Edwin D. Becker
National Institute of Arthritis and Metabolic Diseases National Institutes of Health Bethesda, Maryland
NMR Spectra: Appearance of Patterns from Small Spin Systems
W i t h the rapidly expanding use of high resolution nuclear magnetic resonance (NMR) it is becoming increasingly important that both graduate and undergraduate students in chemistry be aware of the significance and means of interpretation of complex NMR spectra. While there are a number of excellent introductions to NMR (1, 8) and several detailed treatments of the interpretation of NMR spectra (5-5), it has nevertheless become obvious to the author in the course of teaching NMR to graduate and postdoctoral students that there is considerable confusion in terminology and a widespread lack of understanding of the means by which NMR spectra may be readily interpreted. It is my hope that this article will clarify some otherwise confusing points and thus make NMR more generally useful in the determination of molecular structure and other chemical properties. Notation for "Equivalent" Nuclei
It This article deals only with nuclei of spin has become conventional to designate each nucleus by a letter of the alphabet, using letters close to each other for nuclei whose chemical shifts are small relative to the coupling constants (6 I -6J), and letters a t opposite ends of the alphabet for nuclei separated by large chemical shifts. Thus the molecule H F would be designated as an AX system, while the molecule CHCl = CHBr would be designated as an AB system (C1 and Br may he treated as nonmagnetic). This system of notation can easily be extended so that we have for example ABXY, ABRX, etc. Difficulties arise with this notation when two or more nuclei have identical chemical shifts. The obvious extension is to use subscripts to designate such equivalent nuclei; for example fluoroform, CHF,, is designated as an A 3 (or AX3) system. Ambiguities may arise, however, as evidenced by the proton magnetic resonance spectra of the molecules CH2F2and CH2= CF2, shown in Figure 1. Both of these molecules would be logically designated as A2X2systems; yet the spectra obviously have little in common. The spectrum of CHzF2 follows the usual "first order" rules (1, 8), which predict that the proton magnetic resonance spectrum would he a 1:2:1 triplet. The complexity of the spectrum of CH2 = CF2 arises [as was pointed out a number of years ago by McConnell, et al. ( 6 ) ]from the difference in coupling constants between the hydrogen and fluorine that are cis to each other and those that are trans. In CH2F2both protons are coupled equally Based on a leetore presented at the Georgetown Univesity NMR Workshop, April, 1964.
to each fluorine, while in CH? = CF2 the cis and trans H-F couplings are different. The two situations have been distmguished by using the term chemica!ly equivalent for nuclei which merely hare the same chemical shift (e.g., the protons in CH2 = CFz) and the term magnetically equivalent for nuclei which not only have the same chemical shift but which are coupled equally to every other nucleus in the molecule (e.g., the protons in CH2Fe) (7). Unfortunately t,he dist,inction is not always made clear in books and arbicles, and the term "equivalent nuclei" has been defined and used in some cases to denote merely chemically equivalent nuclei (9) and in other cases to denote magnetically equivalent nuclei (4). I t is now becoming more common to denote nuclei which are only chen~icallybut not magnetically equivalent by the use of additional symbols. Thus, CH2 = CF2 might be designated AA'XX'. Even this system can lead to some difficulties,but, we shall use it in this article and shall use the adjectives "chemical" and "magnetic" to denote the types of equivalence. First Order Spectra
Many spectra can be analyzed quite adequately by the well-known first order approximation, in which the spectrum is seen to be made up of simple mukiplets, such as the triplet in the CH2F2case. I t is generally 'ecognized that the first order approximat~ionholds only when the ratio 6 / J is large (1 -6). What is sometimes not appreciated, however, is that the first order approximation holds only when the groups of
Figure 1. Proton resonance spectra of CH?F? and CH*=CF* The i°F resonance spectra would be identical.
at 60 mc.
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nuclei being considered as coupled are in fact magnetically equivalent, not just chemically equivalent. Sometimes nonmagnetically equivalent nuclei give rise to spectra that appear to be susceptible to first order analysis but really are not; this is actually an example of the "deceptively simple spectra" discussed below. Analysis of Spectra
I n general, in order to analyze an NMR spectrum one must first set up and diagonalize the appropriate Hamiltonian matrix. Rules for setting up this matrix have been described in detail by Pople, Schneider, and Bernstein (3) and need not be repeated here. I n some cases (e.g., ABX, AA'XX') the resultant equations are simple, and the spectrum may be analyzed by recognizing certain spacings. Even in more complex cases, however, the rules for setting up the Hamiltonian are basically very simple and amount largely to bookkeeping. This is just the sort of problem that can easily be programmed for a digital computer; and there now exist several programs (8-lo), differing in degree of sophistication, which can be used as an aid to analysis of actual spectra or as a simple means of generating synthetic spectra to demonstrate the effect of systematic variation of one or more chemical shifts or coupling constants. It has been my experience that many chemists who are not very familiar with the theory behind complex NMR spectra can nevertheless profitably approach the analysis of complex spectra by carefully utilizing one of the more straightforward computer programs.' It might also be worth pointing out that NMR spectra have sometimes been analyzed by use of perturbation theory (11). This technique was used more widely in the earlier days of NMR and is employed less frequently today. However, the terminology is still used in cases where a spectrum can be recognized as consisting primarily of first order multiplets with additional small splittings or distortions of intensities. These latter splittings or distortions are often referred to as "second order effects." If one uses the approach described in the preceding paragraph, the calculated spectrum will be the exact solution for the input parameters (chemical shifts and coupling constants) employed. I n that case, one need not try to make the arbitrary distinction between first order and second order effects.
