I Edgar W. Garbisch, Jr.
University of Minnesota Minneapolis, 55455
I
Analysis of Complex NMR Spectra for the Organic Chemist 11. Three spin systems of the ABC, ABX, ABK, and AB, types
A
general perturhation approach for deriving qualitative eigenvalues and eigenfunctions for proton spin systems was developed in Part I of this article." The approach was applied t o the simple two spin system. However, there are only two states having the same M value that mix for this system and the use of eqns. (25), (31), and (32) was shown to lead to exact eigenvalues and eigenfunctions and ultimately to exact expressions for the transition frequencies, relative transition intensities, IJABI, and 6;~. Any spin system having no more than two states of the same M value that mix may he solved exactly in a similar way. Otherwise, the perturbation approach leads t o approximate solutions. The general three spin system has up to three states of the same M value that mix. I n this limit the system would he classified as an ABC type. These systems are quite common and approximate analyses of their spectra, including the relative signs of the three couplings, often may be accomplished by the perturbation approach. An ABX system has only two states of the same M value that mix. Legitimate ABX proton systems are encountered occasionally and their spectra can he analyzed completely, except for the sign of J A B relative to JAX and Jsx. ABIi systems are encountered a t least as often as ABX ones. An ABIi system is defined as one that approaches the ABX limit but yet may not be treated accurately as such because of weak mixing of a third state with two that strongly mix. I t may he accurately treated as a perturbed ABX system and under favorable circumstances its spectrum can be solved completely, including the relative signs of all three couplings. A system of three spins two of which are equivalent by symmetry is classified as an A2B or AB2 type, depending upon whether the equivalent spins resonate at lowest or highest field, respectively. These systems have only two states of the same M value that mix. Their spectra are readily analyzed to ~ . spin systems give exact values of S i n and IJ A ~Three having no states that mix are of the AlVIX type. They are rare for protons, but when encountered their spectral analyses may be accomplished following first-order rules. Spectra resulting from the three spin systems disJ. CHEM.EDUC.,45, 311 (1968). The numbers of figures, tables, structures, equations, footnotes, and literature cited in this paper follow consecutively those in Part I. 402
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Journal of Chemical Education
cussed above may be systematically analyzed by the perturhation approach. The results will be approximate when more than two states having the same M value mix. Rules for and examples of the analyses of spectra arising from ABC, ABX, ABIi, and AB2 (A2B) proton spin systems will be discussed in this paper. Part I11 will discuss four spin systems. Three Spin Systems
A nuclear spiu system of three hydrogens has z3 states and a maximum of 15 single quantum (AM = I), 6 double quantum (AM = 2), and 1 triple quantum (AM = 3) transitions. Under conditions of low rf power only single quantum transition normally will be observed. States
No. of Transitions
15 Total
Designating the three hydrogens A, B, and C and taking VA > vB > V C , the second-order energies for the eight states may he written using eqn. (21). These energies are given in Table 7. The second-order transition energies and corresponding first-order relative transition intensities (from eqns. (22) and (24)) are listed in Tahle 8. Corrections t o the second-order perturbation terns P i n Table 7 and 8 may be obtained from Figure 4 or corrected perturbation terms P', obtained from eqn. (25), may be directly substituted for terms P. Better than first-order intensities may be obtained after reference to the section on Transition Intensities in Part I. As seen from Tahle 8, the general three spiu system allows 4-A, 4-B, and 4-C fundamental transitions (those arising from the changing of the spiu state of nucleus A, B, or C ) and three combination transitions (those arising from the simultaneous changing of the spin states of the three nuclei). The combination transitions are of low intensity for weakly coupled systems, hut for tightly coupled ones they may occasionally become more intense than the fundamental transitions. The following transition intensity relationship, fre-
Table 7.
n
M
Nuclear Magnetic States for the Three Spin System (vn
"9
ARC:
A t h n r d ~ r -
quently called the intensity s u m rule, holds exactly for any three spin system (IS) (I,
+ II , + I,,)+ I =+( II, ++I8I ++114 = I +I +I +I +I
(39)
Equation (39) holds for the first-order intensity expressions given in Table 8. The intensity relationship given by eqn. (40) for the fundamental transitions is only approximate, but nevertheless will be useful in making line assignments.
Repeated spacing relationships (14) exist for all three spin spectra and are invaluable for extracting the 4-A, 4-B. and 4-C fundamental transitions. The reueated spacing relationships given in eqn. (41) apply in general. Here, the numbers correspond to those for the transitions in Table 8. Equation (41) shows that each group of 4-i transitions contains two groups of two repeated spacings and that each of these groups appears again in one of the two remaining groups of 4-i transitions.
A
2nd-order energy -1st-order
-
-
that addition of the expressions corresponding to the three repeated spacings leads to the cancellation of all perturbation terns to give the sum of the three exact couplings. Equation (42) applies to all systems of three nonequivalent nuclei.
ABC System
When confronted with the task of analyzing an nmr spectrum arising from three nonequivalent and nonmagnetically isolated hydrogens, a first-order analysis of the spectrum should be obtained. In doing this, a correct set of 414, 4-B, and 4-C transitions and correct values of J$) should result after application of eqns. (40) and (41). A line assignment (correlation of each observed transition with a transition listed in Table 8) should then be made by applying eqn. (39). In using the expressions in Table 8 for making a line assignment all perturbation t e r m should be neglected. If an unambiguous line assignment cannot be made, assume one
-
Anv ~ermissiblesnectral assignment must exhibit the set of repeated spacing given in eqn. (41).2Q It must be remembered, however, that the three groups 4 4 transitions may not appear as three quartets. For example if (q - v3) = 0, 4-A and 4-B appear as doublets with spacings (n - vp) and (v5 - %),respectively and 4-C appears as a quartet with spacings (u, - v z ) and (va - us). If ( n - va) = ( n - u J , 4-A appears as a triplet with spacing (u, - v3),4-B appears as a quartet with spacings (v1 - v3)and (va - us), and 4-C appears as a quartet with spacings (u, - v 3 and ( v s - ue). Also, one or more of the transitions in the 4-i groups may have negligible intensity. If this is the case it will be found that a complete set of repeated spacings is found, i.e., eqn. (41) is satisfied, only if one or more "phantom transitions" are included. Each of the repeated spacings in eqn. (41) represents a first-order coupling J#). From Table 7 it is seen "
> UB > uc)
by taking the expected set of relative signs for the first-order couplings. First-order UA"), V B ( ' ) , and vc(') are taken as the mean frequencies of the 4-A, 4-B, and 4-C resonances, respectively. Using the first-order parameters, calculate values for the perturbation terms M,, N* and 0, in Table 7. If three or more of these perturbations are significant (i.e., 3 0.1 cps) the specThere are 15 independent sets of repeated spacings for all three spin spectra. For each set, the three couplings can be assumed to have the same sign, JABof opposite sign from those of JAO and Jac,Jnc of opposite sign from those of JABand JBC,or JBO of opposite sign from those of JABand JAC. Consequently, there are 60 independent line assignments. Each line assignment leads to a unique solution, so there are 60 solutions for a given spectrum (14). Generally, the correct set of repeated spacings and the correct line assignment can be chosen after application of eqns. (39) and (40). Volume 45, Number 6, June 1968
/
403
Table 8.
Transition
No.
