JOURNAL OF MAGNETIC RESONANCE, Series A 123, 135–139 (1996)ARTICLE NO. 0226

The Tolerance to J Mismatch and the Symmetry Propertiesof INADEQUATE CR

NIELS CHR. NIELSEN* AND OLE WINNECHE SØRENSEN†

*Department of Chemistry, University of Aarhus, DK-8000 Aarhus C, Denmark; and †Novo Nordisk A/S, DK-2760 Maløv, Denmark

Received August 20, 1996

We have recently introduced the INADEQUATE CR I01 I02 Å {2QC}x 0 i{2QC }y [2](composite refocusing) experiment (1, 2) for tracing out the

with {2QC }x Å I1x I2x 0 I1y I2y and {2QC }y Å I1x I2y / I1y I2x .carbon skeleton of molecules by NMR. It has about a twofoldIn the case of ideal RF pulses, the effect of the antiecho– bsensitivity advantage over the conventional INADEQUATE

INADEQUATE CR mixing sequence (henceforth referred toexperiment (3–6) . This advantage comes about by an effi-as M) on these two terms may be writtencient tailored conversion of double-quantum coherence

(2QC) to observable single-quantum coherence (1QC) fordetection.

{2QC }xM 1

2√2

[(2I1y I2z / 2I1z I2y) 0 s(I1y / I2y)Ideally, either the echo or the antiecho coherence transferpathway is selected exclusively and transfer occurs only to

/ c(2I1x I2z / 2I1z I2x)] [3a]either the left or the right doublet lines of two-spin systems,e.g., for polarization of b lines in the antiecho,

{2QC }yM s

2√2

[(I1x/ I2x)

I01 I02 r1√2

{I01 Ib2 / Ib1 I02 }, [1]0 s(2I1x I2z/ 2I1z I2x)0 c(I1y/ I2y)]

0 c

2[c{2QC }y0 {Z QC }xrepresenting one of the four variants of the INADEQUATE

CR experiment. Equation [1] employs the usual definitionsI{ Å Ix { iIy and Ia /b Å 1

2(1 { 2Iz) ./ 2I1z I2z0

s

2(I1z/ I2z)] [3b]This extreme selectivity hinges on exact matching of the

delays in the mixing sequence to the J coupling constant. It isthe purpose of this Communication to analyze how sensitive with the coefficients c Å cos(pJt) and s Å sin(pJt) andINADEQUATE CR is to mistuning of delays. Furthermore, the zero-quantum operator defined as {Z QC }x Å I1x I2x /using the symmetry properties of the four INADEQUATE I1y I2y .CR experiments we describe generation of pure 2D absorp- Linear combination of Eqs. [3a] and [3b] according totion spectra and so-called ‘‘merged’’ spectra providing ap- Eq. [2] yieldsparent homonuclear decoupling.

The INADEQUATE CR pulse sequence with pulsed fieldI01 I02

M 1

2√2

[2I1y I2z/ 2I1z I2ygradients is outlined in Fig. 1 while phase cycles for thefour individual variants of the pulse sequence can be foundin Table 1. For the sake of simplicity we ignore the gradients 0 s(10 ic)(I1y/ I2y)and the p pulses in the d delays in the present analysis; these

/ (c/ is 2)(2I1x I2z/ 2I1z I2x)0 is(I1x/ I2x)]two p pulses have the combined effect of interchanging themagnetizations of the right and left doublet lines. Further-

/ ic

2[c{2QC }y0 {Z QC }xmore, since the spectra resulting from the four individual

experiments are symmetry related, we restrict detailed analy-sis to the antiecho– b variant of INADEQUATE CR, ideally / 2I1z I2z0

s

2(I1z/ I2z)] [4]

producing the coherence transfer in Eq. [1] .It is most convenient to calculate the effect of the six-pulse

mixing sequence in the Cartesian product-operator basis (7) . which upon transformation into a basis spanned by single-element operators takes the formThe {02QC} operator takes the form

135 1064-1858/96 $18.00Copyright q 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

