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Multiple quantum NMR studies of solutes in liquid crystals Rendell, John Charles Thomas 1987

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MULTIPLE QUANTUM NMR STUDIES OF SOLUTES IN LIQUID CRYSTALS by JOHN CHARLES THOMAS RENDELL B.Sc.(Hons.) (M.U.N) A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department of Chemistry We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1987 ® John Charles Thomas Rendell, 1987  In  presenting  degree  this  thesis in partial fulfilment  the  requirements  for  an  of  department  this thesis for scholarly or  by  his  or  her  I further agree that permission for  purposes  permission.  Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  extensive  may be granted by the head of  representatives.  It  is  understood  that  publication of this thesis for financial gain shall not be allowed without  DE-6(3/81)  advanced  at the University of British Columbia, I agree that the Library shall make it  freely available for reference and study. copying  of  copying  my or  my written  ABSTRACT  The  spectrum  crystalline  of  solvents has  magnetic  resonance  considerable  technique  tremendously  of  been analysed  (NMR)  problem  orientation of the  The  l,3-dichloro-2-ethenylbenzene  to  and  difficulties  in  multiple  proved  problem  using  the  are  split  available  in  quantum  very  the  number  and  liquid  of complimentary nuclear  analysis of  MQNMR  NMR  the  this  spectrum  uncertain  in  the  experiment  poses  geometry  multiple  small  by  dipolar  quantum  a  a  few  couplings  spectrum  and  makes  the  capability spectrum  unavoidable  complete  While  only  the  quantum  analysis but  alone.  is dominated  has  mutiple  prevented  spectrum  relatively  (MQNMR)  spectra. The  useful  this  by  The  its complexity  complex  l,3-dichloro-2-ethenylbenzene lines  a  oriented in nematic  a and  molecule.  simplifying  measured  using  techniques.  due  partially  was  resolution  solution  of  the  spectrum  large  interactions,  the  of  limited  the of the  resolution  determination  of  the  smaller couplings difficult.  To  overcome  quantum the  spectrum  relatively  solvent,  this  spectrum measuring the  this  larger  difficulty was  simple  adapted spectrum  selective  of  a  frequency and of  experiment  developed.  was  small couplings in the  interactions.  This  excitation  After  testing  1,1,2-trichloroethane applied  l,3-dichloro-2-ethenylbenzene  the  selective  where  spectrum  information  ii  to it  proved  contributed  the  dissolved  the  without  of  much  the  experiment in  nematic  more  complex  of  interference from to  on  a  capable  greatly  multiple  the  directly any  of  eventual  analysis of the spectrum.  MQNMR  experiments  spectroscopist  can  be  must frequently  very  time  consuming  and  as  make do with very limited time  a  result  the  domain signals  from which the spectrum must be extracted. This creates a number of difficulties when  the  signals  are  analysed  with  the  fast  Fourier  transform  (FFT),  the  standard method of spectral analysis used in NMR. With these problems in mind, the  suitability  of MQNMR  time  domain signals  for analysis  by  a method of  spectral estimation due to Burg, commonly called the maximum entropy method (MEM),  was examined. By testing Burg's M E M with the MQNMR spectra of a  number of different solutes partiallj' oriented in nematic phases, it was found to be  a  useful  adjunct to  the  F F T when  dealing  with  MQNMR  interf'erograms.  While some care is required in its application, this method of spectral analysis should find important uses in the estimation of MQNMR spectra.  Solution of the spectrum yielded information on the molecular geometry and the orientation of the l,3-dichloro-2-ethenylbenzene  in the nematic solvents used. While  an extensive analysis of molecular geometry proved impossible, the information on molecular  orientation  was  examined  in  orientation data shows excellent agreement  terms  two  different  models.  The  with a recently developed model for  orientation based upon the shape of the solute.  iii  of  TABLE  OF  CONTENTS  Abstract  ii  List of Figures  vi  List of Tables  viii  Abbreviations  ix  Acknowledgements  •  x  I. Liquid Crystals and Nuclear Magnetic Resonance A. Liquid Crystals B. The NMR Experiment 1. NMR in Oriented Systems a. The Chemical Shift b. Scalar Spin-spin Coupling c. The Dipolar Spin-spin Coupling d. The Quadrupolar Interaction 2. Pulsed Fourier Transform NMR Spectroscopy C. Liquid crystals as NMR solvents a. Mechanism of Orientation b. Description of Orientation D. Multiple Quantum Nuclear Magnetic Resonance a. The Advantage of Multiple Quantum NMR b. History c. Two Dimensional NMR Spectroscopy d. The Multiple Quantum NMR Experiment e. Density matrix preliminaries f. The Three Pulse Experiment g. Product Spin \ Operators E. Summary and Aim  1 1 4 4 5 6 6 8 8 11 12 14 16 16 17 18 19 20 23 28 31  II. Experimental methods and Analysis of Spectra A. Materials B. Spectroscopic Details C. The MQNMR experiment - Variations on the Theme 1. Limitations on Resolution 2. Improvement of Resolution a. Odd/Even Order Excitation b. Selective Detection of Particular Orders c. Order Selective Excitation 3. Separation of Coherence by Order 4. Frequency Selective Irradiation D. Computer Methods Used 1. Spectral Simulations 2. Density Matrix Simulations E. A Pedagogical Example  34 34 35 36 36 38 38 39 40 41 44 45 45 47 50  iv  III. Maximum Entropy Spectral Analysis A. Introduction B. The Role of the Fourier Transform in NMR C. The Maximum Entropy Method of Burg 1. The Spectral Estimate 2. The Prediction Error Filter 3. A Sample Spectrum D. Pros and Cons of Burg's MESE 1. The Strong Points 2. The Weak Points E. Selection of Filter Order F. The Reliability of Burg's MEM in MQNMR 1. The Problem of Peak Shifting 2. The Problem of Spurious Detail G. Reduction of Experiment Duration using Burg's MEM H. The Maximum Entropy Method of Daniell and Gull I. Some Practical Examples J. Conclusion  54 54 60 64 64 65 67 69 70 72 74 78 78 81 84 89 90 94  IV. l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR A. The Problem 1. The Single Quantum Spectrum 2. The Multiple Quantum Spectrum B. The Single Quantum Spectrum — Elementary Considerations C. A Frequency Selective MQNMR experiment a. Motivation b. Preliminary Results 1. A Theoretical Description - Weak Coupling 2. Strong Coupling 3. The Frequency Selective Experiment D. Results Using DCEB 1. The Long Range Couplings  97 97 97 100 104 107 107 109 113 116 119 123 128  V. l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation A. Analysis of the 4 and 5 Quantum Spectra B. A Modified Strategy C. Spectra of DCEB in an EBBA/1132 Mixture D. Refinement of Geometry E. A Comparison of Inertial and Shape Ordering Models 1. Ordering Due to Inertial Properties of the Solute a. The Empirical Findings of Anderson b. Samulski's Model for Orientational Order 2. Ordering Due to Solute Size and Shape  132 132 134 141 146 150 150 150 153 155  VI. Conclusion  160  VII. Bibliography  163  v  List of Figures  Figure 1.1 The nematogen EBB A  2  Figure 1.2 An idealized depiction of the nematic phase  3  Figure 1.3 The basic pulsed NMR experiment  9  Figure 1.4 The spectrum of benzene in EBB A  12  Figure 1.5 The MQ spectrum of benzene in EBB A  17  Figure 1.6 The two dimensional NMR experiment  19  Figure 1.7 The multiple quantum NMR experiment  20  Figure 2.1 The DANTE pulse sequence  45  Figure 3.1 The single quantum FID and spectrum of TCE in EBB A  56  Figure 3.2 The MQ interferogram and spectrum of TCE in EBB A  57  Figure 3.3 Fourier transformation of truncated time domain signals  58  Figure 3.4 Comparison of MEM and FFT spectral estimates  69  Figure 3.5 MEM spectral estimate of TCE in EBB A - Filter order 50  75  Figure 3.6 MEM spectral estimate of TCE in EBBA - Filter order 250  76  Figure 3.7 MEM spectral estimate of TCE in EBBA - Filter order 500  77  Figure 3.8 MEM spectral estimate of TCE in EBBA - 512 data points  82  Figure 3.9 MEM spectral estimate of TCE in EBBA - 384 data points  86  Figure 3.10 MEM spectral estimate of TCE in EBBA - 512 data points  88  Figure 3.11 MEM spectral estimate of TCE in EBBA - 640 data points  89  Figure 3.12 MEM and FFT spectral estimates of 1,1-dichloroethane in EBBA ..92 Figure 3.13 MEM and FFT spectral estimates of benzene in EBBA  93  Figure 3.14 MEM and FFT spectral estimate of DCEB in EBBA  95  Figure 4.1 The single quantum spectrum of DCEB in EBBA  97  Figure 4.2 The molecule l,3-dichloro-2-ethenylbenzene  98  vi  Figure 4.3 The five quantum spectrum of DCEB in EBBA  101  Figure 4.4 The four quantum spectrum of DCEB in EBBA  103  Figure 4.5 The 200 MHz. spectrum of 1,1,2-trichloroethane in EBBA  110  Figure 4.6 Multiple quantum spectra of TCE in EBBA  113  Figure 4.7 Two quantum expectation values for TCE in EBBA  119  Figure 4.8 Two quantum expectation values for TCE in EBBA  121  Figure 4.9 Attempted frequency selective excitation of TCE in EBBA  122  Figure 4.10 Frequency selective MQ spectrum of TCE in EBBA  124  Figure 4.11 Frequency selective MQ spectrum of DCEB in EBBA  127  Figure 5.1a Experimental spectrum of DCEB in EBBA  139  Figure 5.1b Simulated spectrum of DCEB in EBBA  140  Figure 5.2 The single quantum spectrum of DCEB in 55 wt% 1132  143  Figure 5.3 The simulated spectrum of DCEB in 55 wt% 1132  145  vii  List of Tables  Table 3.1 MEM vs. FFT of TCE interferograms.  1024 data points  79  Table 3.2 MEM vs. FFT of TCE interferograms.  1024 data points  81  Table 3.3 MEM vs. FFT of TCE interferograms.  1024 data points  89  Table 5.1 Dipolar couplings and chemical shift of DCEB in various solvents  nematic 137  Table 5.2 Chemical Shift Differences in Oriented and Isotropic DCEB  138  Table 5.3 Order Matrix of 20 mol% DCEB in EBBA  141  Table 5.4 Predicted Order Matrix of 20 mol% DCEB in 55 wt%1132  144  Table 5.5 Order Parameter Data for DCEB  149  Table 5.6 Principal Moments of Inertia and Principal Axes of DCEB  151  Table  5.7 Principal Values and Principal Axis  Systems  of the Order  Matrices of DCEB  152  Table 5.8 Predicted Principal Order Parameters of DCEB  155  Table 5.9 Predicted Order Matrix of 2 mol% DCEB in 55 wt% 1132  158  Table 5.10 Predicted Principal Values and Principal Axes of the Order Matrices of DCEB 158  viii  ABBREVIATIONS  C CW DANTE  7B /27rkT. 0  continuous wave. Delays Alternating with Nutations for Transient Excitation —a method of frequency selective irradiation.  5j Dy o-DST EBBA 1132  chemical shift of nucleus i direct dipolar coupling constant of nuclei i and j . l,3-dichloro-2-ethenylbenzene N-(4-ethoxybenzylidene)-4'-rc-butylaniline Merck ZLI 1132: Commercial liquid crystal —a mixture of phenylcyclohexanes.  FT FFT  Fourier transform. Fast Fourier transform.  Jjj  scalar spin-spin coupling of nuclei i and j  K  1024  LC MEM MESE MQ NMR PEF  Liquid crystal. Maximum entropy method. Maximum entropy spectral estimation. Multiple quantum. Nuclear magnetic resonance. Prediction error filter.  SQ  Single quantum.  Sjj  ij'th element of the order matrix S  TCE  1,1,2-trichloroethane iz  ACKNOWLEDGEMENTS First  and foremost I must thank  Dr. E. Elliot Burnell  for his support and  patient guidance. I would also like to thank him for encouraging me to satisfy my  curiosity  about  some widely  disparate  areas  of science, even  though I  sometimes strayed rather far from the topic of solutes in liquid crystals.  I also wish to thank most heartily my labmates Jim Delikatny and Art van der Est  with whom I have shared the  years.  Their  friendship, help,  and  academic travail during the humor  will  surely  be  one  past several of the  most  memorable aspects of my time at U.B.C.  I  would also like to acknowledge the friendship and help of various visitors,  postdocs and students who have passed through the lab during my stay here. In chronological order of appearance these people are: Dr. Peter Barker, Alex(andra) Weaver, Dr. Peter Beckmann, Dr. John Ripmeester, Mei Kok, Dr. Gina Hoatson, and  David Gin. Thanks also to Bill Nickerson for helpful suggestions in the early  stages of the M E M work.  Many thanks to Dr. Myer Bloom and the occupants of Physics Rm. 100. Myer, especially, must be thanked for teaching me an enormous amount of NMR and what to order at the Ho-Ho.  The  support  acknowledged.  of the Without  technical staff the  cheerful  of the and  Chemistry  patient  department  expertise  of  those  must  be  in  the  Electronics shop much of what is described here would have been impossible.  x  I also wish to thank my family, in particular my father,  Charles Rendell, for  their unflagging support and confidence.  Finally, the financial support of NSERC is gratefully acknowledged.  xi  I. LIQUID CRYSTALS AND NUCLEAR MAGNETIC RESONANCE  A. LIQUID CRYSTALS  In certain respects nuclear magnetic resonance (NMR) and liquid crystals have an almost symbiotic relationship. NMR has proven itself to be an extremely useful tool in the study of liquid crystalline systems and liquid crystals have served as unique solvents for NMR structure determination for over twenty years. [1,2,3,4].  The liquid crystalline state is frequently described as an 'interesting' state of condensed matter because it has many of the bulk properties of the liquid state while retaining some characteristics, notabty long range orientational order, usually considered to be properties of crj'stalline solids. Liquid crystals are not merely interesting curios, however. They are an important state of matter, and not at all uncommon. It has been estimated that 0.5% of all organic compounds possess a thermotropic liquid crystalline phase [5]. As for their importance, one need only note how the terms 'liquid crystal display' and 'LCD' have entered our everyday jargon even though most people are uncertain of their meaning. Perhaps no more convincing argument for the importance of liquid crystals can be made than simply to note that the biological membranes are composed chiefly of molecules in a liquid crystalline state [6,7].  To briefly summarize the structure and properties of liquid crystals is a daunting task. The variety of molecules which exhibit liquid crystalline behavior and their myriad  phase  behavior  makes  a  compact  general  classification impossible.  Therefore, only the most general features of liquid crystals pertinent to this 1  Liquid Crystals and Nuclear Magnetic Resonance / 2 thesis will be presented here through the use of a representative example.  The compound N-(4-ethoxybenzylidene)-4'-n-butylaniline, or EBBA  (figure  1.1) is  what is known as a nematic liquid crystal, or a nematogen. Structurally, this molecule exhibits features common to many liquid crystals, chiefly its elongated shape and its structure being composed of a rigid aromatic core and one or both ends of the molecule consisting of floppy alkyl chains.  F i g u r e 1.1  C H 0 2  Figure 1.1  5  The nematic liquid crystal  EBBA.  This compound is a pale yellow solid up to its melting point of 35°C. As it is warmed through this temperature it melts to a translucent  yellow liquid. At  79 °C it abruptly clears to what looks like the isotropic melt of any other organic compound. It is the phase between 35° and 79°C which is the nematic liquid crystalline phase and it was exactlj' this type of behavior observed in cholesteryl benzoate by Reinitzer in 1888 [8] which led to the discovery of liquid crystals. Although discovered by Reinitzer, the term 'liquid crj^staP was coined by Lehman [9], who showed the phase exhibited birefringence, a common propertj' of crystalline materials.  Liquid Crystals and Nuclear Magnetic Resonance / 3 The structure of this phase is often depicted in the idealized manner of figure 1.2. Figure 1.2  Figure 1.2  Although the  An  idealized picture of the nematic phase.  molecules have  great  translational  and rotational freedom their  average orientation is not random, as it would be in an isotropic liquid. In the nematic phase, the mean orientation of the long molecular axes is described by a vector, ft, called the director [10].  In the absence of external fields the director varies continuously from region to region within a bulk sample of a liquid crj'stal, i.e. it describes local order. When a nematic liquid crystal is subject to an external magnetic field, however, the molecular orientation throughout the sample may be described by a single director. This is important for the NMR experiment, as will be seen shortly, for it defines a single fixed direction in space.  Liquid Crystals and Nuclear Magnetic Resonance / 4 B. THE NMR EXPERIMENT  When a nuclear spin with spin angular momentum I is placed in a magnetic field, B , the energy of interaction between the nuclear magnetic moment and 0  the external field is quantized into 1/2,  (21+1)  energy levels. For a proton with spin  we have two levels with the energy difference between them being  7hB /27T 0  where 7 is the gyromagnetic ratio of the nucleus and h is Planck's constant. For values of B  0  typicalty attainable in the laboratory, the energy difference  between these levels lies in the radiofrequency (r.f.) region of the electromagnetic spectrum. By application of r.f. of the proper frequency, transitions between these levels may be induced and detected. This is the essence of the NMR experiment.  1. NMR in Oriented Systems  Why should NMR be so useful in the study of liquid crystalline systems? The reason is that in addition to being a sensitive probe of molecular structure, NMR is very sensitive to any motional or orientational anisotrop}'. The static NMR Hamiltonian can be written quite succinctly and completely as the sum of four terms  H = H + Hj + H + H Z  d  (1.1)  Q  The Zeeman interaction, H , is by far the largest interaction (a strong magnetic z  field is an implicit assumption throughout this thesis) and, indeed, is the term responsible for the magnetic resonance phenomenon. H describes the interaction a z  system of N nuclear spins with an applied magnetic field.  Liquid Crystals and Nuclear Magnetic Resonance / 5 1 z = " I 7B I (1.2) 2TT i  H  0  i7  1 =  £ u I.  (1.3)  0  Here, 7j is the gyromagnetic ratio of nucleus i, B  0  is the applied magnetic field  and 1- is the spin angular momentum operator of nucleus i. u> = — yB 0  0  is the  angular frequency of the precession, the Larmor frequency. Another common convention has a> = +7B 0  [11,12]. The convention used throughout this thesis  0  will be that of (1.3) [13,14]. Note that (1.3) has units of frequency. All Hamiltonians appearing in this thesis will be written in frequency units.  The remaining terms in the Hamiltonian are relatively small and it is nearly always possible to treat them as first order perturbations  of the Zeeman  Hamiltonian.  a. The Chemical Shift  If a nucleus is embedded in a molecule the magnetic field felt at the nucleus differs from the applied magnetic field by an amount which is termed the chemical shift o, and the Hamiltonian is modified thusly: 1 H  z  =  I "wjd 2ir  ap Ij  (1.4)  1  This shift is a second rank tensor and the value observed is dependent on a molecule's orientation with respect to the magnetic field. Rapid reorientation in isotropic solution averages this interaction over all possible orientations to a single sharply defined value. This is the familiar and dominant term in high resolution  Liquid Crystals and Nuclear Magnetic Resonance / 6 NMR. It should be borne in mind though that orientation of the bond may result in an effect quite different from that seen in isotropic solution.  b. Scalar Spin-spin Coupling  The  indirect  spin-spin  coupling H j is  typically the  smallest  of the  four  interactions and is the second mechanism responsible for the structure of high resolution spectra. In its most general form it may be written  Hj  = Z  Z  i. i<j  I1 . - J . L  (1.5)  It represents the coupling of two nuclei and is mediated by the electronic structure  of  any  intervening  bonds.  This  interaction  also  depends on  the  orientation of the molecule with respect to B , and, like the chemical shift is 0  averaged to a single well defined value in isotropic solution. Fortunately, what anisotropy there is in this interaction is small, at least for protons, and values of this coupling in anisotropic solution may be assumed to be equal to values found in isotropic solution [15].  c. The Dipolar Spin-spin Coupling  Effects of the dipolar, or direct, spin-spin interaction, H j , are not seen in high resolution NMR spectra. Rapid isotropic motion averages the dipolar Hamiltonian to zero. Anisotropic motion, however, results in a non-zero average value for this interaction,  which  may  be  quite  large.  In  high  magnetic fields at  high  temperature it is usually sufficient to deal with the so called 'secular' form of the dipolar Hamiltonian  Liquid Crystals and Nuclear Magnetic Resonance / 7 1 7-7ih 3cos 0.j-l - Z . 2 — i — (31 l z  Hd=  J  1  ] <  J 2 4rr  r-  2  zi  3  1  -  2i  J  I-1-). 1  (1.6)  J  The assemblage of constants, angles and distances in front of the parenthesized spin operators in (1.6) is referred to as the dipolar coupling constant, D^. Of course a small molecule in a liquid crystalline environment undergoes vibration and rapid anisotropic motion so the geometric parameters 6^ and r-j must be averaged over all possible values. It is often assumed that r and 6 may be averaged separately and so the dipolar coupling is most often seen in a form in which it is separated into two parts, 1.  a displacement part which depends on the internuclear distance as 1/r . 3  7i7jh —  . 1 . (  (1.7)  >  and 2.  an angular part which depends on the angle the internuclear vector makes with the applied field  i<3cos 0jj z  The constant 7-7jh/47r  2  1>.  has a value of 120.067 kHz A  (1.8).  3  for two protons.  To summarize, the strength of the direct dipole-dipole coupling between two nuclei depends on both the internuclear distance and the orientation of the internuclear vector with respect to B  0  in a purely geometrical fashion. The strength of this  Liquid Crystals and Nuclear Magnetic Resonance / 8 interaction  coupled with  the  precision with which NMR line  positions  may be  measured makes this interaction a sensitive and relatively easily interpreted probe of molecular geometry and orientation.  d. The Quadrupolar Interaction  The  H Q , is  quadrupolar interaction,  also  averaged  to  zero by  rapid isotropic  motion. This interaction is the result of the interaction of a quadrupolar nucleus with an electric field gradient and has the form  HQ=  I.V-I.  (1.9)  2I(2I-l)h This  interaction  may  be  extremely  large,  often  in the  range  of hundreds of  kiloHertz to megaHertz, depending on the nucleus, its electronic environment and orientation with respect to B . The quadrupolar interaction is a sensitive probe 0  of a bond's electronic structure and of its orientation, again with respect to the applied magnetic  field, but it lacks the  simple  and direct interpretation of the  dipolar coupling. The quadrupolar interaction is only possible for nuclei with spin greater than one half, and will not play a role in this thesis.  2. Pulsed F o u r i e r Transform  Although  much  extraordinarily  NMR  Spectroscopy  useful  and  fundamental  NMR spectroscopy  has  been done by irradiating the nuclear spin system with r.f. carefully selected to match the difference experiments (FTNMR).  in energy  levels between spin states, most modern NMR  utilize the technique of pulsed Fourier Transform NMR spectroscopy The  foundation  of  this  technique  will  be  discussed  more  fully  in  Liquid Crystals and Nuclear Magnetic Resonance / 9 chapter 3, but for the moment a more qualitative description with the aid of figure 1.3 will give some feeling for the basics of the experiment. Figure 1.3  Figure 1.3 The basics of the pulsed NMR experiment, a) At equilibrium, before the pulse is turned on. b) Precession in the rotating frame during the pulse, c) The effect of a fjr/2) pulse, d) Subsequent evolution due to H-..  At equilibrium in a static magnetic field B , the nuclear spins precess about the 0  applied field, conventionally taken to be along the z axis. Now consider the effect of an intense r.f. pulse applied perpendicular to B  0  for a time t . The process  is more easily understood if viewed in a frame rotating at u>, the angular  Liquid Crystals and Nuclear Magnetic Resonance / 10 frequency of the applied r.f. (figure 1.3a) If we assume I7B, | > > |CJ — CJ|, then 0  in this rotating reference frame the magnetization appears to precess about B , , which for argument's sake is applied along the rotating y axis. If B , is left on for a time tp the magnetization will precess through an angle 8 = — 7 B , tp as indicated in figure 1.3b. If 6 = 90°, at the end of the pulse the magnetization will lie along the rotating x axis. The notation used to describe such a pulse is (7172)^  or (90°)y. The rotating magnetization now evolves under the internal  Hamiltonian (chemical shifts and couplings) and, of course, in the laboratory frame this evolution occurs at the r.f. frequency. This rotating magnetization may be detected by the voltage it induces in a coil surrounding the sample. This voltage deca3s with time and is called the free induction decay, or FID. As will r  be shown in chapter III, the detected signal may be Fourier transformed to yield the frequency spectrum of the spin system.  To reiterate, all terms in the NMR Hamiltonian are dependent on orientation with respect to the applied magnetic field. Of the possible interactions, H Hj may be present in either isotropic or anisotropic media.  z  and  and H Q are  present staticallj' only in directionally ordered systems. The transition from an isotropic environment to an anisotropic one therefore  represents a qualitative  change in the NMR spectra observed, and quite a drastic change at that. Conventional NMR spectroscopy of pure nematogens is, however, both difficult and rather uninformative. The dipolar coupling is a long range interaction and in principle each nucleus in a molecule is coupled to every other nucleus, ln a molecule like EBBA which contains 2 2 protons, the dipolar coupling gives rise to 2  2 2  energy levels. The proton NMR spectrum of a pure nematogen is a broad  Liquid Crystals and Nuclear Magnetic Resonance / 11 and intractable continuum.  C. LIQUID CRYSTALS AS NMR SOLVENTS  Where NMR spectroscopy shines is in the spectroscopy of small solute molecules dissolved in liquid crystal solvents.! This technique is sometimes referred to as LCNMR or LXNMR. When a small molecule is dissolved in a liquid crystal the solute molecule takes up some preferred average orientation due to the anisotropic environment it is in. This intramolecular  dipolar  anisotropic average  Hamiltonian is  no  longer  orientation means averaged  to  that  zero.  the  Rapid  translational diffusion averages the intermolecular dipolar Hamiltonian to zero so for a system of six spins, for example, we have only 2 of the broad featureless  6  energy levels. Instead  spectrum of a pure nematogen we now see a rich  spectrum of resolved lines. This is illustrated in figure 1.4 which shows the spectrum of benzene dissolved in EBBA. Because of the dependence of the dipolar couplings on molecular geometry and orientation this spectrum may be analyzed to yield accurate structural parameters. An important difference between molecular structures determined by LCNMR and many other methods of molecular structure determination is that LCNMR is especially good at locating the positions of protons in the molecule. Since protons have the largest magnetic moment of all stable nuclei LCNMR is especially good at locating them. Many other methods of structure  determination such as X-ray crystallograplry, electron diffraction, or  microwave spectroscopy have some difficulty determining structural parameters for protons due to the low electron density around the proton or its small mass. t Small in this context refers to the size of the spin system of the small molecule and may typically be taken to mean spin systems of 8 spins 1/2 or less.  