1
4-1 6-2 7 +3 5 8 3-1 5 2 744 8 46 7-2 6-3 2-1 5 +3 6-4 8 +7 5 4
-
Transition Energies ond Relative Transition Intensities for the Three Spin System
Nucleus
A
A A
v* VA
PA
A
PA
B
VB
B B B (Comb.) (Comb.) C C C C (Comb.)
l s h r d e r intensity
2nd-order energy
vs re YB
+ '/r(Jna + J A C )+ Oi + N + + ' / n ( J m - J A G )+ M + - M - + N+ + O+ '/*(JAG- J A B )- M + + M - + O+ + N - '/*(JAB+ J A O )f
+
+ N-
0-
f ' / ) ( J A B J B C )f M + - 0+
+ ' / % ( J A -B J B O )+ N + - N - + M + - 0+ l / ~ ( -J J~A B~ )+ N- - N + + M - - O+ - JAB + J B C )f M - - 0pbs - P C + M + + M - + N + +
Nvc+6~~-M--M++O+f0'/dJac J A O )- M + - N + vc vc ' / l ( J ~c Jac) O+ - 0- - M + - N vo ' / n ( J ~ o- JAO) 0- - 0+- M- - N + vo - '[a(Jec + J A G ) - M - - N v o - SA, - O+ - 0- - N + - NPA+
+
+
+
+
(1
(1 (1
+ +
- G + - E+Y
+ F+F- + E+F- - F+G- - G - + E,)'
+ F+F- + F+E- - F-G+ + G + - E-)a +
(1 G _ + E _ ) ' ( 1 - F+ G+)' E+GE+EF+ G-)I E+F_ G + F - - G + - F_)a (1 - G F-)' E+ - E - - F a - - E+F_)a G+ - G F+GG+F_)' (1 F+ E+I2 G+EG + G - - F+ E-)a G + G _ - G + F _ - E+ F-)' (1 - E W - F_)2 G - - G + - E + G - - G+E-)'
+
+ F+E- + + + + + E+E- + + + (F+ - F - + ( F - - F+ + + + + + (1 - F+G- + + + + (1 + E+G- + ( E - - E+ + (1 (1
trum may be treated as one arising from either an ABC or ABK system. Although the ABK analysis (see later) normally leads to a set of approximate parameters that corresponds most closely to the exact ones, it has the disadvantage of being more time consuming than the approximate ABC analysis. The approximate ABC analysis may be performed following the steps outline below. Step 1. Obtain a permissible set of repeated spacings using eqns. (40) and (41). Correlate hydrogens A, B, and C with the hydrogens on the molecule using firstorder couplings and chemical shifts.
correct line assignment using eqn. (39) if two or more transitions significantly overlap. Step 3. Using first-order parameters JSj(') and calculate values for MI', N*', and 0+'using the uncorrected perturbations P and Figure 4 or the expressions for the corrected perturbations P' given in Table 7. It is essential that the corrected perturbs tions P' he used. Otherwise, the following iteration procedure may not converge. 6,(') = ( v p - v j ( ' ) ) where v,(') is the first-order frequency for the ith nucleus and is given as the mean frequency of the group of 4 4 fundamental transitions.
Step 8. Make a line assignment using the expected set of relative signs for the couplings and the transition frequency expressions (neglecting perturbation terms) given in Table 8. Check this assignment using eqn. (39). Make three new line assignments using the three remaining sets of relative signs for the couplings.21 Check these new assignments using eqn. (39). If all checks are satisfactory (within 10%) it is likely that the spectral analysis will not distinguish the correct set of relative signs of the couplings. In this event, use the first assignment. If only one of the lme assignments gives a satisfactory check, use that assignment. If no line assignment gives a satisfactory check, use the first assignment. It is not always an easy matter to obtain reliible relative transition intensities (see Part I). If two transitions overlap significantly (Av,, < 1 cps) their relative intensities may be dependent upon the sweep direction if the spectrometer is not perfectly tuned, and their intensities relative to those of nonoverlapping transitions mormally will be too high. Consequently, it is diicult to obtain a satisfactory check of a
Step 4. Calculate corrected values for JAB,JAG,and JBC using eqn. (43), J#), and the values of P' obtained in step 3. Use these corrected couplings, the experimentally measured values for (vr - vs) and (a na), and the values of P' obtained in step 3 to obtain corrected values of 6:j from eqn. (43). This is itera-
21
See footnote 18 Part I .
404
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Journal of Chemical Education
Step 6. Using the correctedvalues for J l j and 6:, ohtained in step 4 calculate new and better perturbation terms M,', N*', and O*'. Step 6. Using eqn. (43) and the improved perturbations P' obtained in step 5 calculate still better corrected values for J , and then as was done in step 4. This is iteration-1. Step 7. Reiterate until the values for the corrected do not change significantly upon further jlj and iteration. ~t this pointthe problem has converged toa line assigment adopted second-order solution for in step 2, ~h~ of iterations required for convergence is an indication of how closely the firsborder solution corresponds to the second-order one. For
moderately tightly coupled systems, two to four iterations are sufficient. Step 8. Using Table 8 and the values for Jij, M*', N,', and 0,' that result in step 7 calculate the spectrum and compare it with that observed. The firsborder intensities should not be expected to agree well with those observed when more than one iteration is required. The best calculated relative intensities will always result by using coefficients obtained from eqn. (30) (see the section on Transition Intensities in Part I). The solution resulting from step 7 always will lead to a frequency fit as the iteration procedure effectively forces this. Whether or not the solution is acceptable will depend upon the agreement between the observed and calculated relative intensities. If the agreement is poor, or if there is any doubt about the correctness of the line assignment adopted in step 2, second-order solutions for the three remaining line assignments should be calculated. The solution that gives the best agreement between the observed and calculated relative intensities may be taken as the correct one. Step 9. A second and convenient method for confirming the solution resulting from step 7 involves redetermining spectra in different solvents until a spectrum substantially d i e r e n t from the original one, with respect to transition spacings, is obtained.22 The line assignment that leads to essentially the same set of Jij values for both spectral analyses may be concluded to be the correct one. As an illustration, the approximate ABC analysis procedure will be applied to the nmr spectrum of divinylsulfone, 11.
The spectrum of I1 might be expected to resemble one arising from an AA'BB'CC' system. Figures 8 and 9 show the spectra of I1 neat and in carbon tetrachloride, respectively. These are typical spectra of ABC systems and indicate that the set of A, B, and C hydrogens are magnetically isolatedzafrom the equivalent A', B', and C' set. The spectra therefore represent exactly overlapping transitions arising from the two equivalent and magnetically isolated ABC systems. If the two equivalent ABC systems were to weakly interact so that they no longer are magnetically isolated, the transitions in Figures 8 and 9 would be seen to broaden, together with the appearance of new transitions. This situation is observed in the spectrum of I1 in benzene. The groups of 4-A, 4-B, and 4-C transitions in Figures 8 and 9 were characterized through the use of ¶'It is now well recognized that coupling parameters may be significantly (-3~0.3 He) dependent upon solvent and concentration. Generally the effect is minimized when solvents of comparable dielectric constant are used and the concentration of solute held constant. As the dielectric constants of benzene and carbon tetrachloride are comparable (2.26 and 2.20, respectively at 35"C.), these solvents are recommended. See, WATTS,V. S., AND GOLDSTEIN, J. H., J. Chem. Phys., 42, 228 (1965) for a thorough discussion on this subject. See footnote 5 in Part I.
