I
Edgar W. Garbisch, Jr. Universitv of Minnesota Minneapolis, 55455
I 1
I
Analysis of Complex NMR Spectra for the Organic Chemist 111. Four spin systems of the ABC, ABX,, ABK,, AA'BB', a n d AA'XX' types
A11 too often nmr parameters appearing in the scientific literature must be considered of questionable reliability either because the details of the spectral analyses were insufficiently presented or because the parameters were derived from unjustified first-order analyses. Organic chemists are the principal contributors to this state of affairs. Many organic chemists are genuinely interested in understanding the basic theory and problems associated with nmr spectral analysis; others, perhaps, would prefer a recipe to follow. This article was written so as to nurture both groups of chemists. A general perturbation approach for the analysis of complicated nmr spectra of nuclei having I = was developed and applied to the simple AB system in Part I of this article. The more complicated ABC, ARK, ABX, and AB2 systems were discussed in Part ILZ1 Here in the final part of the article the perturbation approach will be applied to the ABX2, ABK2, AA'BB', and AA'XX' systems containing I = nuclei.
state diagram for the ABCD system shows that a maximum number of 56 single quantum (AM =' 1) transitions may arise in a four spin ~ystem.~a No. of
M
ABCD States ABCD
If a four-hydrogen system contains two hydrogens that are equivalent either by symmetry as in
or by conformational averaging as in
Four Spin Systems
The general four spin (ABCD) system has 16 (24) states. The number of states having the same M value is given by the coefficients of the variables in the expansion of (x y)' where P equals the number of M = spins, starting with M = p / 2 and followed ( p / 2 ) - 1, ( p / 2 ) - 2 . . . - ( p / 2 ) . The condensed
+
H\/c=c /H
Y
'CH,Y,
it is properly designated ABC2. This spin system is most conveniently discussed after applying its sym-
Part I: J. CHEM.EDUC. 45,311 (1968). Part 11: J. CHEM. EDUC. 45,402 (1868). The numbers of figures, tables, structures, equations, footnotes, and literatare cited i n this paper follow consecutively those in Parts I and 11.
+
Pascal's lriangle resrdts if the coefficientsof (z y)" are arranged as shown below. Here, p i s given by the second numeral in each line, starting with p = 0. Each coefficient is equal to the sum of the two coefficients which are nearest to the right and left of it, in the line above. And the sum of the eoetlicients is equal t o ZP.
Using Pascal's triangle and n d n g that each coefficient along a line is separated from its nearest neighbors by A M = + I , the maximltm nnmhor of transitions ( A M = 1) for any spin system (neglecting symmetry) e m be derived rapidly. For example, the maxin~um number of transitions in five, six, and twelve spin s y s t e m m e 2 1 0 , 792, and 2,496, 144, respectively.
480 / Journol of Chemical Education
Table 23.
Nuclear Magnetic States for the ABCz System 2nd-order Energy ABC?'
4no
M
(ABCC)
-
-
7---
2nd-order Energy ABX:
'/2(u*
+ v. + 2vo) + ' / , ( J A B+ 2 J ~ c+ W B C+ J C C )
+ + ' / J A B+ 'IIJCC + + ' / & ( - J A B+ ZJK - ~ J B +C J C C ) + + 2 v c ) + I / < ( - J A B - 2Jnc + W e c + J c c ) l/z(v~ + - 2uc) + ' / , ( J A B- W A C - 2Jao + Jca) ' ~ ( v A
' l 2 ( v ~-
VB)
~ Y C )
YB
ue
'/z(-PA
va
+ IIJcr + + 2vc) + JAB - WAC- WBC+ J C C ) - - 2.c) + ' / , ( - J A B - WAC+ 2Jsc + J c c ) + - 2 w ) + ' / r ( - J A B + WAC- W s c + J c c ) 'h(-v~+ ' / , J A B+ ' / J c c - + JAB + ~ J A +C ~ J R +C J C C ) + + - '/Jcc - '/JAB 'ILJJcc + - ' / J A B- ' / d c c + ' / J A B- 3 / J ~ ~ -
' ~ ( v A- VB) '/A '/.(-PA re) -
+
vs
'/s(--A
'/Q(VA
JAB
'/JAB '/Jco
VB
'/.(-"A
TB
i/l(-v~ '/dx*
YB
YB)
'/.(VA
ye)
VB)
'/Z(-VA
VB)
'/s(-vA
VB)
M i and N' =
~ Y C )
I/&R
S* - [(denominator-l',/4)' 2
+ Jas1l1/*- (denominstor-PJ8)
" Symmetry designation: s indicates symmetric state and a indicates antisymmet,rie state.
