Progress in Nuclear Magnetic Spectroscopy Volume 1 Issue 1

FOREWORD IT IS less than 20 years ago that nuclear magnetic resonance was first observed in bulk matter. This new form ...

Author: R.E. Richards

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FOREWORD IT IS less than 20 years ago that nuclear magnetic resonance was first observed

in bulk matter. This new form of spectroscopy quickly proved to have useful applications in physics and chemistry, and with the discovery of the “chemical shift” in 1949 it became clear that this was to be a most important field of study for the chemist. As the design of magnets reached quite remarkable performances, more and more subtle effects became apparent, and to&y nuclear magnetic resonance is applied to a quite astonishing range of chemical and physical problems. The most extensive applications to chemistry come from the so-called “high resolution spectroscopy”, and this important part of the subject is reviewed most thoroughly in this book. Although much useful information can be derived from purely empirical applications of NMR, a great wealth of data can often be derived only from a proper analysis and understanding of the spectra. A detailed account of the theory and analysis of high resolution spectra is given at length in this book, which will be of great value to all magnetic resonance spectroscopists, There is also a full discussion of other factors which can affect the spectra, profusely illustrated with examples from the literature. In addition to the theoretical part, this book contains a remarkable collection of data for reference purposes. A considerable proportion of the work done so far has been summarised in compact form, which is most useful for reference. Nuclear resonance work is now being published at an ever-increasing rate, which has now reached about 2000 papers a year, and the authors are to be congratulated for having encompassed the work so far in a single book. R. E. RI-S

xvii

PREFACE IN WRVINGthis monograph we have been bold enough to attempt to provide a detailed account of the basic theory underlying high resolution nuclear magnetic resonance (NMR) spectroscopy and also to present a survey of the major applications to problems in physics and chemistry. We have aimed at being as comprehensive as possible with the intention that almost all the text will interest everyone actively engaged in NMR spectroscopy. Because the subject is so vast, the authors are not competent to be critical throughout. Indeed, the phenomenal increase in the published work on NMR will soon make impracticable the task of containing full coverage of the topic within a single text. Already, several aspects of the subject are so well developed that they merit individual presentation in monographs written by the appropriate specialists. It has been necessary to divide our monograph into two volumes, the first one of which is concerned primarily with basic theory and spectral analysis (Chapters 1 to 9) while the other contains most of the published work on struo tural applications of high resolution NMR spectroscopy (Chapters 10 to 12). From the extensive cross-referencing, the reader will be aware that the book has been written as a single entity. A prerequisite for the successful application of NMR is the ability to analyse a spectrum in order to obtain the chemical shift and spin coupling constantparameters, Therefore, we have made the chapter on the analysis of spectra, Chapter 8, the mainspring of the book and we have covered the subject as fully as possible. This section has been written keeping in mind the reader who does not have a strong background in quantum mechanics,hencea ratherdetailedaccount has been given of the theory behind spectral analysis. Nearly all the spectral systems which have been analysed are described and the analyses of some of the spectral types are discussed in great detail to illustrate the general method. This chapter provides an introduction to the analysis of NMR spectra and also serves as a reference source for the practising spectroscopist. The basic theory of NMR is dealt with in Chapters 2,3 and 9, and the aim has been to cover the background of all major applications of NMR rather than to attempt a unified and mathematically rigorous treatment. Chapters 4 and 5 review the progress made in relating chemical shifts and coupling constants to the electronic structure of molecules, while Chapter 9 covers the applications of NMR to kinetic processes. Chapters 10, 11 and 12 contain a survey of the applications of NMR to the determination of molecular structure: Chapter 10 deals with hydrogen resonance, Chapter 11 with fluorine resonance and Chap ter 12 covers all the remaining suitable magnetic nuclei. All three chapters hold many reproductions of spectra and many compilations of chemical shifts and coupling constants. This particular volume should be invaluable to those who

XX

PREFACE

have some knowledge of the fundamentals of NMR and to wish use the technique as an analytical tool. The practicalaspects of NMR spectroscopy are dealt with in Chapters 6 and 7. Chapter 6 covers the theory of the instrumentation required for the measuremeat of high resolution NMR spectra and also describes the types of spectrometer available commercially. In Chapter 7 is given a discussion of all the practical factors which need to be considered in obtaining NMR spectra. This chapter is intended primarily for those new to the subject and includes preparation of the sample, tuning the spectrometer and measurement of spectra. The material for this chapter has been drawn not only from the authors’ own experience but also from many discussions with other workers in the field. In particular, much valuable information has been obtained from the contributions to the monthly letters on NMR (MELLONMR) edited by Drs. A. A. Bothner-By and B. A. Shapiro of the Mellon Institute, Pittsburg, U.S.A. We would like to acknowledge the many helpful discussions we have had with Drs. R. J. Abraham and J. Lee. We are also indebted to Drs. R. J. Abraham, T. B. Grimley and G. Skirrow, who have given up much of their time to read the manuscript, for their suggestions and constructive criticisms of the text. In the course of preparing this monograph we have been particularly fortunate in the help given by those who have supplied us with unpublished data and with manuscripts prior to publication. Their help has enabled us to cover the scientific literature up to the end of 1963. We would like to thank especially Dr. G. V. D. Tiers for allowing us to reproduce his compilation of t-values and also Professors H. S. Gutowsky, G. G. Hall and J. D. Roberts and Drs. R. J. Abraham, J. D. Baldeschwieler, C. N. Baawell, A. A. Bothner-By, F. A. Bovey, P. L. Corio, R. Freeman, P. C. Lauterbur, A. Melera, W. G. Schneider, G. W. Smith, F. C. Stehhng, G. V. D. Tiers and the Office of Naval Research of the U.S.A. We wish to thank Professors C. E. H. Bawn, C.B.E., F.R.S. and W. K. R. Musgrave for their encouragement at all times. We also thank all the authors and editors who have given us permission to reproduce figures and diagrams from their publications. We would be grateful if readers would draw our attention to any errors which they may encounter in the text.

CHAPTER

1

INTRODUCTION HISTORICAL UNTILrecently, our understanding of atomic and molecular structure was based to a large extent on the findings of optical spectroscopy. New sources of structural information have been forthcoming from the development of experimental techniques which have extended spectroscopic measurements to the microwave (roughly 10’ to lo3 MC see-‘) and the radiofrquency (roughly lo2 to IO-” MC set-r) regions. In these regions, radiation is absorbed and emitted by the same basic process as at other wavelengths in the electromagnetic spectrum, namely a quantum of energy is transferred tb or from a system undergoing a transition between two levels. The separation of the levels is equal to the amount of energy transferred, the values encountered being about a tenmillionth of an electron volt and about a ten-thousandth of an electron volt for the radiofrequency and microwave regions respectively. In two of the principal branches of radiofrequency spectroscopy, nuclear magnetic and electric quadrupole resonance, this separation is associated with different orientations of, respectively, the nuclear magnetic dipole moments in applied magnetic fields and of nuclear electric quadrupole moments in non-spherical molecular electric fields. The occurmnce of nuclear moments was first detected from a study of the hype&e structure observed in the electronic spectra of certain atoms using optical spectrographs having very high resolving power’l). The presence of the hypetie structure in atomic spectra led PatW) to suggest, in 1924,that certain nuclei possess angular momentum and thus a magnetic moment which interacts with the atomic orbital electrons. This was confirmed later by spectroscopic work which enabled values of the angular momentum and magnetic moment to be determined for many nuclei. In the presence of an applied magnetic field, such magnetic moments take up specific orientations and it is possible to observe transitions occurring between the mrclear energy levels associated with these orientations by irradiating with energy of a suitable frequency. The nuclear energy levels are quantised as a direct consequence of the quantised nature of the angular momentum operator of a nucleus which has 2Z+ 1 values. The spin number Z can assume any value which is a multiple of +; the highest value known is that of ‘?f Lu having Z = 6 (see Appendix A). The largest measurable component of the angular momentum is Zh, where li = h/2rc and h is Planck’s constant. Values of Z for a particular nucleus (see Appendix A) cannot be predicted but it has been noticed that isotopes having mass and atomic numbers which are both even have Z = 0, and isotopes having odd mass numbers have half integral spin values. Even and equal numbers of protons HB8.1

1

2

HIGH

RESOLUTION

NMR

SPECTROSCOPY

and neutrons in a nucleus (I= 0) may be regarded as a state of complete “pairing” analogous to the complete “pairing” of electrons in a diamagnetic molecule. In 1921, Stem and Gerlacht3), using a beam method, showed that the measurable values of an atomic magnetic moment are discrete, corresponding to the space quantisation of an atom when placed in an inhomogeneous magnetic field. In later experiments a beam of hydrogen molecules was directed through a steady magnetic field thus enabling the small magnetic moment of the hydrogen nucleus to be measured. The method has been developed further by subjecting the beam to an additional magnetic field oscillating at the frequency required to induce transitions between the nuclear energy levels corresponding to the quantised values of the nuclear magnetic momentC4). For a nuclear spin number I there will be 21+ 1 equally spaced energy levels, and in a steady field of strength H, the separation between the highest and the lowest of these levels is 2,~H where p is the maximum measurable value of the nuclear magnetic moment. Thus the separation between successive sub-levels is pH/I and the frequency of the oscillating magnetic field which can excite transitions between these levels is pH/Ih. In the molecular beam experiment, only the molecules which do n,ot change their energy reach the detector, hence the frequency of the resonant exchange between levels is determined by sweeping through a range of frequencies. At some particular frequency value there is a sudden reduction in the number of molecules arriving at the detector. These first successful demonstrations of nuclear resonance were accomplished with main magnetic fields of several kilogauss which lead to oscillating field frequencies in the range lo5 to lo8 cycles set-i. The resonance exchange of energy is not restricted to molecular beams and is detectable in all forms of matter. In 1936, GorteP) attempted to observe the resonances of ‘Li nuclei in lithium fluoride and ‘H nuclei in potassium aluminium sulphate. A further fruitless attempt was made by Gorter and 3roeP) in 1942. In the two attempts, the detection of the absorption of radiofrequency power was to be made by calorimetric methods and the occurrence of anomalous dispersion respectively. The use of unfavourable materials was the main cause of failure. It was not until late in 1945 that the fist actual nuclear magnetic resonance signals were observed. Bloch (‘I) at Stanford University observed the hydrogen resonance absorption in water and Purcell(*) working at Harvard University was successful in detecting hydrogen nuclear resonance absorption in paraffin wax. They received jointly in 1952 a Nobel prize for their discovery. It is interesting to consider why the nuclear paramagnetic susceptibility is too small to be of value in static susceptibility measurements. An important fact is that the nuclear contribution to the paramagnetic susceptibility of a substance is negligible compared with that from the electrons, as the following discussion will show. The magnetic moment of an electron assembly in an atom or molecule is given by (1.1) where g, the Landd or spectroscopic splitting factor, is a measure of the contri-

INTRODUCTION

3

bution of the spin and orbital motion of the electron to its total orbital angular momentum and has the value of 2.0023 for a spinning free electron. J is the quantum number representing total angular momentum and fl is the Bohr magneton defined by B = eh/4n MC (l-2) where e is the charge on the electron, M is the electron mass and c is the velocity of light. In a similar way the magnetic moment of a nucleus is given by PN = gN

MH ,$,J[I(1 N M

+

1)]“2

(l-3)

/IN, the nuclear magneton, is defined in terms of the mass of the hydrogen nucleus bN = eh/4n MHc. (1.4) Now an assembly of paramagnetic nuclei when placed in a steady magnetic field will orientate themselves according to the Boltxmann energy distribution, and the resultant volume susceptibility XNwill be given by the classical hq$vinBrillouin relation XN =

N/JN2/3

kT

WI

where N is the nuinber of magnetic nuclei in 1 cm3. Since g, and gNare approximately equal and ,$# = l/1838 then the macroscopic nuclear susceptibility XNis less by a factor of about (1838)2 than the susceptibility of a substance having paramagnetism originating from unpaired electrons. & is also masked by the electronic diamagnetic susceptibility even in the neighbourhood of the absolute zero of temperature. These considerations indicate the importance of eliminating the electronic diamagnetic factor by means of a resonance experiment employing an oscillating magnetic field. However, despite the intrinsic dif&ulty, in 1937 Lasarev and SchubnikovCg) were able to detect the static nuclear susceptibility of solid hydrogen by using a non-resonance technique in the temperature range 1.76 to 422°K. They were able to separate the diamagnetic electronic contribution from the paramagnetic nuclear contribution because of the temperature dependence of the latter. These experiments led to the determination of the magnetic moment of the hydrogen nucleus to an accumcy of about 10 per cent. Nuclear magnetic resonance absorption is proportional to the static nuclear paramagnetic susceptibility but is far easier to detect. From the above discussion it will be clear that a similar resonance effect will occur for atoms or molecules possessing one or more unpaired electrons. Since the electron has spin + there are basically only two energy levels but these are much farther apart (for a given applied magnetic field) than those of say the hydrogen nucleus, because their magnetic momenta have the magnitudes 9270 x 102*and 14-l x 1O24erg gauss-’ respectively. This means that for applied magnetic fields of the order of kilogauss, the oscillating field required for 1*

4

HIGH

RESOLUTION

NMR

SPECTROSCOPY

electron resonance must have a frequency in the microwave region. It is not surprising then that electron spin resonance (ESR) has been known for as long a period as nuclear magnetic resonance (NMR). The first successful demonstration of the former type of resonance was made by Zavoisky(lo) who used cupric chloride dihydrate; the first free radical spectrum was observed by Kozyrev and Salikhov(l i) who studied pentaphenylcyclopentadienyl. The hyperfine splitting that can be observed in the electron spin resonance spectra of some free radicals arises because some of the nuclei present possess a magnetic moment. Electron spin resonance(12 -I’) now forms one of the main branches of microwave spectroscopy. Nuclear magnetic resonance studies are divided into three main types (i) high resolution (ii) low resolution and (iii) spin echo measurements. These divisions arise mainly from the very different experimental arrangements employed. High resolution measurements are confined to liquids and occasionally gases and the spectra occupy less than O-1per cent of the applied static magnetic field. Low resolution techniques are applied to solids and occasionally to liquids and the spectra occupy between 1 and 10 per cent of the steady magnetic field. This monograph is devoted entirely to high resolution NMR spectroscopy but an outline of spin echo methods is included. It would not be out of place, however, to draw the reader’s attention to the considerable amount of information which is obtainable by the low resolution method”*). The method provides a useful adjunct to X-ray crystallography in the determination of the position of hydrogen atoms in crystaWg’. The onset of various types of molecular motion can be detected as solids are taken through their transition points, thus providing insight into the molecular structure and solid state of small molecules and of po1ymers(20).

HIGH

RESOLUTION

NMR SPECTROSCOPY

The power of the high resolution method derives from the fact that nuclei of the same species in different chemical environments absorb energy at di& erent radiofrequencies (for a given applied field) due to the irbeing shielded to different extents from the applied magnetic field. This discovery was made in 1949 when it was found that the resonance frequencies of the nuclei of phosphorus, nitrogen and fluorine depended upon the chemical compound in which they were located. The separations between the various resonance frequencies are called chemicalshifts: the values of the shifts can be used to obtain information about the electronic environment of a particular nucleus in the molecule under examination. If, say, ‘H resonance is being studied then it is found that a mokule containing only one hydrogen nucleus or a set of ‘H nuclei having the same environment (as for example in water, benzene or cyclohexane) gives a single absorption band. The position of the band is a characteristic of the molecule. Since chemical shifts cannot be measured absolutely, that is, from a nucleus stripped of its electrons, the signal from a reference compound is used as an arbitrary zero : a suitable compound for ‘H resonance is tetramethyl-

5

INTRODUCTION

silane. It is customary to quote chemical shifts in tbe dimensionless parameter B (in units of parts per million) defined by

6=

106

where H - HP is the chemical shift difference between the sample and tbe reference, H, being the absolute position of tbe reference signal in tbe main magnetic field. These field measurements cannot be made with certainty hence it is more convenient to calculate 6 from tbe expression 6 =-

( ) 1’ -

v,

106

vo

Benzene

/I -7.3

FIG. 1.1 Chunid

Water

-5.2

Cyclohrxonc

Tetromethylsilanc

-1.4

shifts of benzene, water and cyclohexanc measured from tetramethylsilane

where v - v, is tbe chemical sbift difference between tbe sample and reference in frequency units (cycles se&)-these are the units in wbicb the spectnun is calibrated. Strictly, one should not use vo-tbe radiofrequency (usually tied) at wbicb the spectrometer operates-but v,, the absolute frequency at wbicb the reference signal appears. The error introduced, however, is very small because y. and V, are almost identical. Tbe obvious need for expressing S in dimensionless units can now be appreciated because different NMR spectrometers are operated at a variety of frequencies vo. Figure 1.1 illustrates the relative positions of the signals of benzene, water and cyclobexane measured from tetramethylsilane. A theoretical treatment of tbe origin of chemical shifts is given in chapter 3. Some molecules can give rise to more than one absorption band, each of which is characteristic of the electronic environment of tbe magnetic nucleus under examination. For example, toluene has a ‘H resonance spc&um consisting of two absorption bands 4.85 ppm apart. Figure 1.2(a) shows tbe spectrum obtained at 60 MC set-l, also included is a small band from the small amount of tetrametbylsilane added as a reference. Tbe low field band corresponds to tbe hydrogen nuclei of tbe benzene ring while the other band originates from the hydrogen nuclei in tbe methyl group. Tbe ratio of tbe band areas (intensities) is 5 : 3 corresponding to tbe number of ‘H nuclei in each environment; tbis property can be used for diagnostic purposes.

6

HIGH RESOLUTION

NMR SPECTROSCOPY

Compounds having magnetic nuclei in more than one chemical environment can have absorption bands showing fine structure (see Fig. 1.2(b)). This structure, know nas multiplet splitting, has its origin in the coupling between magnetic nuclei taking place via bonding electrons (a theoretical discussion is present400 I

r

300 I

200 1

I00 I

0

(0)

H

1

!6-O

7.0

6.0

5-O

Tetromethylsilone

4-o

30

I-0

3

mm FIO. 1.2 (a) The ‘H resonance spectrum of toluene at 60 MC see-‘. (b) The ‘H resonance spectrum of isopropylbenzene at 60 MC se& ; the septet of bands has been increased in intensity to fully reveal its structure. A little tetramethylsilane is present in both spectra to provide a reference. By courtesy of Varian Associates

ed in Chapter 3). The magnitude of the coupling can be determined by measuring the separation of the multiplet components providing that the chemical shift difference of the coupled nuclei is much greater than the magnitude of the multiplet splittings. This band separation is independent of the field strength hence there is no need to quote the splitting in dimensionless units-cycles set-’ are the units normally used. The simplest example of multiplet splitting one can encounter is to be found in the resonance spectrum of a molecule having

INTRODUCTION

7

amagnetic nucleus A and a magnetic nucleusX of another isotopic species. Figure 1.3 is a representation of the resonance spectrum that would be obtained if A and X each have a spin number I = 3. The separation of the lines within each daublet is the spin couphg cons&nr JAx. The doublets appear because each nucleus splits the resonance of its neighbour into 21+ 1 components. The energy differences between the various spin states are so small that, at thermal equilibrium, the Boltzmann distribution makes them all equally probable. Hence, all the lines in a multiplet resulting from coupling to a single nucleus are equally intense. If, however, there are n indistinguishable nuclei, the neighbour is split x _____--_________y ~

nucleus K-----------8A, A

nucleus

FIG. 1.3 Repmentation of the NMR spectnun of a system containing the magnetic nuclei A and X each having Z = l/2 and the condition 6m s JAX is fumed

into 2nl+ 1 lines. For nuclei having I = +, the intensities of the lines in a multiplet follow the binomial coefficients given in the table below: n

Relative Intensities of Multiplet Components

1

1

2

12

1

3

13

3

1

4

14

6

4

1

5

1 5

10

10

5

1

6

1 6

15 20

15

6

1

7

1

7

21

35

35

21

7

1

8

1 8

28

56

70

56 28

8

1

1

Figure 1.2(b) shows the hydrogen resonance spectrum of isopropylbenzene. The single band located at 725 ppm from tetramethylsilane (designated 0 ppm) originates from the ring hydrogen nuclei : like the corresponding toluene band, no multiplet splitting is observed. The most intense group of bands (a doublet) is that centred at l-25 ppm so these may be assigned to the two methyl groups. The remaining group forms a septet centred at 29Oppm which originates from the CH group. The doublet is caused by this group since the number of

8

HIGH

RESOLUTION

NMR

SPECTROSCOPY

hyperfine lines is 2 x + + 1 = 2. The magnitude of the splitting (J7 cycles se&) is common to both the doublet and the septet. The septet appears because of the mutual coupling between the CH group and the two methyl groups. The latter have n = 6, thus they split the band from the CH group into seven components having relative intensities 1: 6 : 15 : 20 : 15 : 6 : 1. From the foregoing it is clear that the magnitudes and types of band splittings can be used to decide the relative positions of functional groups in a molecule. Further assistance may be had with the aid of isotopic substitution. For example, if hydrogen is replaced with deuterium (I = l), the number of lines in the spin multiplet will change according to the change in I. Also the spacing will diminish because of the nearly sevenfold decrease in the magnetogyric ratio y. The treatment given here applies only to molecules having the property ~,,b % J; unfortunately, such a condition is not realisable very often in practice. Spectra governed by this condition are referred to as beingfirst order in type. The spectra normally obtained are more complicated and the evaluation of the variousv,,8 and Jvalues can be a lengthy process. The whole of Chapter 8 is devoted to describing the various methods available for doing this. Considerable effort is expended on analysing spectra fully as a consequence of the wide theoretical interest in the way in which chemical shifts and coupling constants vary from molecule to molecule. Kinetic and thermodynamic data can be obtained from line width or line position variations with reactant concentration and with temperature: typical applications are hydrogen bonding and molecular conformation studies. Another useful feature of a NMR spectrum is that the relative intensities of the absorption bands indicate the relative numbers of differently shielded magnetic nuclei. However, the application to quantitative analysis has not been exploited yet. From this brief outline it is clear that nuclear magnetic resonance is of considerable importance to physical, organic and inorganic chemists. The two nuclei which have ideal properties of high natural abundance, high magnetic moment and nuclear spin + are ‘H and lgF and most investigations concern them. Considerable work has also been carried out on 31P llB and 13C resonances. Further improvements in experimental techniques &ll probably result in the method being used more widely for other nuclei, particularly for the 13C nucleus. REFERENCES 1. S. TOLANSKY,H&h Resolution Spectroswpy, Methuen, London (1947). 2. W. PAUL&Naturwisschqfh, 12,741 (1924). 3. 0. STERN, Z. Phys., 7,249 (1921); W. GBRLXH and 0. Smm, Ann. Phys. Leipzig, 74,673 (1924). 4. N. F. RUSEY, MoIeahr Beams, Oxford University Press (1956). 5. C. J. GARTER,Physica, 3,995 (1936). 6. C. J. GARTERand L. F. J. BROW, Physicu, 9, 59 1 (1942). 7. F. BLOCH,W. W. HANSENand M. PACKARD,Phys. Rev., 69, 127 (1946). 8. E. M. PURCELL, H. C. TORREYand R. V. POUND,Phys. Rev., 69, 37 (1946). 9. B. G. LASAIUVand L. V. SCHUBNIKOV, Phys. Z. Sowjetunion, 11, 445 (1937).

INTRODUCTION

9

10. E. ZAVOISKY, J. Phys. (U.S.S.R), 9,245,447, (1945). 11. B. M. KOZYRJZVaud S. G. SNZHOV, Dokl. Akad. Nat& S.S.S.R., 58, 1023 (1947). 12. D. J. E. INGRAM,(1958), Free Radicals as Studied by Electron Spin Resonance, Butterworths, London. 13. D. Ii. Wmmm, Qaart. B-m., 12,250 (1958). 14. D. J. E. INQRAM,Spectroscopy at ZWio and Microwave frequencies, Butterworths, London (1955). 15. B. BLEANEYand K. W. H. STEVENS, Rep. Prog. Phys., 16, 108 (1953). 16. A. CARRINGTON and H. C. LONGUEPHIGC~INS, Qua. Rev., 14,427 (1960). 17. G. E. PAKE,Paramagnetic Resonance, W. A. Bajamin, New York (1962). 18. E. R. ANDREW,Nuclem Mugnetic Be.mnmx, Cambridge University Press (1956). 19. R. E. RI-, Qrum. Rev., 10,480 (1956). 20. J. G. POWLES,Polymer, 1,219 (1960).

BIBLIOGRAPHY 1. High ZZesolution N&ear Magnetic Resonance, J. A P~PL.E,W. G. Scsmamaa and H. 3. Bnunw, MeGraw-Hill, New York (1959). 2. Applications of Nuclear Magnetic R esonance Spectroswpy in Organic Chemistry, L. M. JA-, Pergamon Press, London (1959). 3. NMR and EPR Spectroscopy, VAIUN Assoa~m, Pergamon Press, London (1960). 4. NMR Spectra CataZog, N. S. BHACCA, L. F. JOHNSON and J. N. SEOOLZRY, National Press, New York Volume 1 (1962), Volume 2 (1963). 5. llw Principles of Nuclear Magnetism, A. h3RAt%M, Clarendon Press, Oxford (1961). 6. Nuclear Ztuhrction, A. K. SN-IAand T. P. DAS, Institute of Nuclear Physics, Calcutta (1957). 7. Nuc&ar Magnetic Relaxation, N. BLOEMBEROEN, W. A. Benjamin, New York (1961).

8. La Rpsomaze Paramagndtique NucZeaire, P. Ganm, Centre National de la Reeherehe SeientZqw, Paris (1955); 9. Nuclear Magnetic Resonance Applications to Organic Chemistry, J. D. Rosaam, McGrawHill, New York (1959). 10. Determination of Organic Structures by Physical Met&,

Vol. 2. Edited by F. C. NACHOD and W. D. PRILLIPS,Academic Press, New York (1962). 11. An Zntroduction to the A~Iysis of Spin-spin Splitting in High Resolution Magnetic Z&sonance Spectra, J. D. ROBERTS,W. A. Bmjamin, New York (1961). 12. Interpretation of NMR Spectra, K. B. WIBERGand B. J. Nm, W. A. Benjamin, New York (1962). 13. Principles of Magnetic Resonance, C. P. SL~CHTER, Harper and Row, New York (1963). 14. Appburtions of NMR Spectroswpy to Organic Chemistry: Zllustratt&ts from the Steroid Field, N. S. BHACCAand D. H. Wm, Holden-Day Inc., New York (1964). 15. The Theory of Nuclear Magnetic Resonance, I. V. Am.~mRov, Academic Press, New York (1966).

16. Structure of High Resolution NMR Spectra, P. L. CURIO, Academic Press, New York (1966). 17. Die Rernmagnetische Resonanr undihre Anwendung in der Anorganischen Chemie, E. FLUCK, Springer-&&g, Berlin (1963). 18. Thories Mol&daires de ha ZZ&onance MagnPtique Nucleaire, G. MAVEL,Duuod, Paris (1965). 19. Interpretation of NMR Spectra, an Empirical Approach, R. H. BIELE,Consultants Bureau, Plenum Press, New York (1965).

EBB.

1s

CHAPTER

2

GENERAL THEORY OF NUCLEAR MAGNETIC RESONANCE TWEfirst two sections of this chapter are concerned with classical and quantum mechanical descriptions of the resonance condition. Both approaches are presented because the former is helpful in dealing with transient phenomena, while the latter is preferable when steady-state processes are being considered. 2.1 Tma CLASSICALME-CAL

DESCRIPTION OF TBBRESONANCE CONDITION

A rotating charge can be regarded as a current circulating in a loop and will therefore behave like a magnetic dipole whose moment p is given by B = iA

(24

‘where i is the equivalent current and A is the cross-sectional area of the enclosed loop. A charge q rotating at v/2zr revolutions a second is equivalent to a current i = qv/2xr. (2.2) On converting the charge to electromagnetic units by dividing by the velocity of light c, the magnetic moment can be found from p = qvr/2c.

(2.3)

Now the circulation of themass M of theparticlegeneratesanangularmomentum and having the value Mu r (see Fig. 2.1). The spinning motion is common to both charge and mass hence the magnetic moment vector is collinear with and directly proportional to that of the angular momentum according to the relationship

p, a vector directed along the axis of rotation

iu = W-W

P.

(2.4)

The direction of the vectors is determined by the sign of the charge on the particle. This model, of course, provides no explanation for the magnetic moment of a neutral particle such as the neutron, nor does it explain why some nuclei have negative magnetic moments. The deficiencies of the model suggest that nuclear structure is complex due to nuclear components having orbital as well as spin motions combining in various ways to give a resultant angular momentum for the nucleus, similar to the coupling of electronic spin and orbital moments 10

GENERAL

THEORY

OF NUCLEAR

MAGNETIC

RESONANCE

11

in atoms and molecules. The origin of nuclear spin is outside the scope of this book; for further information the reader is referred elsewhere”‘. The ratio p/p is called the magnetogyric ratio (y) and is a characteristic property of a particle. A gyroscope will precess about a vertical axis due to the

. FIG. 2.T Classical model of a spinning charged particle

HO

/-

------

I

------

t

\ ----__

,

__i_-jp

Fm. 2.2 The Larmor precession of the nuclear spin axis about the direction of the steady applied magnetic hid I&,

12

HIGH

RESOLUTION

NMR SPECTROSCOPY

torque arising from the earth’s gravitational field. Similarly, if a spinning charged particle is placed in a magnetic field of strength H,, with its magnetic moment at an angle 8 to the direction of this field then it will experience a torque L tending to align it parallel to the field. Newton’s law states that the rate of change of angular momentum p with time is equal to the torque, that is

FIG. 2.3 Vectorial representationof the classical Larmor precession

From simple magnetic theory L = p x Ho.

(2.6)

Substituting equations (2.4) and (2.6) into (2.5)

9 dt-

q

--pxH, 2Mc =yp

(2.7)

x H,,.

(2.8)

This equation of motion describes the precession of p about Ho with an angular velocity w,, defined by dp dt=pwo

and hence

w. = yHo.

(2.9)

GENERAL

THEORY

OF NUCLEAR

MAGNETIC

RESONANCE

13

This is the Larmor equation which may be re-written in terms of the precession frequency v. (2.10) An important feature of this equation is that the precession frequency is independent of the angle of inclination of the particle axis to the field direction. If a small magnetic field H1 is applied at right angles to the main field Ho, then at a particular point on the precessional path the nuclear dipole will experience a combination of Ho and H, which will tend to change the angle 8 by an amount 86. When the dipole has moved a further 180” around the precessional path, the combination of the two magnetic fields will change 0 by - 60, hence the total change in 19is zero. In order to change the orientation and thereby the magnetic energy of the particle, the secondary field Hz must rotate in synchronisation with the precession of the magnetic moment about Ho. In other words, the rotation of H1 must be in resonance with the Larmor precession about Ho. A rotating magnetic field of this type is associated with circularly polarised radiation of frequency vo. The sense of the Larmor precession depends upon the sign of the magnetic moment ; for example, a positive moment requires a left circularly polarised H1. For most purposes linear polarisation is quite adequate since a linearly oscillating field may be regarded as the superimposition of two fields rotating in opposite directions. Only the component

having the correct sense will synchronise with the precessing magnetic moment; the other component will have no effect In practice, an alternating current is passed through a coil mounted perpendicular to Ho so as to produce a magnetic field oscillating along the axis of the coil, say, along the x direction. The voltage applied to the coil at frequency a, produces the equivalent of two counter-rotating fields having the magnitudes (H1 COW t + H1 sine t) and (H, cosc0t - HI sina, t). When o corresponds to the resonance frequency, the magnetic dipole will absorb energy from the coil thereby causing the magnetic moment vector to dip towards the x y plane and an e.m.f. to be induced in a second (receiver) coil having its axis along the y direction. A purely classical model, as presented here, cannot predict more than this continuous absorption of energy. Having seen the nature of the resonance condition, some attention should now be paid to the orders of magnitude involved when these ideas are applied to electrons and magnetic nuclei. Spin angular momentum is quantised and

14

HIGH

RESOLUTION

NMR SPECTROSCOPY

has a value of [I(1 + l)p for nuclei and the projection of it on the main field axis is m It. The magnetic quantum number m can have the values 1, I - 1, . . . - I and transition between these levels is governed by the selection rule flm= f 1. On replacingq by the electronic charge e in equation (2.4) the following classical relationship is obtained P

pe

=2Mc*

This can be applied to atomic particles provided that p is replaced by m h and a factor g is introduced. The Land6 or spectroscopic splitting factor, g, is a measure of the spin and orbital motion of a particle in relation to its total angular momentum. For a completely free electron, g has a value of 20023 while g for a hydrogen nucleus is 558490. The magnetic moments of the electron and a nucleus are given by (2.12)

P

=

p=gNz

gN IbN

(2.13)

where 6 and &. are the Bohr and the nuclear magneton having the numerical values of 9.2712 x 1O-21 and 5.0493 x 1O-24 erg gauss-l respectively. The hydrogen nucleus is taken as the standard for calculation of the latter value. Now the magnetogyric ratio y for sub-atomic particles is defined by (2.14) and from equations (2.10), (2.12), (2.13) and (2.14), the resonance conditions for the electron and magnetic nucleus are found to be IIvg = g,B&

(2.15)

hv, = &al% Ho*

(2.16)

For a steady field of 10,000 gauss, the alternating current providing the oscillating field iY1 would need to have a frequency of 28,000 MC se& for electron resonance to be observed while the corresponding frequency for hydrogen nuclear resonance would have to be 42 MC set+. In practice, electron spin resonance is usually studied at lower main magnetic fields to enable X-band (3 cm; 9,000 MC se&) microwave components to be employed. For nuclear magnetic resonance work it is desirable to have a range of frequencies available to enable each magnetic nucleus to be examined at as high a field strength as adequate homogeneity will permit. Typical frequency values for a field of 10,000 gauss are

GENERAL

THEORY

OF NUCLEAR

MAGNETIC

‘H

42576 MC set-’

2I-l IlB

6.536 MC set- l 13.66OMc set-’

‘SC

10.705 MC set-l

l4N

3.077 MC set-’

‘9F

40.055 MC set-i

RESONANCE

15

2.2 THEQUANTUM MECWWALDESCXIPTIONOFNUCLEARMAGNFI~CRESONANCE When a nucleus of magnetic moment EL is placed in a magnetic field the Hamiltonian for the system is given by

%%= -p-H

(2.17)

and since p=yiI,

then

3P-_ - yi H-1.

(2.18)

The expectation values of the operatorI,are m, hence the expectation values of 2, that is the energy levels of the system, are E = y&mH.

(2.19)

In order to induce transitions between the energy states, a perturbation of some kind must be introduced. A suitable perturbation is the application of an oscillating magnetic field, and the necessary direction of this field can be decided from the properties of spin operators and eigenfunctions appropriate to a nucleus of spin I. A set of spin angular momentum operators Z,, I,,, Z, , I2 can be defined for a nucleus of spin Z which are analogous to the more familiar operators for electron spin angular momentum, namely S,, S,, S, and S2. I2 has eigenvalues Z(Z + 1) h, and Z, has eigenvalues m A.Restricting the argument to nuclei of spin Z = + for simplicity, the eigenvalues of Z, are simply f +h, hence the two possible energy levels are f 3y li H and if the spin eigenfunctions are denoted by the symbols ocand 8, then z,a = + *Aa;

z,/3 = - 31 /I.

(2.20)

The functions (Yand #Iare orthonormal so that

s

a*a&

I

=

a*j%h =

I p/M-r s

= 1

/!?*adr=o

(2.21)

The operators Z, and Z, have the properties ZXa==+1t9; Z,a = *ittttl;

I,/?-+ia Iy(? = -+iia

(2.22)

16

HIGH RESOLUTION NMR SPECTROSCOPY

The probability W’ of a transition between the two levels when an oscillating field is applied along the z-axis is given by W’ = yZHfI (411,18)~* = 0 hence transitions cannot be induced with this arrangement oscillating magnetic fields. If an oscillating field is applied along the x-axis, then (alZ,lB) = tn

(2.23) of the steady and

(2.24)

and a similar tite transition probability is obtained if the oscillating field is regarded as applied along the y-axis. The change in energy when a transition takes place is AE=yiH (2.25) thus the frequency of the oscillating magnetic field must be

AE

v=-=_ h

yH 2n

(2.26)

which is the resonance condition. For the general case of a nucleus of spin Z the transition probability between two energy levels m and m1 is given by . W’ =

r*H:l(~mll;l~m~) I*

(2.27)

which is non-zero only when m = ml f 1; that is, transitions between nuclear energy levels are governed by the selection rule Am = f 1. Transition probabilities for the general case of a nucleus of spin Z have also been calculated by a perturbation method (*). However, such an approximation method is not necessary and both steady state and transient phenomena of nuclear magnetic resonance have been treated rigorously by Schwingefi3) and also by Bloch and Rabi(*). 2.3 THE POPULATION OF SPM STATES The absorption coefficients obtained in optical spectroscopy are independent of the intensity of the radiation source. Normally, a rapid return is made from the excited to the ground state, the liberated energy being dissipated as heat. In nuclear magnetic resonance spectroscopy, the intensity of irradiation (that is, the amplitude of the radiofrequency field) may weaken the absorption signals or cause them to disappear entirely. The situation is a consequence of the isolation of a nucleus from its surrounding lattice. In the case of a liquid, the nucleus may remain isolated for periods of a few seconds while this can be extended to hours for solids at low temperatures. However, fluctuating magnetic fields associated with either inter- or intramolecular motion in the sample will have components at the resonance frequency and they will induce a return to the ground state. Whilst for most of the electromagnetic spectrum the probability of stimulated emission (or absorption)induced by irradiation is usuallynegligible,

GENERAL

THEORY

OF

NUCLEAR

MAGNETIC

RESONANCE

17

for nuclear magnetic resonance spectroscopy it is large’“’ and in comparison the probability of spontaneous emission is negligible(6) (the excited state would have a lifetime of about 10”’ years for hydrogen nuclei”)). Because of this, irradiation with an intense radiofrequency field would rapidly equalise the population of the energy levels, A dynamic equilibrium would be set up between the levels but no absorption would be detected. If there is a Boltzmann distribution of nuclei among the spin states, a net absorption of energy may be observed upon application of low intensity radiation at the resonance frequency. For an assembly of weakly coupled identical nuclei of spin value + the energy levels derived for an isolated nucleus may be applied to the assembly as a whole. It must be assumed that the nuclei do not interact with any other part of the system. When there is thermal equilibrium throughout the assembly then the relative populations of the two energy levels is given by (2.28) where N2 and Nt are the number of nuclei in the high and the low energy levels respectively, k is the Boltxmann constant and Tis theabsolute temperature. The separation of the energy levels is given by hence 2PH

(2.30)

-kT*

Thus, hydrogen nuclei in a field of 10,ooOgauss have an excess population in the lower state of

wf

-=7x kT

10-6.

On examination of equation (2.30) the desirability of the main magnetic field being as large as possible becomes apparent : not only are the energy levels more widely spaced but also the sensitivity is increased since the excess population is increased. The observation of nuclear magnetic resonance absorption depends upon the net absorption of energy by this small excess of population. With continued absorption, the fractional excess dwindles. We may still describe the ratio of populations by a Boltxmann factor but now there is a corresponding rise in temperatnre(*). For this purpose, T is defined as the spin temperature and the spin system may be regarded as undergoing radiofrquency heating. (Since it has been assumed that the nuclei do not interact with any other part of the system, the temperature of the lattice will not be altered. In the next section, the nature of spin-lattice interactions will be discussed.) It is worth digressing for a moment on the concept of spin temperature because when radiofrequency cooling occurs and there is an excess of nuclei in the higher energy level then

18

HIGH

RESOLUTION

NMR

SPECTROSCOPY

the spin temperature can have negative values. Ramsey cg) has discussed how the first and second laws of thermodynamics may be formulated so as to include systems characterised by negative temperature. The line dividing positive and negative temperatures corresponds to all energy levels being equally populated; statistically this is an infinite temperature. Therefore, systemsdonotpass through absolute zero when going from positive to negative temperatures. Since the negative absolute temperature may be defined as full occupation of the top energy level, it is just as inaccessible as the usual positive absolute zero. The probabilities of a given nucleus being in either the upper or lower states are +(l - pH/kT) and +(l + ,uH/kT) respectively. Because of the unequal distribution of nuclei there will be a resultant macroscopic moment per unit volume in the direction of the main magnetic field. The mean nuclear magnetic moment in this direction is given by

;+(l

-g)]=g.

+$$+(I

When there are N nuclei per unit volume the paramagnetic bility is

(2.31) volume suscepti-

NP x0 = 7

(2.32) At room temperature x0 has a value of only about 3 x lo-lo hence the nuclear effect is negligible compared with the electronic diamagnetic susceptibility which has a value of about 10e6. For the general case of a nucleus of spin value Z, (I + l)N# (2.33) x0 = 3ZkT

2.4 SPIN-LATTICE

’

hAXATION

Nuclear spins invariably interact with their surroundings but because the interaction is usually small it is permissible to distinguish between spin temperature and lattice temperature. However, the small interaction does enable thermal equilibrium to be established eventually between the two systems. The resultant temperature will be close to that of the lattice since the heat capacity of the spin system is negligible compared with that of the lattice, except at very low temperatures. Although absorption of radiofrequency energy tends to reduce the excess population in the lower state, flow of heat from the nuclei to the lattice will tend to oppose the shift in population. Because spontaneous emission is unimportant the only means of nuclear magnetic relaxation are the induced downward transitions stimulated by magnetic fields oscillating at

GBNBRAL

THBORY

OF NUCLEAR

MAGNETIC

RESONANCE

19

‘the Larmor frquency. Clearly, an important factor is the rate at which thermal equilibrium can be restored. When hydrogen nuclei in ice at - 180°C were exposed to a high intensity radiofrquency field and then this was replaced by a radiofrquency field small enough to allow resonance absorption to be observed, an exponential growth of the absorption signalwithtime was found”*). The time constant for the hydrogen nuclear spin-lattice relaxation process, that is the spin-lattice time Tl, was found to be 10 min. This is the half-life of the excited nucleus. For solids, T1 .has beenfound to vary from 1O-4 to 104 set while for pure liquids, the range is lo-3 to 10 set but the presence of paramagnetic species (see Section 2.5.1) may reduce the lower end of this range to 1O-s sec. In the previous section, it was stated that the transition probabilities of ab sorption and emission are equal; this is no longer true when spin-lattice interaction is present. If the upward and downward transition probabilities are denoted W, and W, respectively, then at equilibrium W,N,

= WIN,.

(234)

Hence

(2.35)

Provided the interaction energy is small compared with the total energy of the system, that is, the temperature.is stationary, then W,/ WI is independent of N2 and N1. Let

w, = (2.36) w, =

and

where W is the mean of WI and W,. The rates of change of the populations

dN1 -= dt

--=

dN2 dt

are N2Wz - N,W,.

(2.37)

On expansion of the exponential in equation (2.37) as allowed by the assumption that @,/kT < 1 we obtain

Wl

- Nz) dt

= -2W

(Nl -Nzl-(N~+Nzlk~.

MO

1

Multiplying both sides of this equation by cc and using the relationships (Nl + N,)p2$$

= XOHO= MO and

AN,-NJ==4

(2.38)

20

HIGH RESOLUTION

NMR

SPECTROSCOPY

the fundamental equation of paramagnetic relaxation is obtained, namely

dMz

-= dt

-2W(M,

(2.39)

- MO)

where x0 is the static volume nuclear paramagnetic susceptibility, MO is the macroscopic nuclear magnetic moment per unit volume and M, represents the component of the magnetic moment per unit volume which is parallel to the steady field Ho. The exponential relaxation process found in practice is described by equation (2.39) and is characterised by the time constant Tl which is equal to (2FV)-‘. Bloch’“) named Tl the longitudinal relaxation time because it determines the approach to equilibrium of the component of M lying parallel to Ho. The factors which influence Tl must now be considered. 2.5 MECHANLW OF SPIN-LAITICERELAXATION When molecules undergo random translation and rotational motion, any nuclear magnetic moment which may be present will experience a rapidly fluctuating magnetic field produced by neighbouring nuclear magnetic moments (and electron magnetic moments if they are also present). The Fourier component of this time-variant field, having a frequency equal to the resonance fre quency yo, causes transitions between energy levels in the same manner as the applied radiofrequency field. Bloembergen, Purcell and Pound(rz) have calculated the probability of transitions induced in this manner, as follows. The energy for each neighbouring magnetic moment involves three functions of the distance rI, between the nucleus i and its neighbourj, the angle &, between rl, and Ho, and the azimuthal angle vl, of rljmeasured from some fixed reference direction. The three functions are Yo, = (1 - 3cos*e*,)r*;3 YlJ

=

sin&J

Y*J = sin*

8,J

cost&J

(2.40)

exp(i(ptJ)ri’,3

exp(2itptJ)r;3

(2.41) (2.42)

The intensities of the Fourier spectra of these three position functions are denoted by and respectively at the frequency Y. The total intensities for all neighbours for the functions YoJ, YlJ and are represented by Jo (Y), J1 (Y) and Jz (Y) respectively. The motion of the neighbour j at the resonance frequency y. produces at nucleus i an oscillating magnetic field of this frequency which is capable of inducing transitions. Furthermore, fluctuations of Y2J at frequency 2 y. can induce transitions ; in the general case, when i and j are not identical nuclei, the frequency is not 2~~ but is equal to the sum of the precession frequencies of i and j. Fluctuations of the remaining position function Y,,, do not induce transitions. The standard methods of handling random phenomena of the type encountered here are reviewed in Appendix C, Chap ter 4 of reference 2. The calculation of the total probability W of transitions JoJ(v),

J1

J(v)

J2J(v)

Y2J

GENERAL

THEORY

OF

NUCLEAR

MAGNETIC

RESONANCE

21

induced by molecular fluctuations is presented in Appendix 3 of reference 13. The resulting expression for the spin-lattice relaxation time Tl is (2.43)

If unpaired electrons or other magnetic nuclei are present then l/T, is given by summation over all types of magnetic moment, the additional terms for other nuclei being

+

%k)

1

(2.44)

and for paramagnetic ions of magnetic moment p,,

Here, the assumption is made that the spatial quantisation of the ion does not change in a time of the order of the correlation time r,-this is not always valid since the relaxation times for paramagnetic ions can be as low as 10Vgsec. Equation (2.43) holds for.1 > + provided that the populations of successive levels are related by the same factor, thus enabling a spin temperature to be defined. In addition, nuclei having I > 3 possess an electric quadrupole’moment which may interact with the fluctuating gradient of the local electric field and thus provide another mechanism for spin-lattice relaxation in liquids. In order to estimate Tl , it is necessary to evaluate the spectral density functions J(Y) which may be calculated as follows Jo(V) = CJ&) J

= cIE;111’ 2r,,(l I

+ 4n5%)-‘.

(2.46)

te is the correlation time which is of the order of the time a molecule takes to turn through a radian or to move over a distance comparable with its dimensions. m is the time average of 1Yol 12.Taking water as an example, there are two interactions to consider: (i) that between nearest neighbours of hydrogen nuclei, that is, the two belonging to the same molecule and (ii) that between hydrogen nuclei of other water molecules. Hence T;’ = T;& f T;lmj. The intramolecular effect(i) produces a random rotational motion and may be calculated by regarding the molecule as rigid. The vector joining the two hydrogen nuclei has no preferred orientation, thus only the angular coordinates vary with time hence Y1 = case sin0 exp(i@r,‘j y2

=

sin20

exp(2iQ ri3

(2.47) (2.48)

22

HIG-H

RESOLUTION

where r. is the interhydrogen integration to be

NMR

SPECTROSCOPY

distance. The time averages of IYl’ are found by (2.49)

-Iy27z= $r;6. Combination

(2.50)

of these values with equations (2.46) and (2.43) gives 3y4b2

rc

1 + 47c%$t5

=x

+

2%

1 + 163t2~:t:

1-

(2.51)

The correlation time rc is related to rn required in the Debye theory of dielectric dispersion of polar liquids”*), rn being the time required for a dielectric to lose its charge in a preferred direction after the electric field has been switched off. -CD.

-Cc=--=_ 3

4nrj a3

(2.52)

3kT

a represents the radius of a molecule regarded as a sphere embedded in a viscous liquid of viscosity q: a can be calculated from the molar volume. For water, tc = 2.7 x lo-l2 set at 20°C; hence at the radiofrequencies usually employed the term in the square brackets in equation (2.51) reduces to 3t,. Since r. is approximately 15 A then (1 /T,),,,, = 0.12 se&. The time elapsing before relaxation is complete can be very long if the nucleus is sterically shielded from other magnetic nuclei within the same molecule. An example is the resonance of 13C at natural abundance (about 1 per cent) in neopentane; in most molecules the central carbon atom will be linked to non-magnetic 12C atoms and thus will be shielded from other magnetic nuclei, that is, the hydrogen nuclei present in the methyl groups. The intermolecular effect arises from fluctuations in the local field caused by random translational motion of neighbours. Nucleus i may be regarded as stationary while nucleus j moves by diffusion in a liquid of diffusion coefficient D. The time taken forj to move distance r in any direction from i is tcj which can be evaluated from the theory of Brownian motiorP5) r2 t,j = 120 327

ar2

=2kT, The intermolecular

contribution

(2.53)

I

is given by tcJ 1 + 4x2 &rc,

+

2% 1 + 167#t,,

1’

(2.54)

GENERAL

THEORY

OF NUCLEAR

MAGNETIC

RESONANCE

23

Because of the factor r6, only close neighbours are of importance hence the term in square brackets reduces to 3rCl. By assuming that all neighbours are independent and by summin g-their contributions by integrating from infinity to the radius of closest approach (r = 2a), (l/Z’,),,,,, is given by = 9nyil2 ?j&

(2.55)

m

where N is the number of molecules in 1 cm3. The value of (1/Ti)i,,er calculated in this manner is 0.08 set- 1 at 20% therefore the total value of l/T, is 0.20 XC-~. An experimental value (M)of 1/TXfor oxygen-free water is 0.27 se&. In the Bloembergen, Purcell and Pound theory outlined above, the main relaxation mechanism for water is assumed to be a nuclear dipole interaction. Anderson and Arnold(“) have tested the validity of this assumption by measuring Tl of hydrogen nuclei in mixtures of Hz0 and D20. The general expression for such a situation is

0

where the primes refer to non-identical nuclei interacting with the unprimed nuclei whose resonance is to be observed; rr’is the number of nuclei in the molecule. Due to its smaller magnetic moment, which enters quadratically into the relaxation process, a deuteron should be about ten times less effective than a hydrogen nucleus in promoting the relaxation of neighbouring hydrogen nuclei. The assumption would be verified if TI increased quantitatively upon the addition of heavy water. This was shown to be so experimentally; Tl depended upon X, the volume concentration of H20 in the mixture, in the following way ($)=

‘i($,[”

+ (1 - ml.

(2.57)

7’ is the ratio of the viscosity of the mixture to that of H20 and R is given by R

ID

=G

+1

.-.-=(-Jo42

z,

zt, Z” + 1

(2.58)

where ,un and pD are the magnetic moments and ZHand Zn are the spin values of hydrogen and deuterium nuclei respectively. l/q’T, was found to be a linear function of x. Bovey’l *) has questioned the importance of the intermolecular contribution to Tl when non-associated liquids are being considered. Liquids of this type

24

HIGH

RESOLUTION

NMR

SPECTROSCOPY

might have negligible intermolecular relaxation because of the r6 factor in the interaction of two dipoles. Bovey measured the spin-lattice relaxation times of mesitylene, tetramethylsilane, cyclohexane and benzene at various concen-, trations in carbon disulphide. From equation (2.55) it may be seen that qT, should be constant if intermolecular dipole-dipole contributions are unimportant-this was shown to be so within experimental error. The above theory of Bloembergen, Purcell and Pound(12) has been re-examined by Kubo and Tomita(lg) who produced a more rigorous theory based upon density matrix formalism. Gutowsky and Woessner(zo) have generalised the results so that they apply to systems of many nuclei. The relaxation time for the ith nucleus may be written

wherex are summations over nuclei df the same type as the ith and C* are over all other magnetic mklei. In equation (2.59), the’summations are over nuclei in the same molecule while in equation (2.60), they are over nuclei of the neighbouring molecule, l/r& being the mean value of l/rll for two molecules in contact. In obtaining these results, Gutowsky and Woessner have assumed that identical nuclei within a molecule are distinguishable : interference effects have been shown to be small in three and four spin systems(21). Mitchell and Eisner(22) have measured Tl for hydrogen nuclei in solutions of C6H6, C6HdCl and C6H12 in CS2 or C&. From their data they were able to choose the best model giving expressions for the correlation times, trot and zlnnr, of solutions. The Hill theoryc23) provided the superior model : in this, the mutual viscosity TAB is deduced from the binary solutions of viscosity q, given by

,.=(,,,,~)+(f~,.~)+(,,~)

(2.61)

where fAand fBare mole fractions of A and B respectively, a, and CBare the average distances between A and B type molecules, and a,,, is the average distance between molecules in the mixture. The rotational correlation time was deduced to be (2.62) Here

K: =

UABIB) . (MA+ mB> + I,) (mAmB)

(?4B

(2.63)

GENERAL

THEORY

OF

NUCLEAR

MAGNETIC

RESONANCE

25

where Z, and Za are the respective moments of inertia of the molecules A and B about their centres and ZABis the moment of inertia of A about the centre of B during collision. Hill’s theory leads to the following expression for T,-

Mitchell and Eisner have pointed out that the important factor of the Hill theory is the K: term. This term is of the form of a “reduced m,oment of inertia” divided by the reduced mass of the solvent-solute system, hence an approximate expression for the Hill correlation time is 2Zqa z, = FkT where Z is the average of the principal moments of inertia of the molecule of interest, a is the average of the semi-axes of the molecule, 3 is the solvent viscosity and p is the reduced mass of the solvent-solute system. The results obtained by Mitchell and Eisner indicate that the translational mechanism for relaxation is dominant for C6Hs and &H&l, while the rotational and translational mechanisms are about equally important for C6Hz2. It has been shown(24* 25), however, that different types of motion are not independent of one another. 2.5.1 The Eflect of Paramagnetic Substances The effects of electronic paramagnetism are so large compared with those of the nuclear moments that nuclear resonance is often undetectable in paramagnetic samples. If an atom has an unpaired electron as well as a magnetic nucleus then the local magnetic field produced at the nucleus by the electron is in the range lo3 to lo5 gauss. Splitting of the nuclear magnetic levels by these local fields may be accompanied by a more efficient spin-lattice relaxation mechanism but fluctuations in the local fields are great enough to split and/or to broaden the absorption line sufficiently to prevent detection. The observation of nuclear magnetic resonance is thus restricted to diamagnetic or weakly paramagnetic systems. Early workers(26m27) used dilute solutions (-1O-3 M) of ferric or manganous salts to reduce the spin-lattice relaxation time in order to obtain stronger signals. Bloembergen, Purcell and Pound (r2) have calculated Ti in terms of the number N of paramagnetic ions of effective magnetic moment bff in 1 cm3 in a solution of viscosity 7j +

= 123~~y’qNp,2,,/5kT.

(2.65)

1

Pople, Schneider and BemsteS2*) have replaced the factor 12/S by 4 in this equation. Further details of the effect of paramagnetic substances on Tr are given in reference(2g) and in the references quoted therein.

26

HIGH

RESOLUTION

NMR SPECTROSCOPY

If the ion forms complexes in solution then the apparent value of peff is concentration dependent and can be used as a means of studying such equilibria. Another effect which may be used for the same purpose is the concentration dependent shift of the nuclear resonance signal caused by the presence of paramagnetic ions(30* 31). It should be mentioned that diamagnetic ions lower the spin-lattice relaxation time of hydrogen nuclei in water by as much as a factor of fou+). 2.5.2 Quadrupole Effects Nuclei of spin greater than + may possess non-spherically symmetrical nuclear charge distributions, resulting in their having quadrupole moments Q. Positive and negative signs associated with Q imply that the shapes of the charge distribution about the spin axis are respectively those of prolate and oblate spheroids. Nuclei do not have electric dipole moments and so the energy of any nucleus is independent of its orientation in a uniform electric field. However, when an electric field gradient exists, the quadrupoles undergo precession which displaces the nuclear magnetic levels. Quadrupole energies range from negligibly small values to those much larger than nuclear dipolar magnetic interactions. For an isolated nucleus having I > 3 placed in a strong magnetic field?, the energy levels are given byCg3) E=

-W& I

e Q[3m2 - I(I+ + 41(211)

l)] a2v dzZ’

.

(2.66)

The magnetic quantum number m has the values - I to + Jand V is the electrostatic potential at the nucleus caused by external charges. The electric field gradient a2 V/i?z” may arise both from within a molecule (along a bond) and from its crystalline environment. In liquids or gases, Brownian motion tends to reduce intermolecular electric field gradients to zero but this is not true for intramolecular gradients as found in covalent bonds. The time-averaged interaction is generally larger than the magnetic interactions and thus provides an efficient relaxation mechanism from the fluctuating electric field gradients(34). Spin-lattice relaxation times for solids may be of the order of hours if quadrupolar effects are negligible but if large, TX has been found to be as small as 1O-4 sec. The fluctuating gradients originate mainly from lattice vibrations. The expression for the relaxation time Tl would be expected to resemble equation (2.51) and takes the form 1 1 1 3 e4q2 Q2t, + 1 + 163z2v;t,2 ~=10’h212(211)’ 1 +4&v;t:

1(2.67)

where

a2v

q = 7 a2- * The most notable results for liquids have been obtained with the hydrogen and deuterium nuclear resonances of mixtures of Hz0 and D20. In a 50 per cent solution of heavy water values of TXfor the hydrogen and deuterium nuclei are 3.0 and 0.5 set respectively. Now the nuclear magnetic moments of hydrot The magnetic field is parallel to the electric field gradient.

GENERAL

THEORY

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NUCLEAR

MAGNETIC

RESONANCE

27

gen and deuterium are 2.79 and O-857nuclear magnetons respectively, therefore, in the absence of quadrupolar effects, the deuterium relaxation time would be the longer. Bloembergen, Purcell and Pound(12) found that the observed Tl for the deuterium nuclear resonance could be accounted for by assuming a reasonable value for a2 V/W, that is, equivalent to a unit charge placed 1 A away from the deuterium nucleus. The quadrupole moment of this nucleus is small (2.77 x 1O-3 e x 1O-24 cm2) but nuclei having large values may have very short relaxation times thus causing the resonance line to be broad. Pound(3s) found absorption lines 10 gauss wide for 79Br and *lBr-whose quadrupole moments are O-33 x e x 1O-24 cm2 and 0.28 x e x 1O-24 cm2 respectivelypresent in aqueous solutions of lithium and sodium bromides. This particular line width is indicative of a relaxation time of 3 x IO-” sec. If the environment of a quadrupolar nucleus is sufkiently symmetrical, the electric Geld gradient will be zero at the nucleus thus eliminating quadrupole coupling effects. Because of this, nuclei havingappreciable quadrupolemoments, such as Na, Rb, Cs, Br and I, can have their magnetic resonance spectra observed in cubic crystals and in ionic solutions. Solvation effects can distort the spherical symmetry of the ions and give rise to line broadening because of quadrupolar contributions to Tl. Quadrupolar relaxation is responsible for wide hydrogen and nitrogen nuclear resonance lines in NH3 and in unsymmetrical ammonium salts, while narrow lines are observed for tetrahedrally symmetrical NH: and (CH3)4N+ ionF) (see Section 12.3.2). In covalent compounds such as Ccl,, the chlorine nuclei have a quadrupo1a.r coupling of about 35 MC set-’ with the field gradient of the C-Cl bond and this prevents observation of chlorine nuclear magnetic resonance. 2.5.3 The Anisotropic Electronic Shielding Efect The electronic shielding of magnetic nuclei is discussed in Chapter 3 but it is pertinent to mention here the relaxation mechanism which can occur when the chemical shielding is not isotropic. Unlike the other mechanisms’of spin-lattice relaxation presented above, this effect requires the presence of an appliedmagnetic field, the magnitude of which determines the relaxation time. The secondary magnetic field due to electronic currents will not generally be parallel to the direction of the external field, hence the perpendicular component so produced will oscillate as the molecule rotates thus causing transitions between nuclear energy levels. McConnell and Holm(37) have derived the relationship .

1 T,=

$ (

Y2H2(46)2T,

(2.68)

>

where da is the difference between the shielding constants parallel and perpendicular to the axis of symmetry. They obtained some evidence for the relaxation process by measuring Tl for 13C in natural abundance in carbon disulphide and in carbon tetrachloride. Under these conditions the dipole interaction is unimportant, therefore the longer relaxation time of 13CCl,, compared with 13CS2 can be attributed to the lower symmetry of the latter molecule.

28

HIGH

RESOLUTION

NMR

SPECTROSCOPY

Gutowsky and Woessner(20) have demonstrated the field dependence predicted by equation (2.68) and they have estimated Au for 1,3,%itluorobenxene and obtained agreement with the theoretical value. They also showed that Tl is longer for’ H than for rgF as expected from the anisotropic shielding mechanism. More recent work of Gutowsky, Lawrenson and Shimomura(38) cor&med this finding and it was also proposed that at room temperature there are three contributions to Tl for lgF nuclei, namely dipole-dipole interactions, Au and a spin-rotation interaction. 2.5.4 The E#ect of Pressure In an investigation of several saturated organic liquids it was found(3g’ 40) that Tl (‘H resonance) decreases with pressure (up to 1000 kg cm-*) much less rapidly than expected from the changes in viscosity. A similar result was found for aromatic liquids unless they are degassed, when Tl varies roughly linearly with the viscosity reciprocal. Available theories of the liquid state are inadequate to allow a detailed theoretical treatment of the effect of pressurt on Tl. Waugh and Johnson;41) have found that for gases, Tl is proportional to pressure (really to density) thus proving that intermolecular magnetic perturbations occurring in collisions are unimportant.. 2.6 SPIN-SPIN RELAXATION In addition to interacting with the lattice, magnetic nuclei can also interact among themselves. Each nuclear magnet is acted upon not only by the steady magnetic field Ho but also by the small local magnetic field H,,, produced by the neighbouring nuclear magnets. Since a magnetic dipole produces a field of p/r3 gauss at a distance r, a hydrogen nucleus will produce a field of about 14 gauss at another hydrogen nucleus at a distance of 1 A from the former. H,, falls off rapidly with increase of r so that only nearest neighbours have an important effect. It will be seen readily that the nuclei will experience different values of the steady magnetic field and hence there will be a broadening of the energy levels of the dipolar assembly and thus of the absorption lines. The re-orientation and diffusion of molecules in liquids, gases and some solids is usually so rapid that the local magnetic field is averaged to a very small value hence a narrow resonance line is observed. Taking HIOEas the spread of the local field, the resonance equation gives the range of frequencies of Larmor precession as

Lf the nuclear dipoles are precessing in phase at a particular instant, then the time required to get out of phase, (A v)-~, is about lO-4 sec. This may be regarded as part of the spin-spininteraction time Tz. In addition, the local fields from

GENERAL

THEORY

OF NUCLEAR

MAGNETIC

RESONANCE

29

magnetic neighbours may have oscillating (H-) and static (H,,,) components, thus nucleus j producing a magnetic field oscillating at its Larmor frequency (see Fig. 2.4) may induce a transition in nucleus k. The energy for the process comes fromj and simultaneous reorientation or flip-flop of both nuclei results, there being interchange of energy while conserving the total energy of the pair. Since the relative phases of the nuclei change in a time (d ~)-l, then a simple time interval will be required for spin exchange. This process causes a further broadening of the resonance line, observed at a tied frequency, by an amount

--. _--_--__ ;-_ ______ A _e--

___----

----_____

I

----_-______----i

H

___---,_--

I

Ii!2 n is
(3.185)

Uoba = 1 +

c

a

exp(-

h vi/k

0

1

where g, is the degeneracy of the vibrational mode of frequency vi. When vi > 200 cm-‘, and T - 3OO”i( equation (3.185) may be written as %bs = uw) + Cgrda”)exp 1

(3.186)

where da”) = uo) - u(O). The chemical shift v. 6 from some reference compound, whose shielding constant is independent of temperature, is given by vo

6 = vo d(O)

where dI = v. cV) - v. tY”). The variation of v. S with temperature

d(vo8)

-

dT

=

(3.187)

is

rgi4(s)exp(%)

(3.188)

so that the effect of each vibrational mode is additive. Petrakis and Sederholm(77) have in fact observed a temperature dependence of v. 6 for several gaseous compounds, with values of d (v. S)/d T of the order of - OXlO cycles set-1 deg-’ . 3.12 TEMP~TURE DEPENDENCE OF SFIIELDING CONSTANTS If the chemical shift is measured relative to an external reference signal then it will be both temperature and concentration dependent since both the diamagnetic and electric susceptibilities are dependent on both temperature

102

HIGH

RESOLUTION

NMR SPECTROSCOPY

and concentration. When an internal reference signal is used then in most cases the observed chemical shifts do not depend on temperature. However, some examples of temperature dependent shifts have been observed(77*g4); the small temperature variation arising from vibrational excited states has already been discussed (Section 3.11). The equations for the shielding coefficient discussed so far have all been for a single electronic ground state n. However, it is possible that other electronic states are occupied in each of which (I is different; as the temperature is varied the distribution between the states varies and the average value is altered. If for states n = 0,l . . . g, the quantity exp (- E./kT) is appreciable, whilst for other states it is negligible then the weighted mean average of u for the case when the molecule is symmetric about the z-axis and when all orientations of the molecule with respect to the field are equally probable is given by(“) 0 = 01 + 62 + Qj (3.189) where 01 =

and

C = [iTP(-$$)]‘-

A particularly simple case would occur if only one low lying excited state were possible apart from the ground state. The shielding coefficient would then be approximately the term a, in equation (3.189), that is

The majority of cases when the shielding coefficient is temperature dependent are a result of some form of molecular association and are not described simply by equations (3.190) to (3.193) since these equations refer to a single molecule. The temperature dependence of the shielding coefficient when the molecule

THE ORIGIN OF CHEMICALSHIFTS AND SPIN-SPIN COUPLING 103 takes part in molecular association depends upon the association of many molecules, and the various theorems for the equilibrium of the association must be used (see Chapter 9). SPIN-SPIN COUPLING 3.13 THEORY OFSPIN-SPINCOUPLING We have already discussed a mechanism for spin-spin coupling in which the interaction proceeds via the electrons (Section 3.2); however, other coupling mechanisms may also play a part and in general all magnetic interactions must be taken into account.(**i) The interaction energy A?& for two nuclei N and N’ in a molecule with a specific orientation Ais given by(5g) E NN'

=

hIN-

Jme ’ IN’ + hJNN. IN - 1~‘.

(3.194)

JNN*is a tensor of the second rank whose trace is zero. For frequent intermolecular collisions JNppaverages to zero, so that (ENN,LL

=

11JNN*&

*IN*

(3.195)

where INis the nuclear angular momentum in units of It. If a Hamiltonian for the system is constructed which represents all magnetic interactions in the molecule, then J?&~,can be obtained by selecting those parts of the Hamiltonian which depend upon both IN and INI. It is -convenient to write the Hamiltonian as *=z,+X*+*Si-pb

(3.196)

and we will discuss each part of equation (3.196) in turn. 3F’, is simply the Hamiltonian used in the magnetic shielding calculations [Section 3.5 equation (3.22)] with the addition of electron-electron interaction terms, i.e.

For simplicity a specific choice of gauge has been made. The terms flu, X Ls, Zss and flsm are respectively electron orbital-orbital, orbital-spin, spin-spin, and spin-external field; these terms may be neglected for singlet molecular states. In the absence of an external magnetic field only two terms in equation (3.197) need to be considered.

’ (3.198)

104

HIGH

RESOLUTION

NMR

SPECTROSCOPY

d is the unit dyadic.

tiZ and ZJ together describe the nucleus-electron-electron-nucleus interaction discussed in Section 3.2. The mutual interaction energy of a nuclear magnetic dipole pN with an electron dipole pLI,is Eh

=

-J+

3@N ’ rkN) hk ’ r&d

’ pk

PN

Substituting

.

(3.200)

dN

rkN

y h IN for pN and 28 h S, for & gives the Hamiltonian 3 (Sk sZ

=

28ficyN

* rkd

(IN

’ sN>

5 rkN

kN

as

(3.201)

Another term Z’S is needed in the Hamiltonian since equations (3.200) and (3.201) do not apply for electrons in S-states, since if the electron density at the nucleus is finite then as rk, -+ 0 the energy becomes infmite. A term &‘3 was first introduced by Fermi (60) for s-states to account for the hyperfke structure interaction, and the form given here has been discussed by Ramsey(2g* 5g). Imagine the nucleus enclosed by a sphere of radius R, where R is large compared with the size of the nucleus, but small compared with the atomic radius. The value of R is such that we may regard the wavefunction for the electron to have a constant value, y. , over the volume of the sphere, that is the electron density within the sphere is uniform. e(0) = (0 IQ&l

0)

where e(O) is the electron density within the sphere, and I!&) d-function. If m(0)is the magnetisation within the sphere then m(0) = -e(O)‘2~&

(3.202) is a Dirac (3.203)

and the induced field at the nucleus is

B(O)=!+(O) Since the

nUdeUS

=

16x --j-do)~sk.

(3.204)

has a magnetic moment y Ir IN, then the interaction energy is (0) E kN

16n/? =jd”)yNASkelN

(3.205)

and the Hamiltonian is (3.206)

THE ORIGIN

OF CHEMICAL

SHIFTS

AND

SPIN-SPIN

COUPLING

105

The last term in the Hamiltonian is simply the internuclear dipole-dipole interaction which gives rise to the broad lines observed in solids

S4

averages to zero when all molecular

orientations

are equally probable.

CALCULATION OF SPIN-SPIN COUPLING CONSTANTS

3.14

3.14. I

Application of Perturbation Theory calculation(5g) gives a second order energy correction,

A perturbation c

ENNI

=

-&

tEa

;

Eel

(0

icfln>

(3.245)

and the unknown functions fLN’ and f p” are to be found by minimising the energy, (3.246) E NN’ = hJaB INo I,., . Jd is a symmetric coupling tensor and is given by J$’ = (Ol~‘lO) (grad&f p’* - grade f p” + grad, f p’ grad&?” +

(3.247)

Minimising equation (3.247) with respect to variations in f p’ and f p” gives these two functions as solutions of the equations, %N+O

+

&

PO

vff P’ + &

grad& ?l’ograd, f p’

a

= 8

(qq

112

.HIGH

RESOLUTION

NMR

SPECTROSCOPY

and a similar equation in IV. Equation (3.247) may be simplified since mkN is Hermitian and fLN) and fF" are purely imaginary functions, and the new equation as derived by Stephen(73) is

The electron-nucleus contributions to Jd can be found in a similar fashion. The Hamiltonian is (S2 + LX?,) (cf. equations (3.196), (3.201) and (3.206)) and is written (S2 + S3) = INa.%‘:’ + ZNta&&=r)

_

za

YN

c

3 @k

+ ZNa.%‘i6’ + ZNra9:‘.

’ IkN)

rky

k

-

6N

..

Sk,

(3.250) (3.251)

rkN

SP 0 is the same as AC4’ I in N’ (3.252) ti=” is the same as Z’i6’ in N’. To determine (JZNNO+ JsNN,) = J$‘, t =

PO

h

+

%(I,,

the wavefunction

hN’

+

&‘a

y is written as (3.253)

dN’))I

CJJ# is an antisymmetric spin function and v, a symmetric triplet spin function. p” and pN’ are unknown functions which are to be determined by the variational principle. With this wavefunction .I$’ can be written as

f$’ =

(JZNW

ii2

= 4M

+

JJNN,)

(grad,p;N’* grad, py” + grad&“‘*

0 (

grad, piN’)+ 0

II k

I>

+

+

(y8

‘ps

I (jly?)

+

dy’>

py

+

;

(y$

p’:

( (Gty

+

2:))

py*

+

(3f$i’

+

+ complex conjugate. The functions piN’ and pLN” are found from

- li2 FYO

c vzPP’

- $;

(xy)

+

fly’)

+

Y/y))

pp’

py

I q.$ yo)

1 cps yo)

(3.254)

1yo gradk y. - gradkpim = 0

(3.255)

k

and a similar equation in piN”. For rapid, random motion of the molecules av(J+& = +(Jxx + J,.,, + J,,).

THE ORIGIN OF CHEMICAL SHIFTS AND SPIN-SPIN 3.15 THE COUPLING CONSTANT

COUPLING 113

AND THE ANISOTROPYOF THE SFHRLDLNG CGRFFICIRNT

The relative orientation of nuclear spins by direct dipolar interaction of the nuclei is destroyed by random motion only when the shielding coefficients of the two nuclei are isotropic; when the shielding coefficients are anisotropic there is a small but significant contribution to the coupling constants. For inter-hydrogen nuclear coupling the dipolar contribution is very small, but is ‘more important when larger nuclei are considered. An approximate relationship between the coupling constant J1 NNeand the anisotropy of the shielding coefficient can be obtained by considering the nucleus N’ as a magnetic probe investigating the magnetic polarisation produced by N(74). The Hamiltonian X1 (see equation (3.198)) is, neglecting the electron-eleo tron terms, 1 St?, = p+ 2+ v. k2M >

c (

.%‘, describes the motion of the electrons in a magnetic field with vector potential &. If & arises from the nuclear moment y h IN, then

that is, the nuclear dipole gives rise to an induced electron orbital current which may be represented by an inctuced orbital magnetic moment &+ The total magnetic moment produced by the induced current is related to the shielding coefficient of the nucleus in an external field. Consider the energy of the system in the combined field of the nuclear dipole and an external field. The vector potential 4, is now &=&+ih; (3.258) where & is the vector potential arising from the nuclear dipole, and g that arising from the external field. Carrying out a second-order perturbation calculation for the energy, the cross terms in & and Ai’ give the interaction energy between the external field and the induced orbital currents, but this same cross term also gives the interaction energy between the nuclear dipole and the currents induced by the external field, which, of course, is simply what is measured by the shielding coefficient. If H, is the external field then the field at the nucleus is (H, - u,&J, where us,, is the shielding tensor. The field at the nucleus arising from induced orbital currents is - o,&, thus Interaction

energy = y h I, u4 H, =

therefore

-t-&b

*I3

&,& = - y h $,#I,,

(3.259)

where lu,, is the total magnetic moment produced by all the electrons in the molecule, but if interatomic currents are negligible then equation (3.259) can

114

HIGH

RESOLUTION

NMR SPECTROSCOPY

be regarded as arising from the electrons localised on nucleus N. The contributions to the orbital magnetic moment from circulating currents on atoms other than that containing the nucleus N is estimated by, p, = y A {3(I * R,) Rt - R:I} R;’ xr.

(3.260)

x, is the magnetic susceptibility of an atom at a distance R, from N. The coupling energy between two nuclei N and N’ via electron orbital polarisationis equal to twice the interaction energy of the magnetic moment of N’ with the currents induced by N. The currents induced by N may be divided into(74) (a) Currents on the atom with N as nucleus (b) Currents on the atom with N’ as nucleus (c) Currents on all other atoms. Currents (a) give a magnetic moment of - y zia,, Ip at N, and the magnetic field at N’ arising from this induced dipole is y fi e,@G@,, I,,, where .Q is the dipole interaction tensor, E,B = {R2 4,

- 3R,Rp) R-5

(3.261)

R is the position vector of N’ with respect to N.

The unscreened field at N’ arising from the dipole y h I, at N is simply - y A t, I,, so that the contribution from induced orbital currents of type (b) is y h cap a&, I,,, where u$ is the shielding coefficient of N’. The contribution to the field at N’ arising from currents on other atoms can be obtained from equation (3.260). If RI and R; are the position vectors of a third nucleus relative to N and N’ respectively, then the moment induced in atom i by the currents on N is - y A &iauI, xr and the field arising from this at N’ is y h xi &a@E& 1,. The interaction energy arising from currents (a), (b) and (c) is of the form(74) E%) = h Jab I, Ia where

*is a summation over all other atoms’ 1. (5: If all molecular orientations are equally possible,

and

, &a8 Eo# = R-5 (R2 a,, - 3R,R,a,,)

is non-zero only if a,, is anisotropic. If each nucleus has an axially symmetric environment then a,, has two principal values ull and o,, corresponding to the field parallel or perpendicular to the symmetry As. A0 =u\i -ul.