Figure 8. NMR spcstrum (60 Ms/secl of divinylrulphone, I!, (neat). Numbers under the rtisk specbum are the observed resononce frequencies down Reld from TMS.
eqns. (40) and (41). In order to get complete sets of repeated spacings given by eqns. (41), "phantom transitions" labeled No. 8 in both figures had to be included. The relative magnitudes of JJ1' led to labeling the hydrogens as shown in 11. The vincinal couplings JAB and JACmay safely be taken to be of the same sign, normally assumed positive (1). However the sign of may be either positive or negathe geminal coupling JBC tive (1). Consequently, the two probable line assignments are those resulting from taking (a) J A ~ JAC, , and JBCand (b) JAB,JAC, and -JBc. Equation (39) does not clearly lead to the correct assignment-probably because of the several overlapping transitions. Taking assignment (a) (all couplings positive) and using Table 8 leads to the line assignments shown in Figures 8 and 9. Assignment @) (JBC negative) leads to inter-
Figure 9. uroted).
NMR spectrum 160 Mclsecl of divinylsulphono II, in
CC4 (sob
changing transition Nos. 5 and 6, 7 and 8, 11 and 12, and 13 and 14. The approximate ABC analyses of the spectra in Figures 8 and 9 were now completed (beginning with step 3). The results are summarized in Tables 9 and 10. All the data necessary to complete the analyses are given in these tables so that interested readers carry out the calculations independently. Although assignment (a) for the spectra in Figures 8 and 9 assumed all positive couplings, approximate Volume 45, Number
6, June 1968 / 405
Table 9.
Approximate ABC Analysis of the Spectrum of II Given in Figure 8 Assignment a
1storder
Parameter
Ob
1'
ZL
3b
Iterative corn uter anahis.
b
-Assignment 1storder Ob
ABK analysisd
1'
2b
3'
= Given in c s
8.;
Iteration LAOCOON 11. d See later discussion. 'This spacing is used for all iterations. 1 This value should repeat after each iteration, otherwise a. computational error is indicated.
Table 10. Aooroximate ABC Analvsis o f the S~ectrumof II Given in Figure 9
-
lstr order
Ob
16
Zb
3'
4b
JAB J AO Jsc
15.75 8.95 1.4
16.0 9.6 0.5
16.1 10.1 -0.1
16.2 10.5 -0.6
16.2 10.7 -0.85
16.25 10.8 -0.95
, 1storder 15.75 8.95 -1.4
8:"
34.8
31.9
30.8
30.3
30.0
29.9
34.8
Assienment a
Assienment \
sin
.~.
~~~
~~
31.5
30.4
29.6
~
Iteration Go. This spacing is used for all iterations. d This value should repeat after eaoh iteration, otherwise a wmputationd error is indicated.
analyses of these spectra proceeded to give a negative sign to JBc. This illustrates the effect of the perturhation terms in the transition frequency expressions given in Table 8. These perturbation terms are neglected in making line assignments. On the other hand the analyses of these spectra with assignment (h), proceeded to give JBCa negative sign as originally assumed. There was an uncertainty a t the beginning of the analyses as to which assignment was the correct one. Actually, it is unnecessary to calculate relative transition intensities, step 8, to make this decision. The data in Table 9 and 10 clearly show assignment (a) to he correct and assignment (b) incorrect. The analyses of the markedly different spectra in Figures 8 and 9 led to essentially identical J , , values for assignment (a) and to substantially different values of JBCfor assignment (b). The JBC values for the assignment (b) analyses mere in fact diverging so rapidly that only two iterations in the analysis of the spectrum in Figure 9 were necessary to conclude that this assignment was incorrect. Table 9 compares the iterative computer and the ABIC (see later) with the approximate ABC analysis results using the correct line assignment (a). The computer analysis compares favorably with the ABIC and less so with the approximate ABC analyses. Although there are examples where the computer and approximate ABC analyses of spectra arising from moderately tightly coupled three spin systems agree very closely, it is difficult to anticipate whether the agreement will be good or poor. The ABK analysis, which will be discussed later, more uniformly yields sets of parameters that corresponds best with the exact ones. 406
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Journal o f Chemical Educafion
A complete spectral analysis always should include the calculation of the transition intensities (step 8). For the spectra in Figures 8 and 9 this will serve to verify the conclusion made in step 9 that assignment (a) is correct. Figure 8 shows a stick spectrum reflecting the computer solution intensities. The stick spectrum in Figure 9 reflects intensities deriving from coefficients ohtained by solving the secular equations as described in the section on Transition Intensities in Part I. For example, the coefficients in &of the ABC system giving the spectrum in Figure 9 are ohtained from
where Hss - Es
= -0-' Hsa - Es = -6:s Hn - E5 = -6 i o IIm = H. = J d 2 Hm = HT. = Jan12
- N-'
= -5.87
++ ''/n(J*s / z ( J A o- J s o ) - 0-' - N-' - J B O )- 0-' - N-' = 8.12
= -15.20 = -27.17
= 5.40
Going along the first row to obtain cofactors All of the Cknthcolumn according to eqn. (30) kn
Ah
and, h = 0.872 c
AP
+ 0.461 & + 0.615 h.
Ct.
Exact Transition Energies and Relative Transition Intensities for the ABX System
Table 11.
No. 1
2 3 4 5 6 7 8
Nucleus
A A A
A B B B B
Enerw PA
vr vn VA
vs
us va YB
9
(Comb.)
A .
10 11 12 13 14 15
(comb.) X X X X (comb.)
vx vx vx vx ux vx
Intensity
+ '/&La + Jnx) + O+ + JAB - Jnx) + 0 + ~ / ~ J -A Jx I B )+ O+
(Cu (Cm (Cm (Cs (Cw (Css (Ca
- JAB + J A X )+ 0 -
+
JAB
+ J B X )- 0 +
+ ' / , ( J A B- J a x ) - 0 + ' / d J s x - J A B )- 0 +
J B X )- 0 - .x 6;. 0+ 0 'IXJBX J A X ) '/z(Jnx - J s x ) O+ ' / d J ~ x- J A X ) 0 - '/dJsx JAX) - ~ A B- 0-
+
+ + +
us
+
+
+
+ Cad' 0
+ +
+
+ Cd' + C4dg +Cda + Ca)*
+CdP (Css + Csds
- JAB f
+
+ Cad2
(c6&+ C & d 2 - 0- 0+
o+
ABX System
If after applying a firsborder analysis to a three spin spectrum it is found that either lVI, Ni, or Oi is substantial (> 0.1 cps) and the remaining perturbations insignificant (< 0.05 cps), the spectrum may be treated accurately as one arising from an ABX system. The A and B nuclei are those that have the smallest 6, and the third nucleus is labeled X. By convention, PA is a t lower field than uB. To convert the ABC states in Table 7 to those reflecting an ABX system, take nucleus C as being X. The perturbation terms M+ and N, vanish and only states 3 and 4 and 5 and 6 mix. Since only two states having the same M value mix, use of the corrected perturbation terms Oi' leads to exact expressions for the transition energies and application of eqns. (31) and (32) leads to exact expressions for the expressions are relative transition i n t e n s i t i e ~ . ~These ~ given in Table 11. Note that the terms 0 in Table 7 are the same as the 0' terms in Table 7 but with V,,, removed (see later). The analysis of an ABX spectrum gives j J A n l , JAX, JBX and 8iB. Only if the A and B nuclei are moderately be tightly coupled will the relative signs of J Aand~JBx determinable. We see from Table 11 that JAB"' = JAB, i.e., the first-order AB coupling is equal to the exact one, as the 0, terms cancel. Any spectrum giving J,,(')= J,,is insensitive to the sign of this coupling. Consequently, only the absolute magnitude of JAB and not its sign relative to JAX and J D xis derived from the ABX analysis. All terms except unity and G . are seen to vanish in the firs& order intensity expressions given in Table 8. If after referring to the seotion on Transition Intensities in Part I these remaining terms are replaced by the appropriate coefficients,application of eqns. ( 3 1 )and ( 3 2 )leads to the relative intensity expressions ziven . in Table 11. By imposing this restriction we are in effect neglecting to in' / ~ ( J Ax JBX)] terchange 1st-order states 3 and 4 when I&B is negative and to int~rchrtngelskorder states 5 and 6 when - l l l ( J * ~- J B X ) ]is negative (see Table 7 ) . This allows the use of one line assignment for the two solutions of the ABX analysis and consequently minimizes room for error in selecting the correct solution.