* N o mixing between states except where n = 39 and 4s: 6s and 7s; 9s and 10s; and 14a and 15a. No mixing between states except where n = 2s,3s, and 4s; 5s, 68,78 and 8s; 98, 10s and 11s; and 14a and 15a.
metry properties to simplify the condensed state diagram shown for the general ABCD system. Considering this state diagram, and replacing D by C, we find that there is no way to distinguish between the firstorder states corresponding to the asap and aapa basic product functions. These states are degenerate. Basic prodnct fnnctions that lead to degenerate firstorder states may be combined linearly to form symmetric and antisymmetric functions. Doing this invariably simplifies the spi~isystem. The aaap and a@a fnnctions of the ABCa system may be combined to form normalized basic symmetry functions (aaorp acc$cc)/fi and (aaap - ororpa)/&, the former being symmetric and the latter antisymmetric with respect to interchange of the nuclei-C spin fnnctions. Similarly for the M = 0 states, basic product functions ap@ and appa as well as paap and papa may be combined to form a pair of symmetric and antisymmetric basic symmetry functions. And for the M = -1 states, functions PBap and PPBa combine to give symmetric and antisymmetric basic symmetry functions. As states having different symmetry properties never mix
+
and as transitions between states of differentsymmetry are forbidden (see Table 3 in Part I), the symmetric and antisymmetric states of the ABCz system may be isolated and considered independently. The condensed ABCD state diagram now converts to No. of
AM = 1
M 2 1
0 -1
-2
AX2 s-slates
ABCS a-stales
----
--
--
---
-
-
transitions
3
14 14
3
34 Total
when D r C. Consequently, 22 of the allowed transitions for the ABCD system become forbidden in the ABCl system. Energy expressions for the sixteen states (up to second-order) may be written for the ABC? system by application of eqns. (21) and (2.5). These are collected Volume 45, Number 7, July 7968
/
481
in Table 23. The 34 allowed single quantum transitions for the ABCz system are listed in Table 24. Expressions for the first-order ABCz transition intensities, obtained from eqns. (22) and (24), are written in Table 25. Tables 24 and 25 may be used as well for AB2C (19) and AzBC systems. If so, care must be exercised in using the correct signs of .,a: For example, an A B L system may be treated as an ABCz system having vn > vc > us. I n this event, 6Ec would be negative. Consequently, perturbation terms 0,(0;) and T+(T;) and coefficients F, and H+ would contain a negative value for &.
Table 24. No. Transition
Transition Frequencies for the A B C System
Nucleus
Zndnrder Freqoency
A A A A A A A A B B B B B B B B C C C C C C C C comb ( C ) comb ( C ) comb ( C ) comb ( C ) comb ( C ) comb ( C ) comb ( C ) comb ( C ) comb ( A ) comb ( A )
ABXz Sysfems
If the perturbation terms O,, Q,, T,, and R, in Table 23 vanish, the resulting system is properly designated ABXz (C has been replaced by X). ux may he greater or less than vA, however, U A > vs. As no more than two states of the same M value mix in the ABX, system, use of corrected perturbation terms N' and ML and application of eans. (31) ~. and (32) , . leads Transitions Effeclive 6 ; ~ to exact expressions for the transition frequencies M+ quartet 1 5 9 1 3 [ah ( J A X- J ~ I and relative transition intensities. As perturbation terms N and M* are independent of the sign of JAB, ah quartet 6 h (61) 2,3 6,7 10,ll 14,15 the ABXz spectrum is insensitive to the sign of this M - quartet 4 8 12 16 [ ~ L R- ( J A X- JRxII coupling. However, M+ and, as a result, the appearance of ABXz spectra are sensitive to the relative signs of JAx and JBX.Analysis of an ABX, spectrum, therefore, leads to ,JAal, JAx, JBX, and Sin. Only the relative signs of JAX and Jsxare determined. If, for convenience, JAB and (JAx J m ) are assumed to be positive, there is only one solution (both frequency and i~t~ensity) of an ABXl spectrum. This differs from an ABX spectrum which, making the same assumptions, has two frequency solutions. Consequently, an ABXPspectrum analysis may be affected by working with the AB part (recall that the correct ABX spectrum was distinguished by comparing the calculatcd X transition int,ensities for the two solutions with those observed. See Part 11). The AB part of an ABXz spectrum consists of three '/W-1 = YDQ V , 6 > v three double quantum transitions, vDQtaa, and V D Q ~ . ~ , , (v+ v o 5 ) = ( v o h - Y - ) = ' / ~ J A x Jnx) (64) that bccome allowed under perturbing conditions of (v+ - Y - ) = ( J A X J H X ) high rf field levels (see Parts I and 11).
+
+
+
+
482
/
Journal of Chemical Edumtion
+
Table 25.
No.
