Two-Dimensional spin-echo multiple-quantum transitions in strongly coupled spin systems. Calculation of spectra

JOURNAL

OF MAGNETIC

RESONANCE

56,479-509

(1984)

Two-Dimensional Spin-Echo Multiple-Quantum Transitions in Strongly Coupled Spin Systems. Calculation of Spectra M. ALBERTTHOMASANDANILKUMAR Department of Physics, Indian Institute of Science, Bangalore 560 012. India Received April 6, 1983; revised August 22, 1983 A general method for computation of two-dimensional spin-echo multiplequantum spectra of strongly coupled spin systems is presented. It has been demonstrated that the 180” pulse can severely modify the multiplequantum spectra of strongly coupled spins and explicit spectral details are given for the spin systems of types AB, ABX, and AB2. In addition to the modified frequencies and additional transitions, it is found that such spectra possessadditional frequency symmetry with respect to the center of each multiplequantum order, even though the multiplequantum spectrum without the 180” pulse may not be symmetric. The ABX spin system is treated in detail and the behavior of the homonuclear and heteronuclear multiplequantum transitions in the presence of selective or nonselective 180” pulses is discussed. It has been demonstrated that strong coupling allows transfer of magnetization from abundant spins to a coupled rare spin even in the absence of excitation pulses on abundant spins. The amplitudes of various coherences created by selective and nonselective pulses and the nature of spin-echo zeroquantum transitions of strongly coupled spins are discussed in appendices. I. INTRODUCTION

There has been rapid progress in recent years in the excitation, detection, and application of multiplequantum transitions in NMR, largely due to the availability of pulse techniques. Multiple-quantum coherences can be excited in a straightforward manner and several schemes utilizing selective and nonselective pulses for selective and nonselective excitation of multiple-quantum coherences have been suggested (I21). The excited multiple-quantum coherence evolves for a time t, and is transformed by a 90” pulse into detectable single-quantum coherence. Both one- and two-dimensional techniques have been utilized for study of multiple-quantum transitions, yielding useful information regarding connectivity of transitions and relaxation in homo- and heteronuclear coupled spin systems (Z-32). In particular, the utility of higher-quantum transitions in the analysis of coupled spins has been demonstrated (7, 8, 10, 27, 28).

The inhomogeneity of the magnetic field contributes proportionately to the widths of higher-order transitions. The use of a refocusing 180” pulse in the middle of the multiple-quantum evolution time eliminates the inhomogeneous broadening leading to much improved linewidths (5, 7, 8). Selective excitation or selective cancellation of orders accompanied with the removal of field offset using a 180” pulse, can also drastically reduce the data processing and storage requirements in two-dimensional (2D) multiple-quantum spectroscopy. 180” pulses have also been utilized in multiple479

0022-2364184 $3.00 Copyright Q 1984 by Academic Press. Inc. All rights of reproduction in any form reserved.

480

THOMAS

AND

KUM:AR

quantum spectroscopy during preparation, mixing and detection periods (5-10, 1923). Recently, it has been demonstrated that for strongly coupled spins, the spinecho multiplequantum spectrum obtained using a 180” refocusing pulse (SEMQT) is different from that without the 180” pulse (MQT), giving additional lines and new frequencies (33). In this paper a general formalism is presented for calculation of two-dimensional SEMQT and MQT spectra of weakly and strongly coupled spins (Section II). The SEMQT spectra of strongly coupled spins show several new features and explicit spectral details are presented for several spin systems for nonselective and selective usage of refocusing 180” pulse in Sections III and IV, respectively. The absolute value projection of the 2D spectrum on to the multiplequantum axis yields one-dimensional MQT and SEMQT spectra which are also discussed. In Appendix A the amplitudes of the multiple-quantum coherences created by different preparation pulse sequences for several spin systems are presented. In Appendix B, the origin and phase of various zero-quantum transitions (ZQT) in the presence of a 180’ pulse are discussed. A result of this analysis is that for a nonselective 180” pulse the SEMQT spectra of all orders except single, for weakly as well as strongly coupled spin systems, contain the same w1 frequencies as the corresponding J-resolved spectra (34). If the various orders of SEMQT spectra are separated, these frequencies are distributed over several orders. In J-resolved spectroscopy each w2 frequency contains only one line at a particular w, frequency in the weak coupling limit and several lines in the strong coupling limit. In SEMQT and MQT, each w2 frequency, in general, correlates to every multiple-quantum coherence, due to the use of a mixing pulse. To obtain additional information from SEMQT spectra, separation of orders is therefore necessary. For the AB spin system, however, the zero- and double-quantum SEMQT frequencies are quite different from those in J spectroscopy (33). In higher-order spin systems, use of a selective 180” pulse also results in nonequivalence of SEMQT and J-spectroscopy spectral frequencies (Section IV). II. THEORY

The experimental schemes for excitation and detection of multiple-quantum transitions are shown in Fig. 1. The multiplequantum coherence, excited either by a pair of 90” pulses spaced by an interval 7 or by a selective saturation or inversion of a given transition followed by a 90” pulse (1) is allowed to evolve during period cl at the end of which another 90” pulse, called the mixing pulse, transforms it into detectable single-quantum coherence which is observed during the period t2. The resulting two-dimensional spectrum in general, correlates each singlequantum frequency of a coupling network to every multiplequantum transition of all orders of the network. The signal in such a scheme is given by, s(t,, t2, T) = Tr {F,e-

i~f2p-le-i~t~/2R-le-i~t~/2

0 (r (7,

~)e~‘l’2Re~rl’2Pe~~*},

[l]

where a’(~, 4) is the density operator at the end of the preparation period and contains the multiplequantum coherence. The values of a0 for various spin systems for d&rent type of preparation schemes are given in Appendix A. X’ is the Hamiltonian of the

2D SPECTRA OF STRONGLY

0

D

0

gay+*

9OY+$

+ru-T-w--

COUPLED

t1

481

(b)

0

'BOY+)'

SYSTEMS

T+p

t1 1-u--tZ--+

FIG. 1. The experimental schemes for two-dimensional spin-echo multiplequantum spectroscopy with preparation of the spin system in a nonequilibrium state of (a) kind I, utilizing a selective inversion or saturation of one or more transitions by a soft pulse representi by the shaded area; or (b) kind II, utilizing general perturbation of the spin system by a 90” pulse followed by time interval, 7.

spin system and P and R are rotation operators associated with the mixing and the refocusing pulses, respectively. Here P = exp(-i#J)P,

exp(irCIJZ>, R = exp(-i~%)R,

expW’K,),

in which R, = exp(i?rK,),

Fx= 5 Z,(l), /=I

Jy = 5 Z,(m),

and

Ky = $5 Z,(q).

m=l

q=l

PI

The sums over 1, m, and q, respectively, run over the nuclei detected during tz, the number of spins on which the 90 and the 180” pulses are nonselectively applied. It can be shown that a nonselective 180” pulse refocuses the evolution of coherence during tl arising from all interactions linear in spin operators, such as field inhomogeneity, resonance offset, and chemical shift, and it does not perturb those due to bilinear interactions such as indirect spin-spin coupling and dipolar coupling, provided each of these terms commutes with the total Hamiltonian. The chemicalshift term commutes with the total Hamiltonian only in the weak coupling limit. Under such circumstances, it can be shown that the SEMQT spectrum is identical to the MQT spectrum except that its center is at zero frequency. In the event of strong coupling, the above simplification is not achieved and the following procedure for computation of the SEMQT spectra is adopted. The rotation operators P,, and R, for spin-l/2 nuclei can be expressed, using Eqs. [4]-[7] of Ref. WI, as R, = fi [Z+(q) - Z-(q)] = fi F$?(q) q=I q=l

]31

P,, = 3, k [I + 2iZ,(m)] = 2-M’2 ,g F$?’

]41

and

D,

S.N.