THE ORIGIN

OF CHEMICAL

The value of Jco*) NN’ is now 2 J(_b) NN’ = - 5 YNYN 0

SHIFTS

Jl2{R-3[Ll

- 3 C+& Ri3(&)-3

AND SPIN-SPIN

115

COUPLING

u(1 - 3 COSV) + d d (1 - 3 COSW)]

(1 - 3 COS20J}

where 8 is the angle between the symmetry axis of the shielding tensor and the internuclear line and 8# is the angle subtended at atom i by N and N’. EFFECTS

IN PARAMAGNETIC 3.16

&WTACT

MATERIALS

Sm

For some paramagnetic complexes, in which diamagnetic ligand molecules are bonded to a transition metal, the NMR spectrum of the nuclei in the ligand molecules sometimes show unusually large chemical sh.ifWo6 - log*l 19) (see Section 10.36). For example, the hydrogen resonance spectrum of an aqueous solution of the complex ion [(H20)4 Ni(NH2CH2CH2NH2)]2+ consists of two groups of chemically shifted peaks separated by N 300 ppm(lo9). These large chemical shifts are thought to arise because of a slight transfer of unpaired electron spin from the paramagnetic ion to the ligand molecule; the unpaired spin is then transmitted to the other nuclei in the l&and molecule by-the contact mechanism. A qualitative description of. the distribution of spin density in the ligand is given in Section 10.36’where it is seen that, in the nickel ethylene diamine complex, the net spin polarisation of the electrons in the N-H bond tends to be anti-parallel to the applied field and produce a shielding field at the hydrogen nuclei so that their resonance shifts to higher applied fields. There is a positive spin density in the C-H bonds, hence these hydrogen nuclei resonate at lower applied fields. The magnitude of the contact shift d H, at the ith nucleus in the ligand is given by(106.107) AH, _=

H

_a‘~Ms+ YN

1)

6kT

(3.265)

in which g is the g-value of the paramagnetic species of spin S and a1 is the hyperfine interaction constant. It can be seen from equation (3.265) that the direction of the contact shift gives the sign of al. The condition for observation of a contact shift is that

in which Tz is the electron spin relaxation time, and T, is the characteristic exchange time for the paramagnetic species. The value of uI calculated from equation (3.265) can be used to calculate the pn electron spin density at the appropriate nucleus. In the case of hydrocarbons the value of a, obtained from hydrogen contact shifts has been shown

116


to be related to the pn electron density ec on the attached carbon atom by the relationship(116) (3.266) an = Qec. For the 6-H fragment, Q has been found to be - 22.5 gauss(ll’), and for the 6-CH, fragment the value of Q is + 27 gauss(“*). Phillips et al.(lo8) have shown that when the nuclear species whose contact shift is being measured can take part in n-electron conjugation, then equation (3.266) no longer describes correctly the relationship between a, and the electron spin densities. They suggest, for example, that in the case of lgF contact shifts in fluorocarbons the value of aF is related to both ec and & by the equation aF

=

Q&c

+

(3.267)

Q&F

where Q& is an-ucontribution to ar similar to that in hydrocarbons, and Q& is the contribution arising from z-u polarisation of the 1s and 2s electrons of fluorine by the spin density centred on a 2p x fluorine orbital. Phillips er al. suggest that eF is determined by the extent of double bonding in the C-F bond so that equation (3.267) may be written as

wherepc, is the double bond order of the C-F constant.

bond and A is a proportionality

3.17 THE OVWIAUSRREFFECT It has been shown already that in solutions of paramagnetic compounds the nuclear and electron spin systems are not independent of one another, and as a consequence the spin-lattice relaxation time of the nuclear spins in such a system is very short, and the resonance lines are too broad for observation. Overhauser(“O) has shown that another consequence of this electron-nucleus interaction is that if transitions between the electron spin energy levels are excited by radiation at the appropriate Larmor frequency, then transitions are also induced in the nuclear spin system. If the electron-nucleus interaction is the dominant spin exchange process for the nucleus, the result is an increased polarisation of the nuclear spins between the possible energy levels, and hence an enhancement of the nuclear resonance absorption signal. The nature of the electron-nucleus coupling and the mechanisms which produce the nuclear spin polarisation in an Overhauser experiment have been discussed in detail by Abragam” II), and also by Anderson(112). The interdependence of the nuclear and electron spin systems can be expressed by the phenomenological equation’“‘* 112)

((I,) - IO) =

(1- $-)e(&)

-

SO)

(3.268)

where (Zz> arid (S,) are the expectation values of the z-components of the nuclear and electron spin momentum operators, and lo and So are their values in

THE ORIGIN

OF CHEMICAL

SHIFTS

AND

SPIN-SPIN

COUPLING

117

the absence of any irradiation of the electron spin system. Tl and T,, are the relaxation times of the nuclear spins in the presence and absence of the paramagnetic material, and e is a factor which depends on the nature of the electronnucleus coupling. The coupling may be of two kinds: a dipole-dipole coupling for which e = + 3, or a scalar coupling, described by the Fermi contact term, for which e = - 1. If the electron spin system is saturated by applying a strong microwave field at the electron Larmor frequency, then (S,) is reduced to zero and equation (3.268) becomes, ((Z,)

- Z,) =

1 - $(

es, 10)

hence the nuclear polarisation is increased by the factor

which can have a value of up to approximately 1000. For most solutions the enhancement obtained by an Overhauser experiment is of the order of 100(112). Richards and White” 13-1 l 6, have investigated the Overhauser effect in solutions containing small amounts of free radicals, and besides showing that it is possible to achieve signal enhancements of the order of 100, they also show that it may be possible to obtain inter-nuclear distances between interacting molecules from the exact behaviour of the nuclear resonance signal. .

REFERENCES 1. W. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19.

20. 21. 22.

D. KNIGHT, Phys. Rev., 76, 1259 (1949). W. G. Pmmo~ and F. C. Yu, Phys. Rev., 77, 717 (1950). W. C. DICKINSON,Phys. Rev., 77,736 (1950). W. G. PROCTOR and F. C. Yu, Phys. Rea., 81,20 (1951). H. S. GUTOWSKYand D. W. MCCAL.L,Phys. Rec., 82,748 (1951). Phys. Reu., 91,476 (1953). T. F. Wmm-r, H. M. MCCONNELL,A. D. MCLEAN and C. A. REULY, J. Chem. Phys., 23,1152 (1955). W. E. QUINN and R. M. BROWN, J. Chem. Phys., 21, 1605 (1953). H. S. GUTOWSKY,D. W. MCCALL and C. P. SLICHTEX,J. Chem. Phys., 21,279 (1953). D. P. Roux and G. J. BJZNE, J. Chem. Phys., 26,968 (1957). J. M. ROCARD,Archives des Sciences, 10, 209 (1957). D. P. Roux, Archives des Sciences, 10,217 (1957). N. F. RAMSEYand E. M. PURCELL,Phys. Reo., 85, 143 (1952). W. E. LAMB, Phys. Rev., 60, 817 (1941). L. D. LANDAUand E. M. Lmcmrz, Chssicai Theory of Fields, 2nd cd., Pergamon Press, Oxford (1959). E. H~LLERMS and S. SKAVLEM,Pbys. Reu., 79, 117 (1950). W. C. DICKINSON,Phys. Reu., 80, 563 (1951). N. F. RAMSEY,Phys. Rev., 78,699 (1950). N. F. RAMSEY,Phys. Rev., 86,243 (1952). H. MAR~ENAU and G. M. MURPHY, The Mathematics of Physics and Chemistry, Van Nostrand, New York (1943). J. H. VAN VLECK,Electric and Magnetic Susceptibilities, Oxford University Press (1932). J. H. VAN VLECK and A. FRANK, Proc. Nat. Acad. Sci., 15, 539 (1929).

118

HIGH

RESOLUTION

NMR

SPECTROSCOPY

23. G. C. WICK, Z. Phys., 85, 25 (1933). 24. H. EYRINO, J. WALTER and G. KIMBALL, Quatzfum Chemistry, Wiley, New York (1944). p. 111.

25. K. ITO, 1. Amer. Chem. Sot., 80, 3502 (1958). 26. K. Iro, Private communication (1959). 27. G. C. WICK, Phys. Rev., 73, 51 (1948). 28. H. BROOKS, Phys. Rev., 59,925 (1941). 29. N. F. RAMSEY,Molecular Beams, Clarendon Press, Oxford (1956). 30. H. F. HAMXA, Mol. Phys., 1,203 (1958). 31. H. M. MCCONNELL, J. Chem. Phys., 27,226 (1957). 32. H. M. MCCONNELL, Ann. Rev. Phys. Chem., 8, 105 (1957). . 33. E. ISHIGIJRO and S. KOIDA, Phys. Rev., 94,350 (1954). 34. J. F. HO~NIG and J. 0. HIRXHFELDER, J. Chem. Phys., 23,474 (1955). 35. B. R. MCGARVEY, J. Chem. Phys., 27,68 (1957). 36. T. P. DAS and R. BERSOHN,Phys. Rev., 104,849 (1956). 37. M. J. STEPHEN,Proc. Roy. Sot., A243, 264 (1957). 38. T. P. Dm and T. GHOSE, J. Chem. Phys., 31,42 (1959). 39. T. P. DAS and R. BERSOHN,Phys. Rev., 115,897(1959). 40. M. F~XMAN,J. Chem. Phys., 35,679 (1961). 41. J. TILLUXIand J. Guv, J. Chem. Phys., 24, 1117 (1956). 42. R. ~HN, Ann. Rev. Phys. Chem., 11,369(1960). 43. J. A. POProc. Roy. Sot., A239, 541 (1957). 44. J. A. POPI& Proc. Roy. Sot., A239,550 (1957). 45. L. D. LANDAU and E. M.LUXHITZ, Non-RelativisticQuantum Mechanics, Peqamon

Press, Oxford (1962). 46. P. A. M. DIRAC, Quanfum Mechanics, Oxford University Press (1935). 47. L. PAIJLING,J. Chem. Phys., 4, 673 (1936). 48. J. A. POPLE, J. Chem. Phys., 24, 1111 (1956). 49. J. S. WAUGH and R. W. FESSENDEN, J. Amer. Chem. Sot., 79, 846 (1957); (see J. Amer. Chem. Sot., 80, 6697 (1958) for a correction). 50. C. E. JOHNSONand F. A. BOVEY, J. Chem. Phys., 29, 1012 (1958). 51. J. A. POPLEZ, Mol. Phys.. 1, 175 (1958). 52. R. MCWEENEY, Mol. Phys., 1,311 (1958). 53. C. A. COULSONand H. C. LONG~JET-HICXXN~, Proc. Roy. Sot., A191,39 (1947). 54.T. W. MARSHALL aad J. A. POPLE, Mol. Phys., 1, 199 (1958). 55. A. D. BUCKINGU, Con. J. Chem., 38,300 (1960). 56. L. ONSAGER,J. Amer. Chem. Sot., 58, 1486 (1936). 57. P. Draru. and R. FREEMAN,Mol. Phys., 4, 39 (1961). 58. A. D. BWKINGHW J. Chem. Phys., 30. 1580 (1959). 59. N. F. RAMSEY,Phys. Rev., 91, 303 (1953). 60. E. Fe~hsl, Z. Phys., 60, 320 (1930). 61. A. D. M&A-, J. Chem. Phys., 32, 1263 (1960). 62. S. ALEXANDER,J. Chem. Phys., 34, 106 (1961). 63. M. KARPLus, Rev. Mod. Phys., 32,455 (1960). 64. C. A. (%JE4oN, Valence, Oxford University Press (1952). 65. H. M. MCCONNBLL, J. Chem. Phys., 24,460 (1956). 66. L. PNLING and E. B. W&SON, Xntroductionto Quantum Mechanics, McGraw-Hill. New York (1935). J. Chem. Phys., 30, 717 (1959). 67. H. S. G~TOWSKY and G. A. W-, 68. H. M. MCCONNELL, J. Chem. Phys., 30,126 (1959). 69. E. AIHARA, J. Chem. Phys., 26, 1347 (1957). 70. G. RUMER, Giittinger Nuchr., 377 (1932). 71. L. PA~~.~NG,J. Chem. Phys., 1,280 (1933). 72. M. KARpLus and D. H. AND-N, J. Chem. Phys., 30, 6 (1959). 73. M. J. STEPHEN,Proc. Roy. Sot., A243, 274 (1957). 74. J. A. POPLE, Mol. Phys., 1, 216 (1958).

THE

ORlGlN

75. N. F. Mom

OF CHEMICAL

SHIFTS

AND

SPIN-SPIN

COUPLING

119

and I. N. SNEDWN, Wave Mechanics and its Applications, Clarerrdon Press, Oxford (1948), p. 94. 76. L. C. SNYDERand R. G. PARR,J. Chem. Phys., 34,837 (1961). 77. L. PEnurcIs and C. H. SEDERHOLM, J. Chem. Phys., 35, 1174 (1961). 78. S. K. Grroarrand S. K. Sm, J. Chem. Phys., 36,737 (1962). 79. D. E. O'REJLLY, J. Chem. Phys., 36, 855 (1962). 80. D. E. O’REILJ+Y, J. Chem. Phys., 36,274 (1962). 81. H. F. -, Niiouo Cimenro, 11,395(1959). 82. W. C. DXKXNsoN, PhyS. Ret?., 81,717(1951). 83. A. A. --BY and R. E. GLICK,J. Chem. Phys., 26,1647 (1957). 84. A. A. --BY and R. E. GLICK,J. Chem. Phys., 26,1651 (1957). 85. W. G. SCHNEIDER, H. J. BERNSTEM and J. A. POPLE,J. Chem. Phys., 28,601 (1958). 86. M. J. STEPHEN,Mol. Phys., 1,223(1958). 87. A. D. BUCKINORAM, T. -atrdW.G.S cxNarom J. Gem. Phys., 3% 1227 (1960). 88. B. B. How-, B. -ER and M. T. -N, J. Chem. Phys., 36,485 (1962). 89. B. LENDER,J. Chem. Phys., 35,371 (1960). 90. C. J. F. B&-rTheory of Electric Potkrisation, Ekvicr, New York (1952). 91. F. Low Z. Phys., 63,245 (1930). 92. S. Goauo~ and B. P. DAILEY,J. Chem. Phys., 34, 1084 (1961). 93. D. F. EVANS,J. Chem. Sot., 877 (1960). 94. R. FRFPMAN,G. R. MURRAY and R. E. RICHARDS,Proc. Roy. Sot., A242, 455 (1957). 95. T. W. MARSHALLand J. A. POP& Mol. Phys., 3, 339 (1960). 96. H. S. Gurowsrcp, J. Chem. Phys., 31, 1683 (1959). 97.0. V. D. Traps, J. Amer. Gem. kc., 79,5585 (1957). 98. G. V. D. TIERS,J. Gem. Phys., 29,963 (1958). 99. N. F. RAMSEY, Phys. Rev., 87, 1075 (1952). 100. T. W. w MoI. Phys., 4, 61 (l%l). 101. G. G. HAL.Land A. HAWSSON, Proc. Roy. Sot., A268,328 (1962). 102. G. G. HALL, A. and L. M. JA~KMAN,Tetrahedron., 19,.suppl. 2,101. (1963). 103. F. LONDON, J. Phys. RaaYum, 8,397 (1937). 104. E. HU-, Z. Phys., 70,204 (1931); 76, 628 (1932). 105. A. SW Molecular Orbital Theory For Organic Chemists, John W&y, New York (1961). 106. H. M. MCCONNELL and D. B. C-, J. Chem. Phys., 29,107 (1958). 107. H. M. MCCONNELL and C. H. HOLM,J. Chem..Phys., 27.314 (1957). 108. D. R. EATON, A. D. JOSEY,W. D. Parryps and R. E. BENSON,Mol. Phys., 5,407 (1962). 109. R S. Mnpra~ and L. Parr, Discuss. Far+ Sot., 34,88 (1962). 110. A. W. Gvastx~uaaa, Phys. Rev., 91,476 (1953). 111. A. ABRAQAM,Principles of Nuclear Magnetism, Oxford (1961). p. 333. 112 W. A. A~~artaoN, NMR and EPR Spcctroswpy, Per8amoh Press (1960). p. 268. 113. R E. RICHARDSand J. W. Wrrrra, Dtkxss. Far&y Sot., 34,% (1962). 114. R. E. Rrand J. W. WHITE,Proc. Roy. Sot. A269,287 (1%2). 115. R. E. Rxand J. W. Writ=. Proc. Roy. Sot., A 269, 301 (1%2). 116. H. M. MCCONNELL,J. Chem. Phys., 24, 632 (1956). 117. s. I. w-, T. R. TUTIZEand E. DEEIOER, J. Pbys. Chem., 61.28 (1957). 118. A. D. MCLXHLAN, Mol. Phys., 1, 233 (1958). 119. D. R. EATON and W. D. PHIUPS, A&mces in Magnetic Resonance, 1, 103 (1965); edited by J. S. WAUOH,Academic Press, New York. 120. J. G. Pow and J. H. STRANOE,Discuss. Faraday Sot., 34,30 (1962). 121. M. Band D. M. GRANT,Advances in Magnetic Resonance, 1,149 (1965); edited by J, S. WAUGH, Academic Press, New York.

CHAPTER

4

THE CALCULATION OF SHIELDING CONSTANTS OF NUCLEI IN MOLECULES THE shielding constant of a nucleus in a molecule depends upon the form of the electronic distribution and can be calculated provided that the form of the electronic ground state wavefunction for the molecule is known. In practice a satisfactory calculation, either by a perturbational or variational method, is possible only for the hydrogen nucleus in simple molecules. For hydrogen nuclei in more complex systems, and for other nuclei even in simple molecules attempts have been made to express the shielding constant for a particular nucleus as a sum of contributions from the different atoms and bonds in the molecule. In this way the variations in the values of the shielding constants in related series of molecules can be predicted, and related to other molecular properties, such as the dipole moment, the electronegativity of a substituent group, and the magnetic anisotropy of the bonds. In this chapter the success of calculations of shielding constants of nuclei in molecules will be reviewed; some calculations of shielding constants in atoms have been reported but will not be discussed here”, 2. 3). 4.1 THE HYDROGENMOLECULE Almost all methods of calculating shielding constants (a) have been applied to the hydrogen molecule. In many cases the calculation of an2 has been used as a test of the validity of both the calculational procedure and the form of the wavefunctions before proceeding to calculations of c for more complex molecules. The hydrogen molecular system provides the best case not only because of its electronic simplicity, but also because a semi-empirical value of a,, can be obtained and compared with the calculated values. This arises because, as Ramsey has shown, for linear molecules the paramagnetic term uDsramay be obtained from the spin-rotational magnetic interaction constant (see Section 3.53). For a linear diatomic molecule equation (3.46) becomes e2

O=G\O

/

(4.1)

where R is the internuclear distance and ,u’ is the reduced mass of the molecule. The quantity @H,/MJ has been obtained experimentally from molecular beam measurementsC4) and is 13.66 f O-20 gauss, and by substituting for the other molecular constants of the hydrogen molecule the second term in equa120

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

121

tion (4.1) gives the valuet5) of e = - 056 + 0.01 x 10-5. This value of OF is for a rigid molecule and, as a further refinement, NeweW) has found the value of e averaged over the zero-point vibrations of the molecule to be - 055 x 10-5. The first term in equation (4.1), the diamagnetic term e, can be calculated from a knowledge of the ground state wavefunction of the electrons. Newell used a wavefunction of Nordsieck”) and obtained e = 3.21 x 10-5, hence the total value of on, is %

= (3.21 - O-55) x 1O-5 = 2.66 x 10-5.

This value of uH1is the best available and can be used to obtain semi-empirical values of shielding constants for hydrogen nuclei in other molecules if their chemical shifts relative to molecular hydrogen are known. In most calculations of shielding constants, the result is for the rigid molecule at the equilibrium internuclear separation, and hence should be compared with the analogous value for hydrogen of 2.68 x 10-5.(5**) The main difficulty in calculating on, from the equation developed by Ramsey (see Section 3.5) lies in the complexity of the paramagnetic term c. To evaluate a”? exactly requires the evaluation of a summation over all the excited states of the molecule, including continuum states. This is a virtually impossible task and hence the term G can be obtained only approximately. One method of calculating e introduces an average energy A E into equation (3.29) to give the equation (3.31) which involves only ground state wavefunctions. However, this simply transfers the difSculty to that of estimating A E since the only way of evaluating A E is by carrying out the summation in equation (3.29). Snyder and Parr(g) have emphasised the dangers encountered in the estimation of A E, particularly those arising from the neglect of the continuum states. The relative magnitudes of the two terms e and c depend on the origin of the coordinate system in which the vector potential A is defined, that is on the gauge of A. When the origin for A is taken at the nucleus, the semi-empirical calculations of Ramsey show that c is - 0.56 x 10-5; however, it is possible to choose the gauge of A so as to minimise the value of G. The total value of oHz is gauge invariant only when the wavefunction is expanded in terms of a complete set, and this condition is not fulfilled when oH1is calculated by secondorder perturbation theory. Hameka(‘O) has suggested that this difficulty can be overcome by using gauge invariant atomic orbitals (G.I.A.O.). If A,, is the value of the vector potential (see Section 3.5) at the nuclear position R, of the atomic orbital QI,,,then the G.I.A.O. t is given by

and the molecular orbital y,,, is formed from the linear combination atomic orbitals xn; that is

of the’

122

HIGH

RESOLUTION

NMR

SPECTROSCOPY

The calculation of uHs with G.I.A.O. functions based on the functions of Coulson(“), Wang(r2) and RoserP3) are(‘O) UHa

x 105

I

Function ~~

Although Hameka’s G.I.A.O. method still requires the use of an estimated average energy, it reduces the effect of an inaccurate estimate of this quantity by reducing the value of G. Ito (r4) has calculated cnl by means of equation (3.36), which requires neither an average energy nor a summation over excited states. Equation (3.36) with the CoulsorPr) molecular-orbital function for v. gives e = 3.22 x 10-5, c = -055 x lO-5, that is crH1= 2.67 x lO-S which is in almost exact agreement with the value found by Ramseyc5). Unfortnnately, the application of equation (3.36) is limited to molecules which are either highly symmetrical or where the constituent atoms have electron clouds which overlap either to a large extent, or very little(r5). Variational calculations of uHzhave the advantage over the calculations using the perturbation method in that a summation over the excited states of the molecule is not required. The variation method needs only a function which can be adjusted to give the minimum energy value, and the majority of calculations of shielding coefficients, both for the hydrogen molecule and for other molecules, employ a variational method. The best results for the hydrogen molecule have been obtained with a variational function similar to that proposed by Tillieu and Guy(16) for calculations of diamagnetic susceptibilities. In its simplest form the function is Y =Yo(l +g*H) (4.2) TABLE4.1 VARLUTONALCALCIJLATION(~‘)OF 0x2 WITH DIFFERENT FOR= OF ‘po @.”

‘PO

V.B. V.B. V.B. V.B. V.B. M.O. M.O. M.O. M.O.

Heitler-London Waw Weinbaum Rosen Hiichfelder-Linnet Coulson (unshielded) Coulson (shielded) Wallis (configuration interaction) Wallis (limited ionic)

059

u x 105

3.03 2.73 2.77 2.82 2.85 2.47 2.85

2.45 3.40 346 346 3.51 3.01 3.54

- 0.75 - 0.80 -090 - 0.84 - 0.57 - 0.82

244 2.46 247 2.45 2.53 2.27 2.55

241

3.49 3.55

- 0.78 - 0.98

2.53 2.33

268

-

where g is a variational parameter whose value is determined by minimising the totalenergy of the system. For the hydrogen molecule there aremany different forms of y. which have been designed to minimise the dissociation energy of

THE

CALCULATION

OF SHIELDING

CONSTANTS

123

IN MOLECULES

the molecule; Das and Bersohn (“‘have used a number of valence bond andmolecular orbital functions with the variational function given by equation (4.2) to calculate oHI. Their results, shown-in Table 4.1, indicate that the best M.O. and V.B. functions give almost identical results (compare the results from the Hirschfelder-Linnet function with those from the Coulson shielded function). The Hirschfelder-Linnet function is(’ a) 5

= a(l) 6(2)[1 + a Z2 G*(l) %(2) + YJl)Y*(2))

+ mw

+ B Z2G(l)

%ml

Cl+ ~~2{~*(2hu) + Y,4(2)Ya(l)) + B Z2 r*(2)%(1)]

+ Y [c(l) a(2) + b(l) W2)l.

(4.3)

a(1) and b(1) are 1s atomic functions with effective nuclear charge Z centred on nuclei A and B. The Rosen (13) function is derived from equation (4.3) by setting OL= y = 0; 1y = /? = 0 gives the Weinbaum(lg) function; putting u = fi = y = 0 gives the Wang functiorP2), and if Z = 1 also, the result is the Heitler-London function(20). The molecular orbital functions may similarly be derived from the Wallis function(21), fp - = u*(l) P(2) + P(1) aA(2) + P(l) uB(2) + uB(l) P(2) N + A[&(l) aB(2) + P(1) P(2) + aB(1) ti(2) + P(1) P(2)]

(4.4)

where tP and bB are 1s atomic functions with effective nuclear charges 2, and 2, centred on nuclei A and B. With I = 1 the function is the Wallio open shell function, that is with configuration interaction; if rZis chosen so as to minimise the binding energy, the function is known as the Wallis limited ionic function. Coulson’P) shielded molecular orbital function results by putting 1~ 1 and z, = Z,, whereas Coulson’s unshielded function has 2, = 2, = 1 P 1. The variational function in equation (4.2) can be made more flexible by adding a term proportional to cc, the nuclear moment w = ~o(l + g.H

+f.~).

(4.5)

Applying equation (4.5) to a calculation of cHI, Stephen(22) (see Section 3.6) obtained slightly higher results than those of Das and BersohrP7), although only one result of Stephen, that obtained with Weinbaum’s function, is strictly comparable (see Table 4.2). It is not certain that the added complexity of the calculation is just&d by the improvement in the results. TABLE 4.2 Vam.cnoN& Function Valence bond Molecular Weinbaum

orbital

CALcu~nON

I

2 1-O l-193 1-o l-193 l-193

OF an2 BY &-mmdz2’

I

ux105 2-41 2.69 2.39 264 291

124

HIGH

RESOLUTION

NMR

SPECTROSCOPY

Zis again the effective nuclear charge, and the value of 1.193 is the value obtained by minimising the binding energy of the molecule. Both the valence bond and the molecular orbital function give a value of a,,which is in good agreement with that of Ramsey (2.68 x 1O-5)(s). It is interesting to note the success of Stephen’s variational procedure when used with a simple molecular orbital function. The function is one suggested by FixmaiP3’ and applied in the calculation of a for the hydrogen molecule and some simple hydrides. The function has the form

Ye) = wYJ;w + (1 - YMiw11’2 where am is an atomic orbital centred on the adjacent atom and v&k) is a hydrogen atomic orbital. For the hydrogen molecule itself the function is w(1) = 2-1’2 [w:(l) +

wm

(4.6)

where (4.7) and 2 is the effective nuclear charge, and r,, is the distance of the electron (1) from the 4 nucleus. 2 is a variational parameter and is determined to be 1.13 by minimising the total energy; with this value of 2 the shielding constant for the hydrogen molecnle is 2.74 x 1O-5. The good agreement shown between the shielding constant obtained with this simple [L.C.(A.0.)2]1 I2 function and that found by Ramsey illustrates that the form of the variational function is a more important factor in calculating o than the form of the ground state wavefunction. This is amply illustrated by comparing Fixman’s calculated value of o = 2.74 x lO-5 with that obtained by Ishiguro and Koida, o = 2.756 x 10-5*(24) using the 13-term James-Coolidge wavefunction(25). 4.2 SIMPLE HYDRIDES 4.2. I Halogen Hydrides The series of diatomic molecules HX, where X is a halogen, is the next step in complexity from molecular hydrogen. One might expect that as the electronegativity of the X atom increases, the shielding of the hydrogen nucleus would decrease owing to the progressive decrease in electron density associated with it. Thus it would be expected that the hydrogen nucleus in the molecule HF would be less shielded than that in HCI, which in turn would be less shielded than HBr etc. In fact, this is‘what is found experimentally(26), the hydrogen resonance in H F being at lower applied magnetic fields than in HI. Semi-empirical values of the shielding constants of the hydrogen nucleus in the molecule HX can be obtained by adding the chemical shift measured from molecular hydrogen to Ramsey’s value of aHz. The chemical shifts of the molecules HX relative to

THE

CALCULATlON

OF SHIELDING

CONSTANTS

1N MOLECULES

125

molecular hydrogen have not been measured directly, but the shifts of the gaseous samples relative to gaseous methane have been measured and are(26) 6

hi.

5

- 41, H CH4

X

F

Cl

Br

I

6 x 105

- 0.25

OGt

044

1.33

These values may be regarded as those for the isolated molecules, but shifts measured in the liquid phase are considerably influenced by intermolecular interactions, particularly in HF where a liquid-gas chemical shift difference of 6.65 ppm has been observed(26). The absolute shielding values may now be obtained by comparison with the value for methane of 3.04 x 10m5,to give X

F

Cl

Br

I

unx x 105

2.79

3.08

348

4.37

The three reported calculations of shielding constants for the halogen hydrides have all assumed a close relationship betweenon x and the charge distribution in the molecule. The calculations of Hamekac2’) and Fixmar~(~~)have been the most successful and both include the dipole moment of the molecule as an integral part of their wavefunctions. Hamekat2’) used the perturbation method of Ramsey (see Section 3.5), hence uHX is split into the two opposing terms.0~ and a=, whose relative magnitude is determined by the choice in origin of the vector potential A. With the gauge invariant atomic orbit& discussed in the calculation of oH1, Pm and G are invariant under a gauge transformation and the origin of the coordinate system may be chosen differently in each calculation. Thus the hydrogen nucleus is used as the origin for those electrons centred on the hydrogen nucleus and also for the bonding electrons, whereas the halogen nucleus serves as the origin for the electrons centred on the halogen nucleus. Hamekac2’) assumes a close relationship between a, x and the part of the electronic charge centred on the hydrogen nucleus, and also on the overlap charge between this nucleus and the halogen nucleus. The wavefunction is written as a product of one-electron functions, each of which is either an atomic orbital or a linear combination of atomic orbitals. The wavefunction also contains a variational parameter A which is adjusted to fit the experimental value of the electric dipole moment of the molecule. If the halogen atomic orbitals are sp3 tetrahedral hybrids, the ground state wavefunction for the HF molecule is VIZ% 1+(1)1+(2) t~F(4) I~F@) t2~(6) bFt7) ~JF@) ~F(g)~F@) where and

& = (1 + 2&d, r4F

=

aF(2r,)

+

+ Af;)-1’2 (s” + & f4F) bF(2PrF).

is the charge transfer parameter, a variational parameter whose magnitude is determined from the dipole moment pm. AF is the overlap integral between

jlF

126

HIGH

RESOLUTION

NMR

SPECTROSCOPY

is a sp3 hybridised atomic orbital, and aF and bF are hybridiand t4F, flF sation constants, which in this case are + and +1/3 respectively. The calculation of a= with equation (3.31) requires a value of A E and in this case only d E for the first excited state was used. This approximation may be considerably in error, but is not so important since the magnitude of a= is small compared with ad&. Table 4.3 shows the magnitudes of O$$, a&? and uHx calculated by Hamekat2’). 8,

TABLE 4.3 CALCULATED VALUES OF SHIELDINGCONSTNTIS FOR THE HALDGEN HYDRDJS””

An early application variation function(28),

of variation theory to the calculation of ai.,, used a Y=“Yo+B~zwo

where OLand p are variational parameters and S?‘: is the Hamiltonian,

xz=

~Cm:,. 2Mc k

In contrast with Hameka’s treatment, the halogen atomic orbitals were considered to be unhybridised with an effective 2 = 1.19. The calculation was performed for a purely covalent molecule and a purely ionic one, and the shielding constant determined from the ionic character of the bond by assuming a linear relationship between the two (2g). The results (shown in Table 4.4) are in reasonable agreement with experimental values except for the HF molecule. TABLE~.~ VARIATIONALCALCULATIONOF SHIELDINGCONSTANTSIN HX MOLECULFS(~~) X F Cl Br I

Covalent

Ionic

Percentage Ionic character

4.40 4.82 4.94 4.86

0.39 0.39 O-81 0.76

85 40 31 21

ax

105

o-99 3-05 3.66 4.00

The large discrepancy here may be caused by an inaccurate estimate of ionic character, but may also result from the choice of an inadequate variational function(30).

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN MOLECULES

127

Better results were obtained using the variational function of Stephen. Fixman(23) has used Stephen’s approach with a wavefunction similar to that employed in the calculation of aHI. YHX(M = [r V:(k) + (1 - r)

Ysw’2

and the total wavefunction is a product of the k functions ; y is determined from the electric dipole moment of the molecule. The calculation has been performed for hybridised and unhybridised halogen atomic orbitals and also with effective nuclear charges on the hydrogen nucleus of 1-Oand 1.2. Further, the calculations were made with two choices for the dipole moments of HF and HI. For H F the value ‘of 2GlD (Debyes) was used in order to compare the result with that of Hameka, as well as a value of 1.74 D(‘r). For HI the values used were 0*38D(32) and - 0*38D(29). The results are shown in Table 4.5. TABLE4.5 VARIATIONAL C,+~XJLATION

OF C7m BY hCMd23’

Hybridised halogen

Unhybridiscd

The results in Table 4.5 show Srstly (column 3) that the method compares well with that of Hameka, and secondly that there is little support for a hybridised halogen atom (column 4) except in the case of iodine. An estimate of the percentage s character of the H-X bond can be made by comparing the results of columns 4 and 6 with the experimental results. Thus the following values are obtained. Percentage s character 6 (/.J = 1.74) HF HCl

5

HBr

8

HI

25 (ru = 0.38)

4.2.2. Group VI HyaYides The experimental values of the hydrogen nuclear shielding constants in the molecules H2X, where X is oxygen, sulphur or selenium again follow the expected correlation with the electronegativity of X, thus the hydrogen nucleus in water is less shielded than that in hydrogen selenide. The experimental values are obtained from the chemical shifts of gaseous samples measured from a methane gas reference(26), except in the case of H2Se where only a measurement

128

HIGH

RESOLUTlON

NMR

SPECTROSCOPY

the liquid sample is avaiiable (33’. The association shifts (see Section 9.4) in these compounds vary from 4.58 ppm for water, 150 ppm for H,S to an estimated value of 1.0 ppm for H$e. Although the series H#,H$, H$e shows the expected trend with electronegativity it is noticeable that the differences in dHzx are much smaller than in the halogen hydride series. Figure 4.1 shows a plot of CHx and bH2x against the eleetronegativity difference of H-X. The much greater change in flHx with elecon

t 0.4

1.2

0.8

Electronegativity

1.6

difference

,3

(fi -Xl

FIG. 4.1 The chemical shift of the hydrogen nucleus in H2X and HX molecules versus the ektronegativity @atiag(34’) difference between H and X

tronegativity than in oHIX is most readily interpreted in terms of induced local currents and a discussion of this will be deferred unti1 Section 4.3. TABLE4.6 EXPERIMENTALSHIELDXNGCONSTANTS FOR GROUP VI HYXXUD~ Hz0

u x 10’ ]

2.98

H,Se (liquid)

H2S

1

3.03

/

3.38

The only complete calculations for uHzx are those of FixmarP) using the simple molecular orbital function discussed earlier. In this case yx is a hybrid orbital of the form yx = (1 + AZ)-I’Z(s - J?&) where pz is centred on X and points away from the hydrogen nucleus. The calculation is divided into the contributions to uulx of electrons centred on

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

129

H and on X, and of non-local&d electrons. The latter contribution arises from the n-electrons in the non-bonding p-orbital on X perpendicular to the plane of the molecule, and is designated a,. The results of the calculation are shown in Table 4.7 and are in reasonable agreement with experimental values, except in the case of H&z. Although part of the discrepancy can be attributed to the effect of hydrogen-bonding in the liquid, most of it is real. TABLE4.7 CALCULATED VALUESOF u X 0 S Se

FOR

MOLBCUUSH2 X

t7H x 105

ux x 10s

a* x 105

u x 105

1.99 2-32 2-36

O-78 O-52 o-45

0.02 O-08 O-08

2.79 2-92 289

A calculation by Das and Ghose(30) of the shielding constant of hydrogen nuclei in water gave results much lower than the experimental value (1.35 x 1O-s). In contrast to the results of Fixman, they find that a, is 029 x 1O-s, but Fixman’s rather low value for u, is supported by the evidence that the shift between ethylene (with “looser” n-electrons) and benzene is accountable by assuming that it arises from the small (or zero) value of a, in ethylene, and a large value in benzene. It is interesting to note that McConnell’s calculation for shielding by electrons in groups Jar removed from a nucleus when applied to these n-electrons gives un = 0.04 x 1O-5-in good agreement with the value found by Fixman even though equation (3.53) is of doubtful validity for electrons only one bond removed from the nucleus. 4.2.3 Simple Hydrocarbons In both the hydrides of group VI and group VII the shielding constants could be correlated with the electronegativity of the X atom, and one would expect the same to be true of simple hydrocarbons. The hydrogen atoms in the molecules CHI , C2Hd and CzHz show increasing acidic properties suggesting that the electron-withdrawing power of the carbon atom increases from methane to acetylene. The hydrogen shielding constants, however, do not decrease from methane to acetylene; in fact the ‘H resonance signal of acetylene lies between the signals for methane and ethylene. The experimental values of u for gaseous samples are(26) %I, = 3.04 x 10-S;

UczH, = 252 x 10-S;

uclHs = 290 x 10-S.

Both Stephen(22) and Fixman(23) have calculated the shielding constants of the hydrogen nuclei in these molecules by the same variation method. In both calculations, the shielding constant for a hydrogen nucleus in an “isolated” C-H bond is calculated, and then the contribution from the rest of the mole cule estimated. Stephen used an L.C.A.O. molecular orbital function

HIGH RESOLUTION

130

NMR SPECTROSCOPY

where yc is a carbon hybrid-orbital, that is, sp3, sp2 or sp for the three molecules CH,, , C2H4 and C2H2. il is a parameter which allows for the difference in electronegativity between C and H atoms. The effective nuclear charge on the hydrogen nucleus is taken as 1.0. The contribution to u from the electrons not centred on the C-H bond was calculated from equation (3.53) with diamagnetic susceptibilities calculated by Tillieu (3s). The results for the three molecules are(72) @CH,

=

2.65 x lo-“;

uczu, = 2.61 x lo-‘;

uc’CzHz = 2.74 x lo-”

which predict the C2H4 and C2H2 resonances in the correct order, but not that of the CH,. The main error in the calculation lies probably in the estimate of the contribution from the electrons not centred on the C-H bond, as equation (3.53) is valid only for electrons at large distances from the nucleus. Fixman(23) calculates uC+ withy as 05 (equivalent to 1= 1) but with 2 = 1.2. It is assumed that the contributions from the rest of the molecule are negligible in methane and ethylene, but that in acetylene the contribution from the arelectrons is equivalent to that from a pair of nonbonded px and p,. orbitals and is equal to 0.52 x 1O-s. The results are = 3.00 x 10-s; uc2uz = 3*53x 10-5. ocn. = 2.97 x 10-s; a,,, In comparing these with the experimental values, Fixman(23) adjusts the value of y to make the agreement exact. The values of y required are

Y

CH,

C2I-b

C2H2

0.473

0.613

0.662

which are in the order expected from relative acidities. 4.3 THE CALCULATION OF SHIELDING CONSTANTS BY THE INDUCED CURRENT MODEL The calculatiotis of shielding constants described so far have treated the molecular system as a whole, but for more complex systems this type of calculation

becomes virtually impossible, and attempts have been made to divide the shielding constant into contributions from electrons localised on atoms and in chemical bonds. One method of doing this is the induced current model for calculating u, in which the shielding constant cA of a nucleus A in a molecule is regarded as arising from induced electronic currents in the atoms and bonds comprising the molecule. The shielding constant 0, is written as

The term flu arises from induced diamagnetic currents on the atom A, and its magnitude depends upon the electron density around nucleus A, and hence upon the electronegativity of groups attached to A. UE is the contribution to CA of induced paramagnetic currents on A, and arises as a consequence of the mixing of the ground and excited electronic states by the applied magnetic

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131

field. Pople(36) has shown that an excited state contributes to the local paramagnetic current on an atom only if it corresponds to the transfer of an electron between p or d-orbitals, hence G is zero when the electrons localised on A are in pure s-states. a*,# is the contribution to uA arising from local induced currents on the atoms other than A in the molecule, and may be positive or negative depending on whether the induced currents on the atom B are diamagnetic or paramagnetic in character. The magnitude of o,,B depends solely on the nature of B and if the atom, or bond, B is at a distance RAB from A, which is large compared with the radius of A, it is given by aAB =

3R: N Kl - 3 cos2U An 0

%xX+ (1 - 3 COS~6,)%,~+ (1 - 3 COs~ez)%*z] (4.9)

where No is Avogadro’s number; xXx, xrr and %=Iare the three principal axes of the magnetic susceptibility tensor x of the atom or bond B; 8,, 13,,0, are the angles between the principal axes and the AB internuclear vector. If&% 8 %,r * Xst, and the y or x axis can be chosen such that RAB lies in the zy (or zx) plane, then equation (4.9) becomes QAB =

3N01R~ (2A%1- A%i-

A%l3ws2ez)

where AxI = %tr - xrY and Ax2 = xzz - x;*. If the group B is axially symmetric, then xXx = ty * xzz, and equation (4.9) becomes the equation (3.53) of Section (3.5.4), GAB= 3N01RL AXB (1 - 3 cos’k)

in which

(4.11)

Axe = &- xi and 8, is the angle between the anisotropy axis (the z-axis) and the internuclear vector R,,. The equations (4.9) to (4.11) show that uABis zero unless the magnetic susceptibility tensor xe is anisotropic, and for this reason oAB is sometimes referred to as the neighbour anisotropy effect. The last term in equation (4.8) arises from induced currents involving electrons which are not localised on any one atom or bond in the molecule, and is particularly important in aromatic hydrocarbons (see Section 4.5). Equation (4.8) is a good approximation to the value of o, only when the electrons localised on the atom A do not have any appreciable overlap with those on atom B (see Section 3.5.4), and hence is not strictly valid for calculating the shielding constant of small molecules such as the simple hydrides discussed in Sections 4.2.1 to 4.2.3. However, applying equation (4.8) to these hydrides to estimate the relative magnitudes of the local contributions to a,, does help in interpreting the observed variations in the values of the shielding constants, even though the magnitude of a, can be obtained more accurately by the direct methods. When A is a hydrogen nucleus the magnitude of a= is small compared with eU, because the p character of the hydrogen bonding-orbital is negligibly small. The term e” in these hydrides is zero, hence the shielding 6*

132

HIGH

RESOLUTION

NMR

SPECTROSCOPY

uH is the sum of the terms &A and aAn. If uz % aAB for a related series of molecules, then a, should be closely related to the electronegativity of the atom B, but when eAa is comparable in magnitude to flti the variation of err in the series will be more complex. For the halogen hydrides HX it has already been noted that a, decreases as the electronegativity of X increases, and it is tempting to conclude that u$$ is the dominant term in equation (4.8) for these molecules. That this is not the case is apparent when it is noted that the shielding constant of the hydrogen nucleus in the molecule HF is almost the same as that in water, and also that the shielding constants eH in the isoelectronic series CHI, NH3 , OH1, FH are practically identical”@.

constant

u x 10s

CH*

NH3

OH2

FH

3.04

3.05

2.98

2.78

The magnitudes of ci., for the other hydrogen halides are all greater than that for methane, whereas they would be expected to be smaller on the grounds of electronegativity. This anomaly can be explained by the magnitude and sign of o,.,&&. The diamagnetic susceptibility tensor x.x is axially symmetric about the direction of the H-X bond, hence the magnitude of u,x is given by equation (4; I 1). When the applied field lies along the H-X bond direction there is no hindrance to induced electron currents about this direction and xl1 is large and negative. When the field lies perpendicular to the H-X bond direction the electrons on X cannot circulate freely and xL is small. Thus Ax in equation (4.11) is negative, and since the angle ox is zero, the value of uHx is positive (the same sign as 03 and its magnitude increases as the size of the atom X increases. In the hydrogen halides both’ c&&and uxx increase as the electronegativity of X decreases, and this accounts for the high values of a, in this series. In the isoelectronic series CH4, NH3 , OHz , FH only FH is a linear moIecule with a large value of A xx and a maximum value of (l-3 cos2 0). The other molecules in the series are non-linear and give smaller values of GA,. Pople has estimated the magnitudes of bAB in these molecules by the point dipole approximation to be c3’) CH* NH3 OH, FH CAB x lo5

0’~

0.03

0.28

030

GABincreases along the series whereas fl= should decrease, thus the total change in u, is small. Other linear molecules will have large positive values of GAB, for example, acetylene. When the applied field lies along the C-C bond direction there is less hindrance to electron circulations on the carbon atoms than when the field is at right angles to this direction, hence d x is negative. The two other molecules in the series CH*, CzH4, C2H2 are non-linear and thus have smaller values ofeA,,. Pople(37) has calculated that in acetylene aABis of the order of 1.0 x 10q5 and is large enough to make the shielding constant Us greater than uH for ethylene.

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4.4 SHIEI.JXNGCONSTANTSANDTHEMAGNE~CANISOTR~PYOFCHEM~CALBONDS The divisionof the shielding constant into local contributions has more significance for larger molecules than for the simple hydrides. The equations (4.8) to (4.1 I) are particularly useful when in the same molecule two or more hydrogen nuclei have the same immediate environment, and hence have identical but differ in their relative orientation to some distant values of a& and s, group B in the molecule. The chemical shift between these nuclei can be calculated if the principal components of the magnetic susceptibility of group B, and the geometry of the molecule, are known, or conversely, the anisotropy of the magnetic susceptibility may be calculated provided that the chemical shift can be measured. Some examples of such calculations are given in Sections 4.4.1 to 4.4.3.

4.4.1 The Internal Chemical Shift in Amides The two hydrogen nuclei labelled 1 and 2 in the amide

are magnetically non-equivalent and at low temperatures give rise to two chemically shifted resonance peaks. At higher temperatures hindered rotation about the C-N bond with a frequency greater than the chemical shift difference causes the two peaks to coalesce (see section 9.5.2). ‘Now, the electronic environment immediately surrounding the hydrogen nuclei 1 and 2 in the two N-H bonds must be ahnost identical, so that the chemical shift d1_z must arise because the internuclear vectors Ror.r(1Jand Rou,,, and the corresponding RRH vectors are different in both magnitude and direction. The difference in the shielding constants Acrlz for the two hydrogen nuclei is given by the difference between the two terms a,,B in equation (4.8), where B refers to the electrons localised in the C = 0 and C-R bonds. For formamide, R is H and the C-H bond is axially symmetric, thus grrcrr is given by equation (4.11); the C= 0 bond is not axially symmetric and aH,c_o is given by equation (4.10). Acll is given by(jg) 1 (24x1 - A& - AXI 3 COS28~~) , + ~XCH(1 - 3 co&?,,) ArJl2 = R3Wl R& 3% 4

-Ax1 (24x1 Ax2 -[

1

3 COS28~2) + Axcr.l(l - 3 cosV&

Rt2

R3HZ

11

(4.12) whereA%, and Ax2refer to the C=O bond. The values ofAx and Ax2 are not known, hence equation (4.12) cannot be used to calculate Aai2. Narasimhan and Rogers(3g) have used equation (4.12) to calculate values of Ax1 and Ax2 by equating Acl2 to the observed chemical shift difference. They assume that the three principal components of x for the C = 0 bond do not change on going

134

HIGH

RESOLUTION

NMR SPECTROSCOPY

from one amide to another, nor upon N-methyl substitution, and therefore they use equation (4.12), together with a similar equation for the shift between the two methyl groups in N, Ndimethyl formamide, and the average value of the susceptibility

in order to calculate xXx, xYYand xzz for the C=O bond. There are two main difficulties in such a calculation. First, it is not possible to decide on the sign of the chemical shift difference Ao,~. Secondly, the values of Rol and R,,2 depend upon the location of the induced dipole along the C = 0 bond. The dipole is probably best located at the charge centroid of the bond, which depends on such properties as ionic character and hybridisation of the bond, and these are not known for C = 0 bonds in amides. Narasimhan and Rogers calculated the principal susceptibilities of the C = 0 bond for the two cases of do, negative and positive, and in each case the calculations were also carried out with the induced dipole at various positions along the C = 0 bond, starting at O-81 A from carbon and then moving it towards the oxygen atom. The calculated magnetic susceptibilities were then applied to the calculation of do in acetamide and dimethylacetamide, and it was found that da was not affected by the position of the dipole provided that the same choice of dipole position was made throughout. No serious error occurs even by locating the dipole on the oxygen atom. The calculated values of Au for acetamide and dimethylacetamide are shown in Table4.8. The two values of d xC_Hare those calculated by Tiieu and GuyQ6) with two different wavefunctions. Tau4.8 CALCULATEDINTERNAL Sm-rs IN ACETAMIDE ANDD IMETHYLACETAMIDE WITH ILITW OBTAINED FROM Fo RMAMIDEAND DIMETHYLFO-~~~) MACBIETIC SUSCEFTD Au x 10’ Ax=_~ x lo6 Ao(Acetamide

+ 0.26 + 0.49

Dimethylacctamide

+@26 fo.49

ve)

- 1.63 - 1.78 1.43 -1.53

Au ohs. Au(+ ve) + 3.10 + 3.24 + 2.31 + 2.41

f 221

In comparing da calculated with da observed, it is important to note that the measured shifts are for pure liquids and do not represent the values for the isolated molecules. Hydrogen bond shifts would be expected in these molecules, particularly in formamide and acetamide which show evidence of strong hydrogen bonding in the solid state c40). The calculation also assumes that the whole of the observed chemical shift arises from the neighbour anisotropy effect, whereas part of the shift in these polar molecules will arise from the electric field effect discussed by Buckingham (see Section 3.9).

THE CALCULATION

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135

4.4.2 The Calculation of Internal Chemical Shifts in Saturated Hy&ocarbons and the Magnetic Anisotropy of the C-C mtd C-H Bona3 Thesimplest saturated hydrocarbon molecule to show more than one resonance peak is propane, CHJCH2CHJ. The hydrogen resonance spectrum is that of an A2Bs system with a chemical shift between the two groups of hydrogen nuclei of 6 = 044ppm(Aa = 4.4 x 10-3. Now, for eachhydrogennucleus in propane the term (a”H;I+ G) in equation (48) depends on the charge density in the C-H bonds and can be calculated from a simple molecular orbital of the form Vc-Ii = N(% + 2 0 assuming all the C-H bonds to have the same degree of hybridisation. j! is a parameter which depends on the charge distribution in the C-H bond, and is assumed constant for each bond in propane as the molecule has a zero dipole moment. Thus uHu = (e + c) should be almost the same for each hydrogen nucleus in propane, and the observed chemical shift difference can be considered to arise from the difference in the terma,, for the methyl and methylene hydrogen nuclei(42). The group B refers to electrons locahsed in the C-C and C-H bonds, and since both these bonds are axially symmetric the two terms a,, c_c and a,, c_H can be calculated from equation (4.11). From the dimensions of the molecule, and taking into account the rapid re-orientation of the methyl groups, Narasimhan and Rogers(42) found that the differencf in the shielding constants of the methyl and methylene hydrogen nuclei is Aa I 0106A~=-=

- 0.124Af-H

(4.13)

in which Aa = cc,, - cc&. Substituting for Ax=-= and AxCeH values calculated by Tillieu(43) of Axcwc = 1.21 x 10m6,andAfmH = 0.24or 1.50 x 10m6, gives for Au, do = 0.98 x lo-’ (A xc-” = 0.24 x 10-6) = -053

x lo-’

(Ax=-~ = 150 x 10-6).

Comparing these values of do with the observed chemical shift of 4.4 x lo-’ suggests that the term Aa HHis not zero, and in fact provides the major contribution to Au. A value for AcrHHin propane can be estimated either from the electronegativity - chemical shift correlation suggested for CHsCHzX compounds(44*45), or from the electronic charge distribution in propane as calculated bySandorfy(73). Both methods give similar values forda,,; the 8rst method gives 3.17 x IO-’ and the second 3.38 x lo-‘. With AaHH = 3.38 x lo-’ and AaH, = 0.98 x IO-‘, the calculated difference in shielding constants, Au, = 4.36 x lo-‘, in excellent agreement with the observed shift. If it is assumed that the observed chemical shift between the methyl and methylene hydrogen nuclei in propane can be attributed entirely to the effect of the magnetic anisotropy of the C-C and C-H bonds, then the value of Axe -c calculated from equation (4.13) is (4.3 to 5.6) x 10m6cm3 mole-‘.

136

HIGH

RESOLUTION

NMR SPECTROSCOPY

This value of dxc-c

is much larger than that calculated by Tillieu(43), but it is supported by other evidence, such as the chemical shift between the axial and equatorial hydrogen nuclei in cyclohexane (74). At room temperature the hydrogen spectrum of cyclohexane shows a single peak, but on lowering the temperature the spectrum(7s) is that of an AB system with a chemical shift difference of 4.6 x lo-‘. The chemical shift is that between hydrogen nuclei in the axial and equatorial positions in the chair form of the molecule

< The charge distribution and degree of hybridisation should be the same for all the C-H bonds, and the observed shift can be attributed to the anisotropy of the C-C and C-H bonds. The bonds, C1-C6, Cl-C2 and the equatorial C--H bonds at carbon atoms C2 and Cs have the same relative orientation to the hydrogen nuclei H, and H,, and thus do not contribute to the difference in shielding constants do,,, . Neglecting the effects of electrons more than there bonds removed from H,, and H, means that the principal contributions to do,, come from the magnetic anisotropy of the bonds C&&, CS-C6, and the axial C-H bonds at Cz and Cs. With rc_c = 1.54 A, rc_ H = I.10 A, assuming ‘tetrahedral angles, and locating the induced dipoles at the midpoint of the C-C bonds, and 0.77 A from the carbon atom in the C-H bonds, gives for do,,, da,,

= 01073 x @lx=-c - Llxc-H).

Equating da, with the observed shift in cyclohexane of 4.6 x lo-’ means that must be 4.2 x lO-‘j cm3 mole-‘. A similar calculation for (Ax~-~ - Af-H) the isobutane molecule(74) gives 4.15 x 1O-6 cm3 mole-’ and both these values are in agreement with that calculated from the internal chemical shift in propane. Musher(46) has calculated Ax c -c from the chemical shift between axial and equatorial hydrogen nuclei in a number of derivatives of cyclohexanol and decalol. He found A f x to be 5.0 + 0.6 x lO-‘j cm3 mole-‘, in excellent agreement with the values found in cyclohexane, propane and isobutane. However, in these molecules containing polar groups it is possible that the chemical shift will be strongly influenced by the electric field effect(76* “). 4.4.3 Chemical Shifts in Alkyd-X Compoun& and the Magnetic Anisotropy of the C-X

Bond

The change in the shielding constants of the hydrogen nuclei in the molecules CH3X and CH3CHzX as X changes depends on changes in both aHH and $,, c x. is linearly dependent on the electronegativity of X, whereas A&,, cx A~HH depends upon the magnetic anisotropy of the C-X bond, and upon the geometry of the molecule. It is possible to obtain an estimate of the relative magni-

THE CALCULATION

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137

tudes of eHHand a,, cx for both the methyl and methylene hydrogen nuclei h CH3X and CHjCHIX by following the change in the chemical shift with the electronegativity of X. Table 10.2 shows the hydrogen chemical shifts relative to methane gas (internal reference) of both CHs- and CHz-groups in a number of gaseous alkyl compounds(45), and in Figs. 10.2 and 10.4 the shifts are plotted against the electronegativities co’)of X. These figures show that a linear correlation of the chemical shift with electronegativity of X is found when X is H, C, N, 0 or F, but that when X is Cl, Br, I or S there is a deviation from the straight line. The magnitude of the deviations for the hydrogen nuclei in both methyl and methylene groups in the series CHIX and CH&HzX increases in the order Cl < Br < I, S, and are all negative. Both the magnitude and sign of do,, cx can be explained in terms of induced currents in the electrons localised in the C-X bond(45). Consider the molecule CH&H?X shown in Fig. 4.2, where X is a large atom. When the applied field, Ho, is along the

(b)

(0)

FIG. 4.2

Inducedmagneticfields

in CHaCHtX.

Spie#ckeand S&t&&(4’)

C-X bond direction there is free diamagnetic circulation of electrons about the bond, hence a large induced dipole opposing Ho. The induced magnetic fields at the methyl and methylene hydrogen nuclei are both in the same direr tion as H,,. When Ho is perpendicular to the C-X bond direction the electrons cannot circulate freely, and this can be represented by placing an induced dipole in the same direction as Ho, producing induced fields at the hydrogen nuclei which oppose Ho. Thus both for H,-, parallel and H,, perpendicular to the C-X bond direction there is a reduction in the shielding constant for both ocand p hydrogen nuclei. The magnitude of uH,cx is given by equation (4.11) and has been calculated by McConnelS3*)for the case X = halogen. He assumed Ax to be 20 per cent of the total susceptibility of the halogen negative ions, and locating the induced dipole on the X atom gives for au, =the values, + 01 x 10m6,f O-3x 10s6, f 0.3 x 10B6,f 0.4 x 1O-6for X = F, Cl, Br, I. Figure 10.2suggests values of eH,cx considerably greater than these; for example ou, cr is estimated from the graph to be - l-5 ppm(45).The error may arise as a result of locating the induced dipole on the halogen atom. An even more serious discrepancy between experimental and HRS

6s

138

HIGH

RESOLUTION

NMR

SPECTROSCOPY

calculated values of the shielding constant is apparent on examining the 13C chemical shifts in these compounds (45): the results are plotted against the electronegativity of X in Figs. 12. I5 and 12.16. Again the halogens show large deviations from the straight line graph, and this time the shielding constants of the cPC nucleus in CHJX and CH&H2X compounds are larger than expected; that is, cc,cx is positive, whilst for the /?J3C nucleus in CHJCHzX it is negative. This is consistent with the induced dipole model illustrated in Fig. 4.2, since the induced field at the a-carbon nucleus is in the opposite direction to that at either the p-carbon, or any of the hydrogen nuclei. If the deviations from the straight line graph (Fig. 12.15) are considered to arise only from magnetic anisotropic effects of the C-X bond then, in CHJ, the value of o,, ,-r would be 54 ppm. The /?-carbon nucleus in CH3CH2X and the hydrogen nuclei in CH3X are in almost the same position relative to the X atom, hence the magnitude of on, cx and oc, cx for these nuclei should be identical since they depend solely on the magnetic anisotropy of the C-X bond and the values of Bx and R in equation (4.11). In fact, for CH&H,I and CHjI the two values are respectively @-carbon of CH&H,I) UC,Cl = 18 ppm %, CI

=

1.5 ppm

(hydrogen in CHJ.)

suggesting that factors other than magnetic anisotropy are important in determining the chemical shifts in these molecules. The large unexplained contributions to the 13C shielding in compounds containing C-X bonds may arise from the effect on the shielding constant of permanent and induced electric dipole moments in the molecules. Buckingham has shown that the shielding constant of a nucleus in an X-Y bond is influenced by the field component E, along the bond and by EZ, the square of the total electric field at the nucleus, thus 0, = -AE,

- BE=.

(4.13a)

In addition to the electric field arising from permanent dipole moments in the molecule there is also a contribution to E2 from the Van der Waals interaction between atoms (see Sections 3.9 and 3.10). Time dependent dipole moments in an atom give rise to a non-zero value of the square of the electric field (E2) whose magnitude at a distance r is given approximately by

where a is the electron polarisability of the atom and I is the first ionisation potential. The values of A and B depend upon the nature of the atom and its environment. For hydrogen in hydrocarbons A is of the order of + 2 x lo-l2 and B is around + 1 x 10-18. For other nuclei A and B can be considerably greater, and whereas Bis always found to be positive, A may be of either sign. Values of A and B for i3C have not yet been determined, but values of 13C solvent shifts suggest that the electric field effect is an important factor in determining the magnitude of ‘“C shielding constants(92).

THE

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139

4.4.4 The lgF Chemical Shtjlt Diflerence in Perfruorocyciohexane The chemical shift difference between axial and equatorial hydrogen nuclei in cyclohexane is 0.46 ppm and can be explained in terms of the diamagnetic anisotropy of the C-C and C-H bonds. But the chemical shift da, in perfluorocyclohexane between lgF nuclei is f 18.2 ppm and this cannot be attributed entirely to the diamagnetic anisotropy of C-F and C-C bonds; in fact the contribution to Au, from this source can hardly be much greater than 0.46 ppm. Emsley (g3)has shown that this large value of da,,, can be explained in terms of the values of E, , E2 and (E2) at the axial and equatorial lgF nuclei which arise from the polar C-F bonds and the polarisabilities of the electrons in the C-F and C-C bonds. The values of E,, the electric field component in the direction of the C-F bond, and E2, the square of the total field at the lgF nucleus arising from permanentbond dipole moments,were calculatedassuming that each C-F bond has a dipole moment of - 1.4 D located at the centre of the bond. Perfluorocyclohexane was assumed to exist entirely in the chair form and the bond lengths were assumed to be C-F

= 1.32 A

C-C

= 15s

A

and all angles were assumed tetrahedral. (E2) was assumed to arise from the electrons in the C-F and C-C bonds, and in calculating its value from equa4 tion (4.14) the following values of oLand I were used (**), UC-F

= 0.683 x lo-‘* cm3

ac+ = 058 x 1O-24 cm3

I c_F = 28.5 x lo-l2 ergs Ic_c = 18.03 x SO-l2 ergs The value of r was taken as the distance between the lgF nucleus and the midpoint of the bond concerned. The values of A E,, A E2 and d <E2>obtained are A& = E, (axial) - E, (equatorial) = -0.0484 x lo6 e.s.u.

AE2 = E2 (axial) - E2 (equatorial) = 0.0933 x 1012e.s.u.

.

A(E2) = (E”) (axial) - (E2) (equatorial) = O-3033x 1012e.s.u. Substituting in equation (4.13 a) gives

do;, = (-0.0484

x 106) A - (O-4266 x 1012) B.

(4.15)

Experiments on the pressure dependence of the chemical shift of 1gF in fluorine containing compounds suggest that A is negative and of the order of - 10 x 10-12, and that B is positive and of the order of (l&40) x 10-l*. If A and B are given the values found for CHFJ oftg4)- 10 x IO-l2 and + 15 x lo-l8 respectively, then equation (4.15) gives Au, = - 7.88 ppm which leaves a shift of - 10.3 ppm unaccounted for. If it is assumed that the whole of the ob6&*

140

HIGH

RESOLUTION

NMR

SPECTROSCOPY

served shift of - 18.2 ppm can be attributed to the diamagnetic anisotropy and the electric fields present then with a value of - 10 x lO-12 for A, equation (4.13a) gives for B the value of + 44.9 x lo-I*, which is not an unreasonable value. 4.5 CHEMICALSHIFTSOF HYDROGEN

Nucm

IN AROMATIC MOLECULES

The main feature of the shielding constant of a hydrogen nucleus in benzene is that it is considerably different from the shielding constant of ethylenic hydrogen nuclei, the ‘H resonance signal in benzene being 15 ppm to low field of that in ethylene. The contribution to the shielding constant oH from electron circulations localised on the hydrogen atom depends upon the state of hybridisation of the C-H bond and therefore should be the same in both cases. The effect on a, of the anisotropy of the magnetic susceptibility of the C-C and C-H bonds will be different in the two molecules, but this contribution to d Q~is small and can be largely eliminated by comparing benzene ring hydrogen nuclei with the ethylenic hydrogen nuclei in cyclohexadiene- 1 : 3 where the chemical shift difference is 1.48 ppm. This chemical shift difference has been interpreted as arising from the delocalised n-electrons of the benzene ring which, in the presence of a magnetic field along the hexagonal axis, are free to move in a closed loop (see Section 3.8). In benzene, the induced current gives rise to a magnetic field which opposes the applied field inside the current loop and enhances the applied field outside the loop. This is illustrated by the chemical shifts of the methylene groups in the 1&polymethylene benzenes I

I

I

I

In the lower members of this series, the CHI groups are held in positions close to the centre of the aromatic ring where the induced ring currents cause increased shielding; thus, for 1,4-polymethylene benzenes where n = 10 and 12, the central CH2 groups resonate 6.50 ppm and 6.30 ppm respectively to the high field side of benzene, whilst in 1,2_hexamethylene benzene, the CH2 groups farthest from the benzene ring resonate at 5.8 ppm to high field of benzene, a value characteristic of the methylene groups in saturated cyclic hydrocarbons. The c&HZ groups in the 1,4_polymethylene benzenes resonate in the same region as the CHI group in ethyl benzene which suggests that the induced ring currents deshield both groups to the same extent. The magnitude of the ring current shift in benzene can be calculated with the aid of a purely classical model (see Section 3.8) where the n-electrons are considered to move in two circular loops above and below the plane of the carbon atoms, with a radius equal to the C-C bond length (1.39 A) and separated by 1.28 A. Using this value for the current ring separation and equation (3.83), Johnson and Bovey(4*) have calculated the ring current chemical shift contribution for a hydrogen nucleus at any position relative to the benzene ring; their results are shown in Appendix B.

THE

CALCULATION

OF

SHIELDING

CONSTANTS

IN

MOLECULES

141

4.5. I Polycyclic Aromatic Hydrocarbons In the fist attempt to calculate the effect of ring currents in polycyclic aromatic hydrocarbons, it was -assumed that the current flowing in each hexagon is equal to that in benzene(4g). The effect ofthis current was then found by means of Pople’s point dipole calculation (see Section 3.8). Because the spectra of these compounds were obtained from neat liquid samples, the observed chemical shifts are not those of the isolated molecules and hence they will be considerably Muenced by intermolecular interactions. Some of the compounds were re-examined by Jonathan, Gordon and Dailey’so’as dilute solutions in Ccl4 or CS2 and the ring current shifts relative to benzene obtained. The same authors have calculated the ring current shifts in these compounds by lirst calculating the current intensity in each hexagon by means of the molecular orbital calculation of Pople (see Section 3.8) and then using this current intensity in conjunction with Johnson and Bovey’s tables (Appendix B) to obtain the ring current shift of a particular hydrogen nucleus. Table 4.9 gives the current intensities found from the molecular orbital calculation and shows that the assumption of equal currents in each hexagon is a poor one when the number of rings increases. The experimental and calculated ring current shifts are shown in Table 4.10; wherever possible the results are those of Jonathan et uZ.(~O)otherwise they are from the earlier work of Bernstein, Schneider and Pople(4Q). It can. be seen from Table 4.10 that the calculated shifts for the polycyclic aromatic molecules are in the cotiect order, but that most are too large by about O-5 ppm. It is interesting to note that agreement between the calculated and experimental shifts is best for the positions where it is known that steric factors are important; thus in the 4,5 positions in phenanthrene osl,, - o”&, is only - 0.06 ppm. In this molecule the steric interaction between the hydrogen atoms in positions 4 and 5 reduces the shift because of a strong Van der Waals interaction and also because the molecule is not planar. Reidt5’) has calculated that the shift arising from the Van der Waals interaction in phenanthrene is of the order of 0.5 ppm. Differences between theory and experiment are the largest for hydrogen nuclei in rings having two or more benzene rings attached to them, and also for nuclei which are not sterically hindered; thus 9,lO positions in anthracene have u_lC - u,,~ of + 0.79 ppm, and the hydrogen nuclei in coronene havea,,,, - Q,, = + 1.17 ppm. One method of improving the agreement between experiment and theory would be to alter the separation of the current loops, p, in benzene. Johnson and Bovey(4*) varied the value of p to obtain exact agreement of the observed and calculated shifts between the benzene hydrogen nuclei and the ethylenic hydrogen nuclei in cyclohexadiene- 1 : 3. Jonathan et aZ.(50)haveinvestigated an alternative procedure of adjusting p so as to obtain exact agreement between benzene and naphthalene. Figure 4.3 shows the ring current shift for benzene and for the LXand /I hydrogen nuclei in napht+alene as functions of the ring current separation ; it is seen that in order to obtain agreement for the LXhydrogen nucleus the value of p required is 2.09 A and with this value the cal-

142

HIGH

RESOLUTION

NMR

SPECTROSCOPY

TABLE~.~ CALCULATED RATIO OF Ruw CURRENT INTENSITIESCOMPARED WITH BSNZNE(~O’

-u

/CA &

current IllteLlsity z

A

1mo

A

l-093

A B

1.085 l-280

A

B

1.329 0.964

A B

1,133 0.975

A B

1.111 o-747

A B

o-979

A B

l-460 1.038

I,AU

ro A A

fB

0

O-247

'J

THE

CALCULATION

OF

SHIELDING

CONSTANTS

IN

143

MOLECULES

T~~m4.10 Rma CURRENTSmm IN Po~~cycr~c AROMA~CH~DMXAREONRELATMiTo BENZENE

a”(expG

Molecule

1

03 ti

10

B&ma

Ppm

O-54

o-19

O-85 0.42

1 2 9,lO

O-64 o-12 l-04

l-14 0.55 1.83

50

1 2 3

0.72 0.89 o-79

l-12 l-55 l-40

50

O-44 144 @16

50

0:3

O-99 1.38 0.61 O-56 O-99

i-29 O-34

l-34 091

0.82 O-14 O-33

1.03 O-50 O-84

1 2

2

a”@w

Ppm

50

1

63 0 3

3 2 r 6 7

9.10 4, 5 3, 6 2. 7 1. 8

50

50

I

423 /J

l-57

2.74

50

144

HIGH

RESOLUTION

NMR

SPECTROSCOPY

Table 4.10 (continued) Molecule

13 12 11

10

5

4

1

0 ci

6

pm

3 4 5 6 7 8 9 10 11 12 13 I 14

6 2 3 4 5 2

fJJ=(expt.)

HY~=w=

0.57

O-57 I.95 157 0.57 1.57

I

1.20

I

o-37

1 , 2 3 4 *

220 0.47

z,

oc1 (cak.)

mm o-93 l-03 @84 0.35 0.37 o-91 1.82 1.34 o-79 O-36 0.43 1.45 l-35 1.45 0.42 O-36 0.80 O-98 261 O-90 O-40 ON 0.95 o-99

Reference 49

49

49

culated shift for benzene is only 0.5 ppm. Similarly, agreement between c& and ccalc for the @hydrogen nucleus requires p = 2.78 A, and this would reduce u,~,, for benzene to almost zero. It is possible that the large values of a,,, - u~,,~~ mean that the observed shifts between the hydrogen nuclei in polycyclic aromatic compounds and benzene are not entirely caused by differences in ring current shifts. Thus to obtain an experimental value for the ring current shift in naphthalene would require a comparison with the shifts in the 1,2-dihydronaphthalene ovl”

a HZ

Jonathan et aZ.‘50)have shown that only about 0.05 ppm can be attributed to the different “neighbour-anisotropy” shifts in fused aromatic systems. An improved calculation of ring currents has been made by Hall(s2-54) using a self-consistent field (S.C.F.) molecular oFbita1 theory for aromatic systems, instead of the Htickel model adopted by Pople and McWeeney. The S.C.F. theory gives better results for alternant hydrocarbons, but still leaves a large

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

145

difierence between ual, and a,,. This is seen in Table 4.11 which shows the results of calculations on naphthalene by the Htickel method and by the S.C.F. method(s2).

Cur&

loop

seporotion

m benzene

tdii

units

Fm. 4.3 Them&al ring current shifts of csand fi hydrogen meld of naphthakm with respect to bermme and of absolute ring current shiik of bmmnc. Jonathan, Gordon and Dailoycso)

TABLE~.~~R~NGC~RREN~SHIF~~N~RELA~ BY ‘I’kiEHUCKELANDS.C.F. ~BLWZENEC ALCULA-KD METHODS

Hydroeen 1 2

Hiickel

S.C.F.

0.78 @44

0.72 O-37

Expt.““’ O-54 0.19

4.5.2 Heterocyclic Aromatic Molecules All the molecules discussed so far in which ring currents are important have been altemant hydrocarbons where the n-electron charge densities at the carbon atoms are equal. The introduction of a hetero-atom into the ring introduces a polarity in the carbon u-bonds and the n-electron charge densities at the carbon atoms are no longer equal to one another. In these circumstances the S.C.F.

HIGH

146

RESOLUTION

NMR

SPECTROSCOPY

calculations of Hall et uZ,(~~)give very different results from the Hiickel theory, which tends to overestimate the ring currents. Table 4.12 compares the Htickel and S.C.F. calculations of ring currents in the four molecules 0

P ObO

Q

I (Xi

Q La‘0

LHa

I

b

II

L Iv

JII

En this

study, the rings were assumed to be regular polygons with sides equal to 140 A, the C-H bond was taken to be 1.09 A and the resonance integrals were considered to be equal to the value for benzene. Since the values of the coulomb, OL,integrals involving the 2p n atomic orbitals of the nitrogen and Tmm4.12

CALCULATED

Parameters t

_-

0.19 0.49 0.19 0.19 0.49 0.79 0.19 0.49 0.79

Comu-rxo~s HETJsocYcLFs (I-IV)

@nnw

CURRENT

I

0)

-I

q-r1.02 1.02 1.02 1.52 I.52 1.52 2.02 2.02 2.02 1

-r

Rrm

T

00

TO BBNZENE) IN m

(III)

1Hiickel 1 S.C.F. 1Hiickel

S.C.F.

Hiickel

S.C.F.

Hiickel

0.33 0*49(‘s’ 0*62”3’ O-24 0.37”” O&11,

0.53 0.65 0.73 O-44 0.56 0.64

0.18 0.37

O-82

0.56

0.83

074

O-42

0.75

0.34

0.16 0.28’12’ 0.38”“’

0.36 0.48 @56

0.65

0.31

0.67

O-26

0.52 0.64 0.71 0.42 0.55 0.63 0.34 0.47 0.55

1

1 S.C.F. .

044

.0.52 0.14 0.29 0.41 o-11 0.23 0.33

= (aNOX- d/8 t &N(O)

oxygen atoms are not known, a range of values was taken (& and B,, in Table 4.12). The available experimental data of such compounds(54) suggest the following orders of the ring currents relative to benzene, I > II III > IV III > I which is consistent with the S.C.F. calculations but not with those of the Hiickel type. 4.5.3 Azulene The simple classical calculations of ring currents and the molecular orbital theories of Pople, of McWeeney and of Hall have also been applied to the nonaltemant hydrocarbon, azulene. Experimental evidence shows that the hydro-

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN MOLECULES

147

gen nuclei in the five-membered ring resonate at lower field values than those in the seven-membered ring and that the hydrogen nucleus at position 2 in the five-membered ring resonates at a lower field than those at positions 1 and 3. However, all the calculations predict hydrogens 1 and 3 to be at a lower field than 2, (cf. the results of Hall and HardissorW) shown in Table 4.13).

TABLE~.~~

RING

CURRENT

%im~

IN AzumNERELA~To

BENZENE’Hydrogen IlUClCUS

2 193 498 597 6

Ring current shift dative calcUlatcd

1.37 1.23 l-32 1.18 1.16

to benzene @pm) Observed(79’

1.04 1.37 1.61 O-82 l-11

Although the large discrepancies between theory and experiment for axulene can be ascribed in part to the assumption of equal hybridisation of the carbon atoms in the five- and seven-membered ringst5*), they arise mainly from the presence of different charge densities of the n-electrons on the carbon atomsCs3) (see Section 4.5.5). 4.5.4 Porphyrins Porphyrin molecules (Fig. 4.4) have a large cyclic conjugated path and it has been shown that some of their hydrogen nuclei can experience very large ring current shifts. The peripheral hydrogen nuclei HI, Hz, H4 and Hb show a shift of 2.7 ppm to the low field of benzene, whereas the N-H hydrogen nuclei are + 3.89 ppm to the high field side of tetramethylsilane~s5~.Comparison of the shifts of N-H hydrogen nuclei in porphyrins with that of pyrrole suggests a ring current shift of 1l*Oppmto high field of benxene(56).The ring current model of 18 electrons circulating in a loop of 3.3 A radius gives approximately twice the observed chemical shifts and furthermore does not account for the different shifts of M.W and B-hydrogen nuclei. Abraham(s6) has calculated the ring current shifts by an extension of PaulingWs7) theory for the diamagnetic anisotropy of aromatic molecules (see Section 3.8). In this theory, the induced e.m.f. in any loop is proportional to the area of the loop enclosed, and the resistance of the loop is proportional to the number of bonds. Regarding the porphyrin ring as consisting of four pyrrole rings and a large inner ring, the e.m.f.‘s are calculated to beCs6), pyrrole e.m.f. = KT = R(3il - 2iz) inner ring e.m.f. = KS = R[8(il + iJ + SiJ

148

HIGH

RESOLUTION

NMR

SPECTROSCOPY

and for benzene, e.m.f. = KS, = R(6i,) where T, S, and S, are the areas of the rings and K is aconstant. The C-C and C-N bonds are assumed to have the same resistance. From these equations and the dimensions of the porphyrin and benzene molecules the currents il and iz are . 11 = l-91 iB; i2 = 0.89 iR

where iBis the ring current in benzene.

FIG. 4.4 The porphyrin skeleton (drawn to scale). Abrahamcs6)

The effective magnetic field at the hydrogen nuclei has been calculated in the following way. The shifts of the peripheral hydrogen nuclei H1, H,, Hd, H, were calculated by the induced dipole model of Pople, taking into account the currents in the pyrrole rings and in the four hexagons (area S/4, current il + iJ. For the N-H hydrogen nuclei, the shifts arising from the ring currents in the pyrrole rings were calculated by the dipole approximation, but as the N-H groups lie inside the current loop of the large inner ring (area S) the shift arising from this ring was calculated by the method of Waugh and Fessenden, making use of Johnson and Bovey’s tables (see Appendix B). The ring S was considered to have a ring current divided between two loops as in benzene, with area 78r2 = S giving r = 2.81 A. At a point p inside the ring and in the plane of the molecule the field was obtained by looking up the value ofp/r in Johnson and Bovey’s table for benzene shifts and then correcting for the dilferent sizes and currents of the benzene and porphyrin loops. The results of these calculations are shown in Table 4.14. It can be seen that the magnitude of the calculated shifts are about 50 per cent too large, and to some extent this reflects the many approximations in the calculation, and the limitations of the classical model(s6).

THE

CALCULATION

TABLE~.~~ RING

cuRRENT

OF

SHJELDJNG

~~NIIuBUTIONS

CONSTANTS

TO

THE

IN

P~RPENRIN

‘H

MOLECULES CHEMICAL

WA:

L?wF-Is(~~’

VPpm)

230 5.24 6.70 8.16

233 020 0.09 045

3.25 3.25 715 7.15

0.83 0.83 0.08 0.08

Contributions 346 from bexa- 450 gons 643 7.04

1.88 0.85 0.30 0.21

2.50 5.05 5.05 6.70

497 0.60 0.60 0.26

Coaributions from pyrrole rings

149

216 3.25 3.25 406

Contribution from large ring

281 0.83 0.83 0.43

3.15 574 7.55 8.95

048 048 0.06 0-M

4.15 5.32 7.10 7.85

207 0.39 O-39 0.19

- 21.2

Total shift

5.91

8.25

- 16.3

3.15

4-31

Calculated shift x 213

3.9

5.5

- IO.9

21

29

ObMWd

4.2

5.5

-11.0

1.8

2.9

shifts

1

4.5.5 The Chemical Shift and Chat-g2Densities in Aromatic Molecules The introduction of a substituent group into a benzene ring gives rise to a change in the shielding constants of the hydrogen nuclei at the ortho, meta and para positions relative to the value in benzene. Electron withdrawing groups reduce the shielding constants, that is the ortho, meta and para hydrogen nuclei resonate at lower applied fields than in benzene, whereas electron releasing substituents shift the resonances to higher fields. This has been interpreted as arising from changes in the n-electron charge densities d e at the carbon atoms (see Section 10.12.7), and it has been suggested that the shift produced, 6, is directly proportional to A e, i.e.@l* 82*U) 6 = kde.

(4.16)

Not all the observed shift can be attributed to changes in e since there may also be large contributions from the magnetic anisotropy of the substituent, from permanent and induced electric moments, from intermolecular association and from solvent effects. Equation (4.16) may also be applied to aromatic ions, and to non-altemant aromatic molecules and in this case the observed value of 6 must be corrected for the different ring current shifts in these molecules. A value for k in equation (4.16) may be obtained empirically by comparing the observed shift relative to benzene of some molecule whose charge density

HIGH

150

RESOLUTION

NMR SPECTROSCOPY

is known.

The most reliable value obtained in this way is that of Schaefer and SchneideP3) who compared the shifts in the symmetrical molecules C5H;, CsHs, C,H$, giving a value fork of + 10.7 ppm (electrons)-’ with a probable error of If: 1 ppm (electrons)-‘. This value of k can n*ow be used to evaluate experimental values of g and Table 4.15 shows the results obtained for some representative aromatic molecules. Substituents with large magnetic anisoCOMPARISON OFOBSERVED AND CALCULATED ELECTRON DENS-

TABLE 4.15

IN SOMEARO-

MATICMOLECULES

w

‘osition

(Pm)

&0,, * @pm)

@ohs

ecak. tt

Reference

Aniline

N&i 0

0

m

In

P

P

+ 0.75 + 0.20 + 0.62

1,071 1.019 1.059

+ 0.42 +0*04 i 0.37

1.039 1.004 1.035

l-081 0.997 1.064

(85)

0.988 1.059 0.954 1.011 0.969 0.998

(86)

0.952 1.004 0.981

(87)

Anisole OCHZ.

