, w i t h the atomic or group e l e c t r o - n e g a t i v i t i e s £ Y. Thus the use of even a s i m p l i f i e d LCA0-M0 model f o r the d e s c r i p t i o n of hindered r o t a t i o n s i n amides i s j u s t i f i e d i f a s e m i - q u a n t i t a t i v e c o r r e l a t i o n i s to be attempted us i n g r e l i a b l e experimental data. The s e l f - c o n s i s t e n c y of the 22 -18 1 QJ O E u # 14 10 © / I © / / / / I ' / I / I / / t I / / X SCN CN N 3 NCS F Cl Br OCH. NCO 6 O C I i 2 / NH. N(CH 3 ) 2 T r 1 r-2.4 3.2 4.0 4.8 X F i g . 4.16 C o r r e l a t i o n o f f r e e energy of a c t i v a t i o n f o r hindered r o t a t i o n w i t h group e l e c t r o - n e g a t i v i t y , e , of the X- s u b s t i t u e n t i n N,N-dimethyl amides' 140. Huckel c a l c u l a t i o n i s i l l u s t r a t e d by the trends i n the ir-bond orders and ir-charge d e n s i t i e s l i s t e d i n Table 4.9. As shown i n F i g . 4.17, an e x c e l l e n t c o r r e l a t i o n i s a l s o obtained between AG and the N-C Tf-bond order P J ^ Q - The c a l c u l a t e d bond-orders p^ ^  are c o n s i s t e n t l y h i g h e r than Q those f o r the carbonyl bond i n the amido ground s t a t e , VQ-Q> but t h i s i s o n l y a consequence of the p a r t i c u l a r p a r a m e t e r i s a t i o n chosen f o r the N atom and does not a f f e c t any of the r e s u l t s presented f o r the r e l a t i v e e f f e c t s o f the X s u b s t i t u e n t s . As the simple Huckel TT-MO model de s c r i b e d above gives a con-s i s t e n t estimate o f the d i f f e r e n t i a l energy AE^ f o r the ground and hindered r o t a t i o n t r a n s i t i o n s t a t e s f o r s u b s t i t u t e d N,N-dimethyl amides, such a model may be u s e f u l i n a more general d e s c r i p t i o n of TT-bonding w i t h i n a conjugated amide system. In general terms, the d e t a i l s of an e l e c t r o n i c charge d i s t r i b u t i o n f o r any given molecule are contained i n the expansion c o e f f i c i e n t s f o r the L C A O molecular o r b i t a l s , c f . 191 Eq. (4.2.1). The p r o b a b i l i t y d e n s i t y , D, at a p o i n t i n space a s s o c i a t e d w i t h an e l e c t r o n i n a molecular o r b i t a l ii. i s and 3 3 3 hence i n accordance w i t h Eq. (4.2.1): D = 2 P 4>*<j> + I I P.J>*<J> > ( 4 - 2 . 1 3 ) where p and p are defined f o r doubly occupied o r b i t a l s i n Eqs. (4.2.8) and (4.2.9), r e s p e c t i v e l y . Thus the bond order p i s the predominant f a c t o r determining the d i s t r i b u t i o n o f Tr-electron charge i n the space between the bonded atoms w i t h 2P Z-A0's c|> and cj>^  . Although an approximate a n a l y s i s i s p o s s i b l e u s i n g bond o r d e r s , an e l e c t r o n i c O E # O <] 22 -18 1 n 6 2 -HoC 3 \ X H 3C / N--C / ¥ x o 9 i © 9 © 0.66 0.70 0.74 P M -N-C X S C N CN N 3 NCS F Cl Br .OCH, NCO OCHo NH N ( C H 3 ) 2 0.78 F i g . 4.17 C o r r e l a t i o n o f f r e e energy of a c t i v a t i o n f o r hindered r o t a t i o n w i t h N-C TT-bond order, p , obtained from Huckel MO c a l c u l a t i o n s . 141. 192 193 charge d e n s i t y map ' i s the most e x p l i c i t and complete means of showing the form o f the 'rr-charge d i s t r i b u t i o n i n a conjugated amide system. A FORTRAN-IV computer program CNTR has been developed to a u t o m a t i c a l l y p l o t contours f o r constant n-charge d e n s i t i e s by d i r e c t . . 194 numerical c a l c u l a t i o n of the p r o b a b i l i t y d e n s i t y D usi n g S l a t e r type atomic o r b i t a l s . A t o t a l charge d e n s i t y i s obtained as a normalised sum o f c o n t r i b u t i o n s from a l l of the occupied molecular o r b i t a l s . The Huckel u-MO e l e c t r o n i c charge d e n s i t y maps obtained f o r (a) an uncon-jugated s t a t e and (b) the conjugated ground s t a t e o f formamide (or N,N-dimethyl formamide) as de f i n e d by the molecular o r b i t a l s generated u s i n g the parameters l i s t e d i n Table 4.9 are shown i n F i g . 4.18, f o r the o plane and d i s p l a c e d by 0.6 A. Comparison o f these d e n s i t y maps shows the r e d i s t r i b u t i o n o f charge i n both the N-C and C=0 bonding regions i n the conjugated ground s t a t e , the formal lone p a i r e l e c t r o n d e n s i t y f o r the N atom being i l l u s t r a t e d i n F i g . 4.18 ( a ) . The change i n the Q T-bond order P Q _ Q on forming the un-conjugated hindered r o t a t i o n t r a n s -T i t i o n s t a t e , as given by VQ-Q> 1 S r e f l e c t e d i n the charge d e n s i t y maps as the double bond c h a r a c t e r i n c r e a s e s i n the t r a n s i t i o n s t a t e and t h i s i s shown to correspond to an a d d i t i o n a l d e l o c a l i s a t i o n of charge from the carbonyl oxygen atom. This i s a l s o c o n s i s t e n t w i t h a decreased c o n t r i b u t i o n from the resonance form I I given i n (4.2.10). S i m i l a r d e n s i t y maps f o r carbamyl f l u o r i d e are shown i n F i g . 4.19 f o r the geo-metry given i n F i g . 4.22. In the conjugated ground s t a t e f o r t h i s p a r t i c u l a r molecule the Tr-charge d e n s i t y i s c l o s e to being symmetrical (a) 7 ' / • s / / //,-;. I ' M V V > 1 1/ i 1 w v \\ \ \ \ \ V / N \ S V 1 \ V N \ \ \ x \ \ N s v © 'In; i / / / (b) , — „ / ^ ^ ^ \ / / 7 / 'V~"^N ^ — \ \ \ \ M ® :> !/; . v\--v//// F i g . 4.18 Huckel MO e l e c t r o n i c charge d e n s i t y maps f o r the (a) un-conjugated and (b) conjugated s t a t e s o f formamide (a) / ft// • ^  N. I • / f . V , / , / / ' / ' i ' i ; : 1 1 1 ! ' I ------ '-xV \/\ V - / , > x \ ( V x \ \ \ \ \ ^ / / ' I I I " \ \\\^:-''// i f \ V >. - x ' / \ N " - - ^ / . G O III,''(?){],,)) I f ',' 1 1 i : / / / { ' ' ' I l l \ s \ P i g . 4.19 Huckel MO e l e c t r o n i c charge d e n s i t y maps f o r the (a) un-conjugated and (b) conjugated s t a t e s of carbamyl f l u o r i d e 142. with respect to the N-C bond and t h i s f e a t u r e may be the dominant f a c t o r l e a d i n g to very s i m i l a r NMR s h i e l d i n g r e g i o n s , due to the d i a -magnetic a n i s o t r o p i c s of the C-F and C=0 groups, f o r the methyl groups i n NjN-dimethyl carbamyl f l u o r i d e , c f . s e c t i o n 4.1.4. Comparison of the contours f o r the same charge d e n s i t y i n F i g s . 4.18(b) and 4.19(b) shows' the s i g n i f i c a n t l y d i f f e r e n t charge d i s t r i b u t i o n s i n the N-C bond re g i o n f o r formamide and carbamyl f l u o r i d e , r e s p e c t i v e l y . These d e n s i t y maps are c o n s i s t e n t w i t h an i n c r e a s e d bond order p^ ^ and a c o r r e s p o n d i n g l y higher b a r r i e r f o r hindered r o t a t i o n i n formamide, c f , Table 4.9. 4.3 Semi-empirical SCF-LCAO-MO c a l c u l a t i o n s The e l e c t r o n i c s t r u c t u r e of formamide, the s i m p l e s t molecule c o n t a i n i n g the amide group N-C=0, i s of b a s i c importance i n any spec-t r o s c o p i c study of a s e r i e s of s u b s t i t u t e d amides. The amide l i n k a g e i s a l s o c h a r a c t e r i s t i c of the p o l y p e p t i d e s and hence a d e t a i l e d molecular o r b i t a l c a l c u l a t i o n f o r formamide gives a b a s i s f o r the development o f a c o n f i g u r a t i o n a l model f o r these more complicated systems of b i o l o g i c a l i n t e r e s t . The hindered r o t a t i o n about the N-C bond i n 1 5N-formamide has been s t u d i e d e x p e r i m e n t a l l y i n s e c t i o n 4.1.5, and s i m i l a r r o t a t i o n s i n s u b s t i t u t e d amides have been e x t e n s i v e l y s t u d i e d by NMR methods as d i s c ussed i n s e c t i o n s 4.1.1 to 4.1.4 and as summarised i n a recent 195 review . However, very l i t t l e a t t e n t i o n has been p a i d to a t h e o r e t i c a l determination of the form of the t r a n s i t i o n s t a t e s or to a semi-q u a n t i t a t i v e c o r r e l a t i o n of the measured f r e e energies of a c t i v a t i o n w i t h data a v a i l a b l e from a molecular o r b i t a l d e s c r i p t i o n of s u b s t i t u t e d 1 4 3 . amides. E m p i r i c a l estimates o f the b a r r i e r to r o t a t i o n have been made f o r formamide*^ ' and simple. Huckel "rr-MO c a l c u l a t i o n s f o r a s e r i e s of s u b s t i t u t e d amides have been o u t l i n e d i n the preceding s e c t i o n o f t h i s t h e s i s ; but very r e c e n t l y , C hristensen et a l . have published a 196 f u l l ab i n i t i o molecular o r b i t a l study o f formamide i n the ground and hindered r o t a t i o n t r a n s i t i o n s t a t e s . Thus i t i s of p a r t i c u l a r i n t e r e s t to supplement these r e s u l t s w i t h a d e t a i l e d s e m i - e m p i r i c a l SCF-LCAO-MO c a l c u l a t i o n f o r the hindered r o t a t i o n i n formamide, i n the 7 S 76 CNDO/2 approximation " ' . As p r e l i m i n a r y CNDO/2 c a l c u l a t i o n s showed th a t the ground-197 s t a t e geometry proposed by Co s t a i n was l e s s s t a b l e than a correspon-198 ding p l a n a r c o n f i g u r a t i o n , the b a s i c s t r u c t u r e determined by Kurland , and shown i n F i g . 4.20, i s used i n the f o l l o w i n g c a l c u l a t i o n s . By i n c l u d i n g d-functions i n a Gaussian type o r b i t a l b a s i s s e t , the ab 196 i n i t i o c a l c u l a t i o n i n d i c a t e s t h a t the non-planar s t r u c t u r e may be more s t a b l e , but the very small energy d i f f e r e n c e precludes a co n c l u s i v e r e s u l t . The geometry of the N-C"^ group i s f i x e d i n the molecular xy-plane i n F i g . 4.20, wh i l e a v a r i a b l e geometry f o r the amino group may be d e f i n e d by the HNH angle 2a, the d i h e d r a l angle 6 and the angle <j) determining the r o t a t i o n about the N-C bond. A l l bond lengths i n ° °198 F i g . 4.20 are given i n A, and the N-H-bond length i s f i x e d at 0.995 A To date, the most r e l i a b l e MO c a l c u l a t i o n s are based upon 199 a f u l l Hartree-Fock s e l f - c o n s i s t e n t f i e l d (SCF) theory , i n v o l v i n g a l l e l e c t r o n s i n a given molecular system, an u n l i m i t e d b a s i s set of atomic wavefunctions and an e x p l i c i t f o r m u l a t i o n of a l l e l e c t r o n i n t e r a c t i o n s . Roothaan^^^ considered a f i n i t e b a s i s set o f orthonormal atomic o r b i t a l s , 4.20 S t r u c t u r e of formamide used i n CNDO/2 SCF-LCAO-MO c a l c u l a t i o n s 144. (fy, and modified the f u l l SCF theory to give a more t r a c t a b l e f o r m u l a t i o n based upon a l i n e a r combination of atomic o r b i t a l s (LCAO) i n an approximate d e s c r i p t i o n of molecular o r b i t a l s , c f . Eq. (4.2.1). The a p p l i c a t i o n o f the Roothaan method, however, i s s t i l l l i m i t e d by computational complexity and hence, i n ge n e r a l , i t i s necessary to consider f u r t h e r approximations w i t h i n the context o f the SCF theory. The CNDO method, as r e c e n t l y developed by Pople and coworkers, i s a semi-e m p i r i c a l SCF-LCAO-MO theory.which t r e a t s only valence e l e c t r o n s e x p l i c i t l y and s i m p l i f i e s the b a s i c Roothaan equation by use of the complete n e g l e c t o f d i f f e r e n t i a l overlap (CNDO) approximation. The Roothaan SCF equation then reduces to the form where F_ i s the Fock Hamiltonian matrix and i s the column v e c t o r of MO expansion c o e f f i c i e n t s C ^ as given i n Eq. (4.2.1). E^ i s the energy eigen-value corresponding to the MO I J K . Overlap between atomic o r b i t a l s i s only i n c l u d e d i n the e v a l u a t i o n of c e r t a i n bonding i n t e g r a l s i n v o l v e d i n t h i s approximate SCF theory, but i n t e g r a l s d e s c r i b i n g two-electron r e p u l s i o n s are i n c l u d e d i n the Fock matrix elements F . The fundamental yv assumptions used i n the d e r i v a t i o n of the Fock matri x elements have 75 76 been described i n d e t a i l by Pople , and i n the CNDO/2 approximation these matrix elements are given as - P.P. (A % 1 ~l F yy = - h(i + A ) + [(P y y- L V AA ^(P - 1)]Y *yy 1 J \ AA 145. and F = ktf°. + 3°) S -hp Y»„, J. , (4.3.2) yv ^ A yv • -^ yv AB y+v ' ^ J f o r valence s h e l l atomic ( S l a t e r ) o r b i t a l s cj) and <j)^  on atoms A and B, r e s p e c t i v e l y . In the above f o r m u l a t i o n , one- and two-centre e l e c t r o n r e p u l s i o n i n t e g r a l s are r e t a i n e d w h i l e p e n e t r a t i o n i n t e g r a l s are excluded. The non-zero two e l e c t r o n i n t e g r a l s , y ^ and Y^g> are only dependent upon the atoms w i t h which the o r b i t a l s (j) and <j>^  are a s s o c i a t e d and not upon the type of atomic o r b i t a l . Thus the atomic i n t e g r a l s y^g represent an average i n t e r a c t i o n between an e l e c t r o n i n a valence AO on atom A and another e l e c t r o n i n a valence o r b i t a l on atom B, and such 201 i n t e g r a l s are r e a d i l y evaluated u s i n g formulae d e r i v e d by Roothaan The charge d e n s i t i e s and bond-orders i n Eq. (4.3.2), namely p and p , are those p r e v i o u s l y defined i n Eqs. (4.2.7) and (4.2.8); and i s the t o t a l valence e l e c t r o n d e n s i t y f o r atom A: P A A = j j . P W ' ( 4 ' 3 - 3 ) where Z. i s the core charge f o r atom A. S i s the overlap i n t e g r a l f o r A b yv atomic o r b i t a l s <j> and 4>^  and may be evaluated u s i n g the a n a l y t i c ] 88 expressions d e r i v e d by M u l l i k e n et a l f o r S l a t e r o r b i t a l s . The 76 s e m i - e m p i r i c a l bonding parameters, 3^ , and the parameters h(l^+ A ) f o r 2s- and 2p- atomic o r b i t a l s determining the atomic core Hamiltonian 75 matrix elements as used i n the f o l l o w i n g c a l c u l a t i o n s are l i s t e d i n Table 4.10 along w i t h the S l a t e r exponents C . A FORTRAN-IV computer program s i m i l a r to that a v a i l a b l e from the Quantum Chemistry Program Table 4.10 Parameters f o r CNDO/2 c a l c u l a t i o n s H N 0 h{i + A ) v s s h(i + A ) p p -3 A 1.20 7.186 9.0 1.625 14.051 5.572 21.0 1.95 19.316 7. 275 25.0 2.275 25.390 9.111 31.0 F 2.60 32.272 eV 11.08 eV 39.0 eV 146. Exchange has been developed f o r use on an IBM 360 system and a l l r e s u l t s reported here have been obtained u s i n g t h i s program. I n i t i a l l y , the hindered r o t a t i o n was considered as a simple 1^8 r o t a t i o n of the p l a n a r NH 2 group w i t h f i x e d N-H bond length and 2a = 119° " so that the t r a n s i t i o n s t a t e corresponds to 6 = 0° and cb = 90° ( s t r u c t u r e I I ) i n F i g . 