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A qualitative schematic 2D spectrum resulting from appli-cation of M with the four possible peaks detected on spinI1 is shown in the upper left panel of Fig. 2. In practice, aphase cycle or pulsed field gradients could eliminate theecho part of this particular spectrum. The remaining threepanels of Fig. 2 contain schematic spectra for the other threeINADEQUATE CR mixing sequences pMp, pM , and Mp

FIG. 1. INADEQUATE CR pulse sequence for optimum conversion of predominantly giving rise to the antiecho– a, echo– b, and2QC into 1QC through selective transfer into either the a or the b line in echo– a peaks, respectively. The nomenclature for the threeeither the echo or the antiecho spectrum. The gradients in the second d experiments is based on the fact that their mixing sequencesdelay are of double amplitude compared to those in the first d delay. Phase

may be derived from M by insertion of p pulses immediatelycycles for the six pulses in the mixing sequence of the four experimentsbefore or after it. The intensity expressions for all four vari-are given in Table 1. In combination with these, standard INADEQUATE

phase cycles may be applied using the same cycles for the entire mixing ants are summarized in Table 2.sequence as one would for the conventional single-pulse mixing sequence. The quantitative effect of mistuned delays for the mixingUnless otherwise indicated above the pulses, filled and open bars represent

sequence M is illustrated in Fig. 3 with separate plots ofp /2 and p pulses, respectively. The delay t is ideally tuned to an oddreal and imaginary parts of the single-quantum coefficientsmultiple of (2J)01 .in Eq. [5] as a function of the deviation DJ /J0 Å (J 0 J0) /J0 , where J is the actual value and J0 the value used insetting the delays, i.e., t Å (2J0)01 . The two most dominantunwanted signals introduced upon mistuning of the delaysI01 I02

M 1

2√2

[[c(1 0 s) / i(1 0 s)2]are those in the echo part of the spectrum which can besuppressed by gradients or phase cycling. Fortunately, the1 {I01 Ia2 / Ia1 I02 }neighboring doublet line (in this particular case, the anti-

0 [c(1 / s) / i(1 / s)2]{I01 Ib2 / Ib1 I02 } echo– a signal) exhibits a very small amplitude. Taylorexpansion of the transfer functions around DJ /J0Å 0 reveals/ [c(1 / s) 0 ic 2]{I/1 Ia2 / Ia1 I/2 }that transfers from I01 I02 into I01 Ia2 / Ia1 I02 , I/1 Ib2 / Ib1 I/2 ,

0 [c(1 0 s) 0 ic 2]{I/1 Ib2 / Ib1 I/2 }] and I/1 Ia2 / Ia1 I/2 vanish to second, second, and zeroth order,respectively, for the real part and third, first, and first order,

0 c

4[c(I/1 I/2 0 I01 I02 ) / i(I/1 I02 / I01 I/2 respectively, for the imaginary part. As is evident in Eq. [5]

and Fig. 3, it is the imaginary part that is relevant and asuppression of the neighboring doublet line to third order is0 Ia1 Ia2 0 Ib1 Ib2 / Ia1 Ib2 / Ib1 Ia2 )] . [5]

TABLE 1Phase Table for the Mixing Sequence of INADEQUATE CRa

M pMp pM Mp

f1 f2 f3 f4 f3 f4 f3 f4 f3 f4 f5 f6 frec

x y 0y x y 0x y x 0y 0x x 0y x0x x 0y x y 0x y x 0y 0x x 0y x

x x y 0x 0y x 0y 0x y x y 0y x0x y y 0x 0y x 0y 0x y x y 0y x

x y 0y 0x y x y 0x 0y x x y x0x x 0y 0x y x y 0x 0y x x y x

x x y x 0y 0x 0y x y 0x y y x0x y y x 0y 0x 0y x y 0x y y x

x y 0y x y 0x y x 0y 0x y y 0x0x x 0y x y 0x y x 0y 0x y y 0x

x x y 0x 0y x 0y 0x y x x y 0x0x y y 0x 0y x 0y 0x y x x y 0x

x y 0y 0x y x y 0x 0y x y 0y 0x0x x 0y 0x y x y 0x 0y x y 0y 0x

x x y x 0y 0x 0y x y 0x x 0y 0x0x y y x 0y 0x 0y x y 0x x 0y 0x

a The phases f1–f6 are assigned in Fig. 1 and frec is the receiver reference phase.