= 1 = 1 = 1 = 1 = 1 = 1f = 1 = 1
J*sI2D+ JABI~DJ~aa/2D+ JAB/ZDJ~ae/2D+ Jd2DJAB/~D+ - Jns/2D-
+ + +
-
=
1
0
JiB(a-
- a+)l/(a-* + &)(a+z
+
&B)
1
+ C U C ~=) (~ J ~ +B a - a+)'/(a-2 + &)(a+s + J L ) + C E & X )=~ ( J ~ +B a - ~ + ) ~ l ( a+- l +J g ) 1 1 (c44cm+ C&da &,(a- - a+)%/(a-'+ &)(a+¶ + J I B )
(CssCss GC,,
=
As in any spectrum analysis, an acceptable line assignment must be made. For an ABX spectrum an acceptable line assignment must agree with the repeated spacings given by eqn. (41) and with the intensity relationships
There are 16 acceptable line assignments for an ABX spectrum, however 14 of these are redundant because the spectmm is insensitive to the signs of JAB and (JAX Jsx). Therefore we may
+
+ J s x ) posdive
assume J A Band ( J A X
(45)
without loss of rigor. After applying eqns. (41), (44), and (45) only four acceptable line assignments remain. Any one of these can be used to complete the analysis. However the analysis is most easily performed if the term Voi, in Oi' (see Table 7) is removed giving terms Oi that are always positive.26 Having done this, we always must adopt the acceptable line assignment that gives posilivc values for
(v,
-
u.)
and
(vz
-
(46)
us)
Let's proceed now to the steps that should be followed to properly analyze an ABX spectrum. Step I. Our initial task is to find the one line assignment that fulfills eqns. (41), (44), (45), and (46). Sometimes this is quite simple. If Table 11 is examined closely, it will be discovered that the AB part of any true ABX spectrum consists of two typical ABquartets each involving JAB.The O+ quartet involves transitions containing O+ perturbations and the 0quartet involves transitions containing 0- perturbations according to eqn. (47).
~
+
Locate the two AB-quartets in the AB part of the ABX spectrum. If this can not be done unambiguously, go to step 2. If the two outside transitions of Volume 45, Number 6, June 1968
/
407
one of the quartets are not seen, don't worry. If the two outside transitions of both quartets are not seen, the spectrum is not analyzable. Place a mark on the spectrum a t the mean frequencies of both quartets and label the highest frequency (lowest field) mark v+ and the lowest frequency mark v-. From eqn. (48), v+ = ( n v - = (n
+ w ) / 2 = (v, + w ) / 2 =
+ m)/Z = + v d / 2 = ("4
(PC
-
Y-)
make up the 0, quartet and t h e 0- quartet is comprised of the transitions symmetrically disposed about v-. Assign transition numbers 1, 3, 5, and 7 consecutively to the 0+quartet going from low to high field and similarly assign transition numbers 2, 4, 6, and 8 to 0- quartet. This completes the line assignment of the AB part of the ABX spectrum. As 2(v+ -
V-)
+ J B X )=
= (JAX
(YII
-
Y I ~
(49)
transitions 11 and 14 are characterized. Transitions 12 and 13 appear a t "11 - (n- vs) and vlr (n - us),
+
respectively. Transitions 10 and 15 are the remaining ones and are readily characterized as ulo > uls. The correct line assignment has been completed and eqns. (41), (44), (45), and (46) should be satisfied. Proceed to step 3. Equation (44) should not be expected to hold if (1) the
spectrum is not representative of a true ABX system (i.e., if perturbation terms M* and Ni in Table 7 are in fact substantial) or (2) if either the A or B resonances are broadened from unresolved long range couplings (e.g., if either A or B is a benzylic hydrogen). If eqn. (44) is not fulfilled precisely, but yet unambiguous assignments of the 0+ and 0- quartets was possible the correct line assignment probably has been made. If you want to be certain of this, check the line assignment by proceeding to step 2. Step 8. The correct line assignment normally can be made by determining the two allowed double qnantum transitions for an ABX system (see the section on the Analyses of an AB Spectrum in Part I). Under conditions of high rf power levels, double quantum (DQ) transitions may be observed a t the mean frequency of any true AB quartet. For the O* quartets of an ABX system and for a given value of HI, IDQi-1/(8;B+)4 and YDQ* VDQ+
=
>
Y*
"DQ-
(50)
It is necessary to observe only one of the two DQ transitions to characterize v+ and v-. This is because one 408
/
Journal of Chemical Education
DQ transition characterizes one quartet and the remaining transitions must belong to the second quartet. After v+ and v- are characterized, complete the line assignment as described in step 1. Figure 10 illustrates the use of the double quantum '/&A
++
' / ~ Y A
YB) YB)
+
+- '/&TAX ++ JJ BB XX )) '/UAX
(48)
= ' / ~ J A x JBX)
OH H* Ph-A-&h
I
I
Hx H e
111
In this instance the double quantum spectrum serves to confirm the line assignment made in step 1, as eqn. (44) is fulfilled. Step 3. Using the correct line assignment determine the parameters 2D+, JAB, and (JAX JBX) given in
+
,Kl,
wnere 2% = 20+
+
16
ia
* '/WAX- J s x ) ] = * '/WAX- Jex)l' + J:B)'/*
Step 4. For any correct line assignment there are twoz6solutions, solution-1 and solution3, that lead to two sets of parameters (JAX- J B X )and 8;~. Equ* tion (52) gives the expressions for these solutions.