0 m P

Azulene 2

2

6

193 498 597 6 9,lO

3 0 4 5

+ -

0.583 0.067 0.945 0.28 0.17

+ 0.29 + 0.55 + 0.52 + 0.17 + 0.12

- 0.29 + + -

o-49 0.43 0.45 0.05

0.97 I.05 0.96 1.04 (::;

Pyridine

; Y

- 1.31 + 0.155 -0.26

1.014 0.976

t 8’ is the correction for the differencein ring current shifts between the molecule and bzene. 4 &0,=6+S tt e talc. is obtained from the molecular orbital method.

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

151

tropies have been avoided and where relevant a correction, P, has been applied for differences in ring current shifts. Schaefer and Schneider(83) have calculated 9 for other non-alternant aromatic molecules, and they &d that equation (4.16) withk = 10.7 ppm (electrons)-’ gives good agreement between observed and calculated n-electron charge densities. They do not find any evidence to support the suggestion that equation (4.16) should include a term proportional to e*. Schug and Deck(84) have obtained the chemical shifts relative to benzene of the hydrogen nuclei in a number of hydroxy benxenes. After allowing for the effects of the permanent dipole moments, and extrapolating the shifts to a medium of unit dielectric constant, they find that the observed shifts can be fitted to an equation 6 = kl el + kz e2 + k3 e3 where ,eI, e2 and e3 are the n-electron charge densities at the bonded carbon, on the two adjacent carbon atoms, and on any oxygen ortho substituent atoms respectively. The values of the constants kl, k2 and k3 were obtained by calculating the n-electron charge density distributions by means of simple Hiickel molecular orbital (H.M.O.) theory. They found that the values of the proportionality constants were markedly dependent on the values of the Coulomb and exchange integrals employed in the H.M.O. calculations, but they suggest that reasonable values are 7.61, l-26 and 0 ppm (electrons)-’ respectively for kl, k2 and k3, 4.6

CALCULATIONS

OF hamx~

&rms

OF

FLUOI&E NUCLEIIN M~LE~~TXS’

The observed range of chemical shifts for the fluorine nucleus in molecules is much larger than the range of hydrogen shifts; for example, the l 9F resonance in the molecule UFs is 955 ppm to the low field of that in HFt5’). These large chemical shifts arise because of the importance in Ramsey’s equation (equation 3.29) of the paramagnetic terms for the electrons local&d on the fluorine nucleus. Applying equation (4.8) to the case where A is a fluorine nucleus, then, in contrast with hydrogen nuclear shielding, G is the dominant term. The importance of u= arises from the presence of valency pelectrons in the fluorine atom, which on molecule formation can depart considerably from spherical symmetry (see Section 3.7). The paramagnetic term will be negligible in the spherical fluoride ion, so that in this case ti; is zero, whereas in the fluorine molecule F2, G is large and negative, with the result that the resonance of F2 gas is 630 ppm to low field of that in liquid hydrogen fluoride(33). Between these two extremes the chemical shift of the fluorine nucleus in binary fluorides is almost linearly dependent on the electronegativity of the atom bonded to the fluorine (see Chapter 11). The exceptionally large values of UF in some molecules, such as UF, , arise because of the presence of low-lying electronically excited states with the appropriate symmetry to give non-zero matrix elements of the operator n&k in equation (3.29)‘59’. Saika and Slichter(60) have calculated the chemical shift difference between the fluorine nucleus in the molecule F2 and in the ion F- by assuming that the observed shift can be equated to a= in the fluorine molecule. Thus the cal-

152

HIGH

RESOLUTION

NMR

SPECTROSCOPY

culation reduces to one of evaluating the matrix elements of the operator mzr between the ground and excited states. The calculation is simplified further by including only p electrons in the summation in equation (3.31) for u:, and the difference in shielding is given byt60) (4.17) where (l/r3)2, is the average value of l/r3 for 2p electrons and A E is an average excitation energy. The ground state of the fluorine molecule is a 1x; state, while the lowest excited state with the appropriate symmetry to give a nonzero matrix element of the operator mzk is a In, state, arising from the transfer of an electron to an unoccupied 2p orbital. The value of the excitation energy for the transition ‘c: to ‘n~iis known to be 4.3 eP1) and should be similar to that for a ‘xi to ‘n, transition. Using this value of A E and a value of (1/r3>21, of 8.89/a: where an is the Bohr radiuP2) gives a shielding difference OF60)

Aa = -20

x 10-4.

The experimental value of - 6.3 x 1O-4 between F2 and HF can be accounted for in part by allowing some covalent character in the H-F bond. Using an ionic character of 0.43 and an average excitation energy of 7~7eV for HF, Saika and SlichtePO)calculate the chemical shift between F2 and HF to be - 14 x lo-‘, in better agreement with experiment but still too large. The remaining discrepancy probably reflects the many approximations made, and in particular the neglect of higher excited states. Karplus and Das(63) haveextendedthe above treatment of lgF chemical shift calculations, and by using a molecular orbital function in Ramsey’s perturbation expression for a= they have been able to relate the change in shielding of the fiuorine nucleus in similar molecules to changes in ionic character, bond order, and degreeof hybridisation of the bond involving the fluorine atom in question. The molecular orbital function for the ground state of the molecule is written as a single determinant of the form YO = ~(2~)!I-“21~~(l)8(l)~~(2)~(2)y,(3)8(3)

. . . %(2n)a(2n)l

(4.18)

where each of the doubly-filled molecular orbitals y,(k) is formed from atomic orbitals v,(k) by the L.C.A.O. method

substituting equations (4.18) and (4.19) into the approximate expression for the shielding constant involving an average energy approximation (equation (3.31)) gives f? &la = _ cp /q(k) rk2m-rk.rk 3 2Mc2 vll ““\ ’ rk

THE

CALCULATION

OF

SHIELDING

CONSTANTS

IN

MOLECULES

153

The summations in equations (4.20) and (4.21) are over the atomic orbitals occupied in the ground state. The p,,, are elements of the charge-bond order matrix. When applying equation (4.21) to a fluorine atom in a molecule only the atomic orbitals centred on the fluorine atom need be considered. Moreover, since s-orbitals give a zero contribution to a,,-, the q, in equation (4.21) are simply the 2p-orbitals (pX,9)*and vz. The integrals may now be readily evaluated since operating on one of the p-orbitals q, with a component of mk gives either zero or transforms CJ+into another p-orbital, i.e. mXk&V

= 0

1

(4.22)

When equations (4.21) and (4.22) are averaged over all orientations of the mole cule then 47

= 00 UPXX + PYY + PZA

-

HP,,

PYY + PYY Pzz

+ HPXY PYZ + Pm PZY + Pm Pxz>>

wherep,,,

are orbital populations,

PYY, andp,,

+ PII PXJ

-

(4.23)

and (4.24)

This is equivalent to the expression obtained by Saika and SlichtePO) for the chemical shift between covalent and ionic fluorine. By specialising the argument even more to a fluorine atom directly bonded to only one other atom, Karplus and Dasor5r were able to show that G

e&l + 3P 1s + (ex+ ey)U+ s - 14 - exeJ)

= ao{[l - S - Z + Hex +

(4.25)

where Z is the ionic character of the bond, s the degree of sp hybridisation, and ex and eYthe double bond characters of the bond

ex =2-p,,

and

e,=2-p,.

When s, Z, eX, and eYare small the terms in equation (4.25) which involve products of these terms may be neglected, hence o?

=a,[1

-s-

Z+

Hex+ e33.

(4.26)

154

HIGH

RESOLUTION

NMR

SPECTROSCOPY

It might be thought a weakness of equation (4.26) that it was derived from an equation for d defined in terms of an average energy approximation; however, Karplus and Das have shown that equation (4.26) may be derived from the full expression for 6. 4.6.1 Chemical Shifts of Fluorobenzenes The most useful applications of equation (4.26) are in calculating the difference between the shielding constants of fluorine nuclei in related molecules, since having calculated a0 for one member of a series one can assume that it has the same value for the others. Karplus and DaP3) have successfully applied equation (4.26) to predict lgF chemical shifts for multifluorobenzenes. When the fluorine atom is part of an aromatic system having its ring in the yz plane, then eYis zero and oga = 00 (2 - PLZ.PXJ and, without any restrictions on the magnitude of Z, s or eX, aFFm= a0 (1 - s - Z + Z s (e/2) + e/2 (s + 1))

(4.27)

where e has been written for g,. The chemical shifts of the multi&rorobenzenes relative to the reference compound monofluorobenzene, can therefore be calculated from dap = tiy - &JWara and since de will be negligible then A@$’ can be equated to the observed chemical shift, (Hi? - &) A4y = (4.28) HR ’ Substituting AZ=Z,-I,; As=s,-sc; Ae=pR-eec into equation (4.26) gives fWY

= co{-

dZ + d@/2 - ds + (ZR - dZ + JR - ds)de/2

+ [(2+ + @R)/21d Z +

[@R/2)

+

IRj

d s

-

ill

d 81).

(4.29)

The degree of sp hybridisation in fluorine compounds is small (s x 0.05) and thus Ascan be neglected. Assuming that a0 remains constant, and neglecting terms quadratic in A Z and il ,o, we get AfJg

= aO{- AZ(1 - eR/2 - sR> + d e/2(1 + ZR + 3,)).

(4.30)

The value of u. was calculated by applying equation (4.30) to the two molecules F, and monofluorobenzene. For the fluorine molecule Z and Q are zero and s has been calculated(64) to be 0.02. For monofluorobenzene, Z was estimated to be 0.75 from the ionic character versus electronegativity relationships proposed by Dailey and TownesP*) and by Gordy(29); s was taken to be 0*05’65’, and the value of e = O-126 was calculated by means of the Hiickel molecular orbital

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

theory. With these values of I, e and s, and the experimental of - 543 ppm(33), a0 was calculated to be, CO =

-0.863

155

chemical shift

x 1O-3

which on substituting into equation (4.30) gives for A-, de?“” = (0*765dl-

0.777de).

(4.3 1)

From equation (4.28) it is seen that a positive value of da means that the resonance in the compound is at lower applied fields than that in the reference, monofluorobenxene; thus a positive value of dl shifts the resonance to low fields (IR > Ic), and a positive value of A e(eR > ec) shifts the resonance to high fields. In order that equation (4.31) may be applied to the calculation of da for the multifiuorobenzenes, it is necessary to calculate values of d Zand A e. Values of A e can be obtained by means of the Htickel theory and estimates of d Z can be found by applying equation (4.31) to the three difluorobenxenes. From the experimental values of da and the calculated values of A e it is possible to obtain values of Al; the results of these calculations are summarised in Table 4.16, where A a(e) has been written for 0.777 de. TABLE~.~~ CALCULA TEDVALUESOF AZ FOR DIFLUOROBENZENB Compound

ortho meta

pore

de oao93 oaoO3 0~0115

I

Au@) x lo5

Au(cxpt.) x 10’

-@72 - 0.02 - 0.89

- 2.@33’ + @uT6a - 0.65I6”’

.

AZ -

o*o%l6

_ -to9042 + OaI31

The values of AZ for the m- and p-difluorobenzenes compare well with d I values for dichloro- and dibromobenzenes obtained from quadrupole coupling constantrP3); however, the large negative value ofd I for the ortho compound seems unreasonable and suggests that there is an added effect when two fluorines are adjacent to one another and for which no account has been taken in deriving equation (4.31). If equation (4.31) is applied to multifluorobenxenes,with calculated values of A g, and d I values estimated by assuming that a fluorine neighbour in a mera position contributes + O-0042, and one in a para position + 00031, then the calculated values of da are in close agreement with experiment. However, fluorine nuclei with o-fluorine neighbours show an anomalous shift of magnitude 1.67 x 1O-5 to high field. Table 4.17 summarises the calculated values of Au for a number of multifluorobenzenes in which the fluorine nucleus has one ortho-fluorine neighbour, and compares these values with those obtained experimentally. The shielding constant differences da in Table 4.17 are between the fluorine nucleus in the position “in bold type” and the fluorine nucleus in monofluorobenxene. The ortho effect appears to be constant in multifluorobenzenes, except that an unusually high value of - 1.88 x 1O-5 was found for ortho-difluorobenzene.

156

HIGH

RESOLUTION

NMR

SPECTROSCOPY

The calculated values of do for the fluorine nuclei in some other multifluorobenzenes(63) are shown in Table 4.18. The values of Au for fluorine nuclei with o&o-fluorine neighbours contain an empirical “o&o effect” contribution of - 1.67 x 10-5. TABLE4 17 CALCULATED VAWB

Au(e) x 10”

AZ

Au(Z) x 10”

x 105

00X8 0.0204 0~0200

- 0.68 - 1.59 - 1.55

0*0@42 0*0031 oaO73

+ @32 + 0.24 + 0.56

- 0.36 - 1.35 - 0.99

TABLE 4.18

Compound

Aa(caIc)

AQ

Compound

1, a4 I,54 kZ4.5

OF A d FOR SOME ortho SUBSTITUTEDM~LTELIJOROBEN~ENW

Aa(obs)@” x 105

- 2.04 - 3.03 - 2.65

Au ortho x 105 - 1.68 - 1.68 - 166

CALCIJLATEDVALUES OF .4 u FOR SOMEMULTIFLUOROBENZENES~~~)

Ae

Ade)

x105

AZ

A NJ

x

Au(calc)

Au(obs)

103

x 103

x 10s

1,395 1,394

0.0006 0.0103

- 0.04 -0.80

0*0084 a0073

+ 0.64 + 0.56

+ 0.66 - 0.24

+ 0.53’33’ - 0.25’33’

1,3,4,5 I,53 1,z 395 1, 2, 394 1,2,3,4.5

0*0109 0.0117 0.0109 OalO 0.0204

- 0.85 -0.91 - 0.85 - 1.63 - 1.59

0.0115 0.0042 00084 oa73 0~0115

-!- 0.88 + 0.32 + 0.64 -i- 0.56 i- 0.88

+ 0.03 -223 -1.88 -274 -238

_ 2.55’67’

The ortho effect is too large to be attributed to neighbour anisotropy, or to polarisation of the electrons by neighbouring fluorine atoms. One possible explanation(63) is that the large size ofthe fluorine atom compared with hydrogen distorts the molecule so that it is no longer planar, and thus reduces the amount of double bond character. This would mean a large positive d,o of magnitude 0,023. However, this steric effect is not supported by the observed shifts in chlorofluorobenzenes. In these compounds there is a pronounced shift to lower fields (N 21.7 ppm) when the fluorine has an orrho-chlorine neighbour, whereas the increased bulk of the chlorine group would be expected to increase dq still further. The decrease in the shielding constant of fluorine produced by an orrhochlorine atom can be explained in terms of the internal electric field produced by the polar C-F and C-Cl bonds?, and also by the Van der Waals interaction (see Section 3.10.1)(g7). The work of Karplus and Das on the shielding constants of lgF nuclei has been extended by Prosser and Goodman (go). Instead of restricting contributions to the lgF shielding constant to electrons localised on the atom in question the latter authors have developed a theory which gives the shift in terms of the n-electron distribution throughout the molecule. In fluoro-aromatic compounds t

This effect seems to be important in a wide variety of molecules(gg).

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

157

the theory predicts appreciable contributions to the shielding constant from n-electron charge densities on the bonded carbon atom, q C,, on the fluorine atom, q F., and in the C-F bond p(C, F,). They show that these quantities are related to the chemical shift between two lgF nuclei by the relationship 6 = 488dE-‘[11.96q(Fz)

+ 3*9dp(C,F,)

+ 0.1 dq(C,)].

Equation (4.32) has been applied to a number of p - X-C&F by Taft and co-workers(gl). 4.7 CHEMICALSHIFT

CALCULATIONS

FOR hkW.ELLANEO~S

(4.32) compounds

NUCLEI

Only a very few calculations of shielding constants have been carried out for nuclei other than hydrogen and fluorine. The Saika and Slichter calculation has been applied both to phosphorus ~3. 6g) and to cobalt(70) nuclei; for both nuclei the dominant contribution to the shielding constant arises from the paramagnetic term in Ramsey’s equation. A completely different type of calculation has been carried out for aluminium shielding constants in which the variational method was used. More recently, Pople and Karplus have presented a theory of carbon chemical ~hifts(~~~~*).Buckingham and Schneidefig6) have predicted lggHg, 205Tl and 207Pb chemical shifts using a simple atom-in-a-molecule model. REFERENCES 1. W. C. DICKINSON, Phys. Reu., 80, 563 (1950). 2. F .T. ORMANLJand F. A. MATSEN,J. Chem. Phys., 30,368 (1959). 3. R. E. GIXK, J. Phys. Chem., 65, 1871 (1961). 4. J. M. B. KEWOO, I. I. bB1, N. F. RAMSEYand J. R. Z~aiuus, Phys. Rev., 57. 677 (1940). 5. N. F. RAMSEY, Phys, Rev., 78, 699 (1950). 6. G. F.Nmv~u, Phys. Reu., 80,476 (1950). 7. A. No RDS~ Phys. k., 57,556 (A) (1940). 8. W. biNDERSoN, Phys. Rev., 76, 1460 (1949). 9. L. C. SNVDER and R G. PARR, J. Chem Phys., 34,837 (1961). 10. H. F. I-~MEKA, Rev. Mod. Phys., 34, 87 (1962). 11. C. A. ~ULSON, Trans. Far&y Sot., 33, 1479 (1937). 12. S. WANG, Phys. Reu., 31, 579 (1928). 13. N. ROSEN,Phys. Rev., 38,20QQ (1931). 14. K. I?o, J. Amer. Chem. SOL, 80, 3502 (1958). 15. Y. KUXUTAand K. ITO, 1. Amer. Chem. Sot., 82,2% (1960). 16. J. TILLIEUand J. GUY, J. C/tern. Phys., 24, 1117 (1956). 17. T. P. DAS and R. BERSOHN,Phys. Reo., 115,897(1959). 18. J. 0. HIRS~~~L.DER and W. LINNET,J. Chem. Phys., 18,130 (1950). 19. S. WEINBAUM, J. Gem. Phys., 1.593(1933). 20.Y. SUGIURA, 2. Phys., 45,484 (1927). 21. R. F. WALLIS,J. Chem. Phys., 23, 1256 (1955). U.-M. J. STEPHEN, Proc. Roy. Sot., A243, 264 (1957). 23. M. FIXMAN,J. Chem. Phys., 35,679 (1961). 24. E. ISHIGURO and S. KOIDA, Phys. Rev., 94,350 (1954). 25. H. M. JAMESand A. S. WuDciE, J. Chem. Phys., 1,825(1933). 26. W. G. S CHNEIDER, H. J. B~~~srerrr and J. A. POPL.E,J. Chem. Phys., 2& 601(1958) 27. H. F. HAMJXA,Mol. Phys., 2, 64 (1959).

158 28. 29. 30. 31.

HIGH

RESOLUTION

NMR

SPECTROSCOPY

B. R. MCGARVEY, J. Chem. Phys., 27.68 (1957). W. GORDY,J. Chem. Phys., 19,792 (1951); 22, 1470 (1954). T. P. DASand T. GHOSE, J. Chem. Phys., 31,42 (1959). M. R. BAKER,C. H. ANDERSON, J. PINKERTON and N. F. RAMSEY, Bull. Amer. Phys. Sot., 6, 19 (1961). 32. C. T. ZAHN,Phys. Rev., 24, 400 (1924). 33. H. S. GUTOWSKY and C. J. HOFFMAN, J. Chem. Phys., 19,1259 (1951). 34. L. PAVLW~,Nature offhe Chemical Bond, 2nd ed., Oxford University Press (1950), p. 59. 35. J. TILLIEU,Thesis, University of Paris (1956). 36. J. A. POPLE,Proc. Roy. Sot., A239, 541 (1957). 37. J. A. POPLE,Proc. Roy. Sot., A239, 550 (1957). 38. H. M. MCCONNELL, 1. Chem. Phys., 27,226 (1957). 39. P T. NARA~~N and M. T. ROOERS,J. Phys. Chem., 63, 1388 (1959). 40. B POSTand J. LADW, Acta Cryst., 7, 559 (1954). 41. D. R. Wun74~~, M. SAUNDERS, L. ONSAGER and H. E. DUBB, J. Chem. Phys., 32, 67 (1960). 42. P. T. N ARASMHAN and M. T. ROGERS,J. Chem. Phys., 31, 1302 (1959). 43. J. TILLIEU,Ann. Phys., 2,471, 631 (1957). 44. B. P. DAILEYand J. N. SHOOLERY,J. Amer. Chem. Sot., 77, 3977 (1955). 45. H. SPIESECKEand W. G. SCHNEIDER, J. Chem. Phys., 35, 722 (1961). 46. J. I. MUSHER,J. Chem. Phys., 35, 1159 (1961). 47. J; S. WAUGHand R. W. FESSENDEN, J. Amer. Chem. Sot., 79, 846 (1957). 48. C. E. JOHNSON and F. A. BOVEY,J. Chem. Phys., 29, 1012 (1958). 49. H. J. BERNSTEIN, W. G. SCHNEIDER and J. A. POPLE,Proc. Roy. Sot., A236, 515 (1956). 50. N. JONATHAN, S. GORDON and B. P. DAILN, J. Chem. Phys., 36,2443 (1962). 51. C. REID, J. Mol. Spect., 1, 18 (1957). 52. G. G. HALL and A. HARDISSON, Proc. Roy. Sot., A268,328 (1962). 53. G. G. HALL, A. HARDISSON and L. M. JACKMAN, Discuss. Farad. Sot., 34, 15 (1962). 54. G. G. HALL, A, HARDI~~~N and L. M. JACK-~, Tetrahedron, 19, suppl. 2, 101 (1963). 55. E. D. BECKW and R. B. BRADLEY,J. Chem. Phys., 31, 1413 (1959). Mol. Phys., 4, 145 (1961). 56. R. J. ABRAHAM, 57. J. N. : HOOLERY and H. E. WEAVER,Ann. Rev. Phys. Chem., 6,433 (1955). 58. B. P. Dand C. H. Tow, J. Chem. Phys., 23, 118 (1955). 59. E. PITCHER,A. D. BUCKM~HAM and F. G. A. STONE,J. Chem. Phys., 36, 124 (1962). 60. A. SAIKAand C. P. SLKHTER, J. Chem. Phys., 22,26 (1954). 61. G. HERDERG, Molecular Spectra and Molecular Structure, Vol. 1,2nd ed., Van Nostrand, New York (1950). Phys. Rev., 84, 244 (1951). 62. R. STEB 63. M. KARPLUSand T. P. DAS,J. Chem. Phys., 34, 1683 (1962). 64. B. J. mNSlL, Rev. Mod. Phys., 32,245 (1960). 65. H. S. GIJTOWSKY, D. H. ANDERSON and P. J. FRANK.J. Chem. Phys., 32, 196 (1960). 66. R. W. TAFI; S. EHRENSON, I. C. Lawns and R. E. GLICK, J. Amer. Chem. Sot., 81, 5352 (1959). 67. N. BODEN,J. W. EM~LEY,J. FEENEYand L. H. S~TCLIFFE,unpublisheddata. 68. N. MULLER,P. C. LAIJTERBUR and J. GOLDENSON, J. Amer. Chem. Sot., 78, 3557 (1956). 69. J. R. PARKS,J. Amer. Chem. Sot., 79, 757 (1957). 70. J. S. GRIFFITHand L. E. ORGEL,Trans. Far&y Sot., 53,601 (1957). 71. D. E. O’REILLY,J. Chem. Phys., 32, 1007 (1960). 72. J. G. POWLES,Rept. Prog. Phys., 22, 433 (1959). 73. C. SANDORPY, Gun. J. Chem., 33,1337 (1955). 74. A. G. MORITZand N. SHEPPARD, Mol. Phys., 5,361 (1962). 75. F. R. JENSEN, D. S. NOYCZ, C. H. SEDERHOLMand A. J. BERLIN,J. Amer. Chem. Sot. 82, 1256 (1960). 76. J. I. MUSHER,J. Chem. Phys., 37, 192 (1962). 77. J. L Musxs~, J. Chem. Phys., 37, 34 (1962). 78. M. KARPLUSand J. A. POPLE, J. Chem. Phys., 38,2803 (1963).

THE

CALCULATION

OF SHIELDING

CONSTANTS

IN

MOLECULES

159

79. H. Sprrsx~~ andW.G.S CHNEIDER, Tetrahedron Letters, 468 (1961). 80. C. A. COULSON, Valence,Oxford University Press, (1952). p. 248. 81. G. Fm, R. E:CARTER,A. MCLACHLANand J. H. Rrcxi~~~s, J. Amer. Chem. SOL, 82, 5846 (1960). 82. I. C. SMITEand W. G. SCHNEIDER, Can. 1. Chem., 39, 1158 (1961). 83. T.Sand W. G. SCHNEIDW,Con. J. Chem., 41,966 (1963). 84. J. C. SWG and J. C. DECK, J. Chem. Phys., 37,2618 (1962). 85. H. BABA and S. SUZUKI,J. Chem. Phys., 32,1706 (1960). 86. R. D. BROWNand M. L. HEFFERNAN, Aurtrdian J. Gem., 13,38 (1960). 87. R D. BROWNand M. L. HEFFERNAN, Australian J. Chem., 12, 554 (1959). 88. H. H. LANDoLTand R. BCIRNSTFXN, Zahlenwerte und Functionen Springer-Vcrla8, Berlin, I. Band, I. TeiI (1950) and III. .Teil (1951). 89. R. J. W. LEF~RE, Jozirn. andl’roc. Roy. Sot. N. S. W. 95, 1 (1961). 90. F. m and L. GOODMAN,J. Chem. Phys., 38,374 (1963). 91. R. W. TAFT, F. PROSSER,L. GOODMANand G. T. DAVIS, J. Chem. Phys., 38, 380 (1963). 92. W. G. mrz~., Private communication. 93. J. W. m, unpublished results. 94. L. PsIIuKEl and H. J. BERW’IBN,J. Chem. Phys., 38, 1562 (1963). 95. G. G. HALL, Private communioation. Discuw. Faraday Sot., 34,147 (1962). 96. A. D. BUCEINWUMand W. G. SCIPIEIDER, 97. N. B~DEN, J. W. EMS=, J. EENEY and L. I-I. SUTCLIPPE, Mol. Phys., 8,133 (1964). 98. J. A. P~PLE,Mol. Phys., 7,301 (1964) 99. J. Fm, L. H. Strrfznz’~ and S. M. WUKER, Mol. Phys., 11,117,129, 137,145 (1966).

CHAPTER

5

THEORETICAL CALCULATIONS OF SPIN-SPIN COUPLING CONSTANTS OF SIMPLE MOLECULES THEREhave been many applications to real molecular systems of the equations for spin-spin coupling constants developed by Ramsey(‘), McConnelP and by Karplus(3*J). Most of the published work is on the calculation of interhydrogen coupling constants, largely because of the simplicity of the molecular wavefunctions, but also as a result of the large body of experimental values available. The calculations have revealed two interesting properties of spin-spin coup ling constants. Firstly, the valence bond approach developed by Karplus and his collaborators has shown that a non-zero coupling constant between nonbonded hydrogen atoms is a sensitive indication of departures from “perfect pairing” structures. Secondly, both the valence bond and the molecular orbital calculations suggest that the values of J HHin some molecules depend upon the inter-bond angles and thus the magnitude of JHH can sometimes give the configuration of a molecule. 5.1 HYDROGEN DEUTERIDE, HD Experiment has shown that spin-spin coupling is not observed between nuclei which are magnetically equivalent and a quantum mechanical justification for this observation is presented in detail in Section 8.12. The spectrum of the hydrogen molecule consists therefore of a single sharp line; but it is important to stress that this does not mean that JHH in the hydrogen molecule is zero. It is in fact possible to obtain an experimental estimateof themagnitude of the coupling constant in the hydrogen molecule by measuring the coupling constant JHD in the molecule HD since the ratio of the two coupling constants JHD/JHHis equal to the ratio of the magnetogyric ratios of the two nuclei Y&J” (= 0-I 54). There is a small correction to this conversion factor arising from the different amplitudes in the ground vibrational states of the two molecules(112), but this can be safely ignored at this stage since the effect is to reduce JHH by only a few tenths of one cycle set-‘. The hydrogen resonance spectrum of HD consists of three lines of equal intensity arising from coupling of the hydrogen nucleus with the deuterium nucleus of spin 1. The coupling constant measured from this spectrum(5~6’ is JHD = 43.0 f 0.5 cycles set-‘. 160

C A L C U L A T I O N S OF S P I N - S P I N

C O U P L I N G CONSTANTS

161

In calculating coupling constants, RamseyC~>considered four types of interaction and he wrote the coupling constant as a sum of four terms (see Section 3.14), that is, JHD = 1(1 ) + 1(2) ",rid "lUD + J2nD + Js.D. (5.1) By far the largest term for bonded nuclei is the contact term, in this case JSHD. The magnitude of JSMD can be obtained from equation (3.215). Assuming that the orbital and spin part of the ground state wavefunction are separable, then after applying the average energy approximation, equation (3.215) becomes / 4 "~['16~zfl h~ 2 .land = -- ~ 3 - - - ~ a J ~ ~ ynyD(V'.ol~(r~.)" t~(r2D)[ ~V~o) × (v,= IS~ • s , I v,+)

(5.2)

where ~o~oand ~oes are the orbital and spin parts of the electronic wavefunction. A Ea is an average energy defined by equation 0.216) with o~Fa in the place o f . ~ . The electron spin terms in (5.2) are readily evaluated since (~,~IS~.S~lw=)

ffi

,~

{s ~ - s~ - s~,} v,~

=

-~-.

Substitution of this result into equation (5.2) gives =

(5.3)

in which [~O,o[~s2D is the measure of the probability of finding electron 1 on H and electron 2 on D. Evaluating equation (5.3)with V,o as the James-Coolidge function gives(1) 55.8 JH D ----~ (cycles sec-~). (5.4)

,dE~

A E3 in the above expression is in Rydberg units. At present it is not possible to evaluate AE3 from equation (3.215); however, one can place limits on its possible values c1~.James, Coolidge and Present tT~ calculated the lowest triplet state of hydrogen to be 0.67 Rydbergs above the ground state for an internuclear separation of 1.4 ao (ao is the Bohr radius), whereas the energy required to doubly ionise the molecule with this internuclear spacing is 3.77 Rydbergs. For the analogous mean excitation energy A E used in the theories of the electronic contributions to the rotational and spin-rotational magnetic interactions, Brooks c8) has selected values of 1-9 and 1-1 Rydbergs respectively. Ramseytl~ assumed that the term J3sD accounts for 40 cycles sec -1 of the experimental value of 43.0 cycles sec -1, and thus from equation (5.4) the value of AE3 is 1-4 Rydbergs which is consistent with the limiting values. Ramsey estimated that J2nD is about 3cycles sec -I and that (J~Ro a) + J ~ez)) is no greater than 0"5 cycles see -1. mrs.

e

162


A variational calculation by Stephen (°) (see Section 3.14.4) gives the magnitudes of (JtX)nD + J~a~m~)and (J2aD + J3aD) as 1"46 and 47'68 cycles see -~ respectively when a valence bond function is used, and 2.08 and 47-49 cycles see -~ on using a molecular orbital function. Although these values differ from those o f Ramsey they confirm that (J3aD and J:aD) contributes the major part o f J n o. Ishiguro(t o)has improved on Stephen's calculation by using a more flexible variational function, and also by allowing for the zero point vibrations in the molecule. The calculation was carried out for values of the internuclear separation of 1.3 ao, 1.4 ao and 1.5 ao and then JnD was averaged over the zero point vibration. This gives (t°) j¢x) - 0-254 cycles see -1 IHD j¢2) tin> -- + 0"354 cycles see -~ J2nD = + 0.202 cycles see -~

JznD = + 36"837 cycles see -~ and a total of Jr~D = 37"138 cycles see -~, which compares well with the experimental value of 43.0 cycles sec-z, particularly as the calculation does not invoke an average energy approximation. The main difference between the results of Ishiguro's calculation and that o f Ramsey is in the magnitude of (J~I~nD + J ~ D + J2nQ. Ramsey assumed that these would c o n t r i b u t e - , 3 cycles sec -1 to JrtD, whereas Ishiguro's calculation suggests that they total ,-~ 1 cycles see -1, which" leaves J3 n o as 42-0 cycles sec-~. With this value of J3 HD equation (5.4) •gives R = 1.35 Rydbergs. An important consequence of J3.D being the largest term in equation (5.1) is that the coupling constant is positive in sign, that is, the preferred nuclear orientation is the one with the spins aligned anti-parallel. Furthermore, since J3NN' would be expected to be the dominant coupling term for other directly bonded nuclei the coupling constants between them should also be positive. However, Tiers (3s) has found that this is not true in at least one case, for in the molecule CHFC12 he has found that J,~ca and J~3cv are of opposite sign (see Section 5.3.3). Buckingham and Lovering (x1)have suggested a method for o btaining the absolute signs of coupling constants but as yet the method has only been applied to the J aortho u value in p-nitrotoluene where the sign was shown to be absolutely positive (~°°). This result is in agreement with the sign predicted by an indirect method based on the assumption that &3c-a is positive.t The method of Buckingham and Lovering consists of noting the effect on a spectrum of applying a strong electric field, E. F o r example, they show that for a system A X the effective magnetic field at nucleus A when the applied electric and magnetic fields, E and H, are parallel to one another is

(

(HA)E = H[1 -- ¢r -- (~rll -- a~) Z] -- 2#x m x \4/~A/~X

3z)

R-~'.

(5.5)

f A recent calculation by Musher(I~6) predicts that Jr31.1D is negative, and a MO calculation by Pople and Santryn17) gives a positive value for l~ac,H, but a negative value for/t3cF.

CALCULATIONS

OF S P I N - S P I N

COUPLING

CONSTANTS

163

where a u and u± are the components of the shielding constant of nucleus A parallel to and perpendicular to the molecular axis. R is the internuclear distance, and Z is given by 2E2 [ p2] Z -- 45k--'-'~ (all - 0~±) + ~ (5.6) in which 0ql and o~± are the components of the static electric polaxisability parallel and perpendicular to the molecular axis and # is the molecular dipole moment. For R = 2 × 10 -8 cm, PA = #x = 10 -~3 erg gauss -~ and Z = 10-" then

12p^gxZ R3

= 2-3 cycles sec -~, which subtracts from the unperturbed

splitting when J is positive and adds to it when J is negative. 5.2 METHANE

The H - H coupling in methane cannot be observed directly since the four hydrogen nuclei are magnetically equivalent; however, an experimental value of J.. has been obtained by examining the IH spectrum of the molecule CH3D, and itis found that J.. ---- 12.4 _+ 0.6 cyclessec-I. It can be inferredfrom other experiments that the absolute sign is negativem7 (see Section 5.3.3).The firstattempt to calculateJ.. in methane was made by McConnelF 2)using the general theoreticalformulation of nuclear spin-spin couFling constants in terms of the molecular orbital approximation. McConnell came to the conclusion that the coupling constant between hydrogen nuclei which are not directly bonded is dominated by the Fermi contact term, so that J.., ~ J3..'. Applying equation (3.228) to hydrogen nuclei gives J , xn, = h-1 (2/~ h) 2 72. ~

L-~-\-~--] ~ n ' 92H(0)9~,(0)

(5.7)

where ~ . . , is similar to the quantity used by Coulson and Longnet-Higgins (~3) to define the bond order between pairs of carbon nuclei directly bonded to one another in aromatic molecules, and 9H (0) is the electron density at the nucleus. AE, the average excitation energy was assumed to be l0 eV, a value which is unlikely t o be in error by a factor of more than five. With this value of A E, and an effective hydrogen nuclear charge of l'00, equation (5.7) gives J H H ' '~

2OO ~ 2 H, 2~

(cycles sec-0.

(5.8)

An approximate L.C.A.O.M.O calculation by Coulson (~4) for the methane molecule gave I~,n,l --0.118, and substitution of this value into equation (5.8) gives Jmv ffi 0.44 cycles sec-t compared with the experimental value of --12,4 + 0.6 cycles sec-L If equation (5.8)correctly describes the magnitude of J . n , then a more appropriate value for ~,n, would be 0"62, which seems very improbable. 6"

164

HIGH RESOLUTION

NMR SPECTROSCOPY

Besides giving a low value for the magnitude of JnH' the molecular orbital treatment of McConnell predicts that the magnitude of Jx n, remains constant as the number of bonds separating H and H' increases, and also that all H - H coupling constants should be positive in sign. The valence bond approach to the calculation of coupling constants as developed by Karplus and Anderson (s) avoids these particular difficulties, and was thought to be more suitable for the calculation of coupling constants in that it deals directly with the electron spins in molecules and avoids the need for accurate wavefunctions. Unfortunately, it gives the wrong sign for J ~ . Application of equation (3.240) to inter-hydrogen coupling gives ) E(eV)/103-! ¢t ( JHH ,= 1"395:": j~lC'j

I

[l "l-2fit(pH H,)]

(5.9)

where the cs and c~ refer to a complete canonical set of valence bond structures The application of equation (5.9) proceeds in two stages; firstly,the ground state wavefunction ~'o is determined as a linear combination of the canonical valence bond structures, and secondly the values of cs are substituted into equation (5.9) together with the values of the other coefficients (see Section 3.14.3). Methane was treated as an eight electron, eight orbital problem, in which the four hydrogen electrons are in I s orbitals (b, d, f, h), and the four carbon electrons are in sp s hybridised orbitals (a, c, e, g) directed towards the hydrogen atoms. Fourteen canonical valence bond structures result under the restriction that each of the eight orbitals is singly occupied. The total wavefunction is formed by taking linear combinations of these canonical structures which form a basis for the irreducible representation of the tetrahedral symmetry group (see Section 8.9). Karplus and Anderson (a) used the symmetrised functions derived for methane from a non-canonical set by Eyring, Frost and Turkevich C15). The AI ground state for methane is then found to be a combination of three functions ~'o ffi 1.08 Wx + 0.001 W2 - 0"028 Y3 (5.10)

where ~p~ is the "perfect pairing" structure

y~ = (a b)(c d)(ef)(g h) and the 1P2 and ~3 refer to non-perfect pairing structures.

~P2 = (a h) (b g) (c f) (d e) + (a d) (b c) (el) ( f g) + (af ) (b e) (e h) (d g) ~o3 ffi (a b)(c h)(d g ) ( f e) + (af)(b e)(c d)(h g) + (a h)(b g)(c d)(f e) + (ab)(ed)(eh)(gf) + (ad)(bc)(ef)(hg) + (ab)(cf)(de)(hg) where the parentheses indicate pairs of orbitals bonded together. A non-zero JHn, arises because of the presence of ~2 and ~P3, the contribution from tpl being zero for non-bonded nuclei. The ground state function given by equation (5.10) cannot be used directly with equation (5.9) because ~/'ois not derived from a canonical set. However, Karplus and Anderson were able to use equation (5.10) by expanding each of the non-canonical structures in terms of the corn-

CALCULATIONS

plete canonical

OF SPIN-SPIN

COUPLING

165

CONSTANTS

set, and they obtained for JHHt the value of -

113

A E(eV)

cycles

With A E as 9-O eV, this gives JHw. = + 12.5 cycles set-‘, which does not compare very well with the experimental value of - 12.4 f 0.6 cycles set-‘. Hiroike(16) has developed the valence bond method for calculating spinspin coupling constants without recourse to the assumption that the atomic orbitals are orthogonal. The ground state wavefunction is expressed as a linear combination of independent canonical structures vJ, that is, set-I.

t = COY0 +ccJY,

(5.11)

J*o

where y. is the canonical structure in which all bonds are chemical bonds. Substitution of equation (5.11) into Ramsey’s equation for JsNNr (equation 3.215) gives

The non “perfect pairing” structures may be divided further into those involving the HH’ bond (groupj), and those which do not (group k), and described respectively by the functions and coefficients vJ, c, , yk, ck; equation (5.11) may now be written as II (5.13) + Tc,y, + cckyks . P = cOyO k>n The HH’ bonded structures are by far the most important (5.12) since T2 wJ c W,) 6 (rmaJ St * % (

terms in equation

I I, m

where T is the order of the overlap integrals between atomic orbitals associated with nuclei which are not chemically bonded to one another, and T 5 10-r. Thus the principal terms in equation (5.12) are

The neglect of terms in yk introduces an error of about 20 per cent when the number of bonds does not exceed ten. Equation (5.14) was evaluated for J HHnin methane adopting a A E value of 10 eV, and using the exchange integrals evaluated by Van Vleck(17’. The value of JHH, obtained”@ was + 11.9 cycles se&; if a A E value of 9 eV is chosen, corresponding to that used by Karplus and AndersorP, then a value of + 13.2 cycles set-’ is obtained. Further discussion of the limitations of the valence bond approach is given in Section 5.3.2.

166

H I G H R E S O L U T I O N NMR S P E C T R O S C O P Y

5.3 E ~

A~'D ETHYUm'~

The H - H coupling constants of interest in these molecules are those between hydrogen nuclei on the same carbon atom (geminal hydrogen nuclei) and between hydrogen nuclei on adjacent carbon atoms (vicinal hydrogen nuclei). Theoretical calculations of these coupling constants made so far are general H

H

in that they consider the fragments \ C - C /

/H and C \ H . Both the valence bond

and the molecular orbital treatments of the problem predict that there is a correlation between the vicinal H-H coupling constants and the dihedral angle between the C - - H bonds. Experimental data for unsymmetrically substituted hydrocarbons are in good agreement with the theoretical relationship. The experimental Jn H' values of most theoretical significance are those of the parent molecules, ethane and ethylene. However, the coupling constants are not obtainable directly because the hydrogen nuclei are magnetically equivalent (see Section 8.12): the required magnetic non-equivalence can be introduced by either deuterium or carbon-13 substitution.

5.3.1 Vicinal H-H Coupling Constants One of the main contributions to the calculation of vicinal coupling constant has been made by Karplus c4' 18~ who extended the valence bond approach. The magnitude of JH n' is againdominated by the contact term J3 n n', and it is assumed that the other terms in equation (3.21 I) can be neglected. If the wavefunction is expressed in the form given in equation (5.13), so that v2ois the perfect pairing structure, ~pj the non-perfect pairing structures involving an H - - H ' bond, and W~ the non-perfect pairing structures without an H - - H ' bond, then only cross terms between ~o and ~j have to be included in the calculation, provided that co >> c~, ck. Terms between v2o and ~h make no contribution to the coupling constant. Among the valence bond structures ~pj, there are some having two " b r o k e n " bonds, three " b r o k e n " bonds and so on. Structures having two broken bonds, and occasionally those with three, can be neglected since their contribution to the ground state function ~pis small. Furthermore, because the calculation of JHH' using equation (5.9) is concerned with evaluation of the coefficients co and cj appearing in equation (5.13) and not with the total energy, the electrons and orbitals associated with the same valence bonds in all of the selected structures can be neglected. The fragment I'Z\C--C'./H' in ethane and ethylene was considered as a six electron, six orbital problem of the form

A\

/B /co,

i

C~

Cbz

where A and B are I s hydrogen atomic orbitals, and Col and Cb~ are directed sp carbon hybrid orbitals. The perfect pairing structure is thus written

~0 = (A, C°t)(C°2, C~z)(C~, B)

(5.15)

CALCULATIONS

OF SPIN-SPIN

COUPLING

CONSTANTS

167

and the two structures with A B bonds are

~ot -- (A, B)(Co,, Csl)(Co2, Cb2)

(5.16)

~2 = (A, B)(C°,, C.2)(C~, Cb2).

(5.17)

There are five canonical structures for a six electron system, the remaining two being ~3 = (A, Cb~)(C°~, C.2)(C~,, B) (5.18)

(5.z9)

~o, = (A, C°,)(Co2, B)(Cbt, Cb2). These canonical structures can be represented schematically as

HA

He

HA"

*,

*o HA~

-He

*a

H8

~3

O,,

Superposition of these canonical forms gives the coefficients ij,~ andfj,~ appearing in equation (5.9) (see Section 3.14.3). The coefficients Co and cj were determined by solving the secular equation ctg> (s~',j - ~,j E) cj = 0

(5.20)

where •~Ijis the integral f ~ot ~ ~j d3, and Stj is the overlap integral f ~ ~oj d3. Karplus c+> included only nearest neighbour atomic integrals in equation (5.20), neglecting integrals other than those of the form [A, C°l], [A, Ca,], [Col, Co2], [CoI, Cb2], [C,1, C~1] and [C,2, Cb2] where IX, Y] = f x ( 1 ) Y(2) .,~'Y(1) X(2) d 3. The exchange integrals [A, Col] and [A, Co2] are dependent on the angle A Co Cb, (called 0) and the other exchange integrals: the integrals involving carbon orbitals only, are dependent not only on 0 but also on 9 the dihedral angle between the A Ca Cb plane and the Ca C~ B plane. For ethanet.

L~.

\

,'

(

B

Fio. 5.1 The molecular skeleton of 2,2-metacyclophane.The hydrogen atoms in the -C'HeCH~- groups are labelled A and B. Gutowsky and Juan(s4)

The opposing signs of the coupling constants in this molecule cannot be attributed to angular distortion or internal motional effects. It is probable that the geminal coupling constants in all substituted methanes and ethanes have the same sign. This is apparent (27' 35) from a consideration of the values of J ~ ' shown in Tables 5.5 and 5.6. The magnitudes of J ~ in the fragments C--CH2---O and C--CH2--Br are similar to those in CH3OH and CH3Br respectively, hence it can be assumed that the effect on J ~ ' of replacing a hydrogen with a carbon atom is small and will not change the sign of J ~ . Thus the J ~ ' values in all these compounds can reasonably be assumed to have the same sign. Furthermore, the gradual change in IJ ~ l in the series CH3F, CH3OH, CHaNO2, CH3CCI3, CH4 suggests that its sign is the same in all these molecules. Of the two possibilities, J ~ positive and J ~ negative, or J ~ negative and J ' ~ positive, the latter has the more convincing theoretical justification. McConnelP e) has shown that applying the Dirac vector coupling model to the problem of spin-spin coupling leads to the conclusion that J ~ is negative in

174


methane, and also that JHH' will alternate in sign, being negative if the twohydrogen nuclei are separated by an even number of bonds and positive otherwise. A negative J ~ in methane means that the vaience bond calculation is in error. If the mechanism of the coupling between two directly bonded nuclei is dominated by the Fermi contact term then the sign would be expected to be positive. Hence, it has been suggested° ' ' 3s. 39) that the absolute signs of J ~ and r°~, "HH could be deduced by determining their signs relative to that of J,3cH between directly bonded nuclei in the same molecule. The double resonance method of determining the relative signs of coupling constants has been used by Lauterbur and Kurland ¢4°)to showthat J13c~ and J ~ are of the same sign in the molecules 1,2-dichloroethy!ene and 1,2-dichloroethane; Anet c4~)has used the same method of sign determination to ~how that J,3c~ and -~rorg"'(and hence J~,~) are of opposite sign in the molecule CH,DOH. This appears to be reasonable proof that . / ~ is positive in ethanes and ethylenes. However, this assignment depends upon the absolute sign of J~cR being positive, and there is some evidence that this is not necessarily correct.t The argument for a positive coupling constant between directly bonded nuclei means that J~cn and J,3c~ should always have the same sign, but Tiers °s) has found that these coupling constants are of opposite sign in the molecule CFHCl2 where the values of the coupling constants are J13c~ = ±220"0 ± 0.I cycles sec -1 Jt3cv = ~=293-8 ± 0.2~cycles s e c - 1 JHF = ~ 53"65 ± 0"16 cycles sec -~. The available experimental evidence points to a negative geminal coupling constant, J ~ , in substituted ethanes which suggests that the predicted variation of J ~ w i t h the HCH angle, 0, is incorrect. But J~'~ may still be angular dependent, changing in sign for some value of theHCH bond angle, since in ethylene (2°) and some substituted ethylenes(42-4") J ~ and r,t, -ma are of the same sign, suggesting a positive geminalcoupling constant. 5.4 COUPLING CONSTANTS BETWEEN HYDROGEN NUCLEI SEI"ARATi~ BY

F o u r a-BONDS Spin-spin coupling through four a-bonds has been observed in a number of molecules, but as yet no detailed theoretical predictions of their magnitude have been made. Karplus (24) has suggested that the magnitude of the spin coupling through four a-bonds should be £ 0.5 cycles sec -t, whereas the majority of the observed values lie in the range 0,3 to 1.5 cycles sec-t (see Table 5.7). There is some evidence that these long range coupling constants are strongly stereospeciflc, thus, in methacrolein dimer The dependence Of Jcn on s-character given by equation (5.48), and the additivity rule (equation (5.57)) for substituents also apply to substituted ethylenes and acetylenes(l°n. The olefine, CH2 = 13CI-IX is regarded as a substituted methane having as substituents X and =CHz; the s-character of the *3C--H bond is 2 H 2 -- 13CHX) = ~1 [ 1 + A =cna + Ax] as(C 1 = as2 (CH2 = CH2) + "z-Ax

(5.64) (5.65)

CALCULATIONS OF SPIN-SPIN COUPLING CONSTANTS

195

w h e r e / I x has the same meaning as before, and ,4-cH~ is equated to ,4c,~ ( = + 0.008). Values o f a~ for some ethylenic compounds are included in Table 5.22, where observed Jca values are compared with values o f JcM calculated from equation (5.48). There is no simple means o f allowing for the substituent effect on the hybridisation in these compounds for which ~ is listed as sp2 or sp. Ax OBTAINED FROM VALUES OF J u c l t IN C H 3 X COMx~OtmDS(GUTOWS~:YANDJu~'~x°t))

TABLE 5.21 StmS'Trrtr~T P ~ a t

CH3X

lc~ (cycles × ~-

/Ix

I)

CH3X

Ax m

d2(CH3)6 ~i(CH3)4 CHs)sSiCN H'I3C(CH3)3 :H, ~'IsCOCHs ,"I-IsCHs ,'I'13CH(CH3)

113 118 122 124 125 126 126

0"226 0"236 0"244 0"248 0"250 0"252 0"252

126 126 127 128 128 130 131 131

0"252 0"252 0"254 0"256 0"256 0"266 0"262 0"262

0.096 - 0"056 - 0.024 - 0.008 0.000 + 0.008 + O.008 -

I

Br H'IsCeHs H'IsCHO H-IsCH2Br ~'HsCH2CI H'IsCOOH H'IsN(CH3)2

+ 0.008 + 0"008 + 0"016 + 0-024 + 0.024 + 0.040 + 0-048 + 0-048

CHsCH2I CHsNHCH3 CHsC~CH CHsNH2 CHsCCI3 CHsCN (CHs)2NCHC CHsSH (CHs),S CH~SOCHs CHsOH CI'I30CsHs (CHsO)~CO CHsF CHsCI CHsI CHaBr

132 132 132 133 134 136 138 138 138 138 141 143 147 149 150 151 152

0"264 0.264 0"264 0"266 0"268 0"272 0-276 0-276 0.276 0-276 0-282 0"286 0"294 0"298 0-300 0"302 0"304

+ O'O56 + 0.056 +O.056 +O.O64 + 0.072 +0.088 + 0-104 + 0"104 + 0-104 +0-104 + 0-128 + O.144 +0.176 +0-192 + O-2OO + 0-2O8 + 0-216

The observed values o f J c . for substituted ethylenes lie consistently 5 to 10 cycles sec -x below the values calculated from equation (5.48). This probably arises because o f a contribution to J c . from o--~r interactions (see Section 5.6). The term Jca (~r) can be estimated from the equation J~0z)--2"1

x 10 -xs a . . ac

(5.66)

where a . and ac are isotropic hypertine interaction constants for the radical fragment H---~ (sp2). With a . ffi - 65 x 10e cycles sec -1, ,4 Es -- 6 eV, and ac =, + 40.7 gauss, equation (5.66) gives Jc~(~r) -- - 2 . 6 cycles sec -x, which is o f the correct order o f magnitude and sign (l°x). Muller and Pritchard (x°s) have suggested that Jcs should be directly related to both the bond distance r e , and the inter-orbital angles since both these quantities depend upon the state o f hybridisation o f the carbon atom. They derive an empirical equation relating J c . to r e , (see equation (12.3)), which predicts bond lengths in fairly good agreement with the experimentally observed values (see Section 12.2.11). 7*

196


The bond angles in substituted methanes m a y also be calculated from the observed JcM values, but the agreement with experimentally obtained bond angles is poor (see Table 5.23). The bond angles in such compounds as CH2C12 cannot be reconciled with any state ofhybridisation based upon a combination ofs and p atomic orbitals, which m a y explain the above discrepancies (t°". TABLE 5.22 VALUES OF JcH A N D a 2 FOR SOME UNSATURATED COMPOUNDS

J c u (observed)

Compound

cycles see-*

Naphthalene( t o ~) [~D72a¢(

TM

)

Mesitylene (ring H)(*°*) 2,3,5,6-Tetrachloronitrobenzene(**o) 2,4,6-TribromophenoF *xo) 2,4,6-Trichlorophenol( t i o) 2,4,6.Trinitrophenolo lO) 2,6-Dibromo-4-nitrophenol(*xo) 1,3,5-Trichlorobenzene(xxo) 1,3,5-Tribrornobenzene(,, o) 1,3,5-Trinitrobenzene(t ~o) ~-Dichlorobenzene(*1o) ~,4,6-Trimethylpyridine( t l o) 7-Dimethoxybenzene(11o) ',CH3)~C = C = ~3CH2 o o , ) ~yclohexene(t o,) =I-I2= CHz(*o*) ~"'I-.ICI~ 13 C H 2 (cis~.~.)( 101 ) ~'-'I'-IC ! =ffi13 C H 2 (tra/./2)( I 0 t ) ~CI2 = C-'I..I2(I o x)

=H2: laCHCl( *o*) :is-CHCI=CHCI

(*o*)

T a n s - CHCI= CHCI(* o l) = C I ~ = CI-ICl( " o ' ) ~H3Cmet 1 3 C H ( l O 1) [.-[-- C ~ C t cis

157 159 160 175"0 176'9 172"5 171"3 175"8 174"1 174"3 179"5 169"0

$p2

158"5 161.1 166 170 157 160 161 166 195 198"5 199"I 201"2

sp 2 sp 2 sp 2

C__--C- H (101)

= 500 ~ cycles sec-J

sp 2 sp 2 sp 2 sp 2 sp a sp z sp 2 sp 2 sp 2 sp 2 sp 2

sp 2

0"336 0"341 0"341 0.349 0"402 0-408 0"408 0.416

248 251 259"4

~6HsC~*3CH (t°*)

JcM (calculated)

2 aH

168 170"5 170"5 174 201 208 208 213

sp sp sp

refers to CI and H across the double bond.

TABLE 5.23 "IwrER-ORarrAU" AZqOt.ES CALCULA'rED FROM a~2 AND a~ for HALOtUmTF,AN~ A N D THEIR COMPARISON WITH OBSERVED B O N D ANGLES (tot) HCX

Difference Calculated

Observed

:I

102"2 101"8 102'6

106.9 107"3 108"0

F

103.0

108.5

Br

HC){ in CI'I2X2(°)

in C H a X (o)

X

4"7 5"5 5"4 5-5

Difference Calculated

Observed

100.5 99-7 99.7 98.5

114.7 112 111.8 108.3

14"2 12.3 12"1 9"8

CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS

197

A number of ~3 C ~ H coupfing constants between nuclei separated by two bonds have been measured, and these also appear to depend upon the state of hybridisation of the ~3C atom c~°s~ (see Table 12.34). An estimate has been made of the magnitude of the Fermi contact term for interacting nuclei of this type by the method used for calculating F - - F coupling in fluorinated ethane and ethylene (see Sections 5.8.3 and 5.8.1). Although such a calculation gives the correct order of magnitude for s p a and s p 3 hybridisation it predicts positive coupling constants, whereas the signs of Ji3c-ca determined so far have been found to be negative c~°~ (relative to a positive J:3ca), except for the anomalously large value of + 49"3 + 0-3 cycles sec -~ found for acetylene ez°). When the ~3C and ~ H nuclei are separated by more than two bonds the Fermi contact term is no longer the only large term in equation (3.211), and the magnitude of the coupling constant does not depend solely upon the s-character of the bonds o°e' ~o7~. As with F - - F coupling constants there are examples where the coupling over three bonds (J~3c_c_ca) is less than that over four bonds (JJ at-c-c-ca) within the same molecule. REFERENCES 1. N. F. RAMSE~,Phys. Rev., 91,303 (1953). 2. H. M. M C ~ O N ~ t ~ J. Chem. Phys., 24, 460 (1956). 3. M. KxgPLus and D. H. ANDL'nSON, J. Chem. Phys., 30, 6 (1959). 4. M. KxP.~Lus, J. Chem. Phys., 30, I1 (1959). 5. H. Y. C ~ x and E. M. PURCELL, Phys. Rev., 88, 415 (1952). 6. T. F. WIMMEr'r, Phys. Rev., 91, 476 (1953). 7. H. M. JAMES, A. S. COOLIDGEand R. D. IhtEs~rr, J. Chem. Phys., 4, 193 (1936). 8. H. BROOKS, Phys. Rev., 59, 925 (1941); Phys. Rev., 60, 168 (1941). 9. M. J. S ~ , Prec. Roy. Soc.,A243, 274 (1957). 10. E. Ismotmo, Phys. Rev., 111, 203 (1958). 11. A. D. B u ~ i ~ and E. G. LOVEmNG, Trans. Faraday See., 58, 2077 (1962). 12. M. KXnPLUS, D. H. ANDERSON,T. C. F~RAR and H. S. GUTOWSKY,J. Chem. Phys., 27, 597 (1957). 13. C. A. COULSONand H. C. LO~UFr-I-hGGINS, Prec. Roy. Sac., A193, 447 (1948). 14. C. A. COULSON, Trans. Faraday See., 28, 433 (1932). 15. H. EYmNG, A. A. FROST and J. ~ C H , J. Chem. Phys., 1, 777 (1933). 16. E. Hmoll~., Prog. Theor. Phys., 21, 943 (1959); J. Phys. Soc. Japan, 15, 270 (1960). 17. J. H. VAN Vt~'K, J. Chem. Phys., 1, 219 (1933); 2, 20 (1934). 18. M. KA~LUS, J. Phys. Chem., 64, 1793 (1960). 19. L. PAUL~G and E. B. WILSON, Introduction to Quantum Mechanics, McGraw-Hill, New York (1935). 20. R. M. LYNDEN-BELLand N. SHSPt'XnD, Prec. Roy. See., A269, 385 (1962). 21. H. S. GUTOWSgY,G. G. BFJ~VOP,D and P. E. McMxHoN, J. Chem. Phys., 36, 3353 (1962). 22. N. S~.PpXXD and J. J. T ~ Prec. Roy. Sac., A252, 506 (1959). 23. R. U. ~ u x , R. K. KuI2~O and R. Y. Morn, J. Amer. Chem. See., 80, 2237 (1958). 24. M. KTa~Lus, J. Chem. Phys., 33, 1842 (1960). 25. H. CONROY,Advances in Organic Chemistry, Vol. II, Interscience, New York (1960). 26. H. S. GtrrowsKv, M. K ~ L u s and D. M. GRANT, J. Chem. Phys., 31, 1278 (1959). 27. N. SH~pPXp.Dand H. J. BEnNSTI~, J. Chem. Phys., 37, 3012 (1962). 28. M. BAv,E~J) and D. M. GRANT,J. Amer. Chem. See., 83, 4726 (1961). 29. G. V. D. TIEns, J. Chem. Phys., 29, 963 (1958), 30. F. K^PLAN and J. D. ROBERTS,J. Amer. Chem. See., 83, 4666 (1961). 31. C. A. REILLYand J. D. SWALEN,J. Chem. Phys., 35, 1522 (1961).

198

HIGH RESOLUTION

NMR SPECTROSCOPY

32. K. A. McLAUCHL~ and D. H. Wmr~e~, Proc. Chem. Soe., 144 (1962). 33. C. J. BROW~, J. Chem. $oc., 3278 (1953). 34. H. S. GUTOWSKYand C. JUAN, Discuss. Faraday Soc., 34, 52 (1962). 35. C. N. BANWeLLand N. SHL~PAZD,Discuss. Faraday Soc., 34, 115 (1962). 36. H. M. MCCOtC~qh'LL,J. Chem. Phys., 23, 2454 (1955). 37. M. B A ~ L D and D. M. GRANT, Private communication. 38. G. V. D. T i r e , J. Amer. Chem. $oc., 84, 3972 (1962). 39. M. KA~LUS, J. Amer. Chem. Soc., 84, 2458 (1962). 40. P. C. L A ~ u I t and R. J. KUP,L A ~ , J. Amer. Chem. Soc., 84, 3405 (1962). 41. F. A. L. Ate'r, J. Amer. Chem. Soc., 84, 3767 (1962). 42. T. SCH~FeR, Can. J. Chem., 40, I (1962). 43. P. F. Cox, J. Amer. Chem. Sot., 85, 380 (1963). 44. W. BItUI~3EL,TH. A~r.L and F. I~U~B~RO, Z. Elektrochem., 64, 1121 (1960). 45. A. A. BOTHeR-BY and C. NA~-CoLr~, J. Amer. Chem. Soc., 84, 743 (1962). 46. F. A. L. A ~ T , J. Amer. Chem. Soc., 84, 747 (1962). 47. D. 1t. DAvls, R. P. LUTZ and J. D. ROSeRTS, J. Amer. Chem. Soc., 83, 246 (1961). 48. F. A. L. A ~ T , Can. J. Chem., 39, 789 (1961). 49. J. MSI~'WALDand A. LeWLS, J. Amer. Chem. Soc., 83, 2769 (1961). 50. A. A. B ~ - B Y and C. NA~-CoLnq, J. Amer. Chem. Soc., 83, 231 (1961). 51. R. W. FE.SSENDL~and J. S. WAUGH,J. Chem. Phys., 30, 944 (1959). 52. A. A. BOTHeR-BY, Private communication. 53. F. S. MORTXM~, J. Mol. Spect., 3, 335 (1959). 54. S. ~ R , J. Chem. Phys., 32, 1700 (1960). 55. R. A. H o F ~ and S. GRoNowrrz, Arkiv. Kemi, 16, 471 (1960). 56. S. FUJXWARA,H. SmMIZU, Y. ARATA and S. ~ o R I , Bull. Chem. Sac. Japan, 33, 428 (1960). 57. R. R. FRASER,Can. J. Chem., 38, 549 (1960). 58. L. M. JACF~MANand R. H. WXLEY,J. Chem. $oc., 2886 (1960). 59. J. A. POPLE, W. G. SCHN~DER and H. J. B~P~ST~, High Resolution Nuclear ~fagnetic Resonance, McGraw-Hill, New York (1959). 60. R. A. HO~MAN and S. GRoNowrrz, Acta Chem. Scand., 13, 1477 (1959). 61. E. I. S~,Del~ and J. D. ROSERTS,J. Amer. Chem. Soc., 84, 1582 (1962). 62. S. L. MANATr and D. D. ~ , J. Amer. Chem. $oc., 84, 1579 (1962). 63. J. N. SHOOL~tY, L. F. JOHNSONand W. A. A~DEP.SON,J. Mol. Spect., 5, 110 (1960). 64. E. B, Wmp~..e, J. H. GOLDsrr.L~and W. E. STEWART,J. Amer. Chem. Sac., 81, 4761 (1959). 65. W. R. VAU¢3~,'~and R. C. TAYLOR,J. Chem. Phys., 31, 1425 (1959). 66. E. B. WH~PLE, J. H. GOLDSTEn~, L. M ~ E L L , G. S. REDDY and G . R . McCLu~, J. Amer. Chem. Soc., 81, 1321 (1959). 67. N. MULLet and D. E. P z r r ~ , J. Chem, Phys., 31, 768 (1959). 68. J. R. HOLM~ and D. K2VF2ZON,J. Amer. Chem. Sac., 83, 2959 (1961). 69. L. PAULINO,The Nature of the Chemical Bond, 3rd ed., Cornell University Press, Ithaca, New York (1960). 70. H. M. M c C o t ~ . L , J. MoL Spect., 1, 11 (1957). 71. H. M. McCoNN~.L, J. Chem. Phys., 30, 126 (1959). 72. C. N. BA.'qWELL, A. D. C o ~ , N. S~Pp^go and J. J. TuR~a~g, Proc. Chem~ Soc., 266 (1959). 73. T. R. TUTrLe and S. I. We~SSMA~,J. Chem. Phys., 25, 189 (1956). 74. A. SAu¢~, Physica, 2~, 51 (1959). 75. G. V. D. TIERs, Proc. Chem. Soc., 389 (1960). 76. D. F. EVANS,Moi. Phys., ~, 183 (1962). 77. P. T. N ~ and M. T. ROGERS, J. Chem. Phys., 33, 727 (1960). 78. B. BP~LON, Private communication in reference 55. 79. J. A. POPL~, Mol. Phys., 1, 216 (1958). 80. I. SOLOMONand N. BLOEMBERG~,J. Chem. Phys., 2~, 261 (1956). 81. L. H. MEYERand H. S. Gu'rows~Y, J. Phys. Chem., 57, 481 (1953). 82. J. F~ENEYand L. H. Su'rcLrrre, Trans. Faraday Soc., 56, 1559 (1960).

CALCULATIONS OF S P I N - S P I N COUPLING CONSTANTS 83. 84. 85. 86. 87.

199

J. LEE and L. H. SuTCI.IF~, Trans. Faraday Soc., $$, 880 (1959). H. M. McCONN~, C. A. Ren~y and A.D. McLEAN, J. Chem. Phys., 24, 479 (1956). C. N. B~,'qw-E~ and N. S ~ r ~ P ~ , Proc. Roy. Soc., A263, 136 (1961). N. BOD~, J. W. E~L~Y, J. F ~ Y and L. H. S u ' r ~ unpublished work. H. S. GuTowsKY, C. H. HOL~, A. S ~ and G. A. WrLu~MS, J. Amer. Chem. $oc., 79, 4596 (1957). 88. M. KnvKn~ S. MATSUOKA,S. HATrO~Jand K. SEND^, J. Phys. Soc. Japan, 14, 684 (1959). 89. S. L. MANATr and D. D. ~ , J. Amer. Chem. Sot., 84, 1305 (1962). 90. L. M. ~ and C. H. S~'D~OI~, J. Chem. Phys., 33, 1583 (1960). 91. H. S. GUTOWSKYand G. A. Wn.J.~a~, .7. Chem. Phys., 30, 717 (1959). 92. J. D. BALD~SC~V~t~ and S. L. STAFFORD,J. Amer. Chem. Soc., 83, 4473 (1961). 93. S. L. lVtXNATr and D. D. ELt~MA~, .7. Chem. Phys., 36, 1945 (1962). 94. S. L. MANATr and D. D. ~ , .7. MoL Speet., 7, 307 (1961). 95. N. BOD~N, J. W. E ~ Y , J. F~"NEY and L. H. SUTCt.W~ unpublished data. 96. E. Prrc~at, A. D. BuCgJlqo~M and F. G. A. STo~, .7. Chem. Phys., 36, 124 (1962). 97. L. P~ra~a~ and C. H. S~J~et~IOLM,.7. Chem. Phys., 35, 1243 (1961). 98. D. F. EVANS,Discuss. Faraday Soc., 34, 139, (1962). 99. M. T. Roo~.s, Private commuuication. 100. A. D. BucKn~ox~M and K. A. McLAucle..AN, Proc. Chem. Soc., 144 (1963). 101. C. JUAN and H. S. GuTowslcY, J. Chem. Phys., 37, 2198 (1962). 102. E. M ~ m o w s ~ , .7. Amer. Chem. Sot., 83, 4479 (1961). 103. N. MULL~ and D. E. ~ , J. Chem. Phys., 31, 1471 (1959). 104. N. M t n 2 ~ J. Chem. Phys., 36, 359 (1962). 105. G. J. ~ A T S O S , J. D. GRAHAM and F. M. VANE, J. Phys. Chem., 65, 1657 (1961). 106. G, J. KARABATSOS, J. Amer. Chem. Soe., 83, 1230 (1961). 107. G. J. ~ T s o s , J. D. G ~ and F. M. VA~e, J. Amer. Chem. Sot., 84, 37 (1962) 108. R. F p . J ~ . ~ , K. A. MCI.,AUCm.AN, J. I. M U s ~ and K. G. R. PACm.Y~R, Mol. Phys. 5, 321 (1962). 109. J. D. B A I , D F ~ and G. W. FLYNN, J. Chem. Phys., 37, 2907 (1962). 110. H. M. HUTrON, W. F. REYNOLDSand T. ~ Can. J. Chem., 40, 1758 (1962). 111. C. C. COSTAnq, J. Chem. Phys., 29, 864 (1958). 112. H. S. GUTOWSlCY,J. Chem. Phys., 31, 1683 (1959). 113. M. B ~ and D. M. GRANT, .7. Amer. Chem. Sot., 85, 1899 (1963). 114. M. K~PLUS, J. Amer. Chem. Sot., 85, 2870 (1963). 115. A. B. F. DUNCAN, J. Amer. Chem. Sot., 77, 2107 (1955). 116. J. I. M u s t r ~ Private communication. 117. J. A. PoPI.E and D. P. SANTRY,MoL Phys., 8, 1 (1964). 118. R. A. NeW~qK and C.H. S ~ o t ~ , J. Chem. Phys., 39, 3131 (1963). 119. S. NG and C.H. SI~D~OLM, .7. Chem. Phys., 40, 2090 (1964). 120. N. BODEN, J. F ~ Y and L. H. StrrcLm~., J. Chem. Soc., 3482 (1965).

CHAPTER

NMR

6

SPECTROMETERS

AND

THEIR

ACCESSORIES THIS chapter describes high resolution nuclear magnetic resonance spectrometers, their operation and the equipment commonly used in conjunction with them. The necessary theory is developed wherever it seems appropriate. 6.1 BASICREQUIREMENTS Although several types of high resolution N M R spectrometer are available commercially, they all possess the following components: (i) a magnet capable of producing a very homogeneous field (usually in the range 1000-25,000 gauss) ; (ii) a sample holder (probe) which also contains the radiofrequency receiver coil; (iii) a sweep unit to allow the main magnetic field to be varied slightly in a controlled manner; (iv) a radiofrequency oscillator operating in the range 4-100 Mc sec-~; (v) a radiofrequency receiver and amplifier; (vi) an oscilloscope and/or a pen recorder for permanent presentation of the spectrum.

Radiofr

Requl(:le yT~' d--t-"

(6.29)

NMR SPECTROMETERS AND THEIR ACCESSORIES

227

6.5 THE RADIOFREQUENCYOSCILLATORAND REGEIVER The upper limit of about 53,000 gauss for the main magnetic field sets the highest radiofrequency required at about 220 Mc sec -1 (for the 1H and 19F nuclei): the lower limit is about 3 Mc sec -1 (for I"N nuclei). All nuclei could be studied with the latter frequency but the sensitivity relationship(4s) given below dictates that the highest value of the static field should normally be used. S =

C N H 2 7'31(I+ I)

(6.30)

7

(o)

{c)

FIG. 6.22 The hydrogenresonancespectrumof the CH2 group in ethyl alcohol at (a) 12Mcsec-I (b) 30Mcsec -t (c) 40Mcsec -I. By courtesy of Varian Associates

S is the signal strength, N is the number of nuclei under consideration, H is the applied field strength and T is the absolute temperature at which the experiment is being performed. The analysis of spectra is aided considerably by the use of more than one irradiating radiofrequency for a given nucleus (see Section 8.19.5). Since the magnitude of the chemical shift is directly proportional to the strength of the static magnetic field whereas spin-spin inter8*

228


action is independent of it, we have a means of distinguishing between them by obtaining the spectrum at more than one field strength. It is common to encounter molecules in which the chemical shift is of the same order of magnitude as the spin-spin coupling constant. When this occurs a complex spectrum is obtained and it can be difficult to assign completely the fine structure in such a spectrum. Some of the multiplet splittings are inversely proportional to the main field strength. Figure 6.21 shows how the appearance of a given spectrum containing all the above features is affected by the change of radiofrequency (and main field). It is often necessary to examine complex systems at more than one radiofrequency to confirm a spectral analysis. H (a)

f

!b)

Fie. 6.23 The hydrogen resonance spectrum of 5 ~-stigmast-22-en-3-one in

CDCI3 at (a) 40 Mc sec-1 (b) 60 Mc sec-1. Slomp(s°) The improved sensitivity and dispersion afforded by working at as high a field as possible is particularly useful for large molecules having low solubility or available in insufficient quantity. Steroids fall into the latter category(49); they are normally studied by dissolving from about 5 to 50 mg of substance in about 0.5 ml of CDC13. Figure 6.23 is a comparison of the spectra (5°) of 5/~stigmast-22-en-3-one at 40 and at 60 Mc sec -~. The stability requirement of the radiofrequency oscillator is clearly similar to that of the main magnetic field, namely 1 part in l0 s to 109 per minute. The general method of producing frequencies in the range required for nuclear magnetic resonance is to multiply the frequency of a quartz crystal. Some transmitters give different frequencies by controlling the extent of multiplication of the output from a crystal-controlled master oscillator. Others operate by the substitution of different quartz crystals. Some oscillators have their crystals installed in a small oven regulated to within about :t: 0"01°C. In addition to


229

frequency stability the transmitter must have a constant level, usually accomplished by incorporating automatic gain control into the circuit. A means of controlling the radiofrequency power is an essential part of the circuit. So far in this chapter it has been tacitly assumed that a high degree of stabilisation of the radiofrequency oscillator and the magnetic field is necessary. However, since one is only interested in the strict proportionality o f the frequency and the field strength there is no need to stabitise them separately. Baker and Burd c12~(see Section 6.2.1) have controlled the oscillator by the u s e o f a nuclear magnetic ~resonance probe situated in the field, but a more general method which greatly reduces the requirements for time and temperature stabilisation o f the magnet has been devised by Pound and Freeman ¢51). In their method, a constant wave oscillator is phase-locked to a simple super-regenerative oscillator producing a coherent signal at the magnetic resonance frequency of a sample in its coil. This method enables high resolution N M R spectra to be obtained with simpler equipment.

(o}

Ib)

FIo. 6.24 The IH resonance spectrum at 60 Mc sec"1 of 1 per cent ethyl benzene in carbon tetrachloride (a) one conventional spectrometer sweep (b) the computer-average of a hundred sweeps. A filter bandwidth of 0.4 cycles sec-1 was used in both cases. By courtesy of Varian Associates~s~

230


Enhancement of sensitivity. The effective sensitivity of a spectrometer can be improved by a factor of about fifty by time-averaging signals over a long period C95). One way of achieving this is to feed the spectrometer output into a computer of average transients (" CAT")c52' 53). The technique requires that the spectrum be scanned many times in a synchronous fashion. The repetitive experiments then synchronise the averaging process with the result that the desired signal is reinforced because all additions of a positive signal are in phase. Noise signals will add out of phase and thus be averaged out. The noise adds as a root mean square thus giving a signal-to-noise ratio improvement by a factor of l/n, where n is the number of experiments performed. Figure 6.24 shows the improvement which results from a hundred repetitions: trace (b) is the integrated computer output fed into the chart recorder of the spectrometer. A computer having 400 channels only allows about 80 cycles see -a (at 60 Me see -a) to be scanned at a time. A computer having the desirable number of channels (about 4000) required to give a typical scan of 500 cycles see -t costs as much as the spectrometer itself. The full benefits of the method are only realised if the spectrometer has a field-frequency control loop. The main advantage of the method is that it is possible to follow the increase of sensitivity during measurement and to stop the repetitions when the sensitivity seems to be sufficient. A disadvantage of the method is that the resolution is diminished by the finite number of points taken into the calculation. An alternative and essentially equivalent means t54) of increasing the spectrometer sensitivity is to increase the sweep time of the spectrometer to several hours. Again, a field-frequency control loop is essential, and furthermore, it must be of the type using a single probe. (The stability of a spectrometer having two spatially separated probes--one for providing the spectrum and one for the stabilisation--is only satisfactory if the field between the probes is constant.) Ernst and Primas c55~ have achieved a long-term stability of 0.0003 ppm over several days for a spectrometer operating at 25 Me see -1, hence it is feasible to record long-term spectra overnight. The main advantage of the method is that it is simpler than the computer method. 6.5.1 Radiofrequency Power and its Measurement From equation (6.28) it is evident that the application of a large radiofrequency magnetic field, H t , produces a large signal strength. One must bear in mind, however, that saturation (see Section 2.8) and radiation broadening will set in as the magnitude of Ht increases, resulting in a decrease of signal strength: to avoid saturation the condition HI n J is satisfied (see the Bloom and Shoolery treatment (81) given later in this section) then the multiplet structure disappears and the fundamental spectrum has the appearance of Fig. 6.30B. This procedure enables the value of Hx needed to saturate the fundamental spectrum to be determined. After this adjustment has been made, a very small audiofrequency field is applied to the sweep coils, and the resulting first order side bands are observed with a phase-sensitive detector locked to the audiofrequency. The frequency coa and its amplitude Ha are adjusted to the


243

magnitudes calculated from the following relationships derived from Anderson's treatment 's')

~,H. < CO.l~372

(6.56)

COo2 = (~ COo10-6)2 + 7'2Hf

(6.57)

where 6 is the chemical shiftin p p m between the two groups of non-equivalent hydrogen nuclei whose spins are to be decoupled. The two sideband spectra

FzG. 6.31A: the I H resonance spectrum of the CH2 group in propionaldehyde, with markers of 10 cycles sec -~ separation. B: the same spectrum but now the aldehydic hydrogen nucleus has been irradiated strongly. C: spectrum calculated withJ/~ = 0.087 andnocoupling~fthealdchydichydrogennucleus. Freeman (as)

(Fig. 6.30C and 6.30D) can be observed in the output of the phase-sensitive detector. Providing that the reference voltage to the phase-sensitive detector has been adjusted to be in phase with the audiofrequency field, the fundamental spectrum will be suppressed by the phase-sensitive detector and the side band spectra will be in the V-mode display. A slow sweep superimposed upon the static main magnetic field will then result in a spectrum of the type shown in Fig. 6.30E in which the fundamental bands are de.coupled. One set of experimental conditions which will produce the acetaldehyde spectrum shown are COo = 2~z x 56"4 Mc sec -I, CO. = 2~ x 427 cycles sec -1 and HI ~, 10 milligauss. Freeman (st) has described a circuit for this method of double irradiation. Figure 6.31 shows the spectra he obtained for propionaldehyde with and without spin decoupling: the important feature of this experiment is that tedious

244

HIGH

RESOLUTION

NMR SPECTROSCOPY

conventional analysis of the spectrum is eliminated. Another useful circuit for this type of work is that of Elleman and Manatt(*‘). Commercial units are available from Nuclear Magnetic Resonance Specialities, Inc., 145, Greensburg Road, New Kensington, Philadelphia. A drawback to the field modulation methods described above is that they produce a symmetrical pair of sidebands. The unwanted sideband does not interfere with the experiment if the chemical shifts in the spectrum are large; however, in cases where the chemical shift difference between the two groups of nuclei to be decoupled is small, say 50 cycles set-‘, it may happen that an unwanted sideband overlaps another part of the spectrum which then becomes superimposed on the part being recorded. Turner@*) has overcome the difficulty by using a single sideband suppressed carrier modulation of the spectrometer fixed radiofrequency. The latter frequency provides both H, and the single sideband Hz. Another way of eliminating the unwanted sideband employs two modulation frequencies(r4); one sideband, having a high modulation index, provides the decoupling field Hz at a frequency Ye, the other sideband, of low modulation index, provides the field HI at frequency rl. The advantage of the method is that although 1v1 - v2 1 may be only 50 cycles se&, both v1 and v2 are separated from the centre band by a much larger frequency so that the centre band and any unwanted sidebands fall well away from the recorded spectrum. In all sideband decoupling experiments there are two possible methods of traversing the spectrum. The field sweep method keeps (vl - v2) a constant and traverses the spectrum by varying the applied magnetic field H,, . It has the advantage of experimental simplicity but has the disadvantage that the whole of the uncoupled spectrum cannot be recorded in one experiment. For example, in an APX type spectrum (see Section 8.2 for definitions) the effect on the P and X spectra by decoupling nucleus A can be observed only by performing two experiments, one in which (vr - IQ = (vs - YJ, and the other in which (% - YJ = (VP- vJ. In the frequency sweep method, the applied field H,, and decoupling field at frequency y2 are maintained constant and the spectrum is recorded by varying vl. The frequency sweep experiment is simple to carry out with spectrometers having a field/frequeficy control loop (see Section 6.2.1). Bloom and Shooleryc81) have made a mathematical analysis of the effects of radiofrequency fields on nuclear spin coupling, and they were able to predict the various spectra obtained, intermediate between unperturbed multiplet splitting and collapsed multiplets, as the second radiofrequency was tuned and changed in amplitude. The procedure they adopted was to make the Hamiltonian stationary by transforming to a system of rotating coordinates (see Section 2.12.3). On applying a steady magnetic field Ho in the z direction to a system of two nucIei with the magnetogyric ratios y1 and yz (an AX spin system), then their resonance frequencies will have the values y1 H,, and y3 H,, . The magnetogyric ratios used are those observed experimentally which thus takeinto account the effects of chemical shifts. The Hamiltonian for the system is

NMR S P E C T R O M E T E R S AND T H E I R A C C E S S O R I E S

245

where J is the coupling constant in radians sec- ~. Under the usual experimental conditions, only the expectation values for spins in the z direction are effective in the spin-spin coupling. Thus, the allowed transitions between energy levels occur at ~ I H o + m2J and ~'2Ho + m l J where m - - I , I - 1 . . . 1. Hence each nuclear resonance spectrum is split into a field-independent multiplet. Performing now the double irradiation experiment, the first nucleus is investigated by application of a weak radiofrequency field H1 having a frequency of7'1 Ho. The second nucleus is subjected i

I

1 ~o I

g

.