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parent homonuclear decoupling. For convenience we referonly to the signals detected on spin I1 of a two-spin system;the results apply of course to all signals in INADEQUATECR spectra. The combined I1 intensity function for the fourexperiments is

F(v1 , v2) Å FM(v1 , v2) / FpM(v1 , v2)

/ FpMp(v1 , v2) / FMp(v1 , v2) [6a]

with

Fq(v1 , v2)Å f abq d(v1/S)d[v2/ (VI1

/ pJ)]

/ f aaq d(v1/S)d[v2/ (VI1

0 pJ)]

/ f ebq d(v10S)d[v2/ (VI1

/ pJ)]

/ f eaq d(v10S)d[v2/ (VI1

0 pJ)] [6b]

and S Å VI1/ VI2

( the coefficients f pq are defined in Table

2 and d is a delta function). For the complex peakshapeFIG. 2. Qualitative schematic 2D spectra resulting from application of function the definition is

the antiecho– b (M) , antiecho– a (pMp) , echo– b (pM) , and echo– a(Mp) INADEQUATE CR pulse sequences (see text) with mistuned delaysto a two-spin system of which only the signals detected on I1 are shown. L(Dv1 , Dv2) Å a(Dv1)a(Dv2) 0 d(Dv1)d(Dv2)The symmetry operations within the C2£ symmetry group interrelating theintensity functions for the four sequences are indicated. / i[a(Dv1)d(Dv2)

/ d(Dv1)a(Dv2)] [7]

more than adequate in practical applications. Clearly, INAD-with a(Dvi ) Å (1/T2) / [(1/T2) 2 / (Dvi )2] and d(Dvi )EQUATE CR is extremely robust toward mistuning of the ÅDvi / [ (1/T2) 2/ (Dvi )2] denoting Lorentzian absorptiondelays. Even for a severe case of 40% mismatch the ampli-and dispersion profiles, respectively, in the vi dimension oftudes of the wanted and unwanted doublet lines (ideally 100the 2D spectrum. The four 2D spectra may then be character-and 0%, respectively) are 86.02 and 2.95%, respectively,ized by the convolutionwhich still ensures a significant sensitivity enhancement rela-

tive to conventional INADEQUATE.S(v1 , v2) Å F(v1 , v2)∗L(Dv1 , Dv2) , [8]The symmetry relationship between the intensity functions

of the four INADEQUATE CR variants and the symmetryproperties of the 2D peakshape function (9) are important where the components of F(v1 , v2) and L(Dv1 , Dv2) trans-

form according to irreducible representations of the symme-for the generation of pure absorption 2D peakshapes andfor the so-called ‘‘merged’’ spectrum (1, 8) where all four try groups for the intensity (GF) and peakshape (GL) function,

respectively. For each of these functions the symmetry prop-INADEQUATE CR subspectra are combined to provide ap-

TABLE 2Intensities f p

q as a Function of t for All Detectable Peaks Resulting from the Four Versions(q Å M, pMp, pM, and Mp) of INADEQUATE CRa

q/p ab: I01 I02 r I01 Ib2 / Ib1I02 aa: I01 I02 r I01 Ia2 / Ia1I02 eb: I/1 I/2 r I01 Ib2 / Ib1I02 ea: I/1 I/2 r I01 Ia2 / Ia1I02

M 0b/ 0 ia/ b0 / ia0 0b0 0 id b/ / idpMp b0 0 ia0 0b/ / ia/ b/ 0 id 0b0 / idpM b0 / id 0b/ 0 id b/ / ia/ 0b0 0 ia0Mp 0b/ id b0 0 id 0b0 / ia0 b/ 0 ia/

a The coefficients are a{ Å (1 { s)2/(4√2), b/ Å c(1 { s)/4

√2), and d Å c2/(4;

√2) with s Å sin(pJCCt) and c Å cos(pJCCt). ab, aa, eb, and ea denote

antiecho–b, antiecho–a, echo–b, and antiecho–a, respectively. Bold entries in the table indicate the respective main peaks.