D: DL
E' 0
=
0
= = =
0 0 0
H
Cdr= - C a d Cm,
Cs=
6 h
=
-CO.IO
-Csr=Cis.u=
-Ca.t6
C,, = -C,,
C,,.,O=
-Ci~.ll
= -c21 ell.$ = -c8,11 CSS= -Css c 4 2
= =
1st-order Equivalent (ABCX) = D+
Coeficients
=
= =
a
Exact ABXI Intensitv
Transition
Ezocl Equivalent (ABXzl
Relative Transition Intensities for the ABXl and ABC2 Systems
= = =
DE F+
=
F-
= = =
G+
a
1st-order ABCI Intensitv
Ezact Equivalent (ABX,)
0 0
= =
0
=
B+
=
1
=
Coeficients
Cs Ca
= =
BA
1st-order Equivalent (ABCz) = H= I+ = 1-
= -cis
= -Csr C, = -Css Csr = C4r Cqq= C l o . ~ = = c11.11 = c11.15 = Css = CLI.,,= c~1.11 = c11.1= 1c
cw cii
C*>= Css
i m
= = = =
1
1
1 1
H+
+ 2C
Jec - 4 6 k c =t( J A B- Jac - J e c I / d 2
Volume 45, Number 7, July 1968
/
483
With thc general informatiou given above, an ABX, spectrum may he nnalyzed according to the following steps. Step 1 . From the AB part of the spectlwn identify the three v + > van > u. as required by eqn. (64). Label consecutively the component transitions of qoartet M + 1,5,9, and 13 going from low to high field. Likewise label the transitions of qmrtet ab(2,3).( 6 , 7 ) ,( 1 0 , I l ) :and (14,15)and those of qunrtet 11-4,8,12, and 1 G . If the outside transit,ions of either of the M, qwwlets are not ahserved, the inside transitions probably will be closely degenerde and not resolved. Should t,his s i t ~ ~ a t i o n nrise,proeeed tostep4. If the outside transitions of both of the M, quartets have vnniaherl, the spectrum cannot be mnlyi~ed. If diffirulty is eneonnte~edin locating the AB quartets, i t may be helpfnl to apply the intensity and repented I.1.m spacing relationships given in eqns. ( 6 6 ) and ( 6 2 ) . Often, as will be illuatrated later, u * , a." and the tl~sociatedqunrtets may he readily identified by locating t,he douhle q ~ ~ m t ntransit,ions m given in eqn. (63).
AR qoartels and assign
what is called an ah suhStep f. The ab qnavtet eonstit~~les s p e d r ~ r n . ~This ~ subspect,rum now s h o ~ ~ lhe d anal.vsed to give IJABI and 6 ; n usingeqns. ( 6 6 )and (6R).
IJAHI = ( n -
"7)
=
-
(nl
+
(v8 - v n ) = ( v 7 - m ) = 6 : ~ 2N' =
(66)
3,:)
C =
+
I ( X ~ R ) J~ a , w 6:.
=
I(".
-
w1)2
- .I.&I1/~ = [(w VA
= a. .
+ '/~6;u
YR
=
-
Y d
YU)(YI
-
YII)~'/~
interchange transition m~mbers(1 and 9 ) and ( 5 and 13). This gives a correct line msignment for solntion 2 taking V A > v n ; the former change gives S , = &1 and M, 2 0 . The transition frequencies and intensities may be calculated to compare with those recorded experimentally; however, this generally is not neeeqsary. The eqxessiona in Tables 24 and 2.7 may be used for this purpose. The correct solndion pammeters, of course, should be used and N' and M; shnnld repla& N and M in Table 24. Values for N' and M i are obtained by use of eqns. (67) and (70), respectively. Step 4 . If the outside trmsitions of either of the M, quartets are not. observed and the corresponding inside transitions not resolved this means that the effective - 6 : ~ far this quartet, as given in e m . iGlL is small as comoared with JAB.The analvsis results'are'quite sensitive to the spacing between the insidet,ramtions. The inside transitions should be separated by an arbitrary frequency not greater than that which should be resolved. "Phantom" outside transitions then should he marked on the spectrum IJAB~ to the low field side of the low field illside tmnsitian and JAB^ to the high field side of t,he high field illside tranaition so as to give the repeated spacings in eqn. ( 6 2 ) . Label the observed and "phantom" transitions of this quartet as described in step 1 and proceed to steps 2 and 3. Adjust the separation between the inside transitions oi the above discussed quartet until step 3 gives n cowect solution (i.e., until step 3 gives s solution having the same & i nas detevmined fromstep 2 ) .
The AB parts of ABXX'YY' spectra of 4,s-dihydrothiophene-l,lldioxide, VII, that are shown ill Figure 17 typify and may be analyzed as AB parts of an ABX,
(67) (681
(69)
'/~s:R
Step .3. Analysis of the M, quartets leads t,o two sol~it,ions,1 ~ wm and 2 . The solut,ion that leads to the samevalue of 6 ; that obtained in step 2 is t,lre cowert one. Using J u l , ( J a x f J R X ) and 28,b, obtained from eqns. (62), (64), and (70), respectively, cslculate 6 . : ~ , J A X ,and J n x for solot,ions 1 and 2 using eqn. (71).
