Precision of Volume Integrals in Two-Dimensional NMR - Science Direct

JOURNAL

OF MAGNETIC

RESONANCE

98,259-270

( 1992)

Precision of Volume Integrals in Two-Dimensional NMR R. NADJARI AND J.-PH. GRIVET Centre de Biophysique Mokdaire. C.N.R.S. et Universitk d’OrGans, 1.4 Avenue de la Recherche, F-45071 OrGans Cedex 2, France Received May 2 1, 199 1; revised November 4, 199 1 The precision of measurements of peak volumes in a correlation map obtained from a two-dimensional NMR experiment is investigated. It is shown that the noise correlation function in the frequency domain and the volume standard deviation both reduce to a product of one-dimensional factors. The ratio of a peak volume to its standard deviation is derived and is related to the usual signal-to-noise ratio. The effects of sine-bell windowing and exponential muhiphcation are found to be slight. D 1992 Academic Press. Inc.

Many two-dimensional NMR experiments provide qualitative information on the sy,stem under study, such as connectivities between nuclei or multiplicities of resonances. Some experiments, on the other hand, are specifically designed to produce quantitative information as a function of two frequencies. The volume of a cross peak on a NOESY map (1-3) is a known function of the distance between the two relevant nuclei. It is thus possible, at least in principle, to derive the distance from the peak volume. In the EXSY experiment (I, 4-6), the cross-peak volume depends on the kinetic constant of the reaction carrying a nucleus between the sites involved. Here again, knowledge of the volume allows the experimenter to derive chemically useful information. The NMR signal is always somewhat corrupted by noise, so that the two-dimensional integral of a cross peak is never obtained exactly. The experimental value is a random variable, with mean equal to the theoretical value (provided the experiment is accurate or unbiased) and a standard deviation which is small when the experiment is precise. In this report, we generalize the theory of the precision of integrals established in a previous paper to two frequency dimensions ( 7). We are able to estimate the standard deviation of a peak integral and thus to propose limits on the precision of derived parameters, distances, or kinetic constants. THE MODEL

In a typical two-dimensional experiment, one records N, different free-induction decays, each corresponding to a different value of a parameter tl, which is incremented with a step At,. The actual acquisition of an FID uses Np complex data points, a spectral width FV,, and an acquisition time T,; these parameters are related since T, = N,/ W,. The spectrometer records a signal s( t, , t2) and a noise b( t2). We assume that this noise is ergodic, Gaussian, and white, with zero mean. Each FID is corrupted 259

0022-2364/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

260

NADJARI

AND GRIVET

by a different realization of the random process b( t2). The noise can be quantitatively characterized by its variance a: = q W,, where the index t indicates a time-domain quantity, and q is the power spectral density of the noise. The N, FIDs thus collected ( N,NP data points) are considered to be a sampled representation of the two-dimensional signal s( t,, t2), although only the variable t2 corresponds to a physical measurement. It will prove convenient for our work to go through the same reinterpretation of the noise and to introduce notation that treats the two variables on an equal footing. We assume that the spectrometer memory contains a sampled realization of a two-dimensional random process, b( t, , t2). The statistical properties of this process are derived by imposing that the noise power be the same in the two representations. We assume that the signal was acquired using bandwidths IV, and IV2 in the two dimensions, and that the noise spectral densities are respectively ql and q2. W, is identical to the previously defined W,, while W, is fictitious as far as the noise is concerned, since there is no analog filter in the t, dimension. It may happen that W, =S W,. The time t2 evolves from 0 to t2 max, and t, from 0 to 1, max. There are N, digitization points on the t, axis and N2 on the t2 axis. These values are connected to previously defined quantities by N, = N,, t, max= N, At,. We then require that q W, = q, W,q, W, or 4 =

41w192.