+
from eqn. (51) and two Knowing (JAX JBX)and JAB sets of parameters (JAX- JBX)and 6 8 from ~ eqn. (521, calculate two sets of parameters JAB, JAX, JBX, and 8 : ~ (one for solution-1 and the second for solution-2). Step 6. Both solutions-1 and -2 will lead to exact frequency fits, so there is no need to calculate transition frequencies. The correct solution normally can be identified, as (1) one of the solutions may give ridiculous parameters (e.g., vicinal couplings in saturated systems having magnitudes much greater than 20 are unlikely) or (2) only one of the solutions leads to calculated relative intensities for the X-transitions that corresponds well with those observed. The calculated AB transition intensities are the same for both solutions. Using the intensity expressions given in Table 11 calculate intensity ratios Ilo/112 for both solutions. The The two solutions arise because there is no way possible for assignment of the terms 6 ; ~ and '/z(Jnx - J s x ) that appear in D,. Consequently the values for 6 i and ~ Ih(J*x - J s x ) from solution 1 may actually correspond to values for ' / * ( J A x- J B X ) and 6,.; respectively. This second possibility corresponds to solution 2.
n
ms
Figure 10. NMR spectra I 6 0 Mc/sacl of 1-ocetoxy-2-nitro-1.phonylethane, Ill, in CClc The phenyl and asetoxyl proton rerononcer ore not shorn. The middle and bottom spectra of the AB and X hydrogens, rcspsctively, were determined with an rf = 0.03 mg. The top spectrum of the AB hydrogens war determined with on rf = 0.1 mg. Numbers given under thespectro are resononce frequencies downReid from TMS.
solution that leads to the best agreement between the observed and calculated intensity ratio is the correct one. If solution-1 is correct, values for J,,cl)are obtained by applying eqn. (41). If solution-2 is correct, "proper" values for IJfj(" I are obtained by applying eqn. (41) to the AB part of the spectrum after interchanging transition numbers 2 and 6 and 4 and 8. The necessity of this arises because of our requiring O* to always be positive. Step 6. Finally, PA, vn, and vx aregiven by eqn. (53). If transition frequencies are un = (ur = ("1
YB
"X
= ("U
+ v d / 2 - ( J A X+ J s x ) / 4 + 6 b / 2 ++ " d l 2- ( J I X + J a x ) / 4 - 6 d 2 Y S ) / ~
(53)
relative to TMS, then T < = 10 ppm - v,/60 for spectra determined a t 60 Mc/sec. The spectrum of I11 that is shown in Figure 10 may serve as an example of a straight-forward ABX analysis. The correct line assignment, as discussed earlier, and resonance frequencies relative to TMS are given in the figure. With this information, the analysis may be performed starting with step 3 of the recommended steps for an ABX spectrum analysis. The time required for the analysis of an ABX spectrum such as that shown in Table 12.
Results of the ABX Analysis of the Spectrum in Fioure 10"
lstorder
Sol.-1
ABX
Sol.-2
J i j values and S ~ are B given in cps and r values in ppm.
Figure 10 normally is about 30 minutes, starting with step 1. The results of the analysis of the spectrum of I11 are given in Table 12. After comparing the calculated and observed values for I I ~ / I Ii~t ,is clear that solution-1 and JBX are of the same sign, A and is correct. As JAX B are the geminal hydrogens in 111, otherwise these couplings would he expected to have opposite signs ( I ) . The magnitudes of the couplings are also in agreement with this assignment (1). Actually, Figure 10 represents a spectrum of a slightly perturbed ABX system as values for N+ in Table 12 are about 0.2 cps (those for M+ are about 0.05 cps). Solution-1 parameters in Table 12 probably are accurate to better than +0.2 cps. If a higher degree of accuracy is desired, the spectrum should be analyzed by the ABIC approach discussed later. The combination trausitions, UIO and vls, are not observed in Figure 10. This is not always the case. If the AB part of an ABX system is moderately tightly coupled (i.e., if [~AB '/z(J~x - Jnx)] 5 JAB),then (2D, 6An) will not have an excessively large magnitude (( 20, normally). If,in addition, '/z(JAx - Jnx) is of the same magnitude as (2D+ 6AB),then the magnitude of (a- - a+) will be large and the combinations will acquire appreciable intensity a t the expense of the intensities of the fundamentals, v12 and Vl3. This may become clear after examining the intensity expressions given in Table 11. Summarizing the above comments, combination transitions may he expected to be observed - JBXl = (2D+ for ABX systems having GB). The twelve fundamental transitions are clearly observed in the ABX spectrum shown in Figure 10. Often ABX spectra are encounted that exhibit fewer than 12fundamentals. These spectra arise because of transition frequency degeneracies and loss of transition intensities. They sometimes are referred to as "deceptively simple" ABX spectra (15) and a nonthoughtful approach to their analyses can lead to disaster. Some general types of ABX spectra are listed in Table 13 together with conditions required for their observance. Types 1, 4a, 4b, and 5a are illustrated in Figures 10-14.
+
+
+
1v
The spectra in Figures 11-14 are those of IV in different mixtures of benzene and carbon tetrachloride. There must be substantial puckering of Cs above the plane defined by CI-CTC& in IV as JMX is only 10.41 cps (see Fig. 13). A value for lJhlxlof about 1.5 normally is expected (16). The spectra in Figures 12 and 14 cannot he analyzed without prior knowledge of the J , , values. The spectra in Figures 11 and 13 are readily analyzed. For example, the observed AB quartet in Figure 13 may be assigned to the Ot quartet, and the transitions labeled 1, 3, 5, 7 respectwely (see step 2). The remaining AB absorption consists of nearly degenerate va and ve from the 0- quartet, the wings of which ("2 and us) have vanished. "11 and vla are readily characterized, since ' / l ( n l - v12) = '/z(v~ vS) - '/2(v4 v 6 ) . "Phantom" transitions v2 and vs
+
+
Volume 45, Number 6, June 1968
/
409
Table 13.
1
D + # D-and D , > JA@ Da = '/s([&B '/~JA -xJ B X ) ~Jis)'I" ~ JAX = J a x and 6;s # 0 ;.0+'= 0-' 6;e = Oand1/&T*x - Jsxl JAB (This is properly designated an AA'X system)
12(4X, 8AB)
2a 2b
-
+
11(3X, 8AB)
{ (m
nd
-
( Y ~
YE)
( v ~FI
Y.)
4b 5sd 5bd 50'
("11 ("8
4, ("4
I
-
= "l* = ",I= = ~ 4 ("6 ) ~= ~
"I), ("1
6 b = 0 and JAB >> ' / r ( J ~ x
-
~11)
=
Fig. 10 and 11 (common) (uncommon) (uncommon)
al
/"\
0
and 11 = I, 0 9(4X, 5ABY and 1%= I8 a 0
5(3X, 2AB) o),( v n = 12=18=h=17s0 5(1X and 4AB) (n
-
+
J a x - Jexl >>JAB 6;s = 0 and '1% (This is a spec, type of AA'X system the spectrum of which may be analyzed as an AMX type).
10(2X, 8AB) However, Ira = IU = 1 and the spectrum is indistinguishable from that of Type-1.
3
General Types of ABX Spectra
JAX = J e x and JAB >> 6 i R .: D + JAX = -Jex JAB and 6as = 0
"1)
-
- J s x ) :.D, LS.
JAB
JAB
0
(Fig. 14 (uncommon) (uncommon) (uncommon)
8 ( )~ ~ 7= " 8 )
K. A. McLauchlan and T. Schaefer,Can. J. Chem., 44(1966). AB resonances may appear as a 2: 1:1triplet upon going from low ta high field ( I 2 and I8 may be low) thereby giving 7 observable fundamentals. = AB resonances may appear as a 1:1:2 triplet upon going from low to high field ( I , and I , may be low) thereby giving 7 observable fundamentals. d The X resonances appertr as a. 1: 2: 1 triplet and the AB resonances appear as a 1 : 1 doublet with apparent J = '/r(Jax J s x ) thereby =king the appearance of an A& spectrum. T h e spectrum gives the appearance of a three spin system containing a magnetically isolated X-nucleus and an AB system having an apparent 6 h = '/dJ*x - J s x ) .