Liquid Crystals and Nuclear Magnetic Resonance / 12 Figure 1.4  JUIUJIJ  JL/UAJ 1500  1000  500  0  Frequency  ULJU 500  1000  1500  (Hz.)  Figure 1.4 The single quantum spectrum of benzene dissolved in EBB A.  a. Mechanism of Orientation  Although the orientation of small solutes in a liquid crystalline environment has been recognized and exploited for almost  25 years  the reasons for solute  orientation are still a matter of active research. A number of mechanisms have been proposed over the years to account for this orientation. Some of these have invoked specific interactions between the liquid crystal and the solute such as hydrogen bonding [16]. While these may account for orientation in specific liquid crystal/solute mixtures thej' lack generality.  More general models for the mechanism of solute orientation have included  Liquid Crystals and Nuclear Magnetic Resonance / 13 dispersion forces [17] and moments of inertia [18]. The model based on the moments of inertia of the solute is appealing for its simplicity. The model is based on the empirical finding by Anderson that the principal axes of the order matrix and the principal axes of the inertia tensor are coincident, or very nearly so, for a series of 27 substituted benzenes. In order to predict the orientation of a solute one therefore only has to calculate and diagonalize the inertia tensor of the solute, a very simple calculation. This model has met with some success [19] but it does not, for instance, explain peculiarities in the orientation of solutes when the liquid crystal solvent is changed [20].  Recentty, the interaction between a solute's molecular quadrupole moment and the anisotropy in a mean electric field gradient  in the liquid crystal has been  demonstrated to be an important factor in the orientation of some very small solutes such as H  2  and N  2  [21].  As the size of the solute is increased the molecular quadrupole moment  -  electric field gradient mechanism no longer adequately describes the orientation and other factors begin to play a significant role. Obvioushy, repulsive interactions dependent on the size and shape of the solute molecule can be expected to play a major role in molecular orientation in an anisotropic condensed phase. The effect of molecular size and shape on orientation in anisotropic phases has been modelled by considering the liquid crystal to be an elastic tube surrounding the solute and considering the solute to be a collection of van der Waals spheres [22]. The solute then orients so as to minimize the distortion of the elastic liquid crystal.  Liquid Crystals and Nuclear Magnetic Resonance / 14 These mechanisms for orientation will be explored in greater detail in chapter V.  b. Description of Orientation  While the dipolar couplings could be described purely in terms of the the r^ and the  such a description would be clumsy and of limited utility. An obvious  and much more useful description can be had in terms of a molecular geometry and molecular orientation with respect to the magnetic field. The description of a molecular  geometry  needs  no explanation  but  the  description  of molecular  orientation may be less familiar. The most commonly used formalism is that due to Saupe [2], and this is the formalism which will be used here.  What we measure in an NMR experiment is the value of some molecular tensorial propertj , be it chemical shift, dipolar coupling, etc., with respect to the r  magnetic field in a fixed laboratory reference frame (x,y,z). More specifically, we measure only the zz component, T , of some tensor T. In order to relate these zz  values to a reference frame which is fixed in the molecule we need to carry out a coordinate transformation [23]. Saupe found it advantageous to characterize the transformation  in terms of the elements,  S  of a matrix called the order  matrix. The S ^ are defined as  S  where cos0  Q  ab  = i <3cos0  G  cosd  b  -S > ab  (1.15)  is the angle between the molecule fixed c axis and the magnetic  field direction. The angled brackets denote an ensemble average and so the order matrix is a measure of the average orientation of a molecule in the laborator3'  Liquid frame. The  diagonal elements  Crystals and Nuclear Magnetic Resonance / 15  of the order matrix now  have  the same form  as  the angular part of the dipolar Hamiltonian, (1.8).  The  order matrix has  maximum average  3X3  but  it is symmetric  and  traceless  of only five independent, non-zero elements are required orientation  of any  even further [24].t there  dimension  is  only  one  independent  there are two  In  the  S . zz  may  element  in  the  order  If the molecule has  of  to specify the  reduce  matrix,  two  this  number  any  and  this  of the order matrix, and  sA'mmetry  simplifying  is  perpendicular planes of  three independent elements are required  absence  a  a three fold or greater axis of symmetry  independent elements  is one plane of symmetry orientation.  molecule. Symmetry  If a molecule has  conventional^' taken to be symmetry  rigid  so  if there  to describe the there  are  five  independent nonzero elements of the order matrix.  We  now  have  a  reasonably  molecular structure and of  dipolar  interaction single  couplings  observed  the spectrum  is  often  under  offset  assumption  may  that  be the  isotropic  is dissolved  relating  dipolar  by  the  complexity  of benzene  conditions in a liquid  is  of  couplings  high symmetry  described molecule  in terms is a  spectra  split  into  a  of the benzene  trivial  to  of  said  this The 72 that  compared  molecule means that  order parameter  hexagon).  1.4.  total  crystal. It should be  of only one  regular  the  shown in figure  looks imposing, the analysis is relatively  to many other systems. The spectrum  of  orientation, but the simplicity present in the interpretation  transitions when benzene  the  method  generates. Witness the spectrum  line  although  simple  Three  J  (with the  couplings  are  t Symmetrj' here refers to the symmetry of the geometry defined by the nuclei which have non-zero spin, not to the symmetry of the molecule as a whole [25].  Liquid Crj^stals and Nuclear Magnetic Resonance / 16 required to complete the description of the spectrum. Although the spectrum of benzene/EBBA  is governed by only  one order  parameter, the  profusion of  transitions is confusing and the large number of energy levels involved makes a detailed analysis difficult without the aid of a computer.  D. MULTIPLE QUANTUM NUCLEAR MAGNETIC RESONANCE  a. The Advantage of Multiple Quantum  The  NMR  technique of multiple quantum nuclear magnetic resonance, or MQNMR,  [26,27,28] can reduce this spectral complexity considerably. Consider figure 5, which shows the multiple quantum spectrum of benzene dissolved in EBBA. In particular notice the simplicity of the 4, 5, and 6 quantum regions. The number of lines observed is easily rationalized by considering the number of ways to flip a given number of spins. For a six spin system there is, of course only one way to flip all six spins and so we get one six quantum line. Similarly, if one considers the simultaneous flip of  five  of the six spins, it is obvious that the  remaining spin may be 'up' or 'down', hence the single doublet observed in a system of six equivalent spins. This naive interpretation should not be taken quantitatively for could give a misleading appreciation of what is actually going on.  In strongly coupled systems in particular it is incorrect to refer to a  particular isolated spin, but it is a handy picture to keep in mind. Another useful spectra  method for predicting and understanding the is the  analogy between  multiple quantum  appearance of MQNMR coherence  and deuterium  labelling drawn b}' Warren and Pines [29] for situations where chemical shifts are absent.  Liquid Crystals and Nuclear. Magnetic Resonance / 17 Figure  1.5  two quantum  zero quantum  four quantum  five quantum  ~i  0  5000  10000  i  i  six quantum  I — i — i — i — i — 1 — i — i — i — i — I — i — i — i — i — I — i — i — r  15000  20000  25000  30000  F r e q u e n c y (Hz.)  Figure 1.5 The multiple quantum spectrum of benzene dissolved in  EBBA.  b. History  Multiple quantum NMR spectroscopy has a surprisingly long history. Reports of multiple quantum phenomena in NMR go back as far as 1956 [30]. Multiple quantum transitions are, in fortuitous cases, easily observed on a continuous wave spectrometer, all one has to do is turn up the power of the radiofrequency transmitter beyond the range of linear response of the spin system. With this ease of observation many applications were reported. Indeed, a standard text published in the 'golden age' of CW NMR puts multiple quantum transitions on an equal footing with double resonance and spin echo techniques [31].  Liquid Crystals and Nuclear Magnetic Resonance / 18 With the development of pulsed Fourier Transform methods [32,33], however, study of multiple quantum phenomena in NMR lapsed. The fundamental relation which motivates the application of the pulse FT method to NMR obviates any observation of MQ effects in simple one pulse spectroscopy [27].  The advent of two dimensional NMR (2DNMR) resurrected the study of multiple quantum effects in NMR. In the paper which could be said to mark the start of 2DNMR [34], Aue et al. point out the feasibility of observing multiple quantum effects.  Shortly thereafter,  numerous  papers appeared  detailing practical and  theoretical aspects of the problem. This research was centered in several groups, but the work of most relevance for this thesis came from the group of Pines and coworkers at the University of California at Berkeley.  c. Two Dimensional NMR Spectroscopy  Multiple quantum NMR in its modern incarnation is one of a class of what are termed 'two dimensional' experiments [14,35,36]. The salient feature of a 2DNMR experiment is that data is acquired as a function of two time variables. The general form of such an experiment is considered to be of four parts depicted in figure 1.6. The preparation period puts the system into a desired state, which then evolves during the predictabty named evolution period, t,. At the end of the evolution period the mixing period puts the spin system in a state suitable for observation, and the state of the system is sampled during the detection period, t . 2  This experiment is repeated for many values of t,,  transformation with respect to t, as well as t  2  and Fourier  reveals details on the evolution  of the system during t,. The nomenclature of preparation, evolution, mixing, and  Liquid Crystals and Nuclear Magnetic  Resonance / 19  F i g u r e 1.6  Preparation  Figure  1.6  detection  The four  periods  Evolution  Detection  elements of the two  is necessarily  vague.  dimensional  NMR  experiment.  The profusion of 2D experiments  often  gives rise to pulse sequences  which seem to bear little in common to the casual  eye.  can, however,  All  2D  experiments  aforementioned  time  periods.  The  be analyzed  basic  multiple  in terms  quantum  of these  experiment  four is  particularly simple, as we shall see.  d.  The Multiple  Quantum  Multiple quantum  effects  formalism. While with  the spin  Experiment  spectra can be observed with an experiment  appropriately spaced quantum  NMR  ir/2 pulses (figure  in NMR a system  1 case  as simple as three  1.7). The theoretical discussion of multiple  spectroscopy  use of the density  matrix  of 3 level Bloch equations has been developed  to deal  [37], such  demands  a treatment  of higher spin, as will be required.  is not easily  extended  to systems  Liquid Crystals and Nuclear Magnetic Resonance / 20 Figure 1.7  (7T/2)  (TT/2)  (TT/2)  t  T  Figure 1.7 The three pulse multiple quantum NMR experiment,  e. Density matrix  preliminaries  The density matrix provides a powerful formalism for describing the state and the time evolution of complex quantum mechanical systems  [38,39].  Recall that  the state of a quantum mechanical sj'stem may be described by solving the time dependent Schrodinger equation  for the state vector  \fr(t) describes the  system at any time t and may be expanded in a complete set of orthonormal basis functions {$}.  |tf(t)> = I c  n  |0 >.  (1.16)  n  In this representation the time dependence of \p is embodied in the {c }, the {0 } n  n  are independent of time. In order to calculate the value of some observable O, we calculate matrix elements  (1.17)  Liquid Crystals and Nuclear Magnetic Resonance / 21 In calculating the time evolution of a system the matrix representation of the operator O will not change. The time dependence of the expectation value is contained in the c*c„. m n The cnc* m may be arranged to form a matrix and this J  0  matrix may be regarded as the matrix representation of an operator P  cn c* m =  r <<j> \P\4> n< i m Y  >.  (1.18) '  An isolated spin system may be usefully treated in this way. An experimental observation in NMR, however, is typically made on a sj^stem of 10  1 8  - 10 ° 2  molecules. If there are no intermolecular interactions (i.e. the spin S3'stems are non-interacting) as is the case in high resolution NMR or NMR in liquid crystals, we may perform an ensemble average over the spin systems to yield the density matrix, p t (1.19) "  =  c  n m* c  This matrix, with the ensemble average, is known as the density matrix, and a density operator may be defined analogously to the operator P. What we now have in the density matrix is a convenient and compact statistical description of what may be an exceedingly complex quantum system. Properties and applications of the density matrix appear in many excellent references, and so only the most basic properties of relevance to this thesis will be stated here. Important points to note about the density matrix are: 1.  It is Hermitian.  tThe ensemble average is a thornier problem in solids where, due to the dipolar coupling mechanism, the spin system may have to be taken as the entire 1 0 spins. 20  Liquid Crystals and Nuclear Magnetic Resonance / 22 2.  The equation of motion of p is 27Ti  dp/dt =  [p,H].  (1.20)  h If H is independent of time this yields the formal solution  p(t) = e -  3.  2  7  n  p(0) e  m  2 l r [ R t  (1.21)  To calculate the expectation value of any operator  <0>=Tr{pO}.  4.  .  (1.22)  If the density matrix is in its eigenbasis representation the diagonal matrix elements  represent the probability that the system is in state <f>^. The  off diagonal elements p- represent the probability of finding the system in a coherent superposition [40] of states 0j and  While  (1.19)  represents the density matrix one often encounters variants in the  NMR literature which may differ in subtle but significant waj's from its proper definition.  At equilibrium at high temperature the density operator describing a nuclear spin system may be approximated by  7  1  hB  0  27rkT  I. z  (1.23)  Liquid The for  series  expansion  typical  [11]  this  term  and is often  modified  form  Another  a very  (containing dropped.  This is known good  as the high  operator)  The important matrix  point  is that  plays  no  ~  10"  5  temperature  for most  applications. role  to remember dropping  to a traceless rather than  Resonance / 23  as hw /2;rkT o  approximation  the unit  of the density  changes the density matrix is the form  at the first term  temperature.  It is normally  The zero'th  dynamics  may be stopped  fields at room  approximation.  Crystals and Nuclear Magnetic  in the  when  the unit  using matrix  a unit trace matrix. This  of the density operator which will be used in this thesis.  way in which  liberties are frequently  taken  with the density  matrix is  that the constant 7hB /27rkT is often eliminated as a notational convenience, but 0  the  resulting  quantity is still  referred  to as the density  this constant, however, removes the ensemble average is the operator P  with  density  operator.  of (1.18). Strictly  Throughout  this  speaking  thesis  matrix  [41]. Removing  and thus what we are left it should not be called the  the constant  7hB /27rkT  will  0  be  represented by an italicized upper case 'c', C.  f. The Three Pulse Experiment The  mechanism of the multiple quantum  very set  compact  and general way through  for the density  matrix. Spherical  properties are well documented result  of their  easily  handled  manipulations we perform  NMR  experiment  can be explained in a  the use of spherical tensors form  [42,43,44].  on a spin S3'stem  a complete  Their popularity  behavior under  tensors as a basis basis  in NMR  rotations which  set whose  is largely a  are, after  with r.f. pulses. If we choose  all,  the  to use  Liquid Crystals and Nuclear Magnetic Resonance / 24 spherical tensors as a basis, the equilibrium density matrix is represented!  p(0) = C Tj(a).  (1.24)  The first pulse in the multiple quantum experiment (we will assume a (ff/2) pulse) will  result in terms  in T^ and  (a y pulse gives rise to x  magnetization, I ) . X  At the end of the first pulse the density matrix can be written  p( + ) = C I CjTj(a) + c ^ T ^ o ) .  (1.25)  The system then evolves under its internal Hamiltonian for a time r,  p(r) = e -  2 7 r i H  int  p( + ) e ' int  T  2lT lH  T  .  (1.26)  It is during this period that the bilinear terms in the spin operators play an essential role for they bring the system into a state at which the second pulse creates multiple quantum coherence. If H contained onlj' linear terms then the density matrix could only ever possess I in the dipolar Hamiltonian (or I  Z  2  X  and I  V  character. However, terms in  in the quadrupolar Hamiltonian) have a  iz  dramatically different effect. The notation T^ indicates this is the zero'th component of a first rank tensor. The first rank spherical tensor operators have the following relation to the more familiar angular momentum operators: T Q = I ; Tj = I + ily = I ; = I + ily = I . The a is a label required to identify all basis set members needed to span a given rank in multispin systems. Z  X  X  +  Liquid Crystals and Nuclear Magnetic Resonance / 25 In terms of spherical tensor operators, the Zeeman Hamiltonian is  H  z  = Z  (1.27a)  WJTQ,  and the dipolar Hamiltonian may be rewritten  H  d  = Z Z D.. T&ij). i i<j «  (1.27b)  u  In (1.27b) the label a has taken on the form of a pair of nuclei. Expansion of the exponentials in (1.26) yields  p(r) = p( + ) + f r [ p ( + ),H ] + i r [ [ p ( + ) , H ] , H ]  +  2  int  int  int  (1.28)  The first commutator in the expansion is  I.  Tt(T}(ij) + T (ij)), Tj(i)] + Z . Z . D r[(Tj(ij) + T (ij)), T§(ij)]. i i<J 1  1  1  The commutators  ij  1  (1.29)  in (1.29) are easilj' evaluated with the commutation relation  t l , T ] = [ T ^ p T ^ ] = [(l + m)(l±m + l)] 5 T Z  ±  m  The  first  commutator  will  only  yield  l m±r  (1.30)  in (1.29), the one arising from the Zeeman Hamiltonian,  the operators  T^_j and T ^ . lt is the second commutator  in  (1.29), the one arising from the dipolar Hamiltonian, which is essential to the multiple quantum experiment. With the use of (1.30), it is readily evident that  Liquid Crystals and Nuclear Magnetic Resonance / 26 2  2  this commutator will give rise to terms in T ^(ij) and T .^(ij). Evaluation of higher terms in the expansion (1.28) requires considerably more algebra but the net result is that they give rise to terms T + ^ up to Z<N, the number of spins in the system.  Therefore, toward the end of the preparation period, immediately before the second pulse, the density operator may be written  p = I  c  T (a).  (1.31)  Z  / ± 1  ±1  The second pulse now creates multiple quantum coherence, for -iirIy/2 Z T  m  iffIy/2 _  z  D  Z  mm  jl  (1 3 2 )  m  The D , are the Wigner rotation matrices. We now have a density operator in l  m  m  which all operators T^ may be present, for Z,m<N. Such operators represent m quantum coherences. The presence of the factor  TZD|J  in front of the spin  operators in (1.29) means that a preparation time on the order of the inverse of the couplings is required for MQ coherence to develop to an appreciable extent in a group of coupled spins.  The system is again allowed to evolve under its internal Hamiltonian and it is during this period that evolution of the multiple quantum coherences occurs. This evolution  is ' not directly detectable.  The NMR experiment  only  couples to  Liquid Crj'stals and Nuclear Magnetic Resonance / 27 macroscopic vector magnetization and therefore directly detect in the NMR experiment are < I  the only observables we can >  X  and <I >. All other states of V  the system are invisible. An m quantum coherence will evolve under any Zeeman and chemical shift terms as m&u). This is easily seen from the commutation relation  [I T' ] = mT z>  m  (1.33)  l m  and the equation of motion for the density matrix (1.20). Hence the n quantum coherences will appear in the spectrum clustered about a frequency of nAco.  The third pulse in the sequence now redistributes information contained in the various multiple quantum operators in the manner of (1.31), and some of the multiple quantum information is rotated back into single quantum operators. This single quantum magnetization which has been labeled by its multiple quantum precession in t|  is now detected in t . This pulse sequence is repeated for  regular increments in  2  and the resulting time domain signal as a function of  t, is referred to as a multiple quantum interferogram. The time domain signal may be Fourier transformed with respect to one or both of t  y  and t . It is the 2  Fourier transform in t, of the components precessing at nAh) which leads to the experimentally observed n quantum signal. Acquiring the signal for manj' values of t, can be a time consuming process. In MQNMR one must often make do with far fewer data points than one is accustomed to having in single quantum NMR simply because of the limited time available for an experiment. This shortage of data points can have a great effect on the resulting spectrum and is  Liquid Crystals and Nuclear Magnetic Resonance / 28 the subject of chapter III.  g. Product Spin •£ Operators  While spherical tensor operators are useful for a general description of MQNMR, they are not convenient for treating many NMR problems in multispin systems. In choosing a spherical tensor operator basis the connection with the individual spins in the system is lost. For treating systems containing a number of individual spins a product spin £ operator basis proves very useful [45,46,47]. The basis set is easily constructed and the behavior of the members of the basis set follows some very simple rules, at least for weakly coupled spins. To form the members of the basis set, Sorenson et al. use the definition  where N = total number of spin \ nuclei in the spin system k = an index running over all nuclei V  = x, y, or z.  q = number of spin £ operators in this particular product, a = 0 or 1 depending on whether or not spin k is involved in this product.  As a practical example, in a three spin system the longitudinal magnetization of spin 3 would be represented by the product  Liquid Crystals and Nuclear Magnetic Resonance / 29 I  3 Z  = d ) l (D  2  (I ) . z  3  Here the numeral subscripts serve to identify the spin to which a single operator pertains. The matrix representation  of the product is formed by taking the  Kronecker product of the matrix representations of the individual spin operators.  Much of the usefulness of the product spin \ operators lies in the fact that evaluating  the  time  evolution of the  product  operator  merely  consists of  evaluating the time evolution of the individual spin operators which comprise the product. For example consider the effect of a (n ^) pulse on the operator 1  4 I I I y .A (TT/2) 1 x  and 4I  1 x  2 z  I I  2 z  pulse converts I  3  into — I , I  1 x  1 z  is converted into I  2 z  2 x  remains unchanged. The result of the pulse is that the operator  3 V  l3y  is converted to — 4 l I I y . 1 z  2 x  3  Evolution under the chemical shift Hamiltonian is as expected.  exp(-J6J Ikzt) k  I  k x  exp(iw I k  k z  t)  IjjXcos(£Jkt)  =  +  I  fe  (1.34)  sin(tJkt).  (1.35)  Evolution under a weak spin-spin coupling is  exp(-/7rJ 2I I t) I kl  kz  lz  k x  exp(f7rJ 2I I t) = I cos(7rJ jt) + I I sin(irJ t). kl  kz  lz  kx  k  ky  lz  kl  (1.36)  Liquid Crystals and Nuclear Magnetic Resonance / 30 exp(-iirJ 2I I t) I kl  kz  lz  exp(^J 2I I t)  k y  kl  kz  = I cos(7rJ t) - I^sinCffJ^t).  lz  ky  kl  (1.37)  Terms such as I^y'lz are referred to as y magnetization of spin k antiphase with respect to spin 1.  Product operators containing more than a single spin operator can be interpreted by writing the constituent single spin operators in terms of raising and lowering operators. Take, for example, the two spin operator I I . 1 x  2i  1 x  i  = 2(1}  2 y  = 2l\V  The terms  2I,I  2  -  2I,I  2  + i i )( i  2  represent  -  2  2 v  - r) 2  2 i ; i ; + 2I,IJ  - 21115  2Q coherence and  2I,I  2  -  2I,I  represents zero quantum coherence. The linear combination 2 I I  2 v  represents pure two quantum coherence and 2 I I y  represents pure  l x  1x  zero  quantum  coherence.  Product  2  — 2l!yl  operators of arbitrary  2 x  + 21 i y l  2  2 x  complexity can be  interpreted in a similar fashion.  As shown by the preceding explanation the general features of the MQNMR experiment  are easily explained, but an explicit detailed calculation is rather  cumbersome for all but the simplest systems. A spin 1 system is easily treated, as the density matrix for this system has dimension 3X3 and one only has to deal with 8 operators, but as spin numbers increase analytical treatment of the process soon becomes unmanageable. With a system of three spin £ nuclei, the  Liquid Crystals and Nuclear Magnetic Resonance / 31 density matrix grows to be 8X8, and to treat the situation fully one would have to follow the time evolution of 63 operators. Analytical treatment of a six spin problem is obviously out of the question, and one must resort to computer methods. These will be described in chapter II.  E. SUMMARY AND AIM.  The technique of MQNMR appears very promising in the spectroscopy of solutes dissolved in liquid crystals, but apart from some elegant demonstrations  of its  potential it has not found its way into everyday use. The work in this thesis applies the technique of MQNMR to a problem which would not yield to more conventional NMR techniques.  The molecule l,3-dichloro-2-ethenylbenzene is a molecule which, when partially oriented in a liquid crystal gives rise to a spectrum which is extremely difficult to analyze. The difficulty  is not primarily in the number  of lines in the  spectrum, for there are only about 200 of them. This number, while large, is by no means overwhelming by LCNMR standards. Rather, the difficulty lies in the number of unknowns which govern the line positions. First, the molecule has no symmetry, and therefore description of its orientation requires all 5 elements of the order matrix. Second, the molecule has 5 anisotropic chemical shifts which may be significantly different  from the values observed in isotropic solution.  