P. 1. Corio
and R. C. Hirst Mobil Research and Development Corporation Princeton, New Jersey 08540
Magnetic Resonance Spectra of Multispin Systems
T h e applicatiou of high resolution nuclear magnctic resonance spectroscopy to chemical problems frequently requires detailed analyses of the observed spectra for descriptive sets of chemical shifts arid spin-spin couplings. Although the principles of spectral analysis have been thoroughly investigated (1, 8 ) , and many interesting properties of spin-spin systems difclosed, some very useful general coucepts have not B$bn discussed inexpository articles designed for studel& and practicing chemists. The purpose of this paper is to present an elementary description of the theory of irreducible components ( 1 ) of multispin systems and several illustrative examples. The significance of the concept of irreducible compouents is that it reveals the manner in which complex spect,ra arc compounded from the spectra generated by simpler struct~uixlelemeuts; its practical importance stems from the fact that prior lruowledge of these simpler struct,ural elemeuds can often be used to obtain complete or partial analyses of more complex systems. ;\lat,hematical proofs and extensions of the theory t,o systems not specifically considered are given elsewhere ( 1 ) . It will be assumed, however, that the reader is familiar with the elemcnts of high rcsolution nuclear magnetic resonance, the notion of a group of magnetically equivalent nuclei, and the properties of the AB and AIB systems sketched in Figures 1 and 2. This prcrequisitc material has been discussed in several recent publicatio~;n(5).
+
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nuclei A, B,
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The Addition of Angular Momenta
Co~isidera system composed of N spin that admits a decomposition into groups G
C, . . . , of magnetically equivalent nuclei, the number of nuclei in group G being no. The complete spin system will be denoted A,B,,,C., . . .,where the choice of adjacent letters of the alphabet for the several groups indicates, according to the usual convention, that no assumptions are made concerning the relative magnitudes of the internal chemical shifts and spinspin coupling constants. The assumption that the nuclei in group G are magnetically equivalent is logically equivalent to the assertion that the square of the total spiu angular momentum of group G is conserved. This implies that the total spin quantum uumbers describing the angular momenta of the groups do not change with time. For suppose that the angular momentum of group G is known to be described by the total spin quantum number IG a t some time 1, and assume that the total spin quantum number is lo' a t time 1' > 1. Conservation of the square of the total spin angular 1) = momentum of group G requires that Ic(Io Ior(Ic' I), or (Io - To1)(I0 Io' 1) = 0. The lo1 1 = 0 is extraneous, as shown by solution I. the following argument. Thc cperator associated with the square of the angular momentum of group G is ZG2 = ICx2 I G ~ 'where , the hermitian operators IGxt IGy, IGa are the cart.esian components of the angular momentum vector I,. Now the eigenvalues of an operator expressible as a sum of squares of hermitian operators are necessarily nonnegative, so that the eigenvalues of I,2 - IG=', namely Io(Io 1) - mo2, where mo = -Io, - l o 1 , . . . , I o - 1, lo, must satisfy the inequality IG(IG 1) - mo2 2 0 . In particular, when mG = lo,it follows that IQ 0. Thus I. = lo', as asserted. The total spin quantum numbers for group G may be obtained by making use of the fact that the addition of two arbitrary spin angular momenta with spin quantum numbers II and 1%)such that II 12,yields the following sequence of 212 1 total spin quantum numbers for the composite system
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spectrum.
Figure 1.
Itopl Calculated AB
Figvre 2.
(bottom) Calcvl~tedA 8 spectrum.
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I
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1
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2
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(1)
I
These quantum numbers are all distinct, a circumstance that is described by saying that their spiu multiplicities are unity. The addition of several angular momenta often yields total spiu quantum numbers whose multiplicities exceed unity. The multiplicity of a spin quantum number I will be denoted g,. To illustrate the use of eqn. (I), consider the addition of two angular momenta with spin The required I2 = I / , '/, = total spin quantum numbers are It 1, and I, - 12 = ' / 2 - l / 2 = 0,with spin multiplicities g, = g o = 1.