:/ ,

---

L .......

1

~---.-__ ~

i

I~G. 6.32 The forces acting On 12 at different orientations of 11. In this example, /1 ffi/2 ~ ½. Bloom and Shoolery~sx)

to a strong radiofrequency field H~ having a frequency near72Ho. After transference of the system to rotating coordinates, terms including//2, Ho and Ix" I2 are time independent, while Hx can be regarded as a perturbation inducing transitions between otherwise well-defined energy levels. H2 and the spinspin interaction term have only a small effect on the first nucleus and therefore spectrum frequencies calculated in the rotating frame can be transferred back to the laboratory frame by adding to them the factor oJ2/2~. The Hamiltonian is described in the rotating frame by ~o, _- - h [ I ~ - k ( y x H o - w2) + I2" k(~2Ho - oJ2) + Ix "i(y~H2) 4" 12 " i ( ~ ' 2 H 2 ) "1- J-I 1 " I2].

(6.59)

Whereas the first nucleus is quantised only in the z direction the second nucleus is quantised in the direction of its effective field which is at an angle 0 to the z

246

HIGH RESOLUTION

NMR SPECTROSCOPY

direction. 0 may be determined by considering JI~ • 12 in terms of an equivalent magnetic field acting at the nucleus. From Fig. 6.32 we obtain a(mt) = [(y2Ho + J m l - o~2)2 + 7,2H2]÷

(6.60)

cos0(mi) = (y2Ho + J mt - oJ2)/a(mj)

(6.61)

sin0(mx) = Y2 H2/a(ml)

(6.62)

t h e energy levels in the rotating frame are W(mt,m2) = h[ml(ytHo

- -

co2) + m2(y2Ho - 0)2 + mlJ)cosO(mx)

+ m2 7'2//2 sin0(mt)].

(6.63)

For the case o f l t = I2 = ½, the initial states may be defined W(mt = + ½) and the final states W(mt = -½). Only two of the four possible transitions are allowed for the unperturbed situation in w h i c h / / 2 = 0. In the present case, all four transitions may be observed because the angle 0 changes during the transition (see Fig. 6.32) therefore the initial and final states of m2 may not be orthogonal. The transition probability is given by P = cos 2 (9/2)

(6.64)

where qo is the arigle between the initial and final states. The line intensities depend not only upon P but also upon the population distribution amongthe energy levels which is non-Boltzmann because o f / / 2 . Since the thermal relaxation process is not present in the type of spin coupling considered here, the population distribution can be calculated by assuming that the two types of nuclei are non-interacting. Hence the total spin population for each value of mt is governed by the Boltzmann distribution, but at each value of mt the population difference between adjacent levels of m2 is proportional to 3t/.. Here Mz is the slow passage population difference> I, it is a good approximation to use M. = Mo cos20, where Mo is the Boltzmann distribution value. In the general case for spin values greater than ½ there may be as many as 211 (212 + 1)2 lines in the multiplet when H2 is applied. This is because transitions starting from different mt levels may no longer be superimposed since equation (6.63) is not a linear function of m~. Further lines may appear due to the breaking down of the selection rule, d ml = __. 1, caused by the second radiofrequency field. The following detailed calculations made by Bloom and Shoolery ts~) apply only to two nuclei of spin value ½. The four possible transition energies (in the laboratory frame) are A W, = W(½, ½) -

W ( - ½, ½) + h ,o~

A w~ = W(½, - ½ ) -

AWo=

W(½,~) -

w(-½,

w(-½,-~)

,t w~ = W(½, - ½) -

- ½ ) + ha,2

+ h,o~

w ( - ½, ~) + h o,2.

(6.65) (6.66) (6.67) (6.68)

NMR

SPECTROMETERS

AND

THEIR

ACCESSORIES

247

The normal experimental procedure is to adjust the second radiofrequency until 02 = y2H,,, w1 is fixed at y1 H,, and the spectrum is obtained by means of the field sweep d H. Substitution of (Ho + AH) into equation (6.63) and solving for transitions at A W = h o1 in equations (6.65) to (6.68), spectral lines will appear at the positions AH=0

(6.69)

and The intensity of the line at A H = 0 is given by the relative transition probability P, from 4y:Hf P, = Pb = (6.71) (J2 + 4~:.H:) * To find the intensities of the other lines requires the application of equation (6.64). When y1 > y 2, increase of the amplitude of H2 causes the original doublet to spread out and become weak while a new line appears at y1 Ho at the expense of the doublet. When y2 H2 % J only a single line appears due to transitions CIand b, and these are twice as intense as each of the original doublet lines. When yl < y2, the line structure collapses as the radiofrequency power of H2 is increased giving only a single line for the condition 4y2 H2 # J. The theoretical expectations were verified. by Bloom and ShooleryCsl)‘by obtaining the fluorine-19 resonance spectrum of aqueous Na2POjF at 3OW MC Se471 while simultaneously irradiating the sample with the phosphorus-31 resonance frequency of 1291 MC set- l. Freeman and Whiffeno‘g) have applied the above theory to the AX spin system, where A and X are both hydrogen nuclei. They calculated both the field swept and frequency swept spectra of dichloroacetaldehyde and obtained excellent agreement with the corresponding observed ‘H spectra. Baldeschwieler and Ra.ndaW5) have presented detailed quantum mechanical analyses of the double resonance behaviour of AX, A3X and AB spin systems: the frequency of the strong oscillatory field, 02, was considered to be close to the resonance frequency of the X or the B nucleus, while the transitions of the A nucleus are promoted by Hl . In another type of double resonance experiment, namely that involving the Overhauser effect(“0 go*gl), the signal intensity is greatly enhanced by double irradiation (see Section 3.17). Here, the hyperfine coupling between electronic and nuclear spins is removed by irradiating at the electron resonance frequency. For the effect to occur it is necessary that the principal means of thermal relaxation is via spin-spin coupling. In the situation under discussion, however, the thermal relaxation process for each nucleus is mainly through direct interaction with the lattice, therefore the enhancement of polarisation predicted by Overhauser does not occur. Tomita (g2)has developed a general theory of magnetic double resonance which covers all types of coupling.

248


6.9 COMMERCIALNMR SPECTROMm'~'RS The specifications of some of the high resolution instruments currently available are listed below in alphabetical order. It is not practicable to give a price list but as a rough guide the cheapest spectrometer costs about £10,000 while a more flexible instrument having a comprehensive range of accessories can cost more than £30,000. These high performance spectrometers incorporate most of the refinements described earlier in this chapter.

AE1 RS2 Spectrometer (Associated Electrical Industries Limited, Instrument Division, Scientific Apparatus Department, Trafford Park, Manchester 17. England).

Fro. 6.33 RS 2 nuclear magnetic resonance spectrometer. All the electronic components are housedin the one console.By courtesyof AssociatedElectrical Industries Ltd.

The equipment (shown in Fig. 6.33)has an electromagnet capable of providing a homogeneous field from 1000 to 18,000 gauss with guaranteed resolution of 1 in 108. The detection system is of the single coil type. Radiofrequency units can be supplied for any frequency: six units can be built into the console. Many accessories are available including apparatus for controlling the temperature of the sample and units for the observation of wide line spectra of solids. Spectra can be recorded on a 2 in. or a 10 in. chart recorder.

JNM-3H-60 Spectrometer (Japan Electron Optics Laboratory Company Limited, Mitsubishi Naka 28th Building, Marunouchi, Chiyoda-ku, Tokyo). The instrument (see Fig. 6.34) is of the same type as the RS2 above and has roughly the same range of accessories. A spin decoupler is also offered. The


FIo. 6.34 JNM-3H-60 nuclear magnetic resonance spectrometer. By courtesy of Japan Electron Optics Laboratory Co. Ltd.

F£G. 6.35 .INM-C-60 nuclear magnetic resonance spectrometer. By courtesy of Japan Electron Optics Laboratory Co. Ltd.

249

250


maximum field strength attainable with this instrument is 15,000 gauss but a spectrometer is produced (the JNM-4H-100) which has a maximum field of 24,000 gauss. Figure 6.35 shows a version of the JNM-3 H-60 spectrometer which is supplied with the two radiofrequencies 60 and 56.448 Mc sec -t. The spectrometer is designed for routine operation, having pre-calibrated chart paper and a built-in integrator.

PerMn-Elmer NMR Spectrometer (Perkin-Elmer Limited, Beaconsfield, Buckinghamshire, England).

FIG. 6.36 The Perkin-Elmer high resolution NMR spectrometer. The permanent magnet is housed in the thcrmostatted cabinet shown on the right. By courtesy of Perkin-EImer Ltd. Unlike the spectrometers listed above, this instrument (see Fig. 6.36) has a permanent magnet (of field strength 14,092 gauss). The performance, however, is similar to those having electromagnets. Because of the stability of the permanent magnet pre-calibrated chart paper (9½ in. x 16 in.) can be used. The single coil detection system is employed. The radiofrequency units available permit hydrogen, fluorine, phosphorus and boron resonances to be studied. Provision is made for integration of the spectrometer output.

Triib Ttiuber KIS 2 Spectrometer (Triib TAuber and Company, A.G., Ztirich 37, Amperestral3e 3, Switzerland). The spectrometer shown in Fig. 6.37 has a 5 ton electromagnet giving a maximum field of 21,000 gauss (1H resonance at 90 Mc sec -t) with a guaranteed


251

resolution of 5 x 10 -9. A series of probe heads allows radiofrequencies down to 5 Mc sec-1 to be used. The magnetic field is held steady with the aid of a control loop comprising a IH stabiliser unit for the frequencies 30, 60 and 90 Mc sec -~. Thus automatic calibration of spectra can be made at these frequencies. Spot frequencies of 5, 10, 15, 20, 30, 60, 80 and 90 Mc sec -~ can be selected by means of push buttons. A simultaneous pen recording of the spectrum and its integrated form can be made.

Fro. 6.37 KIS2 nuclear magnetic resonance spectrometer. By courtesy of Trilb Ttiuber and Co., A.G. Among the accessories offered are wide line equipment for the examination of solids and an attachment for high resolution studies at high temperatures. The Trfib Tiuber Company also manufactures a high performance but less costly N M R spectrometer (type KIS I/L) in which the electromagnet is replaced by a permanent magnet of field strength 6000 gauss. This spectrometer also has a i H resonance stabiliser and a built-in integrator. Additional equipment may be bought for wide line N M R studies in solids and for variation of the sample temperature in the range - 160° and + 150°C.

Varian A-60 Spectrometer (Varian Associates, Pale Alto, California, U.S.A.). This is a sophisticated routine instrument designed for recording and integrating IH resonance spectra at 60 Mc sec-:, Extensive use of transistors in all

252


the circuitry, including the magnet power supply, has resulted in a compact design as may be seen in Fig. 6.38. A field-frequency control loop introduced via a second reference probe keeps the maximum rate of change of the field homogeneity at I cycle sec -~ hr -z. The resolution claimed is 1 part in 10s. The spectrum is recorded or integrated on the pre-calibrated chart paper (11 in. x 26 in.) of a flat-bed recorder. The main accessories offered with this apparatus are a sample temperature control unit ( - 6 0 ° to + 200°C), a microc¢ll having a capacity of 25/~1, a spin-decoupler, and a time-averaging computer (C.A.T.).

FIG. 6.38 A-60 nuclear magnetic resonance spectrometer. By courtesy of Varian Associates

Varian HR- I O0 Spectrometer The instrument allows hydrogen resonance spectra to be obtained at 100 Mc sec -1. A general view is shown in Fig. 6.39. The electromagnet provides a maximum field of 23,500 gauss hence a 94 Mc sec -1 frequency unit is required for 9F resonance: any radiofrequency can be supplied. Two other versions of this crossed coil spectrometer are available, the HR-60 and the DP-00. Both operate at maximum field of 14,100 gauss. The DP-60 spectrometer can be used for both high resolution and broad line studies, while the HR-00 finds its greatest use in variable-temperature high-resolution work. The DP-60 may be further adapted to electron magnetic resonance spectroscopy or to wide-line, variablefrequency NMR specfroscopy by adding the necessary components for conversion. More recent versions of these spectrometers have a field-frequency control locking system (HA-60 and HA-100).


253

Included in the wide range of accessories are (i) an integrator and base line stabiliser (ii) a variable temperature probe accessory (ill) a microcell holding 25/~1 of sample (iv) a spin-decoupler and (v) a time-averaging computer.

FIG. 6.39 HR-100 nuclear magnetic resonance spe~rometer~ The compommt shown on the extreme right regulates the temperat/tre of the water flowing through the magnet c~oling coils. By courtesy of Varian Asso~ates

REFERENCES

1. MI E. RosE, Phys. Rev., 53, 715 (1938). 2. A. L. BLOOMand M. A. PAC~tJ~, Se~ne~, 122, 738 (1955). 3. M. J. E. GOLAY, R~. Sei. Instruments, 29, 313 (1958). 4. W. A. AND~aSO~, Rev. Sd. Instruments, 32, 241 (1961). 5. B. A. EVANSand R. E. RlcF~,Itm, J. Sci. Instrumems, 37, 353 (1960). 6. A. P. McCANN, F. SmTa, J. A. S. SMitH and J. D. ~ m ~ s , J. Sci. Instruments, 39, 349 (1962). 7. G. SLo~, Rev. Sci. Instruments, 30, 1024 (1959). 8. F. Blra~K, Rev. Sci. Instruments, 37, 342 (1962). 9. H. PRn~tAsand H. H. G ~ , Rev. 8¢i. Instruments, 28, 510 (1957). 10. M. E. PACF,~P~D,Rev. S d . / n s : ~ s , 19, 435 (1948). 11. R. B. W - J x ~ , I.S.A. Proc., 4, 163 (1958). 12. E. B. BAK~ and L. W. BURD, Rev. Sci. Instruments, 28, 313 (1957). 13. I-L PKIto~, 5th European Congress on Molecular Spectroscopy, Amsterdam (1961). 14. R. FR.~F~7.~and D. H. WmFE~, Mol. Phys., 4, 321 (1961). 15. W. A. ANDERSON,Rev. Sci. Instruments, 33, 1160 (1962). 16. F. BLoc~ Phys. Rev., 94, 496 (1954). 17. W. A. ANDERSONand J. T. ARNOLD,Phys. Rev., 94, 497 (1954). 18. F. BLOC~, Phys. Rev., 70, 460 (1946). 19. G. A. WrLLT~tS and H. S. GtrrowsKy, Phys. Rev.,104, 278 (1956). 20. J. I. K~LAN, J. Chem. Phys., 27, 1426 (1957).

254


21. F. BLOCH,W. W. HANSENand M. E. PACKARD,Phys. Rev., 70, 474 (1946). 22. J. T. ARNOLD,Phys. Rev., 102, 136 (1956). 23. N. BLO~ERG~, E. M. PURCELLand R. V. POUND, Phys. Rev., 73, 679 (1948). 24. R. B. WILLIAMS,Annals N.Y. Acad. Sci., 70, 890 (1958). 25. VA~ANASSOCIATF.SSTAFF,N M R and EPR Spectroscopy, Pergamon Press, Oxford (1960), p. 90. 26. S. BROW~STEr~,Can. J. Chem., 37, 1119 (1959). 27. J. N. S~OOLERYand J. D. RoBE~, Rev. Sci. Instruments, 28, 61 (1957). 28. C. FP.ANCONXand G. F~t~o~L, Rev. Sci. Instruments, 31, 657 (1960). 29. A. N. HAMER,J. LEECEand P. G. BENTLEy,U.K.A.E.A. Technical Note, IGR-TN/CA-1048 (1959). 30. W. G. SCHNEIDER,H. J. BV.m~STVZNand J. A. POPLE,J. Chem. Phys., 28, 601 (1958). 31. H. PmMASand H. H. GthCrHAm3, Chimia, 11, 130 (1957). 32. H. PR~tASand H. H. GONTHARD,Helv. Phys. Acta, 30, 315 (1957). 33. E. M. PURCELL,H. C. T o ~ Y and R. V. POUND,Phys. Rev., 69, 37 (1946). 34. E. R. A z ~ , Nuclear Magnetic Resonance, Cambridge University Press (1956), p. 40. 35. W. N. ~ , Proc. Inst. Radio Engineers, 28, 23 (1940). 36. P. Gmvrr, M. SOUTrfand R. G ~ , Physica, 17, 420 (1951). 37. H. S. Gtrrowsg~,, L. H. MeYea and R. E. McCLu~, Rev. Sci. Instruments, 24, 644 (1953). 38. C. E. WAm~o, R. H. SPeN~R and R. L. CUSTER,Rev. Sci. Instruments, 23, 497 (1952). 39. J. B. L E ~ , R. E. R~CHAnDSand T. P. SCHAE~, J. ScL Instruments, 36, 230 (1959). 40. H. S. Gu'rowsKY, Analytical Applications of Nuclear Magnetic Resonance in Physical Methods of Chemical Analysis, Vol. 13, Ed. W. G. Berl, Academic Press, New York (1956). 41. H. G. I-IXRTZand W. SPALTHOFF,Z. Elektrochem., 63, 1096 (1959). 42. V. J. KOWALEWSKXand R. A. HOVFMAN,Nuc. Instruments and Methods, 6, 357 (1960). 43. R. BRADFORD,C. CLAYand E. STruCK,Phys. Rev., 84, 157 (1951). 44. R. FR~temN, Private.communication. 45. B. A. JAcomol-~ and R. K. WAN~SNV.SS,Phys. Rev., "/3, 942 (1948). 46. E. E. SAI.P~XR, Proc. Phys. Soc., A63, 337 (1950). 47. A. M. PORTm,Phys. Rev., 91, 1071 (1953). 48. E. H. ROGERS,NMR-EPR Workshop Notes, Varian Associates. 49. J. N. StlOOLERYand M. T. ROGERS,J. Amer. Chem. Soc., 80, 5121 (1959). 50. G. SLOMP,Private communication. 51. R. V. POUNDand R. FREEMAN, Rev. ScL Instruments, 31, 103 (1960). 52. O. JAP.DErZKY,Private communication. 53. L. C. ALLI~ and L. F. Jormso~, J. Amer. Chem. Soc., 85, 2668 (1963). 54. R. ERNSTand H. PR~AS, Private communication. 55. R. ERNSTand H. PRIMAS,Discuss. Faraday Soc., 34, 43 (1962). 56. L. M. JACKMAN,Applications o f NMR Spectroscopy in Organic Chemistry, Pergamon Press, London (1959). 57. S. WILK~G, Z. Phys., 157, 401 (1959). 58. W. A. ANDERSON,Phys. Rev., 104, 850 (1956). 59. W. A. A~D~RSON,Phys. Rev., 102, 151 (1956): see also reference 25, p. 164. 60. F. C. STEm.~G and R. B. W I L ~ , Private communication. 61. L. A. MoxoN, Recent Advances in Radio Receivers, Cambridge University Press (1949). 62. G. CAMPONOVO,B. MARUGGand L. WEGMANN,Archives des Sciences, 11, 203 (1958). 63. Reference 25, p. 185. 64. VARDaq ASSOCIATES, N M R at Work, No. 84. 65. J. T. ARNOLDand M. E. PACKARD,J. Chem. Phys., 19, 1608 (1951). 66. R. KAS~LUS,Phys. Rev., 73, 1027 (1948). 67. K. HALnACH,Heir. Phys. Acta, 29, 37 (1956). 68. J. H. Btrao~s and R. M. BROW~, Rev. ScL Instruments, 23, 334 (1952). 69. B. SMALL~R,E. YASArrLSand I~. L. ANDERSON,Phys. Rev., 81, 896 (1951). 70. E. YAsArns and B. SMALLER,Phys. Rev., 82, 750 (1951). 71. H. PPaMAS,Helv. Phys. Acta, 31, 17 (1958).


255

72. P. M. MoRsE and H. F~SHBACH,Methods of Theoretical Physics, McGraw-Hill, New York (1953), pp. 619, 1322. 73. H. S. GLrrOWSKYand C. J. HOVFMAN,J. Chem. Phys., 19, 1259 (1951). 74. R. ~ Rev. Sci. Instruments, 33, 495 (1962). 75. J. PrOH and S. SATO,J. Phys. Soc. Japan, 14, 851 (1959). 76. R. KAISER,Rev. Sci. Instruments, 31, 963 (1960). 77. F. Bt~CH, Phys. Rev., 93, 944 (1954). 78. V. ROYD~, Phys. Rev., 96, 543 (1954). 79. J. BRC6SELand F. BrrTE~ Phys. Rev., 86, 308 (1952). 80. T. 1~ CAxv~ and C. P. SucI-rrER, Phys. Rev., 92, 212 (1953). 81. A. L. BLOOMand J. N. SHOOLVmY,Phys. Rev., 97, 1261 (1955). 82. H. S. GwrowsKY, D. W. McCALL and C. P. S L I ~ J. Chem. Phys., 21, 279 (1953). 83. J. A. PoPtm, Mol. Phys., 1, 168 (1958). 84. J. N. SHooI.~Y, Discuss. Faraday Sac., 19, 215 (1955). 85. J. D. ~ D c ~ w z n ~ . ) ~ and E. W. RANDALL,Chem. Rev., 63, 81 (1963). 86. R. F ~ , Mol. Phys., 3, 435 (1960). 87. D. D. l~vi)w~¢ and S. L. M,~A'rr, J. Chem. Phys., 36, 2346 (1962). 88. D. W. T u m ¢ ~ J. Chem. Soc., 847 (1962). 89. R. ~ and D. H. Wl-m,er.a%Proc. Phys. Soc., 79, 794 (1962). 90. A. W. OvmtuAus~, Phys. Rev., 92, 411 (1953). 91. J. KOmPa~GA,Phys. Rev., 94, 1388 (1954). 92. K. Tos~rA, Prog. Theor. Phys., 20, 743 (1958). 93. L. H. P,~LI~, J. D. RAY and R. A. 0oo, J. Mol. Spect., 2, 66 (1958). 94. L. G. At~XAKOS and C. D. C o m ~ x ~ , Rev. Sci. Instrmnents, 34, 790 (1963). 95. M. P. ~ and G. W. B A R ~ , Rev. Sci. Instruments, 34, 754 (1963). 96. P. J. p~)nJm~ and W. D. COOWV,Anal. Chem., 36, 1713 (1964). 97. O. HAWORTH and R.E. R I c I - I ~ s , Progress in Nuclear Magnetic Resonance Spectroscopy, 1) 1 (1966); edited by J. W; EMSt~sy, J. F~NEY and L. H. S t r r ~ Pergamon Press, Oxford.

CHAPTER 7

EXPERIMENTAL

PROCEDURES

IN CHAPTrm 6 is given a detailed description of the equipment used in high resolution NMR studies. However, no mention has so far been made of the overall experimental procedure to be followed in order to obtain the nuclear magnetic resonance spectrum of a sample. This chapter deals systematically with details of the preparation of the sample in a suitable form right through to the recording of the spectrum. A full analysis of the spectrum normally entails evaluation of chemical shifts and coupling constants while a more extensive investigation might require relative intensities of bands and their line widths to be determined. 7.1 PREPARATION OF THE SAMPLE It is usually preferable to examine samples as dilute solutions in an inert solvent such as carbon tetrachioride or cyclohexane, where the intermolecular effects on nuclear shielding will be negligible. The spectral parameters measured under these conditions can be used more reliably for characterisation purposes a n d they will have greater theoretical significance than those obtained from the pure liquid. The pure liquid at room temperature, however, is not to be shunned entirely because it is, at worst, a reproducible standard state. The sample must be a mobile liquid or gas, otherwise incomplete averaging of the direct dipoledipole interaction leads to broad spectral lines and the highest resolution will not be attained. To observe the high resolution spectrum of a solid, the sample must be melted or dissolved to form a mobile liquid. It is important that the liquid is not turbid since the presence of solid particles in suspension can give rise to broad spectral lines from the auisotropic averaging of the field inliomogeneities on spinning the sample. Polycrystalline solids can be made to give fairly narrow lines by spinning the sample about an axis making an angle of 54° 44' with the main magnetic field~1-*~. The rate of rotation should be large compared with the static line width. Using this method the alp resonance spectrum of solid phosphorus pentachloride has been found to consist of two lines 377 ppm apart corresponding to the environments of PC1+ and PC1;. 7.1.1 Gases The spectrum of a gas at normal temperatures and pressures is that of an isolated molecule, that is, all the intermolecular effects on the chemical shifts are negligible (see Section 3.10). The chemical shift obtained from a gaseous sample depends to some extent on the temperature and pressuremfor hydrogen the 256

EXPERIMENTAL

PROCEDURES

257

effect is very small ¢5). Evans te~ has noticed that the fluorine resonance in gaseous carbon tetrafluoride is dependent on pressure, a shift to low fields of 1"15 + 0.15 ppm atm -1 being observed on increasing the pressure. Nevertheless, this is only small compared with the chemical shifts usually observed for fluorine-containing compounds.

7.1.2 Liquids Viscous liquids must be diluted or the sample temperature raised: even dilute solutions of high molecular weight compounds may be highly viscous; for example, a 10 per cent by weight solution of polymethylmethacrylate in chloroform is too viscous to give a good high resolution spectrum unless the temperature of the sample is increased to 90°C ¢7). Chemical shifts obtained from the spectra of liquid samples may be strongly concentration and temperature dependent owing to strong intermolecular interactions (see Sections 3.10 and 10.39). Therefore, in order to obtain a spectrum representative of the isolated molecule it is necessary to use a dilute solution of the compound in an inert solvent. The ideal solvent for nuclear resonance work is one which does not interact with the solute in any way and which is magnetically well-behaved, that is, it is magnetically and electrically isotropie. It is also essential that the solvent does not have a spectrum overlapping that of the solute: in 1H resonance spectral studies deuterated solvents can be useful in this respect. For example, CDCI3 is used extensively as a solvent for organic compounds of high molecular weight. Other useful solvents for hydrogen resonance studies are the inert isotropic compounds carbon tetrachloride, tetramethylsilane and eyclohexane. For fluorine resonance, a convenient solvent is fluorotrichloromethane (CFC13) which, although neither inert nor isotropic, gives solvent shifts small compared with the magnitude of fluorine chemical shiftsc6"s). It is, of course, desirable that the sample be soluble in one of the inert solvents but often this is not the case and a less ideal solvent must be used. The effect of some common solvents on polarisable solutes will now be discussed. It should be remembered that the use of these solvents precludes comparison of chemical shifts from one solvent to another.

Water N M R spectral data obtained from aqueous solutions must be treated with extreme caution. Not only is water a strongly hydrogen bonded liquid but i also may enter into an exchange reaction with H or OH groups present in the solute. Exchange can affect the chemical shifts and may also bring about collapse of spin multiplets in the spectrum. The 1H resonance spectrum of pure ethanol shows a 1 : 2 : 1 triplet for the hydroxyl hydrogen nucleus but in aqueous solution this multiplet becomes a single line whose chemical shift is strongly concentration dependent. The presence of an exchanging group in such a situation may be detected simply by dissolving the sample in D 2 0 and observing the subsequent diminution of the appropriate signal as chemical exchange takes place. HRS.

9

258

HIGH RESOLUTION

NMR S P E C T R O S C O P Y

Aromatic solvents Although aromatic compounds have only weak complexing properties, benzene and other aromatic molecules often produce large changes in the chemical shifts of polarisable solutes due to the large diamagnetic anisotropy associated with the induced ring currents (see Section 3.8). If the formation of a weak complex gives rise to a preferred solute-solvent orientation then a shift in the resonance position occurs relative to that of the non-complexed solute. The shift may be either to high or to low fields depending upon the nature of the complex. Thus the hydrogen resonance signal of chloroform moves by 1"5 ppm to high fields on going from pure chloroform to a 5 per cent solution of chloroform in benzene(9). The upfield shift takes place because the preferred orientation of the complex is as shown below

CI3C--H........ The chloroform hydrogen nucleus thus experiences an applied field which is reduced in magnitude by the induced field arising from diamagnetic ring currents in the benzene molecule. Molecules having hydrogen atoms with some degree of charge polarisation will form complexes of this type. Table 7. I shows the tH chemical shift differences of some representative molecules when dissolved in benzene and in an inert solvent. TABLE 7.1 SOLVENT Sm.vrs IN BENZENE

Solute Chloroform Propargyl chloride (acetylenic hydrogen)

(B)H,, H(A) Vinyl bromide ~C=C (C)H 'Br NO.o H ] ~H(A) p-Methyl~ ~J'~ nitrobenzene ~ J~ H~~ ' H(B) CHs

Solvent shifts~" + 1"35 +0"31 + 0"86(A), + 0"88 (B), + 0"75(C)

+ 0.S7(A), + 0.9603)

t The quoted shifts are measured as the difference in ppm betw~'n the chemical shift of the solute extrapolated to infinite dilution in benzene and that of the solute extrapolated to infinite dilution in an inert solvent.

Substituted benzenes have similar effects on the chemical shifts of solutes, that is, the hydrogen resonance signal is displaced to high fields. However, if the substituent itself has strong complexing properties then a low field shift may be observed. For example, in the case of propargyl chloride (re>, CICH2C = C H . changing the solvent from cyclohexane to nitrobenzene causes a low

EXPERIMENTAL PROCEDURES

259

field shift of 0.71 ppm for the acetylenic nucleus. This is due to complex formation between the nitro group and the propargyl chloride molecule.

Pyridine. This has been found to be a useful solvent for NMR investigations of polar steroids c~). Because of the existence of specific interactions between pyridine and the steroids, the large anisotropic ring current of the pyridine int~uences the shielding of the hydrogen nuclei of the steroids to different extents. The effect can be put to good use in unravelling spectra containing superimposed absorption bands. For example, the 1H resonance spectrum of 4,4,17~trimethyl- 17/~-hydroxy-5-androsten-3-onedissolved in deuterochioroform shows only two bands attributable to methyl hydrogen nuclei: a solution of the compound in pyridine reveals separate absorption bands for each of the five methyl groups. The coupling constants are unaffected by the solvent. Fortunately, the spectrum of pyridine itself is well removed from the region of interest in the steroid spectrum. Trifluoroacetic acid. The nuclear magnetic resonance spectra of amino-adds and peptides can be conveniently measured using trifluoroacetic acid ¢x2~ as solvent rather than water ¢~s. ~4~. Tritluoroaeetic acid is an excellent solvent for these compounds and also has the advantage that, unlike water, it will dissolve sufficient tetramethylsilane for use as an internal reference. Porphyrins exs~ have been examined in both trifluoroacetic and deuterotrifluoroacetie acids--advantage is taken here of both the solvent and strong a d d characteristics. The solvent has the disadvantage of bringing about a certain amount of line broadening. Liquid sulphur dioxide. This is a useful solvent for many organic compounds. Furthermore, it has the useful property of being aprotic. Sulphur dioxide may be kept at the liquid state at room temperature with the aid of the screw-top sample tubes described in Section 7.3.6. Acetone and nitromethane. These and other donor molecules all form specific complexes with molecules having electron acceptor properties; in all cases the resonance signal of the solute undergoes a shift to low fields, the magnitude of which depends on the strength of the solute-solvent complex. 7.1.3 Impurities in the Sample Large amounts of impurities in a sample may obscure a region of the spectrum but in general as much as 5 per cent of a diamagnetic impurity can be tolerated. There are types of interacting impurity which take part in chemical exchange with the sample and need to be removed. For example, small amounts of water may have a large effect by promoting a proton exchange reaction of the type R O H + HsO ~-- R O H + I-~O

Traces of acids and bases can aggravate the situation by catalysing the exchange process. Exchange reactions are discussed fully in Chapter 9. 9"

260


It is necessary to exclude small amounts of paramagnetic materials from the system since they can affect markedly the nuclear spin lattice relaxation times (see Section 2.5.1). Because the Bohr magneton is approximately 103 larger than the nuclear magneton, mole fractions as low as 10 -6 of a paramagnetic impurity can lower Tt considerably. Large m o u n t s of paramagnetic impurities can reduce Tt to such a low value that the line widths broaden sufficiently to obscure the hyperfine splitting. However, the presence of traces of paramagnetic compounds can be an advantage when the value of 7"1 is large, as for example in atp and 13C resonances--paramagnetic species are often introduced deliberately(16) in order to allow the use of higher values of radiofrequency power before the limit mposed by saturation is reached. Dissolved oxygen is always present in samples which have not been degassed but despite its paramagnctism its effect on the spectrum is quite smallcry). If there is any doubt about the effectof oxygen then samples should be degassed in a vacuum system in order that the natural narrow line widths can be observed. 7.2 REFERENCE COMPOUNDS It is not possible to measure absolute values of shielding constants (see Section 3.I) and therefore all chemical shifts are measured from the absorption band of an appropriate nucleus in a pure standard reference compound. The standard compound may be either dissolved in the sample (internalreferencing) or placed in a separate container (externalreferencing). Additionally, the reference compound often serves to provide a locking signal.

7.2.1 External Referencing Relative shifts measured in this way must be corrected for differences in the bulk diamagnetic susceptibility between the sample and reference compound (see Section 3.3), a procedure which presents some difficulties. For cylindrical samples the corrected chemical shift is given by

where A Z = Z, - Z,, the difference in bulk volume susceptibilities of the reference and the sample. It should be noted that Bothner-By and Glick eta' tg~ have found for some systems empirical correction factors some 20 to 30 per cent greater than the theoretical one [(2,-r/3) A ;~]. If both sample and reference compounds were contained in spherical sample tubes then no correction for differences in diamagnetic susceptibilities would be necessary; unfortunately good spherical containers are not easily made. Thus in many cases the method of external referencing is unsuitable because the form of the correction factor may be in doubt and because the accuracy of the corrected shift will rely upon the measured values of the bulk diamagnetic susceptibilities. When external referencing is unavoidable then it is most conveniently carried out with the aid of precision coaxial tubing. Usually, the reference is contained in the central capillary and the sample in the surrounding annulus. Zimmerman and Foster ¢2°~

EXPERIMENTAL P R O C E D U R E S

261

have examined the practicability of such a method and they found that if the sample is magnetically and electrically isotropic then accurate and reproducible results can be obtained. External referencing has the advantage of not contaminating the sample and is often used in N M R studies of nuclei other than hydrogen where susceptibility corrections can frequently be regarded as negligible. In order to evaluate A :~ use can be made of the tables given in Appendix C. Susceptibilities can be measured with the aid of a high resolution N M R spectrometer in the method devised by Frei and Bernstein (21). Figure 7.1 shows the experimental arrangement used in this method: two small reference containers, one spherical and the other cylindrical, are positioned inside a conventional

\

/ Nylon s ~ c r

I

jO'SmmO.D. ~ - 2.5ram QD. sphere

FIo. 7.1 5 mm o.d. sample tube containing a cylindrical and a spherical referencetube. Frci and Bornstein(21) 5 mm o.d. spinning sample tube with the sphere at the centre of the receiver coil. The N M R spectrum of the reference compound contained in the two inner vessels consists of two chemically shifted sharp absorption bands, one from the contents of each container. The separation between the two signals is linearly dependent upon the volume susceptibility of the sample contained in the conventional sample tube and is given by the expression (R) -

--

[go,

- g, h] [z,(R)

- z,(s)]

(7.2)

where ~ is the chemical shift in ppm, ~, is the volume susceptibility in cgs units and g is the geometrical constant which is dependent on the shape of the interface between the sample and the reference, but is independent of the shape of the outer tube. For an ideal case [gcr~ - g, ph] would be expected to be 2~/3 but in practice this constant will need to be determined by calibration with liquids of known volume susceptibility. Some of the volume susceptibilities

262

HIGH

RESOLUTION

NMR

SPECTROSCOPY

obtained from this method are given in Table 7.2. The accuracy is comparable with that of classical methods. TABLE 7 . 2 VOLUME SUSCEPTIBILITIES DEO,~SSF.D LIQUIDS (21)

Si(CHD,. CI-L,CN CsI"Ito CS2

SiCl4 CHaI

OF SOME

-0.543 x 10-6 e.g.s, units - 0.532 - 0.628 - 0.693 - 0-755 - 0.938

Li, Seruggs and Becker(2z) and Douglass and Fratiello (.3) have also reported a method for determining diamagnetic susceptibilities with the aid of a NMR. spectrometer. They used a simpler sample cell than the one described above. In their method, first the spectrum of a sample contained in the central compartment of a precision coaxial cell is recorded, the external reference benzene being contained in the annulus. Second, the same assembly is used for making the susceptibility measurements simply by stopping the spinning, when the benzene then displays two absorption bands whose separation is a linear function of the volume susceptibility of the liquid in the inner cell. The separation (n cycles see -t) is given by the equation (43) n = 4~ % [(21 -- g2) (a/r) 2 + (X2 - Z3) (b/r) 2]

(7.3)

where % is the fixed radiofrequency, a and b are the internal and the external radii respectively of the inner glass tube, r is the mean radius of the annulus, Zt, Z2 and Zs are the volume susceptibilities of(1) the liquid in the inner tube (2) glass and (3) the annular liquid (benzene in this case). With precision bore tubing, a, b and r are constant. The linear plot of n versus the known volume susceptibilities of water, carbon tetrachloride, cyclohexane and methanol was obtained and then used to determine the susceptibilities of samples. Assuming the shape constant to be 2~t/3 the accuracy of the method is within the limits required for correction of spectral measurements, that is, _+0.004 x 106 c.g.s units corresponding to 0"5 cycles sec -t at 60 Mc sec -t. Measurements of paramagnetic susceptibilities have also been made with the aid of a high resolution NMR spectrometer(24). Even when diamagnetic susceptibility corrections are applied, chemical shifts measured from an external reference will often contain an appreciable error because of the uncertainties involved in the correction procedure. It is permissible to compare chemical shifts from an external reference for a series of closely related compounds, as for example a set of geometric isomers. Here the volume diamagnetic susceptibilities can be regarded as constant. Accurate conversion can be made from one external reference scale to another if corrections are made for the difference in diamagnetic susceptibility between the two reference compounds. Some useful approximate conversion factors for external referencing are given in Table 7.3.


263

The effect o f solvents on the position o f fluorine resonance lines cannot be predicted by equation (7.1) even when isotropic solvents are used c2~. 27) (3lick and Ehrenson ~2" have shown that for 1,2-dibromotetrafluoroethane the extent o f the deviation from the behaviour predicted by equation (7.1) is proportional to the polarisability o f the solvent and they suggest that intermolecular influences on the shielding o f ~9F nuclei may be eliminated by extrapolation to zero polarisability. The most commonly used external reference compound in fluorine resonance work is trifluoroacetic acid. Shifts measured relative to the line from this compound are usually quoted uncorrected for differences in diamagnetic susceptibility between reference and sample. Although deviations from equation (7.1) may be as large as 1 ppm even for isotropic solvents, they are small compared with the range of 19F chemical shifts ( ~ 200 ppm). TABX.E7.3 C O - - I O N

FACTORS FOR Ex'rI~NAL RlWl6REN~ C O M P O U N D S ¢2s)

t H Resonance ~C6H6

H20 =6e= +2"49 d~ffi~H6= 6si(c~s'4 + 7"27 ext

6~H6 __ •SH12 -t- 5"83

6 ~ a~ = 6~ c~,c°cH~ + 5"1S CeHe = ~C~C~3 + 0"02

eXg

--egt

6~I~' = o~m 2ct' + 1"94 19]:: Resonance 6Cox~ v3c°°x

"= ~,,,t'sca3F - 78"6

6cF3c°°H ~ = 6ct4Fs + 61.49 13C Resonance ~CH~COOH _- - o.~H6 e=

+

50

Footnote. The conversion factors for xH resonance were obtained from the diffe~mce in the chemical shifts (measured relative to tetramethylsilane) of various compounds in dilute solution in carbon tetrachloride. To convert 6,zt to t~mt, corrections must be made for the difference between the volume stmceptibilities of the solution and the external reference. The conversion factors given in this table are not n__~zes_~rilythose used throughout the t e x t : preference has been given to conversion factors suggested by the original authors. 7. 2.2 Internal Referencing The introduction o f about I per cent of a reference compound into the sample has the distinct advantage that corrections for differences in diamagnetic susceptibility are not required since both sample and reference are surrounded

264

H I O H R E S O L U T I O N NMR S P E C T R O S C O P Y

by the same medium. However, internal referencing contaminates the sample and for this reason it is desirable to employ a reference compound of low boiling point in order that it may be removed readily by distillation. A more serious objection to an internal reference is the possibility of specific solvent-solute interactions which would affect to differing extents the positions of the reference and the sample resonance lines. It is essential therefore that the reference be chemically inert and magnetically and electrically isotropic. Hydrogen resonance. For ~H resonance studies of non-aqueous solutions, recommended reference compounds are tetramethylsilane (TMS) and cyclohexane. Tetramethylsilane, Si(CHa)4, is a volatile liquid (b.p. = 27°C) and it is magnetically and electrically isotropic, reasonably chemically inert and it does not appear to associate with any common compounds (2a). The magnetically equivalent methyl hydrogen nuclei give a single sharp band at fields higher than almost all other compounds: this property also makes it very suitable for providing an internal locking signal. The chemical shift ~ (in parts per million) of a resonance line from tetramethylsilane is defined as

HTMS

]

(7.4)

where H, and H~n~sare the resonance field values of the sample and tetramethylsilane respectively. Defined this way most 8 values are negative: it should be noted that this definition of t~ is used throughout this monograph. Tiers ~2a) has suggested that instead of the resonance line of tetramethylsilane being taken as 6 = 0 ppm it should be assigned arbitrarily to the value of + 10 ppm thus making most chemical shifts positive. Shifts measured for dilute solutions in carbon tetrachiodde relative to tetramethylsilane at 10 ppm are known as values c2s) where • is defined as T = 10.000 + 6.

(7.5)

On this scale almost all hydrogen resonances have a positive value of I=; increasing values of z imply increase in shielding. Table 7.4 shows a comparison of r values obtained using tetramethylsilane both as an internal and an external reference compound (28). The internal chemical shifts were obtained from approximately 5 per cent solutions in carbon tetrachloride, while external shifts were determined by extrapolation to zero concentration of two or more measurements made on carbon tetrachloride solutions of different concentration. There is some evidence (29) that the resonance line of tetramethylsilane is affected by solvent interactions but the changes are small enough (about 0"025 ppm) to he neglected in most cases. In the most accurate work it is advisable to introduce a second internal reference compound and to use the constant separation of their resonance lines as an indication that both are unaffected by the surrounding medium. Cyclohexane is a good reference compound giving a single absorption band at ~ = 8.564, that is, 86.16 cycles sec -t downfield from tetramethylsilane at 60 Mc sec -t . This separation can be used to test for solvent interaction.

265


Tetramethylsilane is virtually insoluble in water, hence a different 1H internal reference c o m p o u n d is required for aqueous solutions. Tiers (s°) has suggested the use of the methyl resonance band in the 1H spectrum o f the sodium salt o f 2,2-dimethyl-2-silapentane-5-sulphonic acid, (CHs)sSiCH2CH2CH2SO~Na" TAI3LE7.4 COMPARmONOF T VALUESOBTAINEDWITH AS A N INTERNAL A N D ~XTERNAL R~,~.RF..NCE(2s)

Compound C6He

c6nsC2Hs (C685CH2)2 p-CeK,(CH3). CsH, (cydooctatetraene) CHsNOs CeHsOCH3 CHsOH (CeHsCH2)2 CeHsCH2CHs CeHsCHs (CH3COhO CHsI CH3COCHs CHsCOOH CHsCN Cell12

• (external refegec~e)

• (internal reference)

2"74 2.89 2.89 3-05

2"734 4- 0.003 2"888 q- 0.002 2"893 4- 0.001 3.053 4- 0"003

4.26 5"69 6"31 6.60 7.13 7"42 7"67 7.81 7.81 7.91 7.90 8.10 8.51

4.309 4- 0.004 5.720 4- 0.002 6-266 4- 0"002 6"622.4- 0-002 7"129 4- 0-001 7.382 4- 0.003 7.663 4- 0.003 7.809 -t- 0-003 7.843 4- 0-004 7.915 4- 0.003 7-930 4- 0.004 8"026 4- 0-002 8.564 4- 0-002

TABLE7.5 A COMPARISONOF SICmLnlNG VALLr~ %"(TMS As ~ A L

REFERENCE) AND T' ('DSS AS ~ A L I~d/:FERENCIE) MEASURED IN SIMILARSOLLrIIOI~O0) .fl

Compound CHsCOOH CHsOH (CHs)sC.CH20H in CHaOH (CH3)3C.CH2OH in (CHs)2SO (CH3)3C.CH2OH in CH3CONHCHs CHsCN CHsN02 (CH3)2SO

7"934 6"650 9"117 9"175 9-109 8"033 5"640 7"457

7"945 6"659 9"133 9"165 9-124 8"022 5"646 7"434

k n o w n as DSS. The salt is prepared by the addition of sodium bisulphite to atlyltrimethylsilane. The three strongly coupled methylene groups give rise to a complex multiplet which at 1 per cent concentration is indistinguishable f r o m the b a c k g r o u n d noise in the spectrum. DSS is stable in acid media and is soluble in concentrated salt solutions. The position o f the methyl resonance band is almost unaffected by the nature o f the solvent or the p H o f the solution. Table 7.5 HRS. 9 a

266

HIGH

RESOLUTION

NMR

SPECTROSCOPY

shows that a x' scale, defined relative to the DSS methyl band at 10"000 ppm, agrees with the x scale to within 0.02 ppm. Aeetonitrile and dioxane have also been shown to be suitable internal reference standards for use in *H resonance studies of aqueous solutions ° t ) . Chemical shifts measured relative to these compounds can be converted to T values using the following relationships: "t" ~ (~acetonlttlle ""~ 7"98

z = ~a~o~,°, + 6"30 where ~ is the chemical shift in ppm measured from the reference compound and it is defined as in equation (7.4). Fluorine resonance. Tiers (') has suggested that monofluorotrichloromethane, CC13F, is suitable both as a solvent and as an internal reference compound (b.p. TABLE 7.6 APPARENT FLUORINE SHIELDING VALUESis) (~o*)

Vol., % Compound

C6HsSO2F CFBra

CFzBr2

BrCF2CF2Br

C6HsCF3

CFCI2CFCI2 CFaCO2H CFaCCI3

C6HsF (CFaCC12)2 (C2Hs)aSiF n-C6HxaF

c o n c h , in

~0 'll

CCI3F

(ppm)

5 20 3 2.O 40 6O 80 2 I0 80 10 4O 80 3 I0 20 40 5 2O 5 2O 3 I0 4O I0 40 2 15 I0 4O I0 4O

-65"509 - 65-547 - - 7"384 - 7.309 - 7-231 - 7"143 - 7"052 - 6"768 - 6.763 - 6"768 + 63"394 + 63"373 + 63"332 + 63"719 + 63"651 + 63"574 + 63"385 + 67"753 + 67"762 + 76"542 + 76-552 + 82'209 + 82"220 + 82-230 + 113-150 + 113"231 + 114"086 + 114"086 + 176.24 + 176.240 + 219-017 + 219-002

Std. dev.

__0.005 ± 0"006 ±0.004 0.002 ---0.007 ± 0.003 ± 0.OO3 0"002 ±0"004 ± 0-O05 ± 0"005 ~ 0.009 ± 0.007 -4-0"003 -4-0"010 ± 0"OO5 ±0"OO9 ±0"OO6 -*-0"006 ± 0"OO5 ±0"OO6 -4-0"002 0"008 -4-0"009 ± 0.002 -4-0"003 ±0"009 ± 0.003 __+.0.02 ± 0.006 ±0.0O6 q- 0-013


267

= 24°C). The chemical shift of a sample relative 'to CCI3F used as internal reference is defined

(, H. -

Bo~l,v

/.

C~.6)

There are small variations of ~* values with concentration, therefore for accurate comparisons the shifts must be extrapolated to infinite dilution (they are then known as 9~values). Table 7.6 shows some ~* values of compounds at different concentrations in CClsF. Evans (~) has asserted that 19F shifts measured at infinite dilution in CC13F are appreciably different from those measured on the gaseous sample. Carbon tetrafluoride and sulphur hexafluoride have ideal characteristics for internal references but the fact that they are gases at room temperature introduces a handling problem. 7.2.3 Tetramethyl$ilane as a Reference for ttll Nuclei A method has been suggested by Jackman (s2) whereby the chemical shift of any magnetic nucleus may be obtained relative to the internal reference tetramethylsilane provided that the sample contains hydrogen coupled to that nucleus. In Section 8.19.5 it is shown how accurate values of chemical shifts can be obtained by means of double resonance experiments. Baldeschwieler(ss) has demonstrated how the the chemical shift of I"N nuclei coupled to hydrogen nuclei can be found by observing the XH resonance spectrum whilst irradiating strongly at the 14N resonance frequency. The frequency of the dccoupling field is locked to the hydrogen resonance frequency by means of a series of frequency dividers and multipliers, final adjustment being made with audiomodulation. In this way the shift difference between the I"N nucleus and the IH nucleus is known to within 1 cycle see -x at 40 Mc sec-1. If the • value of the decoupled hydrogennucleus is known then the shift of the X'N resonance from tetram~thylsilane is known. The method can be applied to any two groups of spin coupled nuclei. 7.2.4 lntereonversion of Reference Scales In the past many internal and external reference compounds havebeenadopted for use in chemical shift determinations and it is sometimes necessary to convert shifts from one reference scale to another. Such a procedure is fraught with difficulties and it is usually impossible to obtain accurate results because of the absence of necessary data for the original sample and reference compounds (for example, concentration of solution, extent of intermolecular effects, diamagnetic susceptibilities and sample shape). Conversion from one internal reference scale to another will only be accurate if both reference materials are ideal and present in small amounts and if the chemical shifts to be converted have been extrapolated to infinite dilution/n the same solvent. The converted shifts will then only apply to solutions of the sample in this solvent. It is impossible to convert accurately chemical shifts measured against an external reference to those on an internal reference scale and vice versa. 9m*

268


7.3 SAMPLE CONTAINERS

Most spectrometers are designed to hold a cylindrical sample container capable of being spun by means of an air driven turbine. For the routine study of hydrogen and fluorine resonance spectra, glass sample tubes approximately 5 mm external and 3 mm internal diameter and about 15 cm long are used. A typical sample volume is 0-3 ml. If sample spinning is employed then it is essential that the outside of the tube is straight and that the wall thickness is uniform. In some instruments good sample spinning is assured by locating the tip of the sample tube in a nylon bush at the bottom of the probe assembly. It is important that the bottom of the sample cell be symmetrical, either rounded or conical. Although some manufacturers of commercial instruments provide standard sample tubes, it is possible to make satisfactory containers in the laboratory. Precision bore sample tubes must be used if the maximum signal-to-noise ratio is required. 7.3.1 Coaxial Sample Tubes A convenient method of external referencing is to use precision coaxial tubing, the reference being contained in the central capillary and the sample in the surrounding annulus. The effect on chemical shifts of imperfections in coaxial cells has been discussed by Zimmerman and Foster ~2°). Coaxial cells can be obtained commercially or they can be constructed in the laboratoryO *). A less satisfactory arrangement is t ° introduce a thin walled capillary tube containing the reference compound into a conventional sample tube. When the reference and sample signals are well separated, Both compounds can be placed in separate standard tubes and measurements made by interchanging the tubes during a single sweep of the magnetic field.

7.3.2 Spherical Sample Cells When discussing external referencing (see Section 7.2.1) it was noted that corrections for differences in diamagnetic susceptibilities need not be applied if both sample and reference compounds are put into spherical sample cells. Perfectly spherical glass sample holders are difficult to make and any deviations from perfect symmetrywill introducediamagnetic susceptibility shielding effects. The Triib Tfiuber N M R spectrometer is supplied with spherical sample cells. The cell devised by Frei and Bernstein ~21> (illustrated in Fig. 7.1) can be used simply as a spherical cell. Spiesecke and Schneider ~16~have designed a spherical sample cell having a large filling factor suitable for the study of t3C resonance spectra. Figure 7.2 depicts the sample container comprising two concentric thinwalled spherical bulbs. The smaller bulb has a capacity of 0"2 ml and it contains the reference compound. The outer sample bulb has a volume of 1.5 ml. The receiver coil is mounted inside the main cylindrical tube and encloses the two bulbs. Richards t3s) has improvised spherical sample holders with the aid of gelatin. A cylindrical sample cell is filled with aqueous gelatin setting at about 50°C :-,nd the sample is introduced into the gelatin by means of a syringe so that the


269

sample is located at the centre of the receiver coil. Finally, to ensure that the globule of sample is spherical, the tube is rotated until the gelatin sets. The main drawback of the method is that the compound under examination must be immiscible with water. Air

I ,bnteol holes

4 0 m*l ~ungste wire

Inner sphere for reference liquid

sphere mpte

Bokelit beoring

F~. 7.2 Spherical sample tube having a large filling factor. Spi~l~k¢ and

7.3.3 Sample Cells with Large Filling Factors In the study of nuclei other than those of hydrogen and fluorine, it is often necessary to improve the sensitivity by increasing the size of the sample within the receiver coil. This is achieved by increasing the diameter of the sample cell and using a correspondingly larger receiver coil. Because it is difficult to spin large sample tubes it is customary to make the measurements on stationary samples and this results in loss of resolution. However, a design of a large sample cell which can be spun is shown in Fig. 7.2. When a large sample is spun under conditions of veryhigh radiofrequencyamplification, avarying coupling between the transmitter and receiver is introduced of sufficient magnitude to overload the detector. The problem can be solved by using a phase sensitive detector capable of rejecting signals having the spinner frequency.

270


An obvious way of improving the filling factor is to use sample tubes having very thin walls, or to mount the receiver coil inside the sample tube. Also, the signal-to-noise ratio can be improved by increasing the number of turns of wire in the receiver coil. Any attempt to improve the sensitivity by increasing either the sample or the coil size will result in a loss of resolution since the main magnetic field homogeneity has then to be maintained over a larger volume element.

7.3.4 Containers:for Small Volumes Although only about a 5 mm length of sample is in the region of the receiver coil, in many spectrometers it is necessary to have a depth of liquid of approximately 2 cm in the sample cell before the receiver coil volume is satisfactorily 80

rl

fl

-AL t ~Id r: 0"0 7 3'!""

"

( snug fit in tube) T

~c~s"

-N\~0.020"

Fro. 7.3 Nylon plugs used to reduce the volume of a conventional sample tube. Shoolery(s~)

Me

'~

H

Me 0 I~..~/C--Me

~

Ho

6-Me iC_I 9

I '

8"0

7-0

6"0

5"0

':,.0

3"0

1

fll

=

,6.M, I

2"0

I'0

ppm

FIe. 7.4 The I H resonance spectrum at 60 Mc sec -z of 6/3, 16~-dimethyl-6 c~hydroxyprogesterone in 25/el of CDCI3 containing tetramethylsilane (located at 0 ppm). Shoolery (3~)

0

EXPERIMENTAL

PROCEDURES

271

occupied. Part of the long length required is due to the formation of a vortex during spinning. W h e n only small quantities of the sample are available it is possible to reduce the volume required by fillingthe dead space at the bottom of the tube either by inserting a nylon or P T F E plug, or by constructing the sample tube so that this portion is made of solid glass. In this way a signal can be obtained with as littleas 5 m m length of liquid.The vortex can be eliminated by trapping the sample between two plugs.The resultingsmall cylindricalsample gives rise to a loss in resolution since the nuclei near the ends of the sample are in a less homogeneous field,owing to the discontinuity in the diamagnetic susceptibility.Shoolery (36)has overcome the difficultyby enclosing the sample between two nylon plugs forming a hemispherical cavity (see Fig. 7.3).Theupper plug has a fine hole bored down itslongitudinal axis for fillingpurposes. The cavity occupies only one third of the volume (25 pl) enclosed by the receiver coil but the most efficientuse is made of a fixed quantity of sample. Figure 7.4 shows the IH resonance spectrum at 60 Mc sec -1 of 6/~, 160~-dimethyl-6o~hydroxyprogesterone obtained using a 560 ~tg sample dissolved in deuterochloroform. If more sample is available the plugs can be spaced a little farther apart.

7.3.5 Sample Cells .for Use in Temperature Studies It is often desirable to observe N M R spectra at different temperatures: several thermostatted probe assemblies are described in Section 6.3.1. If the sample is a solution then it is advisable to have as little dead space above it as possible because changing the temperature will change the concentration. It is also advisable to have as small a volume as possible since the longer the samplethe greater the temperature gradient along it. Petrakis and Sederholm(ST), in a study of the temperature variation of chemical shifts of gaseous molecules, restricted the volume by sealing the tube just above the receiver coil while retaining the upper part of the tube for spinning purposes. When working with a gas or a volatile liquid an increase in temperature can lead to a large increase in pressure. Normal 5 mm o.d. borosilicate glass tubing with 1 mm wall thickness will withstand pressures up to 60 arm (e). When high pressures are expected the sealed end of the tube can be made the weakest part to act as a safety valve.

7. 3.6 Sealing of Sample Tubes Temporary seals can be made with moulded polythene caps. Such seals are useful when one is working with expensive precision bore sample tubes intended for repeated use. A more elaborate temporary seal is shown in Fig. 7.5. This consists of three precision turned pieces of an inert plastic such as PTFE. The sample tube needs to be modified by flanging its open end. The assembly gives a good seal in that it will hold liquid sulphur dioxide almost indefinitely at room temperature. If care is taken in the construction, spinning can be perfect.

272


It is very dangerous to attempt to fill and permanently seal a sample tube simply by immersing an open-ended tube in liquid nitrogen, because oxygen condensed in the tube can cause a violent explosion on warming to room temperature.

~ - \~PTFE

Flanged 5ram O.D. glass sample tube

FIG. 7.5 Polytetrafluoroethylene screw cap for temporary sealing of sample tubes

7.4 RECORDINGTHE SPECTRUM In this section a discussion is presented of the factors which need to be considered if the best resolution of an NMR spectrometer is required.

7.4.1 Homogeneity of the Main ~lagnetic Field The first priority is to obtain a homogeneous magnetic field over the sample volume. The shape of the magnetic field contours in the region of the sample depends upon the geometry of the magnet, particularly the parallelism of the pole caps, and also upon its immediate magnetic history. The shape of the magnetic field may be changed by alteration of the alignment of the pole faces,


273

and in the case of an electromagnet, by magnetic field "cycling" (see Section 6.2.2). Some magnets are also equipped with homogeneity coils (see Section 6.2.1), which change the field shape when controlled currents are passed through them. A typical procedure for obtaining a homogeneous field for an electromagnet is as follows. A sample is chosen which gives a strong sharp single line, and the main magnetic field increased until the resonance signal is observed at the centre of the oscilloscope trace. By moving the sample horizontally across the pole faces it is possible to plot the contours of the magnetic field by observing the direction in which the signal moves on the oscilloscope screen. This experiment is repeated for the vertical axis. The desired field shape is one giving field homogeneity over the sample volume, that is, fiat in both the vertical and horizontal directions. The sample itself and the probe assembly distort the field shape and it is found that an apparently slightly dished field gives the best spectra. Successive cycling can change the field shape from domed to dished but the reverse is not true (see Section 6.2.2). If a magnetic field is dish-shaped then it may be changed to a domed field by reversing the direction of the magnetic field for a few minutes in the case of an electromagnet, or by altering the alignment of the pole caps of a permanent magnet. The final adjustments to the field shape are made with the homogeneity coils if available, the homogeneity being checked first by observing the decay pattern of the transient signals following a sharp resonance line and finally by recorL~RAKISand C. H. SEDERHOLM,J. Chem. Phys., 35, 1243 (1961). 38. J. F~NEY and L. H. SUTct.n~ Trans. Faraday $oc., 56, 1559 (1960). 39. C. A. l ~ n J y, J. Chem. Phys., 25, 604 (1956). 40. J. J. TURNU, Mol. Phys., 3, 417 (1960). 41. D. G. DE K o w ~ . e w s ~ and V. J. K o w , ~ v s r ~ , J. Chem. Phys., 37, 1009 (1962). 42. S. L. MANATr and D. D. F J J ~ , J. Amer. Chem. Soc., 83, 4095 (1961). 43. D. C. DOUGLASSand A. FRATr~IO, J. Chem. Phys., 39, 3163 (1963).

279

CHAPTER

8

THE ANALYSIS OF H I G H R E S O L U T I O N S P E C T R A 8. I INTRODUCTION

WhEN an NMR spectrum consists of more than a single absorption band the spectrum can be expressed in terms of two parameters, namely chemical shifts (1,o6) and coupling constants (J). The purpose of this section is to indicate how numerical values of these parameters can be extracted from an experimentally observed NMR resonance spectrum. Most of the spectra previously

rH Resononce

~gF Resononce

I

~,j-D Fro. 8.1 Diagrammatic N M R spectra of C H s F - t h e tgF resonance giving 2(~) + I = 4 lines and the t H resonance giving 2({) + 1 = 2 lines

considered have consisted of well-separated chemically shifted absorption bands possessing multiplet fine structure due to spin-spin interaction between the differently shielded nuclei (this is always true for nuclei of different species). Under such conditions, the number of multiplet splittings on any absorption band is given by the simple (2/1- + 1) rule and the intensities of the components are distributed symmetrically about the centre of each multiplet (see Chapter I). This type of spectrum is referred to as first order, and it is possible to obtain chemical shift and coupling constant parameters directly from such a spectrum. Thus methyl fluoride, CH3F, should have a X~F resonance spectrum which is composed of a symmetrical quartet due to the fluorine nucleus coupling with the CHa group of total spin, IT = 3/2: the 1H resonance spectrum consists of a symmetrical doublet due to coupling with the single fluorine atom (see Fig. 8.1). 280

T H E A N A L Y S I S OF H I G H R E S O L U T I O N

SPECTRA

281

The necessary condition for the observation of a first order spectrum (i.e. % ~ ~, J) is sometimes satisfied by interacting nuclei of the same species. In the case of 2-bromothiophene, the chemical shift differences between the three non-equivalent hydrogen nuclei at 60.00 Mc sec -1 are much greater than the J values involved and the ZH resonance spectrum (shown in Fig. 8.2) of this molecule closely approximates to a first order spectrum. Here, the three differently shielded nuclei give rise to three separate absorption bands each of which is split into four components of roughly equal intensity due to spin-spin A

X

I]

f"

I

,

H~.S )St

I

I i I

FIG. 8.2 The ~H resonance s p e c t r u m o f 2 - b r o m o t h i o p h e n e at 60 M e sec -~. Cohen and McI.~ueld~O)

interaction with the other two non-equivalent nuclei (the four components being a doublet of doublets). The three coupling constants involved can be obtained directly from the spectrum by subtraction of the appropriate bands as indicated in Fig. 8.2 and the chemical shift differences between the nuclei can be obtained by ass-mlng that the true resonance position for each nucleus is at the centre of its symmetrical multiplet. An analysis of this kind will not yield the relative signs of the coupling constants and this information can only be obtained from a first order spectrum by resorting to further experiments (such as spin-decoupling(~), see Section 8.19.5). When the magnitudes of the chemical shift differences are comparable with the couphng constants, the simple rules for analysing the spectra break down.

282


Such a state of affairs is found in the 1H resonance spectrum of the hydrogen atoms in glyeidonitrile~z),

ri\,

a

shown in Fig. 8.3. There are thirteen absorption bands in this spectrum as opposed to the twelve bands predicted by first order theory for a three spin system of nuclei of spin value ½. The components of the multiplets are of unequal intensity and the chemical shift differences cannot be obtained by assuming that the true resonance frequency for a particular nucleus is at the centre of its multiplet absorption. Chemical shift differences and spin coupling constants can only be obtained from such a spectrum by subjecting it to a full H

M~\,C__c/CN H ,~/',,Oz/

'H ,:

il

~k

5

cycles

sec-i~

i*~ i

iI

,

j'

i

i~' 1 I

Fio. 8.3 The ~H resonance spectrum at 40 Mc sec-t of glycidonitrile. Reilly

andSwalen(2)

quantum mechanical analysis. This requires evaluating the nuclear energy levels and stationary state wavefunctions appropriate to a system of nonequivalent interacting nuclei in a static magnetic field in the absence of the applied radiofrequency field. By considering certain selection rules it is possible to decide which energy levels are involved in transitions when the radiofrequency field is applied at the resonance frequency. The transition energies (which correspond to the measured absorption bands in the spectrum) can then be obtained by subtracting such energy levels. Calculations of the transition probabilities of the various transitions lead to values of their predicted relative intensities. By assigning the transition energies to the absorption bands in the spectrum (paying due regard to the calculated intensities obtained from estimated values of the parameters involved), it is possible to construct a theoretical spectrum which agrees with the observed spectrum and from which numerical values of chemical shifts and coupling constants are obtainable. By this procedure one can also usually ascertain the relative signs of the coupling constants.

T H E A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A

283

The quantum mechanical principles and the evaluation of theoretical spectra form the main part of this chapter. Monographs dealing exclusively with the analysis of N M R spectra have been written (3' 4). 8.2 NOTATION Berustein, Pople and Schneider (s) have introduced a convenient notation for referring to systems of magnetic nuclei occurring in molecules and it is intended to use this notation (with slight modification) in the present text. Non.equivalent nuclei of the same species with resonance absorption bands separated by chemical shift differences of similar magnitude to the coupling constants involved, are denoted bythe letters A, B, C, D etc: such a group is called a basic group of nuclei. Other non-eqnivalent nuclei in the same molecule separated from this basic group by large chemical shifts, but which are separated from each other by chemical shifts of magnitudes similar to the coupling constants involved, are denoted by X, Y, and Z (these need not necessarily be nuclei of a different species from A, B and C). A further set of non -equivalent nuclei separated from the other two sets of nuclei by large chemical shifts, but separated from each other by chemical shifts comparable in magnitude to the coupling constants between them, are referred to as P, Q, R . . . . Thus vinyl fluoride CH2 = CHF, is referred to as an ABCX system. The three vinyl hydrogen nuclei have absorption bands separated by chemical shifts which are not large compared to the coupling constants, at normal values of applied magnetic fields, and are therefore classified as ABC nuclei. The fluorine absorption bands are separated from those oF, the hydrogen nuclei by a chemical shift large compared with the J,v coupling constants and thus the fluorine nucleus is classified as X. Nuclei which are magnetically equivalent are referred to by the same symbol and the number of nuclei in a molecule having the same symbol is indicated by means of a suffix on the symbol. Hence, 1,3-dibromo 2-nitrobenzene H NO= is said to form an AB2 system. Ethyl chloride, CHsCH2CI, constitutes an A2B 3 system on this nomenclature. Nuclei which show chemical shift equivalence but are not magnetically equivalent are referred to by the same letter but are distinguished from each other by means of primes. An ABB' system describes a molecule in which the two B nuclei have the same chemical shift but couple to different extents with the third nucleus A (6). In the molecule 1,1-difluoroethylene, CF2=CH~, there are two pairs of symmetrically equivalent nuclei which are not magnetically equivalent. This is a consequence of the two hydrogen nuclei coupling with each individual fluorine nucleus to a different extent (J~, ~ J~vw). Hence, this molecule is classified as an AA'XX' system. The notation of Bernstein, Pople and Schneider does not distinguish between magnetic equivalence and chemical

284


shift equivalence, therefore it has been necessary to introduce the modification outlined above to clarify this point. Some authors tx'7~ refer to an AA'BB' system as an A 2* B2s system but this convention will not be used in this text. Table 8.1 lists a series of molecules together with their classification in the terminology to be used in this text. TABLE8. l CLASSIFICATIONSOF SOMETYIUCALMOLECt~ Compound H Clf~tiF Clt~Cl

Classification

AX

c1 CH2ffiCCIBr CH2ffiCFCI CHzffiCHBr CH2FCl CHzF CHzffiC=CHCI CF_,ffiCHz H

AB ABX ABC A2X A3 X AB2 AA'XX' AA'BB'

H (~taCH2)PFz

AaB2PX2

ir

o

8.2.1 Equivalent Nuclei Nuclei in molecular systems can exhibit various kinds of equivalence and it is imperative that the meaning of each be clearly defined at this stage: (a) Chemical shift equivalence. Nuclei which have the same chemical shift are said to show chemical shift equivalence. (b) Symmetrical equivalence. Nuclei which interchange their positions in a molecule upon the application of a symmetry operation are symmetrically equivalent: they are of necessity chemical shift equivalent. (c) Magnetic equivalence. Nuclei which have the same chemical shift and which couple to the same extent with all other magnetic nuclei in the molecule are referred to as magnetically equivalent. Hence, if we have a molecule containing a group of nuclei G containing several nuclei G1, G2, Ga ... G~, then these are magnetically equivalent if (i) they all have the same chemical shift: ~ai -- ~a~ -- ~a3 . . . . = ~ , (ii) each nucleus in this group couples to the same extent with each nonequivalent nucleus in the molecule: Jaw'~ -- J ~ ; ffi Jaw'~ . . . . Ja,~j where G~ is some nucleus j in another set of nuclei G'. Whilst it is not necessary for symmetrically equivalent magnetic nuclei to be magnetically equivalent, symmetrical equivalence often does result in magnetic equivalence. Thus, although 1,1-difluoroethylene,

THE A N A L Y S I S OF H I G H R E S O L U T I O N S P E C T R A

285

Hi ~Ft has two pairs of symmetrically equivalent nuclei neither pair is magnetically equivalent: this is because the symmetrically equivalent nuclei are not coupled to the same extent with each non-equivalent nucleus in the molecule e.g. because J.xFx • Jtl2V*, Hx and H2 are magnetically non-equivalent and likewise for the two fluorine nuclei. On the other hand, difluoromethane, CH2F~, contains two pairs of symmetrically equivalent nuclei which are also magnetically equivalent: the two nuclei in each pair have the same chemical shift and both nuclei in any one pair couple with each non-equivalent nucleus in the molecule to the same extent. The same is true for the non-planar molecule 1,1-difluoroallene,

F?C=CfC4H F/

\H

where the symmetrically equivalent nuclei are also magnetically equivalent. Both molecules give spectra of the type A2X2. The requirements for magnetic equivalence can sometimes be satisfied accidentally in a group of nuclei which are not symmetrically equivalent. Magnetic equivalence of this kind cannot be predicted by examination of the symmetry of the molecule but can be detected only by experimental observation. A case of accidental magnetic equivalence is provided by ethylene monothiocarbohate(7)t. Symmetrically equi'valent magnetic nuclei in substituted aromatic compounds are frequently ~nagnetically non-equivalent. Mono-substituted and o- and p-disubstituted benzene derivatives all possess two pairs of symmetrically equivalent nuclei which are magnetically non-equivalent. In the molecules S

S

S

S = Substituent H~

H4

S'

H1 and H2 are magnetically non-equivalent since Jmns ~ J.2as and likewise for Hs and ]Lt. Coupling constants between magnetically non-equivalent nuclei can feature in the spectrum. Whenever the term "equivalent nuclei" is used without further qualification in the present text it is implied that such nuclei are magnetically equivalent.

8.2.2 EquivalenceResultingfrom Con/ormationalMotion Rapid rotation of a group of magnetic nuclei about a molecular axis of rotation can sometimes result in the group of nuclei becoming effectively magnetically equivalent. The methyl and methylene hydrogen atoms in an ethyl 1" The spectrum can be explained also in terms of a "deceptivelysimplespectrum" from magneticallynon-equivalentnuclei.

286

HIGH RESOLUTION

NMR SPECTROSCOPY

derivative form two groups of magnetically equivalent nuclei as a result of internal rotation (see Section 9.5.5). If the methyl groups were held in a fixed position with respect to the methylene group as shown X

Hf~H2 H4~V//'--~H5 H3

then

~1 =~2 :#~3 d3,

=

dl, ~ d2,

and the methyl hydrogen atom would therefore be non-equivalent. At room temperatures, the molecule undergoes rapid internal rotation about the carboncarbon bond, and the averaging process associated with the rotation results in the methyl hydrogen atoms and those of the methylene group becoming two groups of magnetically equivalent nuclei. The spectrum of the fixed rotamer is of the type AA'BB'C while that of the rapidly interconverting molecule is of the type A2B,. If the nuclei of a group in a molecule are rendered magnetically equivalent by rapid internal rotation it is possible that by lowering the temperature of the sample, free rotation will cease (the rotational isomeric forms will no longer be equally populated) and the previously magnetically equivalent nuclei will lose their equivalence (see Section 9.5.5). Rapid interconversion of ring compounds can also result in nuclei which are non-equivalent in the fixed conformational isomer becoming magnetically equivalent (see for example Section 9.7 in which is described the low temperature investigation of perfluorocyclohexaneCS)). An important property of magnetically equivalent nuclei is that the coupling constants between the nuclei in such a set have no effect on the observed N M R spectrum and therefore the coupling constants cannot be measured. Hence, in the molecule SFsCI, which has the tetragonal bipyramidal structure

.... t-CL the four equatorial fluorine atoms at the corners of a square are magnetically equivalent and hence the coupling constants JF2 F3 and JF2 F4 are not observed in the 19F resonance spectrum of this molecule. However, coupling constants

T H E A N A L Y S I S OF H I G H R E S O L U T I O N

SPECTRA

287

between symmetricallyequivalent nuclei which are not magnetically equivalent are observed in nuclear magnetic resonance. For example, in 1,1-d/fluoroethylene, CF2-~-CH2, values of dam and.lvF are obtainable from both the IH and the 19F resonance spectra. In favourable circumstances, suitable isotopic substitution into a molecule can cause a group of previously magnetically equivalent nuclei to become non-equivalent with the result that coupling constants between the nuclei are observed. By enriching a sample of ethylene with lsC atoms many of the resulting molecules have the structure H, /

-

\m

and the four hydrogen atoms (which are magnetically equivalent in ordinary ethylene) form two pairs of magnetically non-equivalent nuclei, and from the i H resonance spectrum it is possible to obtain the coupling constants ./12, ./,s and -/1,- In other molecules containing groups of magnetically equivalent nuclei on adjacent carbon atoms, for example CICH2CH2CI or dioxane, examination of the resonance of the hydrogen nuclei attached to the 13C present in natural abundance (1"1 per cent) gives the desired coupling constants (9). 8.3 QUANTUMMECHANICALFORMALISM It h~s been shown in Chapter 2 that a collection of magnetic nuclei in a magnetic field has a discrete energy spectrum, and transitions between energy levels can be stimulated by applying a secondary magnetic field oscillating with the appropriate frequency (see Section 2.5). The general problem in constructing any theoretical spectrum is that of obtaining expressions for the energy levels of the system and then of computing the energies and probabilities of allowed transitions. No more than an outline of the general method can be given here, and the reader without prior knowledge of quantum theory is recommended to consult referencesl0 to 13. Experiments aimed at obtaining values of the chemical shift and spin-spin coupling constants are carried out with the aid of a stimulating magnetic field //1 of intensity small enough to avoid saturating the system. In this case/'I1 does not change the energies of the stationary states of the system, but simply induces transitions between them. The nuclear system can be regarded therefore as being in an external magnetic field whose intensity does not change with time, and the energy levels are found by solving the time-independent SchrSdinger equation (Chapter 2 of reference 10)

in which ~lff is the Hamiltonian operator and ~ is the stationary state wavefunction. Before discussing the form that ~ and ~ take in the particular case of nuclear resonance it will be useful to look at some of their general properties. It is always possible to find the form of the Hamiltonian but the statefunction is not usually known, except of course that it is a function which satisfies equation (8.1). However, an arbitrary statefunction ~ may always be expanded

288


as a linear combination of the eigenfunctions q~, of some physical quantity f

i.e. ffi E c. ~.

(8.2)

n

where the ~,. satisfy the equation

f~o. = f, ~.

(8.3)

f~ being an eigenvalue o f f . The summation in equation (8.2) extends over all possible values of n, and the set of functions ~o, is known as a complete set (p. 8 of reference 10). The coefficients C, satisfy the relationship IC.[ = = 1

(8.4)

n

that is, the series is normalised. Further, the functions ~o. are orthogonal,

f,~,,, ~.* dr = ,~,..

(8.5)

(d~ symbolises integration over all the possible values of the coordinates). (Sin is the Kr6necker delta: (~,, = 1 i f m = n, zero otherwise. A set of functions satisfying equations (8.4) and (8.5) is known as an orthonorrnal set. If the orthonormal set (8.2) is substituted into equation (8.1) we have X" Z C. ~0. = E E C . ~ ..

(8.6)

Multiplying both sides of (8.6) by ~v* a/~d integrating over all space

f ~*~ F; C. ~. dT = E f ~*. Z C. ~. d~.

(8.7)

The integrals on the L.H.S. of (8.7) will be denoted by the symbol X',.. i.e.

x ' . . = f ~ * x " q~. d'r.

(8.8)

The integrals on the R.H.S. of (8.7) are zero unless m = n because of the orthonormality of ~o.. Equation (8.7) may be written as

Z a~?',.. C. = E ~. C. Or,. n

(8.9)

n

thus (x'.,. - ~..~)Z

c . = 0.