Sohtion 1 correqponds to that for the line assignment made in step 1. If solution 2 is the correctone andif it is desired to calenlate freqnencies and intensities uning Tables 24 and 25, a new line ~ and assignment must be made. If solution 2 gives 6 1 negative (JAX- J R T )positive, change the signs so as to give ~ P \ B positive and (JAX- J n x ) negative5' and then interchange transition nnmbeys (4 and 12) and (8 and 16). If soh~tion2 gives 6:s pmit,ive,
*' Subspectra are sensitive only to those paramet,ers that relate to the subsystems from which they derive (20). For example, an ah srtbspectrum arises from an ah subsystem and only 1.I~nl and 6:n i n f l ~ ~ e mthe e appearance of this nohspectrum. The ah subspectrum of an ABX? system is insensitive to the parameters J A X and J H X . I t may be analya* independently to give exact values f o r j J ~ nand l 8.h. By convention we should take 6 . h > 0 . If Solution 2 gives a negative ~ X B ,change its sign to positive and multiply (JAX J n x ) by -1. Then proceed to calculate J m and Jex. 484
/
Journal of Chemicol Education
Figure 17. IAI NMR spectra (60 Mclsecl of 4,5-dihydrothi~~hene-1.1. dioxide. VII, in benzene.181 in CsHslvllCCle(vl E 20, and (CI in CsHalvl/CCI, 1") 10. Numberr under the spectra are resonance frequencies down field from TMS.
"
Results of the A B X Analyses ~ of the NMR Spectra of VII shown in Figure 17
Table 26.
Spectrum"
1.9 2.7 6.8 -2.25 3.1 13.2 13.4
IJ*xU1l IJsx'''I
IJAB"'I = J AX Jax 6.b
IJABIb
(from M * quartets)
& (from ab ouartet) ' All entries are in units of cps.
1.7 2.5 6.8 -2.25 3.1 9.9 100
1.5 2.3 6.7 -2.25 3.05 7.9 8.1
' This vicinal coupling may be regarded as being positive ( 1 ) .
Transitions va, Y,, vII and Y,: constit,ute t,he ab quartet, (ab subspectrum) from which I.rABland 61, may hc obtained. This is the only true AR q u a r k t with regard to relative ii~tensit,ies. Tmnsit,ions u2, Y O , Y I ~ , and U M may no longer be degenerate with those of the ah quartet. These transitions constitute a new quartct which is called the a'b' quartet. This and t,he AT+ quartets contain repcated spacings but may exhibit, relative intensities t,hat are perturbed from those expected of a true AB quartet. The mean frequencies of these four quartets are given by eqn. ( 7 3 , where corrected perturbation terms P' are indicated. They are also the frequencies of the four double quantum transitions that are sllowed under high rf power levels.
+ = 'h(W + "d= + + '/dJ*x + JAK) + '/dT! + It!) 'h(w + v d = ' h ( n a + n o ) + un) + '/do: + 01 + Q: + Q!) 'h(u f = '/An + = + = + "14 = '/dW + "d= 'A("*+ '/&La + Jm) + '/$(Ti + R:) '/.("5
'/?("A
"8)
VB)
=
P+
= ' ~ ( V A
WE)
This sit,uat,ionarises hccausc ncithcr A nor I3 is coupled detcctahly l,o Y, and X and Y arc wcakly coupled. The chemical shift ,:a in VII is quitc sensitive to the composition of n CsHs-CCI4solvent and the appearance of thc AR resonances may be seen. to vary markedly in I'igurc 17 as tllc rat,io of C6H6:CCll in changcd from m (spectrum A) to 20 (spectrum R) t,o 10 (spectrum C). The three spect,ra shown in Figure 17 were analyzed following thc procedure out,lined in steps 1-3. The resu1t.s are collected in Tahle 26. Transition frequencies rclat,ive to TMS are given in Figure 17 so that interestcd readers may practice the analysis and compare t,heir results with dhow given in Table 26. Thc calculat,etl st,icL spectrum is shown for spcctrum C. Assignment of A and to the olcfriic hydrogens in VII cannot he mntlc wit,h ccrt,aint~y. The assignment shown was made because t,hc niagi~itudesof four-houd vinylall.vlic couplings arc not cxpcct,ed to exceed those of three-bond vinyl-allylic couplings in :i given system (16). As vicinal couplings may be regarded as being positivc (I), JAxis assigned a negative sign. The :u~alysis,howcvcr, only t,ells us t,hat,.TAX aitd J n x are of opposite sign.