4.20. The p l a n a r ground s t a t e i s then defined by 6 = 0° and cf> = 0° ( s t r u c t u r e I) , and the a s s o c i a t e d SCF-LCAO-MO data i s summarised i n Table 4.11. The net atomic e l e c t r o n i c charge (a + IT) f o r the N atom, q^, i s shown to in c r e a s e on forming the t r a n s i t i o n s t a t e ; w h i l e the corresponding charge f o r the 0 atom, q^, decreases. These v a r i a t i o n s are a l s o r e f l e c t e d i n the net atomic TT-charges which are de-TT TT O C C f i n e d as q. = Z. - 2 £ C. , where cb i s a 2P atomic o r b i t a l on atom nA A ^ I U u z TT A w i t h TT-core charge Z. (or a 2P o r b i t a l f o r N i n the t r a n s i t i o n A y st a t e ) and C. i s the a s s o c i a t e d expansion c o e f f i c i e n t f o r the i t h 1 U TT occupied orthonormal LCAO molecular o r b i t a l . Table 4.11 shows th a t q^ TT IT i s approximately constant w h i l e q^ and q^ d i f f e r w idely i n the ground and t r a n s i t i o n s t a t e s . This i s c o n s i s t e n t w i t h an increased c o n t r i b u t i o n from the resonance form N = C - 0 f o r the Tr-system i n the p l a n a r ground s t a t e , i n a s i m p l i f i e d d e s c r i p t i o n o f the amide group, c f . s e c t i o n 4.2. occ TT The N-C TT-bond order, given as p., _ = 2 E C. C. with <j) and <f> 2P N-C I U i v u v z i atomic o r b i t a l s on the N and C atoms, i s c a l c u l a t e d as 0.482 i n the ground-s t a t e and hence represents a s i g n i f i c a n t d e l o c a l i s a t i o n of the N TT-electrons. For comparison, t h i s bond-order i s determined as 0.41 202 u s i n g the Kurland bond-length data i n a formula given by P a u l i n g , 203 and as 0.63 usin g an a l t e r n a t i v e e m p i r i c a l equation proposed by Gordy In the t r a n s i t i o n s t a t e , conjugation between the N P ^ - o r b i t a l and the Table 4.1:. CNDO/2 MO Data f o r Formamide a) a) StrUC- TT TT T T T f T T „ ,-. , , t u r e qN qC qO % qC qO PN-C PC-0 E e E t o t a l m I 0 = 0°, 2 a- 119°, cb = 0° r A ' O O O N / n " r ? ? > r ~ n ' + 0 . 1 8 9 +0.229 -0.418 0.482 0.845 -79.5883 -39.3033 5.92 C-0.292) (+0.561) (-0.477) (4.15) I I 0 = 0 % 2a = 119°, <j) = 90° -0.313 +0.355 -0.262 +0.037 +0.206 -0.263 0.272 0.931 -79.5031 -39.2718 2.96 I I I 6 = 0°, 2a = 117.5°,<J> = 0° -0.236 +0.364 -0.335 +0.190 +0.229-0.419 0.483 0.845-79.5730 -39.3040 3.95 IV 9 = 55°, 2a = 110°, • cb = 90° -0.260 +0.335 -0.273 -- +0.212 -0.283 0.282 0.924 -79.9207 -39.2S83 1.45 V 6 = 55°, 2a = 110°, <j>. =-90° -0.249 +0.343 -0.261 -- +0.213 -0.284 0.284 0.925 -79.6962 -39.2879 4.24 a) E l e c t r o n i c only energy, E s , and t o t a l energy, E , given i n atomic u n i t s b) D i p o l e moment components :ln Debyes 1 4 7 . C = 0 IT-system i s not p o s s i b l e and P ^ _ Q decreases to 0 . 2 7 2 ( t h i s bond-order r e p r e s e n t i n g an e l e c t r o n i c charge overlap between the carbonyl TT-system and the N h y b r i d a-bonding system) w i t h a corresponding i n c r e a s e TT i n P(-_Q J c o n s i s t e n t w i t h an increased c o n t r i b u t i o n from the resonance form N - C = 0 i n t h i s s t a t e . In g e n e r a l , e l e c t r o n i c charge d i s t r i b u -2 0 4 t i o n s are s u c c e s s f u l l y p r e d i c t e d i n the C N D O / 2 approximation and thus i t i s of i n t e r e s t to compare the above ground-state data w i t h that 1 9 6 a v a i l a b l e from the ab i n i t i o • c a l c u l a t i o n without d - f u n c t i o n s , as given i n parentheses i n Table 4 . 1 1 . The r e l a t i v e net atomic charges are i n good agreement, although the C N D O / 2 values are c o n s i s t e n t l y s m a l l e r i n magnitude; but i t i s to be noted t h a t , on i n c l u s i o n of d-functions there are s i g n i f i c a n t changes and the ab i n i t i o q^ value i s given as - 0 . 5 8 4 while q^ and q^' become + 0 . 4 9 8 and - 0 . 4 7 9 , r e s p e c t i v e l y , i l l u s t r a t i n g the s e n s i t i v i t y to the chosen b a s i s set of atomic o r b i t a l s . The net charges f o r the amino hydrogen atoms c i s and trans to the C t carbonyl oxygen i n s t r u c t u r e I , q^ and q , are c a l c u l a t e d as + 0 . 1 3 1 and + 0 . 1 2 4 , r e s p e c t i v e l y , which may be compared w i t h those from the ab i n i t i o c a l c u l a t i o n : + 0 . 1 7 7 and + 0 . 1 6 0 . Thus these charge d e n s i t i e s are s i g n i -i i c a n t l y d i f f e r e n t i n the ground s t a t e and both c a l c u l a t i o n s i n d i c a t e c . t t h a t q^ i s more p o s i t i v e than q^. Therefore, f o r an i s o l a t e d formamide molecule, i n the absence of through-space e f f e c t s from the m a g n e t i c a l l y a n i s o t r o p i c C = 0 and C - H bonds the c i s proton i s l e s s s h i e l d e d and hence would resonate to low f i e l d of the trans proton i n an N M R e x p e r i -ment, assuming the diamagnetic c o n t r i b u t i o n to the proton chemical s h i f t to be predominant. In view of the current i n t e r e s t i n these chemical 2 0 5 s h i f t s t h i s i s a p a r t i c u l a r l y i n t e r e s t i n g r e s u l t . 148. The complete CNDO/2 ground s t a t e d i p o l e moment has a mag-nitude | y | = 3.92 D and i s o r i e n t e d at 3 = 41° wi t h respect to the 198 N - C bond, i n good agremement with the experimental data : | y | = 3.71 + 0.06 D and 3 = 39.6°, whereas the ab i n i t i o c a l c u l a t i o n leads to a higher d i p o l e magnitude and an increased o r i e n t a t i o n angle. In the CNDO/2. approximation, d i p o l e i n t e g r a l s i n v o l v i n g the product o f two atomic o r b i t a l s on the same atom are r e t a i n e d i n the c a l c u l a t i o n of 204 a molecular e l e c t r i c d i p o l e . Thus the d i p o l e a s s o c i a t e d w i t h net charges at nu c l e a r centres f o r the ground s t a t e of formamide i s des-c r i b e d by y = 1.90 and y = 1.42 D, and the c o n t r i b u t i o n due to 1 x y asymmetry of e l e c t r o n i c charge about these centres gives y _ = -1.06 and y^ = 1.15 D. This shows the importance of i n c l u d i n g the asymmetry c o n t r i b u t i o n , which i s e s s e n t i a l l y a h y b r i d i s a t i o n term, i n the d i p o l e c a l c u l a t i o n f o r a p o l a r molecule. The b a r r i e r to hindered r o t a t i o n i n formamide i s simply the d i f f e r e n c e i n t o t a l energy ( i n c l u d i n g n u c l e a r r e p u l s i o n s ) between the ground and t r a n s i t i o n s t a t e s , - E , and f o r s t r u c t u r e s I and I I i s t g' c a l c u l a t e d as 19.8 k c a l . mole . This energy may be compared d i r e c t l y w i t h the free energy of a c t i v a t i o n f o r t h i s r o t a t i o n , G^ , determined by a t o t a l lineshape a n a l y s i s of NMR data f o r d i l u t e s o l u t i o n s o f if N 1 5-formamide: AG = 1 8 . 0 + 0 . 4 k c a l . mole (10 mole % i n acetone) at 25°C, c f . s e c t i o n 4.1.5] and a l s o w i t h 19.9 k c a l . mole 1 from the ab i n i t i o c a l c u l a t i o n * 9 ^ and 20.1 k c a l . mole 1 obtained i n an independent 206 CNDO c a l c u l a t i o n by Scheraga and coworkers . Of course, the t o t a l energies d e r i v e d i n the CNDO/2 approximation are f a r removed from the Hartree-Fock SCF l i m i t and hence, i n such a semi - e m p i r i c a l SCF 149. c a l c u l a t i o n , i t must be assumed that the inherent e r r o r s i n v o l v e d i n an energy c a l c u l a t i o n cancel out i n the energy d i f f e r e n c e taken f o r any two c o n f i g u r a t i o n s of a given molecule. Such an assumption appears to be v i n d i c a t e d i n the cl o s e agreement o f the t h e o r e t i c a l - E t g values quoted above, i n that a CNDO/2 t o t a l energy corresponds to only about 25% of tha t i n v o l v e d i n an ab i n i t i o c a l c u l a t i o n . In g e n e r a l , geometrical changes i n a molecule may occur i n the process o f i n t e r n a l r o t a t i o n ; and i n p a r t i c u l a r , a change i n h y b r i d i s a t i o n at the N atom i n formamide may be expected to have a s i g n i f i c a n t e f f e c t upon the b a r r i e r to r o t a t i o n about the N-C bond. Such a change i n h y b r i d i s a t i o n would be a s s o c i a t e d w i t h the breakdown o f conjugation between the formal N l o n e - p a i r u - e l e c t r o n s and the carbonyl ir-system i n the t r a n s i t i o n s t a t e . Although d e t a i l e d geometrical 196 changes were considered by Christensen et a l . , a l l c a l c u l a t i o n s were r e s t r i c t e d to a pl a n a r 9 = 0 ° ( i n F i g . 4.20) c o n f i g u r a t i o n f o r the NH 2 group. A CNDO/2 energy m i n i m i s a t i o n , through a change i n bond angle o n l y , was checked by showing t h a t the minimum energy c o n f i g u r a t i o n f o r a p l a n a r ground s t a t e corresponds to 2a = 117.5° ( s t r u c t u r e I I I ) , which 198 i s i n good agreement w i t h the experimental value of 119° . The SCF-LCAO-MO data f o r t h i s s t r u c t u r e i s summarised i n Table 4.11, and i t i s seen that the t o t a l energy, E , i s lowered by only 0.44 k c a l . mole 1 from t h a t f o r s t r u c t u r e I. The t r a n s i t i o n s t a t e geometry about the n i t r o g e n atom was then considered f o r <j) = 90° ( F i g . 4.20) i n terms of v a r i a t i o n s of the d i h e d r a l angle 6 i n the range 25 - 75° and the HNH bond angle 2a i n the range 100 - 120°. The c a l c u l a t e d t o t a l energy d i f f e r e n c e s , E = E , are p l o t t e d i n F i g . 4.21, where i t i s shown that the minimum 18 o 6 2 lo 16 o • w o * u U J 111 12 10 75 65 *0 S o --o~~-/ / / o ' ' ! r i / / / f / 3 5 . .o' 8 1 90 100 110 2a 120 F i g . 4.21 T o t a l energy d i f f e r e n c e s f o r the formamide p l a n a r ground s t a t e and v a r i a b l e geometry t r a n s i t i o n s t a t e 150. energy c o n f i g u r a t i o n c l o s e l y corresponds to 6 = 55° and 2a = 110° ( s t r u c t u r e I V ) . A l l c a l c u l a t e d energies are seen t o l i e on w e l l d e f i n e d energy curves which are not, however, n e c e s s a r i l y symmetric i n a about the minimum energy p o i n t . The t o t a l energy f o r t h i s s t r u c t u r e i s 10.4 k c a l . mole 1 lower than t h a t f o r the i n i t i a l t r a n s i t i o n s t a t e considered ( s t r u c t u r e I I ) , and the b a r r i e r t o r o t a t i o n c o r r e s -ponding to the minimal energy molecular c o n f i g u r a t i o n s , as determined i n the CNDO/2 approximation, i s now given as 9.86 k c a l . mole J, This energy i s approximately one h a l f o f the e x p e r i m e n t a l l y determined b a r r i e r , and the above c a l c u l a t i o n i l l u s t r a t e s the n e c e s s i t y f o r c a r e f u l geometrical o p t i m i s a t i o n before comparison with experimental data i s attempted. The above c a l c u l a t i o n o f the b a r r i e r to hindered r o t a t i o n i n formamide, and the d e s c r i p t i o n of the a s s o c i a t e d ground and t r a n s i t i o n states., a p p l i e s to an i s o l a t e d molecule; indeed, the c h a r a c t e r i s t i c s of the e l e c t r i c d i p o l e moment f o r t h i s molecule, as determined e x p e r i -m e n t a l l y from measurements i n the gas phase, have been s a t i s f a c t o r i l y reproduced by the approximate SCF-LCAO-MO model. The experimental data a v a i l a b l e f o r the i n t e r n a l r o t a t i o n energy, however, are obtained from measurements i n s o l u t i o n and i t i s expected that s o l u t e - s o l u t e and s o l u t e - s o l v e n t i n t e r a c t i o n s f o r the p o l a r formamide molecule w i l l have a complex e f f e c t upon the hindered r o t a t i o n . Thus, i n g e n e r a l , i t i s only p o s s i b l e to compare t h e o r e t i c a l and experimental energy data f o r a s e r i e s o f s u b s t i t u t e d amides at the same c o n c e n t r a t i o n (or through an e x t r a p o l a t i o n to i n f i n i t e d i l u t i o n ) i n a sol v e n t that minimises 151. i n t e r m o l e c u l a r i n t e r a c t i o n s . Such a c o r r e l a t i o n of r e l a t i v e energies w i l l be presented i n d e t a i l elsewhere. At t h i s p o i n t , i t i s of i n t e r e s t t o consider f u r t h e r the c a l c u l a t e d d i p o l e moments f o r the parent amide--formamide. As shown i n Table 4.11, the t o t a l d i p o l e moment f o r s t r u c t u r e IV has a magnitude of 1.45 D and i s o r i e n t e d at 6° to the N-C bond. A l a r g e c o n t r i b u t i o n i s due to the asymmetry of the N l o n e - p a i r e l e c t r o n i c charge d i s t r i b u t i o n ( a s s o c i a t e d w i t h an s p 3 - t y p e h y b r i d o r b i t a l ) and i s i n o p p o s i t i o n to that from the carbonyl system and thus leads to a decreased t o t a l moment as compared with t h a t f o r the p l a n a r ground s t a t e , s t r u c t u r e I (or I I I ) . Therefore, i f the s o l u t e - s o l v e n t i n t e r a c t i o n i s e l e c t r o s t a t i c i n form and n o n - s p e c i f i c , the s o l v a t i o n s t a b i l i s a t i o n energy i s decreased i n the t r a n s i t i o n s t a t e and hence the enthalpy o f a c t i v a t i o n AI-i'f (or Arrhenius a c t i v a t i o n energy E ) w i l l be g r e a t e r than a # # AG corresponding to a p o s i t i v e entropy of a c t i v a t i o n AS . That i s , the a c t i v a t i o n parameters AH* and AS* are s e n s i t i v e to i n t e r m o l e c u l a r # i n t e r a c t i o n s , where AG i s predominantly determined by i n t r a m o l e c u l a r i n t e r a c t i o n s due to an e f f e c t i v e compensation of AH and AS i n the thermodynamic r e l a t i o n s h i p AG* = AH* - TAS*. So f a r , i t has been assumed that the amino hydrogens are c i s to the carbonyl oxygen i n the t r a n s i t i o n s t a t e , <£> = 90° i n F i g . 4.20. A v a r i a t i o n a l c a l c u l a t i o n , s i m i l a r to that already d e s c r i b e d , f o r these hydrogen atoms trans to the carbonyl oxygen (ct = - 90°) shows that the minimum energy con-f i g u r a t i o n again corresponds to 0 = 55° and 2a = 110° ( s t r u c t u r e V) and the t o t a l energy i s only 0.25 k c a l . mole 1 higher than t h a t f o r s t r u c t u r e IV. As shown i n Table 4.11, a l l net atomic charges and bond-orders are very s i m i l a r to those f o r s t r u c t u r e IV, but the c a l c u l a t e d d i p o l e 152. moment now has a magnitude o f 4.24 D and i s o r i e n t e d at 70° to the N-C bond. In t h i s c o n f i g u r a t i o n , the s o l v a t i o n s t a b i l i s a t i o n energy i s i n creased i n the t r a n s i t i o n s t a t e and i f the d i f f e r e n t i a l s o l v a t i o n energy exceeds 0.25 k c a l . mole 1 t h i s p a r t i c u l a r c o n f i g u r a t i o n becomes the p r e f e r r e d t r a n s i t i o n s t a t e f o r a s o l v a t e d formamide molecule, and i n t h i s case AS # i s negative. A d e s c r i p t i o n of s t e r e o - s p e c i f i c s o l u t e -s o l v e n t i n t e r a c t i o n s may r e q u i r e a more elaborate computational model, but the r e l a t i v e ease with which the CNDO/2 SCF approximation can be ap p l i e d may allow a r e l i a b l e and comprehensive d e s c r i p t i o n of the form of these i n t e r m o l e c u l a r i n t e r a c t i o n s and t h e i r e f f e c t upon the hindered r o t a t i o n on formamide, which i s o f fundamental importance i n the study o f the more complex b i o l o g i c a l systems c o n t a i n i n g the amide l i n k a g e . However, i t remains to be able to determine the a c t i v a t i o n # # parameters AH and AS wi t h s u f f i c i e n t p r e c i s i o n to warrant such d e t a i l e d c a l c u l a t i o n s f o r comparison w i t h experimental data. To allow a comparison w i t h the experimental data a v a i l a b l e f o r the hindered r o t a t i o n s i n N,N-dimethyl formamide and N,N-dimethyl carbamyl f l u o r i d e , c a l c u l a t i o n s i n the CNDO/2 approximation s i m i l a r to those described above f o r formamide have a l s o been c a r r i e d out f o r the parent amide carbamyl f l u o r i d e . As t h i s p a r t i c u l a r amide i s unstable at room temperature a s t r u c t u r e has not determined from the microwave 145 spectrum , and hence the b a s i c s t r u c t u r e shown i n F i g . 