The calculation of nuclear overhauser effects in coupled spin systems

JOURNAL.

OF

MAGNETIC

RESONANCE

35.95-109

(1979)

The Calculation of Nuclear Overhauser Effects in Coupled Spin Systems JOSEPH

H.

NOGGLE

Department of Chemistry, University of Delaware, Newark, Delaware 19711 Received July 10, 1978; revised November 20, 1978 A theory and computational method are described for the calculation of selective nuclear Overhauser effects for scalar coupled nuclear spins. Sample calculations are included for three spins (two protons and one heteronucleus). The effect of rf power levels is described. Conclusions regarding the utility of selective nuclear Overhauser effects in structural chemistry are drawn.

INTRODUCTION

The discovery of an intermolecular nuclear Overhauser effect (NOE) between a small m o lecule and a macromolecule to which it is attached (I ) has led to an interest in this technique for the exploration of the proximity of m o lecular groupings in biological macromolecules. Investigation of selective NOES in protonheteronuclear systems has shown that resolution is generally quite poor because of the large power levels required and because of long-range polarization or “spin diffusion” in large m o lecules (Z-4). Despite the lim itation on m o lecular weight (estimated by Gerig (4) as 20,000 daltons), selective NOES between naturally occurring protons and “reporter groups” (4) labeled with a heteronucleus such as 19F(4), 13C(5), or 31P(6) show promise as a structural tool in biochemistry. Most theoretical treatments of selective NOES are such that scalar couplings between nuclei cannot be included. Such theories are likely valid in the lim it of complete decoupling. However, the power requirements for saturation and decoupling are quite different. Since the resolution of selective NOE experiments is adversely affected by high rf power and since the J coupling constants between protons and heteronuclei can frequently be as large or larger than proton chemical shifts, it seemsdesirable to have a theory which is valid for spin systemswhich are not completely decoupled. Also, if we wish to calculate accurate “bandshapes” for selective NOES in situations involving overlapping lines and partial, off -resonance, saturation, an accurate theory involving both the transverse and longitudinal magnetizations is needed. In this paper, density matrix methods will be used to provide such a theory. 95

OOZZ-2364/79/070095-15$02.00/O Copyright @ 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

96

JOSEPH

H. NOGGLE

THEORY

A set of “generalized Bloch equations” was derived (7) for elements of the spin density matrix with energy levels labeled Ii), ]j), etc. dUij/dt

=

+

(W

-

Wij)Uij

+

[II

CRij,k[Vkly

kl

duij/dt

=

(0

-

wij)uij

+ dij(aii

-

flij)

+c

Rij,klVkl, kl

-

dCTii/dt

+

CRii,jj(Ujj

-CT:)

=

2CdijUijy

i

PI [31

i

where for the off -diagonal density matrix elements we have and

Uij = Re(oij c iwt)

zlij

=

Im(CTij

e’“‘)

and the power factor is

The relaxation elements R are as given by Redfield (8) These can be solved in the steady state by first eliminating the elements Uij between Eqs. [l] and [2], giving [41

Aijuij +C(Rij,kl/Akl)(dkl(~kk -VU) f CR/cl,mnUmn) = 0, kl

WI”

IIs1

where we have introduced Wij = Riijj with Wii = Riiii = -Cj+iWij, Aij = o -~ij, and xi = aii -(T:, the deviation of the diagonal elements of the density matrix from the equilibrium values ((TO).In addition, we have the condition CXj

=

0.