AII ARK, system m ~ ybe dcfined as n perturbed ABX* system. I t c a ~ be i treated in a manner similar to that adopted for the ABIi system (see Part 11); however, here we will use a somewhat different approach. This approach may he applicable to the approximate analyses of hBC2 systcms, although thc possibility has yet to bepursued. Equation (62) no longer applies for an .ABIiI or ABCz system. Repcated spacings do exist in the AB parts of spectra arising from these systems, but only those involving t,ransitioos within the asymmetric statcs (see Tablr 23) may be equatcd to J A B as / seen from cqn. (72). ("3 - "7) = ( W I - w : ) = I.JAT~I
("1
("4
"I)
=
(YI~
-
"I*)
=
.I*"
'/S(YA
+ 20+ - 20. +' T -T +I
- 1 (72) - " 6 ) = ("8, - v m ) = .Inn+ O + + Q + - T - - R L - V X ) = (913 - m ) = JA.- 0 - QT+ I?+
+
+
YB)
=
Y Z ~
(73)
YB)
'/.(.8
VII
(w -
YU)
'/w+I =
YDQ(+I
=
= YDQ(r.h')
=
Yo'h'
1 / 1 ~ ~ ~ + 1 8= YDQ(.bl
=
Y0l
'/ZYII'?
'/2"rn'a
=
._
Y+
= "DQ(-, = v -
Note carefully from eqns. (63) and (7.1) that in the ABX2 limit only three DQ transitions are expected, as v,.~. = v.,. For reasonably tightly coupled ABIi systcms, i t may be possible to determine the relative signs of all the couplings. It is convenient, however, to assume that (JAIc JBK)is positive SO that v+ > Y S . ~ . > nab > Y - when P A > YR >> "ti
+
and
u+
> v.6 > Y I W >
Y-
when
YK
>> Y A > Y A
(74)
Equation (74) derives from the fact that when V A > un >> ua and when vh >> uA > vn, the perturbation terms in eqn. (73) are positive arid negative, respectively. 1,ater in the analysis, the sign of JIBrelative to either .JAnor Jnn may be explored. The following procedurc is rcconnnendetl for the analysis of ARK, spcct,m. 3 111 t,he ARKx Step 1 . L o d e the f0111. ql~arlelsi n Ihc ~ f part spectrum a d dclewnine the m e w frequency of each. This pl.c,joct may be faaililnted st~bstantially b,v determiniug the foul. douhle q,l~ult,~tm Iran~iliowgive11i u eqn. (73). Lahel I-hese mow, freq~rmciesaccording to e q m (74). Label Itre 1.1wrsilions associated with the M, q ~ ~ n r l eas l s deswibed in t,he step 1 01 (he ABX* analy~ispraeedlire. Label the t,ransit,ions sssoeiated wi1.h the a'b' and ah qnartets 2,6,10, and 14 and 3,7,11, and 13, rospectively, pl.ogressing from ldw ta high field. This prtreedilre for labeling assumes a positive JAB. Thc repeated spacings given hy eqn. (72) must hold wit,hin experimental error. Ot,herwise, the line assignment is probably incorrecl. Slep B. Proceed to step 2 of the ABX2 analysis and calculate and 6": using eqns. (66)-(68). the exact pwnmetelw IJABI Slep O. I'roceed to step :J of thc ARX? anal,vsis. Use the value nf IJAHI ohlained i l l slep 2 (above), IJAK .Jmi) = ( u + - v . ) , and eqns. (70) and (71) with K = X l o caloolstc3' soh&ns 1 and 2. The solrrtion that givw a value of 6.t~that is clascst La lhat ohtained i n st,ep 2 may he ~.egardedas thc correrl one. Slep 4. Calculate VA and V B using eqn. (69) and take u x as the mean frequency of the K resonances. Iletermine values for &a and &. Using these 6& values and the values for JAB, JAK,and JBK obtained in steps 2 and 3, calculatevalues for perturbation terms O;, Q;, Ti and R;. (In doing this, give
+
See footnote 30. Volume 45, Number 7, July 1968
/ 485
J A Btheexpected sign relative t o t h a t of J A K(or J B K )as given hy step 3. No change in the line assignment is necessary.) Expressions for these pertwhation terms are given in Tahle23. ("I - n,) - Q: (v4 (u+
- u-)
-
WZ)
-
+
9'
+ 0: + 0'
+
Results of t h e A
+
B K ~ A ~ ~ I Y ofSt ~h eS NMR
.