+
,,,,
% .,.
m"
,,
, , ,". ,",
< .,,
,m.
Figuro 11. NMR rpsctrvm I 6 0 M c l 4 of 4-bromo.3-t-butylcycIopent-2Tho t-butyl and vinyl hydrogen resonances enme, IV in CCb (10% w/w). are not rhown. Numbers under the spectrum are resonance froquensior downfleld from TMS.
I
I ,568
CB.
m
I ,no I
>
I -2
83.
NMR rpectrvm 160 Mc/res) of 4-bromo-3-t-butylcyclopent-2-
onone, IV, 118% w/w in o 1.5: 1 mixture of CCI4:CsHd. The t-butyl m d vinyl hydrogen resonances are not shown. Numbers under the rpectrvm
are resononcefreqvsncier dpwnfleld from TMS.
II
83%.
,% 3 s m ..
Figure 12. NMR spectrum I 6 0 Mc/rec) o f 4-bmma-3-1-buhllcyclopont-2enone, IV, in CsHa (20% w/w). The t-bvtyl hydrogen resononce is not shown. The vinyl hydrogen reranance istho low fleld doublet. Numbers under the rpectrvm are reranonce frequencies downtleld from TMS.
410
,,
tBI6
Di.8
UB.0
151.8
IS6.0
Figure 14. NMR spectrum I 6 0 Mc/rec) o f 4-bromo-3-t-butylcycIopent.2. enone, IV, (14% w/w in a 2.1 mixture of CsHs:CCld. The 1-butyl and vinyl hydrogen resonances ore not rhown. Nvmberi under the rpedrum ore resonance frequencies d.ownReld from TMS.
Results of Analyses o f Spectra in Figures 1 1-1 4
Table 14.
IJ&
Figure 11 lZL 13 14'
19.2
. .. ...
19.2
First-order* Jzk J% 5.5 4.1 4.5 3.8
2.1 3.5 3.1 3.8
ABXa
&dl'
IJAB
J AX
25.1
19.2 (19.2) 19.2 (19.2)
6.4 (6.4) 6.4 (6.4)
...
20.1 0.0
Jax
6.b
1.2 (1.2) 1.2 (1.2)
16.0 -2.2 5.4 (0.0)
r*
6.97 7.39 7.26 7.37
TB
TX
7.24 7.35 7.35 7.37
4.84 5.36 5.09 5.30
Values of J i j and 6;s are in cps. ~ becalculated. The spectrum is not analyzable. Assuming the parenthetical yalues for Jij appl,y,6 : may T h e spectrum is not analyeable. The speotrum calculated wlth the parenthetmd parameters corresponds to that observed and gives (YI - YE) = ("4 - Y&) = 0.2 cps and YU = "18.
may be marked on the spectrum a t frequencies (11-13) = (12-14) npfield from vl and (11-12) = (13-14) upfield from u,, respectively. Transitions v4 and va appear a t frequencies ( v , - us) = (vs v?) upfield from vz and downfield from vs, respectively, with (va - vs) = 0.2 cps. The analysis may be completed starting with step 3. Results of the analyses of the spectra in Figures 11-14 are summarized inTable 14. The observed and "phantom" transition frequencies relative to TMS are shown in Figures 11 and 13 so that interested readers may analyze these spectra for practice.
-
now mixed to second-order with state 3 and 4 and 5 and 6, respectively, using eqn. (21) where ABX wavefunctions J., and J.* are used to evaluate Hn., and H., = E,cABx, For states 2, 3, and 4 this is tantamount to applying the second-order approximation for solving the secular determinant
for El, E3, and E4. The resulting ABK states are given in Table 15 and expressions for the 15 allowed transitions are given in Table 16. Intensity expressions which represent ABX ones with lstorder correction terms C' included are given in Table 17. The following steps should be followed upon applying an ABK analysis to an ABC or perturbed ABX spectrum. Step 1 . Select the two nuclei that have the largest value of (J,j(L))Z/48Pj(1) and label them A and B with " A > uB. Label the remaining nucleus K. Complete an ABX analysis of the spectrum as described above. Step 2. Keep (JAX J B x ) positive and assign the expected sign of J A Brelative to J A X . If JABis taken as being positive, no change in the line assignment is necessary. If J A Bis taken as being negative interchange transition numbers 1 and 3, 2 and 4, 5 and 7 , and 6 and 8. Whereas an ABX spectrum is insensitive to the
ABK System
Three spin systems that approach the ABX limit but yet may not be accurately treated as such are designated as ABK or perturbed ABX systems (17). An ABK system is indicated when eqn. (44) does not hold. The ABK analysis approach to bedescribed does remarkably well for analyzing spectra arising from moderately tightly coupled ABC systems. Although more time consuming than the ABC analysis approach described above, the ABK analysis generally is the most accurate. The ABK state expressions are written after carrying out perturbations of the ABX states to second-order. This is accomplished by first rewriting the ABC system in Table 7 as an ABX system, replacing O* by O* and adding, in the expression for O* (see Table 11, bottom). States 2 and 7 which do not mix in the ABX limit are Table 15. 11.
1 2 3 4 5 6 7 8
Nuclear Magnetic Energy Stotes for the ABK System
As
M
'/=
-'h - 'h
- 'It
Em,
IABK)
aaa
a/%
'1%
+
'/+A
nag Csra8a f CaBaa Cdna Csda CrsaD8 C&4 CwPaP Caaar88 880
+ +
+
888 D, = A
B C
E
vsrrkB *
'/A"*
Emn
+ + + ' / r ( J ~+s JBK+ J A K ) +
PK)
VB
YB
-
YK)
+ '/dJ*a - JBK- J m ) + A + B
- ' / J A+ BD+
- A D+ -B -'/XYK - '/JAB D C ->/WK - '/JAB - D+E ' / z ( - v A - VB YK) ' / & ( J AB JBK - J m ) - C - E -'/&A f P B f Y K ) ' / I ( J A B JBK J A K ) l / d J ~ J~ B K ) ~ ' J : ~ I ~ / ~ (CuJm C ~ J A ~ ) ' '/WK
'/WK
-
' / ~ A B-
+ +
+
=
+
+
+
+
+
+
- ( J m + J B K )- 4D+ + C s J d = 2 ( 6 ; ~+ ~ B K+ ) 2 J m - ( J m + J B K )+ 4D+ 2 ( 6 k 4- 6 ; ~ )f 25A8 (Cdm
+ D~JBK)' 2 ( 6 1+ ~ 6 ; ~ )- 2 J m + (JAK+ J B K )+ 4 D (Codsn + C ~ J A K ) ' = (CwJnx
=
+ 6 ; ~ )- 2J.m + (JAK+ J B K )- 4D-
2(6:~
See eqn. (55)for corrected perturbation terms P'. Volume 45, Number 6, June 1968
/
41 1
Table 16.
No.