A

and C

MQT

DEPENDENCE

k( 1 - s2e)cau!&

FLIP-ANGLE

and D

z,

[

1 x-x28f s4es2 - LY uS2 2 1

+(l - s2e)caoq,

OF THE

1

A

43)

SEMQT

OF AN ORIENTED

and C

2D SPECTRA

TWO-SPIN

B

(AB)”

and D

c32ea!2 c2es22ea$2 -c22es2e(uq2 - &‘)

+(I - s2&4

SYSTEM

43)

28

“Z, S, D, and T refer to zero-, single-, double-, and triplequantum transitions, respectively, with subscripts giving a serial number and a prime superscript denoting a SEMQT transition. The upper signs refer to S, and S, and the lower signs to S, and S,. Such notation is followed in all tables in this paper. The frequencies of the singlequantum transitions along w2 (S, - . * &) and the various multiple-quantum transitions along W, are as follows: S, = bAB + (1/2)c’ + (l/2) (J + 2D); S, = &a T (1/2)c’ -+ (1/2)(J + 2D); D, = 213~~; 0; = 2A, Z, = c’; Z’, = 0; Zq = c’, Z; = (1/2)c’, where bAB = (f/2)(“* + vB); c’ cos 28 = cvA - ve); c’ r&r = (J - D); c’ = [(VA - Vs)2 + (J - ~)2p*. All intensities to be multiplied by (l/S) sin (Y. Abbreviations used: C = cos; S = sin.

3

s2

D,

TABLE INTENSITIES

u:2 (c2fa+s2f di2e z; a48[(1- s2e) kc22es2e 1(a;2- u:3) +(I+s2ebq,0; +(I+s2eyh4, -t(i+s2e)&, z, c32e[7ca -e2s2es i a]~92 c2eaj2 Z c2es22e [ +caT2s2es2 1a1dj2 +s4e (1+s2e) c2f a- s*i d2e) +c22es2 i a1u82 c22es2e( a42 z-[( +c22es2e cff- 2s2es2 i a1(& z?

2;

D;

S.N.

AND SEMQT

+(l - S28)09,

B

OF THE MQT

s,4 z, C*cd28f .wes* -1(Y~5~ -c2eai2 2 1

S.N.

THE

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

483

etc.

PI

where

FI’.M’ = 5 F:‘)@), m=l

Fyfl)

=

$

FZ”(m)FS)‘(m’),

m=l m’=rn+l

F$?“) =

5

F~)(m)F~)(m’)F(:)(m”),

m=l

m’=m+ 1 m”=m’+l

FIG. 2. Schematic plot of the X part of the two-dimensional multiplequantum spectrum of the ABX spin system calculated for the experimental scheme of Fig. lb, without the 180” pulse and with # = 4 = I$’ = 0. The parameters chosen are JAB = 5.6 Hz, JAx = 14.8 Hz, JBx = 7.4 Hz, 16* - &,I = 5.0 Hz, and 7 = 50 msec. The absolute intensities are schematkdly represented by three contour levels, l-7, 8-20, and above 20, with the dots representing very low (cl) intensity. The numbers indicate the calculated intensity of each peak; given only for the right half of the 2D plot. The left half has symmetric intensities unless stated otherwise. Intensities less than 0.1 are not indicated. The projection on the top shows the X part of the one-dimensional ABX spectrum and the projections on the right show schematically the onedimensional MQT spectrum of the ABX spin system. The singlequantum part of the MQT spectrum is not shown.

484

THOMAS

AND KUMAR

From Eqs. [3] and [S] it is evident that for M := Q, F(YM~M)= Ry and the rotation operator Py for a nonselective 90” pulse is equivalent to a sum of rotation operators, R,, of selective 180” pulses acting on 0, 1, 2, 3, ‘I * - M spins. For example, for an AX spin system

P:X=~(n+R;'fR~-~R~R~~. Taking matrix elements in Eq. [ 11in the basis in which the Hamiltonian one gets

w2

-

P51 is diagonal,

FIG. 3. Schematic plot of the X part of the two-dimensional spin-echo multiplequantum spectrum of the ABX spin system calculated for the experimental scheme of Fig. lb, with 4 = t,A. The projection on the right shows schematically the onedimensional SEMQT spectrum. Other details are the same as in Fig. 2.

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

485

TABLE 2 THE MQT

AND

SEMQT FREQUENCIES

OF THE 2D SPECTRUM

FOR AN ABX

SPIN SYSTEM”

SEMQT

MQT S.N.

w2

S.N.

Ix f D,

a3

T

bx t J

s3

WI

D,

4

03

&e + ; [J + J’ ? D,]

D5 6

4

z

:

z3

2A+J+J’+D,]

28.48 k J

0;

2A k ; [J + J’ + D,]

bAB + ax - f [J + S ? D,]

0; 8

2A 2 f [J + J’ - D,]

Db7

2A+J+J’-D,]

0;

2A + J

6,,-6x+I~J-J’TDl 2

De

D,

6

bAB - dx - ; [J - S T D-1

a D, sin 28, = JAB = J; D, cos 26, = [(a, = (1/2)(D+ - De); 0, = (0, + O-); 0, = (0, - &);

9

6

+

4

S

3A

0; 10

&iB - -:[J-Y&D+]

+J+J’*D-1

T’

dAB + 13~ + ; [J + J’ + De]

rSAB + ; [J - J’ + D-1

d AB

01

+ bx

2

cSx 7 Dp

S.N.

Z’,3

; [J - J’ + Dp]

Z; Zb

Dm

z;8

fD*

4

6,) f (1/2)(J,, - Jsx)]; D, = (l/2)(0+ + D-); J’ = (l/2)(5,, + Jex); ~3~~ = (l/2)(6, + 6,).

and c$ = &, + t,A.

PI

Here N is the order of the multiple-quantum coherence. Fourier transformation of Eq. [7] with respect to t, and t2 gives the required twodimensional spectrum having singlequantum frequencies w,b along w2 correlated with multiplequantum frequencies (wdC+ w&2 along wI , with the centers of respective orders shifted by NA from the carrier frequency. The correlated spectrum has complex signal amplitudes. Using Eq. [8], the real amplitudes are given by (1)

486

THOMAS

AND TABLE

KUMAR 3a

THE MQT AND SEMQT INTENSITIESOF THE 2D SPECTRA OF AN ABX FOR A NONSELECWE

REFoCUSIN(I

MQT S.N.

S.N.

180”

SEMQT

-

B and D

A and C

SPIN SYSTEM

PULSE”

S.N.

A

and C

B

and D

“The spectra along the two pure X single-quantum transitions in the w2 direction (S$ and TQT and DQT along the w, direction are given in this table. For ZQT along the w, direction, see Table 3b. 2, = C0, + SB,; AL\,= CB, - SB,; & = CB, f S8,; $+ = SB, f CB,. Multiply A, B, C, and D coefficients in Tables 3a and b by l/16.

191 TABLE

3b”

MQT S.N. S.N. A and C

s, 6

SEMQT Band

D

S.N.

A

and C

BandD

-5 0 z2 0

f(Z+)uB* z;

0

*sepKI;+)u!, - (2-)&l

+(A+)&

Z;

0

z, -G s z,

0 0 +-)uqJ

z;

0

+cep[(%)u8, -+C~&A+)&

zx4 z;

0 c2ep[c2e+u~~

- C2e-u!:3]

+(A-)&,

Z6

-Se,[C2e+u&

- c2e-4

z;

f S2e,[C2e+(u%

&I

; S2Bp[C2B-(

CZB+a!& -c2e-c4, 0 0

‘See footnote a, Table 3a.

+S@,W+bb 0 0

- u&J - 2S28.&] UL - u:,,

+ 2s2e+l&1

0 0

- (A-)&J - (X)uh] - (A-h&l

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

487

TABLE 4a THE

MQT AND SEMQT INTENSITIES FOR A NONSELECTIVE

OF THE 2D SPECTRA OF THE ABX REFOCUSING 180” PULSE’

MQT S.N.

S.N.