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Volume 46, Number 6, June 1969
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345
particles may he worked The case of three spin angular momentum to out by adding the third spin the results obtained for two spin '/z particles. Since to obtain Il 2 12, one first takes I , = 1 and I2 = 1 = 3/2, and 1 = For the addition of it is necessary to the spin quantum numbers 0 and and I2 = 0. In this case, the addition take Il = 0= yields only one total spin quantum number: appears Since the total spin quantum number and 1, and again in the once in the addition of addition of and 0, its multiplicity is 2. Thus the addition of three spin ' / z angular momenta yields the with spin multitotal spin quantum numbers 3/2, plicities 9.1, = 1, g,/, = 2. The total spin quantum numbers obtained through the addition of n angular momenta with spin '/%are given by (4)
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1 I=-n-k 2
where k
0,1,2,
1 . . . .,-n, if n is an even integer 2
0,1,2,
1 . . .,-(n 2
=
- I), if n is an odd integer
product of the multiplicities of the total spin quantum numbers generating that component. A system in which all n~ are unity is obviously irreducible. In this case, eqn. (4) yields identities such as AB = Al/,BL/,,ABC = Al/,BI/,CL/,,etc. On the other hand, any multispin system with at least one nc > 1 is reducible. The following equations show the reduction of the A,B, AaB, and A4B systems into their irreducible components A,B = AIB~/,+ AoB~/, (5) AaB = As/,Bt/, ~AI/,BI/, (6) AIB = A2Bx11 ~ A L B I / ~ 2AoB5/1 (7) Equation (5) states that the A2B system may be decomposed into two irreducible components. The AoBl/, component generates a single resonance a t the uncoupled Larmor frequency of nucleus B (i.e., v ~ ) The remaining eight lines of the A2B spectrum are generated by transitions of the AIB~/,component. The reduction of the A,B system into its irreducible components is sketched in Figure 3. The mixed or combination transition, whose intensity is very much less than the intensities of all other transitions, is not shown in the figure.
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The multiplicity of a given value of I is (4)
where
Irreducible Components
An irreducible component of the A,B..C., ... system is defined by assigning to each group G a fixed value of the total spin quantum number IG consistent with eqn. (2). A generic irreducible component will be denoted A,ABl,,C1c. . . Each irreducible component is given a weight equal to the product of the spin multiplicities, that is, g,g,&~, . . . . Since the total spin angular momentum of each group is conserved, irreducible components may be treated as independent entities; hence the decomposition of a multispin system into its irreducible components may be symbolically written
(Cl
COMPOSITE
V,
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This decomposition does not entail any special assumptions concerning the time dependence of the applied fields, so that eqn. (4) may also be used in the interpretation of multiple resonance and spin echo experiments. The following discussion will, however, he limited to spectra generated by the single quantum transitions induced by a "weak" rf-field at right angles to the polarizing z-field. For this case, eqn. (4) states . . . system is that the spectrum of the A,B,.C,, obtained by superposing the spectra of its irreducible components. The intensity of each resonance of a given irreducible component must be multiplied by the 346
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Journal of Chemical Education
Figure 3.
Reduction of the A2B system into irreducible components.
Equation (6) shows that the AaB system may also be reduced into two irreducible components. The A,/,B,/, component is just the well-known AB system, so that magnitudes of JABand va - vs may be directly obtained from this component, provided, of course, that the spectrum of the A,/,B,/, component can be identified in the experimental trace. Equation (7) illustrates the decomposition of the A,B system into its irreducible components. The decomposition clearly shows that it is incorrect to assert that the A,B system contains the A,B spectrum. The correct statement is that the AIB system contains differently weighted irreducible components of the A2B system. The foregoing remarks illustrate the practical utility of the theory of irreducible components in the analysis of spectra. Indeed, it is often possible, as illustrated in the following section, to obtain the desired chemical