[61

In practice, if we are irradiating with an rf field with frequency (w) near the proton Larmor frequency (wrr), only elements Vii when i -j corresponds to a single quantum proton transition need be considered. Under these conditions, we can set dij = yHH2/2 for all ij. Equations [4] cannot be directly inverted since the determinant of the coefficient matrix W is equal to zero. However, if Eq. [6] is added to one of the equations [4] multiplied by some constant, we get a coefficient matrix W’ which is the W matrix augmented by the addition of a constant to one of its rows. This constant will ultimately drop out of the calculation so its value is arbitrary; however, for numerical calculations in order to avoid, on the one hand, the disappearance of the constant into the round-off error and, on the other, taking small differences of large numbers with a consequent loss of precision, it is suggested that a coefficient of the same order

OVERHAUSER

EFFECTS FOR COUPLED

SPINS

97

of magnitude as the elements of W be used. Equations [4] can now be solved for the populations as c71

Xi = YHH2C RjC Vjkv k i

where I&’is the inverse of the augmented W matrix, W ’. For Eq. 5 we need only differences of populations. We also should note that, because of the Hermitian nature of the density matrix, vii = - vii. Since the relaxation elements Rijkl with i >j will be zero unless k > 1 also, we will restrict ourselves to the elements Uij with i >j (or vice versa). Introducing this into Eq. [7] we get Xi

-Xj

=

~YI-IHZC

[a

zj,klVkl kl

withi>jandk>I,and zj,k)

= (wik -

wjk

- wi[ +

PI

wjl)/2.

Now, in Eq. [5] we use gkk-ull=Xk--XI-qWH

(remember that kl is a proton transition), where q is a Boltzmann factor as defined by Abragam (9). In practice the value of q does not affect the calculation and can be set for convenience. This gives

Aijvij+ g(&j,/cdAkl)C - (YHH2/2hJHf Rk~,mnVmnI = 0. [lOI Inn[?‘2HHZ~k~,mnVmn Solving for v we get C

QSijmnUmn

=

r111

Cij

with

@ijmn

=

&j:mnAij

+C

(Rij,kl/Akl)(Y~H~Tk,,mn

+Rkl,mn)v

kl

P31

where 6ij:,, = 1 if ij = mn, zero otherwise. Equation [ll] is easily inverted to solve for values of v. To solve for the NOE of the X spin (as the protons are irradiated) we need 77=

I(steady state) - I(equilibrium) I(equilibrium) ’

where I is the intensity of the X transitions. In terms of the spin operator S, of spin X, assuming an experiment such that I cc (S,), 77= Tr S,x/Tr $a’.

[I41

JOSEPH

98

H. NOGGLE

If there are N protons, there will be 2N energy levels with S, of spin X = + 4 and 2N with S, = - 1. Hence, Tr S=(T’= - 2Nqco,,where w, is the X-spin Larmor frequency. Using Eq. 8 to solve for x we get

ml where mn takes values corresponding to all one-quantum X transitions and kl corresponds to one-quantum proton transitions (the only ones for which t)kl # 0). The physical significance of the elements Tii,kl is not immediately apparent. The “diagonal” elements Tab,ab correspond to the quantity T, - Tb introduced by Abragam (10). If we take Eq. [ll] for a single transition, neglecting the crosscoupling elements Tii,k, and Rij,kb we can solve it to give

Vii= (yHzl2)q~HRij,ijl(d + Rz,ij+ yiHgRij,ijTij,ij);

[I61

hence, R,ij = - 11 T2

and

for the ij transition. This is as close as one can come to defining TI and TZ in a coupled spin system in steady-state experiments. We note here that since the only nonzero Uij are those corresponding to the proton transitions, these can be treated as a vector whose length equals 2N+1 for a system of N protons and one spin-:X nucleus. Correspondingly, Rij,kl is a square matrix of dimension 2Nt1. The elements Tij,kl needed for Eq. [ll] to [13] are likewise a square matrix, although some additional elements are needed in Eq. [15]. Both R and T matrices are symmetric with negative diagonal elements. Offdiagonal elements of T are positive or negative; those of the type Tij,jk will be positive if i > j > k (a “progressive” transition). THREE-SPIN