A and X, which diier widely in chemical shift. These two nuclei can be oriented to the magnetic field in four possible ways, ranging from the case in which both are parallel to the field to the case where both are antiparallel to the field. The center portion of Figure 2 shows the energy levels for these four situations when the two nuclei are not coupled to each other. I n this case, the spacing between levels 1 and 2 is eqnal to that, between levels 3 and 4, and likewise the spacing between levels 1 and 3 is eqnal to that between levels 2 and 4. Thus, the two "A" transitions (flipping of spin A, while spin X remains unchanged) give rise to only one lime a t the chemical shift of A, and the two "X" transitions to one line a t the chemical shift of X. Consider now the case when the two nuclei are coupled with a coupling constant which is small relative to the chemical shift. Physically, the presence of a non-zero coupling means that one nucleus "feels" the effect of the other; that is, its energy level is dependent upon the orientation of the second nucleus. We have to distinguish now between two cases: (a) the situation in which the anti-parallel arrangement leads to a lower energy than the parallel arrangement, and (b) the opposite, in which the parallel has the lower energy. Case (a) is by convention referred to as a positive coupling constant and case (b) is a negative coupling constant. As seen in Figure 2, for J > 0, energy levels 2 and 3 are lowered relative to their position in the absence of coupling while levels 1 and 4 are raised in energy. As a result, transitions between levels 1and 2 no longer have exactly the same energy as transitions between levels 3 and 4, and thus the A line is now split into a doublet, the amount of splitting as shown by a quantitative treatment being a direct measure of the coupling constant (6). An identical situation occurs for the X lines. As shown also in Figure 2, a negative coupling constant leads to a different arrangement of energy levels, but the ultimate transitions observed are indistinguishable from the case of positive coupling. Thus for this case we cannot determine from the NMR spectrum the sign of the coupling constant but only its magnitude. The detailed quantitative treatment would demonstrate that ail transitions shown are equally probable so that the observed doublets are of equal intensity as shown in Figure 2. However, if the chemical shift is
Two Coupled Nuclei
I t may be helpful to consider the case of two nuclei, At N.I.H., for instance, extensive use is made of BothnerBy's FREQINT 111, which is a non-iterative program requiring as input the investigator's "guesses" of the correct chemical shifts and coupling constants. The output consists of an ordered listing of line frequencies and intensities (or, in some cases, of an actual plot of the calculated spectrum). For analysis of an experimentally determined spectrum the calculated and observed spectra must be compared and the calculation repeated with new parameters determined by a systematic, but besically trial m d error, method. Considerable insight into the behavior of a simple spin system may often be obtained by systematic varirisi tion of one or more parameters. At N.I.H. the program is nln on a Honeywell 800 computer; the FREQINT series of program* was originally written for the IBM 704 and 7090; and similar programs have been written for smaller computers, snch as the IBM 1620.
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Figure 2. Schematic representation of t h e energy levels and spectrum of on AX system under different condition, of coupling.