A and C

T

0

SPIN SYSTEM

SEMQT B and D

S.N.

A and C

B and D

-(53&

D1

-se&L)ul,

Tce,(z-)o:,

0’1

-semc4t~+b:I

+c2e,(z+)4

02

%(A-)&

Se&A-)o;,

&

semsw+)d,

+ i s2e,(z+)u$,

0;

sems4a+b!,

+ + sze,(A+)ug,

(5*)4 -(5*)& -se,ce,(z-)up3

+c2ep(A+)& 0 0 kc2e,(z_)0:3

D)8

wnS~dA-)u83

T ; S2B,(A+r&

Db

se,sepm)&

D\O

se,cep~a-)&

Dh 0; Db Di

se,ce,(a+b:,

1 T - s2e,(x)& 2 TC2e,(A-)&

’ The spectra for the remaining X transitions along the w2 direction and TQT and DQT along the w, direction are given in this table. For the ZQTs along the w, direction, see Table 4b. All A, B, C, and D coefficients in Tables 4a and b are to be multiplied by (C&,)/16.

The absolute signal amplitude is given by I4 ab,cdpq = (A2+ B2 + C2 + D2)1’2.

[lOI

(a) Symmetry in a Two-DimensionalSEMQT Spectrum The symmetry relation, z nb,cdpq

=

z8a,dcqp

[Ill

and a corresponding relation for the MQT spectrum ensures that both SEMQT and MQT two-dimensional spectra possess inversion symmetry such that S(w,, 02) = S(-o, , -w2). In addition it can be seen that for every element Z&&W, there exists another element z&,&c having o1 symmetric with respect to NA with intensity not necessarily equal, ensuring that SEMQT spectra possess frequency symmetry about NA, even when the MQT spectrum may not. For zero-quantum transitions and for all other orders when A = 0, the inversion symmetry and the additional symmetry mentioned above become identical and coherences u:~ and C& contribute in additive manner to the signals at w1 = +(w& + w&2.

(b) MQT Spectrum The MQT spectrum, obtained in the absence of the refocusing 180” pulse, is obtained by substituting I for R, and zero for A and +‘, resulting in 6, and i&d, in Eqs. [7] and [8] leading to the signal amplitude as

-c2e,(ts)[c2e+u86 s2ep(tT)[c2e+&

M2B,(A-)& +cep(z-)u:6

z6

f s2ep(Mc2e-(

i s2ep(t,)[c2e+(u:6

-se,se,Kz+)uh

0

2

5

se,c4[(~+)&

0

-WnSe,NA+)&

z3

S.N.

-wnce,KA+)u:7

B and D

+ce,(z+)u:,

A and C

z2

S.N.

(Z-b921

-

&)

-

2s2e+u$]

ug5) + 2s2ehd3]

c2e-u:3]

u83-

-

- c2eh:,]

+ (z-)421

- WIUS~I

-

+ @&!,I

A and C

SEMQT

-j

s28,

[(A+)&

0

0

0

0

7 i s2ep[(27+)&

+c2e,[(~+b97

~c2e,KA+)u87

T

+

(A-)&]

+ (x)~:~I

- (A-)~!21

- ~~:-)0!21

B and D

a To obtain the two-dimensional intensities of transitions belonging to S;, the following changes have to be made in Tables 4a and b: (a) The multiplying factor cos 0, has to be changed to -sin 8, for all transitions. (b) For all MQT and for T’, D;, Db, Z;, Zb, Z;, and Z; of the SEMQT spectrum, substitute em = em + 90”, 0, = e, - 90” and for remaining SEMQTs, substitute S&$33, - C&,$.33,, SB,SB, - CB,S0,, CB,CB, - SB,CB,, and CB,SB, - SB,SB,. b See also footnote a, Table 4a.

S.N.

MQT

TABLE 4b”.b

E

s

i

E

2

2D SPECTRA

OF STRONGLY

COUPLED

SYSTEMS

489

TABLE 5a THE 2D MULTIPLE-QUANTUM INTENSITIES ALONG ONE OF THE AB SQ TRANSITIONS (S, ,) IN THE w2 DIREC?TION IN AN ABX SPIN SYSTEM WITH NONSELECTIVE 180” PULSE AT THE CENTER OF THE EVOLUTION PERIOD“

SEMQT

MQT S.N.

S.N.

A and C

B and D

S.N.

A and C

B and D

T

0

(A+)&

T'

0

(A+)&

Dh 0; Dk

-ce,s2e+c4, SBpS20+b~, S4c2e+u~, CBpc2e+u:, (A+)u:, (A+)&

Di

ce,cfl,0~3

Di

-se,senlu83

Db D\O

-ceps8,&

DI

0;

Dl

0;

D3

0;

04 D5 06

-sq2Bmug4

“Given are the TQT and DQTs along the wI direction. For ZQT, see TabIe 5b. All A, B, C, and D coefficients in Tables 5a and b to be multiplied by (A+)/16. s(tI,

t2,

T)

=

z

zubgqei(goN+~N~ei~~*eiw,ll

ab w

with ziib,pg

=

<~x>(lb<~,>pbc~,>,~~~9

0)

t121

and corresponding relations for the real amplitudes A, B, C, D and symmetry relations of Eqs. [9]-[ 111. (c) Phase Cycling for Selection of Orders For a nonselective 180” refocusing pulse, N’ = -2N and for a nonselective 90” mixing pulse N” = N f 1. Consequently, the signal intensities for positive w2 frequencies are given, using Eq. [7], by G = Zd,,d&‘$eiNt [I31 and for negative w2 frequencies by, G =

zba,cdp&-

i$

e

iNt

,

P41

where E = 40 + l$b- 26’. From Eqs. [ 131 and [ 141, it is seen that (i) The phase factor of zero-quantum and axial peaks depends exclusively on the phase of the mixing pulse and is independent of the phases of the other pulses.

Z6

ZS

- 4)

- i s2e,(z+)pze-(

4,

- &j

+ s2e-uq31

+ s3-CT:,:

- f s2ep~z+j[s2e+(&

s2e,(z+j[s2e+a:s

-c2e,(z;)[s2e,&,

- semu&] + ce,&l

-ce,[s2e+& sepIs2e+&

Z4

and C

-G

A

- cemug21

Z

S.N.

ce,[c2e+&

and D

z2

0

B

+ semuS21

and C

C2B+&

A

sep[c2e+u:,

S.N.

a See footnote a. Table Sa.

s,,

S.N.

MQT

TABLE Sb”

- 2c2e+&

- 2c2eb&l

SEMQT

- i s2e,(z,Mag,

- Tj sze,(z+)(&

+ 4)

- d&)

- 4,)

- &j

- ce,d21

and D

+ se,d2i

S2&i(Z+Md25

-C*S,(Z+j(u$

se,b&

-ce,[&

- i s2epu82

f s2epog2

B

$ > P

g

is

2D SPECTRA OF STRONGLY

COUPLED

491

SYSTEMS

TABLE 6a THE 2D MULTIPLE-QUANTUM INTENSITIES ALONG S12 (AB SQT) IN THE w2 DIRECTION IN AN ABX SPIN SYSTEM WITH A NONSELECTIVE REFOCUSING 180’ PLJUE AT THE CENTER OF THE EVOLUTION PERIOD”

MQT S.N.

S.N.

A and C

SEMQT

B and D

S.N.

A and C

B and D

-(A-)&

T

0

a:,

0;

ce,ce,&

f s2e,&

T

0

D,

S2&&

Dl

-c2e-&

0;

-se,ce&,

-s2e,&

D3

0

0;

s4s4dl

- f s2e&!,

0 se,& ad4

Dh 0; Db Di & Db D’IO

cv4d 0 0 ce,s2e& -se,c2e-& -se,s2e-u;4 -ce,c2e-dj4

-c?e,&!, (A-b’!, (A-)& -ce,& 0 seput4 0

04

DS Db

a Given are the TQT and DQTs along the w, direction. 6a and b are to be multiplied by (A-)/16.