NUCLEAR

OVERHAUSER

EFFECTS

Sample calculations for three spins, two proton and one X nucleus, were performed. In most calculations X = i3C or 19F; however, there is no reason it could not be another proton. However, in our calculation, off -diagonal elements Rij,kl between proton transitions (ij) and X-spin transitions (kl) were neglected. Also, it was assumed the protons were loosely coupled, the chemical shift SABD JAB, the protonproton coupling constant. Most important, we neglect the cross-correlation of the three dipole-dipole interactions, AB, AX, BX. The work of Werbelow and Grant (11) shows clearly that cross-correlation effects are not in general negligible. However, their results for NOE of three nonequivalent spins show no effect of cross-correlation. Indeed, except for notation, Eq. [9.23] of Ref. (II) is identical to Eq. [3.28] of Noggle and Schirmer (12). This is apparently due to the presence of a strong rf field since in general cross-correlation cannot be neglected in such cases. Clearly, however, if equivalent spins and/or nonisotropically reorienting systems were considered, cross-correlation should be considered. Finally, for the calculations presented in this paper, a single correlation is assumed for all three interactions. Most

OVERHAUSER

EFFECTS

FOR COUPLED

SPINS

99

of these assumptions are primarily to simplify the calculation of the relaxation elements and are not limitations of the basic theory. Many common approximations are not made. There is no restriction on the rf power level other than the nonviscous liquid yJIzr,<< 1 (ref. (9, p. 511)). The various transitions may overlap or be exactly coincident; i.e., coupling constants can be set to zero if desired. (The proton-proton chemical shift cannot be set to zero because of the loose coupling requirement mentioned earlier.) A minor numerical problem arises due to the offset Akl in the denominators of Eqs. [lo], [12], and [13]. Cases for which algebraic solutions are possible show that there is in fact no infinity in the NOE even when the rf field is exactly on a resonance (A = 0). However, computers are unforgiving about dividing by zero, so it is necessary to add a small constant (say lop4 rad set-‘) to all A values to prevent such an occurrence. The calculated relaxation elements together with related definitions are given in the Appendix. The only elements needed are those corresponding to single-quantum proton transitions; elements for zero or double-quantum transitions do not affect single resonance when the spins are all nonequivalent. Calculations are done on a hypothetical three-spin system with X = 13C spin and for protons at 100 MHz, rAg = 2.0 ii,

rAx= 1.5 ii,

rgX= 3.0 A.

The various correlation time-frequency products used are summarized in Table 1. The dipole-dipole relaxation mechanism is assumed throughout. Figure 1 shows a sample calculation of the 13C{rH} NOE vs ‘H offset with all coupling constants equal to zero (the chemical shift is 4 ppm). At 7, = lo-“set, in the extreme narrowing region, the NOE of the most distant proton (B) is negative, showing the three-spin-effect indirect NOE (Ref. (12, p. 59)). The maximum NOE at the A proton resonance is 1.94, slightly reduced from the maximum 1.98 because of the finite rf field. At 7, = 5 X lop9 set (most ~7, products - 1; see Table l), the NOES from protons are positive, largely obscuring the obvious structural information available at shorter correlation times and clearly demonstrating the “spin-diffusion” effect (2). Also, the maximum NOE is only 0.35, significantly reduced from that found in the extreme narrowing limit. When 7, = 5 X lo-’ set, well into the long correlation time limit, the NOES of the two protons are barely resolved because of the short proton T2 values, and are equal to 0.15, the theoretical value when all 07~ products are >>1. TABLE

1

CORRELATIONTIMESUSEDFORTHREE-SPINCALCULATIONS~ 7, (4

%-rTc (oH+wc)Tc hi-w)Tc WCC

1 x lo-lo

1 x 1o-9

5 x 1o-9

1 x 1o-8

0.06 0.08 0.05 0.02

0.6 0.8 0.5 0.16

3 4 2.5 0.8

6 8 5 1.6

5 x lo+ 30 40 25 8

JOSEPH H. NOGGLE

100

H

FREO

H

FREO

c

FIG. I. Carbon NOE vs proton offset for uncoupled three-spin system. Vertical marks each 10%; horizontal marks each parts per million. For all spectra proton chemical shift Sv = 400 Hz, power u2 = 25 Hz. Correlation time rc = (A) 1 x lo-“, (B) 5 X lo-‘, (C) 5 X 10-s sec. Geometry for all calculations given in text.