[ll

The noise is uncorrelated along the tl axis; this property is a consequence of the fact that the (physical) noise has a small correlation radius, smaller than 1/ W,. In other words, two noise values that lie close together on the t, axis, say b( t, , t-J and b( t, + At,, t2) are recorded so far apart in real time that they cannot possibly be correlated. CORRELATION

The within graph, the vI,

FUNCTIONS

OF NOISE

spectrometer noise is not rigourously white; its spectral power density is constant the pass band, according to the interpretation presented in the previous parawe assume that the noise power density is constant in a rectangular domain of v2 plane; that is, Y(Vl,

v2)

=

q1q2

if -W,/2

0

otherwise.

< vl < W,/2 and -W2/2

< v2 < W2/2

Using the two-dimensional version of the Wiener-Khinchin theorem and the definitions 7, = t, - t ; and 72 = t2 - t ;, we obtain the noise correlation function in the time domain (sine x = sin ( TX)/ TX): R(TI, TV) = qlq2Wl W,sinc( W1rl)sinc( W2T2).

[21

The noise variance can be recovered by placing 7, = 72 = 0 in the preceding equation: a: = ql W,q2 W2. We may safely assume that W, t, maxb 1, W2t2 maxii, 1, so that the noise correlation function simplifies to Rb,,

72)

=

wd(~,)~(n).

131

PRECISION

OF

VOLUME

INTEGRALS

IN

2D

NMR

261

The transition from a physical one-dimensional noise power to a mathematical two-dimensional noise must be done with caution. The physical dimensions of the quantities involved provide a useful guide. Assume that b(t) is a voltage; then B(V), its Fourier transform, is measured in volts per hertz. R, the correlation function of 6, is in volts squared. The power spectral density, y, is the Fourier transform of R: it is measured in volts squared per hertz. This quantity is often a constant, denoted 4. The integral of a line in the frequency domain is expressed in volts, and its variance, a:, in volts squared; a; was shown previously (7) to be of the order of 96, which is dimensionally correct. Similar relations apply in the two-dimensional case: b( t, , tz) is again a voltage, with correlation function R in volts squared and power spectral density y in volts squared per hertz squared. This quantity is called qlq2 in the following, so that the dimensions of the qi are in volts per hertz. It can be seen that R, as given above, has the expected dimensions. Equation [ 31 is also correct, because delta “functions” are in fact of dimension hertz, as can be seen from their Fourier representations. A peak volume in the frequency space is measured in volts, with its standard deviation, a$, in volts squared. When a filter or apodization function is applied in either dimension, the noise is no longer stationary, but the previous formalism is considered to be approximately correct for quasistationary noise. Let .f; (I, ), .fi( t2) be the two filter functions. The function R is now NT, 3 72)

=

[41

4142J’:(t,).f:(t2)s(T,)~(T*).

The function R is separable in a product of two one-dimensional correlation functions, h(r,) and &(Tz). The noise in the frequency domain, B( v, , u2), is the two-dimensional Fourier transform ofthe noise b(t,, t2). We call I’(v, - u,‘. v2 - u;) its correlation function, which is defined as - u;, l3

r(h

-

vr’)

=

uz)B*(Y;,

G{B(V,,

vi,)

[51

if & denotes the expectation value. We show in Appendix A that, just as R, I‘ separates in a product of one-dimensional factors, UVl - y;, v2 -

4)

=

r,h

-

4)r2(4

-

4)

=

s[qlf:(tl)i~[q2S:(t2)1,

if51

where we use 3 to indicate a Fourier transform. We are interested in the real part of the spectrum and in the real part of the noise. We can separate the real and imaginary parts of the noise, B = BR + i&, and of the autocorrelation function, r = rR + il?, . It can be shown (see Appendix B) that &j&(v,

3 V2)~R(V;,v;)} = ${rR(v, DOUBLE

INTEGRAL

+ v;, u2 i- v;) + rRty, - u;, v2 - v;)j. OF

THE

[71

NOISE

When a rectangular region (v, inr = vi0 - 6112 G ~1 < vI sUp= ~10+ 6,/2, v2 inr = ~2~ - 621.2 < u2 G u2sUP= v20 + ?j2/2) of a correlation map is integrated, the noise is summed at the same time: we therefore need to know the statistical properties of K,

262

NADJARI

AND

GRIVET

the double integral of the noise over the same region. G[K] = 0 because the noise itself has zero mean. The variance of K is derived in Appendix C as a product of two similar integrals, which we call a$, and ai*; thus:

a;

=

C[K2]

=

[81

ag,r&.