4.22 has been 20 der i v e d by comparison of the s t r u c t u r a l data a v a i l a b l e f o r acetaldehyde 208 198 a c e t y l f l u o r i d e and formamide . The NCF bond angle o f 112.5° a l s o corresponds to a minimum energy c o n f i g u r a t i o n as determined i n the CNDO/2 approximation. F i g . 4.22 S t r u c t u r e f o r carbamyl f l u o r i d e used i n CNDO/2 SCF-LCAO-MO c a l c u l a t i o n s 153. The minimum energy p l a n a r ground s t a t e c o n f i g u r a t i o n c o r r e s -ponds to 2a = 118° and the t o t a l energy i s given as -66.3134 a.u., R o t a t i o n of the p l a n a r NH2 group to form a hindered r o t a t i o n t r a n s i t i o n s t a t e (cb = 90°) gives a t o t a l energy of -66.2922 a.u. and hence a b a r r i e r to r o t a t i o n of 13.2 k c a l s . mole 1, which i s s i g n i f i c a n t l y lower than that c a l c u l a t e d f o r a s i m i l a r r o t a t i o n i n formamide, c f . s t r u c t u r e I I i n Table 4.11. The t o t a l energies f o r the minimal energy c o n f i g u r a t i o n s corresponding' t o (j) = 90° and cb = -90° are then c a l c u l a t e d as -66.3046 and -66.3062 a.u., r e s p e c t i v e l y . Thus the t r a n s i t i o n s t a t e i n which the amino hydrogen atoms e c l i p s e the f l u o r i n e atom i n carbamyl f l u o r i d e ( cb = -90°), f o r which 0 = 57.5° and 2a = 107.5°, i s more s t a b l e than the a l t e r n a t i v e t r a n s i t i o n s t a t e by 1.1 k c a l . mole 1 and i s t h e r e f o r e the most probable t r a n s i t i o n s t a t e i n s o l u t i o n . The s t a b i l i s a t i o n o f t h i s p a r t i c u l a r non-planar c o n f i g u r a t i o n i s probably a s s o c i a t e d w i t h the non-bonding i n t e r a c t i o n s i n v o l v i n g the f l u o r i n e atom. The a l t e r n a t i v e t r a n s i t i o n s t a t e s t r u c t u r e (cj) = 90°) i s defined by 6 = 60° and 2a = 107.5°. The b a r r i e r to r o t a t i o n , corresponding to the minimum energy t r a n s i t i o n s t a t e , i s then c a l c u l a t e d as AG =5.12 kcals.mole 1 J which i s to be compared with t h a t given above f o r formamide: # AG =9.86 kcals.mole l. Assuming that the N,N-dimethyl and amino groups i n s u b s t i t u t e d amides allow a d i r e c t comparison of the e x p e r i -mental data f o r hindered r o t a t i o n s about the N-C bond, the f r e e energy of a c t i v a t i o n f o r carbamyl f l u o r i d e i s estimated as 15.4 + 0.8 k c a l s . mole The CNDO/2 c a l c u l a t i o n s t h e r e f o r e p r e d i c t the c o r r e c t order f o r these a c t i v a t i o n e n e r g i e s , but the r a t i o of p r e d i c t e d energies i s 1.9 while the experimental r a t i o i s 1.2. Nevertheless i t appears that 154. M O c a l c u l a t i o n s at the C N D O / 2 l e v e l of approximation may be very u s e f u l i n d e t a i l e d s e m i - q u a n t i t a t i v e d e s c r i p t i o n s of the o v e r a l l bonding c h a r a c t e r i s t i c s f o r s u b s t i t u t e d amides, and i n c o r r e l a t i o n s w i t h e x p e r i m e n t a l l y determined d i f f e r e n t i a l energies. As a l l valence e l e c t r o n s are considered and both e l e c t r o n and nu c l e a r r e p u l s i o n s are incl u d e d i n the c a l c u l a t i o n of molecular e n e r g i e s , a l l o w i n g an estima-t i o n o f non-bonding i n t e r a c t i o n e n e r g i e s , the C N D O / 2 c a l c u l a t i o n s have more p h y s i c a l s i g n i f i c a n c e than those u s i n g the simple Huckel TT -MO model p r e v i o u s l y discussed. CHAPTER 5 FOURIER TRANSFORM APPLICATIONS 5.1 B a s i c Formulation. The d u a l i t y of the pulsed and ste a d y - s t a t e NMR methods i s expressed by a F o u r i e r transform r e l a t i o n s h i p between the r e s p e c t i v e response f u n c t i o n s d e s c r i b i n g the nuclear magnetic system under reso-nance c o n d i t i o n s . Indeed, w i t h i n the r e s t r i c t i o n that a nuclear s p i n system may be t r e a t e d as a l i n e a r system, i t can be shown that the f r e e i n d u c t i o n decay a s s o c i a t e d w i t h the tranverse nuclear magneti-z a t i o n as observed i n the p u l s e mode i s a l s o determined, d i r e c t l y as the i n v e r s e F o u r i e r transform o f the unsaturated steady-state spectrum de f i n e d by the same component o f n u c l e a r magnetization. In 1954, Kubo 94 and Tomita d e r i v e d an expression f o r the frequency dependent mag-n e t i c s u s c e p t i b i l i t y of a general nuclear s p i n system i n terms of quantum s t a t i s t i c a l mechanics. This theory provides a general phys-i c a l b a s i s f o r the F o u r i e r transform c a l c u l a t i o n of resonance l i n e -shapes from an a u t o c o r r e l a t i o n f u n c t i o n (or r e l a x a t i o n f u n c t i o n ) f o r nuclear magnetization, and may be considered as a g e n e r a l i z a t i o n of 40 the s t o c h a s t i c model developed by Anderson and Weiss . Lowe and 67 Norberg demonstrated the above d u a l i t y experimentally through an 69 NMR study i n the s o l i d s t a t e , and i n 1967, Ernst and Anderson demonstrated the a p p l i c a t i o n of the general F o u r i e r transform con-cept to high r e s o l u t i o n NMR. 156. A simple model, based upon s e m i - c l a s s i c a l concepts already discussed i n t h i s t h e s i s , may be developed to allow a. concise formu-l a t i o n of F o u r i e r transform methods as a p p l i e d to general f i r s t - o r d e r nuclear s p i n systems. Such a f o r m u l a t i o n forms an adequate b a s i s f o r the numerical a n a l y s i s of more complicated systems i n c l u d i n g second-order J - c o u p l i n g . A f i r s t - o r d e r spectrum may be considered i n terms of n s p e c t r a l l i n e s centred at resonance frequencies OK such that the o v e r a l l lineshape f u n c t i o n , F (co), i s expressed i n the form F(co) = zVf. (co), (5.1.1) where A. i s an i n t e n s i t y n o r m a l i z a t i o n f a c t o r a s s o c i a t e d w i t h the i t h I J 2 2-1 l i n e d efined by the L o r e n t z i a n f u n c t i o n f ^ (co) = [1 + T^(co - O L ) ] , c f . Eq. (2.1.12). TV.; i s the t o t a l t r a n s v e r s e r e l a x a t i o n time, and f o r the i t h s p e c t r a l i i n e . As the normal NMR d e t e c t i o n schemes i n v o l v e r f phase s e n s i t i v e d e t e c t i o n , i t i s convenient to again define an independent v a r i a b l e x by x = co - to such that the f u n c t i o n f. (x) i s d e f i n e d on the i n t e r v a l -°° < x < 0 0 i n a I r o t a t i n g frame of reference as X U ) , (5.1.2) In t h i s way the i t h l i n e p o s i t i o n r e l a t i v e to the r e f e r e n c e frequency co i s given as Q. = co. - co , c o n s i s t e n t w i t h previous d e f i n i t i o n s , c f . O 1 1 o' r Eq. (2.2.24). In g e n e r a l , the i n v e r s e F o u r i e r transform f ( t ) i s defined i n the time domain by +00 f ( t ) = i _ J f ( x ) e x p ( i t x ) d x (5.1.3) 157. The transform p a i r may be considered i n complex spaces and f ( t ) , i n general complex, may then be r e a d i l y evaluated using the contour i n t e -g r a t i o n i n Appendix 1 as Vt} = 2T~ E X P [ _ ( Y _ i f i i ) t ] 2 i 2 i t 1 (5.1.4) = B. exp(- 7f.— ) [cosft. t + i s i n f t . t ] , B. = — 2 i 2 i f o r O ^ t < °°. The observable f r e e i n d u c t i o n decay described by the' f u n c t i o n f ( t ) i s n e c e s s a r i l y d e f i n e d by the r e a l p a r t only. Therefore, i n accordance w i t h Eq. (5.1.4), f ^ ( t ) corresponding to a s i n g l e L o r e n t z i a n s p e c t r a l l i n e , f ^ ( x ) , c o n s i s t s of an o s c i l l a t o r y decay w i t h a fundamental frequency ft. and a c h a r a c t e r i s t i c time constant T„. . As 95 the F o u r i e r t r a n s f o r m a t i o n i s l i n e a r , i t f o l l o w s from Eq. (5.1.4) th a t the o v e r a l l r e a l decay corresponding to F(x) (Eq. (5.1.1)) i s des c r i b e d by S( t ) = Z B. cosft.t e x p ( " f ) { s i s ) That i s , i n general the f u n c t i o n S ( t ) represents a modulated o s c i l l a -t o r y decay with the modulations determined by a l l frequencies |ft^ ± ft_. | . The d e t a i l e d form of an observed f r e e i n d u c t i o n decay a l s o depends upon the r e l a t i v e phases o f the i r r a d i a t i n g and r e f e r -ence r f magnetic f i e l d s i n v o l v e d i n a given phase s e n s i t i v e d e t e c t i o n scheme. This aspect i s considered f u r t h e r at a l a t e r p o i n t . The normal absorption mode spectrum i s defined by F ( x ) . A d i s p e r s i o n mode spectrum i s then defined by a corresponding f u n c t i o n H(x) where 158. n H(x) = E A h.(x) i = l =• ^kiJLjtt-Aj) (5.1.6) c f . Eq. (5.1.1). As shown i n Appendix 1, the inv e r s e F o u r i e r transform o f h (x) i s given as h. (t) = i B.exp[-( ~~ - i«.)t] (5.1.7) i i 2 i A complex lineshape f u n c t i o n ' g ( x ) may now be defi n e d i n the frequency domain as g(x) = f ( x ) - i h ( x ) , (5.1.8) such that the corresponding complex i n v e r s e F o u r i e r transform i s given from Eqs. (5.1.4) and (5.1.7) as g( t ) = B exp(-|-') [cosftt + i sinftt ] . (5.1.9) 2 Conversely, the F o u r i e r transform o f g ( t ) , g ( x ) , i s given by CO g(x) = J g ( t ) e x p ( - i t x ) d t = T _ _ M _ _ _ (5.1.10) o ^ f o r g ( t ) defined on the i n t e r v a l 0$t < m . A complete s p e c t r a l f u n c t i o n G(x) may now be considered i n the form n 1 G(x) = Z A [f (x) - i h (x)]= E A ± (5.1.11) 1 1 1 i = l 1 1 +VT Z L ( / - T A I ) This f u n c t i o n i s equivalent to that p r e v i o u s l y defined i n terms o f a general complex tr a n s v e r s e n u c l e a r magnetization, G_(x, cj)), i n the nor-mal r o t a t i n g frame of reference Ouvz, c f . Eq. (2.4.3) and F i g . 2.2. In d e f i n i n g the nuc l e a r isochromat component G_(x, <j>), the u-axis corresponds 1 5 9 . to the r e a l a x i s o f the complex plane and hence the phase angle <j) i s r e f e r r e d to t h i s a x i s . A l s o , i n a ste a d y - s t a t e NMR experiment, the i r r a d i a t i n g f i e l d v e c t o r i s assumed to be along the u - a x i s , c f . Eq. (2.1.9), and thus the absorption mode lineshape f u n c t i o n V(x) describes the v-component magnetization. In pulsed mode NMR, how-ever, the e f f e c t i v e f i e l d v e c t o r i s considered to be along the v- a x i s of the normal r o t a t i n g frame, such that at time t = 0 ( f o l -lowing a TT/2 p u l s e , $ = TT/2 i n F i g . 2.1) a l l isochromats are a l i g n e d i n the d i r e c t i o n of the u - a x i s . This allows a c o n s i s t e n t d e s c r i p t i o n of isochromat dephasing i n terms of the phase angle 6 f o r the f r e e i n d u c t i o n decay as determined by the u-component magnetization. Therefore, i n the complex plane Ouv, the f u n c t i o n G(x) i n Eq. (5.1.11) i s to be modified by a phase f a c t o r exp(iTr/2), g i v i n g G(x) = I A . ^ X ) + i f . ( x ) ] , (5.1.12) so t h a t i n accordance w i t h Eq. (2.4.3) u^(x) = A^hu (x) and V\ (x) = A^f (x). A l s o , i t i s to be noted that the response s ( t ) corresponds to an i r r a d i a t i n g r f f i e l d H_(t) defined i n a f i x e d frame of reference as H(t) = 2H 1cos(wt + TT/2), c f . Eq. (2.1.5). Thus i t i s seen t h a t the absorption mode lineshape f u n c t i o n f ( x ) d e s c r i b i n g the v-component nu c l e a r magnetization, a s s o c i a t e d w i t h the imaginary a x i s o f a complex plane i n the normal r o t a t i n g frame i s given as the r e a l p a r t of the complex F o u r i e r transform o f the f r e e i n d u c t i o n decay f u n c t i o n s ( t ) . I t i s now necessary to consider the p h y s i c a l consequences o f t a k i n g the r e a l p a r t o n l y , s ( t ) , of the causal f u n c t i o n g ( t ) i n the time domain, c f . Eq. (5.1.9). In accordance w i t h the general i n t e g r a l t r a n s f o r m a t i o n given i n Eq. (5.1.10), the F o u r i e r transform of s ( t ) i s 160. (5.1.12) w i t h s ( t ) = ^ exp(- £ )cosftt 2 2 and (5.1.13) Tz ( % ? ^ 2 j  g + ( x ) — u — Thus i t i s shown f o r s ( t ) d e f i n e d by the s i n g l e frequency ft, t h a t the F o u r i e r t r a n s f o r m a t i o n defines two lineshape f u n c t i o n s centred symmet-r i c a l l y about the reference frequency CO q (x = 0) . That i s , f + (x) and f _ ( x ) (as d e f i n e d i n Eq. (5.1.8)) determine L o r e n t z i a n l i n e s centred at x = ft and x = -ft, r e s p e c t i v e l y . This i s an example of the Faltung 95 property of a F o u r i e r t r a n s f o r m a t i o n and i s of importance i n deter-mining the frequency of the i r r a d i a t i n g r f f i e l d , co = w , i n a pulsed NMR experiment. The F o u r i e r transform of g ( t ) i s given i n accordance w i t h Eq. (5.1.10) as 00 g(x) = ^ g ( t ) cos t x - i s i n t x dt o and hence a cosine F o u r i e r transform i s defined by S C U ) = ^ g ( t ) c o s t x dt (5.1.14) o The cosine transform of the r e a l f u n c t i o n s ( t ) = ^ exp(-t/T )cosftt, 2 c f . Eq. (5.1.9), i s r e a d i l y shown to be 161. g c ( x ) = [1 + T 2 ( x + f t ) 2 ] _ 1 + [1 + T ^ C x-ft) 2]" 1 Comparison w i t h Eq. (5.1.13) shows that t h i s expression corresponds to f (x) + f _ (x) . That i s , i n ge n e r a l , the cosine F o u r i e r transform of s ( t ) determines the abso r p t i o n mode lineshape f u n c t i o n . S i m i l a r l y , a s i n e transform determines the d i s p e r s i o n mode f u n c t i o n , h (x) + h ( x ) . The complex F o u r i e r transform of s ( t ) d e f i n e s both the ab-s o r p t i o n and d i s p e r s i o n mode lineshape f u n c t i o n s i n the frequency do-main as the r e a l and imaginary p a r t s o f the f u n c t i o n g ( x ) , r e s p e c t i v e l y . 96 For a general l i n e a r p h y s i c a l system, the Kramers-Kronig r e l a t i o n s r e l a t e the r e a l and imaginary p a r t s of the frequency response f u n c t i o n s 97 g ( x ) . These r e l a t i o n s are eq u i v a l e n t to a H i l b e r t transform p a i r i n that the r e a l and imaginary p a r t s , f ( x ) and h ( x ) , are r e l a t e d bv -+00 +00 iu) - A V*') , I W ) „ __1 tl+'iAt' (5.1.15) — 00 — OO where the improper i n t e g r a l s are considered as p r i n c i p a l values. These 97 transforms may be compared w i t h those discussed by Abragam . To avoid the improper i n t e g r a l s i n Eq. (5.1.15), the H i l b e r t transform h(x) may 98 be considered i n the form OO L + ' < L * dt (5.1.16) o o For the Lor e n t z i a n lineshape f u n c t i o n f ( x ' ) given i n Eq. (5.1.2), i t f o l l o w s that 00 This i n t e g r a l represents a s i n e F o u r i e r transform f o r a causal f u n c t i o n , 1 6 2 . and hence h(x) = at) [ l+T^t< - t f t.f ] + T x (*+^} [ 1_ + U+^y*-] . Thus the H i l b e r t transform o f f(x) determines the d i s p e r s i o n mode lineshape functions h ±.