COUPLED

THREE-SPIN

NUCLEAR

OVERHAUSER

EFFECTS

Using the same geometry as that used in the previous section, we now set JAx = 200 Hz (presumed to be directly attached to the carbon), Jsx = 6 Hz, and JAB = 8 Hz. The smaller J values do not affect the calculation to any significant extent. Various chemical shifts and correlation times are used.

OVERHAUSER

EFFECTS FOR COUPLED

H

H

SPINS

101

FREQ

FREQ

FIG. 2. Carbon NOE vs proton offset for a coupled three-spin system. Vertical marks each 50%; horizontal marks each parts per million. For all spectra: Sv = 400 Hz, JAx = 200 Hz, Jax = 6 Hz, JAB = 8 Hz, and rc = 1 x lo-“sec. Power u2 = (A) 50, (B) 100, (C) 200 Hz.

Figure 2 shows a calculation with a 4-ppm chemical shift, that is, S = 2J, and 7c = 1 x 10-r sec. At a power level of 50 Hz (J/4), the C-H J splitting is clearly seen. The linewidth is primarily determined by power broadening; the width at half-height is about 100 Hz, or twice the power level. When the power level is raised to 100 Hz (J/2) an effect akin to decoupling is seen, although it could be equally ascribed to a loss of resolution due to power broadening. At a power of 200 Hz, the NOE spectrum is broad and relatively featureless. The maximum NOE at power uz = 200 Hz is only 1.59 compared to the value of 1.94 obtained for the same

JOSEPH H. NOGGLE

102

H

I-

H

H

FREQ

FREQ

FREQ

FIG. 3.CarbonNOEvsprotonoffsetfor acoupledthree-spinsystem. VerticalmarkseachSO% (A), 10% (B, C); horizontal marks each parts per million. For all spectra: 6v = 400 Hz, JAx = 200 Hz, Jax = 6 Hz, JAB = 8 Hz, and v2 = 50 Hz. Correlation time T== (A) 1 x lo-‘, (B) 5 x lo-‘, (C) 1 x lo-* sec. Note wide sweep in (C).

geometry when J = 0 and u2 = 25 Hz (Fig. 1). Indeed, when the power is doubled again (u2 = 400 Hz, or twice J), the maximum NOE is only 1.86. Our first conclusion is thus: it is futile to try to “decouple” large J’s since power levels greater than 4J are needed to obtain the full theoretical NOE. An extrapolation method as an alternative to “infinite” power spectra will be discussed in the next section. Figure 3 shows the same system as Fig. 2, but at a constant power level (~2 = 50 Hz) and various correlation times. As with the uncoupled spin (Fig. l), we see the three-spin negative NOE at short correlation times and spin diffusion at longer correlation times. In this case, the broadening of the lines (note that (C) is a wider

OVERHAUSER

EFFECTS FOR COUPLED

SPINS

103

A

H

H

FREQ

FREQ

I

P v

H .FREO

4. Carbon NOE vs proton offset for a coupled three-spin system. Vertical marks each 10%; horizontal marks each parts per million. For all spectra: 6v = 100 Hz, JAx= 200 Hz, JBx= 6 Hz, JAB = 8 Hz, u2 = 10 Hz. Correlation time 7c = (A) 1 x 10-l’ , (B) 5 x 10m9,(C) 1 x lo-* sec. Note change in horizontal scale. FIG.