The noise variance is the product of two integrals involving the real parts ( rlR, rZR) of the one-dimensional correlation functions ( r, , r2) defined above. A feeling for the meaning of this result can be obtained by looking at a special case, namely the case where a filter is not used; i.e., f, = f2 = 1. The noise in each of the frequency domains is then uncorrelated; that is, T ,R = q, 6 ( V)/ 2 and a similar expression for r2n. Equation [ 81 then yields U$i = qiSi/2;

0:

=

[91

q,Q72~2/4.

Using known quantities, [ 91 can be written as [9'1 When the integration ranges become identical with the spectral widths, [ 9 ‘1 implies that &becomes identical with qW,, the noise power in the time domain, apart from a factor of 4, which reflects the fact that the Fourier transform is computed for positive t, and t2. On the other hand, qW, = a:, which means that the variance of the integral of noise, over the whole frequency map, is one-fourth the variance of noise in the time domain. Equations [ 61 and [ 81 are the main working formulas used in the rest of the text. Up to this point, we have assumed that the signal and noise are subjected to a twodimensional complex Fourier transform. However, this is not often the case in practice, because of the unfavorable peak shapes that are obtained (see next paragraph). Experimental methods have been devised (9, IO) which yield purely absorptive peaks. We must then ask ourselves whether the statistical properties of the noise, as derived above, still apply to these modified experiments. We choose to investigate the case of the Haberkorn-States-Ruben method, which is simpler to explain. According to these authors, two independent NOESY experiments are performed, and the resulting signals s,( t, , t2) and s,( t, , t2) are recorded separately. This notation refers to the fact that the first data set carries an amplitude modulation proportional to cos w,t, , while the second data set is modulated according to sin w ,t , . A complex Fourier transform in the second dimension then yields two complex interferograms: s,(t,,

v2)

=

at,,

v2)

+

mt,,

v2)

s,(t,,

v2)

=

at,,

v2)

+

K(t1,

v2).

The two imaginary parts are discarded and a new complex interferogram is constructed: et,,

v2)

=

Qt1,

v2)

+

wt,,

v2).

PRECISION

OF

VOLUME

INTEGRALS

IN

2D

NMR

263

A complex Fourier transform with respect to t, then yields the desired signal. Consider now the fate of the noise during this data manipulation; s, and s, each comprise an independent realization of the same complex random process, &( t, , t2) and b,( t, , t2). The first Fourier transformation gives the noise parts of the interferograms: bc(tl,

v2)

=

b6(t,

3 v2)

+

wt,,

v2)

bdt,,

~2)

=

bitt,,

~2)

+

ib:(t,,

~2).

Considered as real random variables, b:, b:, b,‘, and bi have identical statistical properties. It follows that the reconstructed complex noise function b( t, , v2) = bA( t, , u2) + ibi( tr , v2) has the same properties as b, or b,. As a consequence, the noise in the frequency domain, b( ZJ],v2), has the same correlation function as that derived above for a single two-dimensional Fourier transform. :Keeler and Neuhaus have shown (II) that the TPPI method, associated with a real Fourier transform in t, , is completely equivalent to the procedure of States et al. The noise properties derived previously therefore apply to this type of experiment also. THE

SIGNAL

SHAPE

The shape of a signal is a NOESY or EXSY experiment after complex Fourier transformations with respect to both time variables is (2) S(Vl,

v2)

=

N&(v,)Sz(v2)

1 1 + i[(~; - voi)/Av,] S;(q) = tAv, 1 + [(Y, - Q~)/Au;]~.