(x) as given i n Eq. (5.1.13). 96 Now by d e f i n i t i o n , h(x) i n Eq. (5.1.15) i s als o given by the c o n v o l u t i o n h(x) = - e ( x ) * f ( x ) (5.1.17) where e(x) = (TTX) * and * denotes the p r i n c i p a l value of a co n v o l u t i o n i n t e g r a l . This c o n v o l u t i o n i s r e a d i l y evaluated i n d i r e c t l y by usin g F o u r i e r t r a n s f o r m a t i o n s , i n that i t i s w e l l known that a convol u t i o n i n the frequency domain corresponds to a simple m u l t i p l i c a t i o n i n the time domain h ( t ) = - e(t ) f ( t ) . The i n v e r s e F o u r i e r transform e ( t ) i s given by a contour i n t e g r a t i o n as e ( t ) = - i , t > 0 (5.1.18) and hence i t f o l l o w s that h ( t ) = i v - ^ i — exp - (^ - i f t ) t } which defines 2" 2 the d i s p e r s i o n mode f u n c t i o n given i n Eq. (5.1.6). I t i s s i m i l a r l y shown th a t f ( x ) = e ( x ) * h ( x ) . With the advent o f f a s t numerical F o u r i e r transform a l g o r i t h m s , i t i s p o s s i b l e to e x t r a c t accurate a b s o r p t i o n and 99 d i s p e r s i o n mode lineshapes from a stead y - s t a t e spectrum with a r b i t r a r y r f phase c h a r a c t e r i s t i c s . This phasing f e a t u r e i s of course an inherent property of puls e d mode NMR data i n the form o f the f r e e i n d u c t i o n decay f o l l o w i n g a TT/2 p u l s e . 163. 5 . 2 X Resonance C o n d i t i o n . In p u l s e d mode NMR, the i r r a d i a t i n g r f magnetic f i e l d , H_ , i s ap p l i e d f o r a f i n i t e time o n l y , i n the form of a pu l s e o f amplitude A and width t w w i t h a frequency to = to rads s e c . - 1 . The f r e e i n d u c t i o n decay a s s o c i a t e d w i t h a component tra n s v e r s e n u c l e a r magnetization may then be observed i n the absence of an r f magnetic f i e l d . Such a pulse may be considered as one of a sequence w i t h a r e p e t i t i o n p e r i o d T , and may be defined i n a f i x e d frame of reference by P(t ) = Acosto t 0 ^  t < t ^ o w = 0 t < t i T . (5.2.1) w J P ( t ) may now be considered i n terms o f a complex F o u r i e r s e r i e s T "f°° ito i f itot P (t) = c n ( t o ) e i a > w i t h the F o u r i e r c o e f f i c i e n t s c n(to) = — j P ( t ) e 1 U ' d t . For to ^  to , i t i s shown that f o r the fundamental frequency to^ = 2TT/T A I siviCwui^fw). 1 ^ ( 5 . 2 . 2 ) such that the frequency spectrum a s s o c i a t e d w i t h the pulse c o n s i s t s of d i s c r e t e frequencies to defined i n terms of to_ as n f to = to + nto,., n = ±1, ±2, ••• . ( 5 . 2 . 3 ) n o f v J The frequencies corresponding to n = 1, 2 , ••• are conveniently r e f e r r e d to as the upper sideband spectrum, and those corresponding to n = -1, - 2 , 1 ••• then form the lower sideband spectrum. The power corresponding to 2 each component frequency i s given by c (to) , maximum power at to = O J Q 2 being (At /2T) . Thus r f power i n pulsed mode NMR i s d i s t r i b u t e d over a s p e c i f i c frequency range (symmetric about CO Q) which may i n c l u d e a lar g e range of resonance frequencies . This frequency range i s det e r mined by the pu l s e width on l y , and from Eq. (5.2.2) i t i s seen that the f i r s t zero c o e f f i c i e n t i s def i n e d by the c o n d i t i o n naut = TT, that i s . ' f w to = 2ir/t . For a 5usec. p u l s e , t h i s corresponds to a frequency d i s t r i -b u t i o n about W q o f ± 200 kHz and the r f power may be considered to be constant over a range ± 2 kHz. In g e n e r a l , i f w (Hz) i s the frequency width o f a resonance spectrum to be i r r a d i a t e d , the pulse width i s de-f i n e d as t - 0.01 w 1 sec. In a d d i t i o n , the o v e r a l l r e s o l u t i o n and w ' r e p r o d u c i b i l i t y o f the resonance spectrum obtained through a F o u r i e r t r a n s f o r m a t i o n of the response f u n c t i o n s ( t ) f o r a nuc l e a r s p i n system f o l l o w i n g an r f pulse are determined by the spacing of the d i s c r e t e i r r a d i a t i n g frequencies to , that i s , T - 1 H Z . The form o f the above pulse frequency d i s t r i b u t i o n determines the centre frequency to to be used i n a given pulsed NMR experiment. As an r f phase s e n s i t i v e d etector i s normally referenced to to , the resonance spectrum d e f i n e d i n terms o f 'the independent frequency v a r i a b l e x i s considered f o r 0 x < 0 0 only. For a s p e c t r a l l i n e w i t h a resonance frequency > to , the detected component f r e e i n d u c t i o n decay s i g n a l may be assumed to have the form s (t) = B. exp (-t/T„.) cosft. t 1 £k 1 w i t h £h = ah - W q . In accordance with Eq. (5.1.13), the r e a l p a r t o f the F o u r i e r transform o f s ( t ) defines L o r e n t z i a n l i n e s d e s c r i b e d by lineshape f u n c t i o n s given as 165. f. (x) = -1 + 1 + ( x - f t j 2 £ ( x ) = _ ± . 1 + T2, (x+ft.) 2 2 i (5.2.4) On the i n t e r v a l 0 < x < °°, the c o n t r i b u t i o n s from f. (x) and f. fx.) at 2 2-1 x = ft^ are 1 and 1 + 4T ft^ , r e s p e c t i v e l y . Thus a general c o n d i t i o n f o r n e g l i g i b l e d i s t o r t i o n o f the lineshape f ^ + due to a c o n t r i b u t i o n from f. i s T„ft. >> 1. The d i s t o r t i o n a l e f f e c t s due to the transform f u n c t i o n l - 2 I f^_ are shown i n F i g . 5.1 f o r v a r y i n g values of the parameter O^T^' i s to be noted t h a t a 0.25% c o n t r i b u t i o n at x = ft. from f. corresponds l . i - 1 to a freouenc.Y s h i f t ft. = LOT.* rad- sec.." 1. An a d d i t i o n a l lineshane 1 2 d i s t o r t i o n may a r i s e through d e t e c t o r r f reference phase adjustment. This leads to the r e a l p a r t o f the F o u r i e r transform g(x) being a mix-tur e of absorption and d i s p e r s i o n mode f u n c t i o n s , and the e f f e c t o f a d i s p e r s i o n f u n c t i o n l u (analogous to f^_) i s more severe i n that the 2 2 -1 c o n t r i b u t i o n at x = ft. i s 2T„ft. 1 + 4T^ft. . In t h i s case, a 0.25% l 2 l 2 l c o n t r i b u t i o n from h. corresponds to fti = 200T*. This form of d i s -l - r l 2 t o r t i o n , however, i s e l i m i n a t e d by a phase c o r r e c t i o n as d e s c r i b e d i n the f o l l o w i n g s e c t i o n . The centre frequency ui^ may be decreased to increase the s h i f t from resonance, ft., and hence decrease the t r a n s -l form overlap d i s t o r t i o n f o r a given s p e c t r a l l i n e ; but i n so doing, the frequency a s s o c i a t e d w i t h the response s ( t ) i s increased. E x p e r i -m e n t a l l y , the f r e e i n d u c t i o n decay s i g n a l i s d i g i t i z e d using a sampling technique and the maximum frequency component that may be analyzed w i t h n . 'T 2 = o i 50 100 400 Figure 5.1 Pulsed mode NMR resonance c o n d i t i o n s 166. N samples over the pulse i n t e i ' v a l x i s N/2x ^9*100^ higher frequencies x > N / 2 T are a p p l i e d to the sampling d e v i c e , they are down-converted i n t o the frequency range 0 < x < N / 2 T and are then e f f e c t i v e as the lower frequencies x' = |x - nN/x|, 1 0 1 w i t h n = 1, 2, Fol l o w i n g a F o u r i e r transformation of s (t) these frequencies may give r i s e to spurious spec-t r a l l i n e s on the i n t e r v a l 0 < x < N/2x. In order to o b t a i n a q u a n t i -t a t i v e lineshape from the f u n c t i o n s ( t ) , i t i s r e q u i r e d i n general that C O q be set w i t h respect to a l l resonance frequencies uh such that the frequencies ft^ are minimized. Simultaneously, the transform f u n c t i o n overlap d i s t o r t i o n must be minimized according to the above co n d i t i o n s on the parameter Q/T^-. In a general p u l s e d mode NMR experiment, the t o t a l spectrum to be obtained as a F o u r i e r transform o f S ( t ) . c f . Eq. (5.1.5). must lip. to one s i d e of the centre frequency C O q . That i s , f o r x > 0 i t i s r e -qu i r e d that a l l resonance frequencies co^  be such that ah > to (ft^ > 0). In t h i s manner only the upper sideband frequency spectrum a s s o c i a t e d w i t h the r f pulse induces resonance e f f e c t s i n a given nuclear s p i n system. The reason f o r t h i s general resonance c o n d i t i o n i s shown by con s i d e r i n g two resonance frequencies oo. and oo. such that to. < co < co. . i l 1 o 1 The a c t u a l s e p a r a t i o n of s p e c t r a l l i n e s i s ' to. - co., or ft. + I f t . I where ft. = co. - to < 0. The response S ( t ) , however, i s defined i n terms of 3 J o r v the frequencies ft^ and |ft^ | and hence the F o u r i e r transform shows spec-t r a l l i n e s centred at x = ft^ and x = |ft_. j w i t h a f a l s e s e p a r a t i o n ft. - ft.II and i n the p a r t i c u l a r case f o r which I f t . I > ft. an i n c o r -1 l 1 j 11 » r 1 j 1 i r e c t frequency o r d e r i n g . The response S (t) being d e f i n e d i n terms of the frequencies ft^ and [ft.. | may be considered to correspond to the 167. resonance e f f e c t s induced by both the upper and lower sideband frequency s p e c t r a a s s o c i a t e d w i t h the r f pulse. In q u a n t i t a t i v e lineshape s t u d i e s , once the general form of the NMR spectrum has been determined, i t i s o f t e n p o s s i b l e to set the centre frequency to minimize the off-resonance s h i f t £L f o r the p a r t i c -u l a r s p e c t r a l l i n e of i n t e r e s t . In t h i s way, the lowest frequency com-ponents i n S ( t ) define the lineshape f u n c t i o n f (x) and these are the components most r e l i a b l y analyzed f o r any given number of sampled data p o i n t s , N. 168. 5.3 F i n i t e Complex F o u r i e r Transform. E x p e r i m e n t a l l y , a f u n c t i o n s ( t ) i n the time domain may be determined f o r a f i n i t e number of data p o i n t s only and hence the com-p l e x F o u r i e r transform f u n c t i o n g(x) i s determined f o r an equivalent number of data p o i n t s . Thus, at t h i s p o i n t , i t i s necessary to d i s -cuss a b a s i c f i n i t e complex F o u r i e r t r a n s f o r m a t i o n . I f the f u n c t i o n s ( t ) i s e s s e n t i a l l y f i n i t e only on the i n -t e r v a l 0 t. £ T, i t may be considered to be p e r i o d i c w i t h a p e r i o d T and may be expressed i n terms of a general f i n i t e F o u r i e r s e r i e s as s ( t ) = E A e x p ( i x t ) (5.3.1) where i s a complex c o e f f i c i e n t corresponding to the harmonic com-ponent w i t h frequency x . The fundamental frequency a s s o c i a t e d w i t h n the p e r i o d i c f u n c t i o n s (t) i s x^, d e f i n e d as x^ = 2TT/T rad. s e c . - 1 . The f i n i t e number of harmonic frequencies x^ are then given by x = n-2-rr/T, and from Eq. (5.3.1) i t f o l l o w s that s ( t ) = I A e x p ( i ^ p t ) . (5.3.2) Thus i t i s seen t h a t the highest harmonic of the fundamental frequency x^ considered i n the f i n i t e s e r i e s approximation to a F o u r i e r a n a l y s i s i s d e f i n e d by N. I t may now be assumed that s ( t ) i s sampled by P data p o i n t s on the i n t e r v a l 0 t T so that the time corresponding to the k data p o i n t i s t ^ = kT/P, where k = 0, 1, • • •, P - l . The c o e f f i c -i e n t s A^ are now given by a f i n i t e F o u r i e r transform o f the form P-J. A = — ^ S Jj_ (-c _EL^.n) (5.3.3) 169 f o r P even, where i s the f u n c t i o n s ( t ) value at t ^ . This r e l a t i o n -s h i p i s v e r i f i e d by c o n s i d e r i n g an i n t e g r a l form o f Eq. (5.3.1): o The f i n i t e complex i n v e r s e F o u r i e r transform of d e f i n e d i n Eq. (5.3.3) i s given i n accordance w i t h Eq. (5.3.2) as tt 2iik s, = E.A exp ( i - n - n ) . (5.3.4) k n ^ j n p I t i s to be noted t h a t the c o e f f i c i e n t A i s given as A* and hence an -n b n e x p l i c i t c a l c u l a t i o n f o r n < 0 i s redundant and s^ i s e q u i v a l e n t l y de-f i n e d as *•-• 2-rrk s k = E A n e x p ( i - p - n ) (5.3.5) *\= o f o r N even. The c o e f f i c i e n t s A^ d e f i n e the f i n i t e complex F o u r i e r transform f u n c t i o n corresponding to g(x) as p r e v i o u s l y considered, c f . Eq. (5.1.8), i n terms of f u n c t i o n values g(x ) at the d i s c r e t e f r e -quencies x^ separated by the fundamental frequency x^. That i s , the complex lineshape f u n c t i o n g(x) i s d e f i n e d on the normal frequency i n -t e r v a l 0 ^ x < r a b y g ( x ) = A . b n - n The F o u r i e r c o e f f i c i e n t A^ as given i n Eq. (5.3.3) may be expressed i n the form 170. . 1 „ r , 2 T r k n . . • . ,2'rrkn. A = — £ S. Icosf ) - 1 s m )] n p k = ( } k L p p (5.3.6) and hence the r e a l p a r t of t h i s complex c o e f f i c i e n t , a , i s given as n' 1 P^ 1 „ . ,2iTkn, a = - E S| cos ( ) . (5 3 7) n . P k=0 P • Thus i t i s seen t h a t a i s determined as the f i n i t e F o u r i e r cosine t r a n s -n form of s. , and t h i s c o e f f i c i e n t i s equivalent to g (x) where the general f u n c t i o n g (x) i s d e f i n e d i n Eq. (5.1.4). A f i n i t e F o u r i e r s i n e t r a n s -form may a l s o be considered to determine the c o e f f i c i e n t b as the ima-n ginary p a r t o f A . A simple c o r r e l a t i o n between the F o u r i e r c o e f f i c i -ents d e f i n i n g s^ i n Eq. (5.3.5) and those i n a standard f i n i t e t r i g o n o -m e t r i c F o u r i e r s e r i e s i s shown hv r.onsi Heri nc s_ in the form k . r , 2 T r k n . , . . , 2 T r k n . - , s = a A + L l a cos ( n) + b s i n ( n) J k 0 i n P n v p J J n=l c x 2P„* „ ,2TTkn. , , 2 P * „ . ,2-rrkn. where a = - E S cos - — ) and b = - E S s m - — ) . i t then f o l l o w s n P k = 0 k p n P k = Q k p that A = % ( a - i b ) and A = %a , showing that a and b are equiva-n v n n o o 6 n n n l e n t to the f u n c t i o n s f ( x ) and h(x) d e f i n i n g the general complex l i n e -shape f u n c t i o n g ( x ) , i n that a = f ( x ) and b = h(x ). In general terms, i t i s not p o s s i b l e to have a s i g n a l of f i n i t e time duration and s t r i c t l y f i n i t e s p e c t r a l bandwidth. However, a given s i g n a l s (t) of d u r a t i o n T may be assumed to have an e f f e c t i v e bandwidth IV and then according to the sampling theorem i n the time do-main**^, t h i s s i g n a l f u n c t i o n i s determined f o r a l l 0 ^ t ^ T to a good approximation by i t s values at 2T1V >> 1 sampling p o i n t s separated by 2W * sec. This i m p l i e s that the f u n c t i o n s ( t ) i s w e l l - d e f i n e d i f the 171. sampling frequency f i s such that 2W < f g Hz. The number of sampled . data p o i n t s , P, obtained over the time i n t e r v a l T i s determined by the number of memory l o c a t i o n s a v a i l a b l e f o r the d i g i t a l data f o l l o w i n g an A-D conversion of the sampled analog s i g n a l data. The maximum sampling frequency u s u a l l y corresponds to the minimum time, t , r e q u i r e d i n the A-D conversion process g i v i n g data to a s p e c i f i e d accuracy (number o f b i t s ) , as the sampling device may operate over a time i n t