Third, the estimated geometry must be regarded as approximate at best. While the geometries of styrene derivatives have been the subject of a number of experimental and theoretical studies,  the determined  been regarded as being fairly uncertain. [48,49,50].  structures have generally  Liquid Crystals and Nuclear Magnetic Resonance / 32 The great majority of molecules studied thus far by LCNMR have been those for which  an  accurate  starting  structure  has  been  available from  some  other  technique and for which symmetry in the molecule reduces the number of order parameters to one or two. This problem starts with an uncertain geometry, 5 unknown order parameters and 5 unknown anisotropic chemical shifts.  Solution of the problem does promise novel information. Aside from the prospect of obtaining a refined molecular geometry there is the information contained in the order matrix. Studies on the mechanism of orientation have generally worked with  molecules  where  the  orientation  is  described  by  one  or  two  order  parameters. The availability of the complete order matrix for an irregularly shaped solute such as l,3-dichloro-2-ethenylbenzene should be a demanding test for any model of orientational ordering of solutes in liquid crystals. This thesis will examine spectra of this molecule dissolved in different liquid crystals at differing concentrations. A comparison of two simple models for the orientational ordering of this molecule will be presented in chapter V.  In addition to the study of l,3-dichloro-2-ethenylbenzene, this thesis will present two developments  which should prove  useful  additions  to the  technique of  MQNMR.  The first of these additions lies in the area of processing MQNMR spectra. Data processing in MQNMR usually requires that the Fourier transform work with truncated time domain datasets. Other fields have had methods of treating this sort of data for some time, but thej' have been slow to filter into the chemical  Liquid Crystals and Nuclear Magnetic Resonance / 33 sciences. Chapter III of this thesis will examine the applicability to MQNMR of a method of spectral estimation originally developed to deal with geophysical data. With certain provisos it will be shown to be a very useful addition to MQNMR data processing.  The second development  was  suggested  by the  nature of the  spectrum of  l,3-dichloro-2-ethenylbenzene in EBBA. The spectrum suggested a modification of the basic MQNMR experiment which should prove useful in certain spin sj stems. r  This is an experiment which is capable of exciting MQ coherences within a selected group of spins in a molecule. In the case of l,3-dichloro-2-ethenylbenzene this selective experiment will be shown to yield important information in a very simple fashion.  II. EXPERIMENTAL METHODS AND ANALYSIS OF SPECTRA  A. MATERIALS  The  liquid  crystals  used  as  solvents  in  this  thesis  were  N-(4-ethoxybenzylidene)-4'-n-butylaniline (EBBA) and Merck ZLI 1132. (figure 2.1) EBB A was prepared by the method of Keller and Liebert [51]. 1132 is a commercial mixture of cyclohexyl cyanophenyl compounds and has the following composition.  The  solutes used are  24% R,  C H , R  36% R,  CgH,  25% R,  C H, , R  2  = CsN  15% R,  C H,  2  = Ph-C=N  3  7  7  5  = C=N  2  R 5  1 t  R  2  = C=N  all commercially available. 1,1,2-trichloroethane (Fisher  Scientific, technical grade), 1,1-dichloroethane (Aldrich Chemical, dioxane stabilized), and  l,3-dichloro-2-ethenylbenzene (Aldrich Chemical, 98%) were all used without  further purification. l,3-dichloro-2-ethenylbenzene will be denoted throughout this thesis as DCEB.  Most solutions described in this thesis were prepared at a concentration of ~20 mol%.  This is fairly close to the maximum concentration EBBA will tolerate  before the  nematic phase is destroyed, but  these high concentrations were  desirable to provide good signal to noise for the multiple quantum experiments.  34  Experimental methods and Analysis of Spectra / 35 B. SPECTROSCOPIC DETAILS All spectra presented in this thesis were run on either a Bruker CXP 200 spectrometer or a Bruker WH400 spectrometer. The CXP 200 operates at a proton frequency of 200 MHz. While this is a machine designed primarily for solid state work a high resolution probe is available. Spectra run on the CXP were obtained with this probe. Linewidths attainable  with this probe were  typically 0.5 Hz. While this resolution is somewhat poor by high resolution standards, it is quite sufficient for the work undertaken here. 90° pulse widths with this probe are typically 5 microseconds.  With the three pulse sequence of the introduction the data in t  2  ma}' be  recorded as a single point or as a series of points. The design of the CXP 200 makes it possible to capture a single data point from the FID. Experiments performed on this machine were run in this manner to eliminate the lengthy data processing associated with a 2D experiment.  The Bruker WH 400 operates at a frequency of 400 MHz. for protons. 90° pulse lengths typically lay in the range of 11.5 - 12.0 microseconds on this machine.  Linewidths  were  typically  <0.1  Hz. on  isotropic  test  samples.  Experiments performed on the WH400 were performed in a full two dimensional manner. Transients consisting of 64 or 128 data points were collected in t . 2  These data were then transferred via an RS232 serial communications link to a Nicolet 1280 data station where 2D processing was performed.  Both the WH 400 and the CXP 200 use superconducting magnets and in the  Experimental methods and Analysis of Spectra / 36 high resolution mode the magnetic field direction is along the axis of the sample tube. This permits spinning the sample without destroying the orientation of the nematic  phase.  While  magnet  stability on the  CXP 200  was  such that  experiments could be performed without locking, long term drift of the WH 400 was such that a field —frequency lock was required for all 2D experiments. This was accomplished with a melting point capillary tube containing acetone —cf  6  mounted coaxially  in the  sample tube with  teflon spacers.  Chemical  shifts  reported in this thesis will be referenced relative to the residual proton content of this acetone.  All spectra were recorded at 30° C.  C. T H E MQNMR EXPERIMENT - VARIATIONS ON T H E T H E M E  While  the  simple experiment  described  in the  preceding  section  will give  experimental results, its usefulness is significantly limited by a number of factors. Modifications to the experiment, ranging from the very simple to extremely complex have been developed to extend and enhance the technique considerably.  1. Limitations  on Resolution  One feature of the basic three pulse experiment which can be undesirable is the fact that it is capable of creating all coherences of all orders. While this might seem initially to be a desirable attribute, it creates problems with resolution of spectral lines. Time and data storage constraints limit the number of data points in t, which may be collected, and if one is forced to cover n times the width of the single quantum spectrum the frequency resolution suffers severely. While  Experimental methods and Analysis of Spectra / 37 one may routinely collect single quantum spectra of solutes in liquid crystals with resolution on the order of tenths of a Hertz, the more usual resolution in the multiple quantum spectrum is on the order of tens of Hertz.  The problem of data storage and processing on the WH400 spectrometer imposes a severe constraint on the software  achievable resolution. Unlike the CXP 200, the  on this machine will  not allow  single point data  acquisition, all  experiments must be done in a full two dimensional manner. This in itself is not a liability as a two dimensional experiment has a considerable signal to noise advantage over the single point acquisition [52]. Where the problem lies is in the manner in which the data is recorded on disk. The Bruker software requires that each FID start on a disk sector boundary. The minimum number of points that can be collected in t  2  is 32. A sector of the Diablo disk drives  used on the WH400 holds 296 data points. Storing only 32 points per sector wastes storage for 264 data points. For example, if one performs a 2048X32 two dimensional experiment (2048 t,  values, 32 points collected in t ), it will 2  occupy 592K words of disk space disk space even though the data amounts to only 64K. This Figure of 592K words is two thirds of a full disk. The limited number of sectors available on these disks limits the size of the two dimensional experiment to 2048 values in t,. Collecting a data set of this size precludes the possibility of doing any data processing on the spectrometer.  Experimental  methods and  Analysis of Spectra / 38  2. Improvement of Resolution The  problem  of limited  resolution may  solution is to collect data length of an WH400  be  approached  in two  obvious  for a longer time in t ^. This, however, increases  already time consuming experiment, and  it is difficult  ways. An  to  increase  the  amount  on  a machine such as  of data  in  t,  past  a  the the  strictly  limited number.  An  alternative approach  to  the  basic experiment so that one narrow  the  spectral width  longer time. A  a.  Odd'Even  One  relative  a  phase  0°  Obviously,  A  modify  to detect fewer coherences. This enables  in F,  and  one  can  therefore detect the  one  of the  first  between the or  the to  signal for a  Excitation  180°  if one  pulses. If H -  two first  only  has be  to  n t  is bilinear  pulses will excite only  two  phase shift  total spectral width can of  resolution is to  simplest modifications of the basic experiment is manipulation  shift of 90° while  has  of limited  number of methods have been developed to achieve this goal.  Order  of the  problem  will excite only detect  halved  and  odd  either odd one  in I  even  improves  then  even order  order  or  z  of the a  phase  coherences  coherences [53, order  coherences  the resolution by  27]. the  a factor  two.  condition on  implementation  this experiment is that unless H of this  preparation  period.  Hamiltonian  such  as  scheme  This  is  chemical  requires necessary shift  or  n t  the  use  to  remove  is purely  of a  r.f. offset. In  bilinear a practical  refocussing pulse  any  linear  strongly  terms coupled  in  the  in  the  systems,  Experimental methods and Analysis of Spectra / 39 however, a single pulse in the evolution period will not completely remove the linear terms in the Hamiltonian and one then has to use a string of closely spaced 180° pulses.  b. Selective Detection of Particular Orders  Selective detection of particular orders of MQ coherence is a simple method of enhancing resolution which is quite general and very tolerant of experimental realities such as strong coupling. It is based on the characteristic phase shift exhibited by an MQ coherence in response to a shift in the phase of the r.f. used in the preparation sequence. In general an n quantum coherence displays an n(j> phase shift when the phase of the preparation period r.f. is changed by <p degrees [54]. By adding and/or subtracting 360/0 transients and incrementing the preparation period phase by <j> on each shot, many orders of coherence will cancel. Consider an experiment where the r.f. phase is incremented by 90° between  transients  and four transients  are  added together.  Single quantum  coherence will be shifted in phase by 0°, 90°, 180°, and 270°. The sum of these four will be zero. A two quantum coherence will experience phase shifts of 0°, 180°, 360°, and 540°. Again, the sum of these four transients will lead to cancellation of two quantum information. Similar reasoning will show that no three quantum coherence will accumulate. With four quantum coherence, however, the phase response will be 0°, 360°, 720° and 1080°. These four signals will add  constructively and so four quantum information will accumulate. Continuing  the argument along these lines will show that this pulse sequence will result in the cancellation of all but 0, 4, 8, 12,  orders of coherence. Wokaun and  Ernst give a convenient tabulation of the phase cycling necessary to selectively  Experimental methods and Analysis of Spectra / 40 detect various combinations of coherence up to 9 quanta [55].  This is a very general result but it does have its weak points. One is that it takes longer. An experiment like the one above to detect 4 quantum coherence will take 4 times as long as a simple nonselective experiment.  A second criticism of the method is that it wastes much of the signal. The fixed  amount of Zeeman magnetization available at the  start of the pulse  sequence is distributed more or less equally throughout all elements of the density matrix by the preparation sequence. When it comes time to monitor the evolution of p, we detect onlj' a small portion of the available information.  All things considered, however, the fact that this experiment does not rely on the absence or artificial removal of chemical shifts makes it very useful in the case of general chemical systems, where chemical shifts play such an important role.  c. Order Selective Excitation  With regard to the small signal arising from each coherence when all coherences are pumped in the simple three pulse experiment, there would be a considerable signal to noise advantage if one could selectively funnel the starting Zeeman polarization into one particular order of coherence. An experiment which does this has been developed and elegantly demonstrated by Warren et al. [56,57,58] This results in a much stronger signal from the selected coherence and now the example of a four quantum coherence is excited in a single pulse train instead  Experimental methods and Analysis of Spectra / 41 of being selectively detected on four consecutive pulse sequences. In one fell swoop we have a stronger signal and a considerable time saving. Regrettabty, this experiment is very difficult to perform and selective excitation of this sort was not used during the course of this research. Selective excitation of this sort can require thousands of pulses of carefully adjusted and reproducible r.f., and the  experiment  experiences  difficulties  with  sample  heating  and  transmitter  instability. In its most general form the experiment appears to be of limited general utility due to these experimental difficulties. Later work has demonstrated excitation of 4 quantum coherence with just 16 pulses [59] but the demands placed on the spectrometer by this pulse sequence, while not nearty as great as in the general case, are still high. Additionally, and more importantly for general work  in liquid  crystals, this pulse sequence does not  perform well when  confronted with a sample containing significant chemical shifts as well as dipolar couplings.  3. Separation of Coherence by Order.  Another point which merits some consideration arises when one considers how the separation of the multiple quantum orders is achieved. In section I.D.f this was obtained with a real transmitter offset, i.e. if the single quantum spectrum was located a distance ACJ Hertz from the transmitter, the n quantum coherences will be centered around nhco Hertz.  Depending on a real transmitter offset in this manner limits the number of experiments one can perform. As an example of this limitation consider the use of a refocussing pulse in the middle of the t,  evolution period to remove the  Experimental methods and Analysis of Spectra / 42 effect of magnet inhomogeneity. Inhomogeneities in B on  NMR lines. Unfortunately, an  n quantum  0  have a broadening effect  coherence  is n times more  susceptible to this broadening than single quantum coherence. An inhomogeneous field which is acceptable for single quantum work may therefore be unacceptable for multiple quantum spectroscopy. A 180° pulse applied in the middle of the multiple quantum evolution period will remove the effect of magnet inhomogeneity but it also has the effect of removing the transmitter offset. If one were to apply this technique to the experiment of figure 1.5 all multiple quantum orders would be piled on top of one another in a tangled confusion of lines.  An effective transmitter offset may be restored through the use of techniques known as time proportional phase incrementation (TPPI) [60,61] or phase Fourier transformation (PFT) [55,62]. These methods work by manipulating the phase of the preparation sequence to produce an order dependent phase shift, and this phase shift may be used to sort out the MQ orders of coherence. The important point is that through the use of TPPI or PFT, MQNMR can be made more versatile as echo sequences may now be included in the experiment.  TPPI is essential to the success of the experiment when one wishes to resolve narrow lines in inhomogeneous fields. However for spectra presented in this thesis this technique was not used. The use of a refocussing pulse was unnecessary and even undesirable.  Refocussing pulses  were  unnecessary  because  of the  experimental setup used. The WH400 is a high resolution instrument and linewidths of tenths of Hertz are  routinely achieved on isotropic solutions.  Linewidths for solute/liquid crystal systems are t3 pically on the order of 1 - 7 r  Experimental methods and Analysis of Spectra / 43 Hz. The predominant line broadening mechanism is probably thermal gradients in the probe. Any thermal gradient in the sample results in a range of order parameters in the sample which leads to a range of dipolar couplings. This has a significant broadening effect and a characteristic of this broadening mechanism is that lines in the wings of a spectrum are broader than lines in the central region.[63,64] Experimental experience showed that while lines in the centre portion of single quantum spectra typically had linewidths in the 1 - 2  Hz.  range, a linewidth of 5 -7 Hz. was more typical of the outer portions. A 180° refocussing pulse will do nothing to remove this source of line broadening.  A more important source of line broadening for the  MQ spectrum  is the  inevitable truncation of the time domain signal due to the manner in which the data is acquired. This difficulty will be addressed in more detail in chapter 3, and for the moment suffice it to say that the inherent linewidth of the FT of a truncated time domain signal is approximately 1/t, where t is the duration of the sampled signal [65]. In comparison to these two major sources of line broadening the linewidth due to magnet inhomogeneity is unimportant.  More importantly, inclusion of a refocussing pulse in the multiple quantum evolution period was undesirable for two reasons. The first of these is that it will remove chemical shift information from the observed spectrum. For the case of DCEB this information is particularly important, as will be seen. Secondly, when the chemical shifts are of the same size as the dipolar couplings, a 180° pulse can generate extra lines in both single quantum [66, 67, 68] and multiple quantum spectroscopy [69]. In principle, these lines yield extra information but  Experimental methods and Analj'sis of Spectra / 44 their practical effect is often merely to confuse the analysis [70]. An experiment known as TCSTES [71,72], which stands for Total Spin Coherence Transfer Echo Spectroscopy, offers removal of inhomogeneous B  0  while preserving chemical shift  effects and suppressing extra lines, but this experiment suffers a very serious signal to noise penalty.  4. Frequency  Selective  Irradiation  In chapter 4 it becomes necessary to selectively irradiate only certain portions of the NMR spectrum of partially oriented molecules. The reasons for this will be described at that time, but the method used is that of a DANTE pulse sequence [73, 74]. While some NMR spectrometers have a facility for selective irradiation using low power or 'soft' r.f. pulses, this is not possible on the CXP 200. On this machine one is limited to sequences of intense, or 'hard' pulses. The DANTE sequence is a simplification of earlier work by Tomlinson and Hill [75] who showed how to create an arbitrary pattern of irradiation by manipulation of the phase and amplitude of the r.f. pulses. The DANTE sequence utilizes a train of n regularly spaced pulses, each of flip angle 6 . The r.f. intensity spectrum produced by such a sequence consists of irradiation at the transmitter frequencj' and sidebands centered at ±k/t where t r  0, 1, 2 , 3  r  is the time between pulses and k =  (see figure 2.2). The selectivity of the excitation envelope, defined  as being the distance between the first zero crossing points on each side of the envelope maximum, is given by 21m [76]. The cumulative flip angle of such a pulse sequence is nd at the sideband center. By adjusting n, t therefore  r  and tp it is  possible to irradiate any desired portion of the spectrum with an  Experimental methods and Analysis of Spectra / 45 Figure 2.2 1/T  T  •• • • r—-  t Figure 2.2 The DANTE  pulse sequence and its effect on the spectrum.  arbitrary flip angle.  D. COMPUTER METHODS USED  1. Spectral Simulations.  The  analysis of spectra contained in this thesis were carried out using the  computer program LEQUOR [77], a program based on the popular LAOCOON family of NMR simulation programs. LEQUOR is a computer program designed to calculate transition frequencies arbitrary chemical shifts,  and intensities in NMR spectra containing  scalar couplings, dipolar couplings and quadrupolar  couplings. It will do a least squares refinement of the simulation parameters when supplied with experimental line positions, and it has provision to take advantage of magnetic equivalence.  LEQUOR, however, only calculates line positions for the single quantum spectrum.  Experimental methods and Analysis of Spectra / 46 To calculate multiple quantum frequencies LEQUOR was modified to calculate line positions by 'selectable' selection rules, i.e. the user may specify whether Am = ±1,±2,...  While a modified single quantum calculate  spectral  frequencies  the  spectral  simulation program will faithfully  comparison  of experimental  and calculated  intensities is more difficult. For the single quantum case transition intensities are calculated by borrowing the expression for transition probabilities from CW NMR.  A  ik  = <  Vk  |I.|*i>  2  -  This is simply the well known result that the transition probability between two states is proportional to the square of matrix element of the transition operator connecting the states.  In the multiple quantum experiment, however, the amount of multiple quantum coherence produced, and hence the intensity detected, depends on the state of p at the time of the second pulse, and is a sensitive function of the internal evolution of the system during the preparation period. This evolution depends on the internal Hamiltonian and the length of the preparation period. An exact calculation of experimentally observed intensities requires knowledge of the internal Hamiltonian, something we generally do not have. Determination of the internal Hamiltonian is, after all, the aim of the experiment.  The simplest and most reliable way to interpret MQ intensities is to perform a  Experimental methods and Anah'sis of Spectra / 47 number of experiments  while varying r,  the preparation period delay, and  average the results. If the number of experiments averaged over is large then all  coherences  averaged  should be excited at some time or other  spectrum should be independent  and the resulting  of r. In practice the number of  experiments required to give a reasonable T independent spectrum [78] may be 10 to 20. Thus, the collection of an MQNMR spectrum in which the intensities may be easily interpreted can consume a great deal of spectrometer time.  The  program SHAPE [79] was used to analyze dipolar couplings in terms of  molecular geometry and ordering. This is a program which does a least squares fit of  the  geometrical and order  parameters  of a molecule to a set of  experimentally determined dipolar couplings.  2. Density Matrix Simulations.  For  all but the simplest spin systems, calculation of the general case NMR  spectrum (i.e. a spectrum with arbitrary chemical shift, J couplings and dipolar couplings) must be done numerical]}'. The case for numerical calculation is even better when one is required to calculate the evolution of the density matrix describing the system as is the case in the multiple quantum experiment. In order to follow the dynamics of these experiments a computer program was written which calculates the evolution of the density matrix describing a spin system when given the parameters describing the spin system and details of the desired pulse  sequence.  While several approaches  to the simulation of spin  dynamics have been described [80, 81, 82, 83] the workings of the program largelj' follow procedures outlined by Meakin and Jesson [84, 85] with one  Experimental methods and Analysis of Spectra / 48 significant difference.  They outline a five step procedure  for calculating the  evolution the density matrix in the manner of  p(t+r) = e  -2iriHr  p(t) e  27nHr  (2.2)  The five steps they describe are: 1.  The Hamiltonian operator is diagonalized.  2.  The density matrix is transformed to the basis in which H is diagonal.  3.  The operator exp( — also  diagonal  *'27rHr)  is calculated.  with diagonal  H  is diagonal so the exponential is  elements exp( —j'27rE t).  The  j;  E - - are  the  eigenvalues of H. 4.  The unitary transformation is carried out in the Hamiltonian eigenbasis.  5.  The density matrix is then transformed back to the original basis where it may be examined to see how the system has evolved.  Using UpU' to represent the the transformation represented by to represent the transformation  (1.21)  and TUT'  to the Hamiltonian eigenbasis, the five steps  enumerated above may be written  P(t+r) =  T  (exp(TUT ))(Tp(t)T ) (exp(TU )T ) T  1  t  t  t  t  (2.3)  However, by a trivial rearrangment of brackets (2.3) becomes  p ( t + r ) = (T  f  (exp(TUT )T))p(t)(T (exp(TU T )) T) t  t  t  t  (2.4)  Experimental methods and Analysis of Spectra / 49 This indicates that the evolution may be calculated by 1.  Diagonalization of H.  2.  Exponentiation of the diagonalized H .  3.  Transformation of the result of 2 back to the original basis.  4.  Using the result of step 3 to carry out the unitar}' transformation of p.  This four step sequence for calculating the time evolution of p results in the saving of one unitary transformation in the course of the calculation. Over the course of simulating a complete MQ interferogram a considerable time saving accrues by using the second sequence rather than the first.  This simulation uses a product spin £ basis set for its calculations. It will calculate the evolution of the density matrix for arbitrary values of chemical shift, scalar couplings, dipolar couplings, r.f. offset, r.f. pulse length, and phase of the r.f. pulse. It will report expectation values for any of the 63 members of the product spin £ basis set for 3 spins  Modification of the program to  calculate the evolution of more than 3 spins should be trivial. The most time consuming part would be modification of the basis set.  The simulations described in this thesis take no account of relaxation, although such a facility could be added [86].  Experimental methods and Analysis of Spectra / 50 E. A PEDAGOGICAL E X A M P L E At this point sample output of the density matrix simulation might serve as a  MQNMR experiment.  useful example of the  Consider a system of three spins $ with a chemical shift of 5000 Hz. and a dipolar coupling of 1000 Hz. between each of the spins. This could be taken as an  example  of  an  isolated  methyl  group.  