[lOI

Introducing the absorption and dispersion lineshapes ,4 ( v) and D(u), S, can be written as Si(Vi) = Aj(Vi) + iDI( The real part of the two-dimensional sR(uI,

u2)

=

signal is therefore

N[A,(V,b42(v2)

-

DI(~I)D2tv2)1.

[lo’1

N i,s a factor which depends on the physics of the experiment; it is a function of correlation times, distances, etc. AZ+and Au2 are the natural half-widths at half-height along each frequency axis. SR shows the well-known “phase-twist” shape (9-12). A pure phase signal [ SR = A, ( v1)A2 ( v1) ] cannot be obtained, whatever the phase correction. However, we note that both DI and D2 vanish at the peak center, so that the peak height is simply given by S,, = N/( aAv, )( wAQ). Further, these two functions are odd with respect to the variables vi - voi. If the integration region is chosen symmetrical about the peak maximum, as we have done, the dispersive part of the signal will not contribute to the integral. The peak volume is then

V, = 1 arctan 5 . 7r i VI1

[Ill

264

NADJARI

AND GRIVET

The units of I’ are those of N. This expression applies also if the basic NOESY experiment is replaced by one of its variants in order to obtain absorption peaks. THE SIGNAL-TO-NOISE

RATIO FOR PEAK VOLUMES

The volume standard deviation has been established as [ 91, and the peak volume is given by [ 1 I] ; the signal-to-noise ratio rv for this last quantity is therefore simply: TV = Nrv,rvz

t121 The quantity rv considered as a function of either integration range, say bZ, behaves as arctan( 6,/ Av,)/ 6. This function has a broad maximum for ?j2 g 1.5Av2. An order of magnitude estimate of r v is obtained when 6i 9 Avi. In that case, we find

On the other hand, the two-dimensional signal is maximum at the peak center where it takes the value S,, = N/a2Av,Av2. Further, the noise variance in the frequency domain, a:, is obtained most simply as the double integral of the noise power in the time domain, for 0 G ti < ti max; thus af = qlq2tl maxt2max. Since the noise is assumed ergodic, cu can be determined from a single realization, i.e., along any slice of the correlation map. Using these relations, the peak height-to-noise ratio is found as rs =

N

WI

[I31 al maxt 2 max* It is of interest to express the volume signal-to-noise ratio as a function of the corresponding quantity for the peak height: n2Av,Av,

t1m’;-axarctan(d,/Av,)

arctan(b2/Av2). 1141 I 2 All the parameters involved in [ 131 are either experimentally accessible ( Avi) or chosen by the spectroscopist. It is thus possible to predict the signal-to-noise ratio necessary to obtain a prescribed precision on the volumes, or, equivalently, the prescribed number of scans. Typical parameters for a NOESY experiment on a small (six base pairs) oligonucleotide are W, = W, = 3000 Hz, N, = 256 (no zero-filling), Nz = 4096, Au, = Au2 = 4 Hz, 6, = b2 = 10 Hz. For these values, r v/rs z 1.4. The precision of volume determination is of the same order as the usual signal-to-noise ratio. THE EFFECT OF FILTERS AND WINDOWS

Since the two frequency variables are effectively independent, the effect of a windowing function or of a filter can be found for vI and v2 separately. The corresponding

PRECISION

OF VOLUME

INTEGRALS

IN 2D NMR

265

calculations have been reported in a previous publication ( 7). We summarize the results here. An exponential filter is realized by multiplying a row or a column of s( t, , t2) by exp( -ti/ r,), i = 2 or 1. The linewidth at half-height is then increased by Af; = 1 / TTi; the effective width is now 2X; = 2Avi + A&. The signal and its volume are given respectively by expressions similar to [LO] and [ 111, where the Au, are replaced by X,. The noise in each of the two frequency domains now becomes correlated; the real parts of the correlation functions (obtained using [ 61) are

1 1 + [(yi - 4)/Af;]2

riR(ui - ‘:) = &

’

[I51

Using [ 8 ] and [ 151, the variance of the volume is found as a:: = ug, u&2 &=:[26,arctan(&)-A/ln(l+$)]. In the limit where & $ Al;, we recover relation [ 91. This approximation is used in the following derivation. The signal-to-noise ratio, after applying the-filter, is

The volume signal-to-noise ratio, under the same hypotheses, is rc. = NrI,,rvz

6, 1 YLJr =-I5 Tarctan 5 ( I-&q.

iI81

Combining [17] and [18], we find that the ratio yV/rs can be written in terms of experimental quantities: f-t/ -=rs

4fl

xi

iT j=1,2 v&i3

6i arctan i 2xi i .

