I Edgar W. Garbisch, Jr.' University of Minnesota Minneapolis 55455
I
I
Analysis of Complex NMR Spectra for the Organic Chemist I.
Second-order approach with specific application to the two spin system
In a recent discussion and painstaking compilation of literature reported values of vicinal and geminal proton-proton coupling constants, BothnerBy commented (1) that about 90% of such reported values had to be rejected on the basis of their being unreliable! This high rejection percentage is testimony to the idea that chemists largely are inclined to apply firstorder principles to the analysis of nmr spectra. The general application of first-order principles is understandable, as an acceptable analysis of even the apparently simple spectrum often is not a trivial undertaking and the average chemist, who is neither an expert in the field nor motivated in that direction, normally is unfamiliar with the subtleties associated with spectral analyses. I t should be recognized, however, that probably the majority of nmr soluble problems concerning molecular constitution and even molecular conformation and configuration (under favorable circumstances) can he and have beeu solved reliably by the first-order approach. In these instances the key to the solution may be simply recognition of the gross multiplicities of the bands (neglecting higher-order splittings which often are undetectable under the conditious which the spectra were determined) or the observation that a coupling constant is large, say 16 cps, or small, say 3 cps. The reliability of the solution of the problem rarely will rest on knowledge that the correct coupling constant is 16.87 or 3.44 cps. On the other hand, it hardly cau be refuted that conclusions deriving from the more thorough and accurate analyses will be the ones most highly regarded by the scientific community. The availability of excellent iterative computer programs for the solution of nmr spectral problems ( 2 ) is greatly assisting the nonexpert in expediently acquiring accurate analyses of his spectra. Unique solutions, however, are not guaranteed. I n addition, an accept-
able computer analysis of a spectrum may he not only laborious hut may demand more intellectually from the user than an ability to correctly punch input data cards. The computer user should have some fundamental knowledge of higher than firstorder analysis of nmr spectra-even if the program is of the noniterative trial and error type. Not all nmr spectra are analyeable. Most organic molecules contain nuclear spin systems of such potential complexity that capacities of current computers are insufficient to handle the problems without prior simplifying approximations. Often, with some knowledge of higher than first-order analysis techniques, a complex spin system may be factored safely into sets of noninteracting and readily analyeable spin systems The combined separate analyses of these simplified systems then may lead to a reliable complete analysis, or, at least, partial analysis. Complex spectra of systems containing up to four spins often can be analyzed satisfactorily by hand. The time required to complete such hand analyses will of course depend largely on the complexity of the spectrum and the experience of the individual who is working the problem. Hand analyses of such spectra clearly cannot compete with computer analyses in terms of time expended. However, hand analyses offer an advantage a t the instructive level. General Background
I n the absence of a magnetic field, a bare nucleus having a spiu quantum number of I exists in (21 1) degenerate states, q. A hare nucleus is one void of surrounding electrons. Upon the application of a static magnetic field,%H,, this degeneracy is removed giving q nondegenerate states having energies.
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The author is grateful for support of this work by the National Science Foundation (Grant No. G.P. 5806). EDITOR'S NOTE: This article develops a second-order perturbation approach for approximate hand analyses of complex nmr spectra. I t will be published in three parts. Part I (above) gives s n elementary introduction to the general principles of nmr spectroscopy followed by s, dicussion of the secand-order approach with specific application to the two spin system. Part I1 (which will appear in the June i s w ) will discuss three spin s y s t e m of the ABX, ABK, ABC, and AB1 types. Part 111 (which will appear in the July iasue) willdiscuss four spinsystems of the ABXZ, ABK2, AA'XX', and AA'BB' types.