(8.10)

n

Equation (8.10) gives a set of homogeneous linear equations in C., and the necessary and sufficient condition for non-trivial solutions is that the determinant of the coefficients should be zero " t ) , i.e. IX'..

- &..El

= 0

(8.11)

Equation (8.11) is a polynomial in E of degree n and is known as the secular equation. The integrals X',,, may be arranged as a matrix which can be regarded as representing the Hamiltonian operator (Chapter 9 of reference 11). Similarly, the matrix formed from the coefficients C. represents the statefunction ~ .


289

The matrix formed from the elements .,~m, obeys the normal rules of matrix algebra. The elements o~?'=, are not unique since the wavefunction W could be expanded in terms o f another complete orthonormal set q,~ so that ~'O

J'

~ e . ffi f ~..,~ ~, dT.

(s.12)

However, the eigenvaiues obtained from the secular equation are unique even though the matrix elements are not. If the functions ~, are in fact eigenfunctions of the Hamiltonian, that is, ~. ffi ~ ~. (8.13) then the matrix representing.~ has all elements ~ , , zero except ~¢o , that is, the matrix is diagonal. This follows from equations (8.8) and (8.12),

but since ~, is an eigenfunction of .~,

~ , . -- f~,_~ ~, dT ffi F ~ f ~ * ~0. dT = E, ~ ,

(8.14)

so that the diagonal elements are the eigenvalues of ~ . The operators in quantum mechanics which correspond to measurable physical quantifies must have only real eigenvalues, so that if () is an operator with real eigenvaiues g, then

fv,*

6wd~ =

fv, 6*w* d~.

(8.1~

6 * is the operator which satisfies the equation 6* V'* ffi g* V'*.

(8.10

If ()' is the transposed operator (), defined as

then for operators with real eigenvalues 6 . ffi 6,.

(8.18)

That is, the transpose of the operator is equal to its complex conjugate. Such an operator is said to be Hermitian. The matrix elements of a Hermitian operator, such as Jff, obey the relationship

which for operators with only real matrix elements means that ~°m, ffi ~ m .

(8.19)

Thus the matrix representing the Hamiltonian operator is symmetrical about the leading diagonal. m~s. 10

290

HIGH

RESOLUTION

NMR

SPECTROSCOPY

8.4 FACTORmINGTim SECULAREQUATION The secular equation (8.11) is a polynomial in g of the order n, so that the exact solution of the equation would become impossible for n > 3, unless the order of the polynomial can bo reduced by factorisation. Factorisation may be achieved by a suitable choice of the functions 9,- If the functions 9n are eigenfunctions of an operatorfwhich commutes with the Hamiltonian, then ( M ' f - f X ' ) = 0.

(8.20)

Consider two functions 9m and 9n, both eigenfunctions of f with eigenvaluesfm and fn respectively, then

f g* fg, dr

= f g*M'f~9, dz = f~j'9*ov:9,dr.

(8.21)

f 9 * f g 9,, d*.

(8.22)

Since ~ andfeommute,

f 9:,.~taf9nd*

=

Now f is an operator corresponding to some measurable quantity and is therefore Hermitian, thus (fgm)* = (f= 9m)* -- fm 9* (8.23) and

(fg~,)* = 9*f.

(8.24)

The R.H.S. of equation (8.22) is seen to be f 9~*f~'~g'9, dz = f m f 9* vYa9. dr

(8.25)

and, equating (8.25) with (8.20

f . f g * x ' 9 , d, = f . f g = a ' 9 ,

d,.

(8.26)

But in general fn ~ f , , , so that f9~* ~ 9, d~ = O, unless 9m and 9n belong to the same eigenvalue off. This means that the matrix ~ = ~ is considerably simplified and the secular determinant contains many zero elements. If the set of functions 9~ can be grouped into sets 9t belonging to the eigenvalue f . then the secular determinant is factorised into sub-determinants of order i 2 and the secular equation is factorised into polynomials of order i,

8.5 THE HAMILTONIAN The form of the Hamiltonian operator for a collection of nuclei in a magnetic field is (I 3) o~ = + ~ y , h H,I=, + ~ Z d u I , .Ij (8.27) i

i /'/B.

As for other A B , systems, the general appearance of the spectrum depends only on the ratio J/~o ~. Figure 8.11 illustrates the changes which occur in an AB~ spectrum as theJ/~o¢~ ratio varies. When J/~o ~ is large and 1,0d has a smaU but finite value, a symmetrical triplet is observed (see Fig. 8.10) and the separations between components of the triplet are given by ~ o d. In only one of the spectra illustrated does the combination band 9 appear (see Fig. 8.10) and its intensity can be seen to be low. A closer examination of the theoretical AB2 spectrum for J/~o ~ •ffi 0"5 reveals the following interesting features: (i) F r o m the transition energies given in Table 8.13 it may be seen that band 3 in Fig. 8.10 gives the true chemical shift for the A nucleus and contains no contribution from spin--spin coupling. (ii) The true chemical shift of the B nucleus is given by the mean position of bands 5 and 7.

326


It is possible to say which of the two types of nuclei in an AB2 system is the more shielded from examination of its resonance spectrum but it is impossible to find the sign of the coupling constant in this way. If the absorption bands in the experimentally observed spectrum are well resolved, then the analysis is achieved simply by assigning the absorption bands of the spectrum to the transition energies given in Table 8.13 and then finding the chemical shift difference between the A and B nuclei by direct measurement

d

i...%8-(c+-C

-

)..~ •

I..-~,o8 +(C+-CJ I I

2

'

3

1

_--;

I I

4

~ =0.5

It

FZG. 8.10 AB2 type spectra having .//~oG = 0-0 and 0.5

of the separation between band 3 and the mean of bands 5 and 7. Values of (7+ and C= can also be obtained from the spectrum by suitable subtractions of the absorption bands, and knowing these values and )'o ~ one can obtain the J value for the system using equations (8.132) and (8.133). An alternative method of assigning the bands in an AB2 spectrum has been used by Bernstein, Pople and Schneider c2°). They have drawn up a table of the transition energies in an AB2 system for various values of J/% ~ (see Appendix D) using a fixed value for ~o 6 of unity in each case. The problem of assigning the observed spectrum to the transition energies is one of finding the 3/),o 6 which most closely reproduces the observed spectrum in general appearance, and then adjusting the J/% (3 ratio so that an exact fit of the theoretical and ob= served spectra is achieved. In order to do this, one must first scale down the separations in the observed spectrum to correspond to a % ~ value of unity: it is relatively easy to derive a value of ~o 8 from the observed spectrum and

THE ANALYSIS

OF HIGH

RESOLUTION

SPECTRA

327

employ it in the scaling-down process. Bernstein, Pople and Schneider (=°) have analysed the 1H resonance spectrum of the ring hydrogen atoms in 2,6lutidine (see Fig. 8.12): neither the hydrogen nuclei in the methyl groups nor the nitrogen nucleus interacts with ring hydrogen nuclei and consequently they can be excluded from the analysis. Because of the symmetry of the molecule the hydrogen atoms in the 3 and 5 positions are magnetically equivalent. The three

3"042

1-333

~

0 6 "59 I] l,,

o.o,

II,

0 • I00

~000

_4LO

i

J -20

0

i

20

1

I 40

Cycles sec t

FIG. 8.11 AB= theoretical spectra for nuclei of spin ½ and a chemical shift difference ~o~ of 10 cyclessec-I ring hydrogen nuclei thus constitute an AB2 type system. The spectrum shown in Fig. 8.12 may be seen to resemble some of the theoretical AB= spectra depicted in Fig. 8.11. An analysis of the spectrum was performed (2°) by fitting the observed spectrum to a theoretical spectrum having J/~o ~ = 0-375. From the resulting theoretical spectrum, the above authors calculated the following numerical values for the chemical shift difference and the coupling constant between the A and the B nuclei: • o ~ = 21"9 cycles s e c - j 3" = 8"2 cycles sec -1. The A nucleus is less shielded than the B nuclei.

328

HIGH

RESOLUTION

NMR SPECTROSCOPY

Relative intensities can be obtained by using equations (8.128-8.131) to provide values for 0÷ and 0_, f r o m which the intensities can then be calculated. Agreement between the observed and calculated intensities can be taken as an indication that the assi~mment is correct. Table 8.14 gives the observed and calculated transition energies and intensities for the 1H resonance spectrum of 2,6-1utidine and the agreement between them can be seen to be fairly good. By examining the sample at a different magnetic field strength and showing that the observed spectrum is consistent with the parameters obtained from the original assignment one can confirm that the analysis is correct. Tables similar to Table 8.13 can be drawn up for all Ap B, systems, where the general appearance of the spectrum depends upon a single parameter, J/vo & Some of these tables are reproduced in Appendixes D - H . F o r systems containing H

C H~.."-.l~f,~2CH.x

4

iO c~cles sec-4

A (Hydrogen 4)

B (Hydrogens 3ond 5)

FIG. 8.12 Example of an AB2 type spectrum-the IH resonance spectrum of 2,6-1utidine obtained at 40 Mcsec -1. Bernstein, Popl¢ and Schneidert2°) TAeLB

8.14 CO~AmSON OF CALCULAX~.DAND OeS~RV~) SPEC'mA FOR 2,6-Ltrrmn~mt=°) Energy relative to line 3

Relative intensity

Line Calc 1

2 3 4 5

-- 9"4 cycles sec-t -- 3.0 0 6.5 18.3" >

19.0. 25"5 27"7 46"7

Obs

Calc

Obs

-- 9"4 cycles sec- t

0"47 0"68 1.00 1"86 2"85 I 2"52 J

0.45 0-75 0.9 1.6

--2.8

0 6"6 19"3 25"6 27"7

5"0 3"3

0"0025

THE ANALYSIS OF HIGH RESOLUT ION

SPECTRA

329

more than one coupling constant and chemical shift separation, it is impossible to provide for all possibilities in tables of this sort since there are several variables involved. It is usual in such cases to restrict calculations of the transition energies and their intensifies to those expected for a few typical examples using reasonable values of the various parameters and varying only one, and that over a limited range, at any one time.

8.13.3 The Four Spin System, AB3 A molecule containing three magnetically equivalent nuclei coupled to a fourth non-equivalent nucleus, which is separated from the other three nuclei by a chemical shift of the same order of magnitude as the coupling constant between them, is described as an AB 3 system. Both the cis and trans isomers of the fluorinated alkene, CFsCCI=CFCI constitute AB s systemsc23). Other molecules giving rise to spectra of t h i s t y p e are methylacetyleneC~'. 2s), CH-=CCH3, methyl mercaptan c7. ~6), CH3SH, and methanol czT. 28), CH3OH, where rapid rotational motion about the C--S and C - - O bonds results in the methyl hydrogen atoms becoming magnetically equivalent. It is necessary to examine methyl alcohol in acetone solution in order to suppress the chemical exchange of the hydroxyl proton, which obliterates the fine structure from / spin-spin interaction ceT). The grouping C H 3 - C H also forms an AB3 system because the three coupling constants JAB are averaged by rapid rotation about the C - - C bond at room temperatures: examples of spectra from this grouping are given on p. 97 of reference .29. As in other ABn systems, the observed N M R spectrum can be described in terms of the two resonance frequencies o~Aand OJBand the coupling constant J. Coupling constants between the three magnetically equivalent nuclei need not be considered in the analysis. The Hamiltonian operator for the AB3 system is

= [O~AIA. + O~BIB. + JI^. IB] = [~'HAXA, + 7'HBIB, + JIA, I , , + ½J(I~,l~ + I ; I ; ) ]

where and

Ie,-- I~, + I~, + I~, I~ = I~ + I Bs + "Jr- I+]S~

The nuclei in an ABs system can be considered, for symmetry arguments, to be at the four apices of a trigonal pyramid £

HRS.

11~

330


the three equivalent B nuclei being at the corners of the basal equilateral triangle. Such an arrangement of nuclei is described by C3o symmetry. The basic symmetry functions for the system can be constructed in such a manner that they remain invariant under operations of the point group C3,. The appropriate symmetrised spin functions are given in Table 8.15 together with the irreducible representations to which they belong. Also given in Table 8.15 are the diagonal matrix elements for the AB3 system. Theonlynon-zero off-diagonal elements are (~,1 ~ 1 ~ ) = ~' (~6[.,Wl~pT) = x/3J/2 (~0xol~l~0,0 = - ½ J

The four E states are all doubly degenerate, and consequently since only one of each is considered in Table 8.15 it is necessary to double the intensities of any transitions involving the E states. The matrix corresponding to the secular equation for this system can now be constructed (16 x 16) and factorised into six(l x 1) and five (2 x 2) subdeterminants. As in all other AB~ Systems, the eigenvalues and the eigenfunctions can thus be obtained from solution of linear and quadratic equations only, as described in Section 8.13.5. The transition energies can be evaluated from the stationary state eigenvalues by subtracting those which are involved in allowed transitions, and the relative intensities of these transitions can be predicted from the eigenfunctions (p. 386 of reference 7). Table 8.16 gives the expressions for the transition energies and their relative intensities. There are sixteen transition energies in all, two of them being combination bands of rather low intensity. The positive quantities D÷, D_, Do, Do' and the angles 0+, 0_, 0o, 00' between 0 and zc have been introduced for convenience and they are defined as follows D+ cos20. = ½~o 6 + ½J D+ sin20+ = 31': J/2 Do cos20o Do sin20o D_ cos20_ D_ sin20_ D~ cos20~

= = = = =

½Vo,3 J ½-Vo~ - ½J

31;2 .1/2 .tv o 3

D~ sin20o = ½J By elimination of the angles in the above expressions one obtains the relationships for the D values in terms of Vo6 and J. D+ --" +½{(%c$) 2 + 2J(~ oc$) + 4J2} 1/2 Do = + ½ {('0 ~)2 + 4./2}1/2

D = +½ {(% ~)2 _ 2J0'o 6) + 4j2} 1/2

THE ANALYSIS

OF HIGH

~,~.,~ ,+

+

+

RESOLUTION

~.~.

_,.

i,,-q

~#

m

I

~

331

SPECTKA

'--,

I

+

!,-i

~+I

"~#

,~I

I

fl

,,~ N

,,~

,@

'

N

..,,.

i

,@

~

0 N I

z

•,T

~I

,,T

,T

',T,,T

,,T

,,T

,,T

~

~

I

I

~

°~

OW

•-I-

11 a*

+

+

+

I

I

332

HIGH

RESOLUTION

NMR

~'1

÷ 0 o

SPECTROSCOPY

~'1

-L

+

-L

~

m

~

(vl

-~- Jl-

+-L ~

"~

0

~

0

~

0

÷

I

I

++

o

c,4

=?
0, indicating that the states with Is - 0 have a longer lifetime than the others. The lifetime of a spin state depends upon the probability that a transition will be induced by the randomly fluctuating magnetic fields produced at the nucleus by the surrounding lattice, and the probability per unit time of such transitions depends upon the total spin angular m o m e n t u m / , decreasing as I increases ~4). All AB~ systems with n > 1 have the mean of the frequencies of the third and fourth B transitions corresponding to vs, hence for n even, the chemical shift difference v^ - vs (that is, Vo 6) can be measured directly from the spectrum. The coupling constant may also be obtained from the spectrum since the mean of the firstA and the first B transition frequencies relative to VA is ½(% 6) +

1(is + ½). (ii) n odd. When n is odd there must be a sub-spectrum corresponding to Is = ½; examination of Table 8.21 shows that for I = ½ the first two A (or B) transitions are separated by the coupling constant d.

8.13.6 The General Features of an ApB~ Spectrum When the A group contains either more than one nucleus or a nucleus with spin IA > ½ then the secular equations to be solved are of order ~ 3. But even in the most complex ApB, systems it is still possible to obtain values of J and 6 from the spectrum. With the general spin functions AtA.mABx~.mB, which are eigenfunctions of I 2, 12 and 1~, as the base functions, the secular determinant factors first into sub-determinants corresponding to the possible values of IA and Is, and then into sub-sub-determinants depending upon the different values of m for a given pair of 1A and Is. Consider a determinant corresponding to particular values of 1A and lB. The maximum and minimum values of m are

mmax=ls+[A m~l, = --IA -- /8 SO that the sub-determinants belonging to the values m=,~ and mmia are of dimensions 1 x 1, and the functions

AIA. IA BtB.~B and ArA. -IA Btn. -IS are eigenfunctions of the Hamiltonian. The second largest value of m is 1A + IB - 1 which can be arrived at in two ways: when mA = IA -- 1 and - - m s = Is, or when mA = 1A and mR ---/13 1. In general, the state with -

m=Ix

+ ls - - r

THE ANALYSIS OF H I G H RESOLUTION SPECTRA

345

can be arrived at in the r + 1 ways m^

mB

I^

/B--r

I^-1

ID--r+ l

IA -- r

IB

NOW the value of mA must lie between -t- IA, similarly mB must be between :i: IB, so that if 1^ _~ IB the maximum value of • is 21^. This means that the value of m common to the largest number of spin functions is mf

l s - - I^

and the highest order of equation to be solved is (21^ + 1). In systems where all nuclei have spin ½ the secular equations are of order equal to or less than (p + 1); that is, A2 B, involves a cubic equation, A3 B~ a quartic equation and so on. The spectrum of an Ap Bn system is made up of a superposition of all subspectra A, B, where • and s have the same parity (that is, even or odd) as p and n; each sub-spectrum is weighted according to the values of gZA and gIB" A nomenclature has been developedcsS, 13s) for the sub-states corresponding to the possible values of I^ and Is in which the state with I = 0, ½, l, ~, 2, ~ . . . is known as a singlet, doublet, triplet, quartet, quintet, sextet etc., (abbreviated to S, D, T, Q, Qt, Sx) respectively. Thus each state is known by the number of magnetic sub-levels in that state. For example, the A~B~ system has the substates

S^TB, T^TB, T^Ss, SAS~ and the AsBz system has the sub-states SBD^,

SBQA, TsD^,

TBQ^.

Dividing the spectrum into the contributions from sub-spectra (sometimes known as the "complex particle" method) shows that an A~B, system contains lines at the same positions as in all systems with smaller values ofp and n, providing that they have the same parity. For example, the A2B6 spectrum of propane c35~ contains the lines of the A2B2 system (see Section 8.13.8), so that if all absorption bands are resolved and can be assigned to particular transitions, then the chemical shift difference ~o ~ can be measured directly from the Sl~'trum. The types of sub-spectra contained in an ApB, spectrum depend on whether n a n d p are odd or even. Ifp is odd then the general features of an AB. spectrum appear in A~B,, hence for n even it may be possible to obtain ~o ~ and J directly

346

HIGH

NMR

RESOLUTION

SPECTROSCOPY

from the spectrum, and for n odd two lines in the spectrum have a separation equal to J. When both n and p are even there are lines at the frequencies v* and vEhaving the relative intensities (l ‘I)2* l P and T-’ n. Also, since the AzB3 system has sub-spectra in common with all systems havingp and n even, then two lines will occur separated by 2 J (see Section 813.8). The number of lines in an APB,, spectrum can be calculated(“) in a manner similar to that adopted for AB,. The number of A transitions for n and p odd or even is given in Table 8.22. The AB, spectrum contains only one type of combination transition, that is when d mA = + 1, d m, = - 2, but in the TABLE8.22 THE NUMBEROF A TRANSITIONS M AN APB. SYSTW

NA

D

II

odd

odd

;(P-t

odd

even

;

even

odd

&p

t 2)(/r + l)(n + 3)

even

even

&P

c Z)n(fl + 4) + 1

ly(n+l)(n+3)

(p + 1)z!2(n f 4) + 1

TABLE8.23 THE NUMBEROF COMBWA~ONTRANSITIONS IN AN A,,B, SYSTEM IN WHICHAm,,

p

I n

- -r, Am, = (r-l)

!

NM

odd

odd

$ (p i l)(p + 3 - 2r)(n + l)(n + 5 - 2r)

odd

even

$(p+l)(p+3-2r)n(nf6-2r)

even

odd

$p(pi-4-2r)(n

even

even

+ l)(n + 5 - 2r)

-2r)n(n+6-2r)

A,& system, mixed transitions involving changes d mA = + 2, d mB = - 3 etc., are possible. In general, a combination transition involves dm, = - r anddma = +(r - 1) (or vice versa) where r may take the values 2,3 . . . 2I,4, assuming that ZA< ZB.The number of combination transitions in which :1 nt,& = - r is given in Table 8.23: the number of transitions with dmB = - r and d mA = (r - 1) can be obtained from this table by interchangingp and 12.


347

In all ApB, systems the transition frequencies and intensities may be expressed in terms of a single parameter J/% ~. The form of the spectrum is determined solely by this parameter (apart from a scaring factor) so that a trial calculation with an approximate value of .l/% 6 enables the spectral bands to be identified with the calculated transitions. The next three sections are concerned with some important examples of ApB, systems. 8.13.7 The Four Spin System, A2X2

A system containing two different pairs of magnetically equivalent nuclei separated by a chemical shift large in comparison with the coupling constant between the non-equivalent nuclei is described as an A2X~ system. The system is included here to emphasise the nomenclature used throughout this monograph. Analysis is of the simple first order type described at the beginning of Chapter3. The spectrumcomprises two symmetricaltriplets (of intensity 1 : 2 : 1); molecules giving rise to this type of spectrum are methylene fluoride, (CH2F~) and 1,1-difluoroallene allene, (CF2 = C ffiCH2). 8.13.8 The Four Spin System, A,B,

A system containing two different pairs of magnetically equivalent nuclei separated from each other by a chemical shift of the same order of magnitude as the coupling constant between the non-equivalent nuclei is described as an A2B2 system. In this system, both the A nuclei are coupled to an equal extent with both the B nuclei, and neither the coupling constant between the A nuclei, JAA, nor that between the B nuclei, J~s, feature in the NMR spectrum. There are only a few molecules which have magnetic nuclei satisfying the above criteria: examples are provided by certain c/s-disubstituted SF6 derivativesc3" (see Section 11.16.2). Sulphur tetrafluoride~3~ (see Section 11.16.3)is a rather exceptional example of an A2B2 system: the molecule has a trigonal bipyramidal structure with one equatorial position occupied by a lone pair of electrons. Although the two magnetically non-equivalent pairs of fluorine nuclei give rise to an AzB, type spectrum at - 100°C (see Fig. 11.19), the chemical shift difference between them is large compared with the coupling constant (III) and 2,3 dihydrofuran°9> (IV) can be explained by an A2B2 analysis. It is now fairly certain that the spectra of these molecules are deceptively simple AA'BB' type spectra. Each of the molecules is cyclic and hence there is no possibility of free rotation about a C--C bond to average the cis- and trans- H - H coupling constants between the A and B nuclei. It was originally thought that since only one coupling constant could be measured from the spectra, that the coupling constants v~eT~t"and J~'~' must be accidentally equal. However, Gutowsky and Grant c'°~ have made some theoretical calculations which show that these

348

HIGH

RESOLUTION

NMR

SPECTROSCOPY

spectra can also be reconciled with an AA’BB’ system (see Section 8.16.2), provided that IJAA’ + &WI z 5IJAB - JAB’1

IJAA’- JBWIk 5IJm - JAWI

and

6

G-d=0

i

b

I

II a

CHBr-CHBr

CH-CHn

CI-Ia III

IV

In the AzBz system, both IA and IB may take the values 1 and 0 so that on the complex particle nomenclature the sub-states are S.&, S,T,, T& and T,TB. The S.& sub-state contains no magnetic sub-levels and hence does not contribute to the spectrum .The S,T, and TASBdeterminants are both of dimensions 3 x 3 but in each case the secular determinant factors into three first order equations corresponding to the m values 1, 0 and - 1. The determinant for the T,T, sub-state is of dimensions 9 x 9 but the secular equation factors into two ht order, two quadratic and one cubic equations, corresponding to the m values of 2, -2, 1, - 1 and 0 respectively. The 3 x 3 sub-matrix in the T,T, sub-state is (-~8fJ) -J 0 s = -J 0 -J (8.148) 0

(

- J (Y,, 6 + Jl )

The eigenvalues El, E2 and E3 are obtained by solving a cubic equation; the eigenfunctions corresponding to El, E2 and Es are of the form El

aIIAI,

1 B1, -1 + azIAIP

o &,

o +

a31A1,

-1

&,

~52

a12Al,

1 &,

o 4,

o +

a32A1,

-1

&,I

E3

a13AI, 1 B~,-l+a23Al,oB1,0+a33A,,-,B,,,

-1

+

a22A1,

1 I

(8.149)

I

where the coefficients ai1 are normalised eigenvectors belonging to the eigenvalues El, E2 and El. There are seven A, seven B and four combination transitions in an A2Bz spectrum: since the lines are disposed symmetrically about the frequency +(v* + ye) this forms a convenient frequency origin, so that v* = *vo 6 where

vg = -9%8 vg 6 = VA- vg

THE ANALYSIS OF H I G H R E S O L U T I O N S P E C T R A

I "~+ I

]

~

i +"~'~.+

I

I < i-

I

+

÷~

+~

~ +

~÷ +~

~+

+

,~

2

z

g~g

aa~aa

+++

++I++

z

~

z

m

[..,

o

2-

0~,1

,11 Ill

3 9

©,6 II , 4

IJl II j;, ,.o

HI

8,5 12,2

I ?

tO

.

,

!

i

I1

4

12,4

it~2,1

12,~1

6

2

6 I0 t2

8

I| 7 2

I

4

IL!!IIL

2"25 0"50

"13.'3

(2,8)

(~,9) (t6,tO,J2)

=l=ilh

/////1\\\\~,, I

|2,8)

f

'

|

i

•

2

2.25 ' i' O

0.75

2(5,8) 1,7,5,9J3,3 JoJ2,6

Ih i,,,

I

II FIG. 8.17 Theoretical A3Bz type spectra. Corio (7)

12"

0"15

o..

9

19,12) 6

e,~e

5,3

9113 5,3

I0

,=

9,1z

11II III~J ~l,,~°

, l tIll6 4 12= 8II 25

I0

3

(3

5-0

5-0

5-0 >10.0

356


factors of the secular equations for each sub-state are indicated in Table 8.25. The two 3 × 3 matrices are

I'~'(-Y- 1'0 3 - J) S ~: =

+

J 0

1

+(2) l/z J

+ (2) tn J

(8.150)

1

-='(-- ~o~ - J)

and the eigenvalues of S + are E t , E2 and E3 with eigenvectors (art, a2t, a3,), (at2, a22, a32) and (at3, a23, a33) respectively. Similarly, the eigenvalues of S- are E4, Es and E~ with eigenveetors (al4, a24, a34), (ats, a2s, a35) and (ate, a26, a36), respectively. The frequencies and intensities of the allowed transitions are given in Tables 8.26 and 8.27, combined transitions are listed in Table 8.28. The quantities R , , R2, R3 and R4 are defined, RI = [0'o~ - ½j)2 + 6j2],/2 R2 = [(Vo~ + ½j)2 + 6j2],:2 R3 = [0'o~ + ½j)2 + 2j2],,,z

(8.151a)

R4 = [(~'o~ -- ½j)2 + 2j2]t,2 Q, = x/6S/O'o ~ - ½J - R1) Q2 = - ~ / 6 J / ( , o

~ + ½J + R2)

03 = x/2J/0'o ~ + ½J - R3)

(8.151b)

Q , = - , / 2 J / ( ~ o ~ - ½J + R , )

All the frequencies are relative to va and vA - ~% is equal to 1,o & The latter quantity can be obtained from the relationship ~'o0=(B4+B5)/2-A6 where B4, B5 and A6 refer to frequencies corresponding to bands within the B and A parts of the spectrum. After 7,o~ has been found, J can be calculated from any of the line spacings involving R , , R2, R3 or R4. Some typical A3B2 type spectra for various J/~'oO values are shown in Fig. 8.17 while in Appendix H are given line frequencies and relative intensities of A3 B2 spectra over a wide range of J/1,o8 values. 8.14 SPECTRA CHARACTEI~/SED BY T w o COUPLING CONSTANTS

For this situation to prevail, a molecule must contain at least three nonequivalent nuclei: such a molecule would then usually have a N M R spectrum characterised by three coupling constants; an exception will now be discussed.


357

Certain molecules contain three magnetically non-equivalent nuclei of the same species, two of which accidentally have the same chemical shift (an ABB' spin system). Two coupfing constants are featured in the spectrum of such a molecule, namely J^B and JAB', and they are comparable in magnitude with the single chemical shift difference present, namely Vo& Richards and Schaefer (~) have examined the XH resonance spectrum of 2,SMichloronitrobenzene and they have shown that the magnetically non-equivalent HB and HB, ring hydrogen atoms have the same resonance absorption frequency, that is they are chemical shift equivalent. They have calculated the transition energies and relative intensities for an ABB' spin system. They showed that it is only possible to obtain the mean of the two coupling constants [½(JAB + J^V)] and not their explicit values from an ABB' type spectrum. Furthermore, one cannot obtain the coupling constant JBB' from the analysis even though the two B nuclei are magnetically non-equivalent. It should be noted that the ABB' system is a limiting case of an ABC spectrum where vB = Vc. 8.15 SPECTRACHAXACTEmSEDBY THREE COUPLING CONffrANTS

8.15.1 The Three Spin System, ABX An ABX system is comprised of three non-equivalent magnetic nuclei, two of which (A and B) have J^B ~ V0~Aa, and a third nucleus X separated from the other two by a chemical shift which is large compared with the coupling constants involved. The latter nucleus need not necessarily be of a species different from the other two providing that the above condition is satisfied. A molecule having an ABX type spectrum is 2-fluoro-4,6-dichlorophenol(*4). OH Cl(/~F

H~,JH Cl

This molecule possesses two non-equivalent ring hydrogen atoms and a ring fluorine atom. Several types of molecules containing three magnetic nuclei of TABLE8.29 BAsic PRODUCT FU~C'nONS AND DL*OONXLMATRIX ELEM~rrs roe T ~ A B X SYSTEM Diagonal matrix elements ~I

~,~.0~

½(~A+ ~'B+ ~ ) + ~(.TAB+ JAX + J~X) ½(~A+ ~ -- VX)+ ¼(J*B -- JAX -- "YSX) ½(~^ -- ~S + ~ + ¼(--JAm + .tAX-- JBX) ½(--~A + ~B + VX)+ ¼(J--AB -- JAX + -laX)

m, ~o

~PP P~P

½("A -- ~'a -- ~'×) + ¼ ( - - J ^ . -- -'r^x + .1.x)

q~,

PPP

~ ( - - ~^ -- ~ . + ~X) + ~ ( . f A - + J^X + JSX)

~-(-"^ + ,'s - "x) + ¼(- J^s + J^x - Jax) ½(- ~^ - vB + ~'x) + ¼(3AS -- 3 ^ x -- JSX)

358

HIGH

RESOLUTION

NMR SPECTROSCOPY

the same species have resonance spectra closely approximating to that expected o f an ABX system: vinyl ethers (*s. 4~) and mono-substituted thiophenes (4~) are of this type,

H\C=C H

H F"X CN

H/

H~'S)H

R

=

~oR alkyl group

Three resonance frequencies and three coupling constants are required to describe an ABX system as indicated by the diagram

JAB

taAAj~Bx

°Js

X ta X

The system has no symmetry properties and consequently the basic product functions (given in Table 8.29) are themselves used as a basis for the formation of the wavefunctions. The Hamiltonian operator for the system is

X" = (~'AIA. + COSI~, + a'XlX, + ,rASlA.I~, + JAxlA=&. + Jsxls, IX, + ½JAB[I~I~ + I~,I~] + ½dAx[Z~,Ix + I~I~] + 3dBx[I~Zx + I~I~]). In Table 8.29 are listed the basic product functions and the diagonal matrix elements for the ABX system. The only non-zero off-diagonal elements are given by H3a ---- Hs6 ---- 3JAB H2, = H57 = ½JAX

H~3 = H6, = 3I~x The basic product functions 9t and 9s (o~o~ and ~/~/~) are stationary state wavefunctions since they do not mix with any others. The remaining elements of the 8 x 8 matrix can be factorised into two 3 x 3 sub-matriees corresponding to IT = + 3 and IT = -- 3. For IT = + 3, the appropriate matrix is

(/7.,2 - E)

½J, x

tJAx

½J, x

(H3~ - E)

½I^,

½J~x

½JA~

(H,,-

=0.

E)

To obtain the exact solutions of this equation it would be necessary to find the roots of a cubic equation. However, since the chemical shift difference

THE ANALYSIS OF HIGH RESOLUTION SPECTRA

359

between nucleus Xand the other two nuclei is large compared with JAX and Jex there will be very little mixing between functions f2 and functions ~vs and ~v4 even though they have the same value of total spin. This is also true for the function ~v7 which does not mix with ~vs and ~ve. Hence, the ½JAx and ½Jnx off-diagonal elements can be neglected and the diagonal elements H22 and H~7 can be regarded as solutions of the s~g~nlar equation and thus are stationary state energy levels of the system. The roots of the remaining 2 × 2 matrices can be readily obtained by solving the appropriate quadratic equation; the energy levels and the mixed wavefunctions are found to be: Wavefunctions

Energy levels ½~X-- ~JAI~+ D+ ½~x -- ¼JAI -- D+

(¢.osO+)~s + (sin 0+)~,

~3

- (sin 0+) ~s + (oasO÷)~, (cosO_)~s + (sinO_)~6

~5

-+,x -+~x

-(sinO_)~s + (cosO_)~6

- ¼JA! + z)_ - t J A i - z)_

The four non.mixing basic product functions (that is, wavefunctions) and their stationary state energy levels are:

Wavefunctions

Energy levels

,p~ ~, ~

½(~A+~+~X)+¼(J~+JAX+J'X)

V2

~(1¢A "Jr- $~B -- $~X) "~ ~(JTAB -- JAX -- J B ~

~192 0t0~fl

The usually positive quantities D, and D_ and the angles 0, and 0_ (lying between 0 and ~) are defined by the following relationships D÷ cos20+

ffi½vo ~^s + ¼(YAx - Jsx)

D+ sin20+ = ½JA~ D _ cos20_ ffi½~o ~^. - ¼(JAX -- Jsx)

D_ sin20_ ffi½JA~. Hence

D+ = {[½~o 6^n + ¼(JAx -- JBx)] 2 + D _ = {[½,o ~

- ¼(J^x

~x)P

¼j~},/2 I

+ t J ~ } ~/~

J

(8.152)

½(J.x - JBx) = [(D+ + ½a^~)(D+ - ½J.~))'/~ - [(D_ + ½J^~) x (D_

½%~ ^ B

ffi ( D +~ -

-

½J.B)] '/~

¼s~)'/~

-

¼(~Ax -

SBx).

There are fifteen possible transitions between the eight energy levels and these are given in Table 8.30 together with their relative intensifies. Three of the allowed transitions are combination bands, one of which (transition number 13)

360


I I++++I '+'÷'~d

I

I÷ ÷I

I I ÷+

÷ + ~ ~ ~

÷÷

l l l i + + + +

+ I I + + I I +

+ I I + + I I +

Z

44444 +

~

Z

~ +

+

~

4

~ +

+

#

~

+

+

+

+ ~+ ~

Z 0

o

8S8

0 f-

tfttttttt~tt~tt

+

THE ANALYSIS

OF H I G H

RESOLUTION

SPECTRA

361

has zero intensity; the other two are of such low intensity that they are seldom observed at the main magnetic field strengths usually employed. Examination of the transition energies in Table 8.30 reveals that a value for the Jâ coupling constant can be obtained by subtracting the frequencies of any one o f t h e following four pairs of bands: J^B = 3 - 1 = 4 - 2 = 7 - 5 = 8 - 6. Similarly, subtraction of transitions 9 and 12 leads to a value for (J^x + Jinx). It is also possible to obtain values for D÷ and D_ directly from the observed spectrum by subtracting the frequencies of the following bands 2D+ = 6 - 2 = 8 - 4 2D_=5-1

=7-3.

By using these values in conjunction with equations (8.152) the three coupling constants and absorption frequencies for this system can easily be calculated. The assignment of the absorption bands in an observed spectrum to the theoretical transition energies can be made in several different ways depending upon the relative signs of the coupling constants involved. Any assignment of bands must, however, obey the same subtraction rules as the transition energies, namely, 3-1= 42= 7--5= 8-6 | 2-

1 ffi 4 -

3 = 11--9=

10-9=12-11= (lO-

11)=(8-4)-(5-

6-5=

12-

10].

(8.153)

8-7

l)

Figure 8.18 shows a typical theoretical ABX spectrum where the significant separations between the absorption bands have been labelled. The first step in assigning the AB transitions is to recognise the symmetrical quartets (2, 4, 6, 8) and (I,3, 5, 7). It is impossible to ascertain the relative sign of the J ^ s coupling constant from the analysis of an ABX spectrum without recourse to further experiments such as spin-decoupling (see Section 8.19.5). Although the positions of the resonance frequencies are not affected by the relative signs of J^x and Jex the relative intensities of the absorption bands do reflect the signs: one can decide on relative signs by comparing a set of calculated intensities with those observed experimentally. However, even this method is limited to cases satisfying the condition ~s~ JAS/~O~^S --~ 0"5. NO information on absolute signs is obtainable since the combination of + -TAxand - Jsx gives rise to a spectrum identical with that from - JAx and + Jex. The two combination transitions possible for an ABX system (transitions 14 and 15) have an appreciable intensity when 0'A -- ~e) becomes small; they appear in the X region of the spectrum (they need not always be situated outside the X lines as shown in Fig. 8.18). Lines 14 and 15 are intense when JAx is of opposite sign to Jex.

36~

H I G H R E S O L U T I O N NMIi S P E C T R O S C O P Y i~ I

I

l

2[)_

Il 2D+

I1 "'

b

JAB

~-ri

I

I l

( Tl

JA~B 7 6

B

4

2

2 21o,+o_I '~----~,

IS

t2

!0

LI I

Io~~ '~xl '[

1

I1

l

9

~4

Fie. 8.18 A theoretical ABX type spectrum showing the separate AB and X parts. The bands are numbered assuming .TAx> 3sx > 0 and ~A > ~,~. Bands (2, 4, 6, 8) and (1, 3, 5, 7) form two symmetrical quartets

H:~(~ c/C~H5

cycles sec "l

5 6

7

B

f ', 2 3

I 4

l 9

E I l 10 II 12

FIG. 8.19 The t H resonance spectrum of liquid styrene oxide at 40.00 Mc sec-l. Reilly and Swalen¢2)


363

The XHresonance apectrum oJ atyrene oxide (2). This molecule has three nonequivalent ring hydrogen atoms and it approximates to an ABX system since the resonance of the hydrogen nucleus situated at the same carbon atom as the phenyl substituent is separated from the other two by a large chemical shift compared with the coupling constant involved. The 40.00 Mc sec -~ spectrum is shown in Fig. 8.19. Table 8.38 fists the measured absorption band positions for this frequency together with the transitions to which they have been assigned. Values of JAB, (JAx + JBx), D, and D_ can be obtained by the subtraction of appropriate band frequencies and then by using equations (8.152) the following values of chemical shifts and coupling constants between the three nuclei can be calculated: JAB = 5"66, JAx = 2"42, JBx = 4"10 cycles sec -1 ~'o~^ = 100"39, ~o~e = 112"25, ~o6x = 144"46 cycles sec -1. These values can be seen to give calculated transition frequencies which are in good agreement with the observed spectrum (see Table 8.38). However, the calculated relative intensifies for the system are found to differ from the observed ones to a large extent due to the fact that the molecule gives a spectrum at 40.00 Mc scc -1 which only approximates to an ABX type. Precise values of the relative intensities can be calculated by treating the molecule as an ARC system (2) (see Section 8.15.4).

Deceptively simple ABX spectra. Sometimes fewer absorption bands than predicted are observed in the spectra of molecules expected to be of the ABX type. Some of the bands coalesce and others weaken when certain relationships between the various parameters prevail. Less information is available from a simplified spectrum of this kind in that only average values of coupling constants can be determined. Abraham and Bernstein ¢4s) have drawn up a series of theoretical ABX spectra by way of illustrating the simplification that can occur in s o m e limiting cases (see Fig. 8.20). When ~:0~Aeis large compared with JAe a normal ABX type spectrum having 12 bands results. If ~o~Ae ---- 0 or ½(JAx -- Jex) = 0, then the X region of the spectrum becomes a triplet and the AB portion consists of a strong pair of doublets together with four weak absorption bands (see Figure 8.20). If *o ~Ae and the ratio ½(JAx -- Jex)/JAe are both small compared with JAn, then further simplification of the spectrum occurs: the X part of the spectrum is a triplet (the central band being an unresolvable doublet) and the AB part comprises a doublet. The latter system is essentially of the type AA'X, an example ofwhichis provided by the XH resonance spectrum of the ring hydrogen nuclei of 2-furfurol at 6{>00 Mc sec -~ (see Fig. 8.21). Although this spectrum can be explained by assuming that JAX = ,/eX ---- 1"4 cycles sec -1, it is certain that such an equality does not exist (see Section 10.16). The deceptively simple spectrum can be interpreted equally well by using more reasonable values for the coupling constants found in the complex spectrum of the substance dissolved in dioxane (-TAx = 1"89, Jex --- 0.91 and JAe --- 3-16 cycles sec-X). The separation of themultiplet components in Fig. 8.21 gives the average value of the two coupling constants, namely ½(JAx +-Tax) -- 1.4 cycles sec -1. 12a*

364


In connection with deceptively simple spectra, Schaefer C5°~has discussed the relationship between the spectra obtained from the systems A2X, AA'X, ABX and APX. x

aB

,, Ill I,,

Iff I'1 ~ 0 " 9

87

ill

12 lO,tl 9

8

12 I0,11 9

8

7

7

654

32

tlrl

64

53

4,6 3,5

t

(ii)

2

I

2

[

FiG. 8.20 Theoretical A B X spectra for the relatiomhips (i)V0aAa iS large compared with J.s(ii) yogas or ~(J^x - Jax) = 0 (i~) V0~Ae -- 0 and ~(JAx -- Jsx)/JAe approaches zero. Abraham and Bcrnstein (4s>

1"40 cvcJes sac -~

--m. I

'

I

2.80

cycles

sec -~

i

J

2

' j

H×

Ha ,

H8

FIG. 8.21 The 1H resonance spectrum of the ring hydrogen atoms of 2-furfurol (25 per cent in benzene) at 60.00 Mc sec -t. Abraham and Bernstein (4e)


365

8.15.2 The General A~BXp System

A system of three groups of magnetic nuclei gives rise to secular equations of order greater than or equal to three. However, when one of the groups (X) has a chemical shift difference from the other two which is large compared with the coupling constants, then to a first approximation there is no mixing between functions with different values of lx., and the order of the secular equation is reduced. The approximation is valid When I~o~Axl, IVo~Bxl ~ [Vo~Asl, l/Asl,

IJ^xl,

[Jsxl

(8.154)

and the Hamiltonian can be written = (~AIA. + ~81~. + ~x/x. + JAsI^ "IB + J^x]A~ "Ix. +

JBxl,,." Ix.)

(8.155) so that the group X can be regarded as a first order perturbation on the A and B groups, and the order of the secular equations is determined by the numbers of nuclei in the groups A and B. When B contains only one nucleus then only quadratic equations are involved and equations for the transition frequencies can be written in a closed form ~'9~. The operators I ] , I~, I~, (1A. + Is.) and Ix. all commute with the Hamiltonian so that the set of spin functions ~IAo mA BI/2. + 112 X l x , ra,x ,

which are simultaneous eigenfunctions of all these operators, are taken as the basis for the Hamiltonian matrix. It follows that there are non-zero matrix elements of ~o only between the functions with the same values of IA, (m,,, + mB): Ix and rex, that is, only between the two functions ~1 =="~IA,mABll2. 112 Xlx.r~

both of which have

and

9~2 ffi AzA.mA+ , B~/z.- ,/a Xzx,.~x

(8.156)

m--m^+mx+3.

The matrix dements of the Hamiltonian are readily evaluated: ( ~ t [ ' ~ l ~1) ffi m^ ~^ + ½~s 4- mx ~'x + ½ m A J ^ B 4- ½mxJsx + m^ m x J ^ x

(w2 I ~ 1 ~2) -- (m^ + 1)~^ - ½~s +

mx~x

--

½(m^ + 1)J^B

--

½mxJsx

+ m x ( m ^ + 1)JAx

(w~ I,~1Wa) = (~a I ~ l ~00 ---- ½J^s[(IA - mA)(I^ + m^ + 1)] ~/2. Solving the appropriate secular equations gives the energy levels ½[(2mA + 1)~A + 2mx Vx - ½JAs + J^x mx(2m^ + I) + RCZA.,A.~,X)]

and

(8.157)

½[(2mA 4- 1) ~^ + 2mx ~x - ½"/AS 4- JAX mx(2m^ + 1) - R(l^.mA.mX) ] where

(8.158)

R,,A.,.A.MX~ = {[re 6^s - ½(2m^ + 1)J^s + (JAx - JBx)rex] 2 2 + J,,e(I^ - mA)(I^ + m^ + 1)} ~/2

(8.159)

HIGH RESOLUTION

366

NMR SPECTROSCOPY

The eigenvalues corresponding to the two energy levels are 2

-1.

[1 + Qct.~.,,,.~.,,,x,] ['4~A.,,,A+~B~/z.-t/Z Xxx.,,,x + Q'~A.,,'A.,"X~

and IX

× A I A , mAB1/2, t/2 X'lx.lw~¢] 2 -1 Jr" ~(IA,mA.r~X)] [~(/'A, mA,mX) AIA.mA+xB1/2 .- i/2X'zx,mx

(8.160) (8.161)

-- ~'~lA, raA B l / 2 , !/2 XIx, mX]

where the constant QCzA.,,A.mx) is defined as JAe[(IA -- mA)(IA + mA + 1)]'/2 Q(IA, mA. mX)

~'0 ~AB -- ½(2mA + 1)JAe + (JAx -- JBx)mx + RcXA.,~.,,x~"

Each energy level has a degeneracy gtA g~X" (8.162) The frequencies and relative intensities of the allowed transitions can be obtained from the above equations, remembering that the selection rules for a group G transition are /Ira~ = - 1 ,

A m~, = A I~ = A I~, = O

and that the relative intensities are obtained from the matrix elements of the operator I- in the manner outlined in Section 8.11. The results are shown in Table 8.31. Note that in this table a function of the type ArA.~,AB1/2. ± l/2Xx~mx vanishes if mA exceeds q- 1A: for the function ArA.mA+XB1/Z,± 1/2Xrx.mx mA may take the values IA -- 1, IA -- 2 . . . -- 1A, -- IA -- 1. It was shown in Section 8.13.6 that it is convenient to regard an ApB. system as composed of "complex particles" ArB, each of which have fixed values of the spins IA and Is, and the number of such particles is given by the allowed values of 1A and Is. The same approach can be adopted for A,BX~ systems, so that the complete spectrum can be regarded as a superimposition of sub-spectra arising from "complex particles" with each sub-spectrum weighted with the factor gx,.g~x. For example, the system AsBX2 will contain sub-spectra arising from the complex particles,

I & SxDT

SxDS QDT

QDS DDT DDS

5 5 2 3 2 3 2 1

2 1 2

Ix 1 0 1 0 I 0

gl A gl X

THE A N A L Y S I S OF H I O H R E S O L U T I O N S P E C T R A

367

,}]~~ ='i~"~"~'~'~ =., o

~

~.~ =~ . I, R~xA,m..mx~ and QuA.-mA-,..~x~ are changed into RuA.-,~..-J,.,x> and Qc,..-A, ~ , , respectively by changing the sign of JAB. The latter two combination transitions fall in the X region and are also asymmetrical about ~'x f o r . > 1. This asymmetry of the X region enables the relative signs of JAS, JAX and Jnx to be determined when n > 1. When mA = IA, the two X transitions have the frequencies ~x - ½ J ^ x ( 2 I ~ + 1) + ½ ( J ~

- 4~)

~x + ½.r~x(2I~ + l ) - ½ ( J . ~ - 4 ~ )

each having the relative intensity F~'x.mx-*) 2 g~Ag~x" The lines are separated by

2I^-rAx + 4 x ; unlike all other X transitions they are independent of the applied field strength. When n is even, states with IA = 0 contribute pairs of lines separated by Je x. Examination of Table 8.31 and equations (8.159) and (8.162) shows that when n ffi 1, it is possible to obtain the relative signs of JAx and Jnx. When n > 1 it is possible to determine the relative signs of JAn, JAX and Jnx from the X part of the spectrum Ca1" 52)

B Transitions. (i) n even. The states with IA = 0 give lines at ~e + Jnx mx and, since mx may take the values - Ix to Ix, there are pairs of lines symmetrical about ~s separated by Jn x and of relative intensity gu,- o" glx. If p is also even, the state with IA ----0, Ix ffi 0 gives a line at ~'n of intensity glA-o • glx-o. (ii) n odd. The pair of lines arising from the states IA -- ½have the separation JAn. There is a pair of such lines for each value of mx, thus there are 21x + 1 pairs for a given Ix and summingover Ix gives (½p + I) 2 forp even and ¼(p + 1) x (2 + 3) for p odd.


369

A Transitions. The m e a n frequency o f the two A transitions listed in Table 8.31 is ~A + JAX mx SO that for mx -- 0 it is vA. The separation o f the two A transitions is ~ ( I A , m A . reX) - - ~ ( I A . m A - 1, reX)

which is equal to JAn f o r n odd a n d I ^ ffi ½, mA = + ½, o r f o r n even a n d IA ---- 0.

Number oftrar~itions. These can be calculated (49) in a m a n n e r similar to that a d o p t e d f o r A B , a n d ApB,; the results are summariscd in Tables 8.32, 8.33, a n d 8.34. TABLE8.32 THE NU~mEROF A TRANSITIONSIN AN A.BXp SPIN SYST~ Number of transitions cvcn

ev~[l

cv¢~

odd

odd

@vcm

odd

odd

o(÷÷,)(++,)" 1 1 y (n + I) 2 (p + l)(p + 3)

TAsLE8.33 Te~ Ntr~m~ oe B TRANSmONSIN AN AwBXp SPIN S~yrmd Number of tmmilions

¢wm

odd

.~-

+ 1

[(p + I) (p + 3)]

odd

even

~ [(n + I) (n + 3)1

odd

odd

-~- (n + 1)(n+ 3)(p+ 1)(.p+ 3)

+ 1

1

TAm~ 8.34 Tim Nv-~mm oF X TR~-~aoss IN AN AmBXp SPIN SYsteM Number of transitions

(÷ /

even

odd

-~- (p + 1)2

odd

even

P P ..~-.(-~-+ 1)(n + l)(n + 3)

odd

odd

-~- (p -i-I)2 (n -{-I) (n + 3)

1

+ 1

370

HIGH RESOLUTION

NMR

SPECTROSCOPY

TXSL~ 8.35 THE Ntr~mn o7 Co)~t~A'noN TZANSmOtZSIN AN A.BXp 8 ~ r ~

Number of transitions

P even

cven

÷(÷+,) [÷(÷+1).(÷+1)]

oven

odd

¥ 1 p

odd odd

odd

_

1)2]

1 - ~ ( p + 1) [(n2 -- 1)(/7 + 3) + 2(n + 1) (/7 + 1)]

The general AnBPqXp system. Although six coupling constants are involved the treatment of this system is a logical development of that for the A,BXp spin system. The effect of the presence of an additional group P having a resonance frequency greatly different from those of A, B or X has been discussed by Pople and Schaefer (St) and by Corio (49). 8.15.3 The Six Spin System, A3B,X When an A.Bp system is further complicated by the presence of a group X leading to the relationships [~'o~^xl,

[~o~nxI'>IJAxl,

IJ~xl,

I~O~ABI, IJAnl

then the analysis of the spectrum is essentially the same as that for an isolated A.B~ system in that secular equations of the same order are involved. The A3B2X system is important because it often occurs in the spectra of ethyl derivatives (C2Hs),X 0 and Jxx' :~ ~ A ' > 0. Pople, Schndder and" Bcrnstein(~7)

TABLE 8,48 TRANSITION EN~zenm eoR AN A A ' X X ' SYSTEM FOR ~ SPECTRUM

No. I 2 3 4 5 6 7 8 9 l0 L1 L2

Transition 1 $1 -'~ s= lso "~ Yzo "~ Ys Ys -*" Ys ~9 -~tP7

¥9 "*"~6

Is_1-~ 3 eo 2L, 4s o 2s-1 2% 2a_l

- , 2al "~ -,- 2sl -~ 3s; "-* 2ao

la~

la;

2at 2 a _ 1 ~ 2,,;

}

}

Transition energy

Relative intensity

~zN

2

-{N

2

½ g + ½(K = + L~)* - ~ K + ~ ( K = + Ls)~

sin = 0s COS=O,

½K - ½(K~ + LS)~r -½K - ½(K= + L=)* ½M + ½(M2 + L=)*

COS20s

--~-M + ½(M=+ Ls)~

COS2 0a COS2 0 a

½M - ~(M 2 + L~)* -½M - ½(M2 + L=)~

Energies are m c a s u r ~ relative to ~'AK, L, M, N are defined as follows f f = JAA' + JXX' L = ,TAx -- ,TAX'

A A ' PORTION OF

M = /Â' --/XX' N = l a x + JAx' and cos20,: sin20s: 1 = K : L : ( K 2 + Lz)½ cos20,: sin20a: 1 ~ M : L : (M z + Lz)~

sin=O, sin=Oa

sina0.

397


Examination of Table 8.48 shows that the A part of an AA'XX' spectrum will always contain: (i) two strong absorption bands (transitions 1, 2 and 3, 4) separated by N -JAx + "/AX'and centred on the resonance frequency of the A nuclei, *O~A; (ii) twO pairs of symmetrical quartets (transitions 5, 6, 7, 8 and transitions 9, 10, 11, 12). Both quartets are centred on the resonance frequency of the A nuclei, and the components of each quartet nearest the centre of the spectrum are always the most intense. Figure 8.25 shows a typical theoretical spectntm of the A part of an AA'XX' spectrum which illustrates the above features diagrammatically.

3,4 ,/~ 7,12 40

-

r 80

1,2 9,6 / ' ~ I

]It--

120

"

I--

, 200

160

Cyctes sec"

FIo. 8.26 The 19F spectrum o f 1,1-difluorocthylene at 40-00 Mc sec -1. McConneU, McLean and Reilly (ldu)

H~"~.C

"C/F

1,2 3,4

6

7

9

12

f

]

20 cycles sec -~

FIG. 8.27 The 1H spectrum of 1,1-difluorocthylene at 60.00 Mc sec-:. Roberts (4)

398

HIGH

RESOLUTION

NMR

SPECTROSCOPY

The intense transitions 1,2 and 3,4 correspond to the degenerate pairs of transitions and they contain half the intensityof the complete multiplet. McConncLl, McLean and Reilly¢~4)have measured the IH and the tgF resonanc~ spectra of 1,l-difluoroethylcne: reproductions of spectra axe shown in Figs. 8.26 and 8.27. In each case only eight of the ten theoreticallypossible absorption frequencies are observed. The tgF resonance spectrum consists of four intense bands and four weaker satellitebands. Transitions 6 and 9 and transitions 7 and 12 have been assigned to the two central bands of the four intense inner bands and the remaining two bands to the strong transitions 1,2 and 3,4. The other four transitions are assigned as shown in Fig. 8.26. This assignment leads to the following set of coupling constants for the molecule JAA' = JHH' ~' 4 cycles sec-I

Jxx' = JFF" = 37 cycles Sec -~ JA x = J ~

~ 1 cycles sec-I

JAx' = J~"* ---- 34 cycles sec -1 (more recent values (t4°) are 4'8, 36"4, + 0.7 and + 33"9 cycles see-l). At the time it was guessed that J ~ " > J ~ : it has since been confirmed that this is so (see Section 11.11.3). The resonance frequencies for the A and X nuclei arc given by the geometrical centre of the respective multiplets (for exampie the mean of subtracting transitions 1 and 3). Values o f the coupling constants are found from values of K, L, M, N which are obtained from suitable subtractions of the observed transition energies: the absolute signs of K, L, M, N cannot b6 extracted from the spectrum and hence only the relative signs of in, and J ~ are attainable from the analysis. However, Flynn and j~l~ Baldeschwieler c~4°) have shown that the relative line widths in the ~H resonance spectrum of gaseous CHzCFz can be used to identify the origin of the lines. The identification of an AA'XX' spectrum allows IK[ and [MI to be distingnished thus giving the relative signs of Jail and JF~. Similar experiments have been made on cis. and trans- C H F C H F : J a a and JFF are of opposite sign in these compounds (~s°).

Ortho-dichloroperfluorobenzene. An excellent example of an AA'XX' system is found in the molecule 1,2-dichloro-3,4,S,6-tetrafluorobenzene. Figure 8.28 shows the zgF resonance spectrum of the AA' fluorine nuclei in this molecule with the band assignments shown on the spectrum. All ten theoretically possible transitions give rise to resolvable absorption bands and analysis of the complete spectrum gives the following set of parameters for the system l a x = +_ 20.8 cycles sec -~ lAX, = -T- 1"7 cycles sec -~ [JAA" 1 =

7"8 cycles SeC-

l/xx.l ---- 19"9 cycles sec-l vo ~ = If00 cycles sec -~ (at 56.4 Mc sec -l)


399

~,4 t~

CL

Fx .

I0 cycles sec -~

8

Fio. 8.28 The low fieldpart (AA' fluorinenuclei) of the XgF resonancespectrum of neat 1,2,-dichloro-~,4,5,6.tetrafluorobenzeneat 56.4 Mc sec-x

8.16.2 The Four Spin System, AA'BB' A molecule containing four magnetically non-equivalent nuclei forming two different pairs of symmetrically equivalent nuclei separated from each other by a chemical shift which is of the same order of magnitude as the coupling constants involved is described as an AA'BB' system. Numerous examples of this system have been investigated by NMR and they include disubstituted aromatic compounds, thiopbene, furan, monosubstituted pyridines, 1,2-disubstituted ethanes and disubstituted cyclohexadienes as indicated in Fig. 8.29. In all the molecules listed in Fig. 8.29 the four nuclei are magnetically nonequivalent because

J.A'x~ , # JHA'"~" The symmetrised spin functions and the diagonal matrix elements are the same as for the A A ' X X " system and they are listed in Table 8.49. Also given in Table 8.49 are the non-zero off-diagonal elements for the system: it is seen that the only non-mixing spin-functions are s2 and s_ 2 corresponding to the functions o~0~ and ~ / ~ respectively. Thus the spin functions s~ and s_ 2 are already stationary state wavefunctions and the corresponding eigenvalues are given

400

HIGH

RESOLUTION

NMR SPECTROSCOPY

e-

,.*,fig o~

r~ Z

II

II II II

~-

II

t

Ill

II

Z

=

O

I

ilii

E

o

+

~+

+

If+

I

Z

Z

E 0

+

I÷+

I

÷I

I

J


R

R

i T~,-~,O;~,,,

B

(i)

40]

(ii)

(iii)

e,,~s?e A.

OC'H,CH,e,

(v)

(vi)

(iv)

R is a n o n - m a g n e t i c s u b s t i t u e n t .

(i) 1,2-Disubstitutedbcnzenes (two identical substituents)

(ii) 1,4-Disubstitutedbenzenes (two differentsubstituents) (iii)4-Monosubstituted pyridines (iv) Furan (v) Thiophen¢ (vi) 1-Chloro-2-bromoethane

Fro. 8.29 Molecules which are typical AAtBB ' systems

TAme 8.50 EXl,L~crr E~rzeoY Ex~mesmo~ AND WAVmn~c-noNs FOR 12 I~.NJ~Oy Energy s2 1st' 2$1'

25_1 ~ $-2

1 ax ~ 2ax' I ao ~ 2o0' 1a': 2a~-x

"A+'~+½N ½('A + ~ ) -- ½(('Oaf + ~Vt)½

½('A + '~) + ½(('Oaf + JVt)½ --½('A + "B) + ½(('Oaf + ~V2)t --½('A + ~ -- ½(('Oaf + Art)½ --~A -- "~ + ½N ½('A + ~ ) -- ½ X 3 ( , . + ,~) - ½K + - ½K + - ½X--

½{('oa + M ~ + Lt} t ½((,oa + ~ 0 t + L t ) t ½ ( M t + Lt)½ ½ ( M t + Lt)½

- ½ ( , ~ + ~ - ½ K + ½{(,oa - M ) 2 + Lt}½ - ½(",, + ' ~ - ½ X - - ½{(~oa - M ) t + Lt} ½

16 A A ' B B '

Wavefunction (St) ( l S i ) CO~V -- (2S,) ~ n e (ls~) sin~ + (2s~) cose (ls_l) cose + (2s_~) sine - (lS_l) sine + (2s_~) cose (s_t) (1 a:) cosy+ + (2al) sin~o+ - (1 al) sin~+ + (2al) cosy+ (1 ao) cos0o - (200) sin0, (lao) sin0, + (2ao)cosO,, ( l a _ l ) cosy_ - (2a_1) ldnv_ ( l a _ l ) sine_ + (2a_l)cosv_

The angles e, ~p+. 0s and O, am defined by

and

oe T i m

cos2~ : sin2~ : 1 - ~o~ : N : ((,oo')2 + Nt)~ cos20, : sin20, : 1 - K : L : (K 2 + Lt)½ cos20, : sin20a : 1 -= M L : ( M t-I- Lt)½ cos2~+ : sin21p_+ : 1 : ,06 ~ M: L : ( ( ( ' o ~ ± Air)t + Lt} ½ 6 =aB -a^.

402

HIGH

RESOLUTION

NMR SPECTROSCOPY

by the diagonal matrix elements. The 16 x 16 matrix representing the Hamiltonian for the system can be factorised into two 1 x 1, five 2 x 2 and one 4 x 4 sub-matrices. Although one can calculate the roots of the five 2 x 2 secular determinants and express them in explicit algebraic form, the 4 x 4 sub-determinant cannot be solved in this way (it involves the solution of a fourth power equation). Hence, the only explicit eigenvalue expressions and wavefunctions for the A A ' B B ' system are those given in TaMe 8.50. A consequence of this is that explicit expressions for six of the allowed transitions and their relative intensities are not available (all those involving lsh, 2s~, 3sh and 4s~ wavefunctions). Table 8.51A lists the transition frequencies and relative intensities for all the other transitions. Altogether, there are 28 transitions, four of them being combination transitions of very weak intensity which are unlikely to be observed in practice and are therefore not included in Table 8.51 A. Moreover,

TABLE 8.51 A TRANSITION ENERGIES AND RELATIVE INTEN~FI1~ OF A TRANSITIONS FOR

AA'BB' SYSTV.M Transition

4 5 6 7 8 9 10 11 12

l S1 t ~ S 2 1SO' "~ l s x ' S_2 ~ l s _ t ' Is_x' --~ 2So' 3So'

Transition energies relative to ~r(~A + ~e)

Relative intensity

½N+ ]r{(Vo~)z + N2}÷ Not obtainable

1 - sin2q0

- ½ N ' + ~{(Vo~)z + NZ}+

1 + sin2~

Not obtainable

-'~ 2 s t '

2s_t' ~ 4So' [ 4So' ~ 2st' I 2s_t' -+ 3So' J 2ao' ~ 2at' 2a_l '--~ lao' la o' ~ 2a t' 2a_l '--~ 2no'

Explicit algebraic e~xprmsionsfor the~ transitions are not obtainable ½{(rod + M) z + L2}÷ + {(M z + L2)~ {{(Vo8 - M) z + LZ}~r+ ½(Mz + L2)~ ½{(%~ + M ) 2 + L2}~ - { ( M z + L2)~ ~{(POB -- 1~4)2 + Z2}~ -- ~r(M2 + L2)+

sin2 (~, - 'e+) cos2 (0, + v/_) cos2(0, - ~÷) sin2 (0, + ~_)

the A region of the spectrum will always be a mirror image of the B part and consequently only the twelve A transitions need be considered. It should be noted that there are no degenerate energy levels as were found in the A.A'XX' system. The complex relationship between the form of an A A ' B B ' spectrum and the chemical shift and coupling constant parameters leads to the analysis of a spectrum of this type being very difficult. Because of this, several rather indirect methods of analysis have been adopted. F o r example, one can construct a general theoretical spectrum for the system under consideration assuming the chemical shift difference, %~, to be large (that is an A A ' X X ' system) and then, by progressively decreasing the chemical shift difference, the modification in the line positions for each new value of vo3 can be followed. This information enables one to assign experimental bands to


their correct transitions ff the original theoretical AA'XX' spectrum has been constructed using coupling constants not too dissimilar to those in the molecule being examined. When an assignment has been made, the trial values of the parameters are successively modified until a theoretical spectrum in good agreement with the observed spectrum is produced. That the assignment is correct can be confirmed by a series of internal checks on the bands of the spectrum. The problem of choosing approximate values of coupling constants is overcome by examining the spectra of similar bnt more simple compounds of the type being studied. One can make the valid assumption that providing no drastic changes in hybridisation occur on going from the simple molecules to those being analysed then the coupling constants will remain fairly constant. Thus in aromatic molecules the H - H spin-spin coupling constants between atoms in different ring positions are given approximately by the values in Fig. 8.30 regardless of the nature of the substituents.

J° ~ 8"0 cycles sec-I J " ~ 2"0 cycles sec -1 JO < 0"5 cycles sec -1

FIG. 8.30 The H - H Coupling Constants in an Aromatic Ring

If we consider the case of an orthodisubstitutedbenzene (shown in Fig. 8.29 (i)) then from a knowledge of H - H spin-spin coupling it is probable that J ^ s >> JAs. > 0 JBs, >> JÂ' > 0. Assuming all the coupling constants to be positive, the A portion of a theoretical AA'XX' spectrum consistent with the above features can be constructed (see Fig. 8.31) and compared with a similar spectrum for an analogous AA'BB' system. We have already discussed the spectrum of the AA' nuclei in an AA'XX' system (Section 8.16.1). If JxA. = 0, then K = - M and the four pairs of transitions (5, 10), (9, 6), (7, 12) and (11, 8) become degenerate and each pair gives rise to a single absorption band: there will then be 6 bands in all in the A spectrum. When the chemical shift difference decreases to the point where the system becomes AA'BB' in type, the A part of the spectrum is modified as shown in

404


Fig. 8.31 Co). The components of the A multiplet nearest the B spectrum gains in intensity at the expense of the low field part and the transitions 1,2 and 3,4 axe no longer degenerate and give rise to two pairs of resolvable absorption bands in the AA'BB' spectrum. Also, the separations between transitions (9,6) and (7,12) axe increased in the AA'BB' spectrum. It is possible for the bands of the outermost pair of transitions (11, 8) to overlap the corresponding B part of the spectrum but the more intense bands for transitions (3, 4) never cross the bands for the corresponding B transitions. In the actual analysis of a spectrum which approximates to this particular case of an AA'BB' system one can assign the intense transitions (1,2) and (3,4) fairly easily. Lines 1 and 3 can then be manipulated to find the chemical shift of the A nucleus and the parameter N = J^s + ./As. N = v, -

vz

(8.180)

{(~o ~)z + Nz}l/2 = h + ~3.

(8.181)

Lines 9 and 11 can be used to find two of the other parameters

(8.182) (8.183)

( M 2 + L 2 ) 1/2 = ~9 -- ~11 {0'0 ~ Jr M ) 2 + L2} I/2 -- '['9 "31-~"11.

X Part of S~"~'C t rIJ m

',2

3,4

(a) 5 10

,i 8

i

II

i

B ~rt 34

of s p e c t r u m =

~)

Ill

I

II

FIe. 8.31 (a) Theoretical spectrum for A A I nuclei in an A A ' X X ' system having YAx ) ,TAx' > 0 and Jxx' > JAA' > 0 (b) theoretical spectrum for A A ' nuclei in an A A ' BB' system having JAn :~ JAB* > 0 and Jan' > JAA' > O. Bernstein, Pople and Schneider (e~)

A value for K can be obtained from transitions 6 and 7 by an indirect process involving trial and error fitting. If values for K, L, M and N are known then the complete set of spectral parameters for the molecule can be found. Relative signs o f coupling constants in an A A ' B B ' system. Grant, Hirst and Gutowsky(6a) have considered in detail the effects on AA'BB' type spectraofthe relative signs of the four coupling constants. It can be shown that no informa-

THE

ANALYSIS

OF

HIGH

RESOLUTION

SPECTRA

405

tion on the signs of Land M e a n be obtained from the spectrum since an AA'BB' spectrum is independent of the relative signs of these parameters. However, the spectrum is sensitive to the relative signs of K and N, but not to those of either M or L with respect to K and N. When the spectrum is analysed to give the absolute values of K, N, L and M and the sign of K relative to N, it is possible to obtain values of the four coupling constants and also all their relative signs. It should be noted that from an AA'BB' spectral analysis one cannot differentiate between JAA'and J n , nor between JA~ and JAw. To assign these coupling constants to particular pairs of nuclei in the molecule it is necessary to compare the observed coupling constants with values obtained from independent measurements of similarly substituted compounds where the assign. ments are unambiguous. Thus, Gutowsky and co-workers(68) have been able H

I¢

AL' P0 rt

Po r'~ A

r

?,.4

1,2

I T

I r

13.710-28.2

tl 3-70-~

6-4

3.76.4 z 0.8

Cycles

10.2 15.7 ~'2

sec " l

1=Io.8.32 The IH resonance spectrum at 40 Mc sec-I of naphthalene in dioxane. Pople, Schneider and Bernstein(GsT) to show that all four H H coupling constants in thiophene have the same relative sign, that J12 and Jxs in furan have the same sign, and that r~]BUff ~ o s, r,,,,* '~ ] B H a and jv~=oin several disubstituted benzenes have the same sign. Hutton and Schaefer (t49) have calculated theoretical spectra for l,l-disubstituted cyclopropanes. Ortho-disubstituted benzenesmNaphthalene. A typical AA'BB' spectrum of the type we have been discussing is given by the ring hydrogen atoms of naphthalend ~7) HA HA,

The two aromatic rings are identical and the IH spectrum can be considered to arise Rom the four ring hydrogen atoms in either aromatic ring. There is no

406

HIGH R E S O L U T I O N

NMR

SPECTROSCOPY

coupling between hydrogen nuclei in different rings. Figure 8.32 shows the ~H resonance spectrum of naphthalene measured in dioxan¢ solution at 40.00 M c sec -x and it can be seen to be completely symmetrical about the centre of the spectrum. The assignments of the various bands to the transition frequencies are achieved by comparing the observed spectrum with the theoretical spectrum shown in Fig. 8.31 which is known to be typical for an ortho-disubsfitutedaromatic molecule. The assignments are shown in Fig. 8.32 and in Table 8.51n. Although examination of the spectrum cannot predict which of the nuclei is the more shielded, the low field part of the spectrum (the A part) has been assigned to the o~-hydrogen nuclei on the basis of empirical considerations. There are fewer observed bands than predicted for the system and it is necessary to postulate that several of the doublets in the theoretical spectrum (see Fig. 8.31 (13))have coalesced. Based on the assignments given in Fig. 8.32 for the A part of the spectrum one can calculate values for the various parameters of the system. For example N = 7,1 - ~3 = 9.95 cycles sec -~ {0'o ~)z + N:},/z = 1,, q- va --- 17-45. Thus

0'o t~) = 14"34 ( M 2 + LZ) 1/z = v9 -

Now

~'11 = 9"4

{(re t~ + M ) z + Lz} 1/z = ~'9 q- I ' l l =

11-0.

TABLE 8.5111 OBSERVED AND CALCULATED TRANSITIONENERGIES I~LATS~ I~NsrrIES FOR Trm ~H Rr~oN~rCS SPEcrRuM oF N~mTHALE~ ~7) AT 40 MC sec -~

Line

Energy relative to centre of band (cycles see-t) Calculated

11 8

4 3 12 7 6 9 2 1

10 5

0.8 1.6 3"2 3-7 1 6"1 7-5 9"9 t 10"2 13"2 13.7

15"5 15.8

Observed 0.8 3.7 6"4 8"2 10"2 13"7

Relative intensity Calculated 0-52 0.45 1.87 1.57 0.93 0.62 0.53 } 0"49 0.46 } 0"43 0"07 0"07

Observed* 1.1 2.8 0.9 0.6 1.3 1.1

* Normalised to same intensity as calculated values. T h u s M = - 6.03 a n d L = 7-21 cycles sec -1 i f it is a s s u m e d t h a t J A s > J A n ' . I f t h e r e a s o n a b l e a s s u m p t i o n is m a d e t h a t JAA' ~ 0 t h e n K = - M , a n d w i t h

THE ANALYSIS OF H I G H R E S O L U T I O N S P E C T R A

407

this information a complete set of parameters for the spectrum can be obtained, J^B =

8"6 cycles sec -~

J^B, =

1.4 cycles sec -~

JBB' =

6"0 cycles sec- ~

JAA' =

0"0 cycles sec -1

re 5Ae = 14"3 cycles sec -~ . The relativesigns of the coupling constants were not deduced from this analysis. A check on the original assignments can bc carried out by calculating(9) the position of transition 12 with the known parameters and comparing it with the assicmment for this transition. Another check of the assitmments involves calculating the complete theoretical spectrum for the A lines (12 in all) from the parameters extracted from the tentative assitmment. To do thisthe numerical solution for the roots of a fourth order secular equation must be undertaken to find transitions 2, 4, 5, 6, 7, 8. Table 8.51A gives the comparison of the observed and calculated 1H resonance spectrum of naphthalene at 40-00 M c sec-L The IH spectrum of ortho-dichlorobenzene has been analysed in a similar fashion(eV, eg) The most difficult aspect of an AA'BB' analysisis to find a suitable value for K. For many molecules its value cannot be found directly and an assumed reasonable value must be used; good agreement between bbserved and calculated spectra justifies the choice of a particular K value.

Benzofurazan. Dischler and Englert (7°) have conducted a comprehensive study of several AA'BB' type molecules. A particularly good example of such a system with Jew >> J . ^ , > 0 and J^B ~ JAB' > 0 is benzofurazan A N B~J \ B,L I 0 -A%~,~ N / The 1H resonance spectrum at 56-4 M c sec -I for benzofurATsn is shown in Fig. 8.33: all but two of the transitions (I and 2) give rise to resolvable absorption bands. The assignments of the bands are shown on the spectrum and the observed and calculated transition frequencies and relative intensities for a positive K value with (i) N > 0 and (ii)N < 0 are given in Table 8.52. The assignments made by assuming N > 0 were shown to give a better agreement with the observed spectrum, thus indicating that the coupling constants have the same sign (N.B. the relative signs of all coupling constants are available). They used an AA'BB' analysis outlined by Dischler and Maier (71) similar to that suggested by Rao and Venkateswarlu(72). It is convenient to decompose the spectrum into four groups of bands as shown in Fig. 8.33, for the purpose of indicating the significance of the various separations between the bands.

408


Para-disubstituted benzenes. Richards and Schaefer ¢7a~ have applied the AA'BB' analysis to interpret the XH resonance spectra of several para-disubstituted benzene derivatives. They overcame the difficulty of finding a value for K by assoming that the two meta H - H coupling constants are equal and the para H-H coupling constant to be zero, that is, in the molecule R!

H~'..jHB, RH

TABLR8.52 O ~ V E D AND CALCULATEDtH SeECrRAOF BRNZOFURAZAN(7°) AT 56.4 Mc sec-1

Observed frequency

(cycl~ ,~dgC-I)

Relative int~mity (2' =

8)

Calculated assuming N > 0 correct assignment Assignment

N =

J'*

~1i~1=

.

(8.206)

ilvi)~k =l~Ik+j

(8.207)

J.k

The term II~kl 2 may be expressed as N

II~] 2

=

(Ifk)(lfk)* = (Ilk) ~ (I,a-ill

and summing over j and k gives II Jk

l2

= ~ I -Jk Iz~ + = Tr(I- I +) ffi Tr(l + I-).

(8.208)

j.k

The symbol Tr denotes the trace of the matrix, that is the sum of the diagonal elements. The transition frequency ~jk is given by (Ej - Ek) which in turn is equal to oW~ - oW,t where oW~ are the diagonal elements of the matrix representing the Hamiltonian, using eigenfunctions as basis functions, "~

= X ~r ]zr + X J,r Is. It.

(8.209)

Substituting for vjk and II~kl2 in equation (8.206) gives <S"> = ~ (~'k~ -- 3~'~)" 6k I~/Xr I +I-.

(8.210)

432

HIGH

RESOLUTION

NMR SPECTROSCOPY

Applying equation (8.210) to give the first moment ( S ) gives<ss) ( S ) = ~ ( . ~ - ae'a)Is~ l ~ / T r I + lJd¢

= Z (~ Lk

I ~ - t ~ ~gss) l ~ / T r I + I-

= Z ['~,I+]tJb';/TrI+IAk

= Tr[af',I+]I-/TrPI -

(8.21 I)

Similar calculations for the higher moments give ( S 2) = Tr [-.~, I +] [I-, ~ ] / T r I + I-

(8.212)

( S 3) = T r [ ~ , [.~, I+] ] [I-, ae']/Tr I + I-

(8.213)

(S*) = Tr [gf, [,Yg',I+]] [[I-, af'], ,Y~/Tr I + I-.

(8.214)

The diagonal sum equations for the moments have the advantage that the trace of a matrix is independent of the matrix base functions and may be evaluated on any convenient representation without solving a secular equation. As an example of the evaluation of a trace, consider the denominator in equations (8.211) to (8.214) Tr I + I- = Tr

(~ (Iz. + i ly,)) (~ (Ix, - i ly,)) (8.215)

In equation (8.215) only the terms with r -- s contribute to the diagonal elements, h e n c e ~ 2 Tr P / - - - [Tr (" x, + /,2)1 = 2 T r ~ I z, (8.216) r

since the directions x, y and z are equivalent. When all the nuclei have the same spin quantum number, equation (8.216) simplifies to Tr I + I- = 2 N T r I 2 .

(8.217)

The diagonal sum in equation (8.217) may be conveniently evaluated on the product spin function representation, in which I~, is a diagonalmatrix which can be expressed as a product of a matrix involving the coordinates of nucleus r alone, and an identity matrix of dimension (21 + 1)N -t. Tr 15, over the coordinates of nucleus r alone is +I

Tr P,, = Y. m, =

t ( t + 1)(2I + 1)"

--l

(8.218)

hence

2N Trl÷l-

-

3

I(I+

I)(21+

I) •.

(8.219)


433

The commutators in the numerators of equations (8.211) to (8.214) can be evaluated with the aid of the equations (8.32) and (8.36). The results are [~r, I÷1 = ~ ~, I,+

(8.220)

g-, .e3 = Y ,,, I ;

(8.~.1)

f

[~', [~', ~+]]

=

[[z-,~j,~

= E ~,~ I,,- + E : , ( , ,

Z ,2, I, + + E a, (,, - ,.) i, + i,. P

-

,,)z;~..