[I91

This result is very similar to Eq. [ 141 (where a filter is not used). The differences are due: to two facts. (i) The peak is broadened ( Aui replaced by hi). (ii) The noise is effectively recorded during a time TV = 1/rAf; instead of ti max. The influence of an exponential filter can be examined in more detail on a special case, We choose an integration range equal to twice the effective width, 6; = 4hi, and set .X = AA’/ Aui. Starting from the exact expressions [ 12 ] and [ 161, we compute the rati’o K, = r vi (without filter)/ r r/i (with filter). After some algebra, we get Ki =L

’ arctan(?+Z.+[Zarctan(+)-&ln[l+4(+r]} f- x arctan 2

266

NADJARI

AND

GRIVET

As Af; (or x) increases from 0 to a large value, Ki decreases from 1 to 0.625; in the case of a matched filter (Ah = Avi), K = 0.75. Since the effective linewidth and the integration range increase linearly with Ai, a slight improvement in precision is associated with a greater peak overlap. The case of a shifted sine-bell window is handled in a similar manner. The functions f; are in this case A(ti)

= SiIl(

7r ‘i[Tixti)

.

The Fourier transform of qi f ;?yields the two factors entering the noise autocorrelation function in the frequency domain. The noise variance is the value of this function when vi = 0; we find U”2x

.

[211

Two special cases may be noted. When k, and k2 are both equal to either 1 or 2, the variance is one-fourth of the value found without a window. Using [ 6 ] and [ 8 ] and going through the same steps as in the case of an exponential filter, we derive the variance of the peak volume, a$. The conditions 6i 9 1 /ti max always obtain for NMR and allow us to write 1 2

Uf& = - qiFiSiI12

’

;

ki Apart from the sine factors, this expression of ai is identical to that in the unwindowed case [ 91. The peak height of the signal is &I

=

S(VOl,

vo2)

=

N

sin(r/k,) ITAV,

sin(7r/k2) aAv2

so that the signal-to-noise ratio appears as y,~sm=NL U”

J-J sin(P/ki) IT’ i=1,2aAvi=

’

~231

The peak volume that could be measured on the same map reads ,=bsin($arctan(-j-);

V=IVNV,V2,

1241

so that the ratio of the volume to its standard deviation is r v = Nrvlrv2

rvi=5 =e arctan(hlAvi> uKi

r

vz&

~251

.

This expression is identical to [ 18 1, which pertains to the case where no window is applied. The precision of the integral is not affected by windowing, at least under the

PRECISION

OF

VOLUME

INTEGRALS

IN

2D

NMR

267

hypotheses used in the present work. The volume signal-to-noise ratio can be expressed in terms of the usual peak-height signal-to-noise ratio as :==

i-1.2

arCtall(Gi/Au;) Aq\jti,,. V&in( 7r/k,)

Dl

We have assumed that the same type of signal processing was applied for each time variable; when different window functions are used, the complete result can be obtained by combining the individual Vi and OKifactors given above. It must be pointed out that the accuracy (absence of bias) of the volume depends strongly on the lineshape, the window function, and the integration interval. It is well known that a Lorentzian has extensive “wings. ” Thus, in order to recover at least 99% of the area, the one-dimensional integration range must be larger than 32 full widths. Since such a large range would probably comprise many other lines, one must use a smaller interval. The correct method, in principle, would be to determine the width of each peak (for both frequency dimensions) and integrate over a constant multiple of the width. A reasonable alternative is the following. If the signal-to-noise ratio ys is, say, 10, then r y is expected to be about 10 also. We then accept a systematic error of lo%, which implies integrating on 3.5 times the width. Since Gaussian lines have much smaller extensions, a Lorentzian-Gaussian window has some advantage in this res’pect. Unfortunately, there is no analytic expression of the correlation functions in this case. CONCLUSION