e r v a l of 1 usee, or l e s s . The sampling frequency may be e f f e c t i v e l y i n c r e a s e d , 107 however, by using a sample-and-hold device or "boxcar" i n t e g r a t o r to determine data p o i n t s at a time separation t < t . The sampling the-orem i n the frequency domain*^ s t a t e s that the frequency spectrum corresponding to a s i g n a l s (t) of d u r a t i o n T sec. i s completely deter-mined by amplitude values at a s e r i e s o f p o i n t s separated by l/T Hz ; This s e p a r a t i o n frequency i s j u s t the fundamental frequency x^, and hence the sampling theorem provides a b a s i s f o r u s i n g the f i n i t e F o u r i e r s e r i e s given i n Eq. (5.3.2) to d e f i n e the f u n c t i o n s (t) , where the c o e f f i c i e n t s (amplitudes) A^ are determined only f o r the d i s c r e t e frequencies x^ = nx^. In a d d i t i o n , the sampling i n t e r v a l i n the time domain has been determined as 2W * sec. and the maximum frequency x > 0 i n c l u d e d i n the f i n i t e F o u r i e r s e r i e s given i n Eq. (5.3.2) i s N/T Hz. Thus f o r P » 1 i t f o l l o w s that N i s determined as N = P/2. That i s , the maximum frequency that may be analyzed f o r the sampled f u n c t i o n s ( t ) as defined by P data p o i n t s i s the lowered frequency f = P/2T Hz. The frequency range of i n t e r e s t becomes 0 x £ P/2T Hz, as i n accor-dance with the sampling theorem i t i s i n v a l i d to c a l c u l a t e A^ f o r n > P/2 using Eq. (5.3.5). The highest frequency t h a t may be analyzed 172. i s t h e r e f o r e determined d i r e c t l y by the sampling frequency f , now.de-f i n e d as f = 2f . s m In p u l s e d mode NMR using F o u r i e r transform techniques, the s i g n a l f u n c t i o n s ( t ) i s tire f r e e i n d u c t i o n decay f o l l o w i n g a Tr/2-pulse and T i s equivalent to the p u l s e r e p e t i t i o n p e r i o d T, as discussed i n the preceding s e c t i o n . A F o u r i e r transform frequency spectrum may be obtained by n u m e r i c a l l y e v a l u a t i n g the complex amplitudes A^ d i r e c t l y , as given i n Eq. (5.3.3). This i s a time-consuming process i n t h a t P o p e r a t i o n s , each of which may be considered as a complex m u l t i p l i c a t i o n 108 f o l l o w e d by an a d d i t i o n , are r e q u i r e d f o r each A . Cooley and Tukey , however, have developed an a l g o r i t h m f o r a b i n a r y a r i t h m e t i c c a l c u l a t i o n t h a t allows a very s i g n i f i c a n t r e d u c t i o n i n computational time on a d i g -i t a l conrouter. The a l g o r i t h m renlar.es an a r r a v o f comnlex numbers of length P = 2 , r an i n t e g e r , by i t s complex F o u r i e r transform. The 108 t o t a l time i n v o l v e d corresponds to l e s s than 2P£og2P operations , as defined above, without r e q u i r i n g more data storage than i s r e q u i r e d f o r the input data array. The l a r g e s t a r r a y that may be transformed depends upon the computer memory: f o r a 32,768 word memory, the maxi-mum i s P -- 8192. I t i s to be noted that f o r the f i n i t e F o u r i e r transform p a i r as described by Eqs. (5.3.3) and (5.3.5), t h i s a l g o r -ithm generates a number of l i n e a r l y independent s p e c t r a l amplitudes equal to the number of input data p o i n t s . However, i t i s r e a d i l y shown that f o r r e a l input data, the a l g o r i t h m output data i s symmetric about n = P/2 i n that A = A* . This i s c o n s i s t e n t w i t h the data n P-n p o i n t s defined by n = 0, 1, • • •, P/2 determining the transform spectrum over the frequency range of i n t e r e s t . 173. A FORTRAN computer program LGTRN has been w r i t t e n and checked f o r general F o u r i e r transform a p p l i c a t i o n s i n NMR.. This program uses the IBM SHARE l i b r a r y subroutine PK FORT to carry out the b a s i c F o u r i e r t r a n s f o r m a t i o n using the Cooley-Tukey a l g o r i t h m . The r e a l input data i s normalized, transformed and then separated i n t o re-normalized absorption and d i s p e r s i o n mode s p e c t r a l data s e t s . These data sets may be f u r t h e r processed using r e a l a r i t h m e t i c to o b t a i n phase c o r r e c t e d s p e c t r a , en-69 hanced r e s o l u t i o n and f i l t e r e d s p e c t r a w i t h improved s i g n a l - t o - n o i s e r a t i o s , as discussed i n the f o l l o w i n g s e c t i o n . F u l l p l o t t i n g options are a v a i l a b l e f o r a CALCOMP p l o t t e r and s p e c t r a l data may be presented over any s p e c i f i e d frequency range, w i t h i n the l i m i t s 0 and P/2T Hz. The b a s i c output data p o i n t s e p a r a t i o n i s def i n e d as l/T Hz, and hence a subroutine SBHARM has been developed to allow the c a l c u l a t i o n of sub-harmonic data p o i n t s . This c o n s i s t s o f extending the e f f e c t i v e time i n t e r v a l f o r the input data and hence the number of data p o i n t s P, re a r r a n g i n g the r e a l input data as the two p a r t s o f a complex a r r a y , transforming t h i s array and then assembling the s p e c t r a l data over a s p e c i f i c frequency range. In t h i s manner, accurate F o u r i e r transform s p e c t r a may be obtained f o r sub-harmonic m u l t i p l i c i t i e s of two, fou r and e i g h t on a computer with a 32K memory. In high r e s o l u t i o n NMR, the sampling i n t e r v a l T, as normally determined by the s p i n - s p i n r e -l a x a t i o n time 1 , may be 2 - 5 sees, and hence the fundamental f r e -quency i s 0.5 - 0.2 Hz. Data p o i n t s at t h i s frequency s e p a r a t i o n are r e q u i r e d to define a c c u r a t e l y frequency p o s i t i o n s and i n t e n s i t i e s f o r s p e c t r a l l i n e s w i t h f u l l - w i d t h s at half-maximum o f the order of 0.5 Hz. This i s not p o s s i b l e , i n g e n e r a l , and s i g n i f i c a n t d i s t o r t i o n s are 174. apparent i n the r e s u l t a n t F o u r i e r transform s p e c t r a . By using a sub-m u l t i p l i c i t y of e i g h t , however, the e f f e c t i v e data p o i n t s e p a r a t i o n i s reduced to 0.06 and 0.025 Hz f o r T = 2 and 5 s e c , r e s p e c t i v e l y , and an accurate reproduction of the unsaturated steady-state NMR spectrum i s obtained. The program LGTRN has been developed f o r maximum e f f i c i -ency and the computational (CPU) time f o r 1024 input data p o i n t s and an 8192 p o i n t F o u r i e r t r a n s f o r m a t i o n , corresponding to a sub-harmonic • m u l t i p l i c i t y of e i g h t , w i t h phase c o r r e c t i o n s and the s e t t i n g up of f u l l p l o t t i n g data arrays i s t y p i c a l l y l e s s than 20 sec. on an IBM 360/67. The o v e r a l l accuracy o f the numerical complex f i n i t e F o u r i e r t r a n s f o r m a t i o n i s most c l e a r l y shown f o r a s i n g l e s p e c t r a l l i n e w i t h a given off-resonance s h i f t ft, c f . Eq. (5.2.4). The free i n d u c t i o n decay as def i n e d by s ( t ) i n Eq. (5.1.13) may be considered on the i n t e r v a l 0 ^ t 4 0.64 sec. f o r a L o r e n t z i a n system w i t h ft = 100 Hz and = 0.1 sec. The numerical F o u r i e r transform absorption and d i s p e r s i o n mode data are represented by crosses i n F i g . 5.2a, and the f u l l lineshapes as c a l c u l a t e d i n accordance w i t h Eq. (5.1.13) are shown as f u l l l i n e s . The numerical data p o i n t s are separated by 0.39 Hz, as def i n e d by a sub-harmonic m u l t i p l i c i t y of f o u r , the f u l l - l i n e - w i d t h at half-maximum being 3.2 Hz. I t i s seen that the d e v i a t i o n from the exact absorption mode lineshape f o r t h i s 4096 p o i n t transformation i s minimal, the d e v i a t i o n being only 1% and 4% at frequencies o f 103.1 and 105.47 Hz, r e s p e c t i v e l y . An ad d i -t i o n a l check on the o v e r a l l accuracy o f the F o u r i e r t r a n s f o r m a t i o n 109 i s a f f o r d e d by Parseval's theorem , i n that f o r an exact t r a n s -f ( x ) Figure 5.2(a) F i n i t e F o u r i e r transform c h a r a c t e r i s t i c s f o r a L o r e n t z i a n lineshape system. h(x) Figure 5.2(b) F i n i t e F o u r i e r transform c h a r a c t e r i s t i c s f o r a Gaussian lineshape. system. 175. 2 2 formation £ Is, I = P E |A I . The d i f f e r e n c y between these two 1 k 1 1 n 1 J f a c t o r s was c a l c u l a t e d to be less than 2% f o r the above lineshape f u n c t i o n . The corresponding lineshapes and numerical data p o i n t s f o r a Gaussian system are shown i n F i g . 5.2b. The absor p t i o n and d i s p e r s i o n mode Gaussian l i n e s h a p e s , f ( x ) and h ( x ) , are given i n Eqs. (5.4.20) and (5.4.21), r e s p e c t i v e l y . The a s s o c i a t e d response f u n c t i o n , s ( t ) , i s taken as the r e a l p a r t of the f u n c t i o n defined i n Eq. (5.4.22) and i s also shown i n F i g . 5.2b. The Gaussian absorption mode lineshape f u n c t i o n has c h a r a c t e r i s t i c s d i s t i n c t from those o f the L o r e n t z i a n f u n c t i o n , and again the d e v i a t i o n from the exact l i n e -shape i s shown to be minimal. The response s ( t ) f o r a L o r e n t z i a n system may be considered i n the form s( t ) . = Bexp ( - t / T 2 ) c o s f t t , cf. Eq. (5.1.5), and has been assumed above to be f i n i t e only on the i n t e r v a l 0 6 t & T. I t now remains to consider the e f f e c t o f the t r u n -c a t i o n of t h i s f u n c t i o n as compared with the normal d e f i n i t i o n on the i n t e r v a l 0 ^  t < °°. The lineshape f u n c t i o n f ( x ) i s given as the F o u r i e r cosine transform o f s ( t ) , c f . Eq. (5.1.14), and hence T f ( x ) = B [ e x p ( - t / T 2 ) c o s f t t cos x t dt. (5.3.8) o The f u n c t i o n f (x) centred at x = ft i s then defined by T f + ( x ) = | J" e x p ( - t / T 2 ) c o s [ ( f t - x ) t ] d t , o 176. A 1 + T ^ U L ) 2 -(5.3.9) where A i s a n o r m a l i z a t i o n constant. This equation shows that i n the l i m i t T 0 ° , the f u n c t i o n f (x) describes a normal L o r e n t z i a n a b s o r p t i o n mode lineshape on the i n t e r v a l 0 ^  t < °°. For a f i n i t e value of T, how-ever, t h i s lineshape i s modified by o s c i l l a t o r y terms w i t h an o v e r a l l amplitude f a c t o r (0, - x) exp (-T/T2) . For * - Q, the exponential i s dominant and defines an amplitude of < 1% o f the normal L o r e n t z i a n f u n c t i o n maximum f o r T > 7T^. In a d d i t i o n , the cosine and sin e terms i n Eq. (5.3.9) tend to cancel f o r a l l x and hence the o v e r a l l e f f e c t of t r u n c a t i o n i s expected to be n e g l i g i b l e f o r the T l i m i t defined above. This i s shown to be the case f o r the Lor e n t z i a n a b s o r p t i o n mode lineshape f ( x ) considered i n F i g . 5.2a, f o r which T = 6.4T . 177. 5.4 Phase C o r r e c t i o n s . In pulsed mode NMR the f r e e i n d u c t i o n decay s i g n a l i s deter-mined by the t r a n s v e r s e nuclear magnetization and i s described i n terms of a complex f u n c t i o n S ( t ) , c f . Eq. (5.1.4), which i s defined f o r a general f i r s t - o r d e r s p i n system by^ S ( t ) = exp(-t/T 2) I B i [ c o s f t i t + i s i n f l t ] (5.4.1) i n the normal r o t a t i n g frame of reference. i s an amplitude f a c t o r f o r the component decay w i t h the c h a r a c t e r i s t i c frequency £L = ah - co o, c f . Eq. (5.1.2). The t r a n s v e r s e magnetization may be considered i n terms of isochromat components G_(x, cb) i n the r o t a t i n g frame, as shown i n F i g . 2.2, w i t h time-dependent phase angles cb defined w i t h respect to the r e a l u-axis o f the complex transverse plane. A l l isochromat phase i n f o r m a t i o n i s r e t a i n e d experimentally by using an r f phase sen-s i t i v e d e t e c t i o n scheme, i n which the s i g n a l S ( t ) i s mixed w i t h a r e f -erence s i g n a l S and i n t e g r a t e d over a time i n t e r v a l t > I/Aw, where Aw i s the r f a m p l i f i e r bandwidth. This mixing corresponds to a simple m u l t i p l i c a t i o n which may be represented i n general as S Q ( t ) = Re S ( t ) - S r , (5.4.2) where S (t) i s the r e a l output s i g n a l and the reference s i g n a l i s de-f i n e d i n complex form i n a f i x e d frame of reference as S r = b-exp -i(u> ot + 4>r) • (5.4.3) That i s , t h i s s i g n a l has a frequency equal to the r f pulse i r r a d i a t i n g frequency and an a s s o c i a t e d phase angle cb which may be adjusted 178. a r b i t r a r i l y and i s defined with respect to the f i e l d H(t) defined by |HjJexp -i(co Qt + TT/2), c f . Eq. (2.1.5), c o n s i s t e n t with the assumption that the f i e l d v e c t o r Hj_ l i e s along the v - a x i s of the r o t a t i n g frame, as p r e v i o u s l y discussed. The phase s e n s i t i v e d e t e c t o r e f f e c t i v e l y references a l l output frequencies to co = to0 (x = 0) . The time-indepen-dent reference f i e l d v e c t o r may be represented i n the r o t a t i n g frame shown i n F i g . 2.2 such t h a t S r = b exp(i<j>r). (5.4.4) The output s i g n a l f o r a general complex response S ( t ) i s given i n accordance with Eqs. (5.1.4), (5.4.1) and (5.4.2) as -t/T S (t) = Re{b e B.(cosft.t + i s i n f t . t ) coscb + i sind) } o l i. I I J r r r r -t/T = be B. cosft.t cost!) - s i n f i . t s i n * . (5.4.5) I l r r l r For a component decay i n Eq. (5.4.5), the F o u r i e r transform of s (t) takes the form g ' W = 1 + iT 2(x+ f t ) Ccos* r-i sin(})r) + 1 + iT 2Cx-0)- Ccos* r +i sin* r)' (5.4.6) Tl i e r e f o r e , on the frequency i n t e r v a l 0 x < °°, the complex lineshape f u n c t i o n centred at x = Q, i s given i n accordance w i t h Eq. (5.1.8) as t For s i m p l i c i t y , i t has been assumed th a t T 2^ = T2 f o r a l l s p e c t r a l l i n e s . 