For ZQTs,

see Table

-(A-b:,

6b. All intensities

in Tables

(ii) In selection of even/odd orders, for constant ~$0and &, a phase shift of 180” of # causes a change in sign of G and G for all even orders including zero and axial peaks and a coaddition of these experiments cancels them, while a subtraction retains them, canceling all odd orders (3, 38). On the other hand, for constant #, a 90” phase shift of #, or a 180” phase shift of C#J~ causes the amplitudes of all odd orders to change sign and appropriate coadditionjsubtraction can cancel odd/even orders (19, 23).

(iii) In selection of a particular order, coaddition of 2M experiments with the phases satisfying ,$ = n*/M, where M can be 1, 2, 3, etc. and n takes values from 0, 1,2, **. (2M - 1) and with (a) $ = n7r retains M, 3M, 5M, * * orders while with (b) $ = 0 retains 0, 2A4, 4M, * * * orders (3, 19, 23). That is, if all orders below N are to be canceled, let A4 = N in scheme (a). If only zero-order is to be retained, let M > N/2 in scheme (b). For example, if the highest MQT order is 6, let M = 4 and $ = 0, and a coaddition of 8 experiments with E = 0, ~14, 2~14. . -7~14 will cancel all orders up to 7, except zero (3, 19). Further phase cycling may be required to cancel imperfections in pulses. For example, Exorcycle compensates for an inhomogeneous or imperfect 180” pulse (39). Additional phase shifts can be brought about by the use of different phases for the two 90” preparation pulses. (See Appendix B for further discussion.) The above conclusions also apply to 2D MQT spectra with 4’ = 0. l

s12

- ug,

- ; s2ep(z~)[s2e~(u~,

Z’,

- 2c2e+&]

- ~$6) - 2C2e-a:,]

- ; S2e&)[S2e+(u$

.G

+ S2e-&]

S~e,(Z-)[S2e+u&

z; Z

+ s2edg

-s2ed&

a921

46

+

Z6

-S@,[SB,&

-C2e,(Z_)[S2e+uq6

-(U& c2e-a$

-s2e~(zJu:5 Z

r&l

and C

0

f S28,&

z;

-

A

Z5

-(z-b:3

-S2e+(Z-)a$,

z3

-ce,[ce,uX,

- f s2e,a4&

z;

z;

S.N.

SEMQT

Z4

ad2

se,d2

wdJ92

B and D

-72

-Cl+&

Z,

and C

A

S.N.

a See footnote a, Table 6a.

S.N.

MQT

TABLE 6b”

43)

S2e,(2-M& +

- i S2ep(Z)(uS3

- f S2e,(8-)(&

-

- uL)

- u&J

43)

+ s2e-a:21

- c2ehJ3

+ S2BL&]

- C2eLTy2]

and D

-c2ep(z-)(u%

-Se,[ce,ub

ce,[ce,u~,

ce,[se,u&

Se,[Se,uR

B

E

g

% u

P ifi!

2

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

ii

ii vi

ii

-Q

.N Q

4

s”

2t-4 + ‘2

h

G

6

.n 9

b”

6

‘s”

494

THOMAS

AND KUMAR

III. NONSELECTIVE

PULSES

In this section, the results for strongly coupled spin systems AB, ABX, and AB2 for nonselective pulses are given. For direct comparison both MQT and SEMQT 2D spectra are discussed. (a) AB Spin System Table 1 gives the flip-angle dependence of the intensities of the MQT and SEMQT 2D spectra of a strongly coupled two-spin system (AB) for 4’ = I) = 0, 4 = t, A. Tabulated are the absorption A and C and dispersion B and D coefficients for positive wI frequencies. Use of a symmetry relation, Eq. [ 111, allows computation of the negative w, domain from the above expressions. The real parts of a0 contribute to A and B coefficients and the imaginary parts to C and D coefficients in this and all subsequent tables in this paper. In the two-spin case, the SEMQT spectrum has three zero-quantum transitions (SEZQT) as opposed to only one in the MQT spectrum (Table 1). The zero-frequency SEZQT results from nondiagonal elements of the density matrix during tl , as opposed to axial peaks in the 2D spectra, although they are experimentally indistinguishable. In strongly coupled spin systems the 180” pulse in the evolution period causes transfer of axial magnetization into zero-quantum coherence and vice versa, resulting in new zero-quantum frequencies in the SEMQT spectra, such as the new transition at C’/2. Phase discrimination of various zeroquantum and axial components is discussed in Appendix B. The two-spin system has only one doublequantum transition which appears undistorted in the SEMQT spectrum. The intensities of all multiple-quantum transitions (Table 1) are proportional to sin (Y, and the highest quantum transition appears in pure dispersion for (Y = 90”. Other transitions appear, in general, in mixed phases for all values of (Y. The flipangle dependence of the MQT and SEMQT spectra of higher-order spin systems follow similar rules and will not be treated explicitly in the following sections. (b) ABX Spin System The X parts of the calculated 2D correlated MQT and SEMQT spectra of an ABX spin system are shown in Fii. 2 and 3, respectively, along with their projections on the w1 and w2 axes, and the frequencies and intensities are given in Tables 2 to 6, for 4’ = $ = 0 and 4 = t,A, for nonselective excitation, refocusing, and mixing pulses.’ The ABX spin system has one triple (TQT)-, six double (DQT)-, and six zero-quantum transitions in the MQT and one triple-, ten double-, and eight zeroquantum transitions in the SEMQT, each of which is correlated to the 14 singlequantum transitions (SQT) in the 2D spectrum. The refocusing pulse has modified the double- and zero-quantum spectra in this case. In the absence of order separation, the SEMQT frequencies (Table 2) are identical to the J-resolved spectroscopy frequencies of ABX for a nonselective 180” pulse (34). ’ Tables 3 to 6 contain A, B, C, and D coefficients of the 2D spectra for four X and two AB singlequantum transitions along uz. The 2D intensities for the remaining two X transitions can be obtained from Table 4. The intensities for the remaining AB transitions, not relatable in a simple way to Tables 5 and 6, follow a similar intensity pattern (44).

2D SPECTRA OF STRONGLY

COUPLED

495

SYSTEMS

TABLE 8a THE MQT AND SEMQT INTENSITIESOFA~D SPECTRUMOFAN ORIENTED SPIN SYSTEM (AB2) ALONG S, TRANSITION IN THE WeDIRECTION"

MQT S.N.

s,

SEMQT w2)

S.N.

D,

+ ; S%, + f Se,]&

D’,

02

+ ; C%, - ; C%,]uf,

0;

S.N.

Zbl,

03

-cw2082

D4

-W+MA'ih78

Z(w,,

w2)

0;

-C%, ; (1 - C2%+) - fisze,

Db 0; Db D; Db

-se

d

f (1 - C2%+) - Vkze,

(Z;XZ”-)S%,&

1 1 u$

uoq,

W+W')ce,d,,

-(Z’+)(Al)S%,u:, v+Mzxe,uS3

’ Spectra for TQTs and DQTs alon the w, direction are given here. For ZQT along w,, see Table 8b. Z’, = (fk%, + Se,); Z: = (C%, + $ 2S%,); A; = (EC%, - Se,); A; = (C%, - &%,). All Z(w,, 02) in Tables 8a and b are to be multiplied by (2’+)/16.