now made smaller while retaining the same coupling constant, the intensity relationship changes, as shown in Figure 3. This leads to the well-known AB pattern or "nonequivalence quartet" which has been treated in a number of places (1-5). Both the coupling constant and the chemical shift between A and B may readily he obtained from such a spectrum. The necessary simple formulas will not he reproduced here. I n the limit where the chemical shifts of the two nuclei become identical we have an A* case which leads to only a single line, in accordance with the well-known fact that two equivalent nuclei do not result in a splitting even if they are, in fact, coupled. Three Coupled Nuclei
For an AB or AX case the appearance of the spect,mm depends only upon the ratio of J / 6 . With three ~mclei,however, the situation is more complicated; this is the simplest system in which magnetic equivalence becomes an important factor. I n order not to unduly complicate the situation, we shall consider only the ABX system in some detail. For the plots in Figure 4 we have taken 6*, ax, J*B, J A X ,and J s x as constants and demonstrate the effect of variation only of 8% The first ease might more properly be described as an AMX case rather than an ABX case since is large. The first order analysis of an AMX case is indicated, and the actual spectrum shown in Figure 4, spectrum a (calculated exactly by a computer program) shows that the first order approximation is quite adequate, there heing only small distortions of the intensities and virtually no deviations in line positions. the quartets become less recogAs 6~ approaches nizable, and in the situation shown in spectrum c one has a typical ABX spectrum in which the AB pattern can he seen as a superposition of two AB quartets. (Lines 1,3,5,and 7 constitute one quartet; lines 2,4,6, arid 8, the other.) The analysis of such a pattern has been described in several places (5-5). We shall be more concerned here, however, with the situation shown in spectrum d, where 8~ is now quite close to 6& The two overlapping quartets are now recognized only with difficulty. I t is quite significant that the X portion of the spectrum, which heretofore has shown a series of four lines unaffected by the chemical shift of B, now begins to changemost prominently in the center where two lines begin to draw together. Small satcllite lines now appear in the X portion, but their
Figure 3. Colculoted spectrum for a system of two nuclei or the chemical shift between them is changed relative to their covpling conrtant.
iutensity is quite low. Finally, as 6* and 6~ become equal (spectrum e ) , the AB portion collapses to a doublet and the X portion becomes a triplet with intensities 1:2: 1. There are, in addition, very small satellite lines, but in practice the intensities of these lines are so low that they would very probably not be recognized. Since the situation shown in spectrum e describes a system in which there are two nuclei of identical chemical shift and one of quite different chemical shift, this might he described as an AA'X s y ~ t e m . ~ Kote that in this case the two A nuclei are definitely not n~agnetically equivalent, since one is coupled to nucleus X by 7 cps and the other by only 3 cps. If this spectrum were encountered in practice, rather than as the result of a calculated change in one of the parameters, it might very easily be mistaken for a real A2X system in which the two A nuclei are magnetically equivalent and are coupled to X with a 5 cps coupling constant, the average of the two actual coupling constants. This is an example of what has been called a "deceptively simple" spectrum (12, 13). Such spectra are found very commonly in practice. One example is given in Figure 5 . In general, a spectrum which gives the appearance of being susceptible to a simple first order analysis should be suspected of heing deceptively simple whenever nuclei are present which are chemically but not magnetically equivalent, as, for example, in a cyclic system where free rotation can not average out the coupling constants of nuclei in different conformations. The situation described above is not, of course, restricted to three nuclei. A simple bspin example We have been discussing a situation in which lJasl >> J A XJ e x . In the limit in which A and B become chemically equivalent the X p a t of the spectrum reduces to a triplet as shown. Bernstein (personal communication) has pointed out that a different type of deceptively simple spectrum can result from chemically equivalent A and B when J A B / > 1, and the four protons on Cz and Cp behave as a strongly coupled group. The result, as shown in spectrum a, is that the protons on C, and C4 give rise not to a simpie triplet as might be expected from first order analysis, but to the complex low field multiplct shown. The multiplet for the protons on C2 and C3 is likewise complex. This situation can be contrasted with that in 1,5dibromopentane (Figure 7, spectrum b). In this case the chemical shifts of the protons on Cz and Cs are different; for J = 7 cps, J/6 = 0.3. The C1 and C5 protons thus give a first order triplet. Protons on Cz and Ca are, of course, chemically equivalent, but Jzr= 0,so these are not strongly coupled. Anothcr interesting example of virtual coupling is fillown in Figure 8. The spectrum of 2,5dimethylquinone (a) is readily interpreted by first order anal-
' I n systems such as the one shown there is sometimes a small
through four or more single bonds, called a long-range coupling. This real coupling should not be confused with the virlual coupling that we have been discussing. coupling
ysis; that of 2,6-dimethylquinone (b) shows additional splitting6 caused by the "virtual coupling" of the Cz methyl protons with CsH, and by symmetry of the Csmethyl with C3H. Protons on Cgand Cs are coupled via a small long-range coupling interaction, but their chemical equivalence results in a large value of J/S. Often the effect of virtual coupling is merely to bring about an apparent broadening of peaks, since many of the lines are frequently close together. One should always be alert for the presence of virtual coupling in NMR spectra. I t may occur whenever there are two or more strongly coupled nuclci which are chemically but not magnetically equivalent. Molecular Asymmetry
Often two protons which at. first glancc appcar lo be rhemically equivalent (those in an acyclic CH2 group, for example) turn out to be nonequivalent, as shown by complicating features in their spectrum. Consider for example the case CHZ-CPQR, where X, P, Q, and R are different substituents. We can picture three stablc conformations about the C-C bond:
Normally rotation about a C-C single bond is suficiently rapid so that the spectra of the individual conformers are not observed. The observed spectrum will represent the average of the chemical shifts and coupling constants in the three conformations. Because of the asymmetry of the one carbon atom the average chemical shift of HA is not necessarily the same as that of HB. This nonequivalence can occur in principle even if all three conformations are equally probable, since there are no conformations that are mirror images (15-17). In practice one conformation is often favored over others, and this can further lead to nonequivalcncc in the average chemical shifts (18). Whet,her one a(:tually observes separate lines for HA and Hn depcnds, of course, on the magnitude of bhe various effects operative. The establishment of the types of structures and environments that give rise to appreciable now equivalence has been the subject of recent investigations (19). An example of nonequivalence is shown in Figure 9. I t should be emphasized that thc preecnce of asymmetry or dissymmetry in the molecule, usually very near t,he protons in questions, is a necessary condition for nonequivalence when lhcrc is rapid rotation about the C-C bond. Thus, one would not expect nonequivalence in a molecule such as l,4-dibromobutane (the spectrum is shown in Figure 7 ) . Of the three
Figure 8. Proton resonance spectra 160 mcl of la1 2.5-dirnethylquinone and lbl 2.6-dimethylquinone, shoving the effect of "virtual coupling'' in lbl.