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erties may be described using the transformation formula and the character table for the irreducible representationsassociated with the various I1 spin signal components is[e.g., for F(v1 , v2)]

C2(v1 Å 0,C2y E v2 Å VI1

) sn(v2 Å VI1) sn(v1 Å 0)PF

RF(v1 , v2) Å x ( j )F (R)F(v1 , v2) [9]

F(B1)(v1 , v2) Å Re[F(v1 , v2)] 1 01 1 01

F(A2)(v1 , v2) Å Im[F(v1 , v2)] 1 1 01 01and the associated projection operators

C2(Dv1 Å 0,E Dv2 Å 0) sn(Dv2 Å 0) sn(Dv1 Å 0)

P ( j )F Å 1

h∑R

x ( j )F (R)PF

R , [10] L(A1)(Dv1 , v2) Å a(Dv1)a(Dv2) 1 1 1 1

L(A2)(Dv1 , Dv2) Å d(Dv1)d(Dv2) 1 1 01 01

L(B1)(Dv1 , Dv2) Å d(Dv1)a(Dv2) 1 01 1 01

L(B2)(Dv1 , Dv2) Å a(Dv1)d(Dv2) 1 01 01 1

where R is a group element, PFR a group operator, h the

group order, and the set of h numbers x ( j ) (R) Å {1 thecharacter for the j th one-dimensional irreducible representa- with the components classified according to their respective

irreducible representations, i.e.,tion G ( j )F of the symmetry group for F(v1 , v2) (10, 11) .

F(v1 , v2) and L(Dv1 , Dv2) both exhibit C2£ symmetryF(v1 , v2) Å F (B1) (v1 , v2) / iF (A2) (v1 , v2) [11a]

L(Dv1 , Dv2) Å L (A1) (Dv1 , Dv2) 0 L (A2) (Dv1 , Dv2)

/ iL (B1) (Dv1 , Dv2)

/ iL (B2) (Dv1 , Dv2) . [11b]

It is now straightforward to describe the generation ofpure absorption phase 2D INADEQUATE CR spectra. Thesignal function S(v1 , v2) is characterized by the direct-product group GF1L accommodating the commuting opera-tions of GF and GL (11) . Within GF1L , elimination of termsdepending on d(Dv1) corresponds to selecting the signalcomponents transforming according to the direct-productrepresentations G (B1)

F # G (A1)L , G (B1)

F # G (B2)L , G (A2)

F #

G (A1)L , and G (A2)

F # G (B2)L . Using the relation x ( ij )

F1L(RQ) Åx ( i )

F (R)x ( j )L (Q) it is evident that this selection may be ac-

complished using the projection operator

P2D abs Å P (B1A1)F1L / P (B1B2)

F1L / P (A2A1)F1L / P (A2B2)

F1L

Å 12[PF1L

EE 0 PF1Lsn(v1Å0)sn(Dv1Å0) ] [12]

leading to

P2D absS(v1 , v2) Å 12[PF

EF(v1 , v2)∗PLEL(Dv1 , Dv2)

0 PFsn(v1Å0)F(v1 , v2)∗PL

sn(Dv1Å0)

FIG. 3. Graphical illustration of the effect of mistuned delays for the 1 L(Dv1 , Dv2)]antiecho– b mixing sequence of INADEQUATE CR on the real and imagi-nary intensity components for the four observable peaks on one of the two Å F(v1 , v2)∗ [L (A1) (Dv1 , Dv2)spins as a function of the deviation DJ /J0 (DJ Å J 0 J0) , where J is theactual value and J0 the nominal value used in the delay settings [t Å / iL (B2) (Dv1 , Dv2)] . [13](2J0)01] . The curves represent the coefficients in Eq. [5] for transfer ofI01 I02 into I01 Ib2 / Ib1 I02 (solid line) , I01 Ia2 / Ia1 I02 (dashed line) , I/1 Ib2 / In practice, the pure 2D absorption projection operator inIb1 I/2 (dot-dashed line) , and I/1 Ia2 / Ia1 I/2 (dot-dot-dot-dashed line) . All

Eq. [12] is realized by combination of the spectra recordedcurves are normalized through multiplication of the coefficients in Eq. [5]with M and pM to yield the b spectrum or pMp and Mpby a factor 0