The  equilibrium  density matrix for three spins \ in the product spin \ basis is  C 2  i.e. p = C ( I  1 Z  + I  2 Z  +  I ). 32  After a 9 0 ° y pulse the density  matrix is  high  temperature  Experimental methods and Analysis of Spectra / 51 In terms of the spin % product operator notation this is  P = Iix  +  J  2X  +  1  3X  Now the system is allowed to evolve for a time r, say 7.811  milliseconds. The  resulting density matrix is:  c  0  - 0 . 9 9 7 + 0.075i  -0.997 +0.075i  - 0 . 9 9 7 + 0 075i  0  0  0  -0.997-0.075i  0  0  0.941 + 0.339i  0  0.941 + 0.339i  0  0  -0.997-0.075i  0  0  0.941 + 0.339i  0  0  0.941 + 0.339i  0  0  0.941-0.339i  0.941-0.339i  0  0  0  0  -0 720-0.694i  -0.997-0.075i  0  0  0  0  0 . 9 4 1 + 0.339i  0 . 9 4 1 + 0.339i  0  0  0.941-0.339i  0  0  0.941-0.339i  0  0  -0.720-0.694i  0  0  0.941-0.339i  0  0.941-0.339i  0  0  -0.720-0.694i  0  0  0  - 0 . 7 2 0 + 0.694i  0  - 0 . 7 2 0 + 0.694i  -0.720+0.694i  0  2  Decomposed in the product spin \ basis, this matrix is  P = -  12  12  0.0411(I  + I  1X  0.1922(1, I y  + I )  2 X  3X  2z  + I  1 z  I  2 y  + I  1 y  I  3 z  + Iizlay + laylaz +  2Z  + I  1 Z  I  2 X  + I  1 x  I  3 z  + I, I  + I  1 z  zl 3 y)  0.0692(I I 1X  Z  3 X  z' 3 x)  + 0.3240(I I I 1y  2z  3z  I  2 y  I  3 z  + Iiz^z^y)  + I I z + 2 x  3  Experimental methods and Anatysis of Spectra I 52 -  0.8998(1 I I 1X  2Z  + I  3z  I  1 Z  2 X  I  + IizWax).  3 Z  The second pulse in the three pulse sequence now creates multiple quantum coherence, and the density matrix is  C 2  -0.123  0.138 + 0.015i  0.138 + 0.0lSi  0.900 + 0.384i  0.138 + 0.015i  0.900 + 0.384i  0.900 + 0.384i  -0.972i  0.138-0.015i  -0.041  0.900  0.015i  0.900  O.OISi  -0.324i  -0.900 + 0.384  0.138-0.015i  0.900  -0.041  0.0151  0.900_  -0.324  O.OISi  -0.900 + 0.384  0.900-0.384i  -O.OISi  -0.015i  0.041  0.324J  -0.900  -0.900  -0.138 + 0.015  0.138-0.015i  0.900  0.900  -0.324i  -0.041  0.015i  0.015i  -0.900 + 0.384  0.900-0.384i  -0.015  0.324  -0.900  -0.015i  0.041  -0.900  -0.138 + 0.015  0.900-0.384i  0.324i  -0.015i  -0.900  -0.015  -0.900-  0.041  -0.138 + 0.015  0.972i  -0.900-0.384i  -0.900-0.384i  -0.138-0.015i  -0.900-0.384i  -0.138-0.015i  -0.13B-0.055i  0.123  Note that the elements of p which had been zero before the pulse are now nonzero. These elements represent multiple quantum coherence. In the product spin \ basis, the density matrix is now p = - 0.0411(I  -  + I  1Z  0.1922(1 I  12 yl  1x  3  2y  + I )  2 Z  3Z  + Ii I y  2 x  + Iix^y  +  I i y l 3 x  +  ^x^sy +  x)  + 0.0692(I I 1X  2Z  + 0.8998(1 I I 1X  2X  + I  3Z  1 z  I  + I  2 x  + 11 I I X  2Z  3x  1 X  I  3 Z  + I  1 Z  I  + IizWsx  3 X  + I  2 x  I  3 z  +I I ) 2Z  3X  Experimental methods and Analysis of Spectra / 53  + 0.3240(1, I y  This  density  matrix  2 x  now  I  3 x  + I  contains  coherences. As explained in chapter I  1 x  I y 2  I  1 x  2 y  I  3 x  terms  +  IixWay)  which  represent  multiple  quantum  1 these are embodied in terms of the sort  and I i y ^ x l i x - The simulation program also has a facility to collect the  expectation  value of any  operator  as  anj'  of the  time delays is incremented  regularly. Calculations of this sort will prove useful in understanding the results of chapter 4.  III. M A X I M U M ENTROPY S P E C T R A L  A.  ANALYSIS  INTRODUCTION  As noted earlier, the analysis of multiple quantum interferograms strains the capability of the Fourier transform. Consider for illustrative purposes figure 3.1. Figure 3.1a presents a typical 'good' single quantum NMR FID collected after a single 7T/2 pulse. The signal decays smoothly to zero and for all intents and purposes we have collected the entire time domain signal. Fourier transformation of this FID yields the spectrum in Figure 3.1b. The lines have a Lorentzian shape and a width of ~3 Hz.  Consider  next  the  signal  presented  in  Fig 3.2a.  This  is  an  MQNMR  interferogram obtained from the same sample as the FID in figure 3.1a, but whereas the signal in Fig 3.1a has been sampled for —700 milliseconds, the MQNMR interferogram in figure 3.2a has been sampled for onty 17 milliseconds. This savage truncation affects the spectral fidelity in two  waj'S,  both of which  may be clearly seen by comparing figures 3.1b and 3.2b.  One effect is that truncation limits resolution in the frequency domain. Using the Fourier transform, the resolution in the frequency domain for a given time domain signal goes approximately as 1/t, where t is the duration of the time domain signal. As the length of the interferograms referred to in this thesis are t3'pically 10-20 milliseconds, the frequency domain resolution presents problems when dealing with couplings on the order of 10-20 Hz.  54  Maximum  a  Figure 3.1 The FID b.) spectrum.  Entropy  Spectral Analysis  / 55  Figure 3.1  The single quantum spectrum of 1,1,2-trichloroethane The FFT spectral estimate. This is equivalent to the  in EBBA. a.) CW absorption  Maximum Entropy Spectral Analysis / 56 Figure 3.2  a  i o  r  ~i  1  1  1  1  0.005  1  1  1  0.010  1  1  1  1  r  1  0.015  time (seconds)  Lii uw  i—i—i—i—i—i—r 0 5000  "1—i—i—i—i—i—i—i—i—i—r T—i—i—i—i—i—i—r 10000 15000 20000 25000  i  1  I  1  30000  Frequency (Hz.) Figure 3.2 The multiple quantum spectrum of 1,1,2-trichloroethane in EBBA. The MQ interferogram b.) The FFT spectral estimate. The region between 6000 and 13000 Hz. corresponds to the single quantum spectrum of figure 3.1b.  a.) Hz.  Maximum Entropy Spectral Analysis / 57 The second problem due to truncation of the time domain signal is that of 'leakage' or 'sidelobes'. Much of the power contained in the time domain signal 'leaks' to other frequencies, giving rise to artifacts which can be deceiving. NMR spectra are often very crowded and have good dynamic range. Artifacts resulting from this leakage can therefore create confusion in the analysis.  As for the problem of limited resolution, little can be done with the Fourier transform. The limited resolution of a truncated  time domain signal may be  regarded as being akin to the uncertainty principle [87]. The length of time we observe a process places a limit on how accurately we may determine its energy or, equivalent^', its frequency. Short of observing the process for a longer period there is nothing that Fourier transformation can do to overcome this limitation.  In order to alleviate the problem of leakage we must first understand its origin. This is presented diagrammatically in figure 3.3. Here we take the simple example of a single sinusoid and consider the signal that we see experimentally to be the sinusoid 'windowed' (i.e. multiplied) by a function which is one over some finite time interval but zero everywhere else. The signal we Fourier transform is the product of the windowing function and the time domain signal. One of the fundamental relations in Fourier analysis is the convolution theorem which states that the FT of the product of two functions is the convolution of their Fourier transforms [88]. While the FT of an infinite sine wave is a delta function, shown in figure 3.3a, the FT of a rectangular boxcar is the sine function, (sin x)lx. The practical implication of this is that the convolution of the delta function and the sine function smears the delta function so that the FT of  Maximum Entropy Spectral Analysis / 58 Figure 3.3  t ~  f  Figure 3.3 The effect of truncation in the time domain on the spectral estimate, a.) An infinite sinusoid and its FT magnitude spectrum, b.) A truncated sinusoid and its FT magnitude spectrum, c.) A truncated sinusoid windowed by a sine bell and its FT magnitude spectrum. Only the positive half of the spectral estimates is shown. A mirror image of these spectra extends to negative frequency.  Maximum Entropy Spectral Analysis / 59 the windowed function will appear as in fig 3.3b. What we see is a broadened peak with sidelobes. This characteristic sine function shape can be accounted for if we are dealing with a single isolated peak, but when dealing with crowded spectra  with  good dynamic range,  both of which  are  hallmarks of NMR  spectroscopy, such leakage can confuse the analysis greatly.  The traditional approach to this problem is that of windowing the time domain signal by an appropriately chosen function other than a simple boxcar function, (note the  qualifying  common use,  and  'appropriately') the  literature  There are many windowing functions in extolling and damning  various  choices is  voluminous. An example of a windowing function in common use in NMR, the so called sine bell, is illustrated in figure 3.3c. Multiplication of the time domain signal of figure 3.3b by the first half period of a sine wave yields the modified time domain signal shown and Fourier transformation of this signal yields the magnitude spectrum shown on the right. Note that the sidelobes have been verj' greatly reduced by the sine bell window, but the line is now wider than the line in figure 3.3b. A windowing function will not eliminate leakage from an FT spectral estimate, it will only replace bad leakage by less objectionable leakage.  The effect of a good windowing function would therefore  appear to be very  desirable, but we should pause for a moment and consider what we have done to the original data in extracting the spectrum of fig 3.3c. First, we have assumed that the data outside the range 0 •+ t is zero. For the case of many signals, an MQNMR interferogram being a case in point, this is a drastic and untrue assumption. Second, as if an unfounded assumption about the data we do  Maximum Entropy Spectral Analysis / 60 not know were not enough, we then do violence to the data we do know by applying a windowing function. Comparing the original sine wave of figure 3.3a with the windowed truncated function of figure 3.3c, it is perhaps remarkable that the spectrum of figure 3.3c contains an)' pertinent information concerning the original signal at all.  The main problem in choosing a windowing function is that no one window does everything. Inevitably one has to decide which spectral attributes are desirable and which are expendable. A windowing function will not eliminate leakage, it merely replaces bad leakage by less objectionable leakage.  The seemingly single minded adherence of NMR spectroscopists  to the Fourier  transform may appear puzzling to outsiders, for a brief survey of the spectral analysis literature shows that while the FFT, or periodogram as it is also known [89], is one of the favorite tools of spectral analysis, it is by no means the onlj'  method  of analyzing time domain signals  [90]. Before describing any  alternate methods, therefore, the rather special role of the Fourier transform in NMR should be reviewed.  B. THE ROLE OF THE FOURIER TRANSFORM IN NMR.  The Fourier transform occupies a central position in the analysis of time domain NMR signals. It has long been recognized that the Fourier transform of the free induction decay and the CW slow passage absorption spectrum are equivalent. This relationship was first recognized by Lowe and Norberg [91] in 1957. That this is so is easily demonstrated for single quantum  spectra  of the  type  Maximum Entropy Spectral Analysis / 61 described in this thesis.  Take the case of a single intense (1*12)^ pulse applied to a spin system initially at thermal equilibrium. Immediately after the pulse, we have  (3.1)  p(0 + ) = C I . X  If our time domain signal comes from observation of <I >, we know that X  <I > X  = Tr{p(t) I },  (3.2)  x  where  p(t) = exp(-zH t) p(0 + ) exp(iH t) int  int  = C exp(-iH t) I Jnt  In the eigenbasis of H j  exp(iH t).  x  int  (3.3)  the operator evolution is  nt  pit) = C Z<j|I |k> exp(-i(Ej-E )t). x  k  (3.4)  Using the above to evaluate the trace of (3.2) we have  <I > X  = C L |<j|l |k>| x  Fourier transformation of (3.5) with the aid of  2  cos(c;j-cJ )t k  (3.5)  Maximum Entropy Spectral Analysis / 62 1 6(CJ -GJ) 0  =  J* exp(i(cj -cj)t)dt.  (3.6)  6(&)j - w  (3.7)  0  finally yields  f(co) = C Z <j|l |k> x  2  k  - CJ).  However, from the theory of CW NMR [13]  f(co) oc Z <j|I |k>  2  x  S(a>j - a> - co).  (3.8)  k  Here, f(cj) is the shape function describing the lineshape of the CW absorption signal. This relationship holds for a great many NMR experiments and relies on the quantum mechanical properties of nuclear spin systems rather than depending on results from the field of spectral analysis. The kej' step in the above derivation is the form of the initial density operator, p(0). Our ability to write the equilibrium density operator as being linear in I the  Fourier transform.  temperature form then  If the  initial  density  the Fourier transform  z  is crucial for the use of  operator  is  not  in its high  of the pulse response  is not  equivalent to the CW absorption spectrum. The result is a spectrum in which intensities and phases of the signals may be severely distorted.  The conditions necessary for the equivalence of the pulse FT spectrum and the CW slow passage spectrum have been commented on at length by Ernst et al. [92,93], but these papers are not often quoted in the literature. A fact which seems to have been underappreciated  in the development of two dimensional  Maximum Entropy Spectral Analysis / 63 NMR is that the density matrix at the start of the mixing period is not the equilibrium density matrix. In this case the Fourier transform loses the special status it enjoys in NMR, and one may justifiably look into other methods of signal analysis for such NMR datasets.  There has  recently been a surge of interest in alternative data processing  methods in NMR [94,95,96]. One method which has attracted a good deal of attention is the maximum entropy method (MEM). A very general method of maximum entropy spectral estimation has been introduced into NMR by Sibisi et al  [97,98].  This  method  of  maximum  entropy  spectral  estimation  is  computationally complex and very time consuming. Jaynes notes that there are, in fact, a host of 'maximum entropy methods', many of which remain to be developed [99]. Here we will examine a much simpler and more efficient MEM due to Burg [100] and show that it has useful applications in MQNMR. It would perhaps be much better to say that both of these techniques are simply methods of spectral analysis which use the principle of maximum entropy and not to denote either one as the maximum entropy method. Common usage, however, has made the term 'MEM' so ingrained that there is little hope of altering this terminology. Anyone reading the literature should be aware of the fact that the term MEM may refer to at least two substantial^' different methods.  Maximum Entropy Spectral Analysis / 64 C. THE MAXIMUM ENTROPY METHOD OF BURG.  1. The Spectral Estimate  Returning to the situation depicted in figure 3.2 we should consider what the calculated spectra actually represent.  This is easily considered if we imagine  doing an inverse FT of the calculated spectra. What we recover in the time domain is a signal which is consistent with the windowed data.  It would seem better, common sensibly and philosophically if we had a spectrum which, when inverse Fourier transformed would give the most unpredictable or random signal in the regions where we lack data while remaining consistent with the data we do know. This is equivalent to requiring that the time domain signal have the maximum possible entropy  [101,102,103]. The spectrum we  estimate will assume the least about the data in regions where we have no knowledge. In contrast, the more common assumption that the data we have not measured are zero is a very strong assumption. How do we go about finding such a spectral estimate? There are several approaches which all lead to Burg's MEM. In the brief outline here we will choose the description of prediction error filtering. A common alternative is that of autoregressive modelling of a time series [104].  For the prediction error filter treatment we first make the assumption that the time domain signal we are dealing with arises from a stationary Gaussian process. The entropy ratet of such a process is related to the spectral density, tWe use the entropy diverge as t •» »  rate  because for a stationary process the entropy will  Maximum Entropy Spectral Analysis / 65 S(/), by 1  h = *ln(2B) + — /In S(/) df 4B  (3.9)  where B is the rate at which the process is sampled. The object is to maximize h under the constraint that S(/) remains consistent with the autocorrelation values of the known data. This maximization may be accomplished by a number of standard techniques [105,106,107]. The result is a spectral estimate given by P  m  S(f) =  (3.10) 2B| 1 +1 a exp( - i2 7rmfAt)|  2  The function of (3.10) as a spectral estimator of the data is unclear until the roles of P„, „ are explained. The m a „ are the coefficients of what is m and the am r  known as a prediction error filter and P is the power output of the filter. n  2. The Prediction Error Filter.  The literature concerned with the design and use of digital filters is large and digital filters turn up in a surprising number of places [108]. Here we will be concerned with one specific type of digital filter, called a Prediction Error Filter. The literature concerning this topic uses the acronym PEF. We start with a filter which does 'one step ahead' prediction [103]. This filter operates by taking a number of past values of a time series up to some limit M and predicts the next value in the series. This is implemented by the nonrecursive filter  Maximum Entropy Spectral Analysis / 66 M y = 2 n  k=  1  h x . k  n  (3.11)  k  Here y  is the predicted value for the next datum in the time series and the  n-k  ^  x  a r e  e  P  a s t  v a  l  u e s  upon which the prediction is based. The quantity M is  known as the filter order. The h  k  are calculated by minimizing the mean  squared error between the predicted value and the observed value. The sequence {h } is referred to as the impulse response of the filter and is the output of the k  filter when the input is {1, 0, 0, 0, observes  when a short  evaluation of the h  k  intense  }. It is analogous to the ringing one  pulse  is applied to an  analog  filter. The  has been extensively studied and falls into the realm of  Wiener filtering [109]. The prediction filter is a special case of the Wiener filter.  To arrive at the prediction error filter we denote the error between the actual value x  fi  and the predicted value y  n  by e . The prediction error filter is then  described by  n  n  n  J  M = x„ Z k*n-k h, x n k =  M = I k= 0  1  a x . k  n  (3.12)  k  where {a , a,, a , a , 0  2  3  } = {1, h  1 ;  h , h , 2  3  }  Maximum Entropy Spectral Analysis / 67 By minimizing e  n  we get a filter which filters out the maximum possible amount  of predictable signal and passes only noise. It is also known as a prewhitening filter [110]. The first step in calculating a Burg maximum entropy spectral estimatet  involves calculating a set of {a } from the data. Efficient recursive m  algorithms have been developed to calculate the a  m  from a time series and are  documented in a number of places [103,104,111,112,]. Having calculated the filter coefficients and the output power of the prediction error filter, it is a simple matter to calculate the spectral estimate according to (3.10). Ironically, (3.10) is most  efficiently  evaluated  if we use  an  FFT algorithm to  evaluate  the  denominator.  3. A Sample Spectrum  Figure 3.4 provides a very clear illustration of the principal characteristics of the MEM vs. the FFT. Figure 3.4a is the square root of the power spectrum calculated from 1024 points and zero filled to 8k before Fourier transformation. Although the spectrum is somewhat noisy, all the expected peaks are present with good intensity. The important features to note for comparison are the linewidth, ~70 Hz., and the considerable amount of leakage. The leakage and >95% of the linewidth can be attributed to truncation of the time domain signal.  The maximum entropy spectrum, figure 3.4b, looks quite different. The most striking features are the narrowness of the lines and the absence of leakage, two salient features of Burg's MEM. The relative intensities of the lines, however, have changed somewhat from those of the FT spectrum. t The acronym MESE can be used for maximum entropy spectral estimate, or estimation. The acronj m MEM is used for the maximum entropy method. r  Maximum Entropy Spectral Analysis / 68 Figure 3.4  i—i—r  —i—i—r 5000  ~i 10000  i  i i—i—i—i—i—r 15000 20000  "I—I—I—I—I—I—I—I 25000 30000  Frequency (Hz.)  J  1,  i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—r 0 5000 10000 15000 20000  i  r  i  i r 25000  1—1—I 30000  Frequency (Hz.) Figure 3.4 Principal characteristics of the MEM vs. the FFT. The MQ spectrum of 1,1,2-trichloroethane in EBBA. a.) The FFT spectral estimate, b.) The MEM spectral estimate.  Maximum Entropy Spectral Analysis / 69 The line marked with an asterisk in the FFT spectrum at 8245 Hz. is actually a triplet. That this is so is certainly not obvious, even when the peak is expanded, when compared to the leakage around the bases of the other peaks. In  the Fourier transformed  spectrum the largest peak (at 14865 Hz.) has  sidelobes which are five times larger than the weakest clearly identifiable real peak in the spectrum, the peak at 6629 Hz. Artifacts such as these can play havoc with attempts at automated peak picking and, if the spectrum were more crowded, could confuse the wariest eye.  The  spectrum  remarkably  processed  by the maximum entropy  clean. The triplet centered  at 8245  method  is, in contrast  Hz. is now unequivocally  identified as such and leakage is now nonexistent. In addition, the peaks are much narrower with a linewidth of — 40 Hz. compared to the —70 Hz. linewidth observed in the FFT spectral estimate. The linewidth has been improved by about a factor of two. In light of such results further  investigation of this  method of processing MQNMR is certainly warranted.  D. PROS AND CONS OF BURG'S M E S E .  No present method of spectral estimation is an overall panacea for the problems in spectral estimation. Each has its own particular strengths and weaknesses, and Burg's MESE is no different in this respect.  Maximum Entropy Spectral Analysis / 70 1. The Strong Points  MESE offers greater resolution than traditional methods of spectral analysis based on the Fourier transform. One of the problems in MQNMR is the hindered resolution due to the nature of the experiment. The above results suggest that the Burg MEM might offer a route to much improved spectral resolution without consuming large amounts of machine time. As it stands with the Fourier transform,  if one desires a factor of two improvement in the resolution, it  requires an experiment twice as long.  No windowing functions are required with Burg's MESE, and leakage is reduced to the point of nonexistence. The use of windowing functions with the FFT, while able to suppress leakage artifacts, inevitablj' leads to a loss of resolution. Even if Burg's MEM were to prove incapable of significantly improving resolution, the example in figure 3.4 suggests that Burg's MEM will suppress artifacts due to leakage without degradation of resolution.  A great advantage of Burg's MESE is that it is fast and simple. This is a result of the recursive nature of the filter coefficient calculation. This is an important  point and makes it feasible to implement Burg's MEM on older  minicomputers which are common in NMR labs. Pulsed NMR spectrometers generally have small computers associated with them for experiment control, data acquisition and data processing. The processing power of the machines is small and the compactness and efficiency of the Burg algorithm is favorable for its implementation on such machines.  Maximum Entropy Spectral Analysis / 71 A machine typical of those found in NMR labs is the Nicolet 1280. When a Burg MEM coded in FORTRAN is run on this machine it is found that it runs much more slowly than the manufacturer's FFT, but this is only to be expected as the FFT is coded in assembly language and uses integer arithmetic whereas the Burg calculation is coded in FORTRAN and uses double precision arithmetic.  All the MEM spectra presented here, however, were calculated using an IBM PC XT. The Nicolet 1280 does not execute FORTRAN code efficiently and it was found that the analyses could be carried out roughly 10 times faster on the IBM PC than  on the  Nicolet  1280. (The PC is equipped  with  a math  coprocessor chip.) FORTRAN listings to carry out an evaluation of the filter coefficients are available [103,111], but for the spectral analj'ses carried out on the IBM PC a program was written from scratch following the scheme outlined by Andersen [112]. On the IBM PC a spectral estimate using 1024 data points and a filter order of 400 takes about 2 minutes  A feature of the calculation which may be used to advantage is its recursive nature.  This means  that to  calculate a spectrum  of filter  order M, one  necessarily has to calculate the filter coefficients for all lower order filters. It is therefore possible to temporarily save any or all of these sets of lower order filter coefficients as the calculation goes to the selected filter order. When the calculation stops one then has the filter coefficients necessary to calculate the spectral estimate resulting from any filter order less than or equal to M, and such a calculation takes only seconds. Therefore, one has to wait minutes for the filter coefficient calculation, but then spectral estimates from many filter orders  Maximum Entropy Spectral Analysis / 72 are available in seconds. The IBM PC is, of course, a rather limited computer and use of a more powerful computer would speed things up dramatically.  2. The Weak Points.  There are difficulties with Burg's MEM as there are with any other spectral estimator. Points to note about Burg's MEM which may cause difficulty in the analysis of time domain signals are: 1.  The Burg MEM requires stationarity.  2.  The method has been demonstrated to produce variable intensity estimates.  3.  The method has been shown to produce frequency shifts in the data.  4.  Burg's MEM has been known to split peaks.  5.  The method does not perform well in low signal to noise situations.  6.  The determination of the correct filter order to be used in the analysis of a signal is uncertain.  Happily, not all of these are applicable to MQNMR. The first two points are easity dealt with. As for the issue of signal stationarity, this is required by the form of the expression for the signal entropy (3.9) and the PEF calculation is also based on the stationary nature of the signal. Stationarity can therefore be expected to be of central importance. Luckity, MQNMR interferograms in dipolar coupled systems of the type discussed here will nearly always be stationary to second order statistics. The high sampling rates necessary in these experiments coupled with the  limited  amount of data  which can be acquired and the  relatively slow signal decay rates for these systems will nearly always yield 1 or 2K of quasi-stationary data, so this requirement is not a serious obstacle.  Maximum Entropy Spectral Analysis / 73 Burg's MEM has been applied to nonstationary single quantum FID's with varying results. Viti et al. [113,114] and Ni and Scheraga [115] have applied the method to decaying FID's and report no ill effects, but Hoch has applied the method to synthetic data consisting of sinusoids with differing decay rates and has found that the shorter  lived components completely disappear from  the  spectrum when the signals are processed with Burg's MEM [116]. The fact is that important assumptions Burg made in his derivation of the spectral estimate (3.10) are based upon signal stationarity. One should not be surprised if Burg's method occasionally malfunctions when fed data as grossly nonstationary as an NMR FID. In such a case one should refrain from blaming the method, the problem lies in the fact that the data is not well suited to the method.  