The SEMQTs result from transformation of various MQTs to others within the same order by the nonselective 180” pulse. In weakly coupled system, this transformation is one-to-one between transitions having complementary spin states; for example, a CUY(Y- CYC@ transition is transformed into a /3&3 - P/3a transition. For strongly coupled spin systems this transformation is not one-to-one, except when both the states involved in the transition are pure. In 2D spectra, any transition which involves only pure states in either o1 or w2 directions cross-correlates to all other transitions in pure absorption or in pure dispersion, for a mixing pulse of angle 90” (I). An ABX system, has four “pure” states which give two single-, two double-, and one triple-quantum transitions, both in MQT and SEMQT spectra. All the MQTs and SEMQTs cross-correlate to the two “pure” single-quantum transitions in either pure absorption or pure dispersion (Table 3). The triple-quantum transition crosscorrelates to all single-quantum transitions in pure dispersion (Tables 3 to 6). The two double-quantum transitions involving only pure states, namely D3 and D4 in MQT and 0; and Db in SEMQT, also cross-correlate to all the single-quantum transitions either in pure absorption or pure dispersion (Tables 3 to 6). The SEZQTs Z’, to Z’, arise from transformation of ZQTs into themselves giving rise to new SEZQT frequencies. Z$ and Z’, on the other hand arise from transformation of axial magnetization into ZQTs and vice versa, resulting in SEZQT frequencies which are half of ZQT frequencies (Table 2, see also Appendix B). There is no

496 THOMAS

AND

I

I

D B Y ? fI

. 1. d L

.N

KUMAR

uc

Z2

02

Z,

D,

S.N.

2&e

2&j

D-

D+

- J’

+ J

WI

AND

0

and C

1 + - s4e-CT:5 2

0

* 5 s4e+aq,

1

A

MQT

SEMQT 2D SPECTRA

+c2e+&

f S26+)&

_+c2e-a&

-( 1 f s2e-)a$

-(l

Band D

OF THE AB

PART

D+

Z;

0

Z$

Zi

2A

++

Z;

D)2

D+

0

2A

WI

SPIN SYSTEM

Z;

z;

0;

S.N.

OF AN ABX

0

and C

T 2C’29S28-a~5

+-c=2e-s22e-(o~5

0

2c32e+s2e+lT:,

fC’29S2@-a& +C2e-S’2e-a&

i

_tC~2e+S2e+(a~3

kc32e,s2e+& 7c2e+s32e+&

A

AB

PULSES”

- u”,,)

- &)

SEMQT

FOR SELECTIVE

and D

+C22e-s2e-(&

TC’20-u!& +c2es=2e&og5

(1 + s2e-)a$

+C22e+S2e+(&

+c32e,(r403 TC2e+S~2e+&

(1 + s2e+)&

B

-

-

u&J

CT”,,

’ ST, S,, , Se, S12, etc., belong to AB SQTs whose frequencies are given in Table 2; all the intensities to be multiplied by l/4. To obtain the above spectra for SI, from S,, & from S,, , S,, from Sg, and SIO from S ,2, change the sign of B and D for the double-quantum transitions and the sign of A and C for the zeroquantum transitions.

&I*

II

s,

S.N.

MQT

TABLE 9

498

THOMAS

AND

KUMAR

zero-frequency SEZQT in ABX, unlike the AB case, due to coupling with the third spin (40). The five forbidden three-spin-singlequantum transitions (3-spin- 1QTs) of ABX, four of which are degenerate with allowed 1-spin- 1QTs of the X spin, also appear in the MQT spectrum. In the SEMQT case, the four degenerate transitions give rise to 16 new frequencies (33), while the nondegenerate transition, emanating from pure states is focused and appears in the center of the single-quantum spectrum, at a1 = A. (c) AB2 Spin System Tables 7 and 8 contain the MQT and SEMQT frequencies and intensities of an oriented symmetrised AB2 spin system, for $J’ = $ = 0, 4 = t, A and a mixing pulse of angle 90”. The tables contain the intensities for one cross section of the 2D spectrum at w2 = bAB+ 3[J + 20 + D’]/4 - C+/2. The complete spectrum can be computed in the manner outlined in Section II and has features in common with the above cross section. An AB2 spin system gives rise to two ZQTs, four DQTs, and one TQT, which in the spin-echo experiment become four, eight, and one, respectively. The zero- and double-quantum spectra are modified by the 180” pulse. Two of the SEZQTs are

I II II I I I (b)

FIG. 4. Schematic plot of the AB parts of two-dimensional spectra of an ABX spin system for selective preparation and selective mixing AB pulses. (a) MQT spectrum, (b) SEMQT spectrum for selective refocusing pulse on AB spins. Four levels of contours have been utilized for indicating the absolute intensities. The intensities are symmetric within each AB sub-part such that &,, St,, S,, and St0 have intensities equal to S,, S,, S14, and &, respectively. The preparation scheme for the above intensities was of Fig. la with selective inversion of two single-quantum transitions (S, and SlO) followed by a selective 90°(AB) pulse. Inversion of any one of the singlequantum transitions excites only one ZQT and one DQT, which are correlated to four transitions, one of which is the inverted transition. Other details are same as in Fig. 2.

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

499

due to transformation of ZQTs among themselves, while the other two are due to transformation of ZQTs with axial magnetization, during the evolution period. All the SEDQTs are due to transformation of various DQTs among themselves, none of which involves only pure states. In addition to the transitions listed in Tables 7 and 8, the multiple-quantum spectrum of AB2 appearing along wI contains in the symmetric part eight I-spin-1QTs and four degenerate 3-spin-l QTs. In the spin-echo case all these frequencies are modified to 16 new frequencies (34). In absence of order separation the SEMQT spectral frequencies (Tables 7 and 8) are identical to those in J spectra (34). The AB2 spin system has two antisymmetric eigenstates, giving one single-quantum transition and a degenerate 3-spin- 1QT along w , . The antisymmetric transition does not cross-correlate with any of the symmetric transitions in the 2D spectrum. In higher-order spin systems, such as AA’BB’, there are several antisymmetric states giving rise to higher-order MQTs between them (41). All such antisymmetric transitions cross-correlate only among themselves in the 2D spectrum and lead to identification of these transitions and a simplification of spectral information. Each order of the SEMQT spectrum of AB2 along wl, is frequency symmetric about its center frequency, unlike the corresponding MQT spectrum. IV. SELECTIVE

PULSES

This section deals with selective pulses in the MQT and SEMQT 2D spectroscopy. Selective pulses have found applications in heteronuclear coherence transfer experiments (36, 37) and the aim of the present section is to investigate the influence of strong coupling on the heteronuclear multiplequantum coherence transfer process, with and without a selective 180” pulse at the center of evolution period. An ABX TABLE 10 2D MQT AND SEMQT SPECTRA OF THE X PART OF AN ABX

S.N.

w,

A and C

X PULSES’

SEMQT

MQT

S.N.

SPIN SYSTEM FOR SELECTIVE

BandD

S.N.

w,

A and C

a FOG X SQTS, &, S,, Sz, and S,, the frequencies are tabulated in Table 2; all intensities (1/(8@)S2fJ,. The upp~ signs refer to S, and S, and the lower ones to S, and S,.

B and D

to be multiplied

by

500

THOMAS

AND KUMAR

heteronuclear spin system, where a i3C or 19F nucleus (X) is coupled to two strongly coupled protons (AB), is dealt with in detail. In the ABX spin system, the use of a selective 180”(X) pulse leads to heteronuclear multiple quantum transitions (HMTs) with spin-echo frequencies identical to the corresponding J-spectroscopy frequencies, while the use of a selective 180”(AB) pulse does not. Unlike nonselective pulses (Section III), a selective 180” pulse causes coherence transfer between different orders for heteronuclear coherences which involve at least one of the A and B spins and the X spin. To facilitate description of various possible selective excitation schemes, the following notation is used. The preparation, refocusing, and mixing pulses are designated by P, R, and A,$ respectively, the detection by D, and the phases, if required, by a subscript. If the two preparation pulses are not identical, they will be designated by P, and P2. Case (i). Selective Pulses on Protons The experimental schemes considered in this section are P(AJ3) M(AB) D(AB) and P(AB) R(AB) M(AB) D(AB) for MQT and SEMQT, respectively. P(AB) excites two ZQTs and two DQTs involving only AB spins. The MQT and SEMQT 2D spectra in this case are superpositions of two AB subspectra, one for each + l/2 and - l/2 states of the X spin (Table 9 and Fig. 4). One ZQT and one DQT are correlated to

FIG. 5. Schematic plot of the X part of the two-dimensional spectra of an ABX spin system for selective preparation and selective mixing X pulses, (a) MQT spectrum, (b) SEMQT spectrum for selective X refocusing pulse. Only zero-quantum coherence is excited in this scheme which is correlated only to transitions involving mixed states. The two “pure” transitions, shown with broken iines in the o2 projection, do not carry any intensity. Absolute intensities are indicated with three contour levels in the right half of the spectrum with the lefi half being symmetric to it. The preparation scheme utilized was of Fig. la with selective inversion of S,, followed with a 90’(X) pulse. Other details are the same as in Fig. 2.