(73)
+ JHK)
(76)
ABX2 IJm
Using e q n ( 7 3 ) rnleulate corrected values for 2S+b+ and ( J A K J d . Idenlify um of the K resonances. Use the coweeled value for S+b+, the v a h e for C given hy eqn. ( 6 7 ) and eqn. ( 7 6 ) tn cnlnllate a ron.eeletl vxhle for ur from whirl, rolxrt,ed values f o r 6:~ and 6 : ~are obtained. These are 11sed to obtain heitel. values for pert,n.hation terms whirh in turn m e used in eqns. ( 7 5 ) rind ( 7 6 ) to ohtxin better corrected vahles for 2S,b,, (Jm J m ) , and mi. This pmceaa should he iterated ~ u ~ tvisl h e r n m c ~ invariant i y o n reiteration, R e p 5. Using the best corrected parametela 2S,b + and ( J A ~ . : J a n ) from step 4 , retun, to step 3 and ealenlate a correcled ABK? sol~ition. This upre resents the OfDiteration. If corrected vallles for J A Kand Jna differ sr~bstantixllyfrom those ohtaiuod in step 3, step 4 should be iterated usirrg the corrected values of J A K and Jnn. This rcs~rltsin yet hetter corrected pnlxmetms 2S,h + and ( J A K J B K )which are used t o calculate a hal.ter col.rected ARK9 solution. This represents the 1st-iterntion. Iteralions shottld he continued ilntil the ABK, snh~tionht.eomm iwnl.ian1 upon reiter?tio~t. the magnit,,des of t h e pel.t,,rhation terms c+,,lated in step 4 cal. when sign of J~~ is ,.hanged, then c,,lated from stop 3 shoi,ld converge the i n step 2 ,lpon it,el.xtion if the J~~ wL9 in step 4 is con.ect,, ~f 6;n from st,ep 3 ,,Sing the hest parametel.s snhstantiallg fl.om it, st,ep 2, repeat st,eps al,d ;,,it,, having opposite sign, ~h~ sollltion that gives the closest hetweell the bwo values of 6in may he regarded as the correct, cine. Step fi. If it is desired to cnlcdate the spectrum lo iompare i l with t,hat ~ecorded,me the transition frequency expressions i n Table 24 together with the hest pertmhation terms P'ohtained in
JAK
Ba If designations v + and u had been interchanged, the above ARK* analysis would have given - J B K and + J A K And if J A B and J B K are expected t o have the same sign, the final signs att,arhed t o the parameters c o d d be either -JAB,- J B K , + J m or J m , J R K ,- J I K The appearance of the spectrum is sensit,ive onl?~ t o relative signs of the conplings. As vicinal couplings may he taken as heing positioe, t,he latter relative sign combination would he the preferred one.
18,
+
+
+
+
JaK
61, (from M * quartets) ab quartet)
"( YB uxb
Spectrum
of Vlll
Present Analyses"
2S+h+ = 2s-h=
- '/dT' - T: R' - R:) = (J*a ur; = ~ $ 1- S-hC/2 0' - T I R:
+
T a b l e 27.
--
ReportedD
0-Iteration ABC. ABKI (81)
13.1 -1.1 7.2
13.1 -1.1 7.2
13.1 -1.2 7.3
15.4
15.7
15.7 378.9 363.2 240.5
15.7 378.9 363.2 240.6
ABC* (18) 13.1 -0.5 7.1
" All entries are in units of cps. "elative
t o TMS.
Spectrum taken a t 60 Mc/sec.
step 4 and vn~,lcsfol. M; a,ld N, ohtxined fl.om eqI,s, ( 7 7 ) and (67),
mi
+ 61"
i (JAS -
&I)
=
%,I>
(77)
2S,h, given b,veqn. (75). To e x l n h t e the transilio~cintensilies that are hotter than first-ovdev convert the ARCt in tens it,^ expl.essionsiin Tnhle 23 t o ones rontaining s l m s of p r o d ~ ~ cof t s coetlcients following directions given in the section on Transition Tntensities in Part I. If the ABKg system is a weakly perturbed o v a l ~ ~ 101. e s the cocfiicients. When t.hesc AB?L one, iheu ~ s ABX, are zero, use first-order ARC-. values. >Lxp~esriansfor the AEXr and fitst-ordpr ARC* coeflicionts are piven in Tahle 22. I f the ARK, system is lightly r o ~ ~ p l e(i.?., t l more clnsely an ABCI systern), the best val~tesfor the coofiicients may he ohtained itsing ecln. ( 3 0 ) .