(8.2u) (8223)

r~8

Evaluating the diagonal sums and substituting in equation (8.211) gives the first moment as <S> = N -~ ]~ ,,. (8.224) r

It is convenient to set the first moment equal to zero, that is the axis about which the first moment is taken passes through the "centre of gravity" of the spectrum. Frequencies measured from this arbitrary zero will be denoted as 3 ~,, and the moments as (A S~). In this notation the first four moments are ffi N-I ~ A , , = O

(8.225)

ffi N-~ Z (A ~,)2

(8.~0

> IJl O.Id'

t

(a)

4

0.2J

0.3J

' 0"4J

0"60

O'8J

lJoJ

~'~

2.0J

1

I

l,l ,li ,,I I dl

A 0

(~)

JY~

~

I

,11 il ti11111LII I It

FIG. 8.57 The A part of the IH resonance spectrum of dichloroacetaldehyd¢ (AX spin system) when a second radiofrequency field (YxH2/2.~s JAX/2) is applied near the X resonance. H2 is offset intentionally from the centre of the X doublet by a quantityA orA' expressedin units of J ( - JAr.). (a) The field sweep method for various values of ,4' =. ~x -~ t'2 - 6AX. Co) The freqmmcy sweep method for various values of . i . 0,xHo/2=) ~,,. -

T h e calculated p a t t e r n s a r e 8iven u n d e r e a c h set o f spectra. ~2 ffi 60 M c sec =I JAx -- 2"7 ~ 0"1 cycles sec -~, (~AX"ffi 3"27 p p m . F r e e m a n a n d Vfl'h~en (1111)

when the multiplet then takes on its simplest form. Note that this condition is quite different from the condition for saturation of the nucleus in question, where

~2R]TIT~ ~ 1. Saturation can be shown to be a side effect in the decoupling process (11"). When (c) is to be used for precise chemical shift determinations and for

462

HIGH RESOLUTIONNMR SPECTROSCOPY

nearly all examples of (d), H2 has to be kept small and a compromise must be reached with the condition ~ H2/2~ ~

I~1.

Freeman and Whiffen(~s) have developed a theory for the AX spin system (both nuclei i~-ing XH) to enable spectra to be predicted for conditions slightly 3

-I -

2

~

-2 ~ 5 -3 2.0 1,6 1.2 L

0-8

~

0.4 0 -3

I~o. &58 The t ~ Q o n

I -2

I -i

0

1

2

f ~ . Q ~ c ~ Q ~-d m ~ i f i ~ L for the A resonance of

an AX2 spinsys~m plotteda~!n~ the offsetparameterA for y H 2 / 2 ~ - 0 . 2 J'AX, whereA - - ( ~ 2 - - mx.)/(2nl J',x I) a n d Q - (co x - c o ~ / ( 2 ~ z I'TAX I). Anderson and

Freem(a1n21)

different from those required for best decoupling. They expressed their results in graphical form for both the field sweep and the frequency sweep method (see Section 6.8) of carrying out the double resonance experiment. The latter method offers a simpler set of conditions for spectral interpretation and there are no experimental difficulties using a field/frequency locked spectrometer: the main magnetic field Ho is held constant, ~2 is set near resonance for the nucleus to be irradiated while ~1 is varied uniformly over the range necessary to record the resonance ofthe other nucleus. Figure 8.57illustrates the two types


463

of spectra recorded under the two methods of sweeping. For most instances, there are four transitions for the frequency sweep method and three for the field sweep method. Freeman and Whitfen (11s) concluded that there is no advantage to be gained in using the frequency sweep method when spin deeoupling is to be used for the determination of the relative signs of coupling constants or for L~

I

O

-1

-2

-3 2.0 ~.6 1.2 L 0"8

3,4

0.4

o

-2

-!

0

I

2

3

FIG. 8.59 The transltion frequenciesD and inte~ifies L for the A rmonanoe of an AX2 spin system plotted against the offsetpm'ameter ,4 for 7Ha/2~ - ?AX. /1 and .O are defined in the caption to Fig. 8.58. Anderson and Freeman(1al) the accurate determination of chemical shifts c12°). There may be an advantage in the method, however, in the simplification of complex spectra (but see reference 154). Anderson and Freeman c1~I) have extended the calculations made on AX spin systems, to A , X , spin systems, where m, n ~ 3 and the X nuclei are subjected to the strong field H~. The spin decoupling behaviour was again presented in graphical form--this time in such a way that A and X may be identical or different nuclear species. Examples of the AX2 spin system are shown in Figs. 8.58 and 8.59 (these diagrams suffice in fact for aI1A~X2 molecules). The experimental variables are o>1, oJ2 and Ho. In the frequency sweep experiment,

H I G H R B S O L U T I O N NMR S P E C T R O S C O P Y

¢o2 and Ho are held constant or locked together and o~1 is swept through the A resonance. The resulting spectrum may be predicted by drawing a vertical line on Figs. 8.58 and 8.59 atthe requisite value ofA. The field sweep experiments entail holding cox and ~o2 constant while slowly varying Ho, corresponding to varying ~2 and A simultaneously. On the figures a line can be drawn of slope 7A/TX and the intercept on the A axis is called A', corresponding to the deviation from the resonance condition of o~z when o~, = O~A. For 1H-XH decoupling the slope of the line is very close to 45 ° and A' = [2~ c~AX -- (C01 -- C02)1 1 ( 2 ~ IJl)

that is, the discrepancy between the chemical shift and the frequency difference between the two oscillators. Figure 8.58 relates to a weak perturbing field from oJ2; here the frequency sweep method produces spectra which are easy to interpret but the field sweep technique gives complicated results. Double irradiation of AXm type spectra with intermediate values of H2 gives an easily recognisable pattern for A' = 0 (essentially a singie line in the field sweep experiment); for A,X,, spin systems there is a residual splitting which makes recognition of the exact resonance condition considerably more difficurt. Strong//2 fields applied at exact resonance cause muRtiplets to collapse to a single line in AX~ type spectra: A,X,, systems having n > 1 have a residual splitting of the resonance line that persists as/-/2 is increased. The splitting is (2#JAx) 2 (mA -- ½)/7H2 tad sec -1 for 7H2 >> 2.~ IJ]. When the fields are sufficiently strong to satisfy the condition )'H2 > ½ IOJA -- OXl this splitting begins to increase with increase of H2, reaching 2Z~JAx rad sec -1 in the limit. These results illustrate that caution shourd be applied when using the simple pictures of the spin decoupling process--usuaUy the strong field H2 is regarded merely as causing rapid transitions of the X nuclei thus "washing out" the A group muRtiplet ct22). Electric field perturbation or moduRation of a high resolution N M R spectrum appears to be a promising technique o5 x): the theoretical spectra from spectra of spin ½ nuclei, where quadrupole effects are absent, have been calculated for the AB, AB2 and AX3 spin systems. An example wiU now be given of the use of double irradiation in evaluating the chemical shift difference between two coupled groups in a complex spectrum. Turner cx2s) has reported a singie sideband technique (see Section 6.8) which gives decoupled spectra free from unwanted sideband resonances. The modulation frequency ~m is fixed and the spectrum is recorded by sweeping the applied magnetic field; under these conditions only those nuclei are decoupled whose resonances are separated exactly by ~'m. Turner c~2a) demonstrated the method with clerodin, which has the partial structure

c-cI}(


465

The ~H resonance spectrum is shown in Fig. 8.60. The lowest field group is a triplet (3 ffi 3.80) assigned to the ~-hydrogen; the next group of bands is a doublet (3 = 4.09) arising from the hydrogen nucleus flanked by two oxygen atoms, and which is coupled to the p-hydrogen. The resonance of the/~-hydrogen nudens could not be separated from the complex spectrum but the signal would be expected to appear at either ~ _~ 8 or at about z = 6 to 7, depending on whether the/~- and ),-hydrogen nuclei are equivalent or not. The position of the p-hydrogen band was found by observing the low field doublet with a H

I3~' cycles sec"

/~ #Yl

(o)

t" = 3"80 4"09

-f'130

136

1.31

137

6" 53

132

138

13,:t

139

135

(b)

140

Cycles sec "z

FIG. 8.60 (a) The XHresonance spectrum at 56.4 Mc sec-z of clcrodin, (b) the low field portion of (a) in the presence of the indicated ~deband froqumcies. "~r1~123)

radiofrequency ~i while applying a strong field at ~I + ~,. The doublet will show optimum decoupling when ~m is equal to the chemical shift difference between this hydrogen nucleus and the /~-hydrogen nucleus. Figure 8.60 Co) shows the low field spectrum for different values of ~=, from these it was found that the chemical shift difference is 137 cycles sec -~ giving T = 6.53 for the ~-hydrogen.

Signs of coupling constants. A valence bond treatment of spin-spin coupling constants predicts that coupling constants may be either positive or negative depending on the molecule considered. The simple nuclear resonance experiment is unable to provide the absolute signs of the constants; this may be seen by observing that changing the sign of the Hamiltonian leaves the theoretical spectrum unchanged"='). It is possible in certain cases, however, to obtain the relative signs of the coupling constants by (i) a full analysis of the spectrum of a strongly coupled spin system, (ii) spin-decoupling of nuclei in a particular spin

466


state c12s), (iii) examination of double-quantum transitions (see Section 8.19.4). All three methods require that there are at ]east three groups of non-equivalent nuclei(1 s. 126).

(i) Signs of coupling constants from spectral analysis. The details of the analysis of common systems have been discussed fully; the information obtainable on the relative signs of coupling constants in a particular spin system may be found by reference to the relevant section in this chapter. The relative signs affect only those systems in which at least two groups are strongly coupled so that the simplest system of interest is ABX. Here it is possible to obtain the relative signs of J^x and JBx, but not of JAn with respect to JAX or JBx, by comparing t h e distribution of line intensities. In general, the systems A , B ~ and A.BpCq can yield the relative signs of all three coupling constants but the differences between spectra computed with different sign combinations may be very small. (ii) Signs of coupling constants from spin-decoupling experiments. Weakly coupled spin systems are particularly suited to the spin-decoupling method because unequivocal results are obtained c~3°). Between the extremes of weak and strong coupling lie many spectra of intermediate complexity in which some of the observed transitions are recognisable in terms of first order theory. Here, and in the case of strongly coupled systems, double resonance cannot be used in its simple form. The necessary refined experiments are described later in this section. The spin-decoupling method t o b e described now aims at decoupling only those nuclei in a particular spin state, for example, 0c or/~ for nuclei of spin 3. It is essential that there should be no mixing between different product spin functions, that is, the interactions must be first order. The simplest system is therefore APX which contains three weakly coupled nuclei; systems such as ABX introduce further complications. Section 8.19.1 describes how spin systems may be converted to the weakly coupled type. For an APX system, the Hamiltonian may be written ,~ ffi~A[A, "I-1'p[pz -l-~x/xz 4- JAplA, Ipz -I"JAX[A~[Xz "~ YpxIp, lxa and the transition energy for a group A transition is ~A + m x J A x +

mpJ~.

(8.277) (8.278)

Similar expressions can be written for X and P transitions. W h e n IA = Ip = IX = ½ then mA, rap, m x can have the values + 3, that is, o~ or : states, and there are four transitions in each group. If all the coupling constants have the same sign (positive)and J^p > JAx, then the four .4 transitions in order of

decreasing frequency are


467

In transition A 1, mx and m~, are both + ½, corresponding to the nuclei P and X both being in the spin state o~. Table 8.63 shows the spin states of P and X for each A transition, also of A and X for P transitions, and A and P for X transitions--assuming that all the coupling constants are positive. The transitions in each group are numbered so that number 1 is at the highest applied field. T~SL~ 8.63 S ~ STAT~Sor N~OHBOLraS~ AN A P X S~'r~M Neighbouring nucleus

Al A2 A3 A4

A X

~

~

Pz P2 P~ P~

Xz X: X3 X~

=~p~

~

~

~

~

In Table 8.63 it has been assumed that JAP > JAx J'~P > Jpx YPx > YAXNow it is possible to irradiate at a frequency midway between transitions AI

and ,42 with a radiofrequency field strong enough to decouple the A nuclei but without disturbing transitions A3 and A4. This has the effect of decoupling . only those nuclei in the A group which belong to the spin states having P nuclei in the spin state ~, so that if the X group transitions are simultaneously observed then transitions X2 and X1 are.seen to collapse to a single line, while transitions Xs and X4 remain unaffected. If all the coupling constants have a negative sign then o~ must be interchanged with/~ everywhere in Table 8.63 and there is no difference in the de,coupled spectra. If, however, J ~ and Jl, X are of opposite sign such that J~F is positive, then in Table 8.63 the0~ and/~ must be interchanged in r o w P of column X, and row X of column P; irradiation of transitions A1 and Az causes transitions X3 and X, to merge. Hence it is possible to ascertain the relative signs of JAP and JPx. Similarly, by irradiating at a frequency midway between transitions P1 and P2 and observing the X transitions one can decide the relative signs of Jxx and JAr; like signs cause X1 and X3 to coalesce while unlike signs cause X2 and X, 4o merge. In any particular APX system the ease with which the signs of coupling constants may be determined depends upon the proximity of transitions At, A2, As, A4 etc., and upon how good is the first order approximation for the interactions. Consider the IH spectrum of 2-furoic acid at 60Me sec-' ; Fig. 8.61 shows that the spectrum is almost first order. The magnitudes of the chemical shifts and coupling constants obtained directly from the spectrumc~°) are cycles sec -1 cycles sec -1 IJAXl = 0"8 cycles sec-1 ~A -- *F ----39"2 cycles see-' ~'x -- ~A ----31"6 cycles see-1. IJ^PI =

3.5

IJ~xl =

1.8

468

HIGH RESOLUTION

NMR SPECTROSCOPY

If JAe and Jl, x are of the same sign then irradiation midway between transitions 5 and 6 with a radiofrequency of strength y H I > = IJAx[ should collapse the transitions 9 and 10. If JAe and Jex are of opposite sign the same experiment would collapse lines 11 and 12. It was found that lines 9 and 10 coalesce (see Fig. 8.62). The relative signs of JAx and JAP may be obtained by irradiating midway between transitions 1 and 2 and observing the X part of the spectrum. For like signs lines 9 and 11 would coalesce and be superimposed upon line 10,

H x~ 0 / , ~ C OOH

I0

8(111

4 .

FIG.8.61 The IH resonancespectrum of 2-furoicacid at 60 Mc sec-2. Freeman and WhilTen 0"27) H

(Q)

9 I0 II

(b)

(c)

I0 II

J2

(a)

FIo. 8.62 The X part of the spectrum shown in Fig. 8.61 but with simuJtaneous in~diafion of (a) transitions 5 and 6 Co) transitions 7 and 8 (c) transitions I and 2

(d) transitions 3 and 4. Freeman and Whiffon(127) hence the spectrum would appear as two lines with tine 10 three times its normal intensity. If JAP and JAx are of unlike signs then line 11 would appear to be three times its normal intensity. Figure 8.62 shows the observed X part of the spectrum under various conditions of irradiation and it is evident that JAX and dAe have the same sign. When the interactions in a molecule are not entirely first order, for instance in an ABX system, then the interpretation of the spin-decoupled spectra is more difficult. It is not possible to separate the transitions into pure A, B or X types; attempts to decouple A and X nuclei by irradiation of A transitions also decouples B and X. trans.Crotonaldehyde provides an example of a limiting


469

case a n d illustrates h o w sensitive spin-decoupling can be (.28). The ~H spectrum at 60 M e sec -x is o f the A B P ~ X type. H(X) (A)I-L,\

~C=O

/C=C, cH3 \H(B) (PD H

S

X

t

v

A

X

B

yz

P

F]~. 8.63 The 1H resonance spectrum of liquid trans-crotonaldehydeat 60 Me sec-1. The vertical arrows indicate the location of the strong field H2 used in the double irradiation experiments. Freeman (12s) Pople and Schaefer (sl) have analysed the spectrum and obtained the following coupling constants and chemical shifts:

IJABI = 15..6 cycles see-*

]JAe] =

6"8 cycles sec -1

IJBP] =

1-6cycles sec -1

IJexl =

7"8 cycles sec -1

[JA x I =

0.0 cycles sec- 1.

It was established t h a t JBP a n d JAP have opposite signs. The frequencies measured f r o m the centre o f the X doublet are s = 144.0 cycles sec-* t -- 159.6 cycles sec-* u = 193.7 cycles s e c - ' z~ = 201.4 cycles sec -a w = 209.1 cycles sec- 1 x = 216.9 cycles sec-* y = 450.3 cycles see=' z -- 451.7 cycles sec -1.

All frequencies are precise to + 0"2 cycles sec-*.

470

HIGH

RESOLUTION

NMR

SPECTROSCOPY

Applying the same method as that used for 2-furoic acid, it would seem most convenient to locate the strong de.coupling field/-/2at s or t and to observe the P part of the spectrum: a and t are separated by 15.6 cycles sec -~ and are easily distinguishable. If JAB and JsP are of like signs, applyingH2 at a would collapse lines 1 and 3, whilst the same two lines would collapse for unlike signs with//2 applied at t. However, irradiating at s or t does not produce such a simple pattern at P since besides eliminating the coupling Jm, it also affects the coupling between the B and P groups. The problem may be overcome by applying

.,.

,,

/~, x!> 27

(9.96)

(a to ~p/2) 2 ~ 1

(9.97)

~'p ~, ~p

(9.98)

A simple Lorentzian line shape then results 6 =

- i tol M o

v~,

+ ( T ; l)V -- iA to

(9.99)

This expression leads to a transverse relaxation time (T~)ve for a mixture of diamagnetic and paramagnetic environments such that (r;%p

= ~;x + (7;')D.

(9.10o)

From this equation and the relation zv --- (k[P]) -x

(9.101)

506

H I G H R E S O L U T I O N NMR SPECTROSCOPY

one obtains the expression /6

~.

[p]-1 [ ( y ~ l ) D p __ (T~I)D]

(9.102)

which is the same result as that derived by physical arguments for the Cu(I) - Cu(II) example presented above. A more general, less stringent condition than those of equations (9.96-9.98) is (½3 ~ o~)2 >> 1

(9.103)

providing that z is appreciably smaller than 3p, Condition (9.103) can be checked experimentally for most systems since it must be satisfied if the nuclear hyperfine splittings due to X are to be resolved in the electron magnetic resonance spectrum of the DP mixture. The case of 3t) ~ 3p is a difficult one to treat since allowance must be made for the Boltzmann population distribution between the P+ and the P= states so that the line shape formula gives both the broadening effect and also the paramagnetic shift of the nuclear resonance signal. A simple application of the NMR method to the class of reactions under consideration is that of the TI(1)=TI(III) electron exchange system(33) in the form of molten TI2Br4, Rowland and Bomberg found two absorption lines for the nuclear resonance of 2°5TI and ascribed them to the TI(1) and TI(III) valence states at high and low fields respectively. Both states are diamagnetic but the broadening of the lines with temperature enabled the chem/cal lifetimes of the states to be estimated (I0 =5 sec at 500°K), Here the line shape was cal= culated for a chemically exchanging system of two species having equal con= centrations. The validity of the method has been checked by studying the kinetics of the MnO~-MnO~- exchange reaction (in sodium and potassium hydroxide solu= tions) both by tracer (2i) and by NMR (22) measurements. The two sets of results were found to be in agreement. The NMR experiments were based on SSMn resonance which gives no signal in the paramagnetic manganate environment. A further set of results, in agreement with the others, was obtained by making measurements of the relaxation times TI and T2 with the aid of spin echo techniques(23). Johnson 04°) has suggested that a full treatment of the problem investigated by McConnell and Weaver(3°) should take into account both the hydrogen nucleus-=electron dipole-dipole interaction and the isotropic exchange inter= action. The former is not very important for free radicals because the correla= tion time 3# for the interaction is of the order of I0 =11 sec. However, the corre= lation time 3p for the isotropic exchange interaction is equal to the lifetime 3A of the paramagnetic species of the spin-lattice relaxation time ~ of the unpaired electron, depending on which is the shorter, and may be of the order of 10 -4 sec. Johnson applied the exchange broadening theory developed by Kaplan (I 7) and by Alexander(334) to derive an equation for the transverse relaxation times of the nuclei. It was shown that under certain conditions, when more than one NMR signal is observed, the rate constant and the concentration of

THE EFFECTS OF CHEMICAL EQUILIBRIA

507

paramagnetic molecules can be determined from the line widths if the concentration of the diamagnetic species is known. 9.3 ION AND GROUP EXCHANGEREACTIONS At the beginning of the previous section it was pointed out that the detailed mechanism of an electron exchange reaction might involve atom or group transfer. This section, however, deals with those reactions in which valency changes are absent and in which there is little doubt that ions or groups participate in the chemical exchange process. Obviously, a wide variety of equilibria fall into this category; these will be presented in turn, except for hydrogen bonding which will be deferred until the next main section.

9.3.1 Protolysis Reactions One of the main applications of the basic theory given at the beginning of this chapter is to the study of protolysis reactions. Dissociation of protie acids is included in Section 9.3.2. Most protolysis reactions studied by the NMR method have been concerned with amines or amides. However, interesting exceptions are the protolyses of methanol(34) and ethanol (35). The rapid exchange of the hydroxyl proton in alcohols is accessible by the method. The reaction is both acid and base catalysed and is accelerated to a rate corresponding to a lifetime of the alcoholmolecule of the order of 10-? sec by the addition of acid or base in the concentration range 10 -6 to 10 -5 M. Figure 9.12 shows typical spectra used in the rate determinations. The methyl group resonance is not included because it is unaffected by the proton exchange process. For simplicity, only the methylene and water hydrogen resonance signals were utilised in the interpretation. The width of the methylene line is a measure of the total rate of proton exchanges (i) between alcohol hydroxyl groups (ii) between water and alcohol hydroxyl groups. The width of the water line is a measure of the rate of the second process only. Luz, Gill and Meiboom defined the s p e c i e rate of exchange Rx of a given molecular species X as the reciprocal of the mean lifetime, ~rx, of a molecule X between successive exchanges Rx = ~x I -

l -IX]

d[X] (9.104) at

The full width of the water line at half height, zl ~,, was measured as a function of acid or base concentration in the range where separate water and hydroxyl resonances were observed. A 1, is related to the specific rate of exchange of a water proton by the expression Rw = ~w1 = ~A 1, - T~"1

(9.105)

508


where/'2 is the spin-spin relaxation time in the absence of exchange and can be determined from the decay of "wiggles" after rapid passage through the water line. The total rate of exchange RT was found from the methylene resonance using the procedure for a quartet, outlined in Section 9.1.2.The specific rate of exchange RA between ethanol and water was obtained from the broadening of the water line using the equation RA . .

Rw

rw . .

~A

Ci z

2 [HzO] . [EtOH]

H20

i

(9.106)

OH

,o,

H FiG. 9.12 Hydrog¢n resonance spectra of ethanol containing 4"3 per cent w/w wator in (a) basic solution (b) neutral solution (c) acidic solution. Luz, Gill and M¢iboom caS)

Since RT ---- RA + RB (where RB is the rate of direct exchange between ethanol molecules), RBcan thus be determined. The following reaction scheme was found to be compatible with the kinetic data kt ROH + O H - ~--- R O - + H 2 0 k2 R O H + R O - ~ RO- + R O H k3 ROH + HaO+ ~- ROH~ + H 2 0 k4 R O H + ROH~ ~ ROH~ + ROH k~ ROH + (H)OH ~ RO(H) + HOH

THE EFFECTS OF C H E M I C A L E Q U I L I B R I A

509

Paths 1, 2 and 5 apply to the base range, while paths 3, 4 and 5 apply to the acid range. At 22°C, the mean values of k l , k2, k3, k4 and ks were found to be 2"8 x 106, 1-4 x 106, 2-8 x 106, 1"1 x 10~ and 0.8 iv1-1 sec -I respectively. The rate of proton exchange in neutral anhydrous ethanol (reaction 5) was found to be 2.7 M-1 sec -1 at 42 ± 1°C by a direct method(~37): a graphical procedure was devised which gave rapid evaluation of the average lifetime of a hydroxyl proton between exchange events. In the methanol experimentsc3~ measurements were made on buffered solutions. The results obtained at 24.8°C after extrapolation to zero buffer concentration can be summarised by the equation: rate of proton exchange = 8.8 x 10l° [MeOH,+] + 1.85 x 101° [MeO-]

(M-I see-l). These measurements have been extended to rate determinations for proton transfer between methanol and the components of benzoic acid-sodium benzoate buffers in dilute solution°6' 351.3s2). T w o main terms appear in the rate law; one is a termolecular process involving benzoic acid, benzoate ion and a methanol molecule, the other involves one benzoic acid molecule and two methanol molecules. The number of methanol molecules taking part in each kinetic process was determined by comparing the O H proton exchange rate, as determined from the IH resonance of the C H ~ group, with the exchange rate between methanol and benzoic acid, as obtained from the carboxyl or from the coalesced carboxyl-hydroxyl resonance. Proton exchange in various other alcohol-water systcrns has been investigated(3S7. 3ss) F r o m rapid passage experiments, Mciboom (37)has made a detailed examination of proton transfer in water. Formally, the proton cxchangc reactions can be written H~0

kj + (I'hO) + ~ - ( H 2 0 H ) + +

H20

k2

OI-h + OH" ~ OH- + HOH. The transverse relaxation rate (l/T2) of the hydrogen resonance was found to be pH dependent due to the partially exchange-averaged spin-spin splitting of the resonance by the magnetic nucleus 170 (spin ~) .The increase of relaxation rate observed in natural water (0"037 per cent 170) is greatly increased in 170 enriched water. The conclusions were confirmed by a study of the dependence of the width of the 170 resonance signal on pH. The rate constant kl has a value of (10.6 ± 4) x 109, while k~ is (3.8 -k 1.5) x 109 ~-1 sec-L The spin-spin coupling constant between ZH and 170 was found to be 92 ± 15 cycles sec -1. A more accurate value of 73.5 ± 2-1 cycles sec -1 has been obtained for JJToHfrom the 170 resonance spectrum by reducing the exchange rate °sl. This was accomplished by diluting the water (containing 8 atom per cent ~70) with a large excess of carefully purified acetone (see Section 12.4.3). The quadrtlpole moment of the 170 nucleus prevents observation of multiplet structure in the IH resonance spectrum. Loewenstein and SzSke °91 have re-determined kl and k2 at 25°C and they have obtained values similar to those found by Meiboom. Also they measured

510

HIGH RESOLUTION

NMR SPECTROSCOPY

the respective heats of activation and found them to be 2-6 4- 0.3 kcal mole- 1 and 4.8 + 0.5 kcal mole -1. The rates of inter-water proton exchange have been measured(3sl) for water in various organic solvents. Another system to be investigated in which hydrogen is attached to an oxygen atom is the exchange of hydrogen between water and hydrogen peroxide (4°). The spectrum is never divided into two absorption lines, because the rate of exchange is too rapid at room temperature. However, the line width was seen to be dependent on pH and the hydrogen peroxide concentration. In the pH range 2.5 to 4.5, the reaction involves H30 ÷ and H202 while in the range 4.5 to 6-5 the participating species are HO~, H202 and H20. Protolysis kinetic studies have been carried out on ammonium(2~' ~1-43), methyl, dimethyl and trimethyl ammonium ions (5. ~8.,s. 44. 3 5 3 , 3 s 4 . 3 5 5 ) A kinetic study of the related protolysis reactions of the trimethylphosphonium ions in aqueous solution has also been made ('5). The procedure used is similar to that indicated in the example described above. The addition of diethyl- and triethylamine to acetylacetone has been shown to alter the keto-enol equilibrium position(4~). Proton exchange takes place between the O H and = C H groups of the enol tautomer. Kinetic data has been obtained for amides: the rate of proton exchange for N-methylacetamide and N-methylformamide has been measured in solutions at different pH values (7), the reaction being kA

RCONHCH3 + H30 + ~ (RCONH2CH3)+ + H20. From the N-methyl band (a doublet for pH values 1 to 4) in the NMR spectra of these compounds at 25°C, kA was found to be 200 + 70 M-~ sec -~ and 10 _ 3 M-~ sec -I for N-methylacetamide and N-methylformamide respectively. The former value is in good agreement with the results of other workers (47). Saika (4s) has determined the heats of activation for the acid hydrolysis to be 15 + 3 kcal mole -~ and 13 4- 3 kcal mole -z respectively. In the presence of alkali the reaction becomes kB R C O N H C H s + O H - ~- ( C H 3 C O N C H 3 ) - + H z O

k~) is known only for N-methylacetamide(4:), having a value of 5"2 × 10~ M-t se¢ - 1.

GiUespie and Birchall(49) have investigated the site of protonation for acetamide, N, N-dimethylacetamide, formamide and N, N-dimethylformamide. They used fluorosulphuric acid as the source of protons and they varied the temperature from - 9 8 to 25°C. At the lower temperatures the proton exchange rate decreased sufficiently to permit observation of a new band in the 4-

H resonance spectrum. The band was assigned to the group > C = O H , indicating that protonation had occurred on the oxygen atom. A similar conclusion has been reached by Herbison-Evans and Richards (5°) who measured the line widths in the ~'N resonance spectra of N, N-dimethylformamide in trifluoroacetic acid. Increase of line width (a measure of T~ 1 here) with increase of acid concentration provides evidence for increase in the electric qua-


511

drupole coupling constant. Measurements have been extended to include other amides, thioamides and sulphonamides13s6~. Evidence has been found for diprotonation of thiourea and N-methylthiourea in fluorosulphuric acid. A series of interesting proton exchange reactions between hydrogen fluoride and the weakly basic, substituted benzenes has been studied c5D. In the instances of hexamethylbenzene and durene it was found that the proton may exchange intramolecularly. CH3 CH3 ~I"13, I jH CH3. [ < C H 3 \/'~/ \/~/H I + t\CH ~ --~ i + lJ "CH3 CH3

CH3

H CH3\/[x./CH3 I+1

cs/) zB only a few hydrogen nuclei are influenced by the paramagnetic ion in a time during which the magnetisation is established. Fast rates of exchange and slow rates of relaxation lead to a small effect on each 1H nucleus. Pearson and his co-workers have analysed mathematically

THE EFFECTS

OF C H E M I C A L

EQUILIBRIA

525

the system of a magnetic nucleus having the three environments, a diamagnetic ion labelled A, a paramagnetic ion (possessing a single unpaired electron) with spin up labelled B and spin down labelled C. They adopted McConnell's procedure (s) described at the beginning of Section 9.1. Assuming that the solution is dilute and that there is rapid spin exchange for the electron, then the experimental width of the line 2"~1, unshifted from the pure solvent signal, is given by T~ 1 -- T ~ + Pe/(T~.e + za) (9.119) where Pe is the mole fraction of water molecules coordinated by the paramagnetic ion and it has the value 6 M/55 for a coordination number six and an ion concentration of M moles litre-L The equation shows that the relative values of the exchange lifetime ze and the paramagnetic relaxation time T2 B determine the effect of the paramagnetic ion. If zB >> 7'2 B then the exchange is rate determining, the line width being given by T~ x -- T ~

-1 + "rA1 ffi T2A "{- PB ZB 1"

(9.120)

If Tae >> ze, then relaxation predominates and 2"2 is obtained from T21 ---- T ~ + Pe T 2-1 B.

(9.121)

The relaxation time T2 A of a nucleus in a diamagnetic environment (bulk solvent) is not the same as that for pure water since dipole--dipole interaction occurs through several layers of water molecules surrounding the ion. A concentration dependence is present of the form • 1 T~2A ---- T]~ + Be T],~A

(9.122)

where 2"2o and T2,A are respectively the spin-spin relaxation times for pure water and the relaxation time for the second and higher layers of water molecules. Since T ~t is about ten times greater than T~2,xA,T~2A t can be estimated without serious error. Experimentally, the relative importance of ~ and 2"2 can be assessed in several ways. If the chemical exchange is rate determining then the addition of other reagents will have an effect. Another method is to study the temperature dependence of the experimentally-determined T2. Increasing the temperature will reduce ~e since there will be associated with it a heat of activation according to the expression z~ t o~exp ( - ._4H / R 7")

(9.123)

where/! H is the heat of activation. By contrast, T2 e increases with increase of temperature ix°e) if molecular rotation limits the correlation but if electron spin exchange limits the correlation then 2"2a will be reduced slightly by increase of temperature. A difficulty arises from the fact that zl H is normally rather small. Pearson et al. applied their treatment to the line widths of the hydrogen resonance signal from solutions of a variety of chromium (HI) complexes at ,several acidities. The rate constant k for the proton exchange reaction Cr~NH + OH- --* Cr~N- + H 2 0

526


was calculated on the assumption that this reaction is responsible for an line broadening in strongly alkaline solution. Table 9.5 fists some of the results obtained from the equation ~gt _- k[OH-]. (9.124) The value of T2A was found by using the results for acid solution where exchange is negligible. Cr(en)s(CIO,)s, Cr(NH3)6(CIO,)3, [Cr(NH3)sH20] (CIO,)s, and K[Cr(NHs)2(SCN),] were chosen for study because independent measurements have been m a d e at low p H values~t°-"),where the rates of exchange are slow enough to be estimated by conventional methods. G o o d agreement was found; for example, Cr(en)] + proved to have a rate constant of TABLE9.5 N - H EXCH~OE RAT~S ~OR Cr(en)s(CIO,0s

WATERAT25°C [OH-I

I"2(see)

pH 2"5 pH 6"1 pH 9"5 pH 9.8 0.0044 0.022 0.044 0.044

0-040 0.044 0-043 0"030 0.024 0"0090 0"0052 0.0066

TB (s~)

1

×

5"7 × 1'2 × 6"5 × 8"6 ×

10-2 I0-s I0-s 10-6 10-6

k (M-~ sec-~)

2 × 3"8 x 3"7 x 3"5 × 2"6 ×

102 106 10~ 10¢ 10~

2"3 x 10s M-t sec -t. There was not good agreement in the values of the heat of activation: unforeseen effects in the N M R work were held to be responsible. Pearson et aL also accumulated data for some paramagnetic ions [Cr(ITI), Fe(HX), Mn(II), Co(H), Ni(1I), Cu(II), Ce(III) and Gd(II1)] dissolved in methanol acidified with hydrochloric acid. By observation of the nuclear magnetic resonance signals for both the CHs and the O H groups, the two rate constants were seen to be almost equal. This is evidence that Ts is being measured and in the instances of Ni(II), Cu(H) and possibly Fe(III) it seemed that the rate of exchange of methanol molecules had been measured. Comparison with results obtained for aqueous solutions (99' to4. tos) showed that the rates of methanol exchange are slower than those of water molecule exchange--as expected. Ethanol solutions gave less clear-cut results but generally the exchange of ethanol molecules is faster than that of methanol. It is worth mentioning in passing that line width measurements on electron magnetic resonance signals given by paramagnetic ions can give useful information about the symmetry of the electric fidd generated by the complexing groups (tee). Connick and his co-workers (z°°' tot) have made measurements of t~O resonance fine widths in order to obtain T2; they also found Tt. They employed solutions of Mn 2 *, Fe 2 *, Co 2 *, Ni 2 * and Fe z * in 0" I M HCIO4 in water enriched with t e e . The rates of exchange (and the corresponding thermodynamic quantifies) of water molecules between the bulk of the solution and the first coordi-


527

nation sphere of these paramagnetic cations were determined(1°1). The rates at 25°C are: 3.1 x 107 sec -1 (Mn2÷), 1.0 x 104 sec -1 equatorial and 2 × l0 s sec -1 axial(Cu2÷),2.7 × 104 sec -x (Ni~÷),l.13 × 10s sec-l(Co2÷)and3.2 x 106 sec-l(Fe2÷). The two exchange rates shown by the cupric ion were interpreted in terms of a distorted octahedral hydration shell. The line broadening of :4N resonances has been used to study the exchange of N H 3 with Ni(NH3)~ + in liquid H 2 0 and NI-I~°'3). In the aqueous exchange, the reaction is first order with respect to the concentration of the complex: itwas found that the enthalpy and entropy of activation are 9.5 + I.I kcal mole -I and - 5 + 4 cal mole -x deg -I respectively. The rate of exchange in anhydrous ammonia is similar at 25°C and so are the energetics (/II-I* = 10 + I kcal mole -x, ,4S* = - 3 4- 4 cal mole -x deg-1).

Results obtained from pulse methods. As discussed in Section 2.5.1, a small concentration of electronically paramagnetic ions dissolved in water causes a marked reduction in the spin-latticerelaxation time T~ of the hydrogen nuclei. Thus TI values can provide information about paramagnetic ions or complexes dissolved in water or any other solvent containing a suitable magnetic nucleus. The most convenient means of measuring TI is by pulse techniques (Section 2.12.3). A n early application of the method was to the study of the rate of reduction of the aqueous Eu(III) ion by zinc(I°7).Most measurements of 2"I and T2 of paramagnetic solutions have been made with a view to providing detailed verificationof the current theories of hydrogen nuclear relaxation or information on the electronic structure of the central metal ion from its effective magnetic mo m e nt.Examples are theions of Fe(TI)(108) ,Fe(III) (109) ,Co(II) (110) ,Nd(III) (110) , Cr(III) (11°), Cu0T) (111), Gd(III) "°2), Ni(II) c112) and Mn(I~ "12). Aqueous solutions of these ions have T~/T2 ratios near unity except for Mn(II) which has 2"1/2:2 ~ 10 for hydrogen resonance at 40 Mc sec-l. King and Davidson (~~3) noticed that the addition of ethylene diamine tetra-acetate or nitrilotriacetate increased T2 to give 71/72 ~ 1. The results obtained for manganous ions in water show that in some cases the temperature dependence of Tx and Te provides information about the hydration of paramagnetic ions. From such data for Mn~÷aq Bemheim and his co-workers (1°2) found the heat of activation for the dipolar relaxation mechanism to be 5.5 kcal mole -1, and that for the chemical exchange to be 8.4 kcal mole -~. The results did not indicate whether chemical exchange involves only the protons or whether water molecules as such participate. However, the low activation heat of 8.4 kcal mole -~ suggests that the mechanism mainly comprises proton transfer. Aqueous solutions of Cu(II), Co(II) or GdffH) proved to have negligible chemical exchange contributions to T1 and 7"2 hence only a dipolar heat of activation can be determined. The values are 2.7, 1"45 and 2.4 kcal mole -1 respectively. The addition of a variety of simple anions to the cupric ion solutions did not produce a significant effect, but the addition of ethylenediamine(en) or 2,2'-dipyridine does have a marked influence on T1 and Te "~'). The data enabled the first order rate constant of 2.4 x 107 M-I sec -~ at 27°C to be found for the reaction C u ( e n ) 2 ( H 2 0 ) i + + en -*.

528


Measurements of Tt and T2 at various temperatures (~tS) led to a heat of activation of 4.5 kcai mole -~. The effective barriers to tumbling for the ions Cu(H20)+2+ , Cu(en)(H20)~ + and Cu(en)2(H20) 2+ were also obtained having the values 5"1, 4"5 and 3"9 kcal mole -~ respectively. Spin-lattice relaxation time studies c~~e) on aqueous solutions of substitutioninert chromium (III) complexes showed that the ions having greatest tendency to interact with solvent water molecules (by virtue of possessing exchangeable protons, for example, Cr(H20)~ +, or by participating in hydrogen bonding with the solvent, for example, CrF~-) have the smallest spin-lattice relaxation times. The order of increasing TI is Cr(H20)~ +, CrF~-, Cr(NHs)~ +, Cr(C20+)~-, Cr(en)] + and Cr(CN)~-. Results obtained from Cr(H20)6Cls are dependent on the thermal history of the solutionsc117). Gasser and Richards c91) added ethylenediamine to aqueous Co(NH3)~ + and observed ligand exchange according to the overall equilibrium Co(NH3)t+ + 3 en ~ Co(~a)~+ + 6NIt3. The low resolution SgCo resonance spectrum showed two separate absorption bands corresponding to the two Co(IH) entities. The exchange process is sufficiently slow to enable the growth of one absorption band, at the expense of the other, to be measured on the addition of ethylenediamine. The experi= mental data gave a value of 22 4- 2 kcal mole -~ for the overall heat of activation of exchange. The first order kinetics show that the first substitution reaction is rate determining and that the subsequent substitution reactions are rapid. The two 59Co resonance lines recorded for sodium cobaitinitrite in water were attributed, one to a nitrite form of the complex and a hydrolysis product and the other to the nitro isomer. The enthaipy change associated with the isomerism was estimated to be 8 + 1 kcal mole -t.

Addition compounds. Nuclear magnetic resonance has proved effective in the study of complex formation between the strong accepter molecule boron trifluoride and donor molecules containing oxygenc1t s. 121.3~1. a62). The fluorine resonance of a liquid mixture of boron trifluoride and two donor molecules, for example methanol + ethanol, consists of two absorption signals when sufficiently low temperatures are employed. Band area measurements showed that reversible equilibria are established between the two donors and their respective complexes: it is possible to calculate the relative concentration equilibrium constants. Hydrogen resonance spectra were found to give similar data. From the results it was evident that affinity for boron trifluoride decreases in the order water, methanol, ethanol, n-propanol and n-butanol "~s" 1~9) When the temperature was raised to S0°C, the two lines merged to a single line because of rapid exchange of BFs among the donor molecules. The rate of exchange depends upon the boron trifluoride concentrationcIx9) and leads therefore to the mechanism BF3 + CH3OH.BF3 - * B2F¢ + CHsOH B2Fe -+ 2 B F s

slow

rapid

C H 3 O H + BF3 - * CHsOH.BF3

rapid.


529

From a study of the IH resonance of these addition compounds, Di©hF 1~°) has obtained evidence for aggregates greater than 1 : l for alcohols; some may have the structure

The second alcohol molecule is held by a hydrogen bond. Diem also studied the more pronounced changes in chemical shift in the fluorine resonance spec= tra of complexes of boron trifluoride with alcohols, diethyl ether, water and acetic acid. Line width data were used to evaluate the chemical exchange times which in turn allowed the heat of activation to be calculated: 7"3 + 1 kcal mole -1 for the exchange of BF3 between methanol and ethanol. Observations of the XgF spectra of the water-boron trifiuoride system (BF3 concentrations from 0"5 to 11 ~) led to the identification of the species H B F , , HBF3OH, HBF2(OH)2 and HBF(OH)3. One can expect N M R investigations to be extended to other donor molecules such as ketones, and carboxylic acid anhydrides. Nitrogen-containlng donor molecules are also amenable to study ff interference from the nitrogen nuclear quadrupole moment is avoided by application of the double resonance method (see Section 6.8). Happe c122) has made measurements on the association betwcen pyrrole and pyridine from the 40Mcsec -1 XH resonance spectra with simultaneous saturation of the nitrogen nuclei by irradiation at 2.9 Mc sec -~. The large low field shifts of the N H multiplet of pyrrole on the addition of pyridine are good evidence for a N - - H m N linkage. The enthalpy change associated with the equilibrium was found to be - 4-3 kcal mole -~.

Aromatic complexes. These complexes are encountered in nuclear magnetic resonance spectroscopy whenever a polar solute is dissolved in an aromatic solvent. Hydrogen bonding can be a factor cx23) in determining the extent of the solvent-solute interaction. However, there undoubtedly exists a weak specific solvent-solute interaction ct23' ~27). The presence of a molecular complex according to the rapidly established equilibrium solute + solvent ~ molecular complex implies that the chemical shifts of the solvent magnetic nuclei are not only concentration dependent but temperature dependent as well. The latter effect has been studied in detail by Hatton and Schneider c12". It is believed that in complexes between polar solute and non-polar aromatic hydrocarbons, the solute hydrogen nuclei tend to be located above the plane of the ring and ad= jacent to the symmetry axis. This topic is discussed more fully in Sections 3.10 and 10.39 under the general classification of solvent effects. A brief study has been made (12s) of aqueous silver ion complexes of cyclohexene, c/s-2-butene, benzene and toluene. The olefmes appear to form a single non-labile l : 1 complex. Evidently rapid exchange is present in the aromaticsilver solutions since every aH resonance spectrum featured only a single

530


absorption band due to a particular aromatic hydrogen nucleus. Both 1 : 1 and I : 2 complexes are formed with the former predominating.

9.3.4 Miscellaneous Atom and Group Transfer Reactions The largest group of compounds failing under this heading are the inorganic fluorides which are particularly suited for nuclear magnetic resonance studies. The first work of this kind to be reported is that of Muetterties and Phillips (129) who observed the temperature dependences of the xgF spectra of chlorine trifluoride, bromine trifluodde, bromine pentafluoride and iodine pentafluodde. Impure CIF3 gave a single relatively broad absorption band affected little by temperature variation. The C2, symmetry of the molecule (xa°' 1ai) leads to the expectation that the spectrum should originate from two sets of non-equivalent fluorine atoms and should consist of a doublet and a triplet for an irradiating radiofrequeney of about 40 Me see -1 or higher (a6~). Extreme precautions in the removal of fluoride impurities resulted in this type of spectrum being observed at - 60°C. Raising the temperature gave the characteristic feature of the onset of rapid chemical exchange, namely only a single broad concentration-dependent band was recorded at 60°C. The lifetime of the exchanging fluorine nuclei was estimated to be 2.4 x 10 -5 see at the latter temperature and the heat of activation was estimated to be 4.8 kcal mole -1. The 19F resonance spectrum of bromine trifluoride differed in that above its melting point (8.8°C) only a single sharp line was exhibited. The inability to detect even line broadening was ascribed to a heat of activation lower than that of CIFa. The mechanism put forward for fluorine exchange is

F\ //F,~ / F _ F \ / F \ / F F/CI\F / CI~,F ~-"F/CI,,,F/CI(,,F since association of the halogen halides has been established (132"134). Hamer (I 3s) has cast some doubt on the above results by demonstrating that even greater care in the removal of hydrogen fluoride from chlorine trifluoride raised the temperature at which line broadening occurred. Furthermore, the precise temperature at which multiplet splitting disappeared varied from sample to sample. It may well be that exchange between chlorine trifluoride molecules does not occur in the absence of hydrogen fluoride but even if it does the heat of activation must be considerably greater than 4.8 kcal mole -1. Muetterties and phillips (:29) also collected evidence for fluorine exchange in the N M R spectra of bromine pentafluoride and iodine pentafluoride and were able to evaluate the enthalpy of activation of the latter compound. These workers also extended their investigations to a wide range of inorganic fluorides (~3~). They found that the spectra of all fluorides that are of tetrahedral or octahedral symmetry are relatively insensitive to fluoride impurities by contrast with fluorides of lower symmetry. Sulphur tetrafluoride, of C2, symmetry, shows exchange behaviour and the temperature dependence of the spectrum of this compound is illustrated in Fig. 9.16. Fluorine atoms appear to exchange between the two non-equivalent fluorine environments of the pseudo trigonal

THE EFFECTS

OF CHEMICAL

531

EQUILIBRIA

bipyramid. The activation heat for the process was calculated to be 4.5 + 0.8 kcal mole -1 in good agreement with earlier results "3~). A dimer was suggested once more as a means of effecting exchange F

F

F

F

23°C

SOF:-- , / . ~ . ,

.

-~O"C

H II SO~

SF~

;

tl'SOE

-50~C

-45°C

l-soF:

I I

I_SOFo It

-58°C "w

~

j

-98°C $ = 52 I:}prn d = 78 cycles sec-L

Fxo. 9.16 The temperature dependence of the 19F nuclear magnetic resonance spectrum (30 Mc sec-1) of sulphur tetrafluoride. Muetterties and PhillipsO3e)

Lone pair p electrons occupy one of the coordination positions. A similar set of findings was obtained for selenium tetrafluoride, while teUurium tetrafluoride gave a single sharp line down to - 100°C. The rates of fluorine exchange follow the order SF, < SeF, < TeF, which is the same as that for increasing molecular association as evidenced by melting and boiling points. Some evidence was obtained for the acid-catalysed exchange between F - and SiFt-. Myers (lss) has used the quadrupole broadening of the lzTI resonance line of the iodide ion brought about by the presence of iodine molecules to investigate the kinetics of the tri-iodide equilibrium. In aqueous solution the iodide ion suffers a strong quadrupole pulse as it undergoes chemical reaction with an iodine molecule: there is no longer a spherically symmetrical charge distribution in the tri-iodide ion. Quadrupole couplings in the iodine molecule and

532

H I G H RESOLUTION NMR SPECTROSCOPY

the tri-iodide ion are so strong that the 12~I resonance line is broadened beyond detection. On the basis of the simple equilibrium

I- + I2.q ~ I~ the rate constants for the forward and backward reactions at about 35°C were found to be (4-1 + 0.4) x 101° M-1 sec -1 and (7"6 + 0"8) x l0 T sec -1. The former rate constant is one of the largest reported for a reaction in aqueous solution involving a neutral molecule. Nevertheless, the lifetime of an iodide ion is many microseconds. The heat of activation must be small since the rate approaches the collision frequency. An interesting exchange system is that of allyl magnesium bromide which has a bond switching mechanism within the moleculec~39). It was noticed that the ~H resonance spectrum is much simpler than is normally encountered with allyl compounds due to the merging of the signals from the hydrogen atoms in the 1 and 3 positions thus giving a spectrum of the AX, type. The spectrum can be interpreted reasonably in terms of the rapid equilibrium (tl/, _~ 0.001 sec) B r M g - - C H 2 - - C H ~ C H 2 ~- C H 2 ~ C H - - C H 2 - - M g B r

Other allylic Grignard reagents show this behaviour (1.°). Aluminium trialkyls exist as dimers due to the formation of two A1--C--A1 bridges. Two separate IH resonance signals have been detected(1.~' ~42) from the bridged and outer methyl groups of aluminium trimcthyl at about - 7 5°C: at about - 20°C the mean lifetime of a configuration is about 3 x 10 -3 sec. In mixtures of aluminium trimethyl and aluminium tri-isobutyl an exchange of alkyl groups takes place (~43). The resultant mixed aUcyls are strongly associated as dimers by moans of methyl group bridges. From the concentration dependence of the chemical shifts, Hoffmannc~43) has estimated the T values of the bridge and outer methyl groups to be 9"69 and 10-59 ppm respectively. As the proportion of aluminlum tri-isobutyl is increased, the stage is reached when only one methyl group is available for dimer formation, the structure being i Bu\

/iBu,.

/ i Bu

iBuz

~M¢/

\iBu

/AI,:I

)AI\

Hoffmann has also investigated exchange reactions between AIMe3 and AIMe2H, A1Me2C1, A1Me,OMe, A1Me,OEt, A1Me2t-Bu and A1Me3 • NEt3. The remainder of this section will be taken up with exchange processes sufficiently slow to give distinct spectra of the species taking part. When a mixture is made of boron tri-fluoride, chloride, bromide and iodide rapid equilibrium is established between all possible halide species c144, 3~v). Rapid disproportionation to the BX3 forms prevents isolation of the mixed halides but observation of the 19F resonance allowed the spectrum of the new compound BBrC1F to be recorded( ~ and its chemical shift to be measured (see Section 11.19.1). Intermediate species have also been reported ~45) from the ~~9Sn spectra of mixtures of SnCl,, SnBr, and SnI, (see Section 12.8.2). Koski, Kaufman and Lauterbur(.46) supplemented their infrared(~4~) and mass spectrometric(~48)

THE EFFECTS

OF C H E M I C A L

EQUILIBRIA

533

studies of the exchange of deuterium between diborane and pentaborane with hydrogen nuclear magnetic resonance measurements. In these, the exchange of deuterium between BzD6 and BsH9 was measured by allowing mixtures of them to equilibrate for several hours before examining their N M R spectra. Exchange was found to occur preferentially in the terminal hydrogen positions of pentaborane. The rate of exchange of the apex hydrogen atoms appeared to be within __ lOper cent of that of the base terminal hydrogen atoms: bridge hydrogen atoms in pentaborane did not participate in the exchange. A similar investigation has been carried out on the deuteration ofdecaborane by deuterium oxide (149). The results of previous work else) were confirmed in that rapid exH

t

Reoctlon time

O" O0 hr I I

I I

[ i

0. 15hr

0' 5 0 h r I

I

I • O0 hr

^'teSJ'l ,.,

I

I

1[ ,5,74

1241 o

I,

tL ~01

Fro. 9.17 ThetlB nuclear resonance spectrum of dccaborane in DzO as a function of reaction time. Shapiro, Lustig and Williams(149) change in the bridge positions occurred followed by slower exchange at the terminal sites. Figure 9.17 shows the I*B resonance spectrum at various times; the deuteration of the terminal positions becomes apparent at long reaction times. The 1H resonance spectrum showed similarly that exchange takes place at the bridge positions in less than 0.15 hr. The nuclear magnetic resonance technique has been applied to the study of the well-known hydration reaction of aldehydes ('sl)

H20+R--C. The electrostatic description does not account for all the phenomena associated with hydrogen bonding, the more important discrepancies being( xs 3) (i) the increase in intensity of the infrared band corresponding to the stretching mode A - - H ; this increase is much too large; (ii) the absence of correlation between hydrogen bond strength and the dipole moment of the base; (iii) the decrease in intensity of the infrared absorption band corresponding to the bending mode of A--H.

536

HIGH

RESOLUTION

NMR SPECTROSCOPY

Attempts to calculate the covalent contribution to the hydrogen bond have not been conclusive (1~°" 1~). A qualitative molecular orbital description has been put forward by Pimentel (162) for hydrogen bonding in the ion HF~. One pair of electrons takes part in bonding in an orbital extending over two bonds (three-centre orbital), each bond then being relatively weak. The other electron pair occupies a non-bonding orbital which places the excess charge on the fluorine atoms. For the general case of any two atoms separated by a hydrogen atom, the location of the charge is dictated by the relative electronegativities of the two atoms. Hydrogen bonding will only be favoured if these atoms are sufficiently electronegative for the non-bonding orbital to accept an electron pair. Pimentel and McClellan (153) have made the intriguing suggestion that this description also applies to atoms of low electronegativity providing that the compounds are electron deficient thus leaving the non-bonding orbital unoccupied as in the case of the boron hydrides. The carbonyl group in benzophenone, for example, has different hydrogen bonding properties when dissolved in hexane as opposed to ethanol. In the latter case a hydrogen atom of the solvent behaves as ff it has a good deal of mobility whereas in the former case it is strongly bound. Thus, the hydrogen atom in the C - - H groups of hydrocarbons is in a valence state involving mainly 1s atomic orbitals. Hydrogen atoms in hydrogen-bonded groups like N - - H or O - - H can have contributions to their valence state from 2s and/or 2p atomic arbitals (tes' le,~). In his molecular orbital treatment, Paolini (~ts) has used this idea of n = 2 orbitals in conjunction with the Lippincott-Schroeder (~ts) potential function description of the hydrogen bond. The strength o f this bond is regarded as being determined by the distance apart of the atoms A and B between which the hydrogen atom establishes a link. The theory was developed for (i) the weak hydrogen bond normally encountered in liquids or solutions and (ii) the strong hydrogen bond present in many crystalline substances and having rather small A B distances. A suitable choice made for evaluation was the O--H. . . . O system. Starting with two orthogonal atomic orbitals on the hydrogen atom, one was selected as being an excited (Rydberg) orbital to help to emphasise the covalent character of the H B bond. This extra orbital on the hydrogen atom is more favourable to bonding, due to its location in space, than is the alternative possibility of an anti-bonding orbital on the A - - H group. The atomic orbitals of A and B were defined as hybrids built from s and p orbitals. The calculations indicated that hydrogen bonds can be formed with very little charge migration. It was also shown that the theory can account for the increase in width and intensity of the infrared absorption band of the O - - H group caused by hydrogen bonding. The large variation of the electronic charge on the hydrogen atom (assuming the O O distance remains unchanged) implies that there is a large probability of the bond being broken within the period of several vibrations. However, there is a series of O - - - O distances permitted by the various parameters so that a hydrogen bonded liquid probably corresponds to a certain statistical distribution of these distances. Since each distance corresponds to a given bond strength, there will be a given shift of the O - - H stretching frequency. The broadening of the bands is due to the overlapping of the various


537

shifted bands. Paolini's theory is also capable of predicting the observed increase of dipole moment brought about by association caused by hydrogen bonding. Calculated values of the bond dissociation energy followed the trend of the experimental values only if the amount of s character of the atomic orbitals of the oxygen atom Ob in O~----H----Oe increases as the bond becomes stronger. Because of the interdependence of the lone pair hybridisation, the increase in s character of one of them brings about a decrease in the other. The unsymmetrical distribution of s character produces Oo--H and H ....Ob bonds of different strengths: the final situation is governed by the groups attached to Oo and O~.

9.4.3 The Origin of Chemical Shifts due to Hydrogen Bonding A comprehensive coverage of the theory of chemical shifts is given in Chapters 3 and 4 but the effect due to the extra complication of hydrogen bonding was omitted: this section attempts to fill the gap in our treatment. The hydrogen resonance chemical shift accompanying hydrogen bond formarion is towards lower fields except when an aromatic compound is the base. This apparent reduction in electron density surrounding the particular hydrogen nucleus is somewhat surprising if one argues from the standpoint of electronegativity of the electron donor. The lack of correlation between the shift and the electronegativity (position in the Periodic Table) may be seen from Table 9.6 which lists the experimental data by Schneider, Bernstein and Pople tle6~. TAaL~ 9.6 NET OmMXCAL S~tS DUE TO THE LIQUID STATE (166) Compound

Liquid shift ppm (to low fields)at t°C

t°C

m.p. °C

CH4 C2H6

0.00

- 98

--184

0-00

-

¢2I'~

0.43

- 60

--172 -170

C2H2 NFI3 H20 I'IF PHa HC1 HI HCN

1-30

- 82

-

1"05

- 77

--77

4.58

6.65 0-78 2"05 2"55 1-65

88

82

0

- 6O -- 90 - 86 - 5 - 13

0 --92 -

133

-

112 50

-

14

They measured the chemical shifts of a series of gaseous hydrides from a methane gas internal reference: shifts of the same hydrides in the liquid phase were measured from an external cyclopentane reference. The liquid measurements required correction for the bulk susceptibility, then the net shift attributable to the liquid state was calculated by subtracting the shift in the gas phase from that in the liquid, both being referenced to methane gas. However, these authors found little correlation between the gaseous chemical shifts and the electro-

538


negativity of the central atom' For first row elements of the Periodic Table, the trend of the hydrogen resonance signal shift was to lower fields but elements of the second row showed the reverse trend. The net shifts of the liquid hydrides given in Table 9.6 are substantial for all except the saturated hydrocarbons. One can readily appreciate that the N M R method lends itself to the study of association. Liquids exhibiting strong hydrogen bonding properties like H F and H 2 0 have much greater shifts to lower fields than other substances. Another interesting feature of the table is its demonstration of hydrogen bonding of a hydrogen atom in one molecule with the ~ electrons of a neighbouring molecule in the cases of acetylene and ethylene. Earlier studies of a variety of hydrides c1~7" zts) produced no clear correlation of chemical shift with ionicity or hybridisation. The experimental data just presented leads, in the absence of anisotropic or electric field effects, to the conclusion that the electron density probably increases in the neighbourhood of the hydrogen atom responsible for intermolecular association. The electron distribution is such that the magnetic shielding decreases. The remainder of this section PrOvides an outline of the theoretical treatments of the factors influencing shielding of hydrogen bonded magnetic nuclei. Schneider, Bemstein and Pople (z66)considered the difference between the secondary magnetic field due to induced currents both for a gaseous molecule containing the A - - H bond and for a hydrogen bonded system A m H . . . . B to consist of two contributions: (i) the magnetic field experienced by the hydrogen nucleus due to the direct effect of induced currents in B; (iii the polarisation of the electrons in the A - - H bond caused by B, which in turn affects the shielding of the hydrogen nucleus. The theory of effect (i) is very similar to that of the induced current model discussed previously in Section 4.3: the theory gives a general understanding of the main features of hydrogen shielding. Effect (i) depends upon the nature of B and for linear molecules forming linear hydrogen bonds, increased screening is caused by hydrogen bond formation. This means that, except for aromatic systems, the observed chemical shift results from the predominant effect (ii). Treating the hydrogen bond as being electrostatic, effect (ii) manifests itself as a paramagnetic contribution from B thus providing an asymmetric electric field in the vicinity of the A - - H bond. Schneider, Bernstein and Pople (1~6) estimated that hydrogen bond shifts can be interpreted largely in terms of the reduction of the diamagnetic circulation in the A m H bond by the electrostatic field of the donor Y. Hameka ¢~9) has also regarded hydrogen bonding as affecting shielding constants by the opposing intermolecular and polarisation effects. He applied his method of calculating hydrogen resonance shifts (see Section 4.2) to assess the effect of hydrogen bonding on the hydrogen shielding constants of the ammonia molecule, taking the two contributions separately in order to determine their relative importance. (i) The intermolecular chemical shift.

THE EFFECTS

OF CHEMICAL

EQUILIBRIA

539

The N - H axis of one ammonia molecule was assumed to be collinear with the lone pair orbitals of another molecule, the N----H distance being taken as 4.5 ao

H

N

J The intermolecular shielding constant a' of the hydrogen bonded hydrogen nuelens was shown to be given by [0.1985(1 + 2A) - 0-0093(2' + 2A)] × 1.775 x 10-5 2(1+22A +22 ) where A ( = 0.5955) is the overlap integral for tetragonal symmetry and 2 is an electronegativity parameter found to have the value of 0.724 by calculation from the dipole moment of the ammonia molecule. Hencecr' = - 1"02 x 10 -6. (ii) The polarisation chemical shift. The polarisation of the ammonia molecule from liquid association was represented by the change of 2 caused by hydrogen bonding. The dependence of the shielding constant on the variation of ~ was then estimated to be (0-55432 z + 2.96042 + 2.1704) x 1.7748 x 10-s 2 z + 1.19104 + 1 ;t was found to change by 0-046 because of the polarisation shift giving a corresponding change in the shielding constant of - 0.45 x 10 -6. The total change in the shielding constant is thus - 1.47 x 10 -6 as compared with the experimental value ~166~ of 1-05 x 10-6. Hameka's results show that, in the case of ammonia, the intermolecular shift is the main effect whereas those of Schneider, Bernstein and Pople c166~placed the emphasis on the polarisation shift for linearly-associated molecules. Although the two situations may be quite different, Hameka believes that the intermolecular contribution predominates in all liquid association shifts.

9.4.4 Interpretation o:[ N M R Data General Pimentel and McClellan¢1s3~have drawn attention to a set of practical rules cssential to a thorough investigation of hydrogen bonding. These are (i) A really pure solvent should be selected paying due regard to its hydrogen bonding properties. (ii) The concentration should be varied and the sensitivity of the spectrum to this variable determined. Sufficiently low concentrations should be used to

540


enable control over intermolecular association to be exercised and, if possible to give complete dissociation to monomer. (iii) The chemical shifts should be measured at different controlled temperatures. (iv) Deuterium should be substituted for hydrogen to verify the assignments. The earlier nuclear magnetic measurements were made before all of these precautions became evident. One of the first systematic studies was made by Huggins, Pimentel and Shoolery(~7°) on the concentration dependences of the chemical shifts of various phenols in carbon tetrachloride. The data were interpreted in terms of a monomer plus dimer, the observed chemical shift ~ being the weighted mean . Phenols have provided some interesting results, in particular, the intramolecular hydrogen bonding of ortho substituted phenols (19~-199, 336). The hydroxyl group of phenol itself has a chemical shift of 3"93 ppm to low fields of cyclohexane (internal) at infinite dilution in carbon tetrachloride at 25°C "~3. 2ol). Dimer formation is said to occur at high concentrations of phenol (xl°' 202) but Saunders and Hyne (lv3) contend that the monomer-trimer equilibrium is finite--the trimer probably having a six-membered ring structure. An investigation of several substituted phenols dissolved in carbon tetrachloride showed(XTo) the concentration dependence of the chemical shift of the hydroxyl group in ortho-chlorophenol to be less pronounced than for other phenols. This is certainly caused by intramolecular hydrogen bonding of the monomer reducing the tendency to form higher aggregates. The chemical shift at infinite dilution is about 1 ppm to low fields of the corresponding phenol signal. Dilution chemical shifts have been measured "~9" ss6) for the ~H resonance of the OH group in all the ortho-halophenols over a concentration range of 1 to 5 mole per cent in CS2 and over a temperature range of - 53 to + 107°C. In ortho-fluorophenol no evidence was found for intramolecular hydrogen bond interaction between the hydroxyl and the ortho.fluoro groups presumably because the H - F separation is too great even for the c/s configuration. The N M R results led to the evaluation of the equilibrium constants and their associated enthalpies for cis-trans conversion for the remainder of the orthohalophenols. As expected, the c/s form predominates. The rates of exchange between a solvent proton and one in an intramolecular hydrogen bond have been found to depend upon the strength of the internal hydrogen bond (se°). The activation heat for proton transfer between dry methanol and ortho-chlorophenol was found to be 4.58 kcal mole -1. B:RS.

18

546

HIGH RESOLUTION

NMR

SPECTROSCOPY

Sterio hindrance by ortho substituents such as methyl groups also causes the hydrogen bond of polymers to be weaker measured from the observed spectrum of the internally rotating molecules is given by

{., J, +

=

J'o. C F~Br CF Br CL

66 °C

J,L__j

L

i

I

li

If

tI

-120%

ill

,,

Yrr ]1-

F x"

i "CL Br

I

III

"

II

B~

I!

t "St Br II!

i [I

,,

F×

CL"

(o)

CL

8r"

I "F x Br II

FIG. 9.24 The ZgF resonance spectrum at 56.4 Mc sec -1 of CF~Br CFBrCl in CFCI3. The individual rotamers are AB X spin systems but one approximates to an AA' X spin system. Ramey <se4)

The rotamer fractions nt and n o are temperature dependent: the functions describing the resulting temperature dependence of < J > are given later. An actual example of this type of behaviour is shown in Fig. 9.24. The 19F spectrum of CF2BrCFBrC1 at room temperature is of the A B X type, an average of one

564

HIGH RESOLUTION

NMR SPECTROSCOPY

from each rotamer. The spectral parameters are different in each case. The isomers have the following coupling constants ¢269) { I, } = - 11 "8 cycles sec -1 I Jg = - 14.Scyclessec -x II HI

½(2g + Jr) -- - 14.0 cycles see -I { J, } = - 2 0 " 3 cycles see -1 $g = - 2 1 . 3 c y c l e s s e e -~

Manatt and Ellemanc27s) used a double irradiation technique in a study of the compound CF2BrCBr2F at - I10*C. They found the F - F geminal and vicinal coupling constants to be opposite in sign. The spectrum indicated that the most populated conformation is the one having two of the bromine atoms trans to one another. (e) The three rotamers are equivalent. The slowly and the rapidly interconverting molecules give rise to spectra containing the same chemical shifts but more spin-spin coupling constants characterise the spectrum of the slowly re-orientating molecule. The rapidly interconverting form has a spectrum which is temperature independent. There is only one possibility, namely o

I

b

The two halves of the molecule have C3° symmetry and all the atomic substituents have magnetic nuclei. From the foregoing it is evident that the symmetry of rotamers and their time-average is all-important in determining the NMR spectrum. Gutowsky(27~) has stressed the central role of asymmetry in the occurrence of magnetic nonequivalence. He has given an algebraic analysis of the (d) behaviour listed above. Taking the three dl pairs b

o

y

b"

I I

8

7

"a"

a'"

(I)

a'

I I

~

7

a

a ~

(2)

8

I I

"b

(3)

the rotationally averaged chemical shifts < ~> are given by

and

=

Zx.,: |

= ~* x. ~' J iI

(9.139)

THE

EFFECTS

OF

CHEMICAL EQUILIBRIA

565

where x. is the mole fraction of rotamer n and ~. is the chemical shift of the ith nucleus in rotamer n. The net chemical shift between a and a' is given by

(,/1~,~>=

#,~,., -- ~,,,,> = ~.,x.O'~

(9.140)

-- ~':')

n

which can be written as

f Y,(x.--'})d~,° + ~,(x.--"})A,. + ~½zt,.. II

II

II

The first term depends upon the chemical shift differences caused by the initial asymmetry of the parent compound (when b = a) and the population differences introduced by having b @ a. The second term depends upon the population differences and the shift changes ,4 ~, caused by b ~ a. The last term is the "asymmetry effect "--the net shift which would result if the rotamer populations were identical. By choosing spin systems which can be analysed relatively easily it is possible to evaluate coupling constants and obtain qualitative information on rotamer populations. Many substituted ethanes (25s, 27o-274, 277-282, 373) have been examined, providing examples of the Various classes of behaviour defined above. A particular example will now be described. Anet (2s2) has examined the 1H resonance spectra of the meso and d! isomers of 2,3-dibromobutane and has analysed the spectra as AA,V ~3~,s v , spin systems having Jxxo = O. The meso isomer exists as a trans and two gauche forms CH3

CH5

H.. I

CH3

I

CH .. I ....B,

I

CH3

I

Br

H

"Irons I

Gouche IT

The d/isomer is a mixture of distinct d and I molecules each of which can have three rotamers CH3 Br,,

I //CH 3

I

H Trr

CH~ H,,

I .p...Br

I

CH3 ~Z

Cl-~ CH3 .

I

I

Br ~Z

..-H

566


From intensity measurements of the multiplets featuring the comparatively large values of JAA' (7.85 cycles see -a) in the spectrum of the meso isomer, it can be shown that rotamer I predominates (66 -t- 10 per cent) but that appreciable amounts of II are also present. Changing the solvent changes the dielectric constant thus affecting the conformer populations (z72, 2s3, 29e~ since I and H have different dipole moments. The low value of JAA" ( = 3"15 cycles see -a) found for the dl isomer indicates that V must be present in small amounts (14 + 10 per cent) while the remainder is mainly IH. A survey has been made c2e') of A_&'BB' type spectra given by hydrogen nuclei in 1,2-disubstituted ethanes (denoted XCH2CH2Y). A large number of such compounds give IH resonance spectra which can be classed as "deceptively simple", that is, seven or three line spectra are observed instead of the possible ten lines for half the spectrum. Ten lines are observed when ~A -- VS (ffi ~0~) and JAA' -- JBS" (ffi M) are small. Five line and three line spectra occur as a consequence of VA -- ~'Band JA A' -- JB 3, being large compared with JAB "F JAS' (---- N). The rotamers are

x H(B).., I H( A )\.,.' ~ / / I y

H(A} H(B)--.. ...H(e~}

..H(B') H(A')

H(A') H(B).. [ ..H(e') X~" " / / I Y

H(A,) ,"/ ~ , .i, ~ \ X

I Y

Trons

H(A)

GaUche

The spectral expectations for these conformers can be obtained under behaviour (c) given above. Assuming the trans (Jr) and gauche (Jg) coupling constants are invariant (but see reference 344) for a given compound then JAB = n, Jg + ½ng(J, + Jg) and It may be shown that and hence

(9.143)

Jan" = JA'B = n, Jt + noJo

N = J + J' = ½[J, + 3J, + n,(J, - Ja)]

(9.144)

L = J-

(9.145)

J' = (½ - ~n,)(Jr - Jo)

2JA~ + Jaw = Jt + 2Jg = I N + ½L

(9.146)

Abraham and Pachler(2s4~ arrived at several important conclusions. First of all, ] N + ½L should remain constant for a given compound and should be independent of the rotamer populations. From the spectrum analysis, only the magnitude of N and L may be found but by assuming N to be positive, the sign of L may be determined under conditions giving different conformer populations (e.g. by varying the temperature or the solvent). Hence Jf + 2 J o can be determined uniquely. Also the sign of L tells us whether the trans or the gauche form is the more stable, because Jt > Jg, so that if for example L > 0 then nf is less than 3, that is, the gauche isomer is the more stable.


567

I t m a y b e seen t h e n t h a t ff the r o t a m e r s are o f e q u a l energy, L = 0 a n d the h a l f s p e c t r u m c o n t a i n s five or three lines. N o t e that the value o f [ N + ½ L is n o t exactly e q u a l to ~(J, + 2Jo), the value for L = 0, TABLE 9.11 H-H COU-PX~GCONSTANTSAND CHEMICALSl'uyl~ IN 1 , 2 - D ~ ETIIAN~¢2a't) Compound

Yo6

M

N

L

~N ± ½L

J, + 2J.

1-Chloro-2bromoethane liquid 1:lin acetonitrile

12-60

0-95

15"00

3"07

24"0 (21"0)

20-21

1"02

14-00

1"58

21"8 (20"2)

2-Odoroethanol Liquid I : 2 i n CCI,t 1 : 1 in acetone

11.24 13.20 10.83

@81 0-76 0-80

11"18 11"82 11"43

1"90 2-15 1"22

17-7 (15"8) 17-3 (15"2) 17-8 (16"5)

17.6

22.8 25"0 25"65 21.8

1"94 1"71 1"77

11"72 11"30 11"64 12-09

1"20 1"48 1"05 0

18"2 (174)) 17"7 (16"2) 18"0 (16"9) 18'1 (18"I)

18.0

11"70 12"20

9"61 9"73 9"66 10"02

2.24 2"80 2"82 2"08

15"5 (13"3) 16"0 (13-2) 15-9 (13"1) 16-1 (14.0)

15-8

11-52

1-53 1.87 1"24 1"52

34"8 35"4

0 0

11"06 11"52

2"10 1-76

17"6 (15"5) 18"1 (16"4)

17"9

36-3 35-8 35.9

0-83 0-89 0"85

9"68 9-80 9"73

2"83 2"79 2"88

15"9 (13-2) 16"1 (13"1) 16-0 (13"2)

I

10.72 15-36

1"10 1.18

13"17 12"90

1"35 1-91

20-4 (19"1) 20-3 (28"4)

}

14.40 13-87 15.39 15-4

1-15 1"08 1"26 1"73

13"07 12"96 13"12 12-78

1"54 1"38 1"88 1"93

20"4 (18"8) 20-1 (18"8) 20"6 (18-7) 20-1 (18"2)

20"4

29"3

2-82

15-89

3-94

25"8 (21-9)

21"9

20.6

2-Bromoethanol liquid 1 : 1 in CHCI3 I : 1 in CS2 1 : 1 in acetone

-

J

2-Methoxyethanol liquid I : 2 i n CC14 I : I in D 2 0

1 : 1 in acetone

8"98

2-Chloroethyl ~tate liquid : 1 in CC14

2-Methoxyethyl ~cetate liquid : 1 in CC14 : 1 in acetone

~aevulic acid : 1 in CHOa : 1 in D20

16"0

Sthyl laevulate iquid : 1 in CHCIa : 1 in MeOH ! : 3 in CFaCOzH

t-Cyano-2,2~methyl 'mtyraldehyde iquid

568


but it is a function of the rotamer proportions. Hence the range of N values observed may be due in part to the substituent effects but it also reflects the rotamer populations. Table 9.11 lists the data obtained by Abraham and Pacifier c2a4). The table shows that the gauche isomer of a 1,2-disubstituted ethane is often more stable than the trana isomer, a result which is opposite to that found in the haloethanes. However, low resolution N M R observations on solid halogenated ethanes c'-93) have shown that CFCla. CFC12 exists mainly in the gauche form while CFzC1. CFzC1 and CF2Br. CF2Br are probably in the trans form. Some of the solvent effects are surprising, for instance, the rotamer fractions of 2-methoxyethanol are virtually identical in CC14 and in D20. From Table 9.11 it may be seen that the value ] N + ½L is not quite constant for a given compound and it increases as the fraction of the trans isomer increases. Hence J, + 2Jr is not the same for the two isomers of each compound. Abraham and Pachler have explained the effect as being due to the dihedral angies of the gauche isomer differing c2ss' zs~) from 60 ° and 180°. The fact that the M values change with the solvent leads to the conclusion that geminal coupling constants are more sensitive to environmental changes than are vicinal couplings. In the above discussion it has been assumed that the two gauche isomers are indistinguishable, but Harris and Sheppard czsl) found evidence contrary to this in their study of the 19F spectra of 1,2-disubstituted perfluoro-ethanes. They utilised the ingenious method of analysing the tsC satellite signals of the main tgF resonance (z~2) (see Section 8.19.2).

Solvent effectS. The discussion of solvent effects presented so far has not included a quantitative treatment. Whipple ¢z87) has reported the variations of cgs-l,3 ~H coupling constants (J) in 2,3-dihalopropenes. The two h"kely conformers were deduced to be z

z¢cH=l

I

I Y rr

J is given by J -- nx Jx + nxx J , where n~ and nn are the mole fractions of the isomers. Also, it was assumed that J~ = ½Y' and Jn ffi IJ', J' being the 1,3 coupling constant in the appropriate 2-halopropene. The equilibrium constant K was calculated from nx and nn thus

K

ffi nl/nlt =

2(ZdZnL)exp(--AE/RT)

(9.147)

THE EFFECTS OF CHEMICAL E Q U I L I B R I A

569

where ZI and Zn are the partition functions of the two isomers. The energy difference/I E betwcen the two isomers in a solvent is given by approximately H E ffi A E o -

where

(8=Nd/3M)[(e

-

1)/(2e + 1)] (p~ - / ~ )

(9.148)

d - density of solute

M = molecular weight of the solute • ffi dielectric constant of the medium Px and Pn -- dipole moments of the conformers. Thus there should be a linear relatiouship between log K and (~ - I)/(26 + 1); this was confirmed by measurements made on the isomer ratios of 2,3-dichloropropene (2e~). Snyder(s'9) has pointed out that . / ~ can be affected by solvent without conformational changes being involvedisis).

Temperature studies. Microwave spectroscopy has provided a good deal of information on the barriers to molecular internal rotation. Unfortunately, the restrictions inherent in the method allow measurement of barriers only of methyl groups rotating against various other groups. Barriers tend to be small and roughly constant at the value 3 kcal mole -I for molecules of this type and it seems that steric repulsions make little contribution to the barrier. Heights of barriers of rapidly rotating isomers have also been measured by ultrasonic relaxation (29e). Enthalpy differences between gauche and trans isomers can be small: the value for n-propyl chloride has been estimated tO be 50 -I- 150 cal mole -1 by spectroscopy(294) and 300 -I- 200cal mole-x by eieetron diffraction(295). Substitution at both carbon atoms in ethane can raise the barrier substantially, for example the barrier (2.°) for CCIsCC13 is 10-8 kcal mole -x. Barriers of thi.q height can be measured in molecules containing hydrogen or fluorine using nuclear magnetic resonance spectroscopy. By measuring the relative areas of the lines due to the three rotamers, the relative energies of the three potential minimst can be obtained. Comparison of the observed spectra at transition temperatures with calculated spectra, assnming likely barriers to internal rotation, can provide much information about the heights of potential maxima. Potential maxima have been measured(2") for CFC1BrCFC1Br where they are all about 10-0 + 0.5 kcal mole -1 above the potential minima, consequently the low temperature 19F spectrum is a superposition of the spectra of three different rotamers of each of the meso and the d/isomers. The estimation of the relative energies of all three forms of ethanes having an asymmetric carbon atom is one of the triumphs of NMR spectroscopy. A method of evaluating the energies in the situation prevailing when separate rotamer spectra can be observed has been reported ¢2s8). In most substituted ethanes, rotational averaging occurs which simplifies the spectrum but reduces the information content. Gutowsky, Belford and McMahon (~sg) have considered how valuable data can be acquired from rotationally-averaged spectra. Their procedure is given below.

570

HIGH

RESOLUTION

HMR SPECTROSCOPY

The experimentally observed average resonance frequency (chemical shift) of the tth nucleus is given by !