179. (5.4.7) Thus i t i s shown that the r e a l p a r t of the complex F o u r i e r transform f u n c t i o n , f ' ( x ) , i n general corresponds to a l i n e a r combination of Lor e n t z i a n a b s o r p t i o n and d i s p e r s i o n mode lineshape f u n c t i o n s f ( x ) and h(x) as defined i n Eqs. (5.1.2) and (5.1.6), r e s p e c t i v e l y . For the p a r t i c u l a r case i n which the reference f i e l d v e c t o r i s i n the d i r e c t i o n o f the u-axis of the r o t a t i n g reference frame, <t> • = 0 and from Eqs. (5.4.5) and (5.4.7) i t f o l l o w s that S (t) = b exp(-t/T_) E B.cosft.t O 1 2 ; 1 1 v and G< (x) = b £ A. [f. (x) - i h (x)] , I x c o n s i s t e n t w i t h the expression d e r i v e d - p r e v i o u s l y (Eq. (5.1.11)) under the assumption that the f r e e i n d u c t i o n decay was determined by u-compon-ent n u c l e a r magnetization. The absorption mode lineshape, f ( x ) , i s obtained from Eq. (5.4.7) as f ( x ) = f' (x)cos(J) + h'(x)sincb r . (5.4.8) Both f 1 ( x ) and h'(x) are a v a i l a b l e from a numerical complex F o u r i e r t r a n s f o r m a t i o n of d i g i t i z e d data i n the time domain and hence the phase 180. angle tf> may be adjusted to define f ( x ) to an accuracy determined by the c r i t e r i o n used to define the c o r r e c t absorption mode lineshape. B a s e l i n e e q u a l i z a t i o n over s p e c i f i c frequency ranges has been shown to give p a r t i c u l a r l y e f f i c i e n t numerical phase c o r r e c t i o n . Ernst 69 and Anderson have used an area r a t i o c r i t e r i o n f o r t h i s r f phase adj ustment. the response S ( t ) that can be analyzed using a sampling technique i s x = N / 2 T where N i s the number of samples i n the time domain and x i s the r f pulse r e p e t i t i o n p e r i o d . Higher s i g n a l frequencies are down-converted and give r i s e to spurious s p e c t r a l l i n e s as determined by a F o u r i e r t r a n s f o r m a t i o n , and high frequency noise s i m i l a r l y down-converted leads to a degradation of s i g n a l - t o - n o i s e r a t i o i n the frequency range of i n t e r e s t , 0 < x < N / 2 T . T O minimize these e f f e c t s , the output from the r f phase s e n s i t i v e d e t e c t o r , s (t) , i s f o l l o w e d by a low pass f i l t e r to attenuate the frequencies x > N/2x. The f i l t e r output s i g n a l s ^ ( t ) i s then f e d to a sampling device and an A-D con-v e r t e r to o b t a i n the f i n a l data i n d i g i t a l form. This f i l t e r , how-ever, introduces a frequency dependent phase s h i f t which complicates the determination of a c o r r e c t absorption mode lineshape. The e f f e c t o f the f i l t e r i n the time domain i s des c r i b e d by a convol u t i o n r e l a -102 t i o n s h i p and hence i t i s more convenient to de f i n e the F o u r i e r transform f u n c t i o n s s fx) and s^fx) and to consider the e f f e c t of the o f f i l t e r i n the frequency domain. In t h i s manner I t has been noted that the hig h e s t frequency contained i n — 00 181. where T(x) i s the complex t r a n s f e r f u n c t i o n d e f i n i n g the f i l t e r char-a c t e r i s t i c s , such that s^ Cx) = T(x)s ( x ) . This t r a n s f e r f u n c t i o n may be considered i n the form T(x) = D(x)exp i 9 ( x ) (5.4.9) so t h a t D(x) determines an a t t e n u a t i o n and 8(x) defines a c h a r a c t e r -i s t i c frequency dependent phase s h i f t . I f i t i s assumed that the sampling device does not introduce a f u r t h e r frequency dependent a t t e n u a t i o n or phase s h i f t , the F o u r i e r transform o f the f i l t e r e d s i g n a l may be considered equivalent to g j (x) . Therefore, i n accor-dance w i t h Eqs. (5.4.7) and Eq. (5.4.9), s^(x) i s given by s f ( x ) = f»(x) + i h 1 (x) = D ( x ) [ f ( x ) - i h ( x ) ] [ c o s ( 6 ( x ) ' + <f>r) + i s i n ( 6 ( x ) + c ^ ) ] (5.4.10) and the absorption mode f u n c t i o n f ( x ) i s now given i n general form as f ( x ) = D _ 1 ( x ) { [ f ' ( x ) c o s 6 ( x ) + h' ( x ) s i n 0 ( x ) ] cos<J> - [ f • (x)sinO(x) - h' (x)cos6(x)] sincf^}. (5.4.11) 102 The t r a n s f e r f u n c t i o n f o r a s i n g l e s e c t i o n low-pass RC f i l t e r i s H U t L l (5.4.12) and hence D(x) = [ l + (xRC) ] , cos8(x) = D(x) and s i n 0 ( x ) = -xRCD(x). For t h i s p a r t i c u l a r f i l t e r , f ( x ) i s independent of D(x) and from 182. Eq. (5.4.11) i t f o l l o w s t h a t f ( x ) = [f»(x)cos<j> + h'(x)sin<J> ] + xRC [ f 1 (x)sin<|> - h 1 (x) c o s c f j . ' (5.4.13) This expression allows a simple e v a l u a t i o n o f the co r r e c t e d absorption mode lineshape i n terms of the complex F o u r i e r transform f u n c t i o n s f 1 ( x ) and h ' ( x ) , the reference r f phase angle cj> and the f i l t e r parameter RC. This l a t t e r parameter may be e m p i r i c a l l y v a r i e d over a small range to o b t a i n an improved lineshape from numerical F o u r i e r transform data ob-t a i n e d e x p e r i m e n t a l l y using a standard sampling device i n c o r p o r a t i n g a low-pass f i l t e r . The e f f e c t o f f i l t e r and r f reference phasing i s i l -l u s t r a t e d i n F i g . 5.3 f o r a r e p r e s e n t a t i v e L o r e n t z i a n lineshape d e f i n e d by ft = 100 Hz and T = 0.1 sec. (3.2 Hz f u l l - w i d t h at half-maximum). The off-resonance s h i f t o f 100 Hz ensures that transform overlap d i s -t o r t i o n , as discussed i n s e c t i o n 5.2 i s minimal. The s p e c i f i c RC f i l t e r time constant i s 0.001 s e c , corresponding to a c u t - o f f frequency of 160 Hz. The r e a l and imaginary p a r t s of the F o u r i e r transform s^(x) are shown i n F i g . 5.3a, the f r e e i n d u c t i o n decay f u n c t i o n s ( t ) being d e f i n e d by 1024 data p o i n t s over a time i n t e r v a l of 0.64 sec. With a computational sub-harmonic m u l t i p l i c i t y of four the data p o i n t s shown i n the frequency domain have a s e p a r a t i o n of 0.39 Hz, the cor-responding fundamental frequency being 1.56 Hz. The f i l t e r frequency dependent phase c o r r e c t i o n determines the r e a l f u n c t i o n f ' ( x ) shown i n F i g . 5.3b, and an a d d i t i o n a l frequency independent r f reference . phase c o r r e c t i o n of-20 determines the f i n a l L o r e n t z i a n absorption mode lineshape shown i n F i g . 5.3c. The s i n g l e s e c t i o n RC f i l t e r gives an a t t e n u a t i o n o f -3dB at the c u t o f f frequency x = (2TrRC)~~ Hz and Figure 5 .3 F i l t e r and r f reference phase c o r r e c t i o n s f o r a L o r e n t z i a n lineshape system. 183. -40 dB at x n = lOOx . An increased a t t e n u a t i o n o f -80 dB at the f r e -£ c quency x^ i s obtained by using a two-section RC f i l t e r , each s e c t i o n being matched and i s o l a t e d , w i t h a t r a n s f e r f u n c t i o n 2 - 1 2 where w = xRC and D(x) = (1 + w ) , cos0(x) = (1 - w )D(x) and sinO(x) = -2wD(x). Again f (x) i s shown to be independent of D(x) and i n t h i s case f ( x ) = (1 - w 2 ) [ f ' (x)cosct r + h'(x)sin<J> ] +• 2w[f' (x)sin<J> - h» (x)cos<J> (5.4.15) In g e n e r a l , e x p e r i m e n t a l l y , i t i s not p o s s i b l e to o b t a i n the response s (t) f o r t -»- 0 because of the f i n i t e r e c e i v e r recovery time f o l l o w i n g a Tr/2-pulse during which the f r e e i n d u c t i o n decay s i g n a l i s not observable, and hence the f u n c t i o n s ( t ) i s to be considered on the 103 i n t e r v a l t Q ^  t < °°. The time s h i f t t > 0 corresponds to a phase s h i f t i n the frequency domain which may be considered i n terms of the F o u r i e r transform o f a modified response p ( t ' ) d e f i n e d , on the i n t e r v a l 0 t 1 < °°, to be equivalent to s (t) f o r t ^ t < °°, th a t i s , s m (x) = [ p ( V ) e ^ , ( - i * i ' ) i f ' , i ' - i - t 0 0 c>Cj>(ut0) I s(f) t*j> [ - U t ) dt 2 ev-p {Ui0) SU) (5.4.16) where s(x) i s the F o u r i e r transform o f s ( t ) as defined on the normal i n -t e r v a l 0 ^ t < °°. By again c o n s i d e r i n g the F o u r i e r transform f u n c t i o n 184. sm ( x ) as equivalent to gj ( x ) i n Eq. (5.4.7), i t f o l l o w s that f ( x ) i s given i n terms of the frequency dependent phase s h i f t x t Q by f ( x ) = f ' ( x ) c o s x t + h ' ( x ) s i n x t . (5.4.17) 0 0 I t i s to be noted t h a t a l l phase s h i f t s i n v o l v e d i n the e v a l u a t i o n of phase c o r r e c t e d lineshapes may now be combined as ^ = $ r + S(x) + x t Q so that g'(x) = f• ( x ) + i' h ' ( x ) f (x) = f (x)cosci) + h' (x)sin<t> and h(x) = fI(x)sin<J> - h'(x)cos<j> . (5.4.18) The d e f i n i t i o n o f th m a . v b e e x t e n d e d t o i n c l u d e a n v a d d i t i o n a l n h a . s e f a c t o r s . In l i q u i d s under high r e s o l u t i o n c o n d i t i o n s the r e l a x a t i o n times T^ and T^ are of the order of seconds and the normal Tr/2-pulse widths used are 10 - 100 usee. . An average r e c e i v e r recovery time i s 10 usee., and hence the phase f a c t o r xt over a frequency range o f 1 kHz f o r *H NMR i s n e g l i g i b l e . For 1 3C NMR i n v o l v i n g chemical s h i f t s (and hence s p e c t r a l widths) of the order o f 10 kHz, however, t h i s f a c t o r may not be neglected. In s o l i d s with much s h o r t e r r e l a x a t i o n times, t h i s phase f a c t o r becomes very s i g n i f i c a n t as the s p e c t r a l frequency width i s extended by up to three orders of magnitude. A l s o , the above approximate l i n e a r phase c o r r e c t i o n may not be adequate when t Q becomes s i g n i f i c a n t w i t h respect to which defines the t o t a l s i g n a l sampling time. In the l a t t e r case i t i s p o s s i b l e to use 185. a Tr/2-Trpulse sequence w i t h a pul s e s e p a r a t i o n T < and to sample the s p i n echo centred at t = 2x. The F o u r i e r transform o f s ( t ' ) i s obtained f o r O ^ t ' < OT, as t ' = 0 now corresponds to t = 2 T , and i t i s w e l l known that the s p i n echo s i g n a l on t h i s i n t e r v a l i s equivalent to the normal f r e e i n d u c t i o n decay. The phase e f f e c t corresponding to the response s ( t ) being undefined f o r 0 4 t < t has been t r e a t e d n u m e r i c a l l y , and i s i l l u s -o J t r a t e d i n F i g . 5.4 f o r the r e p r e s e n t a t i v e L o r e n t z i a n lineshape d e f i n e d by ft = 100 Hz and T 2 = 0.1 s e c , c f . F i g . 5.3. The f r e e i n d u c t i o n decay, s (t) , a s s o c i a t e d with a Tr/2-pulse width of - 1 msec, was de-f i n e d by 1024 data p o i n t s over a time i n t e r v a l of 640 msec, w i t h t = 1 msec. F i g . 5.4a shows the F o u r i e r transform f u n c t i o n f' (x) , cf. Eq. (5.4.17), w i t h the frequency domain sub-harmonic data f p o i n t s separated by 0.39 Hz. A l i n e a r frequency dependent phase cor-r e c t i o n as given by Eq. (5.4.17) determines the absorption mode l i n e -shown i n F i g . 5.4b, where the phase f a c t o r xt takes the value 36° at a frequency o f 100 Hz. The small negative d e v i a t i o n , as compared w i t h the c o r r e c t lineshape shown i n F i g . 5.4d, i s presumably due to the i n t e g r a l approximation i n v o l v e d i n Eq. (5.4.16). I t i s now i n t e r -e s t i n g to compare the above a b s o r p t i o n mode lineshape w i t h that shown i n F i g . 5.4c obtained by using a frequency independent phase c o r r e c t i o n , c f . Eq. (5.4.8). The c o r r e c t i o n angle, as def i n e d by a b a s e l i n e e q u a l i z a t i o n i n a determination of the c o r r e c t l i n e s h a p e , o i s +36 . In ge n e r a l , however, f o r a s e r i e s of s p e c t r a l l i n e s over an extended frequency range, the l i n e a r phase c o r r e c t i o n must be a p p l i e d . f(x) f ( x ) (a) (b) £(x) f ( x ) M K I ' . I M W J S B S Figure 5.4 L i n e a r frequency dependent phase c o r r e c t i o n f o r a Lo r e n t z i a n l i n e -shape system. 186. In g e n e r a l , a phase c o r r e c t i o n as given i n terms o f Eqs. (5.4.8) and (5.4.17) i n v o l v e s an i t e r a t i v e numerical f i t t i n g o f the absorption mode lineshape to s e l f - c o n s i s t e n c y as defined by a spec-i f i c c r i t e r i o n f o r a c o r r e c t lineshape. The phase independent p a r t of the complex lineshape f u n c t i o n g ( x ) , c f . Eq. (5.1.8), may be con-s i d e r e d to be def i n e d by an amplitude f u n c t i o n * ^ A(x) , where g(x) = A(x) exp(10(x)) (5.4.18) w i t h A(x) = [ f 2 ( x ) + h 2 ( x ) ] J s and 0(x) = t a n _ 1 [ - h ( x ) / f (x)] . Now, f o r a general F o u r i e r transform lineshape f u n c t i o n g'(x) w i t h an a r b i t r a r y 2 2 phase f a c t o r , c f . Eq. (5.4.18), i t i s shown th a t f (x) + h (x) = f | 2 ( x ) + h , 2 ( x ) ; t h a t i s , the amplitude f u n c t i o n i s simply d e r i v e d from the numerical F o u r i e r transform f u n c t i o n s f ' ( x ) and h ' ( x ) . Furthermore, f o r a L o r e n t z i a n lineshape as defined i n Eqs. (5.1.2) and (5.1.6), i t i s seen that the absorption mode f u n c t i o n i s given 2 d i r e c t l y as f ( x ) = A ( x ) . In t h i s p a r t i c u l a r case, a very simple e f f e c t i v e phasing c o r r e c t i o n i s obtained by e v a l u a t i n g f ( x ) d i r e c t l y , as i l l u s t r a t e d i n F i g . 5.5 f o r the standard L o r e n t z i a n lineshape p r e v i o u s l y d e s c r i b e d w i t h an a s s o c i a t e d phase angle of 20 at a f r e -quency o f 100 Hz. Numerical F o u r i e r transform f u n c t i o n s are shown i n F i g . 5.5a, i l l u s t r a t i n g the phasing e r r o r i n v o l v e d . As u s u a l , the frequency domain data p o i n t separation i s 0.39 Hz. F i g s . 5.5b 2 and 5.5c show the amplitude f u n c t i o n s A(x) and A ( x ) , r e s p e c t i v e l y , where the l a t t e r has been re-normalized to the maximum value defined by A ( x ) . I t i s seen that A(x) takes the form of a modified L o r e n t z i a n lineshape w i t h an increased f u l l - w i d t h at half-maximum o f 2/3 2 rad. s e c . - 1 . The n u m e r i c a l l y d e r i v e d f u n c t i o n A (x) i s a very good Figure 5.5. Amplitude f u n c t i o n and phase c o r r e c t i o n f o r a L o r e n t z i a n l i n e -shape system. \ 187. approximation to an exact L o r e n t z i a n absorption mode lineshape i n that the d e v i a t i o n at x = 93.6 Hz (two l i n e w i d t h s from the centre frequency) i s l e s s than 2% and t h i s frequency corresponds to an i n t e n s i t y approximately 4% of the maximum. This lineshape may be compared with that shown i n F i g . 5.5d as obtained using the f u l l phase c o r r e c t i o n s p r e v i o u s l y described. s i t y enhancement lea d i n g to increased s p e c t r a l r e s o l u t i o n through numerical methods, modified L o r e n t z i a n absorption mode fu n c t i o n s may be defined i n general by w i t h n = 1, 2, ••• and A a n o r m a l i z a t i o n constant f o r the normal l i n e -shape f u n c t i o n (n = 1). These modified f u n c t i o n s have i d e a l s p e c t r a l lineshape c h a r a c t e r i s t i c s i n that they are symmetric about x = ft, they do not have zeros, the e f f e c t i v e l i n e - w i d t h decreases with i n -cr e a s i n g n and the s i g n a l - t o - n o i s e r a t i o i s improved when t h i s r a t i o i s g r e a t e r than u n i t y . For n = 2 and 4, the f u l l l i n e - w i d t h s are given as l^ST,, 1 and 0.84T" 1, r e s p e c t i v e l y , as compared w i t h the normal L o r e n t z i a n width o f 21^ rad. s e c . - 1 . The numerical reso-l u t i o n enhancement as determined by the f u n c t i o n s f 2 ( x ) a n <3 f ^ M 1 S i l l u s t r a t e d i n F i g . 