As nn illustration, the ABK, analysis approach has heen applied for the analysis of the spectrum of t~ans1,3-dichloro-1-propene, VIII, that is shown in r'g ' 1 ure H*
\
C1
/CITnlKIC1
/C=C\
HB
VIII
To facilitate the ident,ification of the four quartets in the AB part of the spectrum, the double quantum transitions were dca w l I~B.~. termined. These are shown in Figure 18. The line assignment followed readily from eqn. (74) and step 1 of the ARK, analysis procedure. The analysis was then completed following the directions in steps 2-6. The results are given in Table 27 and the ,, calculated "stick spectrum" is shown in Figure 18. The labeling of the A and B hydrogens in VIII follows from the relat.ive magnitudes of the couplings. The perturbation terms discussed in step 4 of the wm& analysis apptoach are practically insensitive to the sign chosen for JA8 with the result that the sign of this coupling relative and JBK cannot be deterto those of JAK mined by spectrum analysis. However, JAB and J a x may be assumed to have the same =., _.I mi A,., . -., =.. =,.I %a *,.* =,., sign as both are vicinal couplings. Since the line assignment adopted gives JsIc Figure 18. NMR spectrum I 6 0 Mc/sec) of bbns-1,3-dichloro-1-propene, VIII, in CClr (30% v/v). NU^^^,. under the .re frequenciesdownfleld from TMS. D O U ~ I ~~ o ~ i t i JAB ~ e ,a h may be assumed to be quontvm tranlitionr, obtained using dn rf = 0.2 mg, are shown in the top left spectrum. positive.32 urn(+,
3b .7
2o
ad.7
486
/
Journal of Chemical Education
The data compiled in Table 27 show that the ABXz and ABIG analyses of the spectrum of VIII give uearly identical results, even though the nondegener&cy of transitions (10 and 11) and (14 and 15) signal a perturbed ABX2 system. Similar results for other ABRz spectra should not be expected. Transition frequencies relative t o ThlS are given in Figure 18 so that interested readers can practice the ABlG analysis approach.
by treating it as a AA'BB' system having .Jas = JAB.. The condensed state diagram for the general four spin system (ABCD) may be factored into symmetric and antisymmctric components by recognizing the syrnmetrv nronerties of an AA'BB' svstem. Talcine
. . .
AA/B$; A', C = B, and D = B', \vc see that t,he a m pa and asap basic product functions may be combined t,o eive svmmetric and antisvmmetric functions. The
B
=
AA'BB' Systems (22)
An AzBz system may be defined as one in which by symmetry there arc two non-equivalent pairs of equivalent nuclei and one JABcoupling. Such a system may be schematically represented as
These systems are not common. Their spectra afford upon analysis only J*B and.,a: The more common AA'BB' system33 is defined as one in which by symmetry or conformational averaging there are tmo nonequivalent pairs of equivalcnt nuclei and two JABcouplings. This system may he represented schematically as AR -
JAB
The analysis of an A2B2spectrum may he accomplisl~ed Table 28.
8"~~clpi having the same chemienl shift hut different coupling constants t o any ehemicnlly shiited nwlens are distinguished Irom earh other by primes. System-i would be properly designated as an AA'X system. The two hydrogens in iaresymmetrically n o w equivalent, but have the same chemical shift, except for a negligible isolope effect. System-j wotlld be designated AA'BB'C..
I-lere, by symmetry, there are three nooequivnlent pairs of eqnivnlent. rn~clei. Primed and nnprimed nurlei of the same leller designation are commonly but improperly referred to as heiug "magnetically nonequivalent" (2.7). This !)sage cunf~sesstudents and professional rhemists alike. The A and A' in i are noneqnivitleot by symmetry and cansequent.ly m,zgnet.ic.zlly nonequivalent,; nueler~sA experiences diflerent n~tclcsrspin couplings than n w &us A', but their chemical shifts are essentially equivalent. Contrariwise, the A and A' nuclei in j are symmetrically and consequenlly magnetically equivalent,-nuclei A and A' experience JAC .LAAS equivalent nuclear spin cooplings [i.e., (JAB JAB,) = [JAW JA'C JA'A J A ~ B )and ] their chemical shif1.s are equivalent.
+
+
+
+
+
+
Nuclear Magnetic States for the AA'BB' System 2nd-order
12
1s 2s 3s 4s
5s 6s
78 8s 98
10s lla 12a
13a 14a 15a 16a
M 2 1 1 0 0 0 0
-1 -1 -2 1 1 0 0 -1 -1
P'
S
= [(denominator P/4)*
2
+ nomerator PI1/?- (dennminator P/X)
S = i l when deltmninntm P 2 0 Volume 45, Number 7, July 1968
/
487
The AA'BB' or AA'XX' spectrum is symmetrical about its midpoint. Table 29 provides only the AA' transitions. The BB' transitions, relative to ve, are enantiomerically related to the AA'ones. All AA'BB' and AA'XX' spectra consist of six quartets made up from the 26 fundamental transitions as given by eqn. (78).
he combined. This leads to the followiug condensed AA'BB' state diagram. A maximum of 28 single quantum transitions are possible in the AA'BB' system. AA'BB' s-states
M
AA'BB' a-states
No. of M = i transitions
Table 29.