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Inst,itutc of Technology for reading thc manuscript and suggesting chnngos. Literature Cited (1) ROBERTS,J. I)., ' ' N I I c ~ ~ Mngnet,ic ~LI. Resonance. Applies; tions t,o Problems in Organic Chemist,~y,"h1cGr.z~-Hill Book Co., Iiew York, 1959. (2) J:LCKMAN, L. hl., "Applicat,ians of Piucleitr Magnet,ie Ilewnance Spectroscopy in Organic Chemist,ry," Pergnmon Press, New Yark, 1959. (3) POI'LE, J. A,, SCHNEIDER, W. G., A N D BERNSTEIN,1.1. J., "High Resolution Yuclear Magnetic Resonance," MeGmw-Aill Book Co., New York, 1959, Chap. 6. (4) Conlo, P. L., Chem. Rm., 60,363 (19GO). (5) ROBERTS,J. D., "AU Intr~ductionto the Analysis of SpinSpin Splitting in High Resulution Nnrlear Maguet,ie Hesonarree," W. A. Benjamin, Inc., S e w York, 1961. (6) MCCONNELL, H. M., MCLEXN, A. U., A N D REILLY,C. A., J . C h m . Phys., 23,1152 (1955). ( 7 ) GUTOWSKY, H. S., A m . A'. Y. Acad. Sci., 70,786 (1958). (8) SWAI~EN, J. D., A K D REILI,Y, C. A., J . Chem. Phws., 37, 21 1lgfi'2). ~....-
Figure 9. Proton magnetic reranonce spectrum (60 mcl of the impropyl group of piperitone in CDCL. Each of the nanequivalent methyl groups, MI and Ma is coupled to the iropropyl CH proton.
couformations (whcre R = CHzBrCHz-), I and I1 are mirror images, the average of which must have 6* = 6~ and 6~ = 6 ~ and ; 111 has a plane of symmetry, so I hat the equalities hold herc too. Acknowledgments
I am indebted to W. B. Moniz of the Naval Research Laboratory for the 100 mc spectra in Figure 7 aud to L. Tsai and H. Ziffcr of the N.I.H., C. F. Hammer of Gcorgctown U~iiversit~y, and B. L. Shapiro of Illinois
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(9) C~STELLANO, S., A N D BOTHNER-BY,A. 4.)J . Chem. Phys., 41, 3863 (1064). 11,. C., Fift,h Experimental NAfR Conference, (10) ~IOPKINS, PitSsbnrgh, Pentrsylvnnia, Fe1,rnar.v 2!1, 1964. W. A,, Phi,% KPV., 102,151 (19%). (11) ANDERSOX, (121 R. J.. . A N D BERVSTEIN. 11. J.. Can. J . Chem... 39.. . . ABRAHAM. 216 (19'61). ' (13) RICHARDS,It. E., A N D SCHAEFER,T., Jfd. Phys., 1, 331 (1958). (14) M u s m n , J. I., A N D C o n m , E. J., Tet~.ahedron,18, i 9 1 (1962). (15) WAUGM, J. S., A N D COTTOX,F. A,, J . Phys. Chem., 65, 562 (1961). (16) GUTOWSKY, 11. S., J. Chem. Phys., 37,2196 (1962). (17) POPLE,J. A., Md.Phys.. 1, l(1958). (18) NAIE,P. M., A N D ROBERTS,J. U., .J. Am. Chem. So?., 79, 4565 (195i). (19) WIIITEBIDES, G. XI., ET ,AT., .I. Am. Ckem. Soe., 87, 10% (1965).