Alfred P. Sloan Foundation Research Fellow. The convention used here will be to take Ho d o n g the negsi tive z direction of a. cartesian coordinate system. ( H o directed down along the z-axis.) Using this convention for the proton, the ~ t a t ewith m = - I / . is of lower energy than the state with m = I/*. If the reader refers to the treatments listed in the Suggested Reading List, he will note that this convention is not uniformly employed. The alternative convention is to take Hn dong the positive z direction (Ha directed up along the e axis). Using this convention, the p states have energies E , = - g ~ i i ~ H m = - y H o m = -hvom and far the proton the state with m = -I/% is of higher energy than that having m = I/,. Volume 45, Number 5, May 7968
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31 1
where nz, the magnetic quantum number, may assume values of I,I-1,I-2, . . . -I; g~ is a dimensionless constant called the nuclear g factor (3) and has a value of 5.585 for the proton; Bw is called the nuclear magneton and has a value of 5.0493 X erg/gauss; y is called the magnetogyric ratio and has value of 2.675 X lo4 radians/gauss sec for the proton. This removal of the (21 1)-fold degeneracy of the nuclear magnetic states in the presence of a magnetic field is called the nuclear Zeeman effect. The nuclear magnetic resonance experiment involves placing a nuclear spin system in a static magnetic field and inducing nuclear spin transitions between the resulting q nuclear energy states by irradiating the system with electromagnetic radiation of the appropriate frequency. The experiment is completed by detecting the resultant absorption of energy. For a single nucleus, the magnetic selection rule for transitions between the p states is
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so that I n eqns. (1) and (2), vo is the frequency (called the Larmar frequency) required to induce allowed transitions in a bare nucleus in the presence of a magnetic field of strength Ho. For the proton, Ha, in a Ho of 14,091 gauss, eqn. (2) tells us that a vo of 60 Rlc/sec is required for magnetic resonance. The transition energy, L% = hvo is 5.6 X lo-%kcal/mole. Even though stimulated Em = (emission) and Em = -112 transitions Em Em = (absorption) are equally allowed, a net absorption of energy is observed as a slight excess population is maintained in them = -I/% state relative to the nz = '/% state at normal temperatures and in the absence of s a t ~ r a t i o n . ~Consequently, we will consider subsequently only transitions from lower to higher energy states. Protons in molecules are shielded relative to the bare n u c l e u ~ . ~This shielding arises predominantly from the bonding electrons which in the presence of an applied
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field produce an opposing (diamagnetic) field. As a result, the net field a t the proton is less than Ho by Hau, where a is a dimensionless shielding or sc~eening constant. Actually, u is a composite of the sum of various chemical contributions Xuo, among which is the diamagnetic contribution mentioned above, and contributions arising from surrounding medium influences Za' accordil~gto n = zvo
+ Xo'
(3)
The terms comprising the sums in eqn. (3) may carry either positive or negative signs, depending upon the origin of the magnetic effect. A positive contribution to u sometimes is said to contribute in a diamagnetic sense (shield) and a negative contribution in a paramagnetic sense (deshield). The task of characterizing the various contributions to n for nuclei in complex molecules is extremely difficult. Values of u for protons range from zero for H+ to around 30 X lo-' (30 ppm) for protons in molecules. Nuclear shieldmg effectively reduces the Zeeman splitting by y6Hau as shown in Figure 1 and consequently, A73, the nuclear spin transition energy, becomes smaller the larger the shielding. For the shielded proton, hue # hHo (1 - a) and our bareproton conditions vo = 60 Mc/sec and Ho = 14,091 gauss, will not effect the proton spin transition. Proton resonance conditions
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Figure 1. The proton Zeemon stater at ,totic uo for: (A) H+ in zero field, 181 Ht in %totic H , IC) Y-H in static Ha-nonresonmse condition, and (Dl Y-H in static IH. f Hod-resonance condition.
8 When two states involved in a transition become equally populated, they are said to be saturated. Saturation may result from ineffective nonradiative transition.(spin-lattice relaxation) and/or an applied radio frequency, v, power (amplitude) that is too high. 'This is not so for all nuclei. The fluorine nuclei in F., for example, are deshielded relative to the bare nucleus due to the dominance of an indnced paramagnetic field arking from the nonbonded electrons on fluorine. Magnetically isolated spin systems are ones where intrssystem J i i Z 0 and intersystem Jii = 0. For example,
may be met by reducing vo by voa (frequency sweep) while keeping the applied field fixed a t Ho giving huo (1 - u) = -&Ho(l - u) or by keeping the rf at vo and increasing the applied field by Hou (field sweep) giving H H ] . The two techniques for hvo = y6[Ha(l - u) effecting nuclear magnetic resonance, i.e., field and frequency sweeps, lead to indistinguishable spectra. Figure 2 shows a hypothetical proton magnetic resonance spectrum of two magnetically nonequivalent protons that are magnetically i s ~ l a t e d . ~By convention, the spectrum is displayed with magnetic field increasing (or frequency decreasing) from left to right. Also, it is customary to label the nucleus exhibiting resonances a t lowest field (highest frequency) with the letter A and progress alphabetically to nuclei exhibiting resonances at higher fields (lower frequencies).
contains two magnetically isolated spin systems, one of the AB type and the other of the AA'BB'CC'DD' type, as the nuclear spin coupling between the olefinic and the ring nuclei are likely to he close to zero. If all Jii = 0 for the aliphatic nuclei, the molecule would contain three Isolated spin systems: two equivei lent ABCD systems and one AB system.