‘We have shown how the precision of volume determinations for two-dimensional NMR correlation maps is affected by noise, filtering, and windowing. The consequences of these results for applications can now be briefly examined. In magnetization-transfer spectroscopy, the volume of a peak located at frequencies v1 = Vk,u2 = V/is proportional to the corresponding matrix element (3, 6), vk/

a

[w-R~rn)lk/r

[271

where R is the relaxation matrix (or the kinetic matrix) and 7, the mixing time. For simplicity’s sake, we assume that the initial rate approximation holds, so that [27] can be replaced by vk,

cc

-Rk/rm.

[281

Th’e precision on Rkl is determined by its standard deviation, which follows from the classical error propagation formula ( 9)

Since absolute intensities are meaningless in NMR, volume ratios are used in practice. The precision of Rkl will therefore be roughly one-half of that predicted by [ 29 1. Furthermore, in the case of a NOESY experiment, Rkl is proportional to the sixth power of rk/, the distance between the two relevant nuclei. Combining the various numerical factors, we find that determining a distance to 10% requires a 30% relative

268

NADJARI

AND

GRIVET

precision on I’, or a signal-to-noise ratio 23 for that peak. The initial rate approximation is now considered unsatisfactory for large molecules, and full relaxation matrix treatment has been advocated ( 10). The volume standard deviations derived above can easily be inserted in a full error propagation calculation (6). APPENDIX

A

We show here that the noise correlation function in the frequency domain is separable. Starting from Eq. [ 51 and using the definitions of B( vl, u2) as 3 [ b( t2, t2)], we obtain rt*,

- *;, V2 - 4) = J-f c

s,r [

G(b(t;,

t;)b*(t,,

t2)}e-2~*(~~~~-~~~~)

x e-2iR(uZfZ-u;t;)dt;dt;dt,dt2 Since the noise is white and shows no correlation between variables 1 and 2, the time correlation function is separable: G{b(t;,

t2)} = R,(t;

t;)b*(t,,

- tl)Rz(t;

- t2) = n

- ti)e

qJj(tj)s(ti

i=l,2

Inserting these functions in the integral for r, we find rt*,

- v ;, v2 - *;) = n

qiff(t;)e-2’“‘i’Yi-O:)dti.

i= I,2

r appears as a product of two Fourier transforms, for the time variables 1 and 2. APPENDIX

B

Here, we derive an expression for the correlation function of the real part of the noise in the frequency domain. The correlation function r is a complex function which we write as I? = IR + iri. The noise in the frequency domain is written as B = BR + iB, ; inserting this form in the definition of r, we find r(*i

-

*;,

~2 -

4)

=

&{BR(*I,

VZ)BR(V;, +

wl

+

vi, -

v2+

4)

*iI>

+

+

&(*I,

VZM~I',

Q)BR(V

i&{&t*l,

v2)W;,

= G{B(v,,

Q)&(vI’,

&(*I,

4)

4))

=

i&{Bd*l,

;,

*;>

G{BR(vI,

VZ)BR(V;,

vi)

-

*;,I BR(~,

YZ)&(V;,

~2)BR(4,

4

+

v~)BI(YI',

BR(VI,

vi)>-

Adding the real parts of these two equations, we find rR(%

6

~2,

vi)

+

rRh

-6,

*2,

-vi)

=

26{B~(h,

v~)BR(v;,

h>>.