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shifts and coupling constants without solving secular equations whose degrees exceed 2. It should not be concluded, however, that a given irreducible component contains all the parameters defining the composite system. Consider, for example, a system where all nc are even integers. Such a system always contains the particular irreducible component AoBoCo. . . , in which the total spins of each group are zero. But if the total angular momentum of each group vanishes, the associated (total) magnetic moment of each group also vanishes; hence, the AoBnCo. . . component does not interact with external or internal magnetic fields. I n other words, the AoBoCo.. . component generates no spectrum a t all. However, the usefulness of irreducible components in which one or more, but not all, groups are described by total spin quantum numbers Ic = 0 should not be underestimated. For since such groups are not coupled to those groups G" for lo' # 0, there is an effective reduction in the number of coupled groups, and the analysis of such irreducible components is correspondingly simplified. Consider the three group system AzBC = AIBB/,CZ/, A,B,/,C,/,. Since group A does not couple to B or C in the AoBt/,Cc/, component, this component is effectively a BC system whose identification in the observed spectrum permits the immediate determination of lJ19cl and luu - ucl. From eqn. (4), it follows that the total intensity of the A.,B,,C,, . . . system is
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Illustrative Examples
The following examples are designed to illustrate the theory of irreducible components. All spcctra were obtained with a Varian A-60 spectromctcr operating at a fixed frcqucncy of GO RIHz (1 Hz = 1 cps), and a constant but unmonitored temperature in the range 2540°C. Since the fine structurc of each system studied was the principal point of interest, no attempt was made to relate the observed chemical shifts to a common reference standard. Thc frcqucncy origin in each displayed spectrum coincides with the uncoupled Larmor frequcncy of ouc of the groups, and the magnetic field increases linearly from left to right. Complete or partial analyses were carried out by assigning some of the lines according to the priuciples outlined in the preceding section. The parameters thus obtained were introduced into a computer program which calculated all transition frequencies and intensities. The program used was that of Cast,ellano and Bothner-By ( 6 ) , modified1 to provide magnetic equivalence factoring and a separatc printout aud plot of each irreducible component. (1) Alelhyl Mereaptan, A& The 60 MNx proton spectrlm of methyl mercaptan, C H I S - H , is shown in Figure 4(a). Figure 4(d) shows the calcdated composite spectrum, and attempts to simulate a real spectrum by using a gaursiw line shape ior each resonance.
It can be shown (5) that the sum of the intensities generated by transitions between states of a generic irreducible component is
If the irreducible component appears grAgrBgro . . . times in the reduction of the composite system, its contribution to the total intensity is obtained by multiplying eqn. (9) by the product gl,gr.grc. . . , as indicated in eqn. (8). Combining eqns. (8) and (9), one obtains (5) InlAnABnnCnc . . . = 2N-'N (10) where
... (11) To illustrate these results, consider again the A,B, AaB, and A4Bsystems. From eqn. (9), one obtains N=n*+na+nc+
AIB: I ~ ~ A I B I = / , 11
IntAoB>/% = 1
A;B: IntAz/xBt/,
IntAx/;Bs/, = 4
=
24
A,B: I ~ ~ A I B I = / , 45
inlA,B~/,
=
11 intAoB~/.= 1
These results, together with eqns. (5)-(7), show that InlAxB = IntA,Bx/,
+ IntAoBx/, = 12 = 2'.3
+ = 1ntAzB~/,+ 3lntA,B1/,+ 21ntAaB~/.
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The A~/,B~/,component may be analyzed as an "ordinary AB" spectrum to give the conpling constant, JAB,and the internal chemical shift v~ - DR. The experimental valoes are2 PA vB = 30.01 Hz and JAR = 7.43 Ha.$ ( 2 ) Isobutyronilrile, A&. The 60 MHz 'I1 spectrum of isabutyronitrile, Figure %a), has been analyzed ss an example of an A6Bspectrum. The rednction of the AeB system into its irreducible components is given by
A6B
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=
Z4.5
in agreement with eqn. (10). The importance of the intensity formulas in eqns. (8), (9), and (10) is that they permit one to estimate in advance the fractional contribution to the total intensity of an arbitrary irreducible component.
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A J B I / , ijA2B~/? OAIBII~ SAoB~/l
The irreducible components are sketched in Figures . X b ) 3 ( e ) . Figwe 5(f) shows the calculated composite spectrum with a linewidth of 0.2 11%. -~
IntAIB = IntAs,nB~/~ 2IntA1/,Bx/, = 32 = Za.4 IntA'B
Figure 4. Experimental and theoretical proton spectra of methyl mercapton at 60 MHz: (01 experimental trace, (bl the A I / ~ B component, ~/~ (4 the AI/.BI/,component, id1 colculoted compo4te spectrum.
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' T h e modification was int,rodueed by 111.. L. G. Alexakas (Mnbil i h e a r c h and llevelopment. Ccxporstion, Paulsbara Laboratory, Paulsboro, New Jersey). 2Speet.ra of the type A.,B,,, are independent of the sign of J A ~which . is arbitrarilv assumed t o be oositive. A more detailed disms.iion of the propel.liei of the A,,R,,, system is given in Chapter 6 of 13efel.ence (I). a Experimental et.l.