With regard to the second point above, it is true that intensities in MEM spectra must be interpreted with caution [117]. For reasons explained in chapter 2, multiple quantum intensities are usually more trouble than they are worth, so this is not a serious objection to the use of Burg's MEM.  While points 1 and 2 may be comfortably put to rest the difficulties of peak splitting and frequency shifting are unsettling. Instances have been documented where MESE will cause splitting of spectral lines [118] and where frequency shifts have been observed in the calculated spectra [119]. These observations are disturbing because the information content of the MQNMR spectra discussed here lies primarily in the splittings measured between peaks. Any possibility that these measurements may be tainted by the data processing must be examined carefulty. This is the subject of section F.  Maximum Entropy Spectral Analysis / 74 In order for the method to function well, the data should have good signal to noise. In the case of low signal to noise, the mathematical model upon which the Burg MEM is based, (3.11), begins to fail for sinusoidal signals, and the method gives poor results [120]. This is most noticeable in markedly decreased resolution [121]. The signals treated here generalty have good signal to noise.  Finally, the determination of filter length appropriate to the time domain signal in question can be a difficult and vexing problem. This is probably the most difficult aspect of applying Burg's MEM and MQNMR is not special in this respect.  E . SELECTION OF FILTER ORDER  This is the most difficult aspect of applying Burg's MEM to MQNMR. In order to carry out a MEM analysis one has to decide upon the filter order to be used to represent the process. The filter order chosen will have a profound effect upon the resulting spectral estimate. As might be expected when fitting data to a model, a low filter order will result in an overly smooth spectrum with few sharp features. Converse^, a high filter order will give a spectrum with very sharp (and possibly spurious) detail.  The effect of filter order on the spectral estimate is illustrated in figures 3.5 to 3.7. Figure 3.5 presents a spectrum of 1,1,2-TCE in EBBA with a low filter order of 50. Note the smooth features and lack of resolution. Figure 3.6 shows the same data processed with a filter order of 250. Resolution is good and all peaks have the proper position. Going to still higher filter order results in the  Maximum Entropy Spectral Analysis / 75 Figure 3.5  l  I  l  l  0  l  l 5000  I  l  l  I  l  l  I  10000  I  l  i  I  l—I  15000  I—i—I—i—i—i—i—i—i—i—i—i 20000  25000  30000  F r e q u e n c y (Hz.)  Figure 3.5 The multiple quantum spectrum of 1,1,2-trichloroethane in EBBA. The MEM spectral estimate using 1024 data points and a filter order of 50.  spectrum seen in figure 3.7. Now there is spurious detail in the spectrum. Some of the noise peaks are growing large but much more serious is the observation that the peak marked with an asterisk is now split into two peaks.  This poses the difficult question 'What is the 'correct' filter order?' This question is not easily answered. Several theoretical tests have been developed to serve as objective measures of the filter order of a particular dataset [122,123,124], but regrettably  these tests do not  perform well  when working  with harmonic  processes. In such cases they tend to significantly underestimate the required filter order [103]. Applications of Burg's MEM to the analysis of harmonic  Maximum Entropy Spectral Analysis / 76 Figure 3.6  A  J  i — i — i — i — i — i — i — i — i — i — i — i — i — i — i  0  5000  10000  i — r  15000  " i — i — i — r  20000  "1—I 30000  n — I — r  25000  Frequency (Hz.)  Figure 3.6 The multiple quantum spectrum of 1,1,2-trichloroethane MEM spectral estimate using 1024 data points and a filter order  in EBBA. of 250.  The  signals have invariably resulted in empirical guesses as to the filter order required when analyzing a dataset. These range from 'N/3-1 to N/2-1' [103], 'N/5 to 2N/5" [117] and '0.4N to 0.65N' [110] to ' almost 100%' of the data [118]. How do we choose a filter order from these wide ranging and overlapping recommendations? These suggest a range of 0.2N to N would be acceptable.  The results of this work suggest a choice of filter order toward the low end of this range. Experience gained from processing many spectra of the type presented in this thesis showed that the most stable and reliable spectral estimates resulted from filter orders in the range 0.2N to 0.4N. The lower limit represents the  Maximum Entropy Spectral Analysis / 77 Figure 3.7  l  I  l  l  0  I  l  l  I  l  5000  I  l  l  I  l  l  10000  l  l  I  l  15000  l  I — i — i — i — i — i — i — i — i — i — i  20000  25000  30000  Frequency (Hz)  Figure 3.7 The multiple quantum spectrum of 1,1,2-trichloroethane in EBBA. The MEM spectral estimate using 1024 data points and a filter order of 500.  minimum higher and  useful  resolution for our purposes  end of the range  runs  the unsettling prospect  a very  of split  great  while risk  peaks. This  use of filter  orders  of introducing spurious still  represents  at  the  detail  a considerable  range of choice, however, and the principle of parsimony [125] should  be kept in  mind. It is advisable to use filter orders greater than 0.4N with caution.  Maximum Entropy Spectral Analysis / 78 F. T H E RELIABILITY OF BURG'S M E M IN MQNMR  1. The Problem of Peak Shifting  As has been mentioned Burg's MEM encounters a number of difficulties in practice. One problem of particular concern for MQNMR in dipolar coupled systems of the type treated here, where accurate frequency determination is all important, is the noted ability of MEM to introduce frequency shifts in the analyzed spectrum. This frequency shift depends primarily on the initial phase of the input signal and is more severe the shorter the data record. Errors in frequencj  7  determination of up to 30% have been demonstrated for some  very  short data sets [119]. Short in this context means less than one full cycle of the sinusoid has been sampled. The MQNMR interferograms presented here have considerably more data than the extreme cases discussed in ref. [119]. The severity of this frequency shift dies off rapidly as the data record lengthens but because of the importance of accurate frequency determination and the complex phase relationships of MQNMR signals, this point should be examined in some detail.  To this end a series of MQNMR spectra of 1,1,2-trichloroethane partially oriented in EBBA were collected and high quality single ' quantum spectra were recorded immediately before and after the multiple quantum experiments. These single quantum spectra were then used to predict precise frequencies which should be observed in the multiple quantum spectra. The spectrum of 1,1,2-trichloroethane was chosen as a test sample because it contains a reasonable number of lines which are relatively well resolved, the smallest splitting in the spectrum being  Maximum  Entropj' Spectral Analysis / 79  -170 Hz.  An  analysis  of 32  different spectra  both the F F T and Burg's M E M  Table  3.1  MEM  vs. F F T  spectral  positions The  analysis  multiple  potentially  has  errors  spectrum  15  for a  dataset. The F F T  peaks  has therefore  expected to have vanishing  data  in these  quantum  of  total missed  1024  data  estimate (Hz.)  1407 3514 6625 6845 8034 8240 8446 10337 11732 11961 14869 18372 18778 20182 26808  of the pristine  and standard  estimates. 250.  MEM  1403.3 3500.6 6629.7 6831.6 8034.0 8234.9 8435.9 10332.2 11736.4 11936.5 14865.5 18366.1 18768.0 20172.3 26802.0  FFT  in E B B A  performed  with  yielded the results summarized in table 3.1.  Predicted Frequency (Hz.)  An  of trichloroethane  +  ± ± ± ± ± ± ± ± ± ± ± ± ± ±  6 6 12 10 9 7 10 6 9 12 5 5 5 7 8  1410 3509 6632 6841 8040 8239 8446 10337 11738 11953 14868 18371 18776 20181 26803  positions  partially of 15X32  -  Filter  order  F F T estimate (Hz.)  identified 408 peak  points  peaks  480  3 4 9 5 10 4 5 3 6 16 2 4 5 3 6  + ± ± ± ± ±  + ± ± ± ± ± ± ±  with  as indicated  oriented =  ±  mean  peak  in the table.  1,1,2-trichloroethane peaks  in the entire  72 peaks. However, some peaks can be  intensity due to the intensity variation among single  Maximum Entropy Spectral Analysis / 80 (unaveraged) MQ spectra, so the FFT can not be held completely responsible for this failure. Results of MEM vs FFT. analysis using the same data and a filter order of 250 are summarized in columns 2 and 3 of the table. The average values differ by as much as 8 Hz. from those of the FFT analysis. The errors are somewhat higher than those of the FFT analysis, but the differences are not enough to recommend or condemn the MEM when compared to the FFT. The standard errors in both cases amount to about one point in the digital resolution when  the  amount  of zero filling  used is considered. The most significant  difference is in the way the MEM identifies small peaks and rejects artifacts. The MEM estimate successfully identified 452 out of a possible 480 peaks. The missing 28 peaks were among those that the FFT missed also and so it may be concluded that these peaks actually did have vanishing intensitj'. Additionally, while this MEM calculation showed no spurious peaks, what hasn't been said to this point is that the FFT spectral estimates were cluttered with over 500 artifacts. These arose because in order to identify many of the weak peaks the cutoff for the peak picking routine during the FFT analysis had to be set so low as to identify large numbers of sidelobes.  The same datasets were also processed using a filter order of 400 yielding results which, while statistically better were qualitatively different in significant ways. The mean peak positions and standard errors are reported in table 3.2. The peak positions have changed slightly and the standard error is now lower, close to the value for the FFT analysis! but the high filter order has had an t It should be noted that these are extremely high filter orders compared to what is found in the geophysical literature, for example. There the filter orders are usually < 10. That this calculation works for these high filter orders is a testament to the quality of the recursion used in the calculation. For the  Maximum Entropy Spectral Analysis / 81 Table 3.2 MEM spectral estimates. Mean peak positions and standard errors. Filter Order 400 Predicted Frequency (Hz.) 1403.3 3500.6 6629.7 6831.6 8034.0 8234.9 8435.9 10332.2 11736.4 11936.5 14865.5 18366.1 18768.0 20172.3 26802.0  MEM estimate (Hz.) 1408 3513 6630 6843 8035 8238 8446 10338 11737 11954 14869 18371 18777 20182 26806  ± 4 ± 5 ± 10 ± 8  + ± ± ± ±  + ± ± ± ± ±  7 5 8 7 7 8 4 6 7 5 7  FFT estimate (Hz.) 1410 3509 6632 6841 8040 8239 8446 10337 11738 11953 14868 18371 18776 20181 26803  ± 3 ± 4 ± 8 ± 5 ± 10 ± 4 ± 5  +  3  ± 6 ± 15 ± 2 ± 4 ± 5 ± 3 ± 6  impact on the appearance of the spectra. The spectra have now generally become more noisy and some of the weak peaks have dropped below the noise with the result that the MEM estimate now misses 39 of the potential 480 peaks. More serious, however, is the fact that there were 12 instances in these calculations where very intense peaks were split into two closely spaced peaks.  2. The Problem of Spurious Detail  The ability to introduce spurious detail into a spectrum is a well documented property of Burg's MEM [90]. An illustration of this is presented in figure 3.8. The peak at 18768 Hz., indicated with an asterisk is now clearly split into two peaks. The details of the analysis (512 data points used with a filter order of t (cont'd) calculations discussed here it was essential to use double precision arithmetic in calculation of the filter coefficients.  Maximum Entropy Spectral Analysis / 82 Figure 3.8  Figure 3.8 A MEM spectral estimate of the MQ spectrum of TCE in EBBA. The spectral estimate was calculated using 512 data points and a filter order of 255.  The most easilj' understood cause of spurious peaks is simply the choice of too high a filter order [103]. This problem is nicely discussed by Komesaroff et al. [126] These authors show that the model used by Burg's MEM tries to fit the spectrum using quasi-Lorentzian lines. All other things being equal, the width of these lines depends inversely on the filter length used in the calculation. Using too high a filter order forces the method to use lines which are very narrow and as a result a broad peak may be reconstructed as a sum of two or more narrow peaks and spurious detail results. This highlights the problem of filter order selection. It is therefore advisable to use the minimum filter order which is adequate for ones purposes.  This guideline comes under  the more general  Maximum Entropy Spectral Analysis / 83 order selection. It is therefore advisable to use the minimum filter order which is adequate for  ones purposes. This guideline comes  under the  more  general  principle of parsimony which should be remembered when fitting data to a model of any sort.  There is another case in which extra peaks may occur in a spectrum. This is a rather special situation and has been termed 'spontaneous line splitting'. Fougere has shown this line splitting to be at its worst when the data contains an odd number of quarter cycles of a sinusoid whose initial phase is 45° [118]. While this may seem to be just a very odd quirk of the method, Jaynes [99] has shown it is only to be expected given the assumptions upon which Burg's MEM is based. A solution to this spontaneous line splitting problem has been published [127] and FORTRAN code for its implementation is available [128] but it adds a great deal to the computational complexity and has not been widely implemented. In doubtful cases of split peaks it is much simpler to vary the length of the data record, if possible, so that the requirement of an odd number of quarter cycles is not met.  It  should be emphasized  that while the  possibility of lines being split is  disturbing, it is not common enough that it should rule out the use of Burg's MEM in MQNMR. Experience gained while processing the spectra presented here indicates that by and large Burg's MEM performs quite admirably in situations where the FFT would be much inferior.  Maximum Entropy Spectral Analysis / 84 G. REDUCTION OF EXPERIMENT DURATION USING BURG'S MEM  Another feature of Burg's MEM which recommends its consideration for use in MQNMR is its ability to extract information from short data records. The accumulation of an MQ interferogram is a time consuming process. The time required for collecting an MQNMR dataset is almost directly proportional to the number of t,  values collected. The time required for data collection in t  2  is  negligible and the time required for data storage to disk and other housekeeping chores the computer must do amounts to a small fraction of a second. In contrast, to avoid saturation of the signal the basic pulse sequence may typicallj' be repeated only everj' 2-3 seconds, and the number of points collected in t, becomes the determining factor in the time required for the experiment. If the number of points required in t,  can be reduced in some way, then the time  required for the experiment can be reduced proportionately.  It  turns  out  that  Burg's MEM can  give reliable MQNMR  results  with  significantly fewer data points than are required by the FFT. To illustrate this capability figures 3.9 through 3.11 present the same spectrum as figure 3.3, but now the number of data points used in the analysis is varied.  The  most dramatic example is shown in fig 3.9. Now, only 384 data points  have been used in the analysis. The FFT spectrum is recognizable, but the resolution is decidedly poor. The maximum entropy spectrum has a striking^ better appearance. This is an extreme case however. In order to achieve this resolution with so few data points, the filter order required for the MESE is ~N/2 and, as already demonstrated, this can be dangerous. It is better to keep  Maximum Entropy Spectral Analysis / 85 Figure 3.9  a  i—i—i—i—i—i—i—\—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i 0  5000  10000  15000  20000  25000  i  i  i  i 30000  Frequency (Hz.)  b  i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i 0  5000  10000  15000  20000  25000  30000  Frequency (Hz.) Figure 3.9 The multiple quantum spectrum of 1,1,2-trichloroethane in EBBA. a.) The MEM spectral estimate using 384 data points and a fdter order of 190. b.) The FFT spectral estimate.  Maximum Entropy Spectral Analysis / 86 the filter order significantly below this limit, if possible.  In figure 3.10, 512 data points and a filter order of 170 have been used in calculating the spectral estimates. The FFT gives a reasonable spectral estimate. All  peaks  are  now resolved, but the linewidths are large and leakage is  extensive. With the MEM resolution is much better and leakage is nonexistent. The filter  order used is N/3. Using a higher filter order to improve the  resolution is unwise in view of figure 3.8. An attempt to process a series of spectra using only 512 data with a filter order >200 had to be abandoned because of an unacceptable amount of spurious detail induced by the high filter orders. We are still only dealing with one half the available data, however.  A much more reasonable expectation of the MEM is shown in figure 3.11. Now, 640 points have been used in calculating the spectrum. Using a filter order of ~N/3 we have good resolution, no leakage, and the estimated line positions are in excellent agreement with predicted values. The FFT using 640 points is shown in fig 3.11b.  The same set of spectra processed in the previous section were reprocessed using only the first 640 points of each dataset with a filter order of 200. The mean peak positions and errors are reported in table 3.3. There is little to choose between  Burg's MEM and the  unwindowed  FFT has  FFT on a  missed 90 peaks  statistical basis.  while  Additionally, the FFT generated nearly 750 artifacts.  the  However, the  MEM has  missed 60.  Maximum Entropy Spectral Analysis / 87 Figure 3.10  a  Frequency  (Hz.)  Figure 3.10 The multiple quantum spectrum of 1,1,2-trichloroethane in EBBA. a. ) The MEM spectral estimate using 512 data points and a filter order of 250. b. ) The FFT spectral estimate.  Maximum  Entropy  Spectral  Analysis  / 88  Figure 3.11  a  I — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r  0  5000  10000  15000  20000  25000  30000  Frequency (Hz.)  b  i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i 0  5000  10000  15000  20000  25000  30000  Frequency (Hz.) Figure 3.11 The multiple quantum spectrum of 1,1,2-trichloroethane in EBBA. a. ) The MEM spectral estimate using 640 data points and a fdter order of 210. b. ) The FFT spectral estimate.  Maximum Entropy Spectral Analysis / 89 Table 3.3 MEM spectral estimates. Mean peak positions and standard errors. Filter order 200 Predicted Frequency (Hz.) 1403.3 3500.6 6629.7 6831.6 8034.0 8234.9 8435.9 10332.2 11736.4 11936.5 14865.5 18366.1 18768.0 20172.3 26802.0  MEM estimate (Hz.)  FFT estimate (Hz.)  1409 3511 6618 6848 8035 8239 8448 10339 11732 11958 14868 18366 18781 20181 26806  1410 3508 6632 6842 8042 8237 8446 10337 11736 11945 14868 18368 18776 20181 26803  + + + + ± ± ± + + + + + + + +  10 9 12 10 5 12 15 9 9 12 8 7 9 7 10  ± ± ± ± + ± ± ± + ± ± ± ± ± ±  4 8 9 11 21 8 15 4 12 14 3 4 6 4 9  H. T H E MAXIMUM ENTROPY METHOD OF DANIELL A N D GULL.  The inner workings of the maximum entropy method of Daniell and Gull [129], introduced into NMR by Sibisi et al., are quite different from those of Burg's MEM. This particular method has generated a good deal of interest in NMR circles  [130,131,132] Unfortunately, both  methods  are  commonly referred  to  simply as 'MEM'. The MEM of Daniell and Gull was originally developed for the purpose of image enhancement in astronomy [133]. In its NMR incarnation it has been adapted to work with time domain signals in the following fashion.  Rather than extract the frequency spectrum directly from the time domain data, it first generates a trial spectrum in the frequency domain. This trial spectrum may simply be noise. It then uses an FFT to transform this frequency spectrum  Maximum Entropy Spectral Analysis / 90 into the time domain. This calculated time domain signal is compared to the observed time domain signal. This comparison is used to adjust the frequency domain signal. This adjusted frequency domain spectrum is again FFT'd to yield a better fit to the time domain signal and the whole process is repeated iteratively until the calculated time domain signal agrees with the observed signal to a desired tolerance. Skilling et al. [98] say that 30 iterations are frequently sufficient to yield acceptable results and that 100 iterations virtually guarantee convergence.  The method of Daniell and Gull has the advantage of being able to deal with arbitrary time domain data whereas a fundamental requirement of the Burg algorithm is (weak) stationarity, although this requirement may sometimes be bent considerably. The price paid for this generality is a very large increase in mathematical and computational complexity [134]. Whereas the Burg method can be usefully implemented on a computer as small as an IBM PC, the method of Daniell and Gull requires much more computing power.  I. SOME P R A C T I C A L E X A M P L E S .  While  the  spectrum  of  1,1,2-trichloroethane  in EBBA  nicely  illustrates  the  application of Burg's MEM to MQNMR and serves to give a measure of the accuracy of the method in dealing with MQNMR signals, the spectrum is rather too simple to be a practical example of the usefulness of the method. This section will illustrate the capabilities of Burg's MEM with some more complex spectra.  Maximum Entropy Spectral Analysis / 91 One step up from 1,1,2-trichloroethane in complexity is the four spin system 1,1-dichloroethane. The multiple quantum spectrum of this molecule in EBBA is shown in figure 3.12.  The classic spectrum of a solute dissolved in a liquid crystal is that of benzene. As illustrated in figure 1.5, the high order multiple quantum spectra of benzene in EBBA are quite simple. The three quantum region of this spectrum is much more crowded and an expanded view of this region is shown in figure 3.13a. The dataset consists of IK points in the time domain. This was zero filled to 2K before Fourier transformation The duration of the signal implies a linewidth of ~60 Hz. and manj' lines are not clearly resolved. The same spectrum after the  application  of  a  sine  bell  window before  zero  filling  and Fourier  transformation is shown in figure 3.13b. Artifacts due to leakage have been supressed but there has been a very noticeable decrease in resolution. The same data processed using the MEM with a filter order of 340 (N/3) is shown in fig 3.13c. Clearly, the MEM spectral estimate is much superior in detail, and the line positions are in good agreement with predicted values.  The best example of the utility of Burg's MEM and one of much relevance to this thesis is the series of spectra shown in fig 3.14. The spectra shown are expansions of the five quantum region of a multiple quantum spectrum of DCEB in EBBA. Figure 3.14a shows the FFT magnitude spectrum. Artifacts due to leakage are a major problem. While at least seven peaks are identifiable, there are  a host of smaller but  still  rather  large  peaks  which  are extremely  misleading. Note in particular the peak marked with the asterisk in figure 3.14a.  Maximum Entropy Spectral Analysis / 92 Figure 3.12  WJ i 0  i  i  i  i  i  i — i — i — i — i — i — i — i — i — i — r  5000  10000  "i—i—r 20000  1—r  15000  1  1  1  1  25000  Frequency (Hz.)  i 0  i — i — i — i — i — i — r 5000  I 10000  I  I  I  I  I 15000  I  I  I  I  I  20000  I  I  I  I I 25000  Frequency (Hz.) Figure 3.12 The multiple quantum spectrum of 1,1-dichloroethane in EBBA. a.) The MEM spectral estimate b.) The FFT spectral estimate.  1 3 0 0 0  1 4 C 0 0  1 5 0 0 0  1 6 0 0 0  1 7 0 0 0  1 8 0 0 0  Figure 3.13 The three quantum region of the MQ spectrum of benzene partially oriented in EBBA. a.) The FFT magnitude spectrum, b.) The FFT spectral estimate after application of a sine bell window, c.) The MEM spectral estimate using a filter order of 400.  Maximum Entropy Spectral Analysis / 94 Figure 3.14b is the FFT magnitude spectrum after the data has been massaged by a sine bell window. By and large the artifacts have been taken care of and in particular the peak marked with the asterisk in figure 3.14a has disappeared. The resolution has suffered, though, and two peaks which previously were well resolved have now merged. One peak is now present only as a shoulder on the more intense peak. Finally, the spectrum presented in figure 3.14c shows the spectral estimate resulting from a MEM analysis using a filter order of 400. The seven peaks present stand out clearly and the resolution is much better. Again, this spectrum indicates that the dubious peak marked with the asterisk in 3.14a is not real. Most conclusively, when the final analj'sis of the spectrum was accomplished it was found that the nearest real peak was —150 Hz. away from the asterisk marked peak.  J. CONCLUSION  This chapter has demonstrated that Burg's Maximum Entropy Method may be profitably applied to the analysis of MQNMR data. The Burg maximum entropy method is very fast compared to other MEMs and it is particularly well suited to the analysis of MQNMR signals due to the quasi-stationary nature of the data.  While the use of Burg's MEM in the analysis of MQNMR data can give very striking results it should not be taken that the results presented here suggest that it replace the FFT. Rather, it should be regarded as a useful adjunct to the FFT.  Maximum Entropy Spectral Analysis / 95 Figure  I  0  1  1  1000  1  1  2000  1  3.14  1  3000  1  1  4000  1  1  5000  F r e q u e n c y (Hz.)  Figure 3.14 The five quantum region of o-DST in EBBA. a.) The FFT spectral estimate b.) The FFT spectral estimate after use of a sine bell apodization. c.) The MEM spectral estimate using a filter order of 400.  «  Maximum Entropy Spectral Analysis / 96 Because of the absence of leakage in MEM spectra it is an extremely handy diagnostic  tool when  dealing  with  crowded  spectra.  Artifacts  which might  otherwise be confused with real peaks are clearly ignored for what they are. This appears to be the method's most useful application in MQNMR. The use of windowing functions to suppress spectral artifacts entails an inevitable loss of resolution. ME spectral estimation easily identifies artifacts due to leakage while often dramatically improving resolution.  As for the method's ability to enhance resolution, this is a useful capabilit}', but the method should not be 'pushed' to high filter orders simply to enhance resolution, as tempting as this might be. Given the method's demonstrated ability to 'enhance' one peak into two, the spectroscopist must exercise restraint in selecting the filter order to be used and fine splittings in MEM MQNMR spectra should be interpreted with caution.  