sweep) and consequent loss of resolution is due to the the decrease of T2 with increasing rc. In Figure 4, the chemical shift is 1 ppm (J/2) so that the 13C{Hs} NOE is coincident with one of the lines of the 13C{HA} NOE. At short T,, this combined line is reduced by the negative Hn NOE, while when spin diffusion is important, the combined line is

104

JOSEPH H. NOGGLE TABLE 2 THENONADDITIVITYOFCOINCIDENTNUCLEAROVERHAUSEREFFECTS

6=2J Al A2 B S=JJ2 Al A2+B

u*=lO

I&=20

v*=50

u*=lO

7,=1x1o-10

7f=1x10-10

7c=1x10-10

7,=5x1o-g

0.84 0.84 -0.64

0.95 0.94 -0.70

1.04 1.01 -0.68

0.20 0.20 0.17

0.22 0.21 0.18

0.24 0.23 0.19

0.84 0.45

0.95 0.57

1.03 0.66

0.20 0.26

0.22 0.28

0.25 0.29

u*=20 7,=5x1o-g

v2=50 7,=5x1o-g

a Table values are the extremes of the selective NOE; v2 in hertz, T, in seconds.

larger than the single line. However, this additivity of NOES is not quantitative, as shown in Table 2. Figure 5 shows the interesting case where S = 50 Hz, smaller than J&2, indicating that the theory is equally useful when J splittings put resonances on the other side of another proton. (However, we still require in this calculation that JAB c SABso the protons are loosely coupled.) This case is not fundamentally different than that shown in Fig. 2. POWER DEPENDENCE

OF NUCLEAR

OVERHAUSER

EFFECTS

Earlier we commented on the desirability of a method for extrapolating NOES to “infinite” power. Such a method has been developed for uncoupled spin systems (13) and used by several investigators (4,5). The NOE at “infinite” power, qco,is given by [I71

$=+[l+s-l], co

where the saturation factor is S = y2H;T~T2=br2v;T~Tz.

WI

Despite the difficulty in defining Tl and T2 in coupled spin systems, we specialized the three-spin program to calculate two-spin NOES to test Eq. [17] on coupled spin systems. The results in Table 3 are calculated for a ‘H (90 MHz) and a “F (84.67 MHz) separated by 2 A for various coupling constants and correlation times. When the J coupling is resolved, one has a choice of NOES, the “center” value at the nominal resonance position or the “peak” value, at roughly f J/2. When appropriate, both values are listed. The “actual” values of TIT2 listed in Table 3 are calculated assuming

TI = -CKjk, kl

and

$ = -zRijkr,

Cl91

where ij is one of the proton transitions and the sum over kl includes all of the proton transitions (including ij itself). This can be shown to be an exact definition for exactly

OVERHAUSER

EFFECTS

H

H

FOR

COUPLED

SPINS

105

FREQ

FREQ

FIG. 5. Carbon NOE vs proton offset for a coupled three-spin system. vertical marks each 10%; horizontal marks eachpartsper million. For all spectra:6~ = 50 Hz,J*x = 200 Hz, Jax = 6 Hz, JAB = 8 Hz, and u2 = 10 Hz. Correlation time 7c= (A) 1 X lo-“, (B) 5 x lo-’ sec.

coincident lines, A, = Akl, etc. Calculated values are by least-squares extrapolation using Eq. [17] and three calculated points (marked by asterisks in Table 3). Comparing the J-coupled values with the J = 0 set included for the sake of comparison, we see that the actual values of the NOE are grossly in error until uz is several times the size of J. However, extrapolation using Eq. [17] gives very good values for qoo albeit poor values for TIT,; i.e., the intercept is correct but the slope wrong. The extrapolation works even with points where the J coupling is partially resolved, in which case the “center” NOE values must be used. Extrapolation of the “peak” NOE values of resolved NOE spectra is seen to give reasonable values of TIT2 but absurd values for qao. All in all, this does not seem to be a satisfactory way to measure relaxation times.