(,,> -- xx "l + x2 ~'~ + x3 ,~

(9.149)

and the averaged coupling constant between the tth and jth nuclei is given by

(a',~>

__ x ~ J ; ~ + x~S~~ + x~ j u

(9.150)

where xt, x~ and x3 are the mole fractions of the three retainers. The mole fractions may be represented in terms of the relative energies of the retainer (see Fig. 9.25) ;these are definedby putting A El = Ex - E3 and A Ez = E2 - E3. The ratios of the mole fractions can then be expressed

xt: xz: x3 = Q't e x p ( - A E t / R T ) : Q ; exp(-AE2/RT): Q'3

(9.151)

E 3

F 0

i

', ~

2~

FIG. 9.25 T h e energy o f a s u b s t i t u t e d e t h a n e as a f u n c t i o n o f t h e dihedral angle between C--a a n d C--b b o n d s in a g r o u p a--C--C-b

where Q" is the partition function for a particular retainer n, excluding the internal rotation coordinate ~. To a good approximation Q~ ffi Q~ = Q~, hence 0 ' , ) -- Q~' [v~ e x p ( - A E , / R T ) + ~ e x p ( - A E a / R T ) + ~,~1 (9.152) where Q~, is the internal rotation partition function for the three retainers at thermal equilibrium

Q~ = e x p ( - d E t / R T ) + e x p ( - d E 2 / R T ) + 1

(9.153)

Also, assuming coupling constants are temperature independent ¢3°~), ij

(Ju) = Q~t[J~4 exp(-AE~/RT) + Jz e x p ( - A E 2 / R T ) + J~].

(9.154)

Thus if the temperature dependence of (r~) and ( J u ) is dependent only on the equilibrium proportions of the rotamers then the relative energies of the rotamers and the individual chemical shifts and coupling constants of each rotatio-


571

nal isomer can be calculated. Coupling constant measurements are preferable because they'are unaffected by molecular association; there may be temperature effects, however (344). The resulting equations are difficult to solve and the magnitude of the effect of temperature on (,/tj~ is only about I cycle sec -I in ~H resonance spectra over a range of 200°C (while (~i) changes by about 2per cent over a similar range). Despite the difficulties it is possible in favourable cases to extract all the unknowns (374). The situation will improve as instrumental techniques advance. The dependence of JHs upon the dihedral, angle has been calculated to be (see Section 5.3.1) d , s -- 9 cos2~ - 0.3 (cycles sec -x) (9.155) but Fessenden and Waugl I(29°) doubt if a similar function can be applied to coupling between hydrogen and fluorine nuclei. Fitting of equations (9.152) and (9.154) to the observed temperature dependence of (vt) and ( d ~ ) is best done with the aid of a digital computer. This assistance is necessary even when the two gauche forms are identical so that the unknown parameters are reduced from five to three, that is,

•4 E ( = E 2 - E 3 = E , - E o )

plus dt

and

do

or vt and ~o. Thus equation (9.154) takes the form

(s)

=

2J, + J, e x p ( - A E / R T ) 2 + exp ( - .4 E]RT)

(9.150

in which it is implicit that Ez > Ea. If a A E value is assumed or obtained from another set of experiments, such as infrared spectroscopy, then the coefficients o l d a and Jt in equation (9.156) can be calculated for each temperature,/'1...Tt at which there is a value for (d). Hence there are a set of k equations of the form (.It> = a,d, + b,d, (9.157) which is linear in the unknowns ,/g and dz, The reduction of the equation to a linear form permits the application of least squares methods to the evaluation of best trial values for d o and $ . With these and/I E it is possible to calculate (d~ as a function of temperature from equation (9.156) and thence obtain the sum of the squares of the deviations of the experimental points from the theoretical curve. The sum is minimised by repeating the process with systematic variation of A E. The above procedure has been applied by Gutowsky, Belford and McMahon to studies on CHC12CHC12, CHCI2CHF2, CF2CICFCI2, CHCI2CI-IFCI, CF2BrCFBrC1 (the dvalues found for this compound are not in agreement with those reported by Newmark and Sederholm (269) who froze out the rotamers) and CFC12CHClz. They found that the procedure could not be utilised in treating data for CH(C6Hs)2CH2(COC6Ha), CHBr2CH2Br, CH2(CeHs)CH~C1 and CH2(CeHs)CH2Br because the low value of A E (35 to 90 cal mole -x)

572


causes ( J ~ ) to be virtually temperature independent. The following approximate expression was used for such systems: of the four isomers of 10-methyldecalol-2 showed that the ring configuration greatly affects the resonance signal location of the angular methyl hydrogen atoms and that there is also a smaller effect due to the hydroxyl group being either c/s or t r a n s to the methyl group.

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584

H I G H RESOLUTION NMR SPECTROSCOPY

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APPENDIX

589

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I

I

II

I

16378 0.17316 0.18447 0.19310 4.36852 0.02493 1.28210 6-81818 2.00000

1"37361 1"10468 0.79501 0.41602 - 0.64792 0.72621 0.32621 -0.40000 0-00000

2.68098

-- 0.88278 -- 0.86728 -- 1"13278 -- 0.84307 -- 1-15693 -- 0.79732 1"20268 -- 1"64039 -- 0.84307 -- 0.79732 -- 1.20268 - 1.13904

10 11 12 13 14 15 16 17 18 19 20 21

4.83622 7"66306 3-16084 8"48153 4-31908 7-30336 3"43838 0"63148 8"07507 9"79297 3"89972 2"18182

- 0.87361 -- 0.85746 - 1"14254 -- 0.83287 1"16713 -- 0.78814 -- 1"21186 1"85208 -- ~82621 - 0"77379 -- 1-22621 - 1-50000

0.00265 0-01345 0-05947 @01740

- 2.88278 - 2.59307 -- 2.23M6 -2.23346

22 23 24 25

0.00294 0"01492 0.06517 0.02521

-

1 2 3 4 5 7 8 9 6

0.18288 0..19718 0-21707 0.24401 4"23949 1-00994 1"43065 6.58077 2.00000

1.25294 0.69240 0.32187 - 0.63058 0.65926 0.27842 - 0.38085 0.00000

0"14744 0"15294 0-15781 0.15439 4"47079 0"84917 1"14775 7'03190 2"0O00O

1"49473 1"21792 0.90036 (>51479 - 0.66238 0.79382 0"37654 -- 0.41728 0.O0OOO

10 11 12 13 14 15 16 17 18 19 20 21

4"81712 7"61994 3-18005 .8"40569 4"36288 7.17604 3"51651 0.76051 7.99006 9"55941 3.98858 2"41923

-- 0.87794 --0.86201 -- 1"13799 -- 0.83751 -- 1"16249 --0.79196 -- 1"20804 -- 1.74442 -- 0. 83426 -- 0.78489 -- 1"21511 -- 1"44415

10 11 12 13 14 15 16 17 18 19 20 21

4"85256 7"69962 3"14443 8"54482 4"28216 7"40564 3"37482 0"52921 8"15083 1"00031 3"82035 1-96810

-- 0.86973 -- 0"85347 -- 1.14653 -- 0"82898 -- 1"17102 -- 0.78545 -- 1.21455 -- 1.96262 - 0.81882 -- 0.76390 -- 1"23610 -- 1-55772

22 23 24 25

0.00282 0.01436 0"06345 O-02136

-- 2"99288 --2-69240 -- 2"32187 -- 2"27842

22 23 24 25

0.00302 0.01521 0.06515 0"02881

-- 3"21792 -- 2-90036 -- 2"51479 -- 2"37655

number

O.6OO

O.50O 3 4 5 7 8

9 6

-

Ia/,oaI

Intensity

number l J/ o~

408808

- -

- -

-

3"10468 2"79501 2-41602 2"32621

0-650

0"550

0.99288

640

APPENDIX F

(continued)

i I'o ]

Frtmition I

In~mi~

F_nqua~

Inte~ty

Frequency

0"11055 0.10874 0"10288 0"08445 4.67061 0.66667 0"82528 7-54539 2.00000

1.86015 1"56431 1"22885 0.83092 -- 0"69377 1.00000 0-54031 -0"45969 0"00000

0"800

0"700 I 2

0"13335 0.13581

1 6

0.13593 0.12484 4.55226 0"78151 1.02736 7"22319 2"O0000

1-00801 0-61730 - 0.67455 0.86203 0"42917 - 0"43286 0"000OO

10 11 12 13 14 15 16 17 18 19 20 21

4-86662 7.73081 3"13032 8.59793 4.25087 7"48836 3.32290 0"44774 8.21849 1.01911 3-74944 1-77681

-- 0"86622 -- 0-84994 -- 1.15006 -- 0"82570 -- 1"17430 -- 0"78359 -- 1"21641 - 2.07545 - 0.81203 - 0.75510 - 1.24490 - 1.61714

10 11 12 13 14 15 16 17 18 19 20 21

4-88945 7.78071 3-10751 8.68087 4-20135 7.61132 3.24494 0"32939 8.33333 1.05081 3.62933 1-45461

--0"86015 -- 0"84401 -1"15599 -0"82054 -- 1.17946 -- 0"78153 -- 1"21847 -- 2"30623 -0"80000 -- 0"74031 -- 1"25969 -- 1"74031

23 24 25

0"00305 0.01526 0.06390 0"03207

-

3-33238 3.00801 2.61730 2-42917

22 23 24 25

0"00304 0"01490 0.05928 0.03734

-- 3.56431 -3"22885 -2.83092 -- 2"54031

1 2 3 4 5 7 8 9 6

0"12118 0"12123 0-11787 0"10210 4-61766 0"72097 0"92021 7.39374 2.00000

1.73804 1.44789 1"11761 0"72286 - 0.68490 0.93078 0"48383 - 0.44695 0.00000

I 2 3 4 5 7 8 9 6

0"10124 0"09799 0"09035 O.O706O 4.71387 0"61785 0"74142 7.68005 2.00000

1"98250 1"68151 1"34151 0.94105 -- 0"70145 1.06965 0.59841 -- 0"47124 0"0000O

10 11 12 13 14 15 16 17 18 19 2O 21

4.87882 7.75759 3"11812 8-64278 4.22420 7"55582 3.28024 0"38234 8.27903 1.03588 3.68605 1.60626

- 0.86304 -- 0"84681 -- 1"15319 - 0-82291 - 1.17709 - 0"78234 - 1.21766 - 2.19010 - 0.80578 - 0.74276 - 1.25272 - 1-67805

10 11 12 13 14 15 16 17 18 19 20 21

4.89876 7.80078 3.09823 8.71344 4.18165 7.65740 3.21553 0"28613 8.38215 1.06407 3-57853 1.31995

-- 0"85750 -- O-84107 -1.15849 -0"81849 -1-18151 -- 0-78105 1"21895 -- 2"42355 -- 0"79465 -- 0.73411 -1.26589 -- 1"80376

22 23 24 25

0-00306 0"01515 0"06184 0-03492

- 3"44789 -3-11761 - 2.72286 -- 2.48383

22 23 24 25

0.00301 0-01456 0.05647 0"03932

--3.68151 -- 3"34151 -2.94105 -2-59841

5 7 8 9

1.61622 1.33238

0"750

0 " 8 5 0

APPENDIX

F

641

(continued) rransitio~ J/~o6

Intensity

Frequenoy

0.09303 0"08867

9 6

0"05962 4.74954 0"57386 0"66746 7.79957 2.00000

2.10508 1-79940 1"45538 1"05291 -- 0.70814 1-13969 0"65796 -- 0"48173 0.00000

10 11 12 13 14 15 16 17 18 19 20 21

4"90697 7"81830 3.09007 8.74145 4"16458 7"69599 3.19085 0"25046 8"42614 1(>75867 3.53296 1"20043

ITransition I

Intensity

Frequemcy

1 2 3 4 5 7 8 9 6

0"07931 0"07348 0"06329 0"04368 4"80409 0"49816 0"54479 8.00000 2.00(300

2.35078 2"03692 1.68614 1"28078 -- 0"71922 1-28078 0"78078 - 0"50000 0"00000

-- 0"85508 -- 0"83925 -- 1"16075 -- 0"81673 - 1"18327 -- 0"78080 -- 1"21920 -- 2"54186 -- 0"78969 --0"72858 - 1"27142 -- 1-86827

10 11 12 13 14 15 16 17 18 19 20 21

4.92069 7"84721 3"07647 8"78676 4"13671 7"75632 3"15223 0"19591 8"5O184 1.09571 3.45521 1"00000

--

0"85078 0.83536 1.16464 0"81386 1.18614 0"78078 1.21922 2.78078 O-78O78 0.71922 1.28078 2.00000

22 23

0"04090

---

0.00284 0"01324 0"04777 0"04295

-

4.03692 3.68614 3.28078 2.78078

0"08576 0"08057 0"07089 0"05081 4"77920 0"53413 0.60227 7-90569 2.O0O0O

2-22784 1-91790 1"57031 1"16623 - 0"71402 1.21008 - 0.71879 --'0"49128 0"00OOO

1 2 3 4 5 7 8 9 6

0"02513 0"01962 0"01327 0"00629 4"95967 0.16987 0.11951 8"74264 2.0OOOO

4.82843 4.47418 4.09524 3.68556 - 0-76393 2.73205 2.14626 - 0"58579 0.O0O0O

13 14 15 16 17 18 19 20 21

4"91424 7-83366 3-08286 8-76568 4"14972 7"72858 3-16999 0"22080 8.46587 1"08636 3.492O4 1.09431

- 0.85284 -- 0.83721 -- 1"16279 - 0"81520 - 1-18480 - 0-78072 - 1"21928 -- 2"66098 -- 0"78508 -- 0"72364 -- 1'27636 -- 1.93372

10 11 12 13 14 15 16 17 18 19 20 21

4.97487 7.95525 3"02361 8"94350 4.03724 7"94320 3"03403 0-04033 8"83013 1"17106 3"13785 2"57359

- 0.82843 -0.81732 -- 1-18268 - - 0"80374 -- 1.19626 -- 0.78568 -- 1-21342 -- 5"23607 - - 0"73205 -- 0"68216 -- 1.31784 -- 3"41421

22 23 24 25

0.00290 0.01371 0.05062 0.04210

----

22 23 24 25

0"00152 0"00599 0"01648 0"03202

-- 6"47418 -- 6-09524 -- 5-68556 -- 4" 1 4 6 2 6

number

number [ .11,oa 1.000

0"9OO

1 2 3 4 5 7

0-07981

8

0.00296 0"01416 0"05354

24 25

3.79940 3"45538 3.05291 2"65796

•

24

25

11-950

1 2 3

4 5 7

8 9 6 10 11

12

~-RS.

21

2.000

3"91790 3"57031 3"16623 2.71879

642

APPENDIX F

(continued) rrensifion number

Intensity

Frequency

0"01211 0-00876 0-00539 0.00226 4"98336 0-08324 0"04722 8"88904 2.000O0

7-31971 6"95075 6"56377 6"15584 - 0.77689 4"21221 3"59822 -- 0-61400 0"0OO0O

Transition number

~/,o6

Intensity

Frequency

2 3 4 5 7 8 9 6

0.00465 0-00314 0-00178 0.00068 4"99437 0.03224 0.01502 8-96156 2.00000

12"31220 11"93107 11"53835 11"13291 -- 0-78657 7"19493 6"55914 - 0"63579 0.00000

5.000

3"000 1

10 11 12 13 14 15 16 17 18 19 20 21

4"98790 7"97913 3"01124 8"97462 4"01683 7"97552 3"01438 0-01664 8"91676 11"86954 3.06374 0-11096

--0.81971 -0"81134 -- 1'18866 - 0"80169 -- 1"19831 -- 0"79038 -- 1"20962 -- 7"72311 --0"71221 -- 0"67379 -- 1"32621 --4.88600

10 11 12 13 14 15 16 17 18 19 20 21

4"99535 7"99221 3-00427 8.99081 4"00611 7"99143 3"00495 0"00563 8"96776 11"95274 3"02343 0"03844

-0.81220 - 0"80667 1-19333 0.80061 1-19939 - 0"79395 1-20605 -- 12"71343 0-69493 -- 0-66927 -- 1"33073 - 7.86421

22 23 24 25

0-00087 0-00316 0.00784 0-01951

-- 8"95075 - 8-56377 --8.15584 - 5"59822

22 23 24 25

04)0038 0-00129 0.00294 0"00882

-13-93107 - 13-53835 - 13"13291 --8"55914

0-00709 0-00492 0-00288 0-00114 4"99010 0-04905 0.02466 8-93898 2.00000

9"81507 9"43855 9.04790 8"64142 - 0-78301 5"70156 5.07384 - 0-62772 0.00000

I 2 3 4 5 7 8 9 6

0-00122 0-00078 0-0(3042 0.00015 4"99866 0-00848 0"00336 8.99073 2"00000

24.80625 24.41577 24"01919 23-61623 - 0.79344 14"68115 14.02962 0"65153 0.00~0

10 11 12 13 14 15 16 17 18 19 20 21

4"99291 7.98799 3.00654 8.98567 4.00953 7"98646 3.00787 0-00901 8"95095 11 "92629 3"03636 0.06102

-- 0"81507 0-80840 - 1.19160 -- 0- 80095 -- 1.19905 - 0-79256 -- 1"20744 - 10.21700 - 0"70156 - 0-67072 - 1"32928 -- 6"37228

10 11 12 • 13 14 15 16 17 18 19 20 21

4"99878 7"99800 3"00111 8"99770 4.00153 7"99790 3.00120 0-00134 8"99152 11"98816 3-00591 0-00927

- 0.80625 - 0. 80327 - 1"19673 0-80015 1-19985 0-79688 -- 1"20312 - 25.20656 -0.68115 - 0-66732 1"33268 1"53485

22 23 24 25

0-00055 0.00193 0"00453 0-01269

- 11"43855 - 11.04790 - 10-64142 -- 7"07384

22 23 24 25

0-00011 0-00035 0-00075 0"00257

- 26-41577 26-01919 - 25.61623 -- 16.02961

- -

-

- -

-

- -

lO.O00

4.000

- -

-

-

- -

-

-

-

-

APPENDIX F

643

(continued) TransitiOnnumber

3/v°61 Intensity 0-00000 ~00000 0-00000 0-00000 5.00OO0 0-00000 (~00000 9.00000

Frequency

oo

o~

- 4.00O00

-

2.00000

21"

10 11 12 13 14 15 16 17 18 19 20 21

5.00000 8.00000 3.00000 9-00000 4-00000 8.00000 3.00000 0-130000 9.00000 12.00000 3.0000O 0.00000

22 23 24 25

0.00000 0-00000 0-00000

1.00000

0-00000 -

2.00000 2.00000 2-00000 2.00000 2.00000 0-00000 - 4.00000

- 1-00000 - 1-00000 - 1.00000

oo -

APPENDIX G

LINE FREQUENCIES AND RELATIVE I N T E N S I T I E S O F HALFA~B= S P E C T R A (P. L. C o m o ,

Transition I number I

J/,o

Intensity

Chem. Rev., 60, 363 (1960)).

Frequency

Intensity I Frequency J,

1 2 3 4

0"00

5 6 7

8 9 1

0"10

2 3 4 5 6 7 8

9 1 2 3 4 5 6 7

0.20

8

9 1 2 3

O-30

0"05

2.21487 2"19901 1-98654 4-00000 1"81445 1-98413 1-80099

0.45013 O.45249 O-50224 0.50000 0.54988 0"50274 0"55249

0-00000 0.00000

1-45512 1"55487

0"51188 0"60990

2-73999 2"57470 1"90650 4"0OO0O 1"51843 1"83459 1"42530

O-35400 O'37202 0"51543 0-5OOOO 0"64741 O'52860 0"67202

0.00007 0.00003

1"42094 1"61896

0-00037 0"00012

1"39803 1"69144

3"04200 2-74278 1"85526 4.00O00 1-40074 1-70049 1-25722

0.30975 0-33852 0-52322 O.5OO00 0"69445 0-55381 0"73852

3"35787 2"89443 1"80516 4"00000 1"29754 1"53606 1"10557

0.26928 0"30902 0"52998 0"50000 0.74024 0"58805 0"80902

0"00122 0"00028

1"38678

0"00288 0"00050

1"38732 1"85828

3"67373 3"02899 1"75801

0"23322 0"28310

3.97544 3"14692 1"71648

0"20182 0"26033 0"53694

2.0O0OO

0.5OO00

2-00000 2.00000 4-00000 2.00O00 2"00000 2"00000

0-50000 0.50000 0.50000 0.500O0 0.50000 0-50000

o.ooboo

1.5oooo

0.00000

1.50000

2.46251 2.39223 1.95243 4.00000 1.65527

0-40113 O.40990 0.50792 0.50000 0-59915

1"92969 1"60777

0"15

0"25

1"77148

0.53474 645

0"35

646

APPENDIX G

(continued) tradition I number I y/v°~

Frequency

J/~o

Intensity I Frequency

0"50000 0"78486 0"63145 0"88310

0-35

7

4.00000 1"20752 1"35451 0"97101

4-00000 1-12774 1"17049 0"85308

0-50000 0"82843 0"68371 0"96033

8 9

0.00547 0.00075

1"39941 1"95105

0.00885 0"00099

1"42247 2"04909

7

4"25184 3'24939 1.68151 4.00000 1.05648 0.99634 0"75061

0"17500 0-24031 0"53644 0"50000 0"87113 0.74418 1.04031

4"49640 3"33793 1"65329 4"00000 0"99244 0"84015 0"66207

0"15242 0"22268 0"53340 0"50000 0"91313 0.81197 1"12268

8 9

0"01262 0-00121

1"45563 2.15176

0"01634 0-00138

1.49778 2.25850

4"70725 3"41421 1-63173 4.00000 0.93449 0"70542 0"58579

0"13356 0.20711 0"52814 0.50000 0.95459 0-88607 1.20711

4"88597 3-47988 1"61631 4"00000 0"88162 0"59229 0"52012

0-11786 0"19330 0"52111 0.50000 0"99567 0.96550 1"29330

0"01956 0.00154

1.54777 2"36881

0"02217 0-00162

1"60447 2"48227

7

5"03592 3"53644 1.60632 4.00000 0.83322 0"49884 0-46356

0.10480 0"18102 0.51270 0.50000 1'08648 1.04935 1"38102

5"16124 3"58525 1-60092 4-00000 0"78863 0.497~2 0"41475

0"09390 0.17006 0"50330 0.50000 1"07713 1"13682 1.47006

8 9

0.02402 0.00168

1.66685 2.59853

0"02520 0"00169

1"73402 2"71725

5-26595 3-62747 1.59935 4.00000 0.74740 0"35985 0-37253

0.08474 0.16023 0-49322 0.50000 1.11773 1"22725 1.56023

5"35366 3"66410 1.60086 4.00000 0-70915 0"30881 0-33590

0.07701 0'15139 0.48270 0-50000 1"15833 1"32007 1"65139

0.02579 0"00167

1.80521 2.83819

0.02588

1"87979 2"96110

4

0"30

5 6

1 2 3

0.40

4

5 6

1 2 3 4 5 6 7

1 2

0.50

0"60

5 6

I 2 3 4 5 6 7 8

9

0.70

Intensity

0"45

0"55

0"65

0"75

0.00163

APPENDIX

G

Frequency

,, J/~o ~

Intensity

0.85

5"48990

647

(continued) Transition J/~o number 1

0.80

2 3 4 5 6 7

1

Intensity [ 5.42746 3.69600 1.60483

0.30400

0-07044 0.14340 0.47195 0"50000 1"19900 1-41484 1"74340

0"02562 0"00158

1-95724 3"08579

5"54302 3.74831 1.61804 0.60929 0.20386 0.25169

0-05996 0"12956 0"45033 0-50000 1.28072 1.60880 1.92956

0.02436 0.00143

2.11909 3.33985

4.00000 0"67356 0-26696

0"90

2 3 4 5 6 7

4.00000

1"00

3.00

0.95

Frequency

3"72387 1-61071 4.00000 0-64034 0.23246 0.27613

0"06481 0"13615 0"46113 0"50000 1-23979 1"51118 1"83615

0.02508 0-00151

2"03712 3"21209

5.58894 3.76984 1.62637 4.00000 0.58020 0.17954 0.23016

0"05574 0-12355 0"43965 0.50000 1-32184 1"70745 2"02355

0-02359 0.00135

2"20283 3-46893 0-02232 0.06155 0-27475 0.50000 2.23552 3"84835 4"06155

5-62808

0.05205

3"78885 1"63556 4.0O000 0-55292

O"11803 0"42914 0"5OOOO 1"36316

0-15950 0"21115

1"80693 2.11803

5.90634 . 3.94029 1.81161 4.00000 0.23917 0.03365 0.05972

0"02267 0-00127

2-28812 3"59923

0-00894 0"00029

4"14542 6"35862

5"95830 3"97279 1"89741

0"01434 0"04138 0"19571 0"50000 3"16867 5"88705 6"04138

5-97655 3-98456 1"93738 4.00000 0"07557 0.00797 0"01544

0"01061 0"03113 0"15066 0-50000 4-13014 7"91160 8"03113

0-002.50 0"00003

8"07287 12.19240

5.99625 3.99750 1.98897 4.00000 0.01311 0.00125 0-OO250

0"00418 0"01249 0"06212 0"50000 10-05380 19-96286 20-01249

4"00000

0-12550 0"01441 0.02721 0"00431 0.00008

2-00

4.00

6"09710 9"25143 ,.,,,

5"00

5.98500 3.99007 1.95828 4-00000 0-05002 O.0O506 0.00993

0"00843 0-02494 0.12207 0.50000 5"10556 9"92780 10"02494

10.00

648

APPENDIX G

(continued) transition

Intensity

number 8

9

5.00

I

Fr~ucacy I J]%

0"00163 0.00001

10"05831 15"15543

6.00000 4.O00O0 2.00000 4-00000 0.00000 0-00000 0"00000

0.00000 0.00000 0.00000 0.50000 co oo co

0.00000 0.00000

~o co

10.00

Intensity 0-00041 0.00(300

]

Frequency 20.02916 30.07878

APPENDIX

H

LINE FREQUENCIES AND RELATIVE INTENSITIES O F A3B2 S P E C T R A Transition number f 1 2 3 4 5 6 A~ 7 i 8 9 10 11 12 = 13 i

1'05383 1"05122 0-95365 1"00250 0"95128 I'00000 1"05103 1"00301 0-94894 1"00244 1"04862

0.00000 0.00000 0.00000 0-00000 0.00000 0-00000 0.00000 0.00000 0.00000 0.00000 O.00(X~ 0.00000

2.30919 4-19229 1.71292 3.80771 4.20721 3"79279 2"33634 2-07388 1.73116 1"87457 1.88577 2"07618

0"07117 0"02378 -- 0"07865 0.02494 0"02494 0"02628 0"07478 0"02035 -- 0"07484

0"00000

-0"90788 2"03370 1-98376 - 1"10713 3"00753 1"93067 2"07930 - - 1"00997 - - 1"00250

2"06044 1.47134 3"76534

1"18515 1-16037 0"88053

1"00000 1-00000 1"00000 1"00000 1-00000

2-00000 4-00000 2.00000 4.00000 4.00000 4-00000 2.00000 2.00000 2-00000 2.00000 2.00000 2.00000

7 8 9 10

111 12

0-00000

'I 2 3

1"00000

-

0"00000 0"00000 0"00000 0"00000 0"00000 0"00000 0"00000 0"00000

.5 6 7 8

.9 0.10

21 a

1.00000 1-00000 1"00000 1.00000 1"00000 1"00000 1.00000

2-69081 1-80771 3-28708 1.98508 2-20721 12.00000 3-61692 2-97284 3-33633 3"97054 2"73116 2"98175 4"41251

3-00000 2.00000 3.00000 2.00000 2-00000 12-00000 4.00000 3"00000 3-00000 4-00000 3.00000 3.00000 4.00000

6

HRS.

0.05

0-(30

5

1 2

Frequency

Frequ~ey

1

A,

Intensity

Intensity

2 3

B

J/~o 6

J#'o ff

2.37259 1"63141 3.54341

-

-

1"00000

0.00001

2.00000 2.00000

0"00001 0-00001

1.00000

0.00000

3-00000 2-00000 2.00000

0-00000 0"00001 0"00001

1"00000

0.00001

-- 1-OOOO0 1"11554 1.10474 0"91410 649

0"15

1.60202 0.95147

- -

-

-

0"03027 0"02824 0"02071

650

APPENDIX H

(continued) Transition' number

J/~o~

Intensity

4 5 6 7

0-10

1"94128 2"42736 12.00000 3-27099 2.87275 3.75441 3"88909 2"51488 2"94153 4"83939

1.00997 0.90523 1.00000 1"10331 1.01419 0"89675 1"00910 1"09415 1-00635 0.90666

2.62741 4"36860 1"45659 3"63136 4"42732 3"57264 2"75454 2"10145 1"51492 1"70399 1"75615

A

8 9 I0 11 12 13 1 2 3

B! 45 ~ 6 7 8

9 10 " 11, • 12

2.08504

fl 2 3 .~ 4

0"00015 0"00012 0.00014 0"00003 0.00000 0"00028 0.00007 0"00008 0.00005

6 7 8

9

1

2 3 4 5 6 7

A'

8

•

9 10 11 12 13

0"20

1.76808 1"32714 3"95330 1"77953 2"89400 12"00000 2-69509 2"38893 4"83922 3-64181 2-19304 2-85171 5.65470

s/v,, ,s

~tenfi~

Frequency

0.15

1"87135 2"65754 12"00000 2"96460 2-67892 4"25941 3"77276 2.33849 2"89540 5"25994

1.02237 0.86201 1.00000 1.15583 1.03669 0.84548 1"01810 1.13642 1.01089 0.86651

0"13446 0-04526 - 0"16410 0.04951 0"04951 -- 0"05523 0.14803 0"03312 0"14891 - 0"07185 -- 0-06110 0-03059

2"93957 4"52866 1"23466 3-47111 4"65731 3"34246 3"26018 2.09490 1"33864 1"50530 1"62624 2.00081

0"18985 0.06463 0-25553 O-07336 0"07336 0.08701 0"21763 0.04139 0-22187 0"12517 0.09634 0.02642

0"83304 2.08438 1.98485 - 1"22710 3"03050 1"87525 2"16435 1.03960 - 1.00997

0"00093 0.00046 0.00062 0"00010 0'00000 0"00170 0.00025 0"00034 0.00023

-- 0"77767 2"15113 2"00257 - 1"35793 3"06992 1"83716 2"25086 1.08802 - 1.02237

1"50515 1.19800 4"11024 1"67103 3"13245 12"00000 2"45798 2"03210

1"34650

Frequency

-

1.26235 1"21780 0.85208 1"03959 0"82170 1.000130 1.20794 1-O7329 0.79765 1.02684 1.17551 1"01431 0.83149

0"25

5"45443 3"51396 2"07115 2-81489 6-00914

-

-

1.27718 0"82793 1-06149 0"78436 1.00000 1"25049

1-12573 0"75560 1"03273 1"21163 1-01593 0"80153

APPENDIX

H

651

(continued) Transition I

1 2 3 4 5 6 7

B'

0.20

8

9 10 11 [ 12 rl 2 3 .~ 4 E ~

5 6

7 8

9

A

1 2 3 4 5 6 7 8 9

10 11 12 13 1

2 3 4 5 6 B~

7 8

9 10 11 12 21a*

0-30

Intensity

Fr~lu~acy

3"23193 4-67286 1"04671 3-32648 4"89333 3"10600 3"84190 2"06969 1"19341 1.29841 1.50233 1.81624

0-23765 0"08211 -- 0-35208 - - 0"09619 0"09619 --0"12170 0"28112 0.04859 -- 0-29372 - - 0"18985 --0"13251 0.00549

0"00329 0"00110 0"00159 0"00019 0.00000 0"00597 0"00057 0-00082 0-00067

- 0"74357 2-23263 2"03565 - - 1-49787 3-12702 1-81900 2"33486 - 1"15385 - 1"03959

1"27627 1"08283 4-24027 1-55150 3.36842 12.00000 2.24844 1"65864 6-05064 3.40196 1-96723 2"78667 6-31618

1"43681 1"33788 0.80737 1"08788 0"75000 1.00000 1-31026 1"19428 0"72076 1"03427 1"24498 1"01562 0"77641

3-72374 4"91717 0-75974 3-08008 5-36568 2-63158 5"06234 2"01299 0"96842 0"92371 1-28190 1"26810

0"31319 0.11212 -- 0-55737 - - 0.1[3788 0.13788 -0.20000 0"38227 0.07067 -- 0"43467 - - 0"34911 - 0-20532 --0"09126

025

Intensity

Frequency

3.49485 4.80200 0"88977 3"19652 5.13097 2"86755 4"46083 2-03914 1"07188 1-38730 1"55627

0"27850 0"09780 - 0"45298 -0"11776 0"11776 -0"15936 0-33638 0-05772 - 0-36458 - 0.26494 - 0.16890 - 0"03374

0-00803 0-00192 0.00300 0"00030 0.00000 0.01443 0-00102 0.00144 0-00149

- 0"73162 2"32740 2"08245 - 1"64545 13"20264 1-82212 2.41329 - 1"23542 - 1"06149

1"08169 0"98039 4-34769 1"42642 3"59766 12"00000 2"06194 1-31588 6"58156 3"31352 1-87783 2.76641 6"57539

1"53240 1.40000 0"78976 1"11855 0"71855 1.00000 1"36075 1"27769 0.69329 1"03091 1"27580 1-01355 0.75540

3"91831 5"01961 0"65231 2-97593 5"59319 2"40235 5"59901 1"99743 0"87963 0"77185 1"18499 0"99756

0.34260 0.12500 - 0. 6 6 4 7 6 -0.15645 0-15645 - 0.24355 0.41896 0-08789 - 0-50419 - 0.44096 - 0-24194 - 0.16545

1"10080

0"35

APPENDIX H

652 (continued)

Transition number

J/Uo

Intensity

Frequency

0.30

--

9

0.00275

-

-- 0"74134 2"43387 2"14115 - 1"79940 3"29657 1-84628 2"48456 1"33084 1"08788

0.35

8

0"01483 0"00276 0"00464 0"00038 0"00001 0-02652 0"00157 0"00207

1.63246 1-46332

0"45

'1 2 3 6

~5

~

6 7

I

1 2 3 4 5 6 A~ 7 8 9 10 11 12

0"40

0"91886 0"88942 4"43649 1"30067 3"81650 12"00000 1"89490 1"03032 7"02442 3"25156

0"77460

1-15322 0.68990 !-00000 1.41118 1.37367

0.67235 1"02335 1.30430 1.01004 0.73790

1"80014 2-75314

•

6"79072 i

tt

3 4 5 6 7 9 10 11 I 12

~

~ 5 6 7 8 9

0.50

- 0"77084 2"55055 2"20992 - 1"95871 3"40739 1"88964 2"54866 1"43831 1"11855

0.00659

0"03307 0"00418 0.00879 0"00052 0"00006 0"05905 0.00385 0"00302 0"00901

0"67218 0"73699 4"57143

1"84307 1"59307 0-75000

0"59450

-

0"55

-

-

0"78380 0.80868 4-51012 1.17820 4"02213 12.00000 1.74410 0-8O666 7-37932 3"21495 1.73203 2.74568 6"96805

- 0"81730 2"67609 2.28709 - 2"12253 3-53278 1"94941 2"60672 1-55613 - 1.15322

0.02866 0"00394 0.00769 0.00048 0-00003 0.05108 0.00298 0.00291

1 2 3

0'02232 0"00342 0'00627 0"00045 0.00002 0'03976 0-00222 0"00254 0"00447

4'21620 5"19133 0"48989 2"79967 6"01313 1"97787 6"40536 2"00663 0"73541 0"54154 1"01323

1"99530 0"80267 0"64529 1"09579 0"77108

8

F~u~

0"36754 0"13668 -0"77460 -0"17343 0"17343 -0"28990 0"44761 0-10876 -0"57334 -0-53916 -0"27907 -0.25371

4"08114 5"11058 0"56351 2"88284 5"80991 2'18350 6"04688

1

I 2

Intensity

0"57995 0"67329 4"62277

1"73623 1"52772 0"76146 1-19163 0-66390 1"00000 1"46185 1"47960

0"65669 1"01267 1"33064 1"00541 0"72330 0"38877 0"14728 0"88646 0" 18882 0"18882 0"33890 0"46977 0"13215 --0"64226 0"64250 -0"31703 - 0"35314 -

--

--

-

0"87768 2-80930 2"37122 - 2"29017 3.67016 2"02250 2"66034 1-68283 1"19163

-

---

1"95246 1"65926 0.73995

APPENDIX H

653

(continued) Transition number

A

,#

i

4 5 6 7 8 9 10 11 12 13

Intensity 0.50

1

2 3 4 5 6 7

B

I 8

9 t0 [1 .. L2 1

2 3 ~

5 7

I 8 9

l 1 2 3 4 5 6 A 7 8

9 I0 :II I 12 13 1

0.60

Frequency

3/~o 6

Intensity

Frequency

0.95377 4"38718 12.00000 1-48239 0"50982 7.87721 3.20204 1-61820 2"74356 7"23363

1.27842 0.61915 1.00000 ! .56495 1.71207 0-63631 0-98627 1.37742 0.99406 0.70076

4.42005 5"32671 0"37724 2.65905 6-37294 1.61282 6.90614

-- 0.75000 - 0"35611 -- 0-46107

2"06166 0"62358 0"38863 • 0"86541 0"36363

0.42255 0"16574 1-11495 0.21511 0.21511 0.44415 0.50043 0-18216 0"77988 0.86083 0-39652 - 0"57533

0"03536 0-00421 0"00960 0"00051 0"00007 0"06340 0"00478 0-00301 0"01160

0"94918 2-94918 2'46107 - 2.46107 3"81718 2-10611 2.71107 1"81718 - 1-23346

0.03590 0.00410 0-01010 0.00050 0.00008 0.06474 0-00580 0.00288 0"01424

1"02948 3.09486 2-55566 - 2.63477 3.97185 2"19791 2-76021 1"95812 - 1-27842

0.50354 0"61662 4.66600 0"85473 4.54546 12"00000 1"36828 0"41427 8"04950 3-21663 1"57026 2.74690 7-33265

2"06394 1-72621 0'73107 1"32621 0.60000 1.00000 1"61775 1"83505 0"62972 0"97215 1"39813 0"98781 0"69201

0'43997 0"56611 4"70264 0.76517 4"68793 12-00000 1-26374 0.34184 8"18647 3.24000 1-52721 2-75213 7.41487

2"17717 1"79382 0"72319 1"37655 0"58272 1.00000 1"67155 1"96085

0.98140 0.68453

4"49646 5"38338

0.43606 0-17379

4.56003 5-43389

0.44783 0-18118

1"06193 4"21268 12"00000 1"60713 0"63702 7"65848 3-20005 1-67178 2"74285 7"11376

1"23346 0.64039 1.00000 1.51304 1.59307 0.64503 1.00000 1.35496 1.00000 0.71107

4"32782 5"26301 0"42857 2.72539 6"20108 1"78732 6.68659 2.02969 0.67612 0"45715 0"93656 0"46188

0.40693 0"15693 -

0.55

1.00000

- 0.20268 0"20268 - 0"39O39 0"48696 0"15693 -

0"71107

-

0-65

-

0-62468 0.95812 1.41720

654-

APPENDIX H

(continued) 1

~amitionnumberI J/*o I 3

i4

0.6O

5 6 7 8

B,

i 9

i L° 12 1

2 3 4 6

7 8

L 9 1 2 3 4 5 6 A 7 8 9 10 11 12 I 13

I

Frequen~

J/~o

Intensity

0-33400 2"59981 6"52865 1-45455 7"07832 2"10005 0-57674 0"33288 0-79920 0.29082

1-23107 -0"22621 0-22621 -0"50000 0"51114 0"20717 -0"84875 -097433 -0"43843 -0"69419

0.65

0.29736 2.54690 6.66873 1.31207 7.21461 2.14250 0.53484 0.28730 0.73766 0.23635

- 1-34819 - 0.23610 0-23610 - 0-55772 0.51978 0.23150 - 0.91773 - 1-08999 - 0.48192 -- 0.81639

0"03520 0-00387 0"01036 0.00047 0.00009 0-06394 0"00686 0-00270 0"01681

-- 1"11675 3"24564 2"65415 --2.81089 4-13263 2.29607 ~80871 -2.10478 - - 1.32621

0.03373 0.00359 0.01040 0.00046 0-00010 0.06177 0.00799 0.00245 0"01921

- 1.20957 3.40089 2.75591 - 2"98911 4"29832 2"39919 2"85725 -- 2"25644 -- 1"37655

0"38680 0"52101 4-73387 0.68491

2.29186 1.86203 0.71616 1-42917 0.56714 1.00000 1.72640 2"08859 0"62076 0"94448 1.43476 0.97494 0"67808

0-34208 0.48064 4"76065 0"61347 4"92916 12.00000 1"07993 0-24290 8-38625 3.30171 1-45322 2.76609 7"54147

2"40776 1"93078 0.70985 1"48383 0.55305 1"000(30 1"78234 2"21766 0"61767 0-93141 1"45091 0.96853 0"67248

- 0"98686 - 1"20737 --0"52704 - 0"94097

4"65793 5"51936 0"23935 2"45737 6.90588 1"07084 7"41208 2.23240 0"46315 0"21879 0-62769 0"16336

0"46724 0"19422 -- 1-58485 -- 0.25272 0.25272 -- 0.67805 0.53276 0"27714 -- 1"05617 -- 1.32616 -- 0"57379 -- 1"06723

1.30685 3.56011

0"02974 0.00294

-- 1-40776 3.72286

4'81546 0.28623 8.29662

3"26911 1"48837 2"75867 7"48362

9 O A L2

2

0"03183 0"00327

6

7

8

-

12"00000 1"16788

4"61320 5.479O0 0"26613 2"49963 6.79409 1"18454 7.32371 2"18709 0"49716 0"24982 0"68057 0"19509

1

2 3 4 5 B I

0.70

Intensity

0-45814 0-18797 - 1"46616 - 0"24490 0"24490 - 0"61714 0"52686 0.25488

--

O.75

I

Frequency

APPENDIX

H

655

(continued) Transition number

Frequency

J/~o ~

Intensity

2"86039 3"16918 4"46801 2"50621 2"90628 - 2"41249 -- 1"42917

0.75

0.01010 0.00039 0.00010 0"05548 0-01023 0.00197 0"02328

2"96717 - 3.35086 4"64101 2"61631 2-95611 - 2.57243 1.48383

0"85

9 I0 11 12 13

4"78375 O'55O19 5"03026 12"00000 0-99926 0.20864 8.46000 3"33609 1-42133 2"77405 7.59049

2"52470 2"00000 0"70416 1-54031 0"54031 1.00000 1-83936 2"34759 0-61521 0"91903 1"46576 0"96223 0"66758

0"27200 0"41190 4-80379 0"49428 5"12004 12"00000 0"92530 0-18117 8"52134 3-37101 1"39232 2"78228 7-63231

2.64250 2"06965 0.69901 1"59841 0-52876 1-00000 1"89745 2"47806 0"61322 0-90738 1.47940 0"95608 0-66328

1 2 3 4 5 6 7 8 9 10 11 12

4"69578 5"55556 0"21625 2-41956 7-00537 0-96974 7-48451 2"27734 O'43238 0"19290 0.57883 0"13858

0-47531 0"20000 -- 1"70416 -- 0"25969 0"25969 - 0"74031 0-53774 0-29820 - 1-12566 - - 1-44609 0-62213 - 1-19464

4"72800 5"58810 0-19621 2"38569 7-09382 0-87997 " 7"54450 2"32114 0"40444 0"17115 0"53379 0"11894

0-48250 0"20535 - - 1"82401 0"26589 0"26589 - 0-80376 0.54200 0"31805 -- 1.19534 -- 1.56694 - 0"67202 -- 1.32284

0-02762 0.OO263 O-OO982 0.00036 0.00010 0"05203 0"01132 0"00173 0"02489

- 1.51165 3-88875 3"07590 3"53398 4"81677 2"72888 3 -00692 - 2-73580 -- 1"54031

0"02556 0"00233 0~349 0-00034 0"00009 0"04863 0.01236 0-00152 0.02622

- 1"61801 4-05745 3-18630 - 3"71836 4"99486 2"84344 3.05880 - 2-90224 - 1"59841

0.22061 0-35609 4"38657 0.40151 5.27046 12.00000

2.88025 2-21008 0-69005 1-71879 0.50872 1.00000

3"/I'o

6

0.70

3 4 5 6 8 9 I 2 3 4 5 6 A ,I 7

0.80

i 8

B

1 2 3 ~

5 6 7 8

, 9

A.

f

1

~

2 3 4

i 5 l 6

0"9O

Intensity 0-01030 0-00043 0-00010 0.05882 0"00911 0"O0221 0"02138

--

0.3O422 O.AAAAA

O-24439 0"38258 4"82126 O-44497 5-19972 12"00000

- -

2-76106 2.13969 0.69433 1"65796 0.50827 1.00000

0.95

F~u~

APPENDIX H

656

(continued) Transition I number I 7 8 9 10 11 12 13

A

J/~o~

Intensity

Frequency

']'/~o ~

Intensity

Frequency

0.90

0"85753 0"15884 8.57286 3.40561 1.36587 2.79061 7.66821

1"95657 2"60881 0"61159 0.89648 1"49193 0"95010 0.65946

0-95

0.79545 0"14046 8"61654 3.43928 1'34172 2"79889 7.69923

2-01667 2"73969 0.61024 0.88631 1"50344 0-94431 0.65606

4-75561 5"61743 0.17874 2.35531 7-17245 0.80028 7.59471 2.36326 0.37900 0.15274 0.49238 0-10315

0.48894 0.21031 1.94433 0.27142 0.27142 0"86827 0-54568 0.33670 1-26523 1.68856 0-72339 1.45154

4-77940 5"64391 0-16343 2.32803 7.24240 0-72954 7.63713 2.40337 0.35577 0-13706 0.45439 0.09029

-

0.49475 0.21492 2"06505 0.27636 0"27636 0"93372 0.54889 0"35418 1.33531 1.81080 0"77618 1"58055

0.02360 0.OO2O6 0.OO913 0.00031 0.00009 0"04536 0-01334 0.OO133 0.02727

- 1"72643

0.02178 0.00181 0.00876 0.00028 0.00008 0.04228 0.01425 0.00116 0.02806

-

0.20000 0-33211 4.85005 0.36319 5.33334 12.00000 0.73859 0.12515 8.65386 3-47168 1.31962 2.80704 7-72619

3"OOOOO 2-28078 0"68614 1"78078 O'5O0OO I'OOOOO 2.07772 2"87056 0.60911

4"80000 5"66789 0"14995 2"30347 7"30470

1

2 3 4 5 I

6 7

B

8 9

10 11 [ 12 1

2 3 ,

5

~

6 7 8 9 1

2 3 4 5 ! I

6

A, I 7 8

9 10 11 12 13 1

2 B

3

4 5

I'OO

-

-

-

4.22868 3"29812 3"90388 5"17493 2"95961 3"11180 - 3.07141 - 1"65796 -

-

-

--

1.83661 4.40218 3.41117 - 4.90042 5.31673 3.07711 3.16594 - 3.24304 - 1-71879

0.87687 1.51402 0.93872 0"65302

0.05051 0-11325 4-95677 0"07967 5"82843 12.0OOO0 0-21674 0-02886 8.91359 3-82280 1.10843 2.91365 7.92684

5"44949 3.73205 0"64575 3-14626 0.41421 1.0OOOO 3.42349 5-44949 0.60202 0.78246 1"61552 0.36329 0.62531

0.50000 0.21922 - 2.18614 -0-28078 0.28078

4"94949 5.88675 0.04323 2.09190 7.80708

0-55051 0-26795 -4.64575 -0"31784 0-31784

2.00

APPENDIX H

657

(continued) Tramition number '

6

I

J/~o 1.00

7 8 B

~

9

10 11 12 f

1 2 3 5 6 7

I

8

, 9 1

2 3 4 •

5

6 7

A

8

9 10 11 12 13 1 2 3 4 5 6 B I 7

189 tO tl ~ [2

I

1 2 4 5

3.00

Intensity

Frequency

0"66667 7-67326 2-44131 0"33453 0"12361 0"41955 0"07969

1"00000 0-55173 0"37046 - 1"40560 1"93356 - 0"83O3O - 1.70972

0-02010 0"00159 0-00838 0'00026 0"00008 0'03942 0"01508 0.00101 0-02863

- 1.94827 4-57:772 3-52530 - 4-27788 5-54OO1 3"19570 3-22118 - 3"41688 - - 1-78078

0-0:'~":'~ 0-05550 4"98000 0-08148 - 5"92603 "12-00000 0"09497 0"01272 8"96163 3"91785 1"05170 2"95387 7"96693

7"93273 5"21221 0-63104 4"59822 0"38600 1"000(30 4"86824 7"98019 0-60087 0-75671 1.64241 O-82687 0.61651

4"97778 5-94451 0-02000 2"04249 7-91303 0-07397 7"96503 2-92485 0"06437 O'O1306 0"04329 0"00704 0"00239 0"00003 0"00176 0"00001 0.00001

J#'o

Intensity

Frequency

0"17157 7"92048 2"82828 0-12613 0"02989 0-10837 0"01661

- 2"41421 0"57651 0"55051 - 2.84275 - 4"42822 - 2-09052 - 4"27096

0"00550 0.0O014 0-00345 0.00005 0.00002 0-01237 0"01773 0-00008 0"02135

- 4"32247 8.32247 5.91671 - 8"13425 9.36148 5"65545 4"48850 - 7"14395 -- 3"14626

0"01240 0"03270 4"98855 0-01644 5"95932 12-00000 0.05225 0-00715 8"97836 3"95328 1"02984 2"97172 7"98116

10-42443 6"70156 0"62348 .6-07384 0"37228 1"00000 6"34128 10.49496 O'60049 0"74681 1"65270 0"80613 O'61228

0"56727 0"28779 -- 7"13104 - 0"32621 0"32621 - 3-88600 0"5843O 0"61473 -4"31234 - - 6"93522 - 3"49680 -- 6"79502

4"98760 5"96730 0-01145 2.02424 7-95087 0"04068 7-98044 2"95883 0"03871 0"00738 0"02242 O'00382

0"57557 0"29844 -- 9-62348 - 0.32928 0-32928 - 5"37228 0"58819 0"64610 - 5"79493 - 9"44082 - 4-94837 - 9"30628

--6-78116 12"23370 8"37860 -- 12"07443 13"29181

0.00132 0.00001 0.00103 0.000(30 0.00000

- 9"26067 16"19014 10"85923 -- 16"04188 17"25465

-

2.00

4.00

658

APPENDIX

H

(continued) Transition number 6

J/Vo ,~

Intensity

Frequency

3.00

0"00578 0.01268 0"00001 0"01301

8"15260 5"89592 --11"04852 -4"59822

5.00

0"00789 0"02150 4-99260 0"01001 5"97437 12"00000 0"03284 0"00458 8"98615 3"96996 1"01933 2.98097 7"98788

12"91948 8"19493 0"61887 7"55914 0"36421 1"00000 7.82550 13.00345 0.60031 0.74206 1"65763 0.79284 0.60977

4"99211 5"97851 0"00741 2"01562 7'96850 0"02563 7"98753 2"97422 0.02576 0.00471 0.01352 0.00241

7 8

1 2 3 4 5 6 A 7 8 9 10 11 ! 12 i. 13 1 2 3 4 5 6

B]

7

18

9 10 11 12

f

1 2 3 4 6 7 8

t

9

1

2

i 34 A

5 6 7 8

Intensity

Frequency

4.00

0"00340 0.00888 0.00000 0.00846

10.65358 7.34788 - 15.00146 - 6.07384

I0"00

0"00195 0"00566 4-99811 0"00224 5-99382 12"00000 0'00784 0.00115 8.99656 3.99243 1"00491 2"99477 7.99698

25-4O967 15.68115 0.60952 15.02961 0.34847 1-00000 15.29514 25.51929 0.60008 0.73555 1.66437 0.76436 0.60484

0"58052 0"30507 12-11887 - 0.33"07.3 0-33073 6.86421 0.59053 0.66448 - 7-28374 - 11.94489 - - 6.41896 11-81261

4"99805 5-99435 0.00189 2-00394 7.99211 0-00618 7-99690 2.99390 0.00693 0.00113 0.00293 0.00061

0.59033 0"31885 -- 24.60952 - - 0"33268 0"33268 -- 14.34847 0.59524 0-69995 -- 14"75966 - 24.45469 - - 13.85965 -- 24-32398

0"00083 0"00001 0"00068 0.00000 0"00000 0"00221 0"00643 0-00000 0"00588

--11.74843 20-16446 13"34751 -20"2148 21.21357 13"15498 8"81865 18"97363 - 7.55914

0.00020 0-00000 0.00019 0.00000 0.00000 0.00054 0.00202 0.00000 0.00172

-- 24.22411 40-11449 25"82388 -- 39.97870 41"18362 25.65960 16"25956 -- 38.91917 -- 15.02961

0-00(300 0-00000 5-00000 0.00000 6.00000 12.00000 0.00000 0.00000

co oo

0.60000 oo

0.33333 1-00000 oo co

APPENDIX

659

H

(continued) Pransition number

91 10 11 12 13

A:

[

1 2 3 4 5

i 6 B] 7 8 9 lO I1 I 12 {

1 2 3

.5

t

5 7 8 9

J/~o 6

Intensity

Frequency

9-00000 4.00000 1"00000 3.00000 8.00000

0.60000 0.73333 1.66667 0.73333 0.60000

5.00000 6.00000 0.00000 2.00000 8.00000 0.00000 8.00000 3-00000 0-00000 0.00000 0.00000 0.00000

0.60000 0"33333

0.00000 0.00000 0.00000 0-00000 0.00000 0.000~0 0-00000 0.00000 0-00000

- - 0 0

--0-33333

0.33333 0.60000 0"73333 --

OO

--

00

--

OO

--

OO

--

CO CO

--

CO

--

OO OO CO CO

--

OO

--

OO

Intensity

Frequency

APPENDIX ]

MISCELLANEOUS SPIN SYSTEMS-NOTES AND LITERATURE REFERENCES ABe XH resonance spectra of isopropyl derivatives: J. RA~wr, Ann. Physik, 10, 1 (1962). A~B~ The ~H resonance spectrum of propane; the "composite particle" method of analysis was used: D. R. WmTMAN,L. ONSAGER,M. SAUNDERSand H. E. Duns, J. Chem. Phys., 32, 67 (1960). A B T T h e 1H resonance spectra of substituted thiophenes: R. A. H O ~ A N and S. GRONO" WlTZ, Arkivfor Kemi, 16, 50 (1960). Here T is a magnetic nucleus of spin value of I ; it is pointed out that the spin system may be treated as ABX2 by ignoring all spin functions which are antisymmetric in the two X nuclei. The only difference between the observed spectra of AB X2 and ABT is one of relative intensifies of bands. X3AA'X~s(i) (Jxx' -- 0) IH spectra of 2,3-disubstituted butanes: F. A. L. ANEr, J. Amer. Chem. Sot., 84, 747 (1962). (ii) AISodone for ethane enriched with two lsC nuclei: R. M. LYNDEN-B~LL,MoI. Phys., 6, 601 (1963). XnAA'X'(Jxx' ffi 0) General expressions have been derived for line positions and intensities: R. K. ~ , Can. J. Chem., 42, 2275 (1964). ABX3 The 1H spectrum of 1-chlorobutadiene-l,2 : S. L. MANATTand D. D. ELL~.~, J. Amer. Chem. Soc., 84, 1579 (1962)..Saturated aliphatic compounds: F. A. L. ANL~T,Can. J. Chem., 39, 2262 (1961). 2-substituted propenes: E. B. WHnn'LE, J. H. GOLDSTE[Nand L. M~'D~.L, J. Amer. Chem. Soc., 82, 3010 (1960). Cis and trans crotonic acids and methyl esters and also N-methylformamide: V. J. KOWAL~WSKIand D. G. DE KOWAL~VSK~,J. Chem. Phys., 33, 1794 (1960). ABCs Perturbation theory has been used to extend the ABXs analysis to ABCs spin systems and the results were applied to the XH resonance spectrum of tr~s-propenylbcnzene (trans-ff-methylstyrcoe): R. W. FL~SV.NDn~and J. S. WAUGH,J. Chem. Phys., 30, 944 (1959). 9F resonance spectra halogenated propenes: J. D. SWALENand C. A. ~ Y , J. Chem. Phys., 34, 2122 (1961). Here the approximate values of the chemical shifts and coupling constants obtained from an ABXs analysis were used to set up trial matrices which were then modified by an iterative process to find the convergent solutions corresponding to the measured eigcnvalues of the system. Theoretically, the relative signs of all three coupling constants can be derived from such an analysis. A B C X Complete analysis made of the XH resonance spectra of the isomers of CHzFCI-IBrCOOCH3 : J. B. STOTH~S, J. D. TAL~&~Nand R. R. Fl~S~t, Can. J. Chem., 42, 1530 (1964). A B P X 1 9 F and IH resonance spectra of halogenated ethanes: J. L~E and L. H. StrrCLIFFE, Trans. Faraday Soc., 54, 308 (1958).*9F and XH spectra of 1-fluoro-2,4-dinitrobenzene :B. D. N. R.AO and P. VENKXT£SWARLU,Prac. Indian Acad. Sci., 52, 109 (1960); J. Sci. and Industrial Res., 20B, 501 (1961). ; B. D. N. RAO, Mol. Phys., 7, 307 (1964). A B X ] r 1H resonance spectrum of ~-picoline: B . D . N . RAo and P. VENKAIT.SWARLU, Proc. Indian Acad. Sci., 54, 305 (1961). R. J. AngAH~ and H. J. BEI~Slam~, Can. J. Chem., 39, 216 (1961). RIGOS has shown that the 4 x 4 secular determinant in the A B X Y analysis can be solved exa~-tly under one of three limiting conditions; the analysis was applied to the tH spectrum at 60 Mc sec-1 of the pyridine ring in nicotine. N. V. RIGGS, Austrah'an ?. Chem., 16, 521 (1963). 661

662

~PPENDIX I

AB2X The tH resonanc~ spectrum of meta-dinitrobenzene: R. J. A a ~ , E. O. BisHoP and R. E. RICHARDS,MoL Phys., 3, 485 (1960). ABaC Explicit expressions for the transition enersies and relative intensities cannot be written down for this system: a computer can be used to obtain convergence on to the correct eigenvalues of the 3 x 3 and 4 x 4 subnmtric~ encountered in the analysis. A B 2 X trial values of the spectral parameters are used initiallyto set up an approximate matrix which is then sue,ce~ively modified until its roots correspond with the observed eigenvalues. The relative signs of all the coupling constants can be obtained. AA~KL and A B K L tH resonance spectra of 2-pyridines: V. J. KowAt~wsru and D. G. D~ KOWALeWSKX,J. Chem. Phys., 37, 2603 (1962). AB2X2 The 19F and alp resonance spectra of triphosphonitrilic~-l,l-difluoride-3,3,5,5-tetrachloride (PaNaCLtF2): M. L. I-IEF~AN and R. F. M. WHrtm, J. Chem. See., 1382 (1961). AB, tX19F spectrum ofpentafluorosulphur hypofluorite (SFs.OF): R. K. HARMSand K. L PACKER, J'. Chem. So¢. 3077 (1962). A B e X tH spectra of isopropyl compounds: J. RAI,n~r, Ann. Phy~ik, 10, 1 (1962). A A ' P P ' X The analysis has been devised for spin systems containing different isotopes: S. MATSUOg.A and S. HATroPs, Sol. Reports Kanazawa Univ., 6, 33 (1958). The tH resonance spectra oftaC satellites in ethylene oxide, ethylene imine and ethylene sulphide: F. S. MORTXM ~ J. Mol. Spect., 5, 199 (1960). AA'XsXs' The tH resonance spectra of 2,3-disubstituted n-butanes: A. A. BoTm,mR-BY and C. N ~ Cot~, J. Amer. Chem. Soc~, 84, 743 (1962). A A ' A " A " ' X X ' tH and 19F resonance spectra of p-dilluorobenzene: S. MATSUOKAand S. I - ~ r o m , Sci. Reports Kanazawa Umv., (1959). Lynden-Bell has tabulated the matrix elements of the hue,lear spin Hamiltonian for this system: it occurs inpara.difluorobenzeneor ethylene having two xaC atoms. The relative signs of the two .fAxconstants can be determined. Also the relative signs of JAA and Jxx can be determined from the X but not the A part of the spe~rum. R. M. LYNDEN-BeJ.~ MoL Phys., 6, 601 (1963). A A' A"Xa Xa' Xa" The t H and t 9F resonancesof the 1,3,5-trisu bstituted benzenes (C Ha)aCell a and (CFa)aCeHs: J. V. Acmvos, Mol. Phys., 5, 1 (1962). The high symmetry of the molecules enabled the group theoretical method of analysis to be used (see E. B. WItaON, J. Chem. Phys., 27, 60 (1957). A2,42'X with strong cross-coupling. The tH spectrum at 60 Mc sec-x of the ring of cyclopropylaminecanbe simplified from being a n A A ' B B~X to an A2 As' Xspin system by choosing the correft concentration in benzene. H. M. HUTrON and T. S c H A ~ Can. Y. Chem., 41, 2774 (1963). AA' P P ' P " P'".1"2 XH resonance spectrum of bicyclobeptadiene:

H(A) H ( p ~ I ) ~ H(P) H(PII)/H~( AI} "~'H(Pt')

F. S. MORTIMER,3".Iv[el. Spect., 3, 528 0959). As in thepreceding spin system, this symmetrical molecule lends itself to the application of the group theoretical method. AA'BB' X tH and tgF spectra at 40 and 56.4 Mc sec-1 of pars-substituted monofluorobenzenes: G. ARULDHASand P. VENKATr~WAKLU,MoL Phys., 7, 65, 77 (1964).

APPENDIX

I

663

A 3 B C X The IH and 19F resonance spectra of c/s and trmts-l-fluoropropene: R. A. BEAUDFr and 3. D. B ~ C m W I E ~ . R , J. Mol. Spect., 9, 30 (1962). A B B ' C C ~The "direct" method of analysis described by D. WHITMAN,.L Mol. Spect., 10, 250 (1963). A BB I C C ' X and A BB'C C'D The 1H spectrum of monofluorobenzene: S. FUnWXRAand H. Sm~azu, J. Chem. Phys., 32, 1636 (1960). 1H resonance spectra of 1, 2, 3, 4, 5, 6-hexachlorocyclohexanes: R. K. HARMS and N. S~PPA~, Mol. Phys., 7, 595 (1964). A B CD ,I": The 1H resonance spectrum of N-benzylthieno [3,2-6] pyrrole: H. S. Gtrrowsgy and A. L. P o n ~ , J. Chem. Phys., 35, 839 (1961). AsBCD The IH resonance spectra of halogenated aliphatic hydrocarbons: H. Fn~GOLD, Proc. Chem. Soc., 213 (1962). A~,Kn... X,. Double irradiation expectations worked out for general case and applied to 40 Mc sec-1 spectra of CH3CHF2 (A3KX2), CH~CH2F (AsK2X) and CF2CHF (AKPX): M. B A R ~ . n and 3. D. BXLDI~nm.gR, J. MoL Spect., 12, 23 (1964).

NAME

INDEXmVOLUME

Abragam, A. 17, 30, 116 Abraham, R.J. 147, 148, 149, 259, 329, 363, 415, 565, 566, 568, 572, 661,662 A~'ivos, J.V. 662 Abroad, M. 575 Aihara, E. 108 Ainsworth, J. 568 Akahori, S. 176 Alberty, R.A. 565 Alder, B.J. 511 Aleksandrov, I.V. 9 Alexakos, L.G. 237 Alexander, S. 106, 108, 176, 451,465, 489, 510 Allan, E.A. 545, 547 Allen, L.C. 230 Allerhand, A. 481 Allred, A.L. 524, 548, 549 Anbar, M. 510 Anderson, D. H. 109, I10, 111, 127, .154, 160, 164, 165, 172 Anderson, E.W. 575 Anderson, H.L. 237 Anderson, J.K. 556 Anderson, J.M. 476 Anderson, M.M. 524 Anderson, P.W. 488 Anderson, W.A. 23, 116, 117, 121, 177, 191,202, 204, 206, 212, 231, 232, 237, 240, 241,243, 309, 329, 335, 344, 347, 350, 351, 388, 391,427, 431, 453, 460, 463, 466, 470, 471, 474, 475, 557, 558 Andrew, E.R. 4, 21, 31, 39, 46, 217, 256 Androes, G.M. 559 Anet, F. A. L. 174,175, 247, 565, 575, 579, 661 Ankel, Th. 174, 387 Arata, Y. 176 Armstrong, A.M. 522 Arnold, J.T. 23, 206, 208, 236, 344, 492, 542, 543 Aruldhas, G. 662 Axtmann, R.C. 519, 523 Ayant, Y. 26

Baba, H. 150 Bacon, J. 347 xxi

1

Baggett, N. 578 Baflar, J.C. 523 Bak, B. 445, 446, 447, 448 Baker, E.B. 203, 223, 229, 244 Baker, M.R. 127 Baideschwieler, J.D. 190, 241, 247, 267, 371, 398, 456, 457, 459, 461,463, 476, 662. 663 Baldwin, H.W. 524 Bannerjee, M.K. 341 Banwell, C.N. 173, 174, 180, 182, 185, 190, 321,358, 374, 387, 423, 435, 436, 437, 438, 439, 440, 441,442, 566 Barfield, M. 103, 171, 172, 663 Barfield, P.A. 552 Barton, G.W. 230 Basolo, F. 526 Batdoff, R.L. 546 Beaudet, R.A. 662 Becker, E: D. 147, 262, 264, 542, 545 Belford, G.G. 168, 169, 190, 569, 571,572 Bell, R.P. 534 Bender, P. 565 B~,n~, G.J. 63 Benedek, G.B. 28 Bensey, F.N. 530 Benson, R.E. 115, 116 Bentley, P.G. 215 Berber, S.B. 504, 510 Berlin, A.J. 136, 575 Bernheim, R.A. 524, 525, 527, 528 Bernstein, H.J. 25, 90, 92, 124, 125, 127, 129, 132, 141,144, 163, 171,172, 173, 176, 216, 261,283, 326, 327, 328, 329, 335, 363, 404, 405, 406, 407, 448, 467, 486, 537, 538, 539, 548, 565, 572, 575, 576, 661 Bersohn, R. 77, 78, 122, 123 Bha~a, N.S. 9, 466, 476 Bhar, B.N. 549 Bible, R.H. 9 Bigeleisen, J. 518 Birc~all, T. 510, 511 Bishop, E.O. 661 Bitter, F. 203, 240 Bleaney, B. 4 Bloch, F. 2, 16, 20, 25, 29, 30, 31,34, 35, 37, 38, 39, 40, 41, 45, 46, 52, 53, 206, 207, 208, 209, 240, 244, 247

xxii

NAME INDEX--VOLUME I

Bloembergen, N. 20, 23, 24, 25, 27, 30, 32, 33, 38, 41, 46, 48, 76, 185, 210, 217, 218, 224, 233, 489, 526 Bloom, A.L. 56, 202, 240, 242, 244-7 Boden, N. 156, 185, 191, 571 Bonner, L.G. 548 Bothner-By, A.A. 90, 94, 175, 176, 260,662 BSttcher, C. J.F. 95, 98 Bottini, A.T. 574 Bovey, F.A. 23, 24, 82, 140, 141, 148, 257, 259, 268, 575 Bradbury, A. 256 Bradford, R. 223 Bradley, R.B. 147 Braillon, B. 177 Brey, W.S. 571 Britt, A.D. 506 Broer, L. F.J. 2 Broersma, S. 26, 524 Bromberg, J.P. 506 Brooks, H. 74, 161 Brossel, J. 240 Brown, C.J. 173 Brown, H . W . 558 Brown, R.D. 150 Brown, R.M. 63, 237 Brown, T.H. 524, 525, 527, 528 Brownsteln, S. 212, 221,263, 519, 556, 576, 577 Bruce, C.R. 502 Bruegel, W. 174, 387 Buckingham, A.D. 87, 88, 89, 90, 94, 95, 134, 138, 151, 157, 162, 191,444, 464 Burbank, R.D. 530 Burd, L.W. 203, 223, 229 Burgess, ~l. H. 237 Burke, J.J. 532 Bushwell, A.M. 548 Byers, H . F . 548 Bystrov, V.F. 548 C,ady, G.H. 347 Cairns, T.L. 574 Calleja, F. J.B. 549 Calvin, M. 559 C,aznploeLl,A.N. 549 Camponovo, G. 236 Cantaguzene, J. 545 Carpenter, D.R. 549 Can-, H.Y. 55, 56, 160 Carrington, A. 4, 522 Carter, R.E. 149 Carver, T.R. 240 Castellano, S. 378, 379, 380, 381,386, 387, 435, 451

Cavanaugh, J.R. 84 Chandrasekhar, S. 22 Chesnut, D.B. 115 Chiarotti G. 23 Choppin, G.R. 519 Chou, L.H. 574 Claeson, G. 559 Clay, C. 223 Clough, S. 256, 425 Clunie, J.C. 534 Cohen, A.D. 182, 190, 281,287, 388, 391, 448, 455, 476, 541, 566 Condon, E.U. 33 Conger, R.L. 526 Connick, R.E. 515, 521,522, 524, 526 Connor, T.M. 481,542, 545 Conroy, H. 170 Coolidge, A.S. 124, 161 Corio, P.L. 9, 285, 294, 304, 307, 321,329, 334, 341,346, 347, 365, 543 ComwelL C.D. 237 Costain, C.C. 554 Cotton, F.A. 531 Coulson, C.A. 106, 122, 123, 146, 163, 536 Cox, P.F. 25, 30, 174, 182, 411, 527, 528 Coyle, T.D. 532 Craig, N.C. 398 Craig, R.A. 528 Crapo, L.M. 185 Crawford, B. 565 Creswell, C.J. 548, 549 Crook, J.R. 548 Custer, R.L. 217 Cutler, D. 56

Dailey, B.P. 98, 135, 141, 142, 143, 144, 145, 154, 408, 413 Das, T.P. 16, 17, 32, 55, 56, 77, 122, 123, 126, 129, 152, 153, 154, 155, 156, 341 Davidson, D.W. 558 Davidson, N. 527 Davis, D.R. 174, 175 Davis, G.T. 157 Davis, J.C. 542 Dearden, L C. 548, 556 De Boer, E. 116 Debye, P. 22 Deck, J.C. 151 De Kowaiewski, D.G. 277, 661, 662 Dharmatti, S.S. 523 Dickinson, W.C. 59, 65, 68, 90, 92, 120 Diehl, P. 89, 95, 368, 528 Dietrich, M.W. 502 Dinius, R.H. 519 Dime, P. A.M. 79, 109

. ° °

NAME INDEX~VOLUME 1 Dischler, V.B. 407 Dixon, J.A. 575, 580 Dobbs, F.W. 341 Dobinson, B. 578 Dodgen, H.W. 527 Dodson, R.W. 522 Douglass, D.C. 262, 515 Dowling, J. M, 554 Downing, J.R. 548 Dravnicks, F. 522 Drinkard, W.C. 510, 542, 543, 580 Drury, J.S. 528 Drysdale, J.J. 565 Dubb, H.E. 135, 345, 661 Dudek, G.O. 550 Duncan, A. B.F. 183 Fades, R.G. 256 Eaton, D.R. 115, 116 Eberson, L 542 Ehrenson, S. 155,263 Einstein, A. 17 Eisner, M. 24, 25 Eliel, E.L. 576 Elleman, D.D. 176, 185, 190, 244, 277, 564, 572, 661 Elvidge, J.A. 461 Emerson, M.T. 95, 96, 97, 98, 99, 510, 548 Emsley, J.W. 139, 156, 185, 191, 571 Englert, G. 407 Ernst, 1',. 230 Ettinger, R. 501 Evans. B.A. 202 Evans, D.F. 26, 98, 184, 186, 191, 257, 260, 262, 263, 267, 271, 466 Eyring, H. 72, 164, 573 Fabricand, B.P. 513, 514 Farrar, T.C. 25, 172 Feeoey, J. 156,185, 191,273, 358, 543, 548, 571, 577, 578 Fermi, E. 104 Feshbach, H. 238 Fe~senden, R.W. 82, 148, 176, 374, 378, 571, 661 Fiat, D.N. 524 Fi~ovich, G. 257, 266 Finegold, H. 476, 565, 569, 663 Fischer-Hjalmars, I. 536 Fixman, M. 77, 124, 125, 127, 128, 129, 130 Fluok, E. 9 Flynn, G.W. 190, 398 Forbes, W.F. 548, 556 Fors~, S. 502, 542, 548, 550 Forsling, W. 549

Foster, A.B.