Finally there is the prospect of considerably reducing the experimental time needed to collect a spectrum. The results presented here suggest that this feature might be best utilized in doing preliminary work or in 'tuning up' an experiment. A time saving of 30% could be most useful when working the bugs out of a new experiment. Before making any controversial conclusions, however, it would be advisable to run a more complete experiment, if it is at all possible, and check the results with more data using both an FFT and MEM analysis.  IV. l,3-DICHLORO-2-ETHENYLBENZENE AND FREQUENCY SELECTIVE MQNMR  A. THE PROBLEM  1. The Single Quantum Spectrum  The single quantum spectrum of DCEB is shown in figure 4.1. Figure 4.1  n  1  1  2000  Figure 4.1  1  T  1  1  1000  1  1  1  1  j  0  1  1  1  1  1  1  -1000  1  1  1  1  r  -2000  The single quantum spectrum of DCEB in EBBA.  Our aim is to interpret this spectrum in terms of the molecular structure and orientation of the molecule shown in figure 4.2. While the spectrum does exhibit distinctive features the analysis promises to be difficult. The spectrum contains a 97  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 98 Figure  4.2  Geometrical Parameters  Bond lengths  H(5)^K  C - C (ring) c -c 2  a  1.3964 1.477A 1.326A  C-H  H(6)'  \  ^  1.084A Bond  -H(2)  ZC C C^  127.6°  ZC C H(3) A  114.2°  Z C f C 2 CJI  123.1°  Zc c c  118.7°  2  HO)  2  Y  Z (out of page)  Figure 4.2  3  X  anglest  a  2  a  i"All unspecified angles are taken to be 120°.  The molecule l,3 — dichloro — 2 — ethenylbenzene.  large number of lines with a resolution of ~ 1 Hz. in the central region.  The  traditional approach to such problems has been to use an approximate  geometry and, by guessing values of the order parameters, generate trial spectra until one is found which gives a reasonable approximation to the experimental spectrum. Lines in the simulated spectrum are then assigned to experimental frequencies and the trial parameters  are adjusted by a least squares process.  This (hopefully) gives an improved spectral simulation and one can then assign more lines in the simulated spectrum for further least squares adjustment . This  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 99 whole process is repeated iteratively, assigning more lines on each cycle, until all lines in the spectrum have been fit. With a spectrum as complicated as that of DCEB, however, this is neither an easy nor a foolproof procedure.  First of all, the geometry we are forced to use as a starting point is crude. While the geometries of the aromatic ring and the ethenyl group may be predicted fairly well, the dihedral angle, defined as the angle between the plane of the aromatic ring and the plane of the ethenyl group is not well known. In 3-ethenylbenzene (styrene) the ethenyl group and the aromatic ring are coplanar [49,50], but in DCEB, steric interactions between the ethenyl group and the chlorine atoms force the ethenyl group out of the plane of the ring. An early estimate using NMR scalar couplings reported this angle as 30° —50°[48] while a more recent photoelectron spectroscopic study places the angle at '~46 ' [49]. Of c  course  the  ethenyl group  will  also experience  hindered  rotation  about  the  ring —ethenyl bond and this will affect the manner in which the dipolar couplings are related to the molecular geometry. This motion was treated by considering the molecule to exist in four conformations. These conformations differ in the dihedral angle being ± c/> and 180° ±<f>, where a dihedral angle of 0° has the ethenyl group lying in the plane of the aromatic ring. The averaging effect of rotation about the ring—ethenyl bond on the dipolar couplings was taken into account by calculating dipolar couplings for each of the four conformations and averaging the D 4=D«6» 2  values  and D  2 5  obtained. This  =D  5 6  averaging serves  to make  D 3=D , 2  3 6  .  The second major difficulty is that the molecule requires nonzero values of all  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 100 five independent elements of the order matrix to describe its orientation. Previous studies of this sort have usually exploited symmetries in the molecule to reduce the number of order parameters needed to describe solute orientation.  Finally there is the problem of chemical shifts. While values for the isotropic chemical shifts are available [48], the chemical shifts can be expected to exhibit significant anisotropy. Due to the strong coupling of the spins and dense packing of the lines there is no simple way to estimate these chemical shifts from the single quantum spectrum. In trying to fit a spectrum in which line positions are measured to less than 1 Hz., a starting parameter whose value may be in error by a hundred Hz. or more is a major problem. It should be noted that other strategies in the analysis of NMR spectra have been reported [135,136] but relatively few applications have appeared  in the literature [137,138,139]. For  complex spectra, these methods still require a reasonable estimate of starting values and can consume very large amounts of computer time.  2. The Multiple Quantum Spectrum  Multiple quantum spectroscopy holds the promise of easing the difficulties due to the spectral complexity. The five quantum spectrum is shown in figure 4.3. While ten lines are expected,! only nine lines are observed.  The spectrum shown in figure 4.3 is the  sum of three experiments with  preparation period delays of 6.5, 7.5 and 8.5 milliseconds. Three different delay tTen lines are expected because there are five distinct ways to flip five spins in the molecule. For each of the ways to flip five spins the remaining spin may be 'up* or 'down', so we should see five doublets.  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 101 Figure 4.3  I  I  32000  l  l  33000  I  34000  i  1 35000  1  1 36000  1  1 37000  F r e q u e n c y (Hz.)  Figure  4.3  The five quantum  spectrum  of DCEB  in  EBBA.  values are not enough to guarantee a T independent spectrum, but looking at figure 4.3 one might surmise that the most intense line is the result of two degenerate lines as it is considerably more intense than the other lines in the central region of the spectrum. This somewhat tenuous assignment is borne out by the eventual analysis.  The quality of the 5Q spectrum of figure 4.3 compared to the spectrum of figure 3.14a is due to the fact that the spectrum of figure 4.3 is the sum of f, projections of three 2D datasets each involving 256 times the data that went  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 102 into figure 3.14a. Figure 4.3 therefore represents 768 times the data involved in the production of figure 3.14a. At the time these experiments were undertaken the MEM calculation was not yet working. If the MEM had been available at that time it is almost certain that the results of figure 4.3 could have been obtained with much less data and effort than was necessary using the FFT. As it was, the MEM calculation proved very useful in re-examining the mountain of data which led to figure 4.3 in an attempt to find the missing tenth line. These efforts were not successful but they did 3'ield the valuable knowledge that the weak peaks in figure 4.3 were only truncation artifacts and were not real. The large amount of data represented by figure 4.3 explains why nine lines are observed while only seven lines are observed in figure 3.14a. Figure 3.14a is the result of a single cross section in the t,  dimension of a 2D dataset in which  the two missing lines have very low intensity.  The four quantum spectrum is shown in figure 4.4. This spectrum was the result of a selective detection experiment achieved by cycling the preparation sequence phase through 0°, 90°, 180° and 270° as described in chapter 2. The dataset for this spectrum consisted of 2048 increments in t,  at a dwell time of  25 microseconds and this places a lower limit of —20 Hz. on resolution in this spectrum. While lines in the wings of the four quantum spectrum are well resolved, the central region is very crowded.  Even with the aid of the multiple quantum spectra, the analysis of this problem is by no means simple. It would seem the simplest place to start would be with the five quantum spectrum, but even this is not appealing. On a strictly  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 103 Figure 4.4  I  1  25000  1  26000  1  1  27000  T  1  1  28000  I  29000  I  I  30000  F r e q u e n c y (Hz.)  Figure 4.4  The four quantum spectrum of DCEB in EBBA.  combinatorial basis there are 10! ways to assign the ten coherences in the five quantum spectrum. As for fitting any one of these assignments, there are 10 lines in the five quantum spectrum and we have 11 parameters to vary, so any one of the 10! possibilities could conceivably be fit. No doubt many of these fits could be rejected immediately, but the number is still frightening. Therefore, rather than expect an easy fit from the multiple quantum spectrum, some more basic work is in order.  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 104 B.  THE  SINGLE  QUANTUM  SPECTRUM  -  ELEMENTARY  CONSIDERATIONS  The first appraisal of the spectrum should be made by looking at the forest rather than the trees. The overall appearance of the spectrum is striking. The central region is composed of a dense and rather nondescript mass of lines, but what stands out are the multiplets flanking this central region.  On the left are three multiplets decreasing in intensity as one nears the center of the spectrum. On the right are two multiplets similar in appearance to their opposite numbers on the left. One supposes that if there is a right side twin to the least intense multiplet on the left it is buried in the central thicket of lines.  A more critical examination of these features leads to the tentative conclusion that the outermost multiplets arise from the aromatic protons of the DCEB molecule. Two observations suggest this. The forest of lines in the middle region is upfield of what one would take to be the center of the distinct multiplet structure. Knowing that ethen)'l protons generally resonate upfield of aromatic protons implicates the multiplets as arising from the aromatic protons and the central region as having its origin in the ethenyl group.  Also, the multiplet structure itself strongly suggests this assignment. The three protons of the aromatic ring of the DCEB molecule might be expected to give rise to an A B  2  type subspectrum. While  some multiplets are undoubtedly  obscured by the ethenyl lines, the multiplet structure which is visible indicates that the subspectrum is more like that of an A A system [140]. 2  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 105 Neither of these reasons for such an assignment is conclusive. Chemical shifts in anisotropic media may be substantially different from those observed in isotropic solution, even to the point of reversing relative values. Also it is. possible that the multiplet structure is due to 'deceptive simplicity' [141] on the part of the ethenyl protons, but this is unlikely.  The assignment  is virtually guaranteed by comparison with a spectrum of  1,3-dichloro-benzaldehyde lines, the  in EBBA. While the  same structure is evident.  There  spectrum are  contains  vastly fewer  too few couplings in this  spectrum to permit an analysis of either geometry or orientation, but it is useful in that it allows some preliminary assignments in the DCEB spectrum.  Proceeding with this  assignment  yields valuable information and  suggests a  fruitful course of action. In a simple AA ' spectrum the splitting between the 2  right- and leftmost lines is 5D^^i If we estimate the splitting in the DCEB spectrum as 3550 Hz. this yields a value of —710 Hz. for D  1 2  (and D ) . 1 6  This number is of course only very approximate as the width of the spectrum is measured from the approximate centers of the multiplets. Similarly, the splitting between two adjacent multiplets in an A A spectrum is 4D^^, From the DCEB 2  spectrum we can estimate D  2 3  to be —160 Hz.  With these estimated values for the couplings and the coordinate system shown in figure 4.2, possible approximate values for the diagonal elements of the order matrix may be found, for  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 106 (4.1)  But 26 is the x direction, so S20  =  S , one of the diagonal elements of the xx  order matrix. With an estimated value of 4.301 A for r ±0.106 for S  xx>  There is an ambiguity in the sign of S  w e 2 6  x x  f" d a value of in  at this point.  0  With this value of S , a value of 2.488 A for r xx  D  1 2  1 2  and the estimated value of  , an approximate value of ±0.040 is found for Sy . y  Using these possible values of the diagonal elements of the order matrix, the only combination which gives a simulation that bears any similarity to the experimental spectrum is S = +0.106 and S = +0.040. These values give xx yy a simulated spectrum which bears an overall resemblance to figure 4.1 in that the general structure is reproduced. The multiplets are present in roughly the right positions and the central mass of lines is where it should be. Fine detail in the spectrum is not well reproduced at all. The internal structure of the multiplets is unrecognizable.  The unknown off-diagonal elements of the order  matrix and the dihedral angle have a great effect on the long range couplings, and as a result these couplings are very uncertain. An experiment which would yield information on these couplings would aid the analysis greatly.  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 107 C. A FREQUENCY SELECTIVE MQNMR EXPERIMENT  a. Motivation  The knowledge that portions of the spectrum arise from a certain group of spins in the molecule suggests an experiment that may selectively elucidate information on the couplings between the aromatic ring and the ethenyl group. Because the outermost multiplets are so well isolated from the ethenyl lines it is possible to irradiate these lines selectively. If the preparation sequence  in the multiple  quantum experiment is made frequency selective for these groups of lines one suspects  that one should obtain the  situation where the  multiple quantum  coherence created is confined to the protons labeled 1, 2 and 6 in figure 4.2. The spectrum resulting from such an experiment should contain at most three quantum coherence and this coherence should show splittings due to couplings to protons of the ethenyl group. These should be the only splittings observed and any chemical shift effects should be suppressed because this multiplet will appear at the sum of the chemical shifts of the aromatic protons. One would therefore have a method of measuring sums and differences of the long range couplings without interference  from  the  much stronger  short  range  couplings or the  chemical shifts.  The term 'selective excitation' finds a number of uses in MQNMR. While this experiment will selectively excite only certain selected multiple quantum transitions it is selectivity of a different sort than other experiments lumped together under the term 'selective excitation' produce. As mentioned in chapter 2, methods have been developed to selectively excite odd or even order coherences,  with the  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 108 ultimate experiment being that of Warren and Pines where one excites just one particular order of coherence. The aim here is to excite MQ coherence only within a particular group of spins. With reference to the analogy drawn between MQ spectroscopy and deuterium labeling [29], this proposed experiment will label the molecule with chemical specificity. The ability to create MQ coherence among a certain group of spins is dependent upon there being a certain amount of 'first orderness' to the spectrum. A first order spectrum in the high resolution context is usually taken to mean that the chemical shift differences between spins are much greater than the scalar couplings between them. Here, the term first order should be taken to mean that the couplings within a particular group of spins are much greater than the couplings between groups of spins. In the case of DCEB, the couplings between the protons of the aromatic ring are much greater than the couplings between the aromatic ring and the ethenj'l group.  The  proposed experiment  would  seem to be a spin £ incarnation of the  experiment of Hoatson et. al. in the spectroscopy of spin 1 sj^stems [142]. There a frequenc}' selective excitation sequence was used to selectively excite double quantum coherence in a spin 1 system. The excitation sequence used was a nonselective 90° pulse, followed after a time 7 by a frequency selective DANTE pulse sequence. This excitation scheme was used to selective^ excite double quantum coherence for selected values of W Q , the quadrupole coupling of the deuterium nucleus. For a spin 1 nucleus in anisotropic media, the dominant perturbation of the Zeeman interaction is the quadrupolar interaction. Couplings between deuterium nuclei are usually so small as to be negligible and so the deuterium nucleus presents a very good example of a three level system and the  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 109 multiple quantum spectroscopy of such a system is limited to zero, one and two quantum spectra. In going to a multispin system such as that of a system of coupled protons there is a significant difference in the experiment, as will be seen shortly.  The proposed  experiment  would appear to have limited utility in the high  resolution spectroscop3 of isotropic solutions. In isotropic solution the limited range 7  of  the  scalar  coupling  mechanism  and  the  fact  that  couplings  between  magnetically equivalent nuclei are not effective give rise to distinctive coupling patterns which have been analyzed by Braunschweiler et a/.[143] These coupling patterns are easily recognized in a contour plot of the 2D MQ spectrum and may be used to identify coupled groups within a molecule in a verj' elegant manner. In partially oriented molecules, however, all couplings are effective and protons separated by large distances which have negligible J couplings may have sizable dipolar couplings. Generally, in partially oriented systems all protons are coupled to each other and coupling patterns are not easily discerned from a contour  plot of the data. While several applications of 2DNMR methods to  partially oriented molecules have been reported [144,145,146], these techniques are unfortunately  much less useful in oriented systems than they are in isotropic  solution.  b. Preliminary Results  In order to test the feasibility of such a frequency selective experiment a series of trial experiments was conducted using 1,1,2-trichloroethane, partially oriented in EBBA. This molecule and its single quantum spectrum are shown in figure 4.5.  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 110 Figure 4 . 5  519.0 Hz.  6,-63  H(2) Cl,  H(3)  D D  1 2  1 3  1734.2 Hz, 95.9 Hz,  H(1)  - 1 2 . 0 Hz 5.9 Hz.  J12 J13  A _j  1—1—1—r  6000  Figure 4.5 oriented in  -1—1—1—r  4000  5000  The 200 MHz. EBBA.  -1—1—1—r  -1—1—1—r  3000  ~ i — r  2000  spectrum of 1,1,2-trichloroethane (TCE)  1000  0  partially  In this spectrum the central triplet is due to the single proton on carbon 1 and the outer doublets are due to the two protons on carbon 2. The minor splitting within the multiplets is |2Djg+J^g|, while the distance between the centers of the doublets, labeled A, is |3D l 12  The nonselective MQ spectrum of TCE in EBBA is shown in figure 4.6a. In the two quantum region the two closely spaced lines are due to the 2Q coherence between HI and H2. This shows a relatively small coupling to H3. The two lines which show the large 'splitting are due to the 2Q coherences of H i and  l,3-Dichloro-2-ethenylbenzene and Frequency Selective M Q N M R / 111 H3, and H2 and H3. These coherences show a large splitting due to coupling to H2 and HI respectively.  With the single quantum spectrum of figure 4.5, it is possible to make one or both pulses in the M Q excitation period frequency selective for the outermost multiplets. This is easily accomplished by setting the transmitter carrier frequenc3' A Hz. downfield (i.e. to the left in figure 4.5) of the downfield doublet. With a pulse repetition rate of 1/A seconds, the first and second sidebands of the pattern depicted in figure 2.1 will irradiate the doublets belonging to H i and H2. This will also provide the offset necessary for separating the M Q orders.  The result should be an M Q spectrum where the only two quantum coherence is that of protons H i and H2. This coherence should show splitting due to coupling to H3. There should be no zero or three quantum coherences observed. The result of such an experiment is shown in figure 4.6b. Figure 4.6b is the spectrum observed when a DANTE sequence set up so as to irradiate the two doublets belonging to protons HI and H2. Clearly the experiment has only excited the double quantum  coherence of HI  and H2. This spectrum was  obtained with a preparation period delay of 7.75 milliseconds. Similar experiments were obtained with a series of ten spectra where, the preparation period delay was varied systematically between 7 milliseconds and 9 milliseconds, and the experiment was initial^' judged a success. These results, however, are deceptive.  Orthodichlorostyrene and Frequency Selective MQNMR / 112 Figure 4.6  a  ^ L J L J L A i  i—i—r 5000  i  < l  i i 10000  r  L  i—i—r  n  15000  i  i  r  20000  i—i—r  "I—I  30000  25000  Frequency (Hz.)  i—i—i—r  J  1—i—i—r 5000  n  i  r T—i—r  i  i  i—i—i—i—r 20000  r  15000  10000  I—I—I—I—I—I—I  25000  30000  Frequency (Hz.) Figure  4.6  in EBBA. EBBA. for  a.) b.)  The  The first  the outer  nonselective  The frequency pulse  doublets  in  MQ  200  selective MQ  the MQ pulse  in figure  4.5.  MHz.  spectrum  spectrum sequence  of TCE  partially  of TCE partially has  been made  oriented  oriented  frequency  in selective  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 113 1. A Theoretical Description - Weak Coupling  An analysis of the details of the experiment reveals a more interesting situation than first imagined. A theoretical description of the experiment is most easily approached using a spin £ product operator basis.  As usual we start with the equilibrium density matrix:  p(0) = C ( I + I 1z  2 z  + I ).  (4.2)  3z  The effect of a pulse selective for protons 1 and 2 is represented  p(t) = exp{-i JL ( I + I y)} P(0) exp{i 1 diy + I y)} 1 y  2  2  = C(I + I 1X  Note that  only  (4-3)  tit  £  2 X  + I ).  (4.4)  3Z  spins 1 and 2 have been affected by the pulse.  We now permit free evolution under the internal Hamiltonian, taken to be weak dipolar coupling with chemical shifts,  H  int = 2 " j l  Evolution for a time r yields  z  + .E. *2D I I . ij2  iz  jz  (4.5)  l,3-Dichloro-2-ethenylbenzene and Frequency Selective M Q N M R  = C exp{ - i . Z  pit)  exp{ iL  The  7 ^ 2 1 ^  -  wjl- + i Z  iZ_ cjjl } ( I iz  1 x  +I  2 x  / 114  + I ) 3z  7T2D..2I. I. }.  (4.6)  evolution under the Zeeman terms and the spin-spin coupling terms may be  calculated separately as the two terms commute.  p(r) = C exp{ -i Z +  7r2Djj 2I Ij } (I cos(w r) iz  z  1x  I COS(CJ T) + I sin(u T) + I ) 2 X  2  2  2y  +  l  3z  I^sin^r)  exp{ i Z . ir2Dj. 2Ij L }. (4.7) i<j J  J  With the weak dipolar coupling Hamiltonian  H  = 7r2D  d  1 2  I I l 2  2 z  +  7r2D I l 3  1 z  I  3 z  TT2D I  +  2 3  2 z  I  3 z  .  (4.8)  we have  p(r) = C{(I, cos(7r2D , r) + 21, yI sin(7r2D, x  2z  2  +  (21! yI cos(7r2D , T) — 41 I  +  (I, yCos(7r2D ,  2  3z  2  T)  1x  —  T)) C O S ( 7 T 2 D T ) costo^r)  2  13  I 3 2 (' '2D, 7 ) ) sin(7r2D T) cos(w, T) sm  2 Z  I  13  2  21 I sin(7r2D, T)) C O S ( 7 T 2 D T ) sin(cj,T) 1x  2  2z  13  — (21, I cos(7r2D, T) — 41 l 2 l 3 s i n ( i r 2 D , r)) sin(7r2D r) X  +  3z  2  1v  Z  Z  2  13  sin^r)  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 115 (I cos(7r2D 2x  ,  2  T)  + 21, I ySin(7r2D z  2  ,  2  T)) c o s ( 7 r 2 D r )  COS(CJ T) 2  23  + (2I yI cos(7r2D , 2 T) - 41, I I s i n ( 7 r 2 D , T)) sin(7r2D r) 2  3z  z  2x  3z  2  + (I yCOs(ir2D , 2 T) - I , I sin(7r2D , 2  z  2x  2  T))  COS(TT2D  23  2 3  T)  COS(CO T) 2  sin(o> T) 2  (2I I cos(7r2D, T) + 41, I yI sin(7r2D , r)) sin(7r2D r) sin(w,r)  -  2x  2  3z  z  2  3z  2  23  + I }.  (4.9)  3Z  Examination of (4.9) reveals a problem with the experimental scheme suggested so far and also provides a remedy for this difficulty.  The problem is that the density matrix described by (4.9) contains latent multiple quantum  coherences  of all orders.  It  4 I , I y I . These terms arise from I z  2  3 z  the spin-spin couplings D ,  3  and D  2 3  nonselective y pulse at an arbitrarj' 2IiyI x  a n  3  d  4I, I yI x  2  3 x  contains 1 x  and I  terms such as 2 x  2I,yI  3z  and  because of evolution due to  as summarized in (1.36) and (1.37). A time  T will serve to turn these into  respectively, and these operators represent multiple  quantum coherences of all orders. It is interesting to note that the desirable terms depend upon r as cos(7r2D, r) while all the undesirable terms depend 2  upon T as sin(7r2D, T). Using this pulse scheme it is therefore 2  possible to  excite only the desired 2Q coherence by choosing an appropriate value of r. The sin(7r2D, T) dependence of excitation of the other orders gives a 'window' during 2  which a nonselective pulse will create MQ coherence between H I and H2. While  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 116 this can work (witness figure 4.6b), it is hardly a satisfactory solution, for selection of an appropriate delay value will require either knowledge of the coupling constants or providential assistance. While a proper choice of T will indeed discriminate against MQ coherences involving proton 3, an improper choice which yields a small value of cos(7r2D, r) will actually inhibit excitation of the 3  very coherences we are seeking to select and favor excitation of those coherences we wish to avoid.  For the case of the second pulse being frequency selective and the first pulse being  nonselective  the  details  are  somewhat  different  qualitatively the same. The first pulse will create I Under the influence of the spin-spin couplings, 1 2I I y. 2z  3  A second pulse which is frequency  convert these into 21, I y and 21 I y. X  3  2 X  3  3x  3 x  but  the  as well as I  result  is  and I .  1 x  2 x  will evolve into 21, I Z  3v  and  selective for HI and H2 will  Again we have excited coherences  which we wish to suppress.  2. Strong Coupling  While the situation in the case of weak coupling is easity treated, the spectra of figure 4.6 were obtained in the case of strong coupling. TCE in EBBA at 200 MHz. exhibits a chemical shift difference of ~500 Hz. while D ~ 1 7 0 0 Hz. 12  Clearly the assumption of weak coupling in this case is unfounded. Analytical treatment of the strong coupling case with arbitrary chemical shift is much more difficult  as different terms in the unitary transformation  of (4.6) cannot be  separated and evaluated in turn. We therefore turn to computer simulation to test the result of the various pulse sequences. These simulations show that the  l,3-Dichloro-2-ethenylbenzene  and Frequency Selective M Q N M R  / 117  results depicted in figure 4.6b do indeed result from a fortuitous choice of T.  Using —  the parameters listed in figure  90° pulse  sequence  are presented  the  amount of double quantum  of  T.  The  produced  lower  as a  diagram  in figure  the amount  produced  of coherence  function of r. The sine/cosine dichotomy  as a function  of H I  and  milliseconds. It is during  weakest  and  the H1/H2  spectrum of 4.6b was  this  period  coherence  that  a complication which plagues  r.f.  the effect pulses  practice. There  many  of imperfect  which  the nature  [147]. A r.f.  such  as  and effect  rectangular.  