106

JOSEPH H. NOGGLE

SATURATIONBEHAVIOUROF

02

0-M

0.1 0.5 1 2 5 10 50 100 Calcd no0 Calcd (Tl Tz)“* “Actual” (TITz)“~

TABLE 3 i9F{‘H} NUCLEAROVERHAUSEREFFECTF?'.~

7c=1x10-10 J=O

7,=1x1o-1o J=2

7,=1x10-lo J=lO

7,=1x10-s J=O

7c=1x10-s J=2

r,=1x10-s J=lO

0.244* 0.505* 0.522* 0.527 0.528 0.528 0.528 0.528 0.528

0.005/0.125* 0.099/0.247* 0.253/0.301* *0.415 *0.506 *0.522 0.528 0.528 0.528/0.280

0.000/0.125* 0.005/0.228* 0.019/0.236* *0.068/0.248 *0.255/0.300 '0.416 0.522 0.528 0.53210.237

-0.006* -0.126* -0.350* -0.631 -0.812 -0.847 -0.859 -0.859 -0.860

-0.005 -0.117* -0.332* -0.615* -0.808 -0.846 -0.859 -0.859 -0.858

-0.002/-0.003 -0.04/-0.068* *0.144/-0.185* *-0.384/-0.387* *-0.717 -0.819 -0.858 -0.859 -0.861/-0.506

1.47

0.15/1.43

0.01/1.68

0.13

0.13

0.07/0.125

1.47

1.47

1.47

0.13

0.13

0.13

D When two values are given the first is the “center” NOE (at the position of the uncoupled resonance) and the second is the “peak” value. If only one value is given, the J coupling is not resolved under those conditions. ’ Values marked with asterisks are used for least-squares extrapolation. Left * extrapolated to left value, right to right. ’ rc in seconds; J in hertz.

Overall, however, the conclusion is rather positive; Eq. [17] can be used to extrapolate NOES in J-coupled spin systems provided at least partial collapse of the J coupling is seen and “center” values are used. CONCLUSION

For three spins this theoretical approach to the calculation of selective nuclear Overhauser effects is quite successful. However, for larger molecules the complexity rises geometrically with the number of spins. However, as we have seen, even a three-spin program is capable of discovering generalizable conclusions. Another utility would be to test less rigorous theories which are more easily extended to multiple spins. Among the other results previously expressed, the results of this paper bear on the utility of 13Clabeling of “reporter groups”. Some may feel that the 13Clabel should be in an unprotonated position, otherwise, the 13C relaxation is dominated by the directly attached proton. Quite the contrary, if no proton is attached, the carbon is subjected to possibly ill-defined relaxation mechanisms other than dipole-dipole. Also, as can be seen in, e.g., Fig. 3, three-spin effects at short correlation times and spin diffusion at long correlation times make the carbon sensitive to protons other than that directly attached. It is, in effect, reporting on the experience of its neighbor proton. Nonetheless, the r3C{lH} NOE is so reduced at long correlation times that 19F is likely to be more useful in large molecules.

OVERHAUSER

EFFECTS FOR COUPLED

SPINS

107

What does this work say about the “resolution” problem in large, slowly rotating, molecules? Not much which has not already been said. However, the existence of an accurate bandshape theory gives rise to the possibility of using deconvolution to enhance resolution. APPENDIX

Basis Set in Order H,AHBX

11)= ~(Y(Y,12)= (Y/G, 13)= @ .YLY, 14)= P&Y, 15)= c&I, 16)= (u/3/3,

17)= PM, 18)= PPP Definitions Dij = yf-y; h2/r;

for

ij = ab, ax, bx

f(w) = 7,/(1 +w27,2) Redfield’s Formula

Spin-Lattice

Elements

Riiji E Wij with Wii s - C W, j#i

V/;z = W34 = Ws, = W,8 = O.lS(Dab +D&(wH)

(B transitions)