In the case of a stationary noise, only frequency differences appear; rR(VI

-

v;,

which is Eq. [ 7 1.

v2-

vi)+

rRh

+

6,

v2 +

~8

=

~@$R(vI,

v~)BR(&

J';,}

v;>>,

PRECISION

OF VOLUME

INTEGRALS

APPENDIX

269

IN 2D NMR

C

Using the fact that the frequency correlation function r = I’n + iI’r is separable in a product of functions relative to each frequency axis (Eq. [ 6]), and writing ri as rjR +r,,(i= 1,2),onefinds

rR(+- u;.u2 r&

-

v;,

-

vi)

=

rlR(Vl

v2 - &) = rlR(u]

-

v;)r2R(u2

- u;)r21(u2

-

6)

-

I‘&

- u;) + rlI(ul

-

v;)r2du2

- u;)r2,(u2

-

Vi)

- Y;).

The quantity of interest is the integral of the real part of the noise:

& { K} vanishes and we ask for the variance & { K2 } . This quantity can be found by first writing K2 as a product of integrals and then taking the expectation value. Thus G{K2}

=

U2)&(

/G(U,(U,,

u

;, v;)} du, du ;du,du;.

Th’e integrand can be substituted according to Eq. [ 71: &{K2}

=~~{rR(VI

+U;,U2fU;)frR(uI

-U;,

u2 - u2’)}du,du;du2du;.

Because of the symmetry of rR, this integral simplifies to &{K2}

= 1 (r&u,

+

U;,

u2 f u;)}duldu,‘duzdu;.

Inserting the expression for rR derived at the beginning of this appendix, one obtains &{ K2} = s { r&u1

- u;)r&2

-

U;)

- rlI(uf

- u;)r&

- u;))duldu;duzdu;.

Each quadruple integral is now written as a product of double integrals involving the variables ul, u; and u2, u;. Because only differences appear, double integrals can be further simplified (8), leading to &(K2}

:= fl s6’ Mc)(~, j=, -6,

- luil)du; - n f’ MuJ(~, 1-I -6,

- b,IWj.

The functions I?; are again Hermitian so that rIi is odd and the last product vanishes. The surviving terms make up Eq. [ 8 1. ACKNOWLEDGMENT

We thank Dr. M.-A. Delsuc for critically reading the manuscript and for helpful suggestions. REFERENCES 1. J. JEENER, B. H. MEIER, P. BACHMAN, 2. S. MACURA

AND R. R. ERNST,

Mol.

AND R. R. ERNST, Phys. 41,95 ( 1980).

J. Chem. Phyx 71,4546 ( 1979).

270

NADJARI

AND GRIVET

3. D. NEUHAUS AND M. WILLIAMSON, “The Nuclear Overhauser Effect in Structural and Conformational Analysis,” VCH Publishers, Cambridge, 1989. 4. B. H. MEIER AND R. R. ERNST,J. Am. Gem. Sot. 101,6441 (1979). 5. S. MACURA, Y. HUANG, D. SUTER,AND R. R. ERNST, J. Magn. Reson. 43,259 ( 1981). 6. C. L. PERRIN AND J. J. DWYER, Chem. Rev. 90,935 ( 1990). 7. R. NADJARI AND J.-ml. GRIVET, J. Magn. Reson. 91, 353 ( 1991). 8. R. PAPOULIS, “Probability, Random Variables and Stochastic Processes,” McGraw-Hill International Editions, Singapore, 1984. 9. D. J. STATES,R. A. HABERKORN, AND D. J. RUBEN, J. Magn. Reson. 48,286 ( 1982). 10. D. MARION AND K. WOTHRICH, Biochem. Biophys. Rex Commun. 113,967 (1983). II. J. KEELER AND D. NEUHAUS, J. Magn. Reson. 63,454 ( 1985). 12. R. FREEMAN, “A Handbook of Nuclear Magnetic Resonance,” Longman Scientific and Technical, Harlow, 1987. 13. P. R. BEVINGTON, “Data Reduction and Error Analysis for the Physical Sciences,” McGraw-Hill, New York, 1969. 14. B. A. BORGIAS AND T. L. JAMES, J. Magn. Reson. 79,493 (1988).