sources  of pulse  instabilities,  timing  of various  rise  and  fall  such  Such  imperfections  times,  the  4.7 assume  are unattainable in  which  will  degrade the  A n excellent discussion is given  by Haeberlen  to the magnitude of the generated  pulse  errors and so forth. Still  of figure  pulses  pulse responses.  pulse  the phase characteristics of the r.f. When and  why  its  experiments, this complication  imperfections  of imperfections are related  pulse  explains  is at  results is hampered by  r.f. pulses. The simulations  are perfectly  are many  number  this  and simulated  multipulse NMR  agreement of theoretical and experimental of  and  coherence  7.0 -  observed.  Quantitative comparison of the experimental  being  the H1/H3  is strongest  H3  of the two traces is  apparent. The experiments of figure 4.6b were carried out in the interval 9.0  r  4.7. The upper trace represents  coherence of H I and H2  shows  —  4.5, simulations of the 90° D A N T E  droop,  other  long  term  imperfections  transmitter  are related to  the transmitter is rapidly  switched on  off phase transients arise during the switching process. The minimization of pulse  imperfections  is  essential  to  the  success  of  many  multipulse  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 118 Figure 4.7  »  - i  i  0.OO5  i  i  0.007  i  i  0.009 time  -  T-  0.005  i  i  0.007  1  1  time  1  1  1  1  0.013  0.015  (seconds)  1  0.009  1  0.CM1  1  1  0.011  1  1  0.013  :  r - —  0.015  (seconds)  Figure 4.7 Density matrix simulations for 1,1,2-trichloroethane in EBBA. a.) The expectation value of the two quantum operator I, I + 11 y^2x function of the preparation period delay. This represents 2Q coherence between HI and H2. b.) The expectation value of the two quantum operator I, I +1, I as a function of the preparation period delay. This represents 2Q coherence between HI and H3. a  X  2 y  X  3 y  y  3x  s  a  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 1 1 9 experiments which involve complicated pulse sequences and various strategies have been developed to minimize their effects, (e.g.  [148,149])  With commercial high  resolution instruments of the tj'pe used in these studies, however, there are no easy solutions. Commercial probes are generally optimized for B  0  homogeneit}'  and sensitivity. Pulse quality is usually subordinated to such considerations and unfortunately  it has a considerable impact on the experiment described here.  Hoatson et al. note that using an MQ pulse sequence with one 30 pulse DANTE sequence gives an experimental intensity which is 25% less than the predicted intensity. While the effects of pulse imperfections will vary from machine to machine and probe to probe, a general rule of thumb was established with regard to the CXP 200. Any coherences for which the predicted intensity was less than —0.2 in a calculation such as that depicted in figure 4.7 were not observed.  3. The Frequency Selective Experiment A simple solution to the problem of exciting all orders of coherence is evident upon slightly more consideration of (4.9) The offending terms are all of the form 2I ^I k  3z  and  ^^X^IM^  3 2  avoid is converting the I  W  ^  3 Z  E R E  ^,1 =  1,2 and X , y = x,y. What we have to  part of these operators into I  3 X  and/or I . This is 3 V  easily accomplished by making the second pulse in the MQ preparation sequence frequency selective for protons HI and H2 just as the first pulse was. This will obviate any MQ coherences involving H3.  This is quite a general method of exciting MQ coherences between selected spins. The fact that both pulses in the preparation sequence are selective for certain  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 120 spins will restrict the MQ coherences to those spins. That is, only those spins which have been irradiated will contribute terms in I^ or 1^ to any spin £ x  products arising during the evolution of p. Those spins which have not been irradiated can only contribute Ij to any spin £ product operators which develop. z  The coherences created may be antiphase with respect to the undisturbed spins, but they will only contain active contributions from the spins which have been irradiated. The resulting MQ coherence may be antiphase with respect to the spins which have not been selected, but it will  only  contain contributions from  the irradiated spins.  The effect of having both pulses frequency selective is illustrated in figure 4.8. This presents the same information as figure 4.7. The effectiveness of having both pulses frequency selective is obvious. The top trace, 4.8a, shows that 2Q coherence between HI and H2 is still excited very efficiently. The bottom trace, however, representing 2Q coherence between HI and H3, is now very nearly zero. Experimental confirmation of these predictions is presented in figures 4.9 and 4.10. Figure 4.9a is a 400 MHz. spectrum of TCE in EBBA with only the first pulse in the preparation sequence frequency selective. A preparation delay of 10 ms. has been deliberately chosen from simulations like those of figure 4.7 so as to enable the excitation of all MQ coherences.  Figure 4.9b presents a  spectrum of the same sample under the same conditions except that now the second pulse in the preparation period is frequency selective. Again, all orders of coherence are present. Finally, figure 4.10 presents the spectrum of figure 4.9 with  both  pulses in the preparation period frequency selective. Clearly, having  both pulses in the preparation period frequency selective has had the desired  0.9 0.8 7 6 5 4 3 2 1 O - 0.1 -0.2 -0.3 -O.A  -0.5 -0.6 -0.7 -0.8 -0.9 0.005  0.007  0.009 time  0.01 1  0.01 3  0.01 5  (seconds)  Figure 4.8 a.) The expectation value of the two quantum operator I, l 2 y + 11 y^2 x cs a function of the preparation period delay, b.) The expectation value of the two quantum operator as a function of the preparation period delay. Now both pulses are DANTE sequences. The upper trace shows that 2Q coherence between HI and H2 is excited very efficiently but the bottom trace shows that 2Q coherence between HI and H3 is now negligible. X  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 122  Figure 4.9  0  5000  10000  15000  20000  25000  Figure 4.9 a.)The 400 MHz. MQ spectrum of TCE in EBBA. The first pulse in the preparation sequence is a DANTE pulse train. The preparation delay is 10 msec. Note the appearance of all two quantum coherences, b.) The same spectrum as (a) but now the second pulse in the preparation sequence is frequency selective  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 123 effect. Qualitatively identical results were obtained with preparation period delays ranging from  10 to  15 milliseconds. The only multiple quantum  coherence  observed was that between protons H i and H2.  The  requirement of two DANTE  sequences  in the MQ preparation sequence  carries a penalty, however. With regard to the previous discussion concerning pulse imperfections and their impact upon the spectrum, having two DANTE sequences in the MQ preparation period compounds the problem. It does not bode well for signal to noise ratios in experiments requiring high selectivity (i.e. many pulses in the DANTE sequence)  D. RESULTS USING DCEB  The  application of  this  frequency  selective  MQ experiment  to  DCEB  is  straightforward in practice but in principle there is a difference. In the case of TCE it was possible to irradiate all multiplets belonging to protons H i and H2, and the density matrix after the first pulse was proportional to l\  x  +  ^2x ^3z+  It contained only single spin operators from all spins.  In the case of DCEB, however, it is not possible to irradiate all transitions arising from the aromatic protons. The central lines belonging to the aromatic protons are obscured by the ethenyl resonances. The safest parts of the spectrum to use a DANTE sequence on are the strong outermost multiplets. This inability to  irradiate  all the  aromatic  resonances  changes  the  theoretical description  somewhat. If we take the case of three equivalent, weakly coupled spins for the sake of simplicity, immediately after the first DANTE pulse the density matrix  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 124 Figure 4.10  I  1  1  1  1  0  1  1  1  1  1  1  5000  1  1  1  1  1  10000  1  1  1  15000  1  1  1  20000  1  1  1  25000  Figure 4.10 a.)The 400 MHz. MQ spectrum of TCE in EBBA. Now both pulses in the preparation sequence are DANTE pulse trains.  will be proportional to  P = C{ I l X  + 4  I  1 X  I  2 Z  I  3 Z  +  * 2  X  + 4I I I ) 1Z  2Z  3X  +  4I I I 1 z  2 x  3 z  + I  3 x  (4.10)  In contrast, if it were possible to irradiate all three spins we would simply have Iix  +  ^2x  ^ax- This  w  i  un  ° t make a substantial difference to the outcome  of the experiment, for an analysis of the evolution of (4.10) under a Hamiltonian  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 125 containing weak dipolar couplings  H  D  =  TT2DI I 1Z  2Z  +  TT2DI I 1Z  + 7r2DI I 2z  3z  (4.11)  3z  shows that at an arbitrary time, r after the first pulse sequence the density matrix will be  p(r)  =  C { I COS(47TDT) 1X  + I cos(47rDr) + I cos(47rDr) 2x  3x  + 2I I sin(47rDr) + 21, I sin(47rD7) 1y  2z  y  + 2I I sin(4;rDr) 2y  3z  + 21, I sin(4 7rDr)  + 2I I sin(47rDr) + 1z  +  3y  Z  2 Z  2 y  2I I sin(47rDr)  4I I I COS(47TDT) 1 X  3z  3 Z  2z  +  3y  41, I I C O S ( 4 7 T D T ) z  2 X  3 Z  + 4I I I cosi;47rD7) } 1z  2z  (4.12)  3x  Obviously, (4.12) contains potential MQ coherences in terms such as 2 I I 1 y  4I I I 1 x  2 z  3 z  3 z  and  and a second pulse at an appropriate time will convert these to MQ  coherences. Note that the form of (4.12) suggests that we may selectively create either 2 or 3 quantum coherence by using a y or an x pulse respectively. This is a result of our neglect of chemical shifts in the Hamiltonian (4.11), and harks back to the even/odd order selective experiment described in chapter 2. Inclusion of chemical shifts in the evolution will lead to a system in which 2 and 3 quantum coherence can be created by a pulse of arbitrary phase at the expense of greatly complicating (4.12).  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 126 It is also worth noting that the spin state described by (4.10) implies that one could selectively create a system containing only 1 and 3 quantum coherence by immediately  following  the DANTE  sequence  giving  rise  to  (4.10)  with  a  nonselective 90° pulse phase shifted b}' 90°. That is, an x pulse applied to the density matrix represented by (4.10) would yield  ^1X  +  4I I I 1 x  2 x  + 3 x  *2X  +  IlX^2X^3X  +  *3X  ^ 1 X^ 2 X^ 3 X  +  This, however, ignores a very real feature of the DANTE pulse sequence. The delaj' periods between the small angle pulses allow antiphase magnetization to develop with an}' other spin couplings in the system. Thus, a phase shifted nonselective 90° pulse at the end of the DANTE  sequence  is capable of  producing undesired coherences. If one only needs to use a very short DANTE sequence or the couplings to other spins in the system are very small, then a phase shifted pulse at the end of the DANTE sequence should work, but here, where the DANTE sequences are reasonably long to provide the selectivity and the couplings are somewhat large, a phase shifted pulse will not produce this result.  The  experiment applied to DCEB is set up so that the DANTE  sequence  irradiates the outermost multiplets in the DCEB spectrum. This provides the offset necessary for the separation of the MQ orders. The spectrum resulting from such an experiment is shown in figure 4.11. The three quantum spectrum consists of 6 resolved lines symmetrically disposed about  CJ +a> '^6' +  1  2  t n e  s  u  m  °f  the chemical shifts of the three aromatic protons. The observed splittings are 103  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR/  0  10000  Figure  4.11  i  r  20000  127  ULXAIJL I 30000  Figure 4.11 The 400 MHz. frequency selective excitation MQ spectrum of DCEB in EBBA. a.) The frequency selective experiment, b.) An expanded view of the three quantum region of (a) c.) The nonselective MQ spectrum of DCEB in EBBA showing all orders of coherence.  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 128 Hz., 235 Hz., and 453 Hz.  1. The Long Range Couplings  With the assumption that the long range couplings are small enough that a 'weak' treatment of the coupling is permissible the expected splittings are:  2D  1 3  + 2D  i a  + 2D,  2D  1 3  + 2D  l a  -  2D  1 3  -  -2D  1 3  2D  1 4  + 2D  1I(  5  + 4D  2 3  + 4D  2D  1 5  + 4D  2 3  + 4D , - 4D  2 5  (4.14)  + 2D  1 5  + 4D  2 3  - 4D  2 5  (4.15)  + 2D  1 5  - 4D  2fl  + 4D  2  2 3  2 a  + 4D  2ll  + 4D  (4.13)  25  + 4D  25  (4.16)  These expressions are written down trivially by considering simple 'up' and 'down' combinations of the coupled spins. Extracting useful information from the above would be difficult if it were not for two additional observations arising from trial simulations using SHAPE. The first is that the couplings are negative with the values of S xx  and S  which have been estimated from the SQ  jy  spectrum.  The second observation is that D, are D  2 3  and D  2 5  3  and D ,  5  are often approximately equal, as  . This is reasonable as protons 3 and 5 have very nearly  identical y coordinates in the coordinate system of figure 4.2. The y displacement between protons 1, 2 and 6 and protons in the ethenyl group is easily the largest of the displacements and so dominates the 1/r  3  term in the expression  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 129 for the dipolar couplings. This serves to make D , D 5 2  and D, , and D 3 and  3  5  2  roughly the same. This degeneracy would explain the observation of only  six lines in the 3Q frequency selective spectrum rather than the expected eight by making (4.14) approximately equal to (4.15). Additionally, couplings between the aromatic ring protons and proton 4 are always smaller due to the extra distance involved. Couplings between the ring and proton 4 are generally about half as big as the couplings involving protons 3 and 5.  With the assumption that all couplings have the same sign, (4.13) can be set equal to —453 Hz. Equating D  1 3  and D  1 5  , and D  and D  23  2 5  reduces (4.14)  to 2D, „ + 4D „. With the expectation that couplings to proton 4 will be the 2  smallest, we can set this equal to 103 Hz. We therefore have  2D, „ + 4D „ ~ -103 Hz. 2  2D  1 3  + 4D  2 3  ~ 2D  1 5  + 4D  2 5  -  -175 Hz.  It is now necessary to find some way of estimating the relative contributions of the individual couplings to the sums above. This was estimated in a crude calculation based simply on the distances involved.  The distances between protons in the aromatic ring and protons in the ethenyl group vary depending on the dihedral angle, of course, but they generally lie between 5 and 7.5 Angstroms. The distances between proton 1 and the protons in the ethenyl group are always —1.5 A longer due to geometry. A simple  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 130 calculation of 1/r  3  using these typical values shows that 1/r  3  to 0.6 as big for r ,  as it is for r , where x =3, 4, 5.  x  2 x  Using this crude estimate the D D,  is only about 0.5  3  -  1 x  are taken to be one half the D  -17 Hz.  D  23  -  - Therefore,  -35 Hz.  Dm  ~- -10 Hz.  D , ~ -20 Hz.  D  ~ -17 Hz.  D  1 5  2 x  2  2 5  ~ -35 Hz.  These values of the long range couplings along with the values of D,  2  and  D 6 estimated earlier give an extremely good simulation of the outer regions of 2  the single quantum spectrum. Most lines can be assigned immediately and after only minor line reassignments the outer multiplets have been fit. A fit of 53 lines with an rms. error of 2 Hz. yields the following: (see next page)  l,3-Dichloro-2-ethenylbenzene and Frequency Selective MQNMR / 131  Table 4.1 Estimated Spectral Parameters of 20 mol% o-DST in EBBA Chemical shifts 6, 6  2420 ± 2517 ±  2  1.6 1.0  Dipolar couplings D  1 2  D, D,, D D 3  1 5  1 6  -705.9 -15.5 -14.9 -23.3 -705.9  ± ± ± ± ±  0.3 1.5 1.5 1.3 0.3  Scalar Couplings[48]  J,  2  J l 3  J ! j, J l 5 J l 6  J 3 2  D D D D  2 3 2 9  2 5 2 6  -36.3 -18.5 -31.2 -165.9  ± ± ± ±  0.9 0.9 0.8 0.9  J n J J 6 J 3H J 2  2  5  2  3 5  J 6 3  J4 J« 6 J 5  5 6  8.06 0.13 0.12 -0.53 8.06 0.00 -0.10 0.39 0.00 11.72 17.82 0.00 1.38 -0.10 0.39  V. l,3-DICHLORO-2-ETHENYLBENZENE - SOLUTION, STRUCTURE, AND ORIENTATION At  this point we have a very good Fit to the outermost  spectrum.  Elementary analysis of the  gross features of the  portions of the single quantum  spectrum have yielded the aromatic dipolar couplings and the frequency selective MQNMR  experiment has greatly simpliFied the task of Fitting the long range  couplings.  An excellent Fit has been made to the outer multiplets. 53 lines have been Fit with an rms error of 0.6 Hz. However the dense thicket of lines in the central region remains a formidable problem. It has proved possible to assign several lines on the low Field side of the central region, but an inspection of the Fit reveals that these lines are predominantly aromatic in make-up. We still have 3 unknown chemical shifts  and  three unknown dipolar couplings. Attempts to  calculate the dihedral angle and/or the off diagonal elements of the order matrix from the long range couplings 3'ielded inconclusive results. If this had been possible it should have yielded reliable estimates for the intravinyl couplings. The long range couplings are simply not sensitive enough to the orientation to allow this.  A. ANALYSIS OF THE 4 AND 5 QUANTUM SPECTRA. Armed with the information gathered so far it is now profitable to turn our attention to the 4 and 5 quantum spectra.  At the outset line assignment in the multiple quantum spectrum is very simple. 132  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 133 Any lines in the four or five quantum spectra which involve the flip of all three vinyl protons are almost immediately assignable. These lines do not depend on the intravinyl couplings. Such lines only depend on the ring and the long range couplings. These lines do, however, depend on the chemical shifts of the vinjd protons. The chemical shift information contained in MQ LCNMR spectra is a point which has not been stressed. In fact work published to this point has almost invariably utilized a 180° refocussing pulse in the middle of the evolution period to remove the linebroadening effects of magnetic field inhomogeneity. Such a refocussing pulse also removes the effect of chemical shifts from the MQ spectrum.  Here, chemical shift effects are of great importance in attaining an eventual fit of the single quantum spectrum. The three vinyl protons are very strongly coupled and  as  a result  of chemical shift  anisotropy have chemical shifts  significantly different from their isotropic values. The technique of LCNMR has at times been suggested as a method of measuring these chemical shift anisotropies [23,150]. Measurement of these quantities is an important application of solid state NMR [12,147] as the information contained in the chemical shift tensor is a sensitive probe of the bonding within a molecule. Considerable effort has been aimed at the measurement of such quantities in the solid state and has resulted in  some  very  important  and  powerful  developments  of NMR theory  and  experiment. Liquid crystal solvents, however, have been shown to be unreliable for the measurement of proton chemical shift tensors [151], and so while values of chemical shift differences for different sites in the DCEB molecule will be reported in this thesis, no attempt will be made to interpret these in terms of  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 134 shift anisotropies.  With considerable effort, what was felt to be a reasonable fit to the four and five quantum spectrum was obtained. This fit had an rms error in the line positions of 15 Hz. This is less than the linewidth in the 4 quantum spectrum and about one third the linewidth in the 5 quantum spectrum.  A  simulation of the  single quantum  spectrum  using these parameters  was  disappointing. The parameters derived from this admittedly marginal fit to the multiple quantum spectra yielded a single quantum simulation which was too broad in the central region and had a rather uniform intensity distribution across that region. While a perfect single quantum fit was not expected due to the resolution limitations in the multiple quantum spectrum, the poor quality of the single quantum simulation was surprising. No small adjustment of the parameters derived from agreement  the  MQ fit would bring the  single quantum  simulation into  with experiment. It was obvious that large adjustments  would be  necessary in order to fit the single quantum spectrum.  B. A MODIFIED STRATEGY.  The large number of lines in the single quantum spectrum makes adjustment of starting values a confusing process. Because of the tight coupling within the vinyl group, adjustment of any one parameter brings about numerous changes in line positions, and it is difficult to follow just what is happening in the spectrum as one adjusts various parameters.  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 135 On the other hand, the spartan appearance of the five quantum spectrum makes it very useful for adjusting spectral parameters in search of possible starting values. The effect of changing a chemical shift is easily distinguished from the effect of a change in a coupling constant. A change in chemical shift will move a number of lines in one direction while a change in coupling constant will move lines in opposite directions. Of course, the same thing happens in the single quantum spectrum but it is much easier to follow the spectral wanderings of ten lines while systematically varying parameters than it is to follow the motions of a hundred lines.  Therefore, rather  than making extensive line assignments  and attempting an  iterative solution of either the single quantum or the multiple quantum spectrum, it was found to be much more efficient to use the five quantum, four quantum and the  single quantum spectra synergisticalty, making changes in the five  quantum spectrum and checking possibilities using the single quantum spectrum and vice versa. This was a very productive utilization of the multiple quantum spectrum.  This strategy eventual^ yielded a very promising set of parameters. The five quantum spectrum was in excellent agreement with the experimental spectrum. The  four  quantum  spectrum  exhibited some problems but  there was good  agreement with most of the line positions. The single quantum spectrum was particularly encouraging. Viewed from a distance the fit looked excellent. The overall shape was very close to that of the experimental spectrum and even minor features ~such as low intensity peaks just upfield of the central region  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 136 were present. Weak peaks such as these often arise from highly mixed states and are extremely sensitive to combinations of parameters. A closer examination revealed some problems but overall the parameters derived from the multiple quantum fit were encouraging, and a sustained effort to fit the spectrum from this starting point was made.  All attempts to fit the spectrum from this starting point failed. While the calculated spectrum had strong similarities to the experimental spectrum, there were some difficulties which no amount of least squares adjustment would solve. The differences were manifest in such things as incorrect intensities and lines being out of position by 5 — 10 Hz. Numerous line assignments (along with the inevitable deassignments)  were made and starting parameters  were varied by  minor amounts, but the spectrum could not be fit. Still, the simulation was tantalizing.  The magnitudes of the couplings seemed right but the signs of the couplings were less certain. Small changes in the off diagonal elements of the order matrix and/or minor changes in the dihedral angle make it possible to change relative signs of the intravinyl couplings almost at will. Simulations carried out while changing the  relative signs of the  intravinyl  couplings reveal the problem.  Simulations to this point had had D < 0 . On changing the sign of D „ u 5  5  the  spectrum is solved almost immediately.  Final analysis of the spectrum yields the parameters given in table 5.1. These parameters come from the final fit of 183 lines in the single quantum spectrum.  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 137 N  N  «  N  «  «  o  ©  ©  d  d  o  r H C S N M - H r H r n c S C S i n C S r H C O r H C S  o  Q o g  -H+l-H-H+l+l  £  ~  © cs cn  cn ©  CS cs  M  (O  01  U)  rt  P3  N  N  N  to •* ** o  "  rt  - H  CO  ©  ->  d  d  d  ©  d  d  #  -H -H -H -H -H -H o o o o o o co m m to eri o _ co CO CS t D CM  —• .s  cs E- co  CO  E  M  to 04  cn co  CO O  co co  CS  CS  CS  m  o  d  o  o  o  o  o  •  • cn  •  •  •  '  Ol  N  -—< CN  ^  O l 1  CM  M  lO  Ul  '  00  '  •^tr-in^'^'cor^tocs^'-cocococ^to m r~ cs o o c s c o r ~ c o e s o © » N » U) . . • t-  N  tomtot^tO'^'T'^'coocn't  ©  O  O  O  O  O  O  O  O  '  f  l  '  C  O  r  H  O  C  S  O  O  ddddddddddddddd - H - H - H - H - H - H - H - H - H - H - H -H -H -H w a i»  co cn to to co to cs to o to es © r i r i l OtOO CS S rr H i (r H O ^ ^r H N CnN CO « i r«H BO r H c n ^ c s c o c n t o c o i n r f i n t o t O ' *  to co rH cn co in co cn o * H  o  - P c s r H « H - H p c o r H © c c r ~ r H c o m r H ©  • H  cs cs  o  - H - H + l - H + l - H - H - H - H - H - H + l + l + l + l  cs  cs  o  ,  -H -H -H +) -H -H cn cs « o  S  o  ddddddddddddddd  CO  rH  d  m m m t ^ i n c o c o f c s c o m c o o e o - f o o o o o o o o * c s « o c s o o  o d d  n  o  CS  o>  co 2  ©  o  - H - H - H + l ' - H - H - H - H - H - H - H + l + l + l + l K j H I O H i o o i t D H i H O l o cn cs to — d in in co cs  B  O  o  tD  cs cs  •  '  i  ^  O  '  »  •  r-1  rH  rH  1  N  CO  hi  Hz  CN CO  CO CO © CO CN CO CO  d d d d  t4_  O o O o cn o d  Q  c o cs  -  .5  to m o o o d d d  to t~ t— o o O d d d  o d  to to to' m o o o o d d d d  to  CS CO  d  *•«  O to m to in in o cs o o d d d d d  -H -H +1 -H -H +)  -H -H -H -H -H -H -H -H +1 -H -H +1 +1 -H +1  O ^  rH cs to es r H r- to cn cn CO CO in r H cn en © CO en cs to CS in in cs d in in d r H r H CO CO r H cs cs cs o o rH 1 1 * • t- 1 t- 1  c~  w  m o o d d  rH  •f  cs  m  cs co  to to cn od co -<r d m rH ^ rH in CO O o  m to oc rH  m  cs cs cs cs cs  to r- cn to en r- cs m t— CO eri ui co d -flo CO1 en cs ee rH  u O u t0)  u o u u 0)  u o u u 0)  uj ui ai e c q  ui  i-  u  C  u o i_ u 0)  d fcc c u  C c c c  rtrtrtrt -C JS -C •i  HJ  -•(** L_ Ln  LH  Ul to UJ m CJ V  bo c  5  a.  J3  8 Q:Q  CO  o o o  #  rH rH +- ++  00  3  ' *© fcO »o *© i o *o  l>  c c c c  Q Q Q Q P Q Q Q Q O Q Q Q  r-(  CO to  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 138 The rms error in the calculated line positions is 0.36 Hz. A comparison of the experimental and calculated spectra is shown in figures 5.1a and 5.1b.  While the use of an external standard in the form of acetone — d  6  makes direct  comparison of chemical shifts somewhat uncertain, it is possible to compare the anisotropic  and  isotropic chemical shift  differences  molecule. These chemical shift differences  for  various sites in the  are listed in table 5.2 and clearly  demonstrate the importance of determining the anisotropic chemical shifts in order to solve the spectrum. Some of the chemical shift differences are as much as 147 Hz. less than the chemical shift differences in isotropic solution.  