WI3 = W24 = Ws, = WC,~= O.lS(Dnb + D,,)~(wH)

(A transitions)

Wls = W,, = W3, = Wz,8 = 0.15(Db, + D,,)f(ox)

(X transitions)

WI4 = W,, = 0.6D,bf(2w,); WI, = W28 = 0.6D,J(w~ W,, = W47 = O.lDJ(

WI6 = W M = O.~D&(WH+

wx)

+ wx); W,, = W6, = O.lD,~,f(o)

wH-o~);

W35= W46=0.1Daf(~~-~x)

remainder zero. Transverse Relaxation

Diagonal

Elements

Elements

Riiii = R; + WiJ2 + Wjj/2 + Dbx)f(0)

for

ij = 12,34,56,78

(B transitions)

Rz = - 0.2(Dab + D,,)f(O)

for

ij = 13,24,57,68

(A transitions)

Rz = -0.2(Dab

JOSEPH H. NOGGLE

108

A-A

Cross-Elements -Dab)f(WH)

R1234=R5678=0.WL

=R3478

R1256

= 0.15(D,x

-Dbx)(f(@X)

o.l&f(oH-&

R3456=

= O.~&~(WH+~X)

RI278

B-B Cross-Elements R1324=R5768=0.15(Dbx-Dnb)f(OH) Rt357

=

R2457

= 0. 1DbxfbH

- OX);

R1368

= 0.6DbxfbH

+ Wx)

R2468

= 0.15(&

-D,,)fbx)

AB Cross-Elements R1213

=

R3424

=

R5657

=

R7868

R1224

=

R1334

=

R5668

=

R5778

= -0.2&f (0)- 0.15&f hi) = -o.l5Dabf(W)

remainder zero (some of the other AB cross-elements would be nonzero if cross-correlation were included) or not needed in the calculation. Note that frequencies w are 2P(frequency in hertz). Power levels given in the paper are v2 = yH2/2~ in units of hertz. ACKNOWLEDGMENTS This work was supported by the National Science Foundation (CHE77-06794). Computations were performed using the Burroughs B7700 Computer at the University of Delaware Computing Center.

REFERENCES 1. P.BALARAM,A. A.BOTHNER-BY,ANDE.BRESLOW, Biochemistry 12,469s (1973). 2. A.KALKANDH.J. C.BERENDSON,J. Magn. Reson.24,343 (1976). 3 A. A. BOTHNER-BY AND P. M. JOHNER, XX Colloquium Spectroscopicam Internationale 7. International Conference on Atomic Spectroscopy, Praha 1977. 4. J.T.GERIG,D.T. LOEHR,AND K.F.s.Lu~,in press. 5. J.J.LED,D.M.GRANT,W.J.HORTON,F.SUNDLY,ANDK.VILHELMSON,J. Am.Chem.Soc. 97,5997 (1975). 5 P. L. YEAGLE,~. C. HUTTON, CHING-HSIEN HUANG, AND R. B.MARTIN, Biochemistry 16, 4344 (1977).

7 J. NOGGLE, J. Chem. Phys. 43,3304 (1965). 0”. A. G. Redfield, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 1, p. 1, Academic Press, New York, 1965. 9. A. ABRAGAM, “The Principles of Nuclear Magnetism,” p. 524, Oxford Press, London/New York, 1961.

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EFFECTS FOR COUPLED

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109

10. A. ABRAGAM, “The Principles of Nuclear Magnetism,” p. 256, Oxford Univ. Press, London/New York, 1961. Il. L. G. WERBELOW, AND D. M. GRANT, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 9, p. 189, Academic Press, New York, 1977. 12. J. H. NOGGLE AND R. E. SCHIRMER, “The Nuclear Overhauser Effect: Chemical Applications,” Academic Press, New York, 1971. 13. D. F. S. NATUSCH, R. E. RICHARDS, AND D. TAYLOR, Mol. Phys. 11,421 (1966).