Transition Frequencies for the ABK System
Transition
Frequency
----
'/*(PA
6-2
'/&A
3
7
4
8 +5
5
3-1 5 2
6 7 8 9 10 11
7
3
'/&A
f '/JAB '/I(JAK
PB)
4- '/JAB - '/~(JAK J B K ) VB) - '/JAB '/~JAK JBK) us)
vs)
'/&K
4
'/*(PA
VB)
'/&A
"8)
'/~JAK
'/~JAK
+ v d - '/JAB - l / , ( J ~+~ J B K )- D - + E m+v~-vx+A+B+C+E '/&
8-6 7 2 (Comb.) 3 (Comb.) 6
VK+D++D--A-E '/~JAK JBK)- A - B PK+D+-D--A-C YK+D--D+-B-E YK - '/I(JAK JBK)- C - E YK-D+-D--B-C
2-1
12 13 14 15
+ + +
+ + J B K )+ D + + B + + D- + A + B - E + + + D+ - A + C + E PA + - '/JAB - '/I(JAR + J B K )+ D- + C 'lz(v.4 f rs) + '/JAB + + J B K )- D + + A 44- '/JAB + J B K )- D - + A + B - C + - '/JAB + + JBK)- D+ - B + C + E
4-1
1 2
YK
5-3
6-4 8 7 5 4 (Comb.)
+
+
+
Table 17. ABK Transition Intensities
No.
Intensityo
" The above expressionshave been simplified by taking all C ; C'' -C' 3 C'I I -- -C'
-
x -
-
$8-
1 and a11terms c;,c;..
=
@AX
+ &XI +
@AX
+ &x) + JAB- I/x(JAx+ J B X )- 2D+
JAB (Cadax
A = 4(vs - u d - 8A - 4B B = 4(v2 - v n ) - 8B - 4A C = 4(vr - v u ) - 8C - 4E E = 4(vs - u u ) - 8E - 4C
/
Journal of Chemical Education
'/WAX+ J B X ) +
+C d ~ x )
2D+
nominators which are in turn used to estimate yet better perturbation terms. One or two iterations generally will be necessary. If a t any point a perturhation term exceeds 10.81 cps, the corrected perturbation P' should be used. Use of eqn. (22) gives P'
=
S
5
[(denominator P/4)'
+ numerator PIL/, denominator P/8 (55)
(54)
As the frequency separations normally will he substantially larger than A, B, C, and E in eqn. (54), approximations to the latter terms result by taking the denominators as being equal to four times the r e spective frequency separations. The resulting perturbation terms then are used to obtain corrected de412
+
( C d ~ x CaJax)
sign of JAB relative to JAXand Jsx, an ABK spectrum may not be. Step 3. Calculate perturbation terms A, B, C, and E, the expressions for which are given in Table 15, using for the numerators the parameters and coefficientsresulting from the ABX analysis and the expressions for the denominators given in eqn. (54) denominator denominator denominator denominator
. . = 0.
where S = *1 when denominator P
2
0
+
Step 4. The parameters 2D,, ( J A X JBX), and JAB used for the ABX analysis now are corrected using eqn. (56).
Step 5. Use the corrected parameters (step 4) to obJAX, JBX, S~B, tain a corrected ABX solution (i.e., JAB, and ABX coefficients). Further refinement of this ABX solution may be acheived by using the results thereof to compute yet better values for the perturbation terms to use in eqn. (56), and thence to a better ABX solution. This iteration process may be repeated until the perturbation terms become invarient upon reiteration. If the perturbation terms are all less than 10.51 cps, the first corrected (0-iteration) ABX solution will be sufficiently accurate for most purposes. , and VK relative to '/Z(VA Step 6. Values for S ~ K 6&, vB) are calculated using eqn. (57) where the right hand terms are the corrected ones obtained upon the final iteration. It is convenient to take '/Z(YA YB) = 0.
+
+
-
Since K is undoubtedly the benzylic hydrogen, a negative J n B is expected ( I ) , and it is most likely that the solution giving the sign of J A B opposite to that of JAK and JBK is the correct one. It should be noted carefully that the ABX analysis of the spectrum of V is substantially a t variance with the correct analysis-even though the spectrum resembles closely a typical ABX one. The ABK analysis was applied to the spectrum of I1 given in Figure 8 as a final illustration of the effectiveness of this approach. The results are summarized in Table 20. I n this instance, as in general, the ABIi analysis results are found to correspond more closely with the exact ones than do those from the 2nd-order ABC analysis.
-
Step 7. The best corrected ABX coefficients (C,, and Ck,) and ABK coefficients (Cnnf)are used in the expressions in Table 17 to compute the ABK relative transition intensities. Theseare then compared with the experimental ones. If desired, the best corrected ABX parameters and perturbation terms may be used to calculate the ABK transition frequencies from expressions given in Table 16. Here, it is convenient to take ' / 2 ( ~ A v B ) = 0. AS the analysis approach effectively forces a frequency fit, the agreement between the calculated and observed transition frequencies should be excellent unless an error was made. Step 8. If the sign chosen for J Ain~step 2 is in question, reverse the sign and repeat steps 2 through 5. The sign that leads to the best agreement between the calculated and observed relative transition intensities is the correct one. However, if a clean-cut distinction is not apparent, other approaches such as double resonance may be used to determine the correct complete solution. As an illustration, the ABK analysis approach has been used to analyze the deuterium decoupled spectrum of 4-phenyl- 1, 2, 3, 3, 6, 6-hexadeuteriocyclohexene, V.
+
This spectrum is shown in Figure 15. The experimental frequencies given in Table 19 can be used to practice the analysis approach and to check the parameters given in Table 18. I n Table 19 the calculated frequencies and intensities are compared with those observed. Tables 18 and 19 show excellent agreement between the ABK and iterative computer results. The ABX analysis of the nmr spectrum of V unambiguously gives as having the same sign (taken as posiJ a K and JBX tive). Whether JAB is negative or positive is not clearly distinguished by the ABK analysis, although the calculated intensities agree somewhat better with those observed when JAB is negative as opposed to positive.
ABz System A three spin system having two nuclei that are symmetrically equivalent is designated AB2. An AB2 system may be treated as a special case of the general ABC system. However, it is more fruitful to approach the problem utilizing the benefits of molecular symmetry. Taking nuclei B and C as being equivalent in = JAG)gives an ABz sysTable 7 (VB= vc and JAB tem. It is seen that under these conditions states 2 and 3 and states 6 and 7 are degenerate to the firstorder. For systems having degenerate first-order states, higher-order states often are most easily obtained by using basic symmetry functions rather than basic product functions. Acceptable basic symmetry functions may be written as sums and differences of (linear combination of) the basic product functions of the degenerate states. The basic symmetry function which is unchanged upon interchange of the spin state functions of the equivalent nuclei in the component product functions is a symmetric (s) function and that which changes sign upon such an operation is an antisymmetric
Figure 15. NMR spectra ( 6 0 Ms/sec) of 1.2.3.3.6.6-hexadeuterio-4phenylcycloheione, V, (neat): normal spectrum (middle), deuterium decoupled spectrum ( b o t t m l and deuterium decovpled spectrum showing DQ+ transitionr (top). The DQ* transitions were determined ot rf = 0.04 on2 0.2 mg, respectively. The spikes in the baseline resulted from rimultaneourly changing the rf and spectrum amplitude retting,.