Figwe 2. hydrogent.
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Hypothetical nmr spectrum of two magnetically isolated
The difference between the resouant fields or frequencies of two nuclei A and B is called the chemicul shift between the nuclei. Referring to Figure 2, Chemical shift (A,B) = H s Chemical shift (A,B) =
VA
- HA = H d a s -
ve = udoa
- CA) = Ho6m
- a d = vodre
(4)
As mnr spectra are most easily calibrated in units of frequency, chemical shifts are generally given as vfiAa."n order to minimize the number of symbols used in subsequent expressions, the equality mill apply. The value of va required for the nmr experiment must be cited when reporting chemical shifts as ~ 5 , ~This ~ . is because resonant frequency separations (us-uj) are directly proportional to vo. Chemical shifts frequently are reported as 6 , where From eyn. ( G ) it is seen that 6,, is dimensionless and independent of uo. As vo is in the Mc/sec (lo6 cps) range for nmr experiments, 6
J ) , they are perturbed to the extent that first-order analyses of their spectra may not afford accurate parameters. When two states mix, the resulting perturbed states have an increased energy separation relative to that in the unperturbed states. I n general, the high-lying state increases in energy and the low-lying state decreases in energy as a result of mixing. The secondorder perturbation treatment overestimates the effect of mixing, particularly when the perturhation terms exceed 1cps. The second-order perturbation treatment drastically exaggerates the effect of mixing as (HkkH,,) 0. Under these conditions eqn. (21) says that
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14 When two states k and n become degenerate, application of mixing rule 4 in Table 3 gives AEt. = 1(H., H d - (H,, Hb,)l = 2 L . Solution of the quadratic equation resulting from expansion of the second-order determinant, rn in eqn. (Is),leads to the exact energies of the mixing states and subsequently to the exact perturbation.
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SECOND ORDER PERTURBATION (cpr) Figure 4. Correction curves for second-order perturbotion terms, P, of the type Jk/146$ =t .). These curves opply when H u (k # I ) = 1id2.
..
corrections should he subtracted from the absolute magnitude of the second-order perturbation, P(i.e., 2nd-order perturbation: -correction). Otherwise, corrected perturbation terms, I",should be substituted for the uncorrected ones, P, in the forthcoming tables using eqn. (25). TransitionIntensities. I n deriving mixing coefficients Cbn,the first-order assumption was made that CC,, = 1 and C k , > jJ,,I and will lead to accurately calculated relative intensities. However, for more strongly perturbed systems where 6tiol = !J&, C x n > 0 and relative first-order intensities will be at considerable variance with experimental ones. A better than first-order approximation of the relative transition intensities in cases where 0.1 < /Cx,l < 0.3 is to take cognizance that funct ons $, each should contain the normalization factor 1/(1 Z: C?,)"' SO
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Table 3 no longer applies and double quantum transitions become allowed. In an AB system, for example, such a condition leads to the mixing of states 1 and 2, 4 and 2, 1 and 3, and 4 and 3. The extent of mixing, or the perturbation, of these states is proportional to H?. Double quantum transition 4 -+ 1 now becomes allowed. From Table 5, the frequency of this transition is 2hvoq = (El - Ed =
~ " D=Qh ( n Figure 6. Nmr spectrum ( 6 0 Mc/recl of threo.2.3-dibrom-3-phenylpropiophenone, 1, in CDCia(roturated1.
- u)(w -
& a = [(PI VA
J
2C IdI,
= ("A
f m)/2
+ '/r6"
=
- n) =
(pa
- v4)
-
("2
-
(DL
= (n =
IdI.