Table 5.2 Chemical Shift Differences in Oriented and Isotropic DCEB. Values are in Hertz. Oriented  Isotropic  402.17 503.72 328.69 35.81  549.76 629.88 398.84 40.76  As it stands there is not enough information in one spectrum to permit a structural analysis. Moreover, it proved impossible to perform a least squares SHAPE analysis of the order matrix if the dihedral angle was included in the fit. Therefore, a number of SHAPE analyses were performed while holding the dihedral angle fixed in the range 0 — 90°. This analysis revealed a broad minimum in the error of the fit in the range 36° — 48°. The minimum in the weighted least squares error was at 39°, and the order matrix calculated for  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 139  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 140  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 141 this geometry is as shown in table 5.3.  Table 5.3 Order matrix of 20 mol% DCEB in EBBA. 0.1436 0.1082 0.0351 0.0102 0.0187  zz xx xz xy  While values of S  z z  and S  x x  are nearly constant across the range of angles  used, the values of S , S and S obtained from the fit are extremely xy jz xz sensitive to the value used for the dihedral angle. For instance, a 1° change in the dihedral angle to 40° brings about a 13% change in the least squares estimate of S  x z  while the error in the fit changes by 0.5%. The geometry must  therefore be regarded as verj' uncertain. More information can be had from the analysis of the molecule in a different liquid crystal. This will provide a different orientation and hence a different  set of dipolar couplings. The liquid crystal  chosen for this purpose was a mixture of 55 wt. % Merck 1132 and 45 wt. % EBBA. C. S P E C T R A OF DCEB IN A N EBB A/1132 MIXTURE.  The reason for this apparently strange choice of liquid crystal is as follows. For small molecules such as H  2  the interaction of the molecular quadrupole moment  with an electric field gradient present in the anisotropic liquid crystal environment has been shown to be the determining mechanism in the orientation of the molecule [21]. The mixture of 55 wt. % 1132 is a mixture which produces a  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 142 zero average electric field gradient as experienced by the deuterium nucleus in D  [152].  2  This removal of the  gradient orientation mechanism has  molecular quadrupole  moment—electric field  led to the development of a model for  molecular orientation based upon the size and shape of the . solute molecule by van der Est [153] which will be discussed shortly.  Therefore, in addition to providing extra parameters for a geometrical analj'sis, the use of this liquid crystal mixture should be a stringent test of this model for orientation due to the irregular shape of this particular solute.  The spectrum of DCEB in 55 wt% 1132 is shown in figure 5.2. A resemblance to the spectrum in EBBA is apparent but the relative values of the dipolar couplings have changed significantly. The method of solution chosen for this spectrum was to take an approximate molecular geometrj' as described in chapter IV. and, using a value for the dihedral angle of 4 6 ° , perform a calculation based on the size and shape of the solute to predict the order matrix.  This requires knowledge of the force constant, k, for the liquid crystal. This was estimated in the following manner. The value of D 6 was estimated to be —140 2  Hz. from the distance between the multiplets as described in chapter IV. The force constant k was adjusted so that the calculated and experimental values of D  2 6  agreed. This is equivalent to adjusting k so that values of S  x x  agreed. The  calculation then yielded the predicted values for the order matrix reported in table 5.4. These order matrix elements have changed significantly from the EBBA fit, and they predict a substantial change in some of the dipolar couplings. In  Figure 5.2  JL  1  2000  J  J.  -ft 1—  1000  i  1  1  1  o  1  r  imj I-  1  n \ L . 1  -1000  Frequency (Hz.) Figure 5.2 The single quantum spectrum of o-DST in 55 wt% 1132.  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 144  Table 5.4 Predicted order matrix of 20 mol% DCEB in 55 wt% 1132. 0.1767 0.0926 0.0059 0.0279 0.0174  zz xx xz xy particular the  values of the couplings within the vinyl group have changed  radically, even to the point of D  fl 5  changing sign. Determination of the signs of  the various vinyl group couplings was one of the major problems in fitting the DCEB/EBBA spectrum.  A trial spectral simulation using these predicted dipolar couplings yields the spectrum shown in figure 5.3. The similarity to the experimental spectrum of figure 5.2 is remarkable.  With the starting point of figure 5.3 it proved possible to obtain a fit to the spectrum using single quantum information alone. The final parameters obtained from this fit are given in table 5.1. The final fit involved 190 lines and the rms. error in the line positions was 0.3 Hz. Note that the model calculation based on the size and shape was correct in predicting the change in sign of  The single quantum spectra of two more solutions of DCEB in 55 wt% 1132 were measured and analyzed in an effort to procure different sets of dipolar  Figure 5.3  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 146 couplings for an analysis of the molecular geometry. One of these was a dilute solution with a concentration of about 2 mol%. The other spectrum analyzed was at an intermediate concentration of 15 mol%. The spectrum of the 15 mol% solution was quite similar to that of the 20 mol% solution and the analysis was straightforward. The spectrum of the 2 mol% solution was 70 % wider than the spectra of the two more concentrated solutions, with a spectral width of roughly 6000 Hz. compared to a width of 3500 Hz. in the more concentrated samples. This spectrum was analysed by linear extrapolation of the dipolar couplings and chemical shifts from the more concentrated solutions to the dilute solution based simpty on the overall spectral widths. This extrapolation gave starting values for the spectral parameters which were sufficiently close to the actual values so -that fitting the spectrum was relatively simple.  The results of the analysis of the dilute solution are presented in table 5.1. The fit involved 150 lines and the rms error in the line positions was 0.6 Hz. The results of the fit to the 15 mol% solution are also reported in table 5.1. Here the parameters result from the fit of 180 lines with an rms error of 0.3 Hz.  D. REFINEMENT OF GEOMETRY.  The starting point chosen for the geometrical analysis of the molecule was a composite structure with features being taken from several sources. The aromatic ring is assumed to be the same as the ring structure of 1,3-dichlorbenzene [154]. Geometrical parameters describing the vinyl moiety are taken from a theoretical study of substituted benzenes which included styrene [50]. The bond angles within the vinyl group are considerably different from from those one would predict  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 147 assuming ideal 120° bond angles around sp  hybridised carbons. Reference [50]  2  finds very little variation in C —H bond lengths at different positions in the ethenylbenzene  molecule, so carbon — hydrogen bond lengths  were taken to be  o  1.084  A throughout  the  molecule. The assumed  geometrical parameters are  indicated in figure 4.2. With these starting values and the experimental data listed in table 5.1, a least squares refinement  of these geometrical parameters was attempted using the  program SHAPE. Due to the strong correlation between the dihedral angle and the off diagonal order matrix elements, the S  x z  in particular, least squares  adjustment of the dihedral angle while simultaneously fitting the order parameters proved very difficult.  Another factor  which weighs against  a refinement of  geometry using these couplings is the fact that the three specta of DCEB in 55 wt% 1132 are linearly related. That this is so is apparent from the way in which the spectrum of 2 mol% DCEB in 55 wt% 1132 was fit. Such a fit was impossible using the dipolar couplings from a single spectrum. By combining the dipolar couplings from all four analysed spectra a stable least squares adjustment was achieved. Even then the number of geometric parameters which could be varied with confidence was limited. Because of the strong dependence of the calculated values of the  order parameters on the dihedral angle a detailed  geometrical analysis vanning many parameters was felt to be unwarranted. The only structural parameters varied were the dihedral angle and the length of the bond connecting the aromatic ring to the vinyl group. The final value of the dihedral angle was 46.5° ±0.4°. The distance from the aromatic ring to the o  o  ethenyl group was shortened by roughly 0.1 A to 1.373 + 0.004 A. It is felt that  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 148 the errors in these values are a bit optimistic. These are the result of the SHAPE error calculation which is based on a statistical analysis of the spectral parameters performed by LEQUOR. While this calculation is statistically correct it was noticed that by performing SHAPE analyses  of any three of the four  available sets of data, the dihedral angle varied by as much as 3°—4° and the o  ring —ethenyl bond varied by as much as 0.03 A. More conservative estimates for these geometrical parameters are felt to be 44° ±3° for the dihedral angle and 1.38 A±0.03 A for the ring-ethenyl bond. The final values of the order parameters arrived at via SHAPE are given in table 5.5. An examination of the data presented in table 5.5 gives an indication of the importance of the molecular quadrupole  moment —electric field  gradient  mechanism in the orientation of this solute. For the three solutions in 55 wt% 1132 the ratio S / S x x  z z  is in the range -0.53 to -0.55. However, for the solution  of DCEB in EBBA, the ratio is -0.76. This change cannot be accounted for by simple  models  based  completely  on  properties  of  the  solute  (e.g. inertial  properties), nor can it be explained by a model where all liquid crystals are treated in the same way (e.g. the size and shape model where all liquid crystals are simply elastic tubes). This difference  in ordering must be due to some  property intrinsic to the liquid crystal, probably the electric field gradient.  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 149  Table 5.5 Order Parameter Data for DCEB. ' 20 mol% DCEB in EBBA  zz S S  X X  xz  S  xy  0.00304  zz S  -0.172 0.0941 -0.00551 -0.0186 0.0027  S  20 mol% DCEB in 55 wt% 1132  S  X X  xz yz xy  S  s  S  15 mol% DCEB in 55 wt% 1132  -0.146 0.111 0.00162 -0.0221  zz xx  S  S  xz yz xy 2 mol% DCEB in 55 wt% 1132  zz X X  5  xz yz xy  -0.187 0.102 -0.00611 -0.0200 0.00316  -0.240 0.127 -0.00871 -0.0238 0.00626  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 150 E. A COMPARISON OF INERTIAL AND SHAPE ORDERING MODELS  1. Ordering Due to Inertial Properties of the Solute  a. The Empirical Findings of Anderson  The model for orientational ordering due to inertial properties of the solute is based  upon  empirical findings  16 years  ago by Anderson that  the  order  parameters and the principal moments of inertia for a series of 27 substituted benzenes  showed  a  strong  correlation  [18]  and  these  findings  have been  corroborated by other workers [155]. His main findings may be summarized as follows. 1.  The principal axes of both the inertia tensor and the order matrix are very nearly coincident.  2.  The axis associated with the large value of the order matrix is found to be perpendicular to the axis associated with the large value of the inertia tensor.  DCEB provides a very simple test of these generalizations. The principal axis system of DCEB is verj' nearly coincident with the Cartesian system defined in figure 4.2. This is because most of the atoms in the molecule are coplanar and the two heavy chlorine atoms are located on an axis parallel to the Cartesian x axis. The principal moments of inertia and the principal axis system of the inertia tensor are given in table 5.6. The principal axes e  e , 2  and e  2  are  given in terms of the x, y and z of the Cartesian coordinate system of figure 4.2. The principal values and principal axis systems of the four order matrices are reported in table 5.7. From table 5.6 we see that the largest moment of  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 151  Table 5.6 Principal Moments of Inertia and the Principal Axis System of DCEB  x y z  Ii  I  I  336.98  619.33  938.91  e, 0.994 0.102 -0.0233  e -0.103 0.991 -0.0876  e 0.0142 0.0895 0.996  2  2  3  3  inertia is about the axis which is very nearly the z axis. Referring to the order parameters in table 5.5 we see that the smallest of the order parameters is that of the z axis, in agreement with point 2 above.  As for the coincidence of the principal axis systems of the inertia tensor and the order matrix this is supported by the results in EBBA and the concentrated solutions of DCEB in 55 wt% 1132, but from the results of the 2 mol% solution the two axis sj'stems differ appreciably.  For all four sets of ordering data, the axes defined by e very nearly identical. The greatest discrepancy between e  3  3  and the s  3  are  and any of the s  3  is 1.7°. The greatest difference between any of these axes and the Cartesian z axis is 7° in the case of DCEB in EBBA.  More substantial discrepancies are noted between e , and e  2  and the s , and  s . For the case of DCEB in EBBA these principal axes of the order matrix A  2  and the inertia tensor differ by at most 4°. As for the results in 55 wt%  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 1  Table 5.7 Principal Values and Principal Axes of Order Matrices of DCEB 20 mol% DCEB in EBBA  S,  0.111  y z A  20 mol% DCEB in 55 wt% 1132  A  A  15 mol% DCEB in 55 wt% 1132 x y z A  A  DCEB in 55 wt% 1132 x y A  A  Z  2  0.0380  S -0.149 3  s1 0.999 0.039 0.003  -0.039 0.992 -0.119  -0.008 0.119 0.993  0.0948  S 0.0782  S -0.173  S  x y z  s  1  2  S 2  3  0.981 0.190 -0.034  -0.192 0.979 -0.068  0.020 0.074 0.997  S, 0.103  S 0.0856  -0.188  s1 0.977 0.212 -0.035  s -0.214 0.974 -0.067  0.020 0.073 0.997  S, 0.130  S 0.112  S -0.242  S1  s2 -0.403 0.914 -0.052  0.022 0.066 0.998  0.915 0.401 -0.047  2  2  2  3  l,3-DichIoro-2-ethenylbenzene - Solution, Structure, and Orientation / 153 1132,  the difference in the 15 mol% case is 6.5° while in the 20 mol% case  the difference is 5.2°. This is very good agreement. For the results in the dilute solution of DCEB in 55 wt% 1132 the two axis systems differ by almost 18°. While  this discrepancy may seem large it is quite understandable  experimental values of S, and S  b. Samulski's  as  the  are very nearly equal.  2  Model for Orientational Order.  Samulski has taken the model for orientational ordering based on the inertial properties of the solute one step further.  Whereas Anderson had noted the  empirical results only qualitatively, Samulski has proposed a method whereby the principal values of the order matrix may be predicted from the inertia tensor of the solute [19,156]. This method has been employed with success in explaining the ordering of flexible solutes such as rc-octane in nematic solvents. To do this Samulski uses the principal moments of inertia to define the semiaxes of the inertia ellipsoid [157] in the.manner  A  a  + I  = ^  ^  " I )  7 7  aa  ] * •  (5.1)  2m  He then defines order parameters based upon the dimensions of this ellipsoid as  S  zz = S  1  " <x A  V  +  2 A  xx = - * + A / 2 A  z  - i + A /2A  z  x  y  z  <-) 5 2  (5.3)  (5.4)  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 154  where the semiaxes are chosen such that  s A  x  < A . These S values are, z  within a scaling factor, the principal values of the order matrix.  A verbatim application of Samulski's prescription for the values of the order parameters leads to some startling predictions in the case of DCEB. Definitions 5.2 — 5.4 force the largest order parameter to lie between zero (for a spherical solute)  and  one  (for  an  infinite  cylinder) while  the  remaining two order  parameters must be between minus one-half and zero. This is at odds with the experimental results for DCEB where one principal value of the order matrix is negative and two are positive.  The  origin of this difficulty lies in the fact that, while not explicitly stated,  Samulski's requirement that the largest order parameter (in the absolute value sense) be positive is more suitable for a prolate ellipsoid than an oblate ellipsoid. Simply reversing the signs of 5.2 — 5.4 rectifies the problem. A calculation using the principal moments of inertia given, in table 5.6 yields the predictions  S,  = 0.440 ; S  2  = 0.133 ; S  3  = -0.574  These should be, within a constant scaling factor, the principal order parameters. By scaling the value S  3  = -0.574 to equal the experimental values of S  in table 5.7, the data presented in table 5.8 results.  3  listed  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 155  Table 5.8 Principal Order Parameters of DCEB Predicted using the Inertial Frame Model. Solution  Order Parameter  20 mol% DCEB in EBBA  Predicted Experimental  S, S S  3  0.114 0.035 -0.149  0.111 0.038 -0.149  S, S S  3  0.133 0.040 -0.173  0.095 0.078 -0.173  S, S S  3  0.144 0.044 -0.188  0.103 0.086 -0.188  S, S S  0.186 0.056 -0.242  0.130 0.112 -0.242  2  20 mol% DCEB in 55 wt% 1132  2  15 mol% DCEB in 55 wt% 1132  2  2 mol% DCEB in 55 wt% 1132  2  3  2. Ordering Due to Solute Size and Shape.  Next we turn to a model for molecular ordering based on the size and shape of the solute. If we know the potential energy of the molecule as a function of orientation the order matrix of the solute is found classically to be [24]  g  J(3cos0 cos0 _ — a  b  -  6 ) exp(-U(fl)/kT)dS2 ab  ( ) 5>5  2fexp(-U(n)/kT)dn  The difficulty lies in determining U. As has already been mentioned, for small  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 156 molecules U may  take the  form of an interaction between  the molecular  quadrupole moment and and the anisotropy in a mean electric field gradient in the liquid crystal solvent.  As the size of the solute increases repulsive interactions dependant upon the size and shape of the solute molecule can be expected to play a dominant role in the orientation mechanism. The effect  of molecular size and shape on the  interaction potential U has been modelled by van der Est by considering the liquid crystal to be an elastic tube surrounding the solute. The z axis of the tube is coincident with the director of the liquid crystal. The walls of the tube are rigid so the circumference of the tube surrounding the solute may be found by looking down the axis of the tube at the projection of the solute in the xj' plane.  As the  solute  reorients  it will,  depending  on its  size, shape and  orientation, deform the walls of the tube. This model considers the tube to resist deformation with a Hooke's law force,  F(O) = -Ac(B),  (5.6)  i.e. the restoring force is directly proportional to the circumferential displacement of the walls of the tube. Here, k is the Hooke's law force constant and c(fl) is the circumference of the XY projection of the molecule held in some orientation fi. The potential energy of the molecule in this orientation is then  U(B) = i*c(0) . 2  (5.7)  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 157 This  potential  is used  in numerical integration  of (5.5)  over  all possible  orientations to predict the values of the various elements of the order matrix. This model has been proved useful in explaining the orientation of a variety of symmetrical solutes in several liquid crystals [158,159].  From the predictions made in fitting of the spectrum in 55 wt% 1132 one is already tempted to judge it a qualified success for even such unsymmetrical solutes such as DCEB. A comparison of the predicted values of table 5.4 with the experimentally determined order parameters for the 20 mol% solution in 55 wt% 1132 listed in table 5.5 shows excellent agreement between predicted and experimental values. Remembering that the S  x x  have been forced to agree by  adjusting k, the agreement between the remaining order parameters is very good indeed. The experimental and predicted values of S__ differ by less than 3% and zz the off diagonal elements of the order matrix, while quite small, have been reproduced very well. Predicted values for the dilute solution of DCEB in 55 wt% 1132 are reported in table 5.9. Again, the force constant k has been adjusted to reproduce the experimental value of S . Comparing the predicted values with the experimental xx  values of table 5.5 we see that the agreement is now slightly worse than in the case of the more concentrated solution, but it is still very good. The values of S  zz  now disagree by about 10%. The off diagonal elements still show good  agreement.  The principal values and principal axis systems of the predicted order matrices  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 158  Table 5.9 Size and Shape Predicted order matrix of 2 mol% DCEB in EBBA/1132. 0.2160 0.1211 0.0074 0.0335 0.0235  zz xx xz xy  are given in table 5.10. Comparison of the experimental and predicted principal order parameters is difficult due to the fact that the predicted values were scaled  so that  the  predicted  S  x x  matched  the  experimental  value in the  nondiagonal reference frame and diagonalization has changed the values.  Table 5.10 Principal Values and Principal Axis Systems of the Size and Shape Predicted Order Matrices of DCEB 20 mol% DCEB in 55 wt% 1132  y A  Z  2 mol% DCEB in 55 wt% 1132  S, 0.1080  S 0.07172  S -0.1797  0.760 0.645 -0.079  -0.650 0.757 -0.069  0.015 0.104 0.994  S, 0.1365  S 0.0831  S -0.2196  -0.536 0.840 -0.081  0.014 0.105 0.994  0  y z A  1  0.844 0.532 -0.068  2  2  3  3  l,3-Dichloro-2-ethenylbenzene - Solution, Structure, and Orientation / 159 Again, the s ' (the prime denoting the predicted value) is very nearly the same 3  as the Cartesian z axis and it differs from s  3  by only ~ 3 ° . The s  and s " 2  exhibit a greater difference from the experimental values. In the case of the 20 mol% solution the predicted axes differ from the experimental axes by 29° while in the dilute solution the difference is 8.5°. While these differences may seem substantial, discrepancies of this size should not be unexpected as the values of S  x x  and Syy are nearly equal. The disagreement of the principal axis systems  should not detract from the excellent agreement of the predicted and experimental order parameters.  The success of this model in predicting the order parameters of DCEB in the 55 wt% 1132 mixture suggests a very practical use for this particular mixture of liquid crystals. The traditional approach to solving complex LCNMR spectra has been to take an approximate  molecular geometry  and then, by using one's  intuition or luck, guess values of the solute order parameters. These guesses are used to simulate spectra until one is found which yields a promising starting point for a least squares analj'sis. This can be a confusing and tedious process. In light of the success of the order parameter prediction which led to figure 5.3, this mixture of liquid crystals could well find use in future LCNMR studies of molecular geometry. In 55 wt% 1132, or some similar mixture where the electric field gradient is zero a calculation based simply upon the shape of the solute might simplify the task of finding a set of starting parameters from which to start a least squares analysis.  VI. CONCLUSION  This thesis has presented several topics concerned with NMR spectroscopy using liquid crystalline solvents. In particular, these efforts have been applied to the analysis of the spectrum of l,3-dichloro-2-ethenylbenzene in two nematic solvents.  Turning first to the problem of data processing in MQNMR,  a method of  spectral analysis due to Burg, known as the maximum entropy method was shown to helpful in processing MQNMR data. Difficulties in processing MQNMR data arise from the fact that one normally only has a very limited dataset. The suppression of spectral artifacts due to truncation in the time domain is probably the method's most valuable attribute as far as its application to MQNMR goes. It also has great powers of enhancing resolution but this capability should be used with care. It is fair to say, however, that a dataset processed with Burg's MEM will fairly routinely give a resolution and quality in the spectral estimate which would probably require twice as much data if the Fourier transform were used to calculate the  spectrum. Burg originally set out to solve problems  encountered in the analysis of truncated geophysical datasets, and the method works very effectively when applied to MQNMR data. Among the types of data typically found in NMR, the MQNMR interferogram seems particularly well suited to analysis by this method and it' should find important future uses in the analysis of MQNMR data. ^  The second distinct subject presented here involved the development of a method of exciting MQ coherence within a particular group of spins in a solute molecule. By making both pulses in the MQ preparation sequence frequency selective for 160  Conclusion / 161 selected transitions, it is possible to exploit features of the spectrum to confine any MQ coherences to certain parts of a molecule. In the application for which it was developed, namely the spectrum of DCEB in EBBA, it proved capable of providing information on small long range dipolar couplings without complications from chemical shifts or any of the much larger short range couplings. This experiment requires some cooperation from the spin system in that a degree of first orderness is required between the selected and nonselected groups of spins but first orderness of this tj'pe is not uncommon in LCNMR. In particular, molecules with much internal motion frequently  fit the  first order criterion.  Motions between different parts of a molecule can average intergroup couplings to small values while the intragroup couplings may remain large. In such cases frequency selective excitation of multiple quantum coherence could simplify the interpretation  of the  dipolar  coupled  spectra  and  facilitate  studies  of  the  intragroup couplings.  Finally, solution of the spectrum of l,3-dichloro-2-ethenylbenzene in EBBA and 55 wt% 1132 demonstrated the efficacy of predicting the orientation of a molecule with an estimate of its shape. For the solution of DCEB in 55 wt% 1132 where the electric field gradient —molecular quadrupole moment orientation mechanism is removed, a prediction of the order matrix based upon an estimated geometry of the molecule yielded a starting point for the analysis which was startlingly close to the final solution. Predictions of orientational behavior of more symmetrical solutes based upon their size and shape have already met with a good deal of success. 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R e n d e l l a n d L.K. Thompson 'Complexes of s u b s t i t u t e d b e n z o t h i a z o l e s . 1. C o b a l t ( I I ) , copper(II) and z i n c ( I I ) c o m p l e x e s o f 2,2'-o-phenylenebisbenzothiazole: a p o t e n t i a l N or S donor 1igand.' Can. J . Chem. 5 2 , 1-7 (1979) Awards. 1987 1983-85 1981-83 1980  NSERC P o s t d o c t o r a l Fellowship. NSERC P o s t g r a d u a t e S c h o l a r s h i p . NSERC P o s t g r a d u a t e S c o l a r s h i p . N S E R C Summer R e s e a r c h A s s i s t a n t s h i p .  1  


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