28 (20-9 and
8-a)
Transition
The parameters that are ohtained from the analysis of an AA'BB' spectrum are:
No. 1 2 3 4 5 6
+
N = JAB J A B S 1, = .I*" - JAW K = .I**. Jnn. M = .I**. - Jnn.
+
Rxpressions for the second-ordcr cnergics of the sixteen st,at,es, t,ransit.ion frequencies, and t,ransition int,ensities that. derive from eqns. (21), (22), (24), and (25) are represented in Tahles 2S, 29, and :30. For conveniencc, expressions for the corrected perturhat,ion terms A' and B' werc added into thc t,ransition frcquency expressions. This leads to new terms 0 and P that appear in Table 29. I n the AA'XX' limit only states (6 and 7) and (13 and 14) mix and conseqnently A' and B' become exact perturbations and t,he transition frequency expressions in Table 29 are cxact. Table 30.
Exact Intensity AA'XX'
10 11
12 13 14
Transition Frequency
3-1 5-3 7-2 9-6 6-2 9-7 8-4 10-8 14 11 16- 13 13-11 16- 14 5 2 (comb) 9 4 (comb)
--
N/2+ R
.A+ VA
v* VA PA VA
VA
PA
++ NK // 22 +- RR ++ S0+-+S+Q++ S- K/2
+ R + 0 + Q+ - Q-
+
+ +
+ K / 2 + R - 0 - Q+ + Q - K / 2 + R - 0 + S+ - S - N / 2 - R + S- + Q- N/2 + R
MI2 P T+ v,-M/Z+P+T"A+ M I 2 - P + T + VA MI2 P+TVA 6:n 'AN R VA 6 : ~ - '/zN R v*
-
-
+
+ +
+ + S+ + Q+ + 4- S - + Q -
7 9
-----
6 9 8 10 141613 16
(aCrs (ad-
2 6 2
-
Journal of Chemical Education
- ac+ + bc+P - af - bf - bd+)Z
+
7
4 8 11 13 11 14 5 2 (comb) 9 -.4 (comb)
Designation
Approximate Intensity AA'KK' (a - b)'
3-1 5-3
/
7 8 9
Transition
Relative Transition Intensities for the AA'XX' and AA'KK' Swtemr
Transition
488
Transition Frequencies far the AA' Part of an AA'BB' System
1st-order
+
(ae ac+ - eb c-b)' eb c+b)% (ae - acbf)a (-ad+ - af - bdacbc-)l (aC,, (a bI2 (ib+ - i g + P (ig- - jh-)= ih+IP (ig+ jg_)z (ih( ~ C S- ac, - bc+Y (ac- - bC4. - b c ~ Y
+
+ + +
+
+
++
AA'XX' and AA'KK'
ab quartet c,d quartet,
s qnartet, 8' quartet a quartet a' quartet
l,U,S',I' 2,7,7',Ir 3,4,5,6 3',4',5',6' 9,10,11,12 9',10',11',12'
6 1 ~ -6:~
JAB
-JAB
-L -L -L
-K -K
(78)
-M -M
-I.
The primes in eqn. (78) indicate R-transitions. If the ah, s, and a quartet.^ can be characterized, the exact analysis of AA'BB' and AA'XX' spectra can be completed according to the following discussions. Analysis o f AA'XX' Spectra
The AA'XX' states and transition frequencies are given in Tahles 28 and 29 where perturbation terms Q*, R, S,, and T+ vanish. Under these conditions the ab and cd quartets coincide and become an AX type quartet, and the s and a qnartets become true AB type quartets with J = K and AT, respectively and 6 O = L for both. As seen from Table 29, uA is the mean frequency of v , . ~and v,.~ and of the s and a quartets. Consequently, the A transit,ions are enantiomerically relat,ed about v*. The AA'XX' spectrum is insensitive to the relative signs of 1\T, I,, I< and A[ because the pert,urbation t,erms 0 and P contain neither K nor non-squared values of 1 , 1 , and I . The import,ant consequence of this is t,hat .IAh, cannot be distingnished from J x x , nor can .TAX be dist,inguished from J ~ x on r the basis solely of spectral analysis. For example, t,he spectrum corresponding t,o JAs= 2, .Inx.= 6, .IA*, = - 5 , .Ixx, = 12 (I, = -4, 31 = -17) is indist,inguishithlc from spectra corresponding to .Inx= 6, J A X ,= 2, J A A ,= -5, J x s , = 12 (1, = 4 , 3 1 = -17), JAX = 2, JAX, = 6, J a n , = 12, .Issr = - 5 (I, = -4, 11 = 17), and JAx= 6, J A X= , 2, J a a , = 12, J x x , = -5 (I, = 4, 11 = 17). Also, A1 cannot be dist,inguished from I> L
K = M (general case)
8
"
+
K=M