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

501

four single-quantum AB transitions while the other ZQT and DQT are correlated to the remaining AB transitions. Each ZQT gives rise to three transitions in the SEMQT similar to the AB case. Case (ii). Selective Pulses on Carbon-13: Use of a “‘Spy” The experimental schemes considered are P(X) M(X) D(X) and P(X) R(X) M(X) D(X). Under weak coupling conditions no MQTs are excited in this scheme. Strong coupling between protons causes the X single-quantum coherence during preparation period to be transferred into proton zero-quantum coherence. This coherence evolves during t, and is transferred to X single-quantum coherences during t2 again because of strong coupling within protons. This scheme thus allows monitoring of proton zero-quantum coherences without the use of any proton pulses through a “spy” carbon-l 3 spin. Earlier a “spy” carbon-l 3 spin was utilized for proton relaxation studies in weakly coupled spin systems (35). Only zero-quantum proton coherences can be excited in this scheme, since they are independent of proton offset and depend only on the chemical-shift difference between the protons and on the carbon- 13 offset (see Appendix A). The two proton ZQTs, Zi and Z,, created in this scheme are correlated with the SQTs of X spin, in the MQT spectrum, provided they share common energy levels (Table 10, Fig. 5). The two “pure” X SQTs do not receive any of the above zeroquantum coherences. In the SEMQT spectrum, the frequencies of these transitions are modified and two additional ZQTs are generated due to the transfer of axial magnetization into zero-quantum coherence and vice versa (Table 10, Fig. 5). The SEMQT frequencies are identical to the corresponding J-resolved frequencies (34). A 2D MQT spectrum with proton decoupling has to be obtained by introducing “delaying tactics” for there is no net magnetization transfer to a multiplet (6). Case (iii). Selective Detection of HMTs via Carbon-I3 The pulse schemes envisaged are (a) Pi(AB) P2(ABX) M(AB) D(X), (b) P,(AB) P*(ABX) R(X) M(AB) D(X), and (c) Pi(AB) P2(ABX) R(AB) M(AB) D(X). The preparation pulses in this scheme excite all coherences in a manner similar to those discussed in Section III, except that the amplitudes of the coherences are independent of 13C resonance offset and all the multiplequantum coherences are created out of proton SQTs only. A selective AB mixing pulse transfers all heteronuclear multiplequantum coherences namely T, D, , 4, D5, 06, Zi , Zz, Zs, Z, (Table 2) and one 3-spin-1QT to every single-quantum transition of X spin (Fig. 6a). These features have been extensively utilized in heteronuclear multiple-quantum coherence transfer experiments (6, 16). A selective 180”(X) refocusing pulse, scheme (b), causes transformation of heteronuclear coherences between different orders. For example, the heteronuclear doublequantum coherences during the first half of tl, are transformed into heteronuclear zero-quantum coherences during the second half and vice versa, and the triple quantum coherence into the three-spin-single-quantum coherence and vice versa, giving w, frequencies different from case (a), Fig. 6b. The even-order coherences, in this case,

THOMAS

502

51

52

53

%

55

s6

51

s2

AND KLJMAR

53

SL

55 56

51

52

53

SL

s5

sfi

FIG. 6. Schematic plot of the two-dimensional spectra of an ABX spin system for selective detection of heteronuclear multiplequantum transitions (HMTs) via the X singlequantum transitions, case (iii). The abscissa represents wr and the ordinate represents w, axes. (a) The MQT spectrum (b) SEMQT spectrum for the selective refocusing X puke, and (c) SEMQT spectrum for the selective refocusing AB pulse. Four contour levels have been utilized for indicating the absolute intensities. S, has intensities equal to .S, in each case. The preparation scheme utilized for the above computation is that of Fig. la with selective inversion of single-quantum transition S, followed by a nonselective 90’ (ABX) puke. Other details are the same as in Fig. 2.

combine to yield eight frequencies, namely aAB t- (1/2)(J & 0,) and dAB + (l/2) X (J +- II,). The triple and three-spin-1Q coherences give a single line at 2dAB. The single-quantum coherences transform into each other, yielding wI frequencies identical to above eight frequencies which in turn are identical to the o, AB frequencies in J-resolved spectroscopy with 180”(X) pulse (34). In the weak coupling limit these eight frequencies reduce to four, namely aA k ( 1/2)J and dB f ( 1/2)J and the 180”(X) pulse acts as a X decoupling pulse in the o, domain. In the A,X limit these reduce to a single line and the even-quantum 2D spectrum of this scheme, for an A,X spin system, when observed under proton decoupling conditions, resembles a heteronuclear chemical-shift correlation single-quantum spectrum (6, 36, 37). A selective 180’(AB) refocusing pulse, scheme (c), also causes transformation of heteronuclear coherences between different orders, in a manner similar to above, Fig. 6c. The even-order coherences combine to yield six new frequencies namely, 6x +- (1/2)J, 6x + (1/2)(J & D-), and 6x - (l/2)(1 + D,). These reduce to 6x f. (1/2)J in weak coupling limit and to a single frequency of 6x in the A,X limit, in conformity with experiments by Miiller (6). The triple and three-spin-single-quantum coherences lead to a single line at 6x. It is seen that the w1 frequencies in the strong coupling limit are different from corresponding J-resolved spectroscopy frequencies, except the common line at 6x (34).

2D

SPECTRA

OF

STRONGLY

COUPLED

SYSTEMS

503

Case (iv). Selective Detection of HMTs via Protons The pulse schemes, under consideration, are (a) Pt(AB) Pz(ABX) M(X) D(AB), (b) PI(AB) P*(ABX) R(X) M(X) D(AB), and (c) P,(AB) P2(ABX) R(AB) M(X) D(AB). Here the preparation pulses are identical to case (iii) and excite essentially all the coherences. The selective mixing pulse on X spin, however, transfers only heteronuclear double- and zero-quantum coherences into the connected SQTs of AB protons. The coherences DI , D2, ZI , and Z, are correlated to the connected &, &, S, ,, and $2 transitions while D5, D6, Z5, and Zs are correlated to transitions S9, Slo, S, 3, and S14, Figure 7a (see Table 2 for the frequencies of these coherences). Such a scheme has the advantage that the HMTs involving a less sensitive nucleus like 13C, “N, etc., can be observed, independent of their gyromagnetic ratios, with enhanced sensitivity (6). A refocusing 180” selective pulse on X spin, rearranges all the HMTs, in a manner similar to case (iii)b, giving rise to eight frequencies in o1 domain, which are correlated to the connected single-quantum transitions of A and B spins, Fig. 7b. A selective 180’(AB) pulse also rearranges the HMTs, in a manner similar to case iii(c), giving six new frequencies, each of which is correlated to the connected SQTs of AB spins, Fig. 7c. CONCLUSIONS

Two-dimensional spectra of spin-echo multiple-quantum transitions have been calculated for various spin systems for selective and nonselective use of 180” pulse. These spectra have been shown to be quite different from that of multiple-quantum transitions for the cases of strongly coupled spins and possess additional transitions and new frequencies. The SEMQT spectra have also been found to be frequency symmetric with respect to the center of various orders, even when the multiplequantum spectra or the one-dimensional single-quantum spectra may not possess such symmetry. Strong coupling also causes transfer of multiple-quantum coherence to a heteronuclear coupled spin and allows the detection of the abundant spins while monitoring

FIG. 7. Schematic plot of the two-dimensional spectra of an ABX spin system for selective detection of HMTs via the AB single-quantum transitions, case (iv). Here the selective mixing X pulse transforms only the double- and zero-quantum heteronuclear coherence to connected SQTs of AB. (a) MQT spectrum, (b) SEMQT spectrum with selective X refocusing pulse, and (c) SEMQT spectrum with selective AB refocusing pulse. This is also an absolute intensity plot with a preparation scheme the same as in Fig. 6.