5.6 f o r two L o r e n t z i a n l i n e s centred at x = -ft and x = ft and f o r v a r y i n g values of the parameter OT^- R e s o l u t i o n i n the above two-line spectrum may be defined i n terms of the r a t i o o f maximum i n t e n s i t y to the i n t e n s i t y at x = 0. For ftT^ = 0.16, cor-responding to a l i n e s e p a r a t i o n of a f u l l l i n e - w i d t h , t h i s r a t i o i s 1.3 f o r n = 1 and 2.1 f o r n = 4, g i v i n g a f r a c t i o n a l r a t i o i n c r e a s e o f 1.6 With regard to e f f e c t i v e l i n e - w i d t h r e d u c t i o n and i n t e n -(5.4.19) 0.32 Figure 5.6 Reso l u t i o n enhancement, lineshape system. modif i e d L o r e n t z i a n 1 8 8 . f o r the modifi e d f u n c t i o n f . f x ) . This f r a c t i o n a l r a t i o i n c r e a s e i s 6 f o r QT2 =0.24 and 18 f o r OT^ = 0.32. To determine the general v a l i d i t y of an e f f e c t i v e phase c o r r e c t i o n through the e v a l u a t i o n of the amplitude f u n c t i o n A ( x ) , the Gaussian f u n c t i o n may be considered, as an example of a lineshape func-t i o n w i t h d i f f e r e n t c h a r a c t e r i s t i c s as compared w i t h the Lo r e n t z i a n f u n c t i o n . The absorption mode Gaussian lineshape f u n c t i o n , c f . Eq. (2.1.13), may be considered i n the form f ( x ) = A exp(-%T 2 2(x - ft)2), (5.4.20) w i t h a l l parameters d e f i n e d as p r e v i o u s l y . The f u l l - w i d t h at h a l f -maximum i s 2(2£n2) 2T 2 rad. s e c . - 1 . The corresponding d i s p e r s i o n mode f u n c t i o n may he evaluated i n terms of the jrwevse F o u r i e r t r a n s f o r m f ( t ) by a contour i n t e g r a t i o n , given i n Appendix 2, as h(x) = A exp (-JgT2 (x-ft) 2) T (x-fi) T 2(x - f t ) r /2 e - * .p (q 2 ) d r j •2 = A " * 1 VT I o i n accordance w i t h the general complex lineshape f u n c t i o n defined i n Eq. (5.1.8). These Gaussian f u n c t i o n s are shown i n F i g . 5.2b, as deter-mined by a numerical F o u r i e r t r a n s f o r m a t i o n of the response s ( t ) , where s ( t ) = B e x p ( - t 2 / 2 T 2 ) [ c o s f t t + i s i n f t t ] (5.4.22) c f . Eq. (5.1.4). From Eqs. (5.4.20) and (5.4.21), i t f o l l o w s that 189. o and hence the absorption mode f u n c t i o n i s not given simply i n terms of f ( x ) = A(x)\ 1 + 1 A ( x ) . As the i n t e g r a l term i s symmetric i n x about x = 0, A(x) defines a m o d i f i e d Gaussian lineshape w i t h an inc r e a s e d l i n e - w i d t h . This i s shown n u m e r i c a l l y f o r T = 0.1 sec. i n that the normal f u l l - w i d t h at half-maximum i s 3.7 Hz ( c f . 3.2 Hz f o r a Lo r e n t z i a n f u n c t i o n ) and the f u l l - w i d t h f o r the lineshape described by A(x) i s 6.8 Hz. Thus, i n 2 ge n e r a l , i t i s p o s s i b l e to use the f u n c t i o n A (x) as an approximation to an abso r p t i o n mode lineshape only f o r s p e c t r a l l i n e s c l o s e l y des-c r i b e d by a Lo r e n t z i a n lineshape f u n c t i o n , as i s the case f o r high r e s o l u t i o n NMR i n l i q u i d s . 5.5 S i g n a l Zero C o r r e c t i o n . In pulsed mode NMR, a s i g n a l voltage f o l l o w i n g p h a s e - s e n s i t i v e d e t e c t i o n , f i l t e r i n g and.dc a m p l i f i c a t i o n may be represented by s' (t) = C + s ( t ) , (5.5.1) where s ( t ) i s the free i n d u c t i o n decay s i g n a l and C i s a constant v o l -tage, i n general d i f f e r e n t from zero. A s u f f i c i e n t c o n d i t i o n f o r the exist e n c e of a F o u r i e r transform o f s'(t) i s absolute i n t e g r a b i l i t y , + C O ^ | s ' ( t ) | d t < 0 0, and the constant C does not s a t i s f y t h i s c o n d i t i o n . — C O I t i s r e a d i l y shown, however, t h a t the in v e r s e F o u r i e r transform o f 2TTC6 (x) (with 6 (x) the D i r a c d e l t a f u n c t i o n ) i s simply the constant C. This i m p l i e s t h a t C i s a s s o c i a t e d w i t h an a n a l y t i c approximation**^ to 6(x) i n the frequency domain. That i s , the F o u r i e r transform o f C, 190. c ( x ) , may be considered i n the form +00 c(x) = C ] e x p ( - i t x ) d t — 00 (5.5.2) The parameter t defines the l i m i t s of i n t e g r a t i o n f o r a f i n i t e com-p l e x F o u r i e r t r a n s f o r m a t i o n as considered i n s e c t i o n 5.3. The above equation defines an o s c i l l a t o r y f u n c t i o n w i t h a maximum value 2 C t m at x = 0 and zeroes at x = mr/t rad. s e c . - 1 f o r n = 1, 2, • • - o n the n m . i n t e r v a l 0 ^ x ^ t . As c(x) i s r e a l , the general F o u r i e r transform of s' (t) may be expressed i n the form g'(x) = [ c ( x ) + f ' ( x ) ] + i h ' f x ) , (5.5.3) c f . Eq. (5.4.18). I f the off-resonance s h i f t s ft. > 0 are minimized as described 1 i n s e c t i o n 5.2, the f u n c t i o n c(x) may give r i s e to an o s c i l l a t i o n w i t h a frequency dependent amplitude superimposed on a s p e c t r a l lineshape of i n t e r e s t . This e f f e c t i s i l l u s t r a t e d i n F i g . 5.7 f o r a r e p r e s e n t a t i v e L o r e n t z i a n lineshape defined by ft = 40 Hz and T = 0.1 sec. The s i g n a l s ' ( t ) as determined by 1024 data p o i n t s over an i n t e r v a l of 0.64 sec. i s shown i n F i g . 5.7a, where s' (t) = 2.0 + 10.Oexp(-t/T )cosftt. The absor p t i o n mode numerical F o u r i e r transform f u n c t i o n , c(x) + f 1 ( x ) , i s defi n e d i n F i g . 5.7b by data p o i n t s separated by the sub-harmonic f r e -quency 0.39 Hz. The s u p e r p o s i t i o n of c(x) i s c l e a r l y shown over the frequency range 0 to 65 Hz, the constant C being 20% of the maximum value o f the free i n d u c t i o n decay s i g n a l . The constant C i s r e a d i l y determined as the mean s i g n a l l e v e l f ( x ) 0 40 x(Hz) Figure 5.7 Numerical F o u r i e r transform d i s t o r t i o n due to non-zero average s i g n a l l e v e l . 191. f o r t •+ T, T being the sampling time i n t e r v a l . A numerical c o r r e c t i o n of -C may then be a p p l i e d to the sampled s i g n a l data to o b t a i n a good approximation to the s i g n a l f u n c t i o n s ( t ) . A numerical F o u r i e r t r a n s -formation o f s ( t ) then gives f ' ( x ) and h'(x) as r e q u i r e d f o r the general phase c o r r e c t i o n s discussed i n the preceding s e c t i o n . This method f o r a s i g n a l zero c o r r e c t i o n has been shown to be both r e l i a b l e and e f f i c -i e n t by using a subroutine DIRCON i n the FORTRAN program LGTRN p r e v i -ously o u t l i n e d . 192. 5.6 High R e s o l u t i o n F o u r i e r Transform NMR The b a s i c concepts and computat ional methods i n v o l v e d i n a F o u r i e r t r a n s f o r m NMR experiment have been c o n s i d e r e d i n d e t a i l i n the preceding s e c t i o n s of t h i s t h e s i s , w i t h p a r t i c u l a r emphasis upon the p o s s i b i l i t y o f us ing t h i s technique f o r q u a n t i t a t i v e 1ineshape: s t u d i e s . As an example h igh r e s o l u t i o n a p p l i c a t i o n , the spectrum o f d imethyl n i t r o s a m i n e has been o b t a i n e d a t a r e l a t i v e l y low H Q f i e l d c o r r e s p o n d i n g to an o p e r a t i n g frequency of 10.0 MHz. T h i s p a r t i c u l a r s p i n system was chosen because o f the s i m p l i c i t y o f the c h e m i c a l l y s h i f t e d equal i n t e n s i t y d o u b l e t spectrum and the magnitude o f the chemical s h i f t , : 45 Hz a t 60 MHz and 30°C. As the p u l s e spectrometer 13 d e s c r i b e d i n s e c t i o n 3 .2 was des igned s p e c i f i c a l l y f o r C s t u d i e s i n a f i e l d o f 9.4 kgauss , i t was a l s o convenient t o use the above o p e r a t i n g frequency f o r NMR. The sample was prepared as an 8% C C l ^ s o l u t i o n , the s o l v e n t carbon t e t r a c h l o r i d e being used to e l i m i n a t e the l a r g e background 1 s o l v e n t ' ' H s i g n a l as o b t a i n e d i n the n o n - s e l e c t i v e pu lsed NMR exper iment . T h i s sample i n a 5mm od tube was degassed u s i n g the usual freeze-pump thaw c y c l e s . The f r e e i n d u c t i o n decay s i g n a l f o l l o w i n g phase s e n s i t i v e d e t e c t i o n was sampled us ing a FABRITEK FT-1064 A-D c o n v e r t e r w i t h a f i l t e r i n g t ime constant of 50 y s e c s , the d i g i t a l data c o r r e s p o n d i n g to 1 2 - b i t amptitude r e s o l u t i o n . The o f f - r e s o n a n c e s h i f t , which i s neces-s a r y to prevent the l i n e s h a p e d i s t o r t i o n s d e s c r i b e d i n s e c t i o n 5 . 2 , was a d j u s t e d to be 27 Hz us ing a f requency s y n t h e s i s e r w h i l e the Ho f i e l d s t a b i l i t y was mainta ined by a VARIAN f l u x s u p e r - s t a b i l i s e r . F i g . 5.8 Free i n d u c t i o n decay and f i n i t e F o u r i e r t r a n s f o r m spectrum f o r d imethyl n i t r o s a m i n e . 193. The r e s u l t a n t f r e e i n d u c t i o n decay s i g n a l s i t , f o l l o w i n g a f u n c t i o n zero c o r r e c t i o n as d i s c u s s e d i n s e c t i o n 5 . 5 , i s p l o t t e d i n F i g . 5 .12(a) over a t ime i n t e r v a l o f 1.5 s e e s . . Using the computer program hGTRN, the r e a l and imaginary par t s o f the f i n i t e complex F o u r i e r t r a n s f o r m f u n c t i o n are shown i n F i g . 5 .12(b) and ( c ) , r e s p e c t i v e l y . Through sub-harmonic a n a l y s i s as d e s c r i b e d i n s e c t i o n 5 . 3 , the f requency spac ing o f the output data p o i n t s i s reduced to 0.08 Hz us ing a sub-m u l t i p l i c i t y o f 8 and 8192 t r a n s f o r m data p o i n t s . In t h i s manner the r e l a t i v e l y narrow l i n e s are w e l l d e f i n e d by at l e a s t twenty data p o i n t s . For the opt imal f i l t e r t ime c o n s t a n t used a f requency dependent phase c o r r e c t i o n , of s e c t i o n 5 . 4 , was not r e q u i r e d . However, the r e f e r e n c e o f phase c o r r e c t i o n used to produce the f i n a l a b s o r p t i o n mode spectrum shown i n F i g . 5 .12(d) was computed as + 2 0 . 5 ° . This spectrum c o n s i s t s o f 288 data p o i n t s on the f requency i n t e r v a l 19.9 < x < 39.9 H z , and the chemical s h i f t i s determined as 7.57 ± 0.08 Hz. The c o r r e s p o n d i n g l i n e w i d t h a t half-maximum i s 0.7 H z , the smal l d i s t o r t i o n o f the l i n e -shape being due to the form o f the Ho f i e l d inhomogeneity which i s very d i f f i c u l t to a d j u s t by o b s e r v i n g the f r e e i n d u c t i o n decay i n the h igh r e s o l u t i o n NMR l i m i t . Thus i t has been shown t h a t a spectrum o f a q u a l i t y which i s at l e a s t comparable w i t h t h a t o b t a i n e d from the t e d i o u s slow passage s t e a d y - s t a t e technique may be very r e a d i l y obta ined us ing the r e l a t i v e l y r a p i d F o u r i e r t r a n s f o r m t e c h n i q u e . Fur thermore , a l l data i s a convenient form f o r f u r t h e r p r o c e s s i n g on a d i g i t a l computer. APPENDIX 1 L o r e n t z i a n F o u r i e r Transform P a i r . Consider the L o r e n t z i a n lineshape f u n c t i o n f ( x ) , d e f i n e d on the i n t e r v a l -°° < x < 0 0 i n terms of the r e a l v a r i a b l e x, expressed i n the form f ( x ) If , k > 0 where k = l/T^. The corresponding complex f u n c t i o n , f ( z ) , given i n terms o f the complex v a r i a b l e z = x + i y , has simple poles at z = fl i i k . These are shown, f o r 0. > 0, i n the z-plane as n The i n v e r s e F o u r i e r transform of f ( z ) , f ( t ) , may be considered i n terms of the r e a l v a r i a b l e t ^ 0. The f u n c t i o n f ( t ) i s then defined by a contour i n t e g r a t i o n f ( t ) = 1 where C i s the contour shown as p a r t o f the closed contour C i n the upper-half-plane. In the l i m i t R ->• 0 0, i t can be shown that 4 0 0 f ( z ) e x P C i t z ) d z = k J K 2 + ( x_ f l )2 . c - 0 0 The r e s i d u e f o r f ( z ) e x p ( i t z ) at the simple pole z = ft + i k i s given as R(ft+ik) = l i m z-»ft+ik \ [ z - ( f t + i k ) ] k 2 e x p l i t z ) I i  1 [ ( z - f t ) + i k ] [ ( z - f t ) - i k ] j ^ k exp [ - ( k - i f t ) t ] and i n accordance w i t h Cauchy's i n t e g r a l formula j 1 f ( z ) e x p ( i t z ) d z = 2iR(ft + i k ) k r 1 y expL-(k - i f t ) t j Thus i t ' f o l l o w s t h a t the in v e r s e F o u r i e r transfoi-m i s given as f ( t ) =T^~ exp [-dr - i f t ) t ] Z 1 2 2 corresponding to f ( x ) = \ j 1 + T 2 ( x - f t ) Z Now consider the Lorentz i a n lineshape f u n c t i o n h ( x ) , defined on the i n t e r v a l -co < x < 0 0 , expressed i n the form k(x-ft) h(x) = -y~ 1 5 — , k > 0. k + (x-ft) The i n v e r s e F o u r i e r transform h ( t ) i s defined by a contour i n t e g r a t i o n s i m i l a r to that described above, where the residue f o r h ( z ) e x p ( i t z ) at the simple pole z = ft + i k i s determined as R(^.+ik) = h k exp [ - ( k - i f t ) t ] . Thus i t f o l l o w s t h a t the i n v e r s e F o u r i e r transform i s given as h ( t ) = i %- exp [-(£• - ityt] 2 l2 corresponding to h ( x ) = T ^ ?-1 + T2(x-Q)Z APPENDIX 2 Gaussian F o u r i e r Transform P a i r . Consider the f u n c t i o n s ( t ) , d e fined on the i n t e r v a l 0 t < °° i n terms of the r e a l v a r i a b l e t, i n the form s ( t ) = [ - C i ^ - t ] where k = l/T^. The F o u r i e r transform o f s ( t ) , g ( x ) , i s given i n terms of the r e a l v a r i a b l e x on the i n t e r v a l -00 < ?( < °° by 00 o OO This i n t e g r a l may be evaluated through a contour i n t e g r a t i o n j" exp(-z ) dz, where the complex v a r i a b l e may be de f i n e d as z = e + in. and the closed contour C i n the z-plane i s considered as shown below: —6- JUJ.\> c 6 -t- e \ For the above contour, Cauchy's i n t e g r a l theorem gives exp(-z")dz = 0, and hence ft C-tfC-t1-) di. + I \ [ - ( (Ulv j ) 7 \ | <LYJ + e^ cp [-(e + d)3-] de ft Thus i n the l i m i t R •> oo, i t i s shown that CO - ° * > The f i r s t i n t e g r a l on the r i g h t hand side i s a gamma f u n c t i o n , and i n the general case i t f o l l o w s that The r e q u i r e d integrand i s now given by the s u b s t i t u t i o n P - 1 V r h - (x " a) and hence The Gaussian lineshape f u n c t i o n s f ( x ) and h(x) are now gi v e n , i n accordance with the d e f i n i t i o n of the general complex lineshape f u n c t i o n g(x) = f ( x ) - i h ( x ) , as f ( x ) = Aexp[-^T 2(x - ft)2] and r \% h(x) = A e x p [ - l 2 T 2 ( x - ft)2].^lexpCn2)dr1) o where A i s a n o r m a l i z a t i o n constant. These f u n c t i o n s define the i n v e r s e F o u r i e r transform s ( t ) i n the form s ( t ) = exp[-t/(2T 2) - i f t t ] . REFERENCES 1. N. Bloemburgen, E. M. P u r c e l l and R. V. Pound, Phys. Rev., 73_, 679, 194S. 2. F. Bloch, Phys. Rev., 70, 460, 1946. 3. H. S. Gutowsky, D. McCall and. C. P. S l i c h t e r , J . Chem. Phys. 21, 279, 1953. . 4. H. S. Gutowsky and A. Saik a , J . Chem. Phys. 21., 1688, 1953. 5. E. L. Hahn and D. E. Maxwell, Phys. Rev., _88, 1070, 1952. 6. H. M. McConnell, J . Chem. Phys. 28., 430, 1958. 7. P. W. Anderson, J . Phys. Soc. Japan, 9_, 316, 1954. 8. R. Kubo, J . Phys. Soc. Japan, 9_, 935, 1954. 9. R. Kubo, " F l u c t u a t i o n , R e l a x a t i o n and Resonance i n Magnetic Systems" D. t e r Haar, ed., O l i v e r § Boyd, London, p. 23, 1962. 10. R. C. Tolman, " P r i n c i p l e s of S t a t i s t i c a l Mechanics", O.U.P., London, p. 424, 1930. 11. U. Fano, Rev. Mod. Phys., 29, 74, 1957. 12. C. P. S l i c h t e r , " P r i n c i p l e s of Magnetic Resonance" Harper, N.Y., p. 127, 1964. 13. A. Abragam, "The P r i n c i p l e s of Nuclear Magnetism" O.U.P., London, p. 276, 1962. 14. J . I. Kaplan, J . Chem. Phys. 28, 278, 1958. 15. J . I. Kaplan, J . Chem. Phys. 29, 462, 1958. 16. S. Alexander, J . Chem. Phys., ,37, 967, 1962. 17. S. Alexander, J . Chem. Phys., 37, 974, 1962. 18. S. Alexander, J . Chem. Phys., 38, 1787, 1963. 20. R. K. Wangsness and F. Bloch, Phys. Rev., 87, 728, 1953. 21. F. Bloch, Phys. Rev., 102, 104, 1956. 