"8)
=
= ("1
"4)
- vr)/(v* -
"3)
(33) (34) (35) (36) (37)
Figure 6 shows the 60 hfc/sec nmrspectrum of threo-2,3-dihromo-3-phenylpropiophenone, I, in CDCla solvent. The two aliphatic hydrogens are magnetically isolated, for all practical purposes, from the aromatic hydrogens and conseqnently constitute an AB spin system. The four transitions arising from this system are seen b e tween T = 4 and 4.5 ppm. The resonances arising from the two magnetically isolated five-spin phenyl systems appear between T = 2 and 3 ppm. HB Br
I
I
I
I
CsHsC-C-COC6H6
(Threo)
The AB spectrum in Figure 7A can he analyzed using eqns. ( 5 ) , (6), and (33-37) and the indicated frequencies. The calcula.ted parameters are 6"n = 9.9 CPS, T A = 4.180, m = 4.34 ppm, JAB = 10.3 cps, and Iz/Il = 13/11 = 6.1. Assignment of HAand HBto I cannot he made rigorously from Figures 6 and 7. However, transitions 3 and 4 are slightly less intense than transitions 1 and 2, respectively. As benzylic protons are lcnown (10) to couple with the COHEprotons ( J generally is less than 1 cps), their resonances in I are expected to be somewhat broader, and as a result less intense, than those arising from the proton on the carbon adjacent to the carhonyl function. Consequently, the B transition (3 and 4) are assigned to the benzylic proton in I. Figures 7B and 7C show the single (AM = 1) and double (AM = 2) quantum transitions for the AB system in I. The observation of a double quantum transition (AM = 2) resulting from absorption of two quanta (2hv) seemingly violates selection rule1 of Table 3. However, this selection rule applies only to spin systems that are unperturbed by an external perturbation such as the rf field, HI, employed to induce normal single quantum transitions or a second irradie tion field, H,, employed in double resonance experiments. When the strength of the rf field, HI, is sufficiently high to begin to saturate" the states involved in normal single quantum transitions, selection rule 1 of 1'
See footnote 3.
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PDQ
=
+
- un)
v,)/2
(YA YB)/~
and its approximate intensity (11) is given by
Equation (38) tells us that for a given value of HI, IDQ increases as the AB system becomes more tightly coupled (i.e., as J / ~ "increases) I and is zero in the A S limit. For a given value of J2/(6")4,IDQ is proportional to HIa, however as H1 is increased the AB states become increasingly saturated and all single and multiple quantum transition intensities vanish. Consequently, in practice IDQ passes through a maximum as H, is increased. Multiple quantum transitions often may aid the analyses of complex spectra. Their use to this end will be discussed in Parts I1 and 111. However, ,ls evidenced by Figure 7C, they may also complicate the interpretation of spectra unless properly characterized. Particularly for tightly coupled spin systems, care must be taken to use as low a rf power (HI) as possible to obtain a decent spectrum. Otherwise, one shonld be
Figure7. Nmr spectra ( 6 0 Mc/res) of the two oiiphdic hydrogensinthreo 2,3-dibromo-3-phenylpropioph~nme, I, in CDCla (roturotedl. Spectra A. 8, and C were determined using rf amplitudes 0.007, 0.05, ond 0.1 mg, respectively.
on guard for appearance of multiple quantum tmnsitions. Whether an analysis of a complex spectrum is acceptable or not will depend often on the agreement between the observed and calculated relative transition intensities. A tightly coupled spin system always mill lead to a spectrum containing both strong and weak transitions. States connecting strong transitions are saturated more readily than those connecting weak transit i o n ~ . ' ~If too high an rf power is used to determine the spectrum, the resulting intensity ratios I(strong)/I(weak) will be lower than the corresponding ones calculated from n correct analysis (which assumes no saturation). I t is desirable, therefore, to use rf power levels below which no change in the experimentally determined relative transition intensities is encountered. Figure 7 illustrates the effect of the rf power level on the relative transition intensities of the AB spectrum of I. The calculated value of 12/11 or 13/11 is 6.1. The observed values of these ratios with rf = 0.007, 0.031 and 0.1 mg are 5.9, '7.3, and 1.5, respectively. Suggested Readings POPLE,J . A., SCHNEIDER, W. G., AND BERNSTEIN, H. J., "Highresolution Nuclear Magnetic Resonance," McGraw Hill Book Co., Xew York, 1959, Ch. 6. CONROY, H., "Advances in Organic Chemistry," Vol. 11, Interscience (a, division of John Wiley & Sons, Inc.), Xew 1-ork, 1960, p. 291ff. la Under resonanre conditions for transitions from state 1, the ground state, to states s, the excess number of nuclei in state 1 is n,. As nt- 0, st&- t m d s becomes equally populated (sat,urattted) and the net ahsorption of energy and the t s transition intensity vanish (1s). Consider eqn. (i)
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Where B is a. factor containing the nuclear gyromngnetic ratio, line shape function, and spin-lattice relaxation time; HI is the applied radio frequency field; and I,+, is the relative intensity of s transition as given by eqn. (24). To a first approximathe t
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those connecting transitions of high &tensity. Loss of transition intensity due to saturdion, therefore, is more pranaunced for intense transitions than for weak transitions.