THOMAS

504

AND

KUMAR

the coupled rare spin, with the rf pulses only on rare spin. The frequencies of such coherences are modified by the use of a refocusing 180” pulse at the center of the evolution period. It is also seen from Tables 2 and 7 that the SEMQT frequencies of ABX and AB2 spin systems, in the absence of order separation (for A = 0 and $J,, = 0), reduce to the respective spin-echo single-quantum frequencies reported earlier (34) for nonselective 180” pulses. To obtain additional information from SEMQT spectra, separation of various orders is therefore necessary. For the AB spin system, however, the zero- and double-quantum SEMQTs are quite different from spin-echo singlequantum spectrum. Selective 180” pulses in ABX spin system also cause transfer of heteronuclear coherences between different orders. Their behavior under strong coupling has been investigated. The amplitudes of various coherences excited by selective as well as nonselective pulses with nonequilibrium conditions of different kinds for weakly as well as strongly coupled spin systems are given in Appendix A, along with an analysis of their behavior under the influence of a 180” pulse at the center of preparation period. From Eqs. [A61 it is evident that in the case of a linear A-M-X spin system, when the scalar coupling between the remote nuclei A and X is vanishingly small, it is possible to excite the triplequantum transitions through the other couplings JAM and JMx, leading to the unequivocal proof that the spins A, M, and X belong to the same coupled network (40, 42). Analysis of the zero-quantum coherence during evolution period, presented in Appendix B, enunciates that in strongly coupled spins the axial magnetization is transferred to zero quantum by the refocusing 180” pulse and vice versa, giving rise to new transitions. APPENDIX

A:

PREPARATION

OF

COHERENCES

In this appendix, the amplitudes of various multiplequantum coherences that are created by two general schemes are outlined for AB, AMX, and ABX spin systems. The schemes considered are (i) selective perturbation of population, followed by a 90” pulse (nonequilibrium states of the I kind) and (ii) two 90” pulses spaced in time (non-equilibrium states of the II kind (43)). Case (i). Population Perturbation

Followed by a 90” Pulse

The state of the system for the scheme of excitation mentioned in case (i), is given by 4

= I2 (PyhdPyh&

Ml

where PII is the population of Hh eigenstate prior to the 90°, pulse. Two situations are considered where the 90°, pulse is (a) nonselective and (b) selective. (i)a. Nonselective 90” pulse. For a strongly coupled two-spin (AB) system, amplitudes of the MQT coherences have been reported earlier (33). For the ABX spin system, various MQ coherence amplitudes with the participating spins in the bracket are TQT (ABX): 88 = K2 - fh

- f&

+ P7,8 - P3$2e+ + P5,d2e-

2D SPECTRA OF STRONGLY

COUPLED

SYSTEMS

505

DQT (AB): = ug ZQT (AB): u$4 = c2e,(P1,, + P*,~ - p5,1 - p6,8 + p3,4s2e+ + c,sz-) 56

DQT (ABX): d8 = wI,Z

- P7.8) +

fw3,4c2e+

-

p5,6m-)

15

d& = A&,z 16

- P7,8) T zk(p3,4c2e+ - p5,6c2e-)

~53 = z&,Z

- &,8) T &(p3,4c2e+

ZQT (ABX): 57

- p5,6c2e-)

U$4 = A*(pI,2 - &,8) f zk(p3,4c2e+ - p5,6c2e-). 67

WI

Here P;,j = Pii - Pj and all amplitudes are to be multiplied by l/8. The subscripts 1, 2, 3, * * * refer to the energy levels of the ABX spin system, which in the limit of weak coupling are arranged in the descending order of energy for 8A > da > 6x. Wherever two coherence amplitudes are given in one expression, the upper signs refer to the upper coherence and the lower signs to lower coherence. In an ABX spin system, a nonselective 90” pulse preceded by selective inversion or saturation of any one of the singlequantum transitions, except the two X “pure” transitions, excites all coherences. Perturbation of the “pure” X transitions excites all coherences involving X spin. In the weak coupling limit, selective population perturbation of any one of the transitions of a particular spin excites all coherences which involve that spin with equal amplitude (11, 14). Thus Nquantum transition in a system of N coupled spins is always excited by selective perturbation of any transitions. However, a nonselective saturation or inversion of all transitions of a given spin does not excite any multiple-quantum coherence, since the system does not reach the nonequilibrium states of the I kind (43). (i)b. Selective 90” pulse. The selectivity is either on the AB or the X part of the ABX spin system. (1) A 90°(AB) pulse excites only homonuclear AB DQTs and ZQTs, whose amplitudes are DQT: 4, = h3 - p4,1 - p3,4s2e+ ZQT:

[A31

506

THOMAS

AND KUMAR

(2) A 90”(X) pulse excites two AB ZQTs with amplitudes

u$ = P3,$2%,.

[A41

All the amplitudes in [A31 and [A41 are to be multiplied by l/4. In weakly coupled spin systems, the use of a selective 90” pulse excites those coherences which share energy level with the selectively inverted or saturated transition and involve only those spins on which the 90” pulse acts. Strong coupling mixes the states and gives additional coherences. Case (ii). Two 90” Pulses Spaced in Time These pulses can be either selective or nonselective. The off-diagonal elements of density matrix created by the first pulse, after evolving for a time T, are transformed into MQT elements of coupled spins over which the second 90” pulse acts. Relaxation during time T is neglected, assuming 7 4 T,, T2 and the amplitudes are given by uzq = 2 (Py)rp(Py)r~Fx)l,e-iYlr(mrmr)

17

WI

where wlr is the frequency of the single quantum coherence during period 7 and ml and m, are the total magnetic numbers of the states I and r. ($a. Nonselective pulses. For the AB spin system, the amplitudes of the coherences have been reported earlier (33).’ Introduction of a 180” pulse in the middle of the two 90” pulses with same phases eliminates the zeroquantum coherence retaining the double-quantum coherence. A shift of 90” in the phase of the second 90” pulse will achieve the opposite. However, in the weak coupling limit in addition to phase shift, the flip angle of the second pulse should be different from 90” for the zeroquantum coherence to have a finite value (21). The amplitudes of coherences in the AMX case, for 90°Y-7-90”, sequence, are given by TQ:

DQ(MX): i a$ = - - sin 2

cos(6x7)cos

+

COS

(8~~17)COS

* The amplitude of the zeroquantum coherenceas reportedin Eq. (1) of Ref. (21) contains an error.

2D SPECTRA OF STRONGLY

COUPLED

507

SYSTEMS

ZQ(MX):

i

& = - sin ; 2 (

( )I

cos ; JAMT JhW cos(6x7)cos ; J.&XT- cos@MT) I ( 1 +fcos(b,r)sin(fJ~~~)sin(~J~x~).