Volume 45, Number
6, June 1968
/
413
Table 19. The Observed and Calculated NMR Spectmm of V Transition No. 10 11 13 12 14 15 3 4 1 7 2 8 5 6
Tramition Frequencies ABK Observed (-JAB) 94.31 85.17 80.84 74.84 70.51 61.35 36.38 32.10 23.79 22.89 19.47 12.59 10.32 O.0O1
94.32 85.16 80.85 74.85 70.53 61.37 36.37 32.06 23.78 22.90 14.47 12.58 10.31 0.OOd
Computer ABC (-JAB)
Observed*
94.31 85.15 80.84 74.83 70.52 61.36 36.39 32.07 23.79 22.91 19.48 12.59 10.32 0.00
0.03 0.77 0.86 1.06 1.20 0.08 0.06 0.44 2.06 2.11 1.50 1.37 0.14 0.31
Relative Transition Intensities ABKb ABKb -ABC (computer)(-JAB) (+JAB) (-JAB) (+JAB) 0.04 0.78 0.84 0.99 1.26 0.09 0.04 0.35 2.13 2.20 1.51 1.40 0.08 0.27
0.05 0.77 0.81 0.97 1.26 0.14 0.02 0.38 2.18 2.21 1.50 1.38 0.06 0.25
0.03 0.77 0.84 1.02 1.25 0.08 0.06 0.36 2.11 2.19 1.52 1.39 0.11 0.25
.
0.05 0.77 0.81 0.98 1.25 0.14 0.03 0.41 2.17 2.20 1.48 1.37 0.08 0.27
The sum of Ilothrough 11,was normalized to 4.00 and the sum of I, through Is was normalized to 8.00. The K-resonance intensities have to be normalized separately as these resonances are broadened by long range coupling hetween K and the phenyl hydrogens. These intensities have been normalized to give a total intensity of 12.00. The normalization factor was 0.993. This is necessary because the first-nrder coeffieienta C;. and C:. are not exact and as a result the ABK wavefunctions are not exactly normalized. 'This transition is 81.8 cps downfield from TMS.
Table 20. ABX'
Results of the ABK Analysis of the Spectrum in Figure 8 siteration
ABK 1-iteration
2-iteration
Iterativeb com~uterABC
Iterativec Znd-order ABC
-. ..
T h e line assignment for the correct ABC analysis was used. LAOCOON 11. See Table 9.
'
See section 1.
(
(a) function. Thus for the ABz system, the normalized basic symmetry functionsfor states 2 and 3 are a(a@ Ba) (I/&$ and a(a@- @a)(I/&-the former being symmetric and the latter antisymmetric. The factor of (l/v'% is the normalization factor which is required so that (&=) = 1. The normalized basic symmetry functions for states 6 and 7 are @(a@ pa) (l/v'5j and @(a@- @a)( l / d @ .
+
+
As states of different symmetry do not mix (Table 3), the two third-order secular determinants for the ABC system factor into two second-order and two first-order determinants when the threespin system becomes one 414
/
Journal of Chemical Education
of the AB2 type. The symmetric and antisymmetric states are isolated. Recalling from Table 3 that transitions between states of different symmetry are forbidden, it may be derived readily that a maximum of nine transitions are possible for the AB1 system. M I/.
ABz a-Slates AB1 o-Slates
No. of Tvansitions
Table 21.
M
n
Nuclear Magnetic States for the
dm
M*' =
ABz
Exact energy lst-orde-
,
$a
1 / X 1 6 ~+ ~
System"
+
l/Ad22JkJ1/' - 'hid ;B
* l/J~~l
-
"No mixine hetweenstates except wheren=2 or 3, and wberen=4 or 5. 6 Symmetry designation, s for symmetric end a for sntisymmetric. Table 22.
Exact Transition Frequencies and Relative Transition Intensities for the ABz System
No.
Transition
1 2 3
3-1 5-2 8-7 6-4 4 -t 2 2-1 3 5 6-5 4-3
4
5 6 7 8 9
-
Nucleus
A A A
A B B B B (Comb.)
Frwruencv VA
v* PA
v*
rs PB
rs IB
PB
Relative intensitv
+
+
('2, fiGda (Cd& ficd2w fi~&& 1 (CuCu .\/ZcuCu)" (v%&z G*CU f i C , C d a (.\/zCs, Cada (&C&m C&a @Cu+n)' Z /\(.cs, C4da Cv'?CuCn 4CmCa CuC2,Y
f JAB M+' f M-'+ Mc'
- JABf
M-'
+ Jm/2 + M+' - M-'
+ JAB/^ - M+' - JA&- M +' - 5*8/2 - M-' -aAB
+
+ M -'
+
- % I + ' - M-'
C66 =
As only two states of the same M value mix, use of eqn. (26) leads to the exact energies of the eight ABz states. These states are shown in Table 21. The exact transition frequencies and corresponding transition intensities (using eqns. (24), (31), and (32)) are summarized in Tahle 22. The AzBspectrum is the mirror image of the ABzspectrum. The transition energy expressions for the A2B system are those given in Tahle 22 with the A's and B's interchanged and all terms following v r B multiplied by -1. The intensity expressions for the AzB system are the same as the AB, system. Examination of Tahle 22 leads to the conclusion that J B Bcannot he determined by analysis of the spectrum. As the ABz spectrum is insensitive to the sign of JAB,only lJaBI can he determined. A correct line assignment of an AB2 spectrum normally can he made rapidly by applying the transition frequency and transition intensity relationships given in eqns. (58) and (59).
+ +
C,, =
+
+
+
+ +
+
+
(0.707)Im I1h(vr - 4'
+ J'I"'
As an ABz spectrum can be analyzed by inspection,
..
After making the line assignment, the parameters V A , v B , and (JABI may be obtained from eqn. (60).
Figure 1 6 . NMR spectrum 160 M c / 4 of 2.6-dirnethoxyphenyl, VI, inCC4. The numbers under the stick spectrum ore obsened resonance froqumcies downfleld from TMS.
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there is no need in practice to compare the calculated and observed spectra.
approximate ABC and ABK analysis approaches to analyze many spectra so as to compare the results with those from computer analyses. Also, I wish to thank Richard Sprecker for determining the spectra shown in Figures 11-14, and Jack Russell for a sample of 111. Finally, I wish to acknowledge support of this work by the National Science Foundation (Grant No. G.P. 5806). Literature Cited
An example of moderately tightly coupled AB, system is illustrated by the aromatic proton spectrum of 2,6dimethoxyphenol, VI, in carbon tetrachloride that is shown in Figure 16. Use of eqn. (60) gives 6;~ = 10.0 cps, lJABl= 8.6 CPS, T A = 3.44, and re = 3.60 ppm. I am most grateful to Ken MacKay for using the
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(13) FESSENDEN, R. w . , AND WAUQH,J. S., J. Chm. P h y ~ . ,31, 996 (19.591.
CAST ELL AN^, S., AND WAUGH,J. S., J. Chem. Phya., 34, 295 (1961); 35,1900 (1961).
ABRAHAM, R. J., AND BERNSTEIN, H. J., Can. J. Chem., 39, 216 (1961). See also Reference (8b), p. 357 ff.(Part I). GARBISW, E. W., JR.,J. Am. Chem. Soc., 86,5561 (1064). REILLY,C. A,, AND SWALEN, J. D., J . Chem. Phys., 32, 1378 R. A,, J. C h a . Phys., 33, 1256 (1960). (1960); HOFPMAN,