√2. Scaling by sin(pJt) from the excitation sequence, which

also scales the conventional INADEQUATE experiment, is not included. to yield the a spectrum (cf. Fig. 2) . This is equivalent to

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the procedure employed in modern gradient-enhanced multi- Smab(v1 , v2) Å iFA2M (v1 , v2)

dimensional NMR spectroscopy for obtaining pure absorp- 1 [a(Dv1)a(Dv2)tion peakshapes.The expressions for the a and b spectra are / ia(Dv1)d(Dv2)] . [16]

Sa(v1 , v2) Å [F (B1)pMp(v1 , v2) / iF (A2)

pMp(v1 , v2)] Finally, we note that the practical approximation to mabINADEQUATE CR spectra involves combination of a and1 [a(Dv1)a(Dv2)b INADEQUATE CR spectra shifted J /2 and 0J /2, respec-

/ ia(Dv1)d(Dv2)] [14a] tively, as described in Ref. (1) .In conclusion, we have demonstrated that INADEQUATESb(v1 , v2) Å [F (B1)

M (v1 , v2) / iF (A2)M (v1 , v2)]

CR is very robust toward variation in J coupling constants.1 [a(Dv1)a(Dv2) This is important for application of the method to molecules

with a large relative spread in J coupling constants./ ia(Dv1)d(Dv2)] , [14b]

REFERENCESwhere for the main peak, F (B1)pMp(v1 , v2) and F (B1)

M (v1 , v2)take the value 0b/ , while for F (A2)

pMp(v1 , v2) and 1. N. C. Nielsen, H. Thøgersen, and O. W. Sørensen, J. Am. Chem.F (A2)

M (v1 , v2) it is a/ and 0a/ , respectively (cf. Table 2). Soc. 117, 11365 (1995).As is evident in Eq. [14], t x (2J)01 causes mixed phases 2. N. C. Nielsen, H. Thøgersen, and O. W. Sørensen, J. Chem. Phys.(see Fig. 3) in the a and b INADEQUATE CR spectra. 105, 3962 (1996).

This is, however, for most practical applications quite insig- 3. A. Bax, R. Freeman, and S. P. Kempsell, J. Am. Chem. Soc. 102,4849 (1980).nificant as even ÉDJ /J0É Å 0.4 results in a phase error of

4. A. Bax, R. Freeman, and T. A. Frenkiel, J. Am. Chem. Soc. 103,only 187.2102 (1981).Pure 2D absorption peaks independent of J can be

5. A. Bax, R. Freeman, T. A. Frenkiel, and M. H. Levitt, J. Magn. Re-achieved by the additional operation of projecting out theson. 43, 478 (1981).

F (A2) term of the intensity function using the projection oper-6. T. H. Mareci and R. Freeman, J. Magn. Reson. 48, 158 (1982).

ator7. O. W. Sørensen, G. W. Eich, M. H. Levitt, G. Bodenhausen, and

R. R. Ernst, Prog. NMR Spectrosc. 16, 163 (1983).P (A2)

F Å 14[PF

E / PFC20 PF

sn(v2ÅVI1) 0 PF

sn(v1Å0) ] [15] 8. M. Sattler, J. Schleucher, O. Schedletzky, S. J. Glaser, C. Grie-singer, N. C. Nielsen, and O. W. Sørensen, J. Magn. Reson. A 119,171 (1996).

which may be reduced to 12[PF

E 0 PFsn(v2ÅVI1

) ] since F(v1 ,9. S. Boentges, B. U. Meier, C. Griesinger, and R. R. Ernst, J. Magn.

v2) only contains components transforming according to Reson. 85, 337 (1989).G (B1)

F and G (A2)F . The relevant operation corresponds to sub- 10. P. W. Atkins, M. S. Child, and C. S. G. Phillips, ‘‘Tables for Group

tracting PFsn(v2ÅVI1

)Sa(v1 , v2) from Sb(v1 , v2) . According Theory,’’ Oxford Univ. Press, London, 1970.to Eq. [14] and Table 2 one obtains for this merged ab 11. M. Tinkham, ‘‘Group Theory and Quantum Mechanics,’’ McGraw–

Hill, New York, 1962.INADEQUATE CR spectrum (mab)

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