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EMSLEY,j.w., FEENEY,J., AND SUTCLIFFE, L.'H.,
"High ~ e s o ~ u tion Nuclear Magnetic Resonance Spectroscopy," Val. I, Pergamon Press, New York, 1965, Ch. 8. DI~CHLER, B., Angew. Chem., 78, 653 (1966). CORIO,P. L., "Stmcture of High-Resolution NMR Spectra," Academic Press, New York, 1967. A , AND MCLACHLAN, A. D., "Introduction to MagCARRINOTON, netic Resonance," Harper and Row, Publishers, New York, 1967. D. W., (Editor) "Nuclear Magnetic Resonance for M.ATHIESON, Organic Chemists," Academic Press, New York, 1967. HECHT,H. G., "3Iagnetio Resonance Spectroscopy," John Wiley 61 Sons, Inc., New York, 1967, Ch. 4.
Literature Cited (1) BOTHNER-BY, A. A,, Advances in Magnetic Resonance, 1, (19651 n. 195. ~G AND OBOTHNER-BY, , A. A,, J . Chem. Phw., (2) C ~ W E L S., 41, 3863 (1964); SWALEN, J . D., A N D REILLY,C. A., J. Chm. Phys., 37, 21 (1962); SWALEN, J . D., ' i P r ~ g r in e~~ Nuclear Magnetic Resonance Spectroscopy," (Editors: EMSLEY,J. W., FEENEY,J., A N D SUTCLIFFE,L. H.), Pergamon Press, New York, 1966, Vol. 1, Ch. 3. (3) SLIGHTER, C. P., "Principles of Magnetic Resonance," Hamer and Row Publishers. New York. 1963. n. 184. (4) STREITWIEGER, A,, JR., "~Olec111ar Orbital yheory for Organic Chemists," John Wiley & Sons, Inc., New York, 1961, p. 33K. (5) MARGENAU, H., A N D MURPHY, G. M., "The Mathematics of Physics and Chemistry," D. Van Nostrand Co., Inc.. Princeton, New Jersey, 1943, p..,313R. (6) BRUGEL.W., ANKEL.T., A N D KRUCKERERO, . F., . Z . Eleclrochm.,64,1121 (1960): (7) PAULISO, L., AXD WILSON, E. B., 'Tntmduetion to Quantum Mechanics." hleGraw-Hill Book Co.. h e . . New York. 19%; and$ANDOiFY, C., "ElectronicSpeetrs. a n d ~ u a n t u i C h e m i s t ~ y Prentice-Hall, ,~ Inc., New Jersey, 1964. H. J., (8) (a) POPLE,J . A., RCHNEIDER, W. G., AND BERNSTEIN, "High-resolution Nnclear Magnetic Resonance," MeGraw-Hill Book Co., 1959, p. 151K; (b) EMSLEY,J. w., FEENEY,J., AND SUTCLIFPE,L. H., ''High Resolution Nuclear .Magnetic Resonance Spectroscopy," Val. I, Pergamon Press, New York, 1965, p. 428ff. (9) Reference (5), P. 49. S., Tetrahed~onLetters, (10) ROTTENDORF, S., A N D STERNAELL, (1963); Bust. J . Chem., 16, 1030 (1963). W. A,, FREEMAN, R., AND REILLY,C . A., J. (11) ANDERSON, Chm. Phys., 39, 1518 (1963); CORIO,P. L., "Structure of High-Resolution NMR Spectra," Academic Press, Ner York, 1967, Ch. 9. (12) See reference (8a) and (8h) p. 3013' and p. 33ff, respect,ively. ~
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