[A61

(ii)b. 90°,(AB)-T-90°,(AB) in theABX spin system.The two zero and two double AB quanta coherences created by this sequence are the MQTs connected to the singlequantum transitions excited by the first pulse. The DQT amplitudes are proportional to ~~0s(1/2){& + 68) + (~/~)(JAx + JBX>} r and the ZQT to sin (1/2){(6, + ~3~)+ (I/ 2)(JAx+ JBx)}7.These coherences are excited in the weak coupling limit as well as on introduction of a lSO”(ABX) pulse in the center of period 7. Use of a selective 180°(AB) pulse, however, causes the zeroquantum coherence to disappear for identical phases of the two 90” pulses; it is again retrievable by introduction of a relative phase shift (21). (ii)c. 90’,(X)-7-90’,(X) in the ABX spin system.Due to the strong coupling of AB spins, this scheme excites two ZQTs which involve the flipping of AB spins. Their amplitudes are equal to sin 28, sin (axr) sin (&r/2) exp(-iD,7/2)/2. Here again, the presence of a 180“(X) pulse at the center of period r causes the ZQTs to vanish which can again be retrieved by changing the phase of the second 90” pulse. APPENDIX

B: ZERO-QUANTUM

SEMQT MAGNETIZATION

The 180” pulse, in the presence of strong coupling causes a transfer of longitudinal (axial) magnetization into zeroquantum coherence and vice versa. Figure 8 indicates the source of the various zero and axial magnetization and their phases in the 90”,,-~-90”,,-t,/2-180”,~-t,/2-90°~-tz experiment. From Fig. 8 it is Seen that all the magnetization which originates from singlequantum coherences during T, acquires a total phase of exp[i(dr - & f I+!)] while that from axial magnetization acquires exp(i$),both being independent of the phase

1

PHASE

FREQUENCY(4)

INTENSITY

ZERO-S

ZERO

AXIAL -

S

ZERO -

S

AXIAL -

S

ZERO -

S

SINGLE AXIAL AXIAL -

t

ZERO

AXIAL

I

AXIAL -

S

ZERO -

S

AXIAL -

S

-do-

0

case

FIG. 8. The phases, frequencies, and intensities of various zero-quantum and axial peaks in a spin-echo multiplequantum spectrum of an AB spin system. For definition of C’ and 8, see Table I.

508

THOMAS

AND KUMAR

of the refocusing 180” pulse. The magnetization which is axial throughout the period t, is termed as “axial” magnetization, and the rest as “zero” quantum coherence. The zero * axial transformation by the 180” pulse, in strongly coupled systems, has w, frequencies which are half of zero-quantum frequencies with intensities proportional to sin 0, where 8 is the angle defining the strong coupling effects. The zero c-, zero transformation, on the other hand, leads to new zeroquantum frequencies, including the possibility of a zero frequency. Coaddition of experiments with change in phase by ?r of $ or 14, - &I or both, can lead to cancellation of all the above, or the first four or last four coherences, respectively. However, by no scheme, it is possible to selectively cancel all “axial” components without canceling the zeroquantum coherence (I 9). ACKNOWLEDGMENTS This work was supported in part by the Department of Science and Technology, India, through Research Grant 1 l( 12)/8 1STP-II. M. Albert Thomas acknowledges with thanks a fellowship from the Department of Atomic Energy, India. We thank Mr. N. Suryaprakash for checking the manuscript. REFERENCES 1. W. P. AUE, E. BARTHOLDI, 2. S. VEGA, T. W. SHATTUCK,

AND R. R. ERNST, J. AND A. PINES, Phys.

Chem. Phys. 64,2229 (1976). Rev. Lett. 37, 43 (1976); J. Chem. Phys. 66, 5624

(1977). 3. A. WOKAUN

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(1979). 8. W. S. WARREN, S. SINTON, D. P. WEITEKAMP, AND A. PINES, Phys. Rev. Lett. 43, 1791 (1979); J. Magn. Reson. 40, 581 (1980); J. Chem. Phys. 73,2084 (1980). 9. A. WOKAUN AND R. R. ERNST, Mol. Phys. 38, 1579 (1979). 10. W. S. WARREN AND A. PINES, J. Chem. Phys. 74,2808 (1981). II. G. B~DENHAUSEN, Prog. NMR Spectrosc. 14, 137-173 (198 I). 12. J. TANG AND A. PINES, J. Chem. Phys. 73,2512 (1980). 13. M. E. STOLL, S. VEGA, AND R. W. VAUGHAN, J. Chem. Phys. 67,2029 (1977). 14. A. BAX, “Two-Dimensional Nuclear Magnetic Resonance in Liquids,” Chap. 4. Delft Univ. Press, 1982. IS. P. BRUNNER, M. REINHOLD, AND R. R. ERNST, J. Chem. Phys. 73, 1086 (1980). 16. A. MINORETII, W. P. AUE, M. REINHOLD, AND R. R. ERNST, J. Magn. Reson. 40, 175 (1980). 17. A. BAX, P. G. DE JONG, A. F. MEHLKOPF, AND J. SMIDT, Chem. Phys. Lett. 69, 567 (1980). 18. RAY FREEMAN, T. A. FRANKIEL, AND M. H. LEVITT, J. Mugn. Reson. 44,409 (1981). 19. D. JAFFE, R. R. VOLD, AND R. L. VOLD, J. Mugn. Reson. 46,475 (1982); 46,496 (1982). 20. D. P. WEITEKAMP, J. R. GARBOW, AND A. PINES, J. Mugn. Reson. 46, 529 (1982). 21. A. D. BAIN ANLI S. BROWNSTEIN, J. Magn. Reson. 47, 409 (1982). 22. S. SINTON AND A. PINES, Chem. Phys. Lett. 76,263 (1980). 23. G. B~DENHAUSEN, R. L. VOLD, AND R. R. VOLD, J. Mug. Reson. 37, 93 (1980); R. L. VOLD et al., J. Mugn. Reson. 38, 141 (1980). 24. RAY FREEMAN, A. BAX, AND S. P. KEMPSELL, J. Am. Chem. Sot. 102,4849 (1980). 25. Y. S. YEN AND W. P. WEITEKAMP, J. Mugn. Reson. 47, 476 (1982).

2D

SPECTRA

OF

STRONGLY

COUPLED

SYSTEMS

509

26. J. H. PRESTEGARD AND V. W. MINER, J. Am. Chem. Sot. 103, 5979 (1981). 27. G. POUZARD, S. SUKUMAR, AND L. D. HALL, J. Am. Chem. Sot. 103,4209 (1981). 28. W. S. WARREN AND A. m, J. Am. Chem. Sot. 103, 1613 (1981). 29. S. EMID, A. BAX, KONIJNENDIJK, J. SMIDT, AND A. PINES, Physica B %, 333 (1979). 30. D. P. WEITEKAMP, J. R. GARBOW, J. B. MURDOCH, AND A. PINES, J. Am. Gem. Sot. 103, 3578 (1981). 31. A. BAX, Toot MEHLKOPF, J. SMIDT, AND RAY FREEMAN, J. Magn. Reson. 41, 502 (1980). 32. D. L. TURNER, J. Magn. Resort 49, 175 (1982). 33. M. ALBERT THOMAS AND ANIL KUMAR, J. Magn. Reson. 47, 535 (1982). 34. ANIL KUMAR, J. Magn. Reson. 30, 227 (1978). 35. Y. HUANG, G. B~DENHAUSEN, AND R. R. ERNST, J. Am. Chem. Sot. 103,6988 (1981). 36. A. A. MAUDSLEY, L. MULLER, AND R. R. ERNST, J. Magn. Resort 28,463 (1977). 37. G. BODENHAUSEN AND R. FREEMAN, J. Magn. Reson. 28,471 (1977). 38. S. MACURA AND R. R. ERNST, Mol. Phys. 41,95 (1980). 39. G. BODENHAUSEN, R. FREEMAN, AND D. L. TURNER, J. Magn. Reson. 27, 5 11 (1977). 40. M. ALBERT THOMAS AND ANIL KUMAR, J. Magn. Reson. 54, 319 (1983). 41. M. ALBERT THOMAS, K. V. RAMANATHAN, AND ANIL KUMAR, J. Magn. Reson. 55, 386 (1983). 42. L. BRAUNSCHWEILER, G. BODENHAUSEN, AND R. R. ERNST, Mol. Phys. 48, 535 (1983). 43. S. SCHKUBLIN, A. H~HNER, AND R. R. ERNST, J. Magn. Reson. 13, 196 (1974). 44. M. ALBERT THOMAS, Ph.D. thesis, Indian Institute of Science, Bangalore, 1983.