XXlll 578

Foster, M.R. 260, 268 Fraenkel, G. 149,215 Franconi, C. 215 Frank, A. 71 Frank, P.J. 154, 572 Fraser, R.R. 176, 180, 661 Fratiello. A. 262, 515 Freeman, R 89, 95, 102, 172, 204, 224, 229, 247, 460, 462, 463,464, 466, 468, 469, 471, 474, 475, 476, 522, 523 Frei, K. 261 Freidman, L. 532 Frost, A.A. 164 Fujiwara, S. 34, 176, 373, 431,545, 663 Fuson, N. 548 Gabillard, R. 43, 217 Garnett, J.L. 445 Gasser, R. P.H. 522, 523, 528 Gassie¢, J. 545 Gates, P.N. 532 George, J.W. 531 Gerlach, W. 2 Ghose, T. 77, 126, 129 Ghosh, S.K. 77 Gill, D. 56, 486, 507 Gillespie, R.I. 347,510, 511,519 Gioumonsis, G. 453 Giulotto, L. 23, 535 Glasel, I.A. 267 Glick, R.E. 90, 94, 120, 155, 260, 263 Golay, M. J.E. 202 Goldberg, S. 513, 514 Goldenson, J. 157.481 Goldstein, J.H. 177, 329, 335, 387, 451,661 Goodman, L. 156, 157 Gordon, S. 98,141, 142, 143, 144, 145 Gordy, W. 126, 127, 154, 548 Gorin, E. 573 Gorter, C.J. 2 Govil, G. 428 Graham, D.M. 565 Graham, J.D. 197 Grab.n, R. 536 Gram, D.M. 103,170, 171, 172, 347, 404, 405 Grant, R.F. 558 Gray, K.W. 256 Gray, P. 558 Griffith, J.S. 157, 523 Grivet, P. 26, 56, 217 Gronowitz, S. 176, 177, 178, 179, 319, 358, 377, 661 Grunwald, E. 482, 490,491,494, 507, 509, 510 Guillano, C.R. 503, 504

xxiv

NAME INDEX--VOLUME 1

Gtinthard, H.H. 203,216, 217, 431 Gutowsky, H.S. 24, 25, 28, 61, 63, 99, 107, 108, 128, 151, 154, 155, 156, 160, 168, 169, 170, 171, 172, 173, 174, 180, 185, 186, 187, 188, 189, 190, 192, 193, 194, 195, 196, 207, 217, 218, 235, 237, 238, 239, 240, 290, 307, 347, 357, 404, 405, 466, 481,482, 485, 486, 487, 488, 517, 522, 524, 525, 527, 528, 538, 545, 546, 547, 552, 555, 564, 568, 569, 571, 572, 573, 663 Guy, J, 77, 78, 122, 134, 146

Hahn, E.L. 29, 40, 52, 53, 55, 56 Halbach, K. 237 Hall, G.G. 83, 144, 145, 146, 147 Hall, L.D. 575 Halpern, J. 502, 522 Hameka, H.F. 75, 85, 121, 122, 125, 126, 127, 146, 538, 539 Hamer, A.N. 215, 530 Hansen, W.W. 2, 25, 46, 208 Happe, J.A. 517, 518, 519, 529, 548 Harbottle, G. 522 Hardisson, A. 83, 144, 145, 146, 147 Harris, R.K. 337, 565, 568, 572, 575, 661, 662, 663 Hartman, J.S. 579 I-Iaszeldine, R.N. 556 Hatton, J. V, 258, 529 Hattori, S. 185, 423, 662 Hawthorne, M.F. 533 Heffernan, M.L. 150, 662 Heidberg, J. 556 Henderson, W.A. 445 Henriques, F.C. 574 Herbison-Evans, D. 510 Hertz, H.G. 222 Herzberg, G. 152 Herzog, B. 29 Hickmott, T.W. 527 Higham, P. 268 Hill, N.E. 24,25 Hindman, J.C. 514 Hiroike, E. 165 Hirschfelder, J.O. 77, 123 Hirst, R.C. 404,405 Hobbs, M.E. 268, 510 Hobgood, R.T. 387 Hoffman, C.J. 128, 151, 155, 156, 238, 538 Hoffman, E.G. 532 Hoffman, R. A. 176, 177, 178, 179, 228,319, 358, 377, 502, 661 Hollis, D.P. 558 Holm, C.H. 27, 115, 185, 187, 357, 482, 486, 487, 552, 555

Holm, R.H. 550 Holmes, J.R. 179, 180, 510, 543, 580 Holt, E.K. 518 Homer, J. 578, 580 Hood, F.P. 576 Hood, G.C. 517, 518, 519 Hornig, J.F. 77 Howard, B.B. 95, 96, 97, 98, 99, 548 Howarth, O. 237 Hubbard, P. S, 24 Htiek©l, E. 83 Huggins, C. M. 540, 542, 545, 546, 548, 549 Hull, R.L. 528, 548 Hume, D.N. 46, 47 Hunsberger, I.M. 545, 546 Hunt, J.P. 527 Hutton, H.M. 196, 405, 662 Hylleraas, E. 67, 68 Hyne, J.B. 540, 541,542, 545 Ichishima, I. 569 Ingrain, D. J.E. 4 Irsa, A.P. 532 Isldguro, E. 77, 124, 162 Ito, K. 72, 122 Itoh, J. 240, 241,517 Jackman, L.M. 83, 144, 146, 147; 176, 230, 267, 329, 461, 575 Jackson, A.H. 259 Jackson, J.A. 523, 524 Jacobsohn, B.A. 41, 42, 224, 226, 558 James, H.M. 124, 161 Jander, J. 556 Jamtsonis, G.A. 556 Jardetzky, C.D. 259 Jardetzky, O. 230, 259, 515, 520 Jarrett, H.S. 549 Jaynes, E.T. 56 Jenks, G.J. 256 Jensen, E. 56 Jensen, F.R. 136, 575 Johnson, C.E. 82, 140, 141,148 Johnson, C.S. 28, 481,506 Johnson, L.F. 177, 191,230, 329, 335 Jonathan, N. 141,142, 143, 144, 145 Jones, A.C. 519 Jones, J. 481 Jones, R. 456 Jones, R. A.Y. 266 Josey, A.D. 115, 116 Josien, M.L. 548 Juan, C. 173, 174, 192, 193, 194, 195, 196 Jumper, C.F. 507, 509, 549

NAME I N D E X - - V O L U M E ] Kaiser, R. 239, 240, 241,242, 475 Kamio, S. 545 Kanda, T. 517 Kaneker, C.R. 428, 523 Kaplan, J.I. 34, 172, 207, 453, 489 Kaplan, M.C. 510 Karabatsos, G.J. 197, 510 Karle, J. 568 Karplus, M. 106, 109, 110, 111, 152, 153, 154, 155, 156, 157, 160, 164, 165, 166, 167, 169, 170, 171, 172, 174, 176, 178, 179, 180, 181, 186, 189, 573 Karplus, R. 237, 239 Kartzmark, E . M . 549 Katayama, M. 545 Katritzky, A . R . 266 Kaufmann, J.J. 532 Kellogg, J. M.B. 120 Kenner, G . W . 259 Khetrapal, C.L. 428 Kimball, G. 72 Kimura, M. 185 Kincaid J.F. 574 King, J. 527 Kirby-Smith, J.S. 548 Kissman, H . M . 574 Kivelson, D. 179, 180, 329, 510, 542, 543, 580 Kivelson, W . G . 329 Klanberg, F. 527 Klein, M.P. 230 Klinck, R.E. 529 Knight, W . D . 59, 60 Koide, S. 77, 124 Komaki, T. 569 Korinek, G.J. 540, 548 Kornegay, R . L . 575 Korringa, J. 247 Koski, W.S. 532 Kotin, L 519 Kowalcwski, V.J. 228, 277, 661,662 Kowalsky, A. 265 Kozyrev, B.M. 4 Krakower, E. 545 Kromhout, R . A . 510 Kruekeberg, F. 174, 387 Kubo, R. 24, 30, 488 Kuchitsu, K. 569 Kuhlmann, K. 476 Kullnig, R . K . 169, 575, 576 Kuratani, K. 569 Kurita, Y. 122 Kurland, R.J. 174 LadeU, J. 134 Lake, K . J . 549

XXV

Lamb, J. 566, 569 Lamb, W.E. 66, 68, 71, 80 Lambert, J.D. 548 Landau, L.D. 66, 67, 79, 80 Lappert, M . F . 552 Lasarev, B.G. 3, 35 Laukien, G. 527 Lauterbur, P.C. 157, 174, 337, 532 Lawrenson, I.J. 28 Leane, J.B. 217 Leztwith, A. 358 Lee, J. 185, 552, 560, 565, 661 L~_,~cc~_,J. 215 LeF/~vre, C.G. 576 LeF6vre, R. J.W. 139, 576 Lemieux, R.U. 169, 574, 576 Lemons, J.F. 523, 524 Lennard-J0nes, J. 535 Lewis, A. 175 Lewis, 1. C. 155 Lezina, V.P. 548 Lhermitte, Y. 545 Li, N.C. 262 Liddel, U. 542 Lifschitz, E . M . 66, 67, 79, 80 Linder, B. 95, 96, 97, 98, 99 Lindstr6m, G. 549 Linnet, W.. 123 Lippincott, E.R. 536 Lipscomb, W. hi. 573 Loewenstein, A. 56, 481,482, 490, 491,493, 494, 509, 510 Lombardi, E. 533 London, F. 83, 84, 85, 95, 96 Longuet-Higgins, H.C. 4, 163 Looney, C.E. 551,552, 553, 556, 557 Lovering, E.G. 162 Lustig, M. 533 Luszczynski, K. 56 Lutz, R.P. 174, 175 Luz, Z. 486, 507, 510 Lynden-Bell, IL M. 168, 170, 174, 192, 197, 284, 448, 661,662 Mackellar, F. 259 Mackor, E.L. 511 Maclean, C. 511 Mahendroo, P.P. 28 Mailer, J.P. 466 Maier, W. 407 Malinowski, E. 193, 194 Manatt, S.L. 176, 185, 190, 244, 277, 564, 572, 661 Mandell, L. 177, 661 Margenau, H. 69, 77

XXVi


Marshall, T.W. 85, 97, 99, 100, 101 Martin, J. 408, 413 Martin, M. 545 Martin, R.J. 529 Marugg, B. 236 Masuda, Y. 517 Matsen, F.A. 120 Matsuoka, S. 185, 423, 662 Matsushima, M. 398 Mattinson, B. L H. 556 Mavel, G. 9, 543 Mayo, R.E. 387 McCalL D.W. 61, 63, 240, 290, 307, 482 McCann, A.P. 202 McClellan, A.L. 534, 535, 536, 539 McClure, G.R. 177 McClure, R.E. 217 McConneIl, H.M. 27, 63, 75, 76, 106, 107, 108, I15, 116, 117, 129, 137, 160, 163, 164, 173, 181,182, 183, 185, 186, 262, 298, 337, 392, 398, 427, 431,482, 497, 503, 504, 510 McGarvey, B.R. 77, 126, 351, 522, 526 McLachlan, A.D. 106, 116, 149 McLauchlan, K.A. 162, 172, 281, 455, 476 McLean, A.D. 63, 185, 298, 337, 392, 398 McMahon, P.E. 168, 169, 190, 569, 571, 572, 573 McWeeney, R. 83, 144, 146 Mecke, R. 541 Meiboom, S. 34, 56, 453, 482, 486, 490, 491,493, 494, 507, 509, 510 Meinwald, J. 175 Meinzer, R.A. 481 Meisenheimer, J. 574 Meisenheimer, R.C. 262 Merril, J.R. 545, 547 Meyer, L.H. 185, 217, 538 Miller, J.J. 533 Miller, R. 576 Milner, R.S. 115 Mitchell, R.W. 24, 25 Miyazawa, K. 569 Mizushima, S. 551,569 Molt, R.Y. 169 Moniz, W.B. 575, 580 Mooney, E.F. 532 Morgan, L.O. 25, 30, 526, 527, 528, 548 Morin, M.G. 268, 519 Morino, Y. 569 Moritz, A.G. 136 Morse, P.M. 238 Mortimer, F.S. 176,388,391,392,451,662 Mott, N.F. 71 Moxon, L.A. 233 Muchowski, J.M. 547

Muctterties, E.L. 520, 530 Muller, N. 157, 177, 191,192, 194, 195, 337, 481, 532, 578 Murphy, G.M. 69 Murphy, J. 25, 30, 77, 527, 528, 548 Murray, B.B. 519 Murray, F.E. 535 Murray, G.R. 102, 523 Murrell, J.N. 266 Murthy, A. S.N. 545 Musher, J.I. 136, 172, 476, 520, 579, 581 Myers, O.E. 506, 531 Naar-Colin, C. 175, 176, 662 Nagasawa, M. 519 Hair, P.M. 562, 565 Narasirrd~m, P.T. 133, 134, 135, 177, 351, 371,372, 392, 442 Nelson, F.A. 460 Newell, G.F. 121 Newmark, R.A. 569, 571,572 Nilsson, M. 550 Nist, B.J. 283 Nolle, A.W. 28, 527, 528, 548 Norberg, R.E. 502 Nordlander, J.E. 532 Nordsieck, A. 121 Noyc¢, D.S. 136, 575

Ogg, R.A. 27, 240, 497, 517, 528, 553, 556, 558 Onsager, L. 89, 135, 345, 661 O'Reilly, D.E. 77, 111, 157, 511 Orgel, L.E. 157, 523 Ormand, F.T. 120 Overhauser, A.W. 116, 247 Pachler, K. G.R. 172, 566, 568 Packard, M.E. 2, 25, 46, 202, 203, 208, 236, 542 Packer, K.J. 337, 520, 662 Pajak, Z. 548 Pake, G.E. 4, 19 PaJko, A.A. 528 Palmer, J.W. 524, 526 Pan/sh, M.B. 530 Paolini, L. 536 Parks, J.R. 157 Parr, R.G. 71, 121,573 Paterson, W.G. 509, 547 Patterson, A. 501 Paulett, G. 268, 519 Pauli, W. 1

NAME INDEX--VOLUME 1 Pauling, L. 81, 106, 109, 110, 128, 167, 180, 553 Pearson, M.J. 558 Pearson, R.G. 524, 526 Petrakis, L. 101, 102, 190, 191, 257, 271 Phillips, W.D. 115, 116, 530, 551, 552, 553, 556, 557, 565 Pierens, R.K. 576 Piette, L.H. 240, 548, 553, 555, 556, 557, 558 Pimental, G.C. 534, 535, 536, 539, 540, 542, 545. 546, 548 Pinkerton, J. 127 Pitcher, E. 151, 191 Pitzer, K.S. 542 Pitzer, R.M. 573 Plovan, S. 550 Pople, J.A. 25, 79, 80, 81, 83, 84, 85, 90, 92, 97,99, 113,114,124,125,127,129,131,132, 141,144, 146, 148, 157, 162, 176, 184, 185, 216, 240, 283, 326, 327, 328, 329, 335, 368. 370, 404, 405, 406, 407, 448, 464, 467, 469, 486, 497, 535, 537, 538, 539, 548, 560, 565 Pone, A.L. 545, 546, 663 Portis, A.M. 226 Post, B. 134 Poulson, R.E. 515, 521,522. 526 Pound, R.V. 2, 20, 23, 24, 25, 26, 27, 32, 33, 38, 46, 76, 210, 217, 218, 221,224, 229, 233 Powell, R.L. 548 Powles, J.G. 4, 50, 56, 63, 130 Pratt, L. 115 Premuzic, E. 576, 577, 579 Present, R.D. 161 Primas, H. 203, 216, 217, 230, 237, 321, 431,435, 436, 437, 438, 439, 440, 441,442 Pritchard, D.E. 177, 191,192, 194, 195, 532 Proctor, W.G. 25, 59, 61 Prohaska, C.A. 572 Prosser, F. 156, 157 Purcell, E.M. 2, 17, 20, 23, 24, 25, 27, 28, 32, 33, 38, 46, 55, 56, 63, 76, 160, 210, 217, 218, 221,224, 233 Purlee, E.L. 510 Quaff, J . w . 347 Quirm, W.E. 63 Rabi, I.I. 16, 50, 120 Ramey, K.C. 563, 571 Ramsey, N.F. 2, 11, 18, 50, 63, 68, 70, 74, 75. 99, 101, 102, 103, 104, 105, 109, 120, 121,122, 124, 125, 127, 151,160, 161, 162, 165

xxvii

Randall, E.W. 241,247, 456, 461 Ranft, $. 661,662 Ransil. B.J. 154 Rao, B. D.N. 407, 408, 411,476, 542, 543, 661 Rao, C. N. IL 545 Ray. J.D. 27, 240, 497, 517, 553, 556, 558 Reddy, G.S. 177, 451 Redlich, O. 517, 518, 519 Reeves, L.W. 258, 481, 510, 542, 543, 545, 547, 548, 549, 550, 551,558, 576, 577, 579 Reid, C. 141, 541, 545 Reilly, C.A. 44, 63, 172, 185, 262, 276, 282, 298, 329, 337, 361, 369, 370, 375, 376, 377, 378, 427, 451, 452, 466, 475, 517, 518, 519, 661 Reilly, E.L. 551, 552, 553 Reynolds, W.F. 196 Richards, J.H. 149 Richards, R.E. 4, 102, 117, 202, 217, 237, 258, 268, 321,357, 408, 417, 418, 419, 420, 421,422, 510, 517, 522, 523, 528, 534, 579, 661 Riggs, N.V. 661 Roberts, J.D. 172, 174, 175, 176, 177, 212, 283. 532, 562, 565, 574 Rocard, J.M. 63 Rodebush, W.H. 548 Rogers, E.H. 227 Rogers, M.T. 133, 134, 135, 177, 351,371, 372, 392, 442, 530, 555 Rollin, B.V. 46 Roper, R. 576 Rose, M.E. 202 Rosen, N. 122, 123, 146 Roux, D.P. 63 Rowland, T.J. 506 Roy, S.K. 56 Royden, V. 240 Ruiner, G. 109 Rutenberg, A.C. 523, 528 Rutledge, R.L. 329, 543 Ryschkewitsch, G.E. 552 Sack, R.A. 488 Sadler, M.S. 549 Saha, A.K. 16, 17, 32, 55, 341 Sa&ka, A. 151, 152, 153, 157, 183, 185, 187, 357, 482, 485, 488, 510, 517, 522, 538, 555, 565 Sallkhov, S.G. 4 Salpeter, E.E. 49, 226 Sandorfy, C. 135 Santry, D.P. 162 Sasson, M. 56

xxviii


Sato, S. 240, 241 Saunders, M. 135,345, 540, 541,542, 545, 661 Schacher, G.E. 511 Schaefer, T. 94, 147, 149, 150, 151, 174, 196, 217, 321, 357, 364, 368, 370, 405, 408, 416, 417, 418, 419, 420, 421,422, 423,444, 469, 549, 662 Schaeffer, R. 456 Schimamouchi, T. 569 Schliiter, J. 527 Schmitz, H. 530 Schneider, W.G. 25, 90, 92, 94, 124, 125, 127, 129, 132, 135, 137, 138, 141,144, 147, 149, 150, 151, 157, 176, 216, 258, 260, 268, 283, 326, 327, 328,335,404, 405, 406, 407, 416, 423, 444, 448, 467, 486, 510, 529, 535, 537, 538, 539, 540, 543, 548, 549, 550, 551, 553, 555, 565, 575, 576, Schroeder, R. 536 Schubnikov, L.V. 3, 35 Schug, J.C. 151, 529, 573 Schug, K. 511, 548 Schumacher, H.J. 530 Schwinger, J. 16, 50 Scmggs, R.L. 262 Searles, S. 550 Scderholm, C.H. 101, 102, 136, 185, 190, 191,257, 271,509, 571,572, 575 Sclwood, P.W. 526, 527 Scnda, K. 185 Shapiro, I. 533 Sheppard, J.C. 506 Sheppard, N. 136, 163, 168, 170, 171,172, 173, 174, 180, 182, 185, 190, 192, 197, 266, 287, 358, 374, 387, 388, 391,423, 448, 565, 566, 568, 572, 575, 576, 663 Shimizu, H. 34, 176, 431, 663 Shimomura, K. 28 Shoolery, J.N. 135, 147, 151,177, 191,212, 240, 242, 244-7, 271, 329, 335, 445, 446, 447, 448, 456, 511,540, 542, 545, 546, 548, 549, 565 Shortley, G.H. 33 Shreeve, J.M. 347 Shuler, W.E. 519 Siddall, T.H. 572 Silvir, B. 510 Sinha, S.K. 77 Skavlem, S. 67, 68 Sliehter, C.P. 9, 63, 151, 152, 153, 157, 240, 290 307, 482, 522 Slomp, G. 202, 228, 259, 351 Smaller, B. 237 Smith, D.C. 532 Smith, D.F. 530 Smith, F. 202

Smith, 1. C. 149 Smith, J. A.S. 202 Sneddon, I.N. 71 Snyder, E. L 176, 177, 569 Snyder, L.C. 71, 121 Sogo, P.B. 533 Sollich, W.A. 445 Solomon, I. 30, 50, 185, 489, 527 Somers, B.G. 547 Soutif, M. 217 Spaeth, C.P. 556, 557 Spalthoff, W. 222 Spedding, H. 5O9 Speirs, J.L. 530 Spencer, R.H. 217 Spiesecke, H. 135, 137, 138, 147, 260, 268 Spurr, R . A . 548 Stafford, S.L. 371 Stanford, S.C. 548 Stehling, F.C. 232 Stejskal, E.O. 25, 482, 490 Stephen, M.J. 77, 78, 90, 91, 92, 93, 94, 95, 111, 112, 123, 124, 127, 129, 162 Stem, O. 2 Sternheimer, R. 152 Stevens, K. W.H. 4 Stewart, W.E. 177, 329. 335 StiUman, A.E. 519 Stone, F. G.A. 151, 191, 532 Stothers, J.B. 529, 661 Stover, E.D. 526 Strange, J.H. 50, 63 Streitwieser, A. 83 Strick, E. 223 Stremme, K.O. 547, 576, 579 Strong, T. 548 Sudhanshu, S. 387 Sugiura~ Y. 123 Sunners, B. 548, 553, 555 Suryan, G. 49 Sut¢liffe, L.H. 156, 185, 191,273, 358, 543, 548, 560, 565, 571,577, 578, 661 Suzuki. S. 150 Svatos, G.F. 337 Swalen, J.D. 172, 282, 329, 361, 375, 376, 377, 378, 427, 451,452, 453, 661 Swift, T.J. 526 Symons, M. C.R. 522 Sz6ke, A. 509 Taft, R.W. 155, 157, 545 Takeda, M. 259, 482, 490, 568 Talman, J.D. 661 Tarbell, D.S. 574 Tarte, P. 556, 557

NAME I N D E X - - V O L U M E 1 Taube, H. 523, 524 Taylor, R.C. 177, 335 Thomas, L . F . 578. 580 Thompson, D . D . 497, 510 Thompson, D.S. 569, 572 Thompson, H.B. 530 Thwaites, J.D. 202 Tiers, G. V.D. 31, 99, 162, 171, 172, 184, 257, 259, 264, 265, 266, 268, 286, 445, 550, 575, 578 Tillieu, J. 77, 78, 122, 130, 134, 135, 136, 146 Tipman, N. 17,. 247 Tolansky, S. 1 Tomita, K. 24, 30, 247 Torrey, FI. C. 2, 40, 52, 53, 217, 221 Tosch, W.C. 578 Townes, C.H. 154 Tsubomuro, H. 536 Turkevich, J. 164 Turner, D . W . 244, 267, 464 Turner, J.J. 168, 169, 182, 190, 277, 287, 448, 565, 566. 576 Tuttle, T.R. 116, 217 Tzalmona, A. 56 Utterbaek, E.J.

548

Vane, F . M . 197 Van Geet, A . L . 46, 47 Van V|eck, J.H. 70, 71,165, 172, 431,432 Vaughan, W.R. 177, 335 Venkateswarlu, P. 407, 408, 411,543, 661, 662 Virmani, Y.P. 428 Wahl, A . C . 502, 506 Walker, S.M. 156 Wallis, R . F . 123 Walter, J. 72, 573 Wang, S. 122, 123, 146 Wangsness, R . K . 36, 224, 226, 558 Waring, C.E. 217 Waugh, J.S. 28, 82, 137, 148, 176, 341,374, 378, 379, 380, 381,386, 387, 435, 451,531, 565, 571,661 Weaver, H.E. 147, 151, 503

I~RS.

22

XXiX

Wegmann, L. 236 Well, J.A. 556 Weinbaum, S. 123 Weinberg, I. 543 Weissman, S.I. 116 Wertz, J.E. 32, 515, 520 West, R. 549 Whatley, L. 549 Wheeler. D . H . 522 Whiffen, D . H . 4, 172, 204, 244, 247, 455, 460, 462, 463, 468, 476 Whipple, E.B. 177, 329, 335, 568, 569, 661 White, J.W. 117 White. R. F . M . 519, 662 Whitesides. G . M . 532 Whitman, D . R . 135, 345, 392, 407, 427, 451,452, 661,662 Whittaker. A . G . 517. 518, 519 Wiberg, K.B. 283 Wick, G.C. 71, 73, 74 Wiley R . H . 176 Williams, D . H . 9 Williams, G.A. 107, 108, 185, 186, 187, 188, 189, 207, 237, 238, 239, 357, 445, 446. 447, 448, 466 Williams, R.B. 203, 211, 224, 225, 226, 232, 233, 234 Williams, R.E. 533, 551 Wilson, E.B. 106, 109, 110; 167, 298, 551, 569, 573, 662 Wimett, T . F . 62, 99, 160 Wishnia, A. 527 Woessner, D.E. 24, 25, 28, 524, 525, 527 Woodbrey, J.C. 487, 555 Yamagata, Y. 517 Yamaguchi, I. 556 Yamamoto, T. 373 Yasaitis, E. 237 Yatsiv, S. 453, 455 Yen, W.M. 506 Yorke, B.A. 517 Yu, F . C . 25, 59, 61

Zacharias, J . R . 120 Zahn, C. T+ 127 Zavoisky, E. 4 Zimmerman, J.R. 260, 268, 329, 547

SUBJECT INDEX The heaviness of the type indicates the importance of an entry: page numbers in bold type can signify that a topic runs on for several pages. Acetaldehyde 273 Acetaldehydes (dichloro-), IH double resonance spectrum 461 hydration of 533 Acetic acid, ~3C spectrum 991. 993 Aoetone, as a solvent 259 Acetonitrile,effect of solwnts on 1H chemical shift 846 Acetoxycholestonanes, IH spectra 706 Acetytenes :3Cc spectral parameters 1001, 1030 effect of solvents on ~H chemical shifts

Alkene.~, 13C-13C coupling constants 1030 cis and trans XH chemical shifts 729 correlation of ZH chemical shifts with group dipole moments 718 Hammett # constants 717 substituent electronegativity 717 correlation of H - H coupling constants with substituent elec~'onegat iviW 714

effect of methyl group sutntitution on XH chemical shifts 733 ~H chemical shifts 727, 731, 732, 736, 749, 848 737, 743, 744, 745 1H chemieal shifts 745 H-H coupling constants 710, 726, 727, H-H coupling constants 746, 748 731,732, 735, 739 Addition compounds of BF3 528 1H spectra 710 ~TAl chemical shifts Alkenes (metal) of alkyls 1094 IH chemical shifts 742 of salts 1094 H-H coupling constants 743 27A1 resonance 1093 Alkyls (metal), coupling constants 689 line widths 1094 Allyl magnesium bromide, IH spectrum 689 Alcohols Amides hydrogen bonding in $42 calculation of internal chemical shift 133 x70 chemical shifts 1045 h~ydrogen bonding of 548 Aldehydes rotational isomerism 553 aromatic proton exchange 510 :aC spectral parameters 1001, 1005 Amines, hydrogen bonding 548 ~H chemical shifts 765 long range H-H coupling constants Amino acids, IH chemical shifts 812 Ammonia 767 calculation of .H-H coupling constant H-x3C coupling constants 1020 183 ~70 chemical sldfts 1045 calculation of hydrogen bond shift 538 Aldopyranoses (acetylated), 1H spectra 704 1aN spectrum 1031, 1093 Aliphatic Ammonium ion acids, 170 chemical shifts 1045 ~H double resonance spectrum 458, 459, fluorocarbons, calculations of coupling 460 constants 183 14N INDOR spectrum 1033 Alkaloids, XH spectral parameters 806 proton transfer 498, 510 Alkanes derivatives, t a n spectra 1038 ~H chemical shifts 666 Analysis H - H geminal coupling constants 677 AB spin system 310 H-H vieinal coupling constants 678 direct method of analysis 438 Alkenes double irradiation (weak field) 472 anomalous IH chemical shifts 719 effect of elecmc fields on 464 x3C chemical shifts 999 xxxi 22*

XXJdi

SUBJECT INDEX

Analysis, energy levels 317 theoretical spectra 320 transition energies 318 AX spin system, double resonance behaviour 247, 462 ABe spin system 320 basic product functions 323 diagonal matrix elements 323 electric field effects 464 energy levels 324 relative intensities 325 theoretical spectra 327, 625 AX= spin system, double resonance behaviour 462, 463 AB3 spin system 329 diagonal matrix elements 331 relative intensities 332 theoretical spectra 333, 629 transition energies 332 AX3 spin system double resonance behaviour 247 electric field effects on 464 AB. spin system 337 eigenfunctions 337 eigenvalues 337 relative intensities 339, 635 spin product functions 336 theoretical spectra 340, 635 transition frequencies 339, 635 ABe spin system 661 AB. spin system 341 general features of spectra 341 relative intensities 343 sub-spectra 342 transition frequencies 343 A2B2 spin system 347 relative intensities 349 theoretical spectra 350, 645 transition frequencies 349 A2X2 spin system 347 A2B6 spin system 661 A3B2 spin system 351 combined transitions 354 relative intensities 352, 353 sub-states 351 theoretical spectra 355, 649 transition frequencies 352, 353 ApB. spin system 344 general features of spectra 344 numbers of transitions 346 A,.X2 spin system, double resonance behaviour

463

AnX,, spin system, double resonance behaviour

463

ABC spin system 372 ABK approximation method

375

Analysis, complete solution 378 diagonal matrix elements 373, 382 double irradiation (weak field) 473 eigenvalues 381 intensity sum rule 380 spectrum (tH) of styrene 381 spectrum (XH) of vinyl bromide 711 subtraction rules 383 Sudhanshu's method 387 trace invariance of sub-matrix 379 ABK spin system 376 energy levels 376 styrene oxide, IH spectrum 377 wavefunctions 376 ABT spin system 661 ABX spin system 357 deceptively simple spectra 363, 364 diagonal matrix elements 357 energy levels 359 relative intensities 360 styrene oxide, XH spectrum 362 subtraction rules 361 theoretical spectra 362, 449 transition energies 360 APX spin system signs of coupling constants 466 triple resonance spectrum 476 AB2C spin system 391,661 off,diagonal elements 392 AB2X spin system 661 ABXz spin system 388 diagonal matrix elements 389 relative intensities 390 spectrum of 2,3-dichioropropene-1 (IH 388 transition energies 390 ABPX spin system 661 ABXY spin system 661 AA'BB' spin system 348, 399 explicit energy expressions 401 1H spectra of benzenes Q~-substituted) 408, 414 ethanes (1,2-substituted) 415, 566 1H spectrum ofbenzofuraTan 407,408, 4O9 benzene (p-bromochloro-) 414 ethane (1,2.bromochloro-) 415 naphthalene 405, 406 matrix elements 400 relative intensities 402, 410 relative signs of coupling constants 404 theoretical spectra 404, 413 transition energies 402, 410 AA'XX' spin system 392 tgF spectrum of benzene (1,2-difluoro3,4,5,6-tetrafluoro) 399

SUBJECT I N D E X Analysis, ~gF spectrum of ethylene (1,1-difluoro) 397 IH spectrum of ethylene (l,l-difluoro-) 397 matrix elements 393, 394, 395 theoretical spectra 396, 404, 413, 449 transition energies 396 AA'KL spin system 662 ABKL spin system 662 ABCX spin system 423, 662 matrix elements 424 ABCD spin system 425 A B K ¥ approximation 427 XH spectrum of glycidaldebyde 428 matrix elements 428 A2A~X spin system 662 ABB'CC' spin system 662 AA'BB'X spin system 416, 662 ~gF spectrum ofpara-fluoroaniline 419 XH spectrum ofpara.fluoroaniline 419 matrix elements 417 relative intensities 419, 420, 421 symmetrised spin functions 417 transition energies 419, 420, 421 wavefunctions 418 AB2X2 spin system 662 AA'PP'X spin system 662 ABC3 spin system 392, 661 ABXs spin system 661 AB4X spin system 662 AA'A"A"'XX' spin system 662 " AsB2C spin system 392 A3B2X spin system 370 XH spectrum of phosphorus triethyl 371 AsBCD spin system 663 AsBCX spin system 662 ABB'CC'D spin system 663 ABB'CC'X spin system 663, 1092 ABC-'DX2 spin sys~m 663 ABCXs spin system, double irradiation of 476

ABPsX spin system, double irradiation of 469 A.BXp spin system 365 general features of spectrum 368 numbers of transitions 369, 370 relative intensities 367 sub-spectra 366 transition frequencies 367 A , . K . . . . X . spin system 663 A.BPcXp spin system 370 AA'X 3 X'3 spin system 661, 662 AA'X.X~ spin system 661 ! II AA I A I I x sXsX 3 spin system 662 A A ' P P ' P " P " 'X2 spin system 662 •

a-es.

224

xxxiii

Analysis, of spectra 280 aids 442 complex particle method 345 direct method 435 double irradiation 455 double quantum transitions

4S3

effect of rotational isomerism 560 electronic computation 4$1 isotopic substitution 445 moment method 431 perturbation theory 428 rules 309 quantitative 211, 234 Anilines (p-fluoro) coupling constants 423 19F spectrum 421 IH spectrum 422 Anisotropy effect of neighbour 131 of shielding constants 94, 113 of C-C and C-H bonds 135, 676, 696 of C-X bonds 136 Annulenes, 1H chemical shifts 745 Anthracene, 1H spectrum 773 Antimony pentafluoride ~9F spectral parameters 928 19F spectrum 929 121Sb resonance of NaSbF6 1097 Aqueous electrolytes 511, 514 Area of bands 234 Aromatic compounds, see Benzenes as solvents 2.58 calculation of chemical shifts 140, 145, 149, 595, 770 calculation of coupling constants 150 charge densities in 149 effect of solvents on IH chemical shifts 851 H - H coupling constants 770 molecular complexes 529 polynuclear, IH chemical shifts 770 ions, 1H chemical shifts 774 Arsenic (VSAs)resonance 1097 Association, see Hydrogen bonding Atomic orbitals, calculation of shielding constant 121 Average energy approximations 71, 106 Azulene calculation of internal chemical shift 146 1H spectrum 778 carbonium ion, XH spectrum 778 11B chemical shifts 971, 974 of boron halides and derivatives 972, 974

xxxiv

SUBJECT INDEX

riB chemical shifts, of boron hydrides and derivatives 972,974 Barrier to internal rotation, nature of 573 Basic product functions 297 symmetry functions 298 consort of 299 of D , point group 304 Benzaldehydes, H-H coupling constants 277 1,2-Benzanthracane (9,10-dimcthyl-), carbonium ion 776 XH spectrum 776 Benzene as a solvent 258 chloro- and deuterochloro-, tH spectra 751 isopropyl-, XH spectrum 6 meta-dinitro-, 1H spectrura in acetone 444 in benzene 444 monofluoro-, tgF spectrum 446 monofluoro-2,3,5,6-D4-, XH spectrum 446 monofluoro-2,4,6-D 3, tgF and ~H spectra 447 ortho-dimethoxy-, 13C spectrum 1008 para-chlorobromo-, XH spectrum 414 para-methyinitro-, effect of solvents o n tH spectrum 853 2,3,5,6-tetrachloro-, tH spectrum with laC satellites 1023 Benzenes xaC-xaC coupling constants 1030 xSC~H satellite spectra 1022 H-H coupling constants 403, 770 ortho-disubstituted, analysis of spectra 403, 405 para-disubstituted; effect of solvents on 1H chemical shifts 852 1H spectra 408 proton exchangewith HF 511 alkoxy, XH chemical shifts 764 disubstituted, 1H chemical shifts correlation with Hammett o constants 758 empirical calculations 754, 1140 fluorinated xgF chemical shifts correlation with Hammett o"constants 897 F - F coupling constants 901 H-F coupling constants 901 halo-, XH chemical shifts 760 hydroxy-, XH chemical shifts 763

Benzenes, monosubstimted 13C chemical shift correlation with Hammett o constants 753 x3C spectral parameters 1002 XH chemical shifts 750 correlation with Hammett ~ constants 752 of meta nuclei 752 of ortho nuclei 752 ofpara nuclei 752 trisubstituted, tH chemical shifts 767 BenzofurnTn,% tH spectrum 407, 408, 409 Binary fluorides, correlation of XgF chemical shifts and coupling constants 880 with electronegativity 882 Biot--Savat law 67, 79 Biphenyls 767 Bloch equations 34, 38 including chemical exchange 482, 505 susceptibilities 39 Boltzmann distribution of nuclei 17 Borane hexa-, l i b spectrum 982, 984 tetra-, t:B spectrum 981 undeca-, XtB spectrum 984 deca-, XlB spectra 982, 984 pentao, 11B spectra 981,985 Borohydrides, xxB spectral parameters 987 Boron alkyls, 11B spectral parameters 975 halides 528, 532, 971, 974, 976, 987 tXB spectral parameters 971, 974 complexes 952, 973 19F spectra of mixed haLides 945 tgF spectral parameters 944 hydrides, ItB spectral parameters 972, 974, 977 deuteration 533 tetr~uoride ion, 19F spe~ral parameters 946 79Br and SlBr resonance of Br~q 517 sxBr resonance of HBr 1096 Bridge circuits 217 Bromine pentafluoride 943 trifluoride 942 Bulk suceptibility correction 260 Butadiene--isoprene copolymer, XH spectrum 832 Butanes (2,3-disubstituted) 1H chemical shifts 690 H-H coupling constants 690 t-Butanol, hydrogen bonding 541 Butene-1

SUBJECT INDEX Butene-1, :H chemical shifts 723 H - H coupling constants 723 i-Butenes (1-substituted) IH chemical shifts 736 H - H coupling constants 737

13C chemical shifts comparison with 29Si chemical shifts 1051 of acetylenes 1001, 1030 of aromatic compounds 1002, 1005, 1030 of carbonyl groups 1009 of ethanes 997, 1030 correlation with substituent electronegativity 996 of metal carbonyls I010 of methanes 992, 995 correlation with substituent electronegativity 996 of monosubstituted benzenes 1002 of phenoh 1006 of simple organic compounds 990, 1030 13C resonance, experimental procedures 988 13C satellites in 1H spectra 448, 450, 475 in 19F spectra of fluoroalkenes 914, 962 Caesium (lS~Cs) chemical shifts of halides 1093 Calibration of spectra sideband method 237, 274 wiggle beat method 276 Carbohydrates (acetylated) contigurational effects on IH spectra 701 Carbonium ions, IH chemical shifts 774 Carbonyl groups, IsC chemical shifts 1009 CAT 230 C-C bond shifts 666, 698 IsC-:3C coupling constants 1029 CDCI3 as a solvent 257 Ceils, see Sample containers lSC-F isotope shifts in fluoro-organic compounds 962 CFC13 as a solvent 257, 266 Character table for D4 point group 301 Charge densities in aromatic molecules 149 Chemical equilibria, effect on spectra 481 exchange 484 Chemical shifts (see Shielding constants), 4, 59, 65 calculation for alkyl compounds 136 amides 133 22 a*

XXXV

Chemical.... aromatics 140, 149, 595, 770 azulene 146 cyclohexane 136 cyclohexa~ (perfluoro-) 139 fluorine compounds 151 fluorob~n=nes 154 heterocyr~ aromatics 145, 789 • paramagnetic contribution for SgCo resonance 1078 paramagnetic contributions for 19~Ig, 2°VPb and 2°sT1 resonances 1098 c/s and trans in olefines 729 contact 826 contributions from ring currents in aro= matics 141, 595, 770 conversion factors for external refe~e~.cin8 263 correction for bulk diamagnetic susceptibility 66, 260 effect of parmnagnetic materials 115, 826 empirical calculations 838 empirical calculations for disubstitoted henzenes 754 equivalence 283 gas-to-solution 97, 841 IH, charts of 1131 IH for organic compounds (Tiers'compilation) 1115 isotope effects 875, 916, 962, 1022, 1092 origin of 59 ortho effect in fluorohenzenes 155 Chemical shift/electron density ratios for hydrocarbon aromatic ions 781 Chlorine 3sCl and 37CI, chemical shifts of inorganic chlorides 1095 3sCl resonance of Cl~ 515 nuclear quadrupole splittings in inorganic chlorides 1096 resonance 1095 trifluoride,19F spectrum 941, 942 Chromium ~ comple~,cs, ligand exchange rates 525 sgCo chemical shifts calculation of paramagnetic contribution 1078 correlation with electronic absorption 1080

of cobalt (HI) complexes 528, 1080 solvent effects 1081 temperature dependence 1081 Cobalt complexes, rate processes 1082 Collapse of doublet by chemical exchange 487, 489

XXXVi

SUBJECT INDEX

Coupling, in fluoroalkanes 1017 Collapse, of triplet by chemical exchange 491 in fluoroalkenes 915, 962 of quartet by chemical exchange 494 in fluoromethanes 883, 962, 1017 2,4,6-Collidine, t3C spectrum 1008 in perfluorocyclobutane 959, 962 Combination transitions 325 . relative signs 1011 Commuting opsrators 291 F-C1 in FCIO3 944 Complex particle method of analysis 345 F-F Computers effect of temperature 878 in the analysis of spectra 451 electron-orbital and orbital-orbital inof average transients (CAT) 230 teraction 184 Configuration of cyclohexanes 703 in aliphatic fluorocarbons 183 Configurational effects in t H spectra of in antimony pentafluoride 928 carbohydrates 701 in aromatic fluorocarbons 183 Conformation in c/~- and trans-l,2-diflunro-l,2-diof fluorinated ethanes 889 chloroethylene 916 of saturated ring compounds 575, 579, m c/s- and trans- N2Fz 948 920 m fluorinated aromatic compounds 901 Conformational motion, effect on spectra m fluorinated cyclobutanes 918 481 m fluoroacyl metal compounds 896 Conjugated polyenes, tH chemical shifts m fluoroalkanes 875, 958 744 in fluoroalkenes 906 Contact shifts 115, 826 m fluoroalkyl metal compounds 895 Copper (eSCu) resonance 1096 in fluorocarbon sulphides 894, 926 Copper(I)-copper(ID exchange reaction in fluoroethanes 886 503, 1096 in fluoropropanes 886 Coproporphyrin-l, IH spectrum 792 in hypofluorites 949 Correlation in interhalogen compounds 940 function 439 tn miscellaneous organic fluorine comapplication to a system of chemically pounds 962 equivalent nuclei 441 in perfluorocyclohexane 921 evaluation of 441 in perfiuoropiperidine 926 time (Te), 21, 31 in perfluorovinyl metal compounds Correspondence principle 291 906 Coupling constants (see Spin-spin coupling) in phosphorus halide derivatives 961 61, 113 in sulphur hexafluoride and its derivaabsolute signs 162, 681 tives 930 angular variation for geminal nuclei 171, in sulphur tetrafluoride and its deriva711 tives 938 angular variation in furam and pyrroles involving perfluoromethyl groups 959 786 long range 879 between H nuclei separated by four a relative signs 888, 914 bonds 174 through-space 190 calculation for HD 160 F-Hg calculation of H - H for methane 163 in mercury fluoroalkyls 897 calculation of v/e/ha/and geminal H - H in in mercury fluoroalkenes 907 ethanes and ethylenes 166 F-M in MFx compounds 880 calculations (theoretical) 103, 105, 106, F-X4N in NTs and c/s- and trana- N2Fa 109, 111, 160 948 dependence on atomic number 63 F-93Nb in NbF~6 1101 B-P in (CH3)~PHBH3 988 F-ZgSi in substituted silanes 1050 xsC--XsC 1029 F--Sn in tin fluoroalkenes 907 F-XXB, in boron halides and derivatives F-P 1061 945 in perfluoroalkyl derivatives 896, 9$9 F_Is c in perfluorotriphosphonitrile 951 additivity effects 1017 in phosphorus halides and derivatives correlation with tgF (t3C-t2C) isotopic 949, 959, 960, 961 chemical shifts 1022

SUBJECT INDEX Coupling..., in phosphoryl halides 1077 H A l in fithium aluminium hydride 1094 H-B in borohydrides 987 in boron halides and derivatives 972, 974

in boron hydrides and derivatives 972, 974, 977, 980, 982, 983 in (CHs)2PHBH3 988 in diborane diammonlate 986 in miscellaneous boron compounds 972, 974 H J 3 C 1011, 1032 additivity rules 193, 1017, 1019 correlation with bond length 1014 correlation with electronegativity 1014 correlation with XH chemical shifts 1020

dependence on hydridisation 1011 dependence on s character 192, 1012 in acetic acid 993 in acetylenes 1001 in aldehydes 1020, 1032 in ~kyl and silyl selenides 963 in alkenes 1028 in aromatic compounds 901, 1003, 1005, 1023 m chloroethylenes 722 m ethanol 993 m ethylene 448 m fluoromethanes 883 m formyl compounds 1014 m heterocyclics 1021, 1032 m hydrocarbons 1012 m methanes 1013, 1014 m monosubstituted benzenes 1003, 1032 in phenols 1006, 1023 in plumbanes 824 in propyne 994 in pyridine 994 in stannanes "824 in unsaturated compounds 196 long range 683, 1024, 1 0 2 7 relation with ~ u and ~ in ethylenes

1020 signs 197, 1011 H-D 1092 H-F in fluorinated aromatic compounds 901 in fluoroacetylene 916 in fluoroalkenes 909, 910 in fluoroethanes 886 in fluoromethanes 883

xxxvii

Coupling..., influoropropunes 8S7 in phosphorus (V) fluoride derivatives 960, 961 in silicon compounds 1050 long range 879, 902 orbital contribution 185 relative signs 888 H-H absolute signs 681, 713 correlation of geminal in alkenes with HCH angle 711 correlation of v/c/ha/ in ethanes with substiment electronegativity 680 correlation with Jc~ 1020 correlation with substituent electronegativity in vinyl derivatives 714 in acetylenes 177, 746, 748 in alkanes, geminal 677 in aikanes, vicinal 678 in alkenes, geminal 711, 714, 722, 728 in alkenes, vieinal 712, 714, 722, 728 in alkenes (metal) 743 in ammonia 183 in aromatic hydrocarbons 180, 682, 770, 1023 m butanes (2,3-disubstimted) 6 9 0 m butene-I 723 m i-hutches (1-substituted) 735 m cyclohexanes 679 m (CH3)ePI-IBHs 988 m ¢yclophane 173 m cyclopropanes 694 ]n esters (~t~.tmsaturated) 739 m ethanes 567, 681 m furans 782 m hexene-I 723 m indene 769 m olefines 710 m picolines 797 m propene-1 723 m propenes (2,3-disubstituted) 732 m propenes (2-substituted) 731, 735 m pyridines 794 m pyrroles 787 m quinolines 800 m silanes, geminal 823 m silicon compounds 1050 m thiazoles 805 m thiophenes 804 long range in acenaphthenes 685 long range in aromatic aldehydes 767 long range in indene 769 long range in olefmes 176, 685, 739, 743 long range in saturated compounds 683

xxxviii

SUBJECT INDEX

Coupling.... signs ofgeminal 172, 682, 695 Crossed coil detection 45 signs of vieinal 172, 682, 695 probe 208 H-Hg trans-Crotonaldehyde in 3-chloromercorffuran 1098 :H spectrum 469 in mercury alkyis 690, 1097 spectral parameters 469 in mercury vinyl 743 Crystal field calculations for SiCo shielding in organomercury compounds 1098 H_t'tN 1078 133~ r~6onanc~ in isouitriles 1040 of aqueous solutions 522 long range 1040 of metal in liquid ammonia 1093 temperature effect 819 of solid hafides 1093 H-xTO in water 509, 1048 H-SXp Cmnene, IH spectrum 830 for directly bonded nuclei 1061 Cyclic compounds, conformation 575 in (CHs)sPI'IBH3 988 Cycloalkanes in dialkyl phosphonates 1072 XH chemical shifts 692 in diphosphine 1074 Cycloalkanones, XH chemical shifts 692 m orgenophosphorus esters 1065 Cyclobutane(l,l-dimethyl-2,2,3,3-tetram pbosphites 1070 fluoro-), XgF spectrum 917 m phmphoryl halides 1077 Cyclobutane(1 -phenyl-2,2,3,3-tetrafluoro-), m phosphorus acids 1069 19F spectrum 917 m phosphorus alkyls 1064 Cyclobutanes, tgF spectral parameters 918 in PjNsCI4(SC2Hs)2 1075 Cyclobexane m symmetrical and unsymmetrical as a reference compound 264 trialkylphosphates 1063 axial-equatorial chemical shift 136 relativesigns 1061 Cyclohexane (bromo-), tH spectrum 576 H-2°TPb in plumbanes 824 Cyclohexane (chloro-), IH spectrum 576 H-19spt in platinum complexes I097 Cyclohexane (10~,3~.dimethoxy-2~acetoxy-), H~gSi in substitutedsilane 1050 IH spectrum 702 H-Sn, dependence on s character Cyclohexane (1~,3/~-dimethoxy-2~-acetoxy-). in stannanes 824, 1086 XH spectrum 702 in tetramethyl tin I013, 1088 Cyclobexane (methyl-), tH spectrum 706 in tin alkyl halides 1088 Cyclohexane (2,2,6,6-tetradeuteromethyl-) in vinyl tin 743 1H spectrum 706 H-TI Cyclohexanes in thallium alkyls 1090 configurations 703 in thallium phenyLs 1091 disubstituted, ~H spectra 708 in thallium vinyl 743 fluorinated 575, 577, 920 relative signs, 1091 XH chemical shifts 696 H-X (group IV element), correlation with XH specU'al line widths 708 atomic number of X 825 H - H coupling constants 700 long range 174 interconversion 575 axp_sIp 1061 monosubstituted, XH spectra 576, 706 in diphosphine 1074 1,2,3,4,5,6-hexachloro-, XH spectra 705 in non-cyclic phosphonitrilic halides perfluoro, interconversion ~$75 1078 Cyclohexyl in perfiuorotriphosphonitrile 951 acetates, XH spectra 703 in phosphates 1071 alcohols, IH specU'a 703 in phosphites 1070 Cyclopentadienyl anion, XH chemical in phosphorus acids 1069 shifts 779 in PsNsCIa(SC~Hs)2 1075 CyclopentadienyLs, IH chem/cal shifts 743, signs from double irradiation 466 780 signs from double quantum transitions Cyclophane, H - H coupling constants 173 454 2°3Tl-2°ST1 in thallium ethoxide 1092 Cyclopropane (l,l-dichloro-2-methoxy-),XH spectrum 693 CoupLing, virtual 814

SUBJECT INDEX Cyclopropane (l-methyl-2,2-difluoro-), XgF spectrum 917 CycloproFanes XgF chemical shifts 916 F - F coupling constants 916 XH chemical shifts 690 H - H coupling constants 694

6, the chemical shift (see Chemical shift) 5 for external refe~er.ce compounds 263 D,t point group, elements and character table 301 De~borane, 11B spectrum 982, 984 DecaUns, 1H spectra 580, 709 D e ~ o l s (10-methyl-), IH chemical shifts 709 Deceptively simple spectra 363 Derivation super operator 436 Deuteration in the determination of polymer ta~cities 836 Deuterium resonance 1092 coupling constants 1092 isotope effects 1092 Deuterochloroform as a solvent 257 Diamagnetic shielding 65 Diazo compounds, 1H chemical shifts 1139 Diborane, xlB spectrum 978 diammoniate, 11B spectral parameters 986 IH spectrum 979 Diboranes, 11B spectra 979 Dienes (conjugated), H - H coupling constants 741 Diethyl mercury, XH spectrum 687 Dioxane, laCH satellite bands 450 Dime delta function 79 Direct method for the analysis of spectra 435 Dished magnetic fields 204 1,2-Dithiane, 1H resonance 559 Double irradiation 240, 446, 447 in the determination of polymer structures 835 of the ammonium ion 458, 459, 460 of trans-crotonaldchyde 469 of dichloroacetaldehyde 461 of 2-furoic acid 468 of heteronuclear systems 455, 1093, 1640 of homonuclear systems 460 of 1,1,2-trifluoro-2-bromoethylene 915 relative signs of coupling constants 461, 466 theory of 244 to measure reaction rates 502

xxxix

Double irradiation, transitory 475, 476 with a weak second radiofrequency field 471 Double quantum, transitions 453 in 1,2-dibromopropionic acid 454 relative signs of coupling constants from 454 Doublet collapse by chemical exchange 487, 489 DSS, for aqueous solution ref~e,acing 265

Electric field effects intramolecuiar on 19F shielding 156, 157 on AB, AB a and AX3 spin systems 464 reaction field effects 88 Electrolyte solutions 511 relative molar chemical shifts 514 Electron densities from chemical shift measurements 781 spin resonance (F.SR) 4 Enthalpy difference between rotamers 570 Epoxides (monosubstituted) analysis of ZH spectra 377 2H chemical shifts 695

Equivalence chemical shift 283 magnetic 283, 308 symmetrical 284 Equivalent nuclei 284, 307 effect of conformational motion 285 Esters 170 chemical shifts 1045 ~fl-tmsaturated, 1H chemical shifts 737 Ethane (1,2-chlorobromo-), IH resonance 415 Ethane (1,2-dibromo-), IaCH satellites 450 Ethane (1,2-dichloro-), x3CH satellites 450 Ethane (1,1,2-trifluoro- l ,2-cfibromo-1chioro-), 19F resonance 563 Ethanes xsC chemical shifts 997 correlation with substiment electronegativity 996 calculation of geminal and vieinal H - H coupling constants 166 enthalpy differences between rotamefs 570

vicinal H - H coupling constants 567, 680 Ethanol 257, 273 13C spectrum 993 chemical exchange 493, 544, 508 IH resonance of aqueous solution 508

xl

SUBJECT

Ethyl derivatives ‘H chemical shifts 672, 673, 686 correlation of 13C chemical shifts with substituent electronegativity 996 correlation of ‘H chemical shifts with substituent electronegativity 670 metal derivatives 13C chemical shifts 999 ‘H chemical shifts 675 H-X coupling constants 688 Ethylene, H-13C coupling constants 448 Ethylene (l,l-dilluoro-) coupling constants 398 ‘H and lgF resonances 397 Ethylene (1,1,2-tritluoro-2-bromo-) lgF decoupled spectrum, 915 lgF spectrum 915 Ethylene [cis- and travel-fluoro-2-(perfluoroisopropyl)-] lgF chemical shifts 913 F-F coupling constants 913 ‘H chemical shifts 913 H-F coupling constants 913 Ethylene oxide, 13CH satellite bands 450 Ethylenes, calculation of seminal and vicinal H-H coupling constants 166 Ethylenes (chloro-) ‘H chemical shifts 722 H-13C coupling constants 722 H-H coupling constants 722 Exchange rate effect on a doublet 487 effect on a quartet 494 effect on a triplet 493 reactions electron 502 ion and group 507

lgF chemical shifts, @+ values 266 calculation of 151,266 correlations with substituent electronegativity 882 effect of solvents on 872 of alkyl and silyl selenides 963 of binary fluorides 880 of boron halides and derivatives 944,952 of fluorinated‘ heterocyclics of S, Se and P 926 of fluorine-containing polymers 954 of fluoroacyl metal compounds 895 of fluoroalkanes 885,957 of tluoroalkyl metal compounds 895, 959

INDEX

lgF chemical shifts, of fluorobenzenes 897 correlation of meta and para shifts with Hammett u constants 897 of fluorocarbon derivatives 879, 890 of fluorocyclobutanes 918 of fluoroethanea 886,958 of fluorohalohydrocarbons 883 of fluoromethanes 883 of fluoronaphthalenes 905 of hypofluorites 949 of inorganic fluorides 530, 880 of interhalogen compounds 940 of miscellaneous fluorine-sulphur containing compounds 939 of metal fluoride complexes 521,952,953 OfnitrogerMuorine containing compounds 946, of phosphorus halides and related compounds 949,959,960,%1 of sulphur hexafluoride and derivatives 930 of sulphur tetratluoride and derivatives 937 lgF (13C-12C) isotopic chemical shit correlation with JcF 1022 Fatty acids, ‘H chemical shifts 812 Fermi contact term 163 Field/frequency control 203, 229 Fig factor 45 First or&r spectra 280 Fluorinated cyclohexanes 920 Fluoroacetylene, lgFspectral parameters 916 Fluoroalkanes lgF chemical shifts 883, 885, 893, 957 lgF coupling constants 883, 885 Fluoroalkenes lgF chemical shifts 906,957 19F coupling constants 907, 909 H-H coupling constants 910 Fluorobenzne (ortLdichloro-) 19F coupling constants 398 lgF spectrum 399 Fluorobenmnes calculation of 19F chemical shifts 154 ortho effect 155 Fluorocarbon sulphides lgF chemical shifts 894 F-F coupling constants 894 Fluorocarbons calculation of coupling constants 183 containing nitrogen, lgF chemical shifts 889 Fluorocyclobutanes 918 Fluoroethanes conformational studies 889 F-13C coupling constants 1017 lgF spectral parameters 886, 958, 1017

SUBJECT INDEX Fluoromethanea F-‘% coupling constants 1017 19F spectral parameters 883, 1017 Fluoronaphthalenes, r9F chemical shifts 905 Fluoropropanes, 19F spectral parameters 887, 892 Fluorosilanes (methyl and ethyl), 19F chemicalshifts 927 Fhtx stabiliser 203 Formamide ‘Hspectrw 241,554 ‘Witradiation 241 Furan and substituted furans, ‘H spectral parametm 782 2-Furfurol, ‘H spectrum 364 ZFuroic acid l-H doubk resonance spectrum 468 ‘H spectral parameter% 467 ‘Hspecmrm 468 g, Land& or spectroscopic splitting factor 14 Gases, medium effects on chemical shifts 857 Gauge invariance 67 Geti coupling constants, see Coupling constants Gemmnes, spectral parameters 823 Glutaric acid (B-methyl-), ‘H spectrum 815 Glycidaldehyde, ‘H specuum 428 Glycidonitrile, ‘H spectrum 282 ‘Hchanicalshifts correlation with Ja 1020 empirical estimation 838 medium&cts 841 of acetylenes 745 solvent effects 749 of alkenes 711, 727,735, 741 anomalous 719 correlation with group dipole moments 718 correlation with Hammett u constants 717 effects of methyl group substitution 733 induced ring currents in annulenes 745 of alkyls 666,670,676, 685 of aromatic compounds 749, 770, 774 aromatic ions 774 correlations with Hammett u constants 752 effect of solvents 749 polyJluclear 770 ring currents 770 of cyclohexanes 696

xli

‘H chemical shifts, of cyclopropanes 690 of ethyl derivatives 670 carrelatioa with substituent electronegativity 670 of heterocychc compounds 782, 787, 794,798 alkaloids 806 amino acids and peptides 812 effect of solvents 798 fatty acids 812 furans 782 pyridines 794 pyrroles 787 quinolines 798 steroids 808 thiaxoks 805 thiophenes 802 of hydrogen attached to atoms other than carbon 816 of metal alkenes 742 of metal alkyls 689 of methyl derivatives 666 of paramagnetic species 826 of polymers 829 of propyl derivatives 676 of water effect of complex formation 520 effect of electrolytes 511 effect of hydrogen bonding 537 for ions at infinite dilution 515 relative molar 514 Hl (radiofrequency field) effect on spectra 230 meawrement 230 Halide ion exchange 511 Halogen fluorides 530, 940 Hamiltonian 290 matrix elements 304 HD, calculation of coupling constant 160 HCtWCpliCS

calculation of rH chemical shifts 145 effect of solvents on ‘H chemical shifts 855 fluorinated derivatives of S, Se and P 926 H-13C coupling constants 1021 ‘H spectra 782 Hexaborane, llB spectrum 982 Hexen* lH chemical shifts 723 H-H coupling constants 723 199Hg and ‘OIHg resonant 1097 chemical shifts of alkyls 1097 coupling constants in alkyls 1097 in 3-chloromercurifuran 1098 in organo-mercury compounds 1088

xiii

SUBJECT

I-U&red internal rotation 551 barrier heights 555,570 nature of barrier to 573 of C-C bond 559 of C-N bond 553 of C-P bond 572 of N-N bond 551 of N-O bond 556 of Ss bond 558 Homogeneity, see Magnets Hydration of ions 1048 Hydrocarbons, calculation of chemical shifts 135 Hydrogen bonding 534,816 chemical shifts due to 537 inamines 548,818 in halogenated compounds 548 in hydroxylic compounds 543 in mercaptaos 548,820 nature of 535 procedures for studying 539 peroxide, proton exchange 510 Hydroxyl groups, ‘H chemical shifts 816 proton exchange 507 Hyperfine interaction constants 115, 177 Hypoffuorites. r9F spectral parameters 948 Z, spin value 1,589 I2 operator 294 Is71 resonance of I’ aq 517,X096 of 1; 531, IO96 Indene ‘H spectrum 769 signs of H-H coupling constants 769 Indoles, ‘H spectra 793,794 INDOR 475,989,1033 Induced current model 79 Inositols. ‘H spectra 706 Integration of spectra 236 Intensities (relative) of multiplet components 7 Intensity measurements of bands 234 sum rule 380,452 transiton 306 Interatomic currents in benzene 82 in ethylene 82 in molecules 81 Interhalogen compounds, 19F spectral parameters 940 Internal rotation, see Hindered internal rotation und Conformation

INDEX

Inversion of nitrogen compounds 574 Iodine pentatktoride, chemical excw 943 Ions aromatic 774 inorganic, infinite dilution shifts 515 Iron carbonyl (pertluoroalkyl-), rgF spectral parameters 926 Irreducible representation 299 Isonitriles, H-i4N coupling constants 1040 Isopropyl benzene, ‘H spectrum 6 Isotopic effect on 19F chemical shifts 875, 916, 962 correlation with Jox 1022 on ‘H chemical shifts 1092 Isotopic substitution as an aid to analysis 445

3-Ketoallopregnrmes, iH spectra 809 Keto-enol tautomerism 549 Ketones, 170 chemical shifts 1045

Lamb diamagnetic shielding 66 Land& (spectroscopic) splitting factor 14 Langevin equation 35 Larmor .equation 13,67 precession 11 Lead (207Pb) resonance 1100 chemical shifts comparison with 19gHg andZosTl 1098 in inorganic compounds 1101 in organic compounds 1101 Lead tetraethyl, ‘H spectrum 687 Leakage 210 optimum value 211 Lime broadening from chemical exchange 485 from sweep rate 224 saturation 34, 40 spin-lattice 31, 32 shape 42 distortion 33 effect of quadrupolar relaxation 496 Lorentzian 505 of collapsing doublet 407. 489 of collapsing quartet 494 of collapsing triplet 493 widths andT2 40 effect of saturation 33 for liquids 30 Gaussian 30

xliii

SUBJECTINDEX Line..., in electron exchange reactions 506 in proton transfer reactions 510 Lorentzian 30,40 mean square 31 measurement of T2 from 49 of ‘*N resonances of ammonia and related compounds 1035 of 14N resonances of nitrites, nitrates and nitro compounds 1036 solvation of ions using 524 variation from quadrupolar relaxation 1102 ‘Li resonance of the metal in liquid ammonia 1093 Long range coupling constants, see Coupling constants Longitudinal relaxation time, Tl (see TJ 19,37 Lorentxian curves absorption (x”) 39 dispersion(x~ 39 Lorentx-Lorentx equation 98 Low resolution NMR 4 Lowering operators 292 Lunacrine, *H spectra 807 Lunine, ‘H spectra 807 2,6-Lutidine, ‘H spectrum 328 3,5-Lutidme, ‘“C spectrum 1008

Magnetic complex susceptibility 36 equivalence 59,283 effect of symmetry on 564 field effect on spectrum 227,442 recurrent sweep 223 shape 204 slow sweep 223 sweep 223 moment of electron 2, 14 of nucleus, see Nuclear moments 589 quantum number, m 14 radiofrequency susceptibility 34.40 screening of nuclei 65 shielding in atoms 66 shielding in molecules induced current model 79 perturbation theory 68 variation theory 77 static susceptibility 35, 65 Magnetisation vector, M 29, 37 Magnetogyric ratio, y 11, 14 Magnets 201 cycling 204

Magnets, field homogeneity 272 shimcoils 202 stabilisatkm 202 Medium efkts on 19F chemical shifts 872 on ‘H chemical shifts 841,857 Mercaptan (methyl), ‘H spectrum 334 Mercaptans, hydrogen bonding 548 Mercurifuran (3chloro-), H-199Hg coupling constants 1098 Mercury alkyls, H-199Hg coupling constants 690, 1097 chemicalshifts 1098 diethyl, ‘H spectrum 687 dimethyl, 19?Hg spectrum 1100 Metal carbonyls 823 ‘JC chemical shifts 1010 “0 chemical sbifts 1047 ethyl derivatives, lH chemical shifts 675 fluoride complexes 9S& 953 Methane calculation of H-H coupling constant 163 effect of solvents on ‘H chemical shift 842 Methanol, ‘H spectrum 544 c&Methyl crotonate, ‘H spectrum 737 Methyl derivatives lsc chemical shifts 992, 995 correlation with substitucnt electronegativity 996 13GH coupling constants 1013,1014 ‘H chemical shifts 666,668 correlation with substitucnt electronegativity 666 of group IVR elements 13C chemical shifts 999 ‘H chemical shifts 685 Methyl halides, ‘H chemical sbifts 667 Molecularbeams 2 Momenta, of spectra 431 Multiplet collapse by chemical exchange 488

‘*N chemical shifts

1033

of nitrates, nitrites and nitro compounds 1036,1037 “N resonant 1031 line widths of ammonia and related compounds 1035 line widths of nitrites, nitrates and nitro compounds 1036 of complex ammines 527

xtiv

SUBJECT INDEX

~4N resonance, of liquid ammonia containing alkali metals 1093 quadrupolar effects 1037 23Na resonance 1092 of N a ~ 515,5.17,520 Naphthalene calculationof chemical shifts 145 IH spectrum 405, 406, 773 H - H coupling constants 407 l,ro~ 19F spectrum 1101 93Nb spectrum 1101 Nickel complexes contact shifts 826 Ni(I1) N,N'-di(6-quinolyl)aminotroponeiminate, XH spectrum 828 Ni(H) ethylene diamir~e, IH spectrum 827 Nitric acid, XH resonance 517 Nitrites (alkyl), rotational isomerism 556 Nitrogen compounds, inversion of 574 Nitromethane as a solvent 259 Nitrosamines, rotational isomerism 5 5 i Niobium (93Nb) F-93Nb coupling constant in NbF~ 1101 spectrum of HbF~s 1101 NMR spectrometers 200 basic requirements 200 commercial 248 cycling of electromagnets 204 magnets for 201 Notation for spin systems 283 Nuclear induction 44 magnetic resonance absorption 45 magnetisation in phase 36 out of phase 36 moments 1, 3, 10, 1102 of IH nucleus 14 paramagnetic susceptibility 3, 35 properties 589, 1109 quadrupole splittings in asCl and ~TC1 resonances 1096 spin, I 1, 589 Nutation 52

lvO chemical shifts 1042, 1044, 1046 correlation with electronic transition energies 1043 of transition metal compounds 1044 effect of paramagnetic ions 1047 of metal carbonyls 1047 lye resonance 1041 aqueous electrolytes 523, 526 170-H coupling constant in water 509

Olefines, see Alkenes and Vinyl derivatives Operators, rahing and lowering 292 Overhauser effect 116, 247

3xp chemical shifts 1053 characteristic 1066 compilation 1143 correlation with bond properties 1056 in perfluoroalkyl phosphorus derivatives 959 in quadruply connected phosphorus compoonds 1059, 1060, 1144 in triply connected phosphorus compounds 1058, 1059, 1143 of cyclic phosphorus compounds 1075 of phosphates (condensed) 1073, 1145 of phosphines 1143 of phusphites 1068, 1143, 1147 of phosphonates 1072, 1149 of phosphonitrilic compounds1075, 1077, 1152, 1154 of phosphorus acids 1069, 1074 of phosphorus-nitrogen containing compounds 1148 of phosphoryi halides 1076 31p coupling constants 1061 relative signs 1061 sip resonance 1051 Paddles 209 Paramagnetic current density 80 susceptibility 35 2°TPb chemical shifts comparison with those of 1~gHg and 2°STl 1098 in inorganiccompounds 1101 in organic compounds 1101 2°VPb resonance 1100 Peaked magnetic fields 204 Pentaborane-9 11B spectrum 981 decoupling of 11B nuclei 456 IH spectrum 456 Pentaborane-ll, ~IB spectrum 981 Pantahorane halides, ~IB spectra 978, 985 Peptides, 1H chemical shifts 812 Perchloryl fluoride and related compounds, ~gff spectral parameters 943 Perfluoroacyl metal compounds chemical shifts 895 coupling constants 896 Perfluoroalkyl metal compounds chemical shifts 895 coupling constants 896

SUBJECT

Perfbtoroalkyl, sulphur hexafiuoride derivatives, tgF spectral pammeters 932 sulphur tetrafluoride derivatives 937 Perfluorocyclohexane axiaJ-eqwtorid chemical shift 139 lgF spectral parameters 920 PerSuorocyclohexanes dihydro-, lgF chemical shifts 922 monosubstituted, lgF chemical shifts 921 lgF chemical shifts 925 PerSuorodecalin (cis- and trans-) conformation 922 lgFspectra 924 Perfluoroisopropyl bromide, lgF spectrum 872 Perfhtoromethylcyclohexane, *‘F spectrum 876,921 Perfhtoropiperhline conformation 925 lgF spectrum 925 Perfluorovinyl metal compounds lgF chemical shifts 906 F-F coupling constants 907 Permanganate-manganate exchange reaction 506 Permutation operator 106 Perturbation theory 68, 428 Phasememory 29 relaxation time, see T2 Phenol (2,4,6tribromo-), ‘H spectrum with 13C satellites 1023 Phenols 1% spectral pammetem 1006 chemical shift/vibration frequency relation 547 ‘H chemical shifts of hydroxyl group 818 hydrogen bonding 545 Phosphate isohypo- ion, 3lP spectrum 1068 tetrapoly- ammonium sal& 31P spectrum 1071 Phosphates 31P chemical shifts 1144 condensed 31P chemical shifts 1145,1073 JlP spectra 1073 triallcyl, H-31P coupling constants 1063 Phosphine (di-), 31P coupling constants 1074 Phosphines, 31P chemical shifta 1143 Phosphite (trimethyl-), 31P spectrum 1054 Phosphites (organic), slP spectral parameters IM8, 1143,1147 Phosphonates, 31P spectral parameters 1072, 1149

INDEX

XIV

tri-Phosphonitrile (hexatluoro-), lgF spectrum 951 Phosphonitrilic (non-cyclic) halides 31P chemical shifts 1152 31P spectra 1077 Phosphorus triethyl-, ‘H spectrum 371 acids, 31P spectral parameters 1069 alkyls, H-31P coupling constants 1064 compounds, compilation of chemical shifts 1143 halides and derivatives, lgF spectral para meters 949,959,960,1055 organic esters, H-31P coupling constants 1064 Phosphoryl halides 31P chemical shifts 1076 jlP spectra 1076 Platinum (lgsPt), H-lQ5Pt coupling constants in platinum complexes 1097 Plumbanes, spectral parameters 823 P3N3Clr(SC2H5)2, 31P spectrum 1075 Polymers ontainiig fiuorine 954 copolymers of butadiene and isoprene 831 segmental motion 829 structure 831, 835 tacticity 833, 836 Poly+l)-methyl-l-alkenes 833 Polymethyhnethycrylate, ‘H spectrum 834 Polypropylene, ‘H spectra of various tatticities 837 Polystyrene 838 ‘Hspectrum 830 Polyvinyl chloride, ‘H spectrum 836 Population of spin states 16 Porphyrins calculation of chemical shifts 147 ‘H chemical shifts 792 Probe (sample holder) 207 Product spin functions 293,297 Propane, ‘H double resonance 476 PropargYl bromide+ ‘H spectrum 747 halides, ‘H chemical shifts 747 Propene-1 ‘H chemical shifts 723 H-H coupling constants 723 lHspectrum 724 eis- and trans-l-fluorolQF chemical shifts 912 ‘H chemical shifts 912 H-F coupling constants 912 2,Edichloro-, ‘H spectrum 388

xlvi

SUBJECT INDEX

Propene-2 (cis- and Mns-2-fluoro-3-chloro-) 1‘F chemical shifts 911 H-F coupling constants 911 ‘H chemical shifts 911 PropeneS

2-substituted ‘H chemical shifts 730 H-H coupling constants 731 2,3disubstituted ‘H chemical shifts 723 H-H coupliag constants 723 Propionic acid l&dibromo-, ‘H spectrum 454 f3-dibromo; doubk resonance 473,474 Propyl (i and n) derivatives, ‘H chemical shifts 672,676 Proppe,

13C spectrum

994

ProtoIysis reactions 507 Proton exchange in alcohols 507 inanlines 1040 Pterocarpin, ‘H resonance spectrum 814 Puke methods 40.50 electrolyte solutions 527 Pyridine as a solvent 259 ‘3C spectrum 994 ‘H spectrum 794 3-bromo-, ‘H spectrum .706 1,2,5,6-tetrahydro‘H decoupled spectrum 798 ‘H spectrum 798 Pylidines 13C spectra 1008 ‘H spectral parameters 794 Pyridinium ion, ‘H spectrum 1040 with l*N decoupled 1040 Pyrrole effect of solvents on ‘H spectrum 856 ‘H spectrum 457,787 ‘H spectrum with ‘*N decoupled 457 Ndeutero-, IH spectrum 787 Pyrroks, ‘H spectral parametez3 787

QUadNpol~ effects in ‘*N spectra 1037 line broadening 970, II02

QMpole

moments 27,589 relaxaton 26,426 effect on ‘H signals of NH groups 819 Quantitative analysis optimum spectr0mete.r conditions 211, 234 of phosphorus compounds 1076

Quantum mechanical formalism 287 Quartet colkpse by chemical exchange 494 Quinoline IH spectral parameters 798 5,7-dichloro-, ‘H resonance spectrum 801 5,7dimethyl-, ‘H resonance spectrum 801 I-methyl-, ‘H resonance spectrum 801 Radiofrequency effect on spectra 227 oscillator 227 power effect on spectra 230 measurement 230 receiver 232 Raising operators 292 Rapid passage experiments 40,497 s7Rb chemical shifts, of solid halides IO93 *‘Rb resonance of the metal in liquid ammonia 1093 Reaction fields 88,95,138,841 in purudinitrobenzene 90 Reference compounds 61,260 scale interconversion 267 Referencing external 260 for ‘“C resonance 991 for lgF resonance 266 for IH rseonance 264 intemal 263 Relaxation quadrupole 26,496 spin-lattice (see uiso TI) 18, 76 spin-spin (see a/so Tz) 28 Representations irreducible 299 reducible 299 symmetry group 298 Resonance condition classical description 10 quantum mechanical description 15 equation 13, 59 Ring currents 141,595, 770 Ringing 40,224 Rotating fields 13 co-ordinates 50,245 Rotational isomerism 551 about GC bond 559 solvent effect on .~ 568 temperature effect on 569

SUBJECT

Rubidium (s7Rb) chemical shifts of solid halides 1093 resonance of metal in liquid ammonia 1093

Sample containers 268 coaxial 268 for small volumes 270 for temperature studies 271 having large filling factors 269 sealing 271 spherical 268 preparation 256 spinning 206 Satellite spectra, 13CH of substituted ben7xnes 1022 Saturation 33,231,461 degree of 38 factor 34 12%b resonance of NaSbFe 1097 screening, see Shielding Secuhu dekrminant 296, 323 equation for NMR 288, 289 factorising 290 of AR spin system 312 Sekction rules 310 Selenium (“Se) chemical shifts of selenous and selenic acids 1097 2gSi chemical shifts 1049 comparisonwith 13Cchemical shifts 1051 2gSi resouance 1048 Sensitivity enhancement 230 SFaOCsH,, lgF spectrum 931 Shape factor 65 fuxtion 30,33 Shielding, diamagnetic contribution

70,75, 78 inalkenes 7% in benzenes 752 in linear molecules 73 in molecules containing heavy atoms 74 paramagnetic contribution 70, 75, 78, 894,895 Shielding constant or ccefkient, u, (see f&o Chemical shift) 61, 66 anisotropy of 94, 113,135, 136 calculation 120 by induced current model 130,595, 770 for halogen hydrides 124 for group VI hydrides 127 for H2 molecule 120

INDEX

xlvii

Shielding. . ., for hydrocarbons 129 effect of, electric field 85 isotopic substitution 99 medium 841, 857, 872 molecular iuteractions 90 Van der Waals forces 95 effects, S, and S,,, in benzenes 754, 1140

of lgF nuclei 874 influence of Iow-lying excited states 894,895 temperature dependence 101 shimcoils 202 Sideband, see Calibration of spectra und Double irradiation calibration of spectra 236 interpolation 275 single 244 superposition 275 Signal shape, see Line shape to noise ratio 233 Signs of coupling constants, see Coupling constants Silanes trisubstituted, ‘H chemical shifts 822 fluoro-, lgF chemical shifts 927 Silver ion complexes 529 Simplification of spectra 277 single coil detection 45 probe 217 “‘Sn chemical shifts 1083 of tin alkyk 1083 of tin halides 1083, 1084 lagSn coupling constants 1086 r ’ ‘Sn resonance 1082 Solvent effects on 13C chemical shifts 992 “‘Co chemical shifts 1081 “F chemical shifts 872 ‘H chemical shifts of acetonitrile 846 aatyknes 749, 848 aromatic compounds 258, 749,851 benrene 258 methane 842 unsaturated heterocyclic compounds 798, 855 rotational isomerism 568 ‘rgSn chemical shifts 1084 spectra 442 Spectral density functions 21 Spectrometers, see NMR spectrometers Spectroscopic

(Landk) splitting factor, g 14

xlviii

SUBJEC TINDEX

spin angular momentum

operators 15,290,291 commutation rules 291 correlation matrix 181 correlation function in benzene and naphthalene 182 decoupling (see uZso Double irradiation) 240 theory 244 densities in tetrahedral Co(U) complexes 828 eigenfunctions 15,293 eigenvalues 15 exchange 29 Spin-lattice relaxation me&anism 18, 76 time (see also Tr) 19 number (or value), Z 1, 589 Spin product functions 293,297 temperature 17 negative 17 system notation 283 Spinning of sample 206 Spin-spin coupling effect of time averaging 65 mechanism 63 molecular orbital calculations 106 valence bond calculations 109 variation method 111 coupling constant, J (see aLso Coupling constants) 7, 61 calculation for HD 160 evaluation 109,111 theoretical calculation of 103, 105, 160 reiaxation mechanisms 28 time (see also T2) 29 Spontaneous emission 18 Starmanes, spectral parameters 823.1086 Stamk bromide and iodide, ilgSn spectra 1084 Steroids, ‘II chemical shifts 808 Strong electrolyte solutions 511 Stytene 'H chemical shifts 719 ‘H spectrum 381 oxide, ‘H spectrum 362,377 Sub-spectra, superposition of 345 Succinic (methyl-) acid, ‘H resonance spectrum 815 Sulphur dioxide as a solvent 259, 519 hexafluoride

Yulphur, disubstituted, lgF spectral parameters 935 monosubstituted, igF spectral parameters 930 tetrafluoride rgF spectrum 531,936 lgF spectral parameters 936 perlkoroalkyl derivatives, rgF spectral parameters 937 Super operators 435 derivation 436 matrix representation 437 Susceptibility bulk diamagnetic corrections 260 diamagnetic 65 pammagnetic 35 volume 3, 262, 605 (compilation) Symmetrical equivalence 284 Symmetrised spin functions 321 notation 335 Symmetry transformation group 298 t values 265 charts 1131 for a variety organic compounds (Tiers’ compilation) 1115 Tl,spin-lattice (or longitudinal) relaxation time 19 contribution from aniso&opy of u 76 effect of anisotropic shielding 27 diamagnetic substances 26 ZH on rH resonance 23 pammagnetic substances 21, 25. 30 pressure 28 quadrupoles 21, 26 viscosity 23 intermolecular contribution 22 intramolecular contribution 22 measurement by direct method 46 measurement by progressive saturation method 47 of electrolyte solutions 527 of ice 19 of liquids 19 of solids 19 T,,spin-spin (or transverse) relaxation time 29, 37 effect of paramagnetic ions 30,505 inter- and intramolecular contributions 30 measurement of 49 measurementsonelectrolytesolutions 527 Tautomerism 549 Temperature

SUBJECT Temperature, control of sample 212 dependence of sgCo chemical shifts 1081 effect on rotational isomerism 569 Tetraborane, rrB spectrum 981 Tetraethyl lead, ‘H spectrum 687 Tetramethyl silane (TMS) 264 as a universal reference 267 Thallium alkyls, H-z05Tl coupling constants 1090 ethoxide structure 1091 203Tl-zo5Tl coupling constant 1092 phenyls, H-20sTI coupling constants 1091 salts, 2osTl chemical shifts 1089 effect of anions 1090 Thiazoles, ‘H spectral parameters 805 Thiophene t-bromo- lH spectrum 281 3-bromo-Zaldehyde, ‘H triple resonance 476 2-bromo-S-chloro-, double resonance 472 deuterated, ‘H spectrum 804 2-methylthio-3-thiophene thiol, ‘H spectrum 311 “Tickling” experiments 471 Tin alkyls rlgSn chemical shifts 1083 llgSn coupling constants 1087 halides 531 llgSn chemical shifts 1083, 1084 1lgSn-H coupling constants 1087 resonance spectra 1082 tetramethyl, llgSn spectrum 1087 205Tl chemical shifts 1088 comparison with those of 1lgHg and 207Pb 1098 of thallium salts 1089 effect of anions 1090 205Tl coupling constants 1090 in thallium phenyls 1091 in thallium trialkyls 1091 signs 1091 205Tl resonance 1088 203Tl and 205Tl resonances of aqueous electrolytes 522 Tl(I)-TI(III) exchange reaction 506 T12Cld, 2osTl spectrum 1089 Toluene, ‘H spectrum 6 Trace of a matrix 431 Transient effects 40 Transition intensities 306,310,315 metal compounds, ‘H spectra 825

INDEX

XliX

Transition, “0 chemical shifts 1044 probabilities 16, 19, 33 Transistory selective irradiation 475, 476 Triaryl carbonium ions, ‘H chemical shifts 774 Trifluoroacetic acid as a solvent 989 r3C I-NDOR spectrum 989 Triple resonance 476 Triplet collapse by chemical exchange 491 Tropylinium cation, rH chemical shifts 779 Twin-T bridge 217 u-mode (dispersion)

42, 210, 218

V-mode (absorption) 41, 210, 218 VO_V(IV) exchange reaction 504 Variation theory 77 Vicinal coupling constants, see Coupling constants Vinyl bromide ‘H spectra for various solvents 443 ‘H spectrum 711 Vinyl chloride ‘H chemical shifts 722 H-l% coupling constants 722 H-H coupling constants 722 Vinyl cyanide and its methyl derivatives, ‘H chemical shifts 719, 734 Vinyl derivatives ‘H chemical shifts 717, 727, 735 anomalous 719 correlation with group dipole moments 718 correlation with Hammett u constants 717 correlation with substituent electronegativity 714, 717 effect of methyl substitution 733 disubstituted ‘H chemical shifts 727 H-H coupling constants 727 metal, H-X coupling constants 743 trisubstituted ‘H chemical shifts 735 H-H coupling constants 735 Vinyl ethers, ‘H chemical shifts 721 Vinyl fluoride, coupling constants 425 Virtual coupling 814 Volume susceptibilities 3,262 compilation 605

Water as a solvent

257

1

SUBJECT INDBX

Water, effect of electrolytes on *H resonance 516 H-*?O coupling constants 1048 proton exchange 509 Wiggle beat method of spectrum calibration 276 Wiggle beats 41

Wiggles

40, 224, 226, 508

Xylenes (substituted) 1H chemical shifts 761 correlation with Hammett a constants 762

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topicCS - Topics in Cognitive Science, Volume 3, Issue 1

Progress in Nuclear Magnetic Spectroscopy Volume 1 Issue 1

Annual Reports on Nuclear Magnetic Resonance Spectroscopy: v. 1

Progress In Optics Volume 1

Nuclear Magnetic Resonance Spectroscopy in Environmental Chemistry

Spatial Economic Analysis - Volume 1, Issue 1

Topics in Fluorescence Spectroscopy, Volume 1: Techniques

Handbook of Magnetic Materials, Volume Volume 1

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Progress in Nanophotonics 1

Progress in infancy research, Volume 1

ADVANCES IN CATALYSIS VOLUME 1, Volume 1

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Physiotherapy, Volume 97, Issue 1 (2011)

Applied Linguistics, Volume 31, issue 1, 2010

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Spatial Economic Analysis - Volume 1, Issue 2

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