22. F. Bloch, Phys. Rev., 105, 1206, 1957. 23. C. S. Johnson, J . Chem. Phys., 4l_, 3277, 1964. 24. C. S. Johnson and J . C. T u l l y , J . Chem. Phys., 40, 1744, 1964. 25. G. Binsch, J.A.C.S., 9_1_, 1304, 1969. 26. R. P. Feynman, Phys. Rev., _84, 108, 1957. 27. D. W. Woessner, J . Chem. Phys., 35_, 41, 1961. 28. L. W. Reeves and E. J . W e l l s , Disc. Far. S o c , 34, 177, 1962. 29. E. L. Hahn, Phys. Rev., 80_, 580, 1950. 30. A. Y. Carr and E. M. P u r c e l l , Phys. Rev., 94-, 630, 1954. 31. S. Meiboom and D. G i l l , Rev. S c i . I n s t r . , 29., 688, 1958. 32. Z. Luz and S. Meiboom, J . Chem. Phys., 39_, 366, 1963. 33. M. Bloom, L. W. Reeves and E. J . W e l l s , J . Chem. Phys., 42_, 1615, 1965. 34. J . G. Powles and J . H. Strange,"Mo 1. Phys., 8_, 169, 1964. 35. A. A l l e r h a n d and H. S. Gutowsky, J . Chem. Phys., 41, 2115, 1964. 36. A. A l l e r h a n d and H. S. Gutowsky, J . Chem. Phys., 42, 1587, 1965. 37. A. A l l e r h a n d and H. S. Gutowsky, J . Chem. Phys., 42, 4203, 1965. 38. A. A l l e r h a n d , J . Chem. Phys. 44-, 1, 1966. 39. H. S. Gutowsky, R. L. Void and E. J . W e l l s , J . Chem. Phys. 43, 4107, 1965. 40. P. W. Anderson and P. R. Weiss, Rev. Mod. Phys., 25, 269, 1953. 41. C. N. Banwell and H. Primas, Mol. Phys., 6, 225, 1963. 42. S. Alexander, Rev. S c i . I n s t r . , 32, 1066, 1961. 43. I. Solomon, Phys. Rev. L e t t e r s , 2_, 301, 1959. 44. E. J . Wells and K. H. Abrahamson, J . Mag. Res., 1_, 378, 1969. 45. R. Freeman and S. Wittekoek, J . Mag. Res., 1, 238, 1969. 46. S. Forsen and R. A. Hoffman, J . Chem. Phys., 39, 2892, 1963: 40, 1189, 1964. 47. F. A. L. Anet and A. J . R. Bourne, J.A.C.S., 89, 760, 1967. 48. H. M. McConnell and D. D. Thompson, J . Chem. Phys., 26, 958, 1957; 31_, 85, 1959. 49. L. P a u l i n g , "The Nature o f the Chemical Bond", C o r n e l l Univ. Press, I t h a c a , p. 281, 498, 1960. 50. S. Mizushima, "The S t r u c t u r e of Molecules and I n t e r n a l R o t a t i o n " , Acad. Press, N.Y., p. 139, 1954. 51. S. Mizushima, T. Simanonti, S. Nagakura, K. K u r a t a n i , M. Tsuboi, H. Baba and 0. F u j i o k a , J.A.C.S., 72_, 3490, 1950. 52. W. D. P h i l l i p s , J . Chem. Phys. 23, 1363, 1955. 53. J . A. Pople, W. G. Schneider and H. J . B e r n s t e i n , "High R e s o l u t i o n Nuclear Magnetic Resonance", McGraw-Hill, N.Y., 366 f f , 1959. 54. A. Loewenstein and T. M. Connor, Ber. Bunsenges. Physik Chem., 67_, 280, 1963. 55. J . Delpuech, B u l l . Soc. Chim. France, p. 2697, 1964. 56. L. W. Reeves, Adv. Phys. Org. Chem., V o l . I l l , p. 187, 1965. 57. C. S. Johnson, Adv. Mag. Res., V o l . I , p. 33, 1965. 58. K. J . L a i d l e r , Reaction K i n e t i c s , V o l . I , Pergamon Press, London, p. 85, 1963. 59. P. T. I n g l e f i e l d , E. Krakower, L. W. Reeves and R. Stewart, Mol. Phys. 15, 65, 1968. 60. R. C. Neuman, D. N. Roark and V. Jonas, J.A.C.S., 89, 3412, 1967. 61. M. R a b i n o v i t z and A. Pines, J.A.C.S., 91_, 1585, 1969. 62. R. C. Neuman and V. Jonas, J.A.C.S., 90, 1970, 1968. 63. A. E. Lemire and J . C. Thompson, Can. J . Chem., 48, 824, 1970. 64. A. A l l e r h a n d , H. S. Gutowsky, J . Jonas and R. A. Meinzer, J.A.C.S., 88, 3185, 1966. 65. C. W. Fr y e r , F. Co n t i and C. Franconi, R i c . S c i . Rend., 8_, 788, 1965. 66. r e f . 13, p. 441, 1962. 67. I. J . Lowe and R. E. Norberg, Phys. Rev., 107, 46, 1957. 68. J . S. Waugh, L. M. Huber and V. Hacberlen, Phys. Rev. L e t t e r s , 20_, 180, 1968. 69. R. R. Ernst and W. A. Anderson, Rev. S c i . I n s t r . , 37, 93, 1966. 70. R. R. E r n s t , Adv. Mag. Res., V o l . I I , 1, 1966. 71. R. E. Lundin, R. H. E l s k e n , R. A. F l a t h and R. T e r a n i s h i , App. Spec. Rev. , 1_, 131, 1967. 72. A. S t r e i t w e i s e r , "Molecular O r b i t a l Theory f o r Organic Chemists", Wiley, N.Y., p. 33, 1961. 73. R. A. Hoffman, J . Chem. Phys. 39, 1397, 1963. 74. L. C. A l l e n and J . D. R u s s e l l , J . Chem. Phys., 46, 1029, 1967. 75. J . A. Pople, D. Santry and G. A. Segal, J . Chem. Phys. 43, 3129, 1965. 76. J . A. Pople and G. A. Segal , J . Chem. Phys., 43, S136, 1965; 44_, 3289, 1966. 77. J . Del Bene and H. H. J a f f e , J . Chem. Phys., 48, 4050, 1968. 78. r e f . 12, p. 16, 1964. 79. F. Bioch and A. J . S e i g e r t , Phys. Rev., CS2, 1940. 80. I. I. Rabi, N. F. Ramsey and J . Schwingcr, Rev. Mod. Phys., 26_, 167, 1954. 81. r e f . 10, chap. 12, 1930. 82. H. S. Gutowsky and C. H. Holm, J . Chem. Phys., 25, 1228, 1956. 83. R. R. Ernst and W. Anderson, Rev. S c i . I n s t r . , _36, 1696, 1965. 84. E. Grunwald, A. Loewenstein and: £ Meiboom, J . Chem. Phys., 27, 630, 1957. 85. B. M u l l e r and M. Bloom, Can. J . Phys., 38, 1318, 1960. 86. M. Takeda and E. 0. S t e j s k a l , J.A.C.S., 82, 25, 1960. 87. M. T. Rogers and J . C. Woodbrey, J . Chem. Phys. 66, 540, 1962. 88. r e f . 53, p. 8, 104, 1959. 89. I. Solomon, Phys. Rev., 99_, 559, 1955. 90. r e f . 13, p. 272, 1961. 91. A. G. R e d f i e l d , IBM J . Res. Develop., _1, 19, 1957. 92. U. Fano, Phys. Rev., 131, 259, 1963. 93. G. Binsch, J.A.C.S., 91, 1304, 1969. 94. R. Kubo and K. Tomita, J . Phys. Soc. Japan, 9_, 888, 1954. 95. R. Br a c e w e l l , "The F o u r i e r Transform and A p p l i c a t i o n s " , McGraw-H i l l , N.Y., p. 98, 1965. 96. E. A. G u i l l e m i n , "Theory of Lin e a r P h y s i c a l Systems", J . Wiley, N. Y., chap. 18, 1963. 97. r e f . 13, p. 93, 1964. 98. Y. W. Lee, " S t a t i s t i c a l Theory o f Communications", J . Wiley and Sons, N.Y., p. 495, 1960. 99. R. R. E r n s t , J . Mag. Res., 1_, 7, 1969. 100. C. E. Shannon, Proc. I.R.E., 37, 10, 1949. 101. A. R. B i l l i n g s , E l e c . Radio Engr., 36, 70, 1959. 102. P. F. Panter, "Modulation, Noise and S p e c t r a l A n a l y s i s " , McGraw-Hill, N.Y., p. 80, 1965. 103. r e f . 95, p. 104, 1965. 104. J . G. Powles and J . H. Strange, Proc. Phys. S o c , 8_2, 6, 1963. 105. G. A r f k e n , "Mathematical Methods f o r P h y s i c i s t s " , Acad. Press, N.Y., p. 323, 1966. 106. J . R e i c h e r t and J . Townsend, Rev. S c i . I n s t r . , 35, 1692, 1964. 107. D. Ware and P. M a n s f i e l d , Rev. S c i . I n s t r . , 37, 1167, 1966. 108. J . W. Cooley and J . W. Tukey, Math. Comp., 19_, 297, 1965. 109. r e f . 105, p. 520. 110. J . C. Buchta, H. S. Gutowsky and D. E. Woessner, Rev. S c i . I n s t r . , 29_, 55, 1958. 111. K. Lu s z c z y n s k i and J . G. Powles, J . S c i . I n s t r . , _36, 57, 1959. 112. P. M a n s f i e l d and J . G. Powles, J . S c i . I n s t r . , 40, 232, 1963. 113. S. Meiboom and D. G i l l , Rev. S c i . I n s t r . , 2£, 688, 1958. 114. M. Sasson, A. Tzalmona and A. Loewenstein, J . S c i . I n s t r . , 40, 133, 1963. 115. R. J . Blume, Rev. S c i . I n s t r . , \3_2, 554, 1961. 116. W. G. C l a r k , Rev. S c i . I n s t r . , 35_, 316, 1965. 117. S. L. Gordon, and J . D. Baldeschwieler, J . Chem. Phys., 43_, 76, 1965. 118. I. J . Lowe and C. E. T a r r , J . S c i . I n s t r . , 1, 604, 1968. 119. L. J . Burnett and J . F. Harmon, Rev. S c i . I n s t r . , 89_, 1226, 1968. 120. D. F. Holcomb and R. E. Norberg, Phys. Rev., 98, 1074, 1955. 121. J . J . Spokas and C. P. S l i c h t e r , Phys. Rev., 113, 1462, 1959. 122. M. Bloom, p r i v a t e communication, 1967. 123. J . Millman and H. Taub "Pulse and D i g i t a l C i r c u i t s " McGraw-Hill, N.Y. p. 246, 1956. 124. J.M. Doyle "Pulse Fundamentals" P r e n t i c e - H a l l , N.Y., p. 1963. 125. K.H.Abramson, P.T. I n g l e f i e l d , E. Krakower and L.W. Reeves, Can. J n l . Chem. 44, 1685, 1966. 126. A.L. Van Geet, Anal. Chem. 40_, 2227, 1968. 127. L.G. P a r r a t t " P r o b a b i l i t y and Experimental E r r o r " , J . Wiley and Sons, Inc., N.Y., p. 126, 1961. 128. R.T. B i r g e , Rev. Mod. Phys. 1_9, 298, 1947. 129. K.J. L a i d l e r , "Chemical K i n e t i c s " , McGraw-Hill, N.Y., p. 67, 1967. 130. W.F.K. Wynne-Jones and H. E y r i n g , J . Chem. Phys. 3_, 492, 1935. 131. R.A. F i s h e r , " S t a t i s t i c a l Methods f o r Research Workers", O l i v e r and Boyd, L t d . , Lond., p. 37, 1946. 132. P.T. Narasimhan and M.T. Rogers, J . Phys. Chem. 63, 1388, 1959. 133. L. Saunders and W.B. Whalley, Tetrahedron 26_, 119, 1970. 134. K.J. L a i d l e r , Trans. Far. Soc. 55, 1725, 1959. 135. (a) R.E. Powell and W.M. Latimer, J . Chem. Phys. 19_, 1139, 1951. (b) J.E. L e f f l e r , J . Org. Chem. 20, 1202, 1955. 136. Z. A r n o l d , C o l l e c t i o n Czechoslov. Chem. Communs. 24_, 760, 1959. 137. W.T. Raynes and T.A. S u t h e r l e y , Mol. Phys. 18_, 129, 1970. 138. R.L. Middaugh, R.S. Drago and R.J. N i e d z i e l s k i , JACS 86, 388, 1964. 139. E. L u s t i g , W.R. Benson and N. Duy, J . Org. Chem. 32_, 851, 1967. 140. A.E. Lemire and J.C. Thompson, Can. J . Chem.'48, 824, 1970. 141. W.W. Hartman and M.R. Brethen "Organic S y n t h e s i s " , V o l . I I , J . Wiley and Sons, N.Y., p. 278, 1943,. 142. I.B. Wilson, J . B i o l . Chem. 235, 2312, 1960; 236, 2292, 1961. 143. H.P. Metzer, J . B i o l , Chem. 238, 3432, 1963. 144. H.A. Scheraga, Adv. Phys. Org. Chem. 6, 103, 1968. 145. J.S. Rigden and R.H. Jackson, J . Chem. Phys. 45_, 3646, 1966. 146. G. Schrader, B r i t i s h I n t e l l i g e n c e O b j ectives Subcommittee, F i n a l Rept. No. 714, B.I.O.S., London, 1945. 147. E. Bock and D. Iwacha, Can. J . Chem. 46, 523, 1967. 148. D.L. Hooper and R. K a i s e r , Can. J . Chem. 43, 2363, 1965. 149. S. Brownstein, JACS 91_, 3034, 1969. 150. E.W. Randall and J.D. Baldes c h w e i l e r , J . Mol. Spec. 8_, 365, 1962. 151. A.J.R. Bourn and E.W. R a n d a l l , J . Mol. Spec. 13_, 29, 1964. 152. A.J.R. Bourn and E.W. R a n d a l l , Mol. Phys. 8, 567, 1964. 153. J.A. Pople, Mol. Phys. 1_, 158, 1958; J.D. Bal d e s c h w i e l e r , J . Chem. Phys. 34_, 718, 1961. 154. L.W. Reeves and K.N. Shaw, Can. J . Chem. 49_, ( i n press) 1971. 155. R.C. Shaddick, L.W. Reeves and K.N. Shaw, Can. J . Chem.49_, ( i n press) 1971. 156. J.H. Freed and G.K. Fra e n k e l , J . Chem. Phys. 39, 326, 1965. 157. H. Shimizu, J . Chem. Phys. 40, 754, 1964. 158. B. Sunners, L.H. P i e t t e and W.G. Schneider, Can. J . Chem. 38, 681, 1960. 159. H. Kamei, B u l l . Chem. Soc. Japan 38, 1212, 1965; i b i d 41_, 2269, 1968. 160. T. Drakenburg and S. Forsen, J . Phys. Chem. 74, 1, 1970. 161. L.H. P i e t t e , J.D. Ray and R.A. Ogg, J . Mol. Spec. 2_, 66, 1958. 162. S. C a s t e l l a n o and A.A. Bothner-By, J . Chem. Phys. 41_, 3863, 1964. 163. K-I.Dahlquist, S. Forsen and T. Aim, Acta Chemica Scand. 24, 651, 1970. 164. S. Nagakura, B u l l . Chem. Soc. Japan 25_, 164, 1952. 165. E.L. Wagner, J . Phys. Chem. 63, 1403, 1959. 166. S. Nagakura, Mol. Phys. 3_, 105, 1960. 167. . F.E. M o r r i s and W.J. Orville-Thoms, J.-Mol. Spec. 6_, 572, 1961. 168. • R.L. Middaugh, R.S. Drago and R.J. N i e d z i e l s k i , JACS. 86>, 388, 1964. 169. J . Sandstrom, J . Phys. Chem. 71_, 2318, 1967. 170. M.J. Janssen and J . Sandstrom, Tetrahedron 20_, 2339, 1964. 171. A. S t r e i t w e i s e r , JACS. 82_, 4123, 1960. 172. H. E y r i n g , J . Walter and G.E. Kimball "Quantum Chemistry" J . Wiley and Sons, Inc., N.Y., p. 99, 1965. 173. H.O. P r i t c h a r d and F.Fl. Sumner, Proc. Roy. Soc. A235., 136, 1956. 174. A. S t r e i t w e i s e r and P.M. N a i r , Tetrahedron 19_, Supp. 2, 88, 1963. 175. G.W. Wheland "Resonance i n Organic Chemistry", J . Wiley and Sons, N.Y., p. 86, 1955. 176. L. P a u l i n g and D.M. Yost, Proc. N a t l . Acad. S c i . U.S. 1_4, 414, 1932. 177. R.S. M u l l i k e n , J . Chem. Phys. 2, 782, 1934. 178. W. Gordy, Phys. Rev. 69_, 604, 1946. 179. J . Hinze and H.H. J a f f e , JACS 84, 540, 1962. 180. J . Hinze, M.A. Whitehead and H.H. J a f f e , JACS 85, 148, 1963. 181. r e f . 72, p. 97, 110, 1961. 182. F.A. Matsen, JACS 72, 5243, 1950. 183. r e f . 72, p. 131, 1961. 184. R. Hoffmann, J . Chem. Phys. 39_, 1397, 1963. 185. C. Sandorfy, B u l l . soc. chim. France 615, 1949. 186. M.J.S. Dewar and H.N. Schmeising, Tetrahedron 5_, 166, 1959. 187. R.S. M u l l i k e n , C A . Rieke and W.G. Brown, JACS 63_, 41, 1941. 188. R.S. M u l l i k e n , C A . Rieke, D. O r l o f f and H. O r l o f f , J . Chem. Phys. 17, 1248, 1949. 189. L.E. Sutton, Chem. Soc. (Lond.) Spec. P u b l i c a t i o n No. 11, 1958. 190. C A . Coulson and H.C Longuet-Higgins, Proc. Roy. Soc. A191, 39, 1947; i b i d A192, 16, 1947. 191. G. Herzberg "Atomic Spectra and Atomic S t r u c t u r e " Dover, N.Y., p. 40, 1944. 192. M. Roux, M. C o r n i l l e and L. B u r n e l l e , J . Chim. Phys. 5_5_, 754, 1958; J . Chem. Phys. 37_, 933, 1962. 193. P.E. Cade, R.F.W. Bader, W.H. Hennecker and I. Keaveny, J . Chem. Phys. 46_, 3341, 1967; i b i d 5£,. 5313, 1969. 194. J.C. S l a t e r , Phys. Rev. 36, 57, 1930. 195. H. K e s s l e r , Angew. Chemie ( I n t e r n . Ed.) 9, 219, 1970. 196. D.H. Ch r i s t e n s e n , R.N. Kortzeborn, B. Bak and J . J . Led. J . Chem. Phys. SZ_t 3912, 1970. 197. C.C. Co s t a i n and J.M. Dowling, J . Chem. Phys. 32, 158, 1960. 198. R.J. Kurland and E. B r i g h t - W i l s o n , J r . J . Chem. Phys. _2_7, 585, 1957. 199. R.G. Par r "The Quantum Theory of Molecular E l e c t r o n i c S t r u c t u r e " W.A. Benjamin, N.Y., p. 21, 1963. 200. C.C.J. Roothaan, Rev. Mod. Phys. 23, 69, 1963. 201. C.C.J. Roothaan, J . Chem. Phys. 19, 1445, 1951. 202. r e f . 49, p. 191, 1960. 203. W. Gordy, J . Chem. Phys. 15, 305, 1947. 204. J.A. Pople and M. Gordon, JACS 89_, 4253, 1967. 205. W.T. Raynes and M.A. Raza, Mol. Phys. 2£,339, 1971; and references t h e r e i n . 206. F.A. Momany, R.F. McGuire, J.F. Yan and H.A. Scheraga, J . Phys. Chem. 74, 2424, 1970. 207. R.W. K i l b , C.C. L i n and E. B r i g h t - W i l s o n , J r . J . Chem. Phys. 27, 585, 1957. 208. L. P i e r c e and L.C. K r i s h n e r , J . Chem. Phys. 31, 875, 1959. PUBLICATIONS 1. L .W. Reeves and K . N . Shaw, Can. J . Chem., £ 8 , 3641, 1970. Nuc lea r Magnet ic Resonance S tud ie s o f M u l t i - s i t e Chemical Exchange. I . M a t r i x Fo rmula t i on o f the B l o c h E q u a t i o n s . 2. L .W. Reeves and K . N . Shaw, Can. J . Chem., 49, 1971 ( i n p r e s s ) . NMR S tud ies o f M u l t i - s i t e Chemical Exchange. I I . Hindered R o t a t i o n i n N , N , - d i m e t h y l carbamyl F l u o r i d e . 3. L .W. Reeves, R . C . Shaddick and K . N . Shaw, Can. J . Chem., 49, 1971 ( i n p r e s s ) . NMR S tud ies o f M u l t i - s i t e Chemical Exchange. I I I . Hindered R o t a t i o n i n Dimethyl Acetamide , Dimethyl T r i f l u o r o - a c e t a m i d e and Benzamide. 4. E . A . A l l a n , R . F . Hobson, L .W. Reeves and K . N . Shaw, J . Am. Chem. S o c . , 1971 ( i n p r e s s ) . Hindered R o t a t i o n i n N , N - d i m e t h y l Amides w i t h Halogen and Pseudo-halogen S u b s t i t u e n t s . 5. L .W. Reeves, R . C . Shaddick and K . N . Shaw, J . Phys . Chem., 1971 ( i n p r e s s ) . A De te rmina t ion o f the Hindered R o t a t i o n B a r r i e r i n Unsym-d ime thy l Selenourea and Comparison w i t h s i m i l a r Compounds. 6. K . N . Shaw and L .W. Reeves, Chem. Phys . L e t t e r s , 19 71. A . S e m i - e m p i r i c a l SCF-LCAO-MO Study o f the Hindered I n t e r n a l R o t a t i o n i n Formamide. 7. K . N . Shaw, Rev. S c i . I n s t r . , 1971 (submit ted fo r p u b l i c a t i o n ) A Simple S o l i d - s t a t e r f - p u l s e gate fo r NMR Spect rometers . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0060141/manifest

Comment

Related Items