Nuclear overhauser effects in exchanging systems

JOURNAL

OF MAGNETIC

RESONANCE

43, 175 192 (1981)

Nuclear Overhauser Effects in Exchanging Systems* M. Bonzo Department

of Chemistry,

Colorado

G. E.

AND

State

University,

MACIELt

Fort

Collins,

Colorado

80523

Received June 26, 1978; revised November 3, 1980 Expressions for the time derivatives of the magnetizations for a system of any number of any types of spin-l/2 nuclei exchanging between two configurations have been developed. Hypothetical examples, illustrating the symbolism and applicability of these expressions, with particular emphasis on possible nuclear Overhauser effect (NOE) experiments, are discussed. Among the examples discussed is the conformational exchange in dimethylformamide due to rotation about the C-N bond. A more complex phenomenon which could be investigated using the formalism presented herein is the NOE observed on the resonance of a solvated spin-l/2 metal nuclide upon irradiation of the solvent resonance (e.g., protons in water). The effect, if any, that an exchange process would have on measured NOE values is discussed for limiting exchange conditions and for various types of exchange. INTRODUCTION

We have used a Solomon-type treatment (1) to find expressions for the time derivatives of the magnetizations for a system of any number of any types of spin-l/2 nuclei exchanging between two configurations. The Bloch equations as modified by McConnell (2) describe the time rate of change of the magnetizations in the case of a single spin-l/2 nucleus (designated here as 11) exchanging between two configurations (labeled A and B) as represented by l/T,

I 1A

I 1B

IT;,

configuration

A

configuration

B

where rA and rB are the first-order lifetimes of I1 in configurations A and B.’ Expressions for the time derivatives of the magnetizations for a system of two spin-l/2 nuclei (I1 and 1.J exchanging between two configurations were presented in 1971 by Combrisson et al. (3). Their equations were for a system of two spin-l/2 * Presented in part at the Third Biennial Rocky Mountain Regional Meeting of the American Chemical Society, Laramie, Wyoming, June 1976, and in part at the Groupe de Contact du PNRS, Liege, Belgium, April 1977; based in part upon a dissertation submitted by M. Borzo to the Graduate School of Colorado State University in partial fulfillment of the requirements for the Ph.D. degree, May 1977. t To whom correspondence regarding this paper should be addressed. 1 As was noted by McConnell (2), the equations to be presented here “can be applied to reactions of any order, but under these circumstances the lifetimes rA and/or Q, are to be related to pseudo first-order rate constants.” 175

0022-2364/81/050175-18$02.00/O Copyright 0 1981 by Academic Rem, Inc. AU ri&s of reproduction in my form reserved.

176

BORZO AND MACIEL

nuclei of equal magnetogyric ratios, i.e., for the special case where y1 = yZ. By extending the existing formalism, we have obtained equations for the time derivatives of the magnetizations for v spin-l/2 nuclei [where each spin in a unit (e.g., molecule) is designated Ik: k = 1, 2, 3, . . . , V] exchanging between two configurations, where the Yk are not necessarily equal for all k = 1, . . . , V. By considering special cases of these equations, we have developed expressions describing a system containing I types of spins (denoted I,: q = 1, 2, 3, . . . , t), where there is a given number of each type of spin (denoted II,: q = 1, . . . , t) in a unit, exchanging between two configurations. The set of spins in a unit (e.g., molecule) is designated here by the symbol (I,),l(I,),,(I,),s . . . (I&; in this notation each factor (I&, represents np nuclei of the same type (e.g., two protons in a CH2 group). We have adopted this symbolism to eliminate the necessity of having a separate label (i.e., value of k) for each spin in a unit (e.g., molecule) in the case when some of the spins in the unit can be considered magnetically as well as chemically equivalent. We begin by presenting the equations we have obtained; the details of the development of our expressions can be found elsewhere (4). Systems to which these expressions might be applied are then described; particular emphasis is given to possible nuclear Overhauser effect (1,5) experiments. EQUATIONS

Consider a system of n units of the type (Il)n,(12)n,(13)n, * * * (It),,. In this system the units can exchange between configurations A and B as represented by l/r* (IlA)~,(I2A)n,(I3A)n,

’ * * (ItAhq

=

(kd&J3)&&,

’ ’ ’ (It&,

l/TB configuration

A

configuration

where rA and TV are the first-order lifetimes in configurations system, the total number of spins, N, is given by

B

A and B.’ For such a

iv = (*=1 c &J(n); and the time derivatives of the macroscopic 11, IZ, I,, . . . , It magnetizations configurations A and B have been found to be given by (4) dMzK( t) = -[R”K dt

+ (TK)-‘](Mp(t)

in

- Mgq

[I] - iv:,“)+ (TJ-‘(Mg(t) - A!fpL), - c (K7)(~qKPK) (; 1(M;K(t) P#P

whereK,L=A,Bandq,p=1,2,3 [I] are defined as

,...,

t.

The relaxation parameters in Eq.

RqK = 2 W$ + 2(n, - 1) WjK*K + C n,( Wfj’@K+ WJKPK) PZ9

NUCLEAR

OVERHAUSER

EFFECTS

IN

EXCHANGING

SYSTEMS

177

and where WgKpK= W$PK and WtKPK= WtKqK. The transition probabilities appearing in the expressions for the relaxation parameters are defined, for example, as follows: WjKPK is the transition probability per unit time via spin-lattice relaxation for a process with Ahm’JK+ AmPK = k2, while WgKpKis the transition probability per unit time via spin-lattice relaxation for a process with AmqK + AmPK = 0. Here fi(AmqK + AmpK) is the change in z component of spin angular momentum for a q&X spin system. We have the following relationships among the equilibrium magnetizations appearing in Eq. [l]: nq ri MQK 0 = --MpK 0 9 PI

np rg

and MqKrL = MqLr 0 o K*

131

In the development of Eq. [l] we have assumed that the isotropic portion of the Zeeman coupling determines the energies of the various states, while the other terms in the Hamiltonian (isotropic as well as anisotropic) are treated as perturbations, and are used in calculating transition probabilities or relaxation times. For the isotropic portion of the indirect spin-spin coupling, for example, this assumption requires that IJ,K,K 1 6 16 I, where JqKpKis the scalar spin-spin coupling constant, and 6 = (wqK - 0.&/2%- is the relative chemical shift (in hertz or cycles per second). Relaxation can be effected by the anisotropic portion of the Zeeman coupling (i.e., chemical shift anisotropy), dipolar coupling, indirect spin- spin coupling, and spin-rotational coupling; clearly, only spin- l/2 nuclei have been considered, and we have restricted ourselves to diamagnetic substances (6). If we let D(rot) stand for spin-lattice relaxation via rotationally modulated dipolar coupling (in the isotropic random motion approximation), and other stand for all other modes of relaxation, then for the transition probabilities in a configuration K we can write W$ = (WKkm,t,

+ ( WKK)other,

wgKqK= (wKqKhrot~+ (wKqK)ottler, WKPK= ( WKPKhmm+ ( w~KPKher, w%KpK= (w%KpKhxrot, + (w%KPKhther,

[44 [4bl 14cl

WI

where (WKhms

= (n, - 1X WKqKhrotj + C nP(WKfKhmtj

PaI

Pf9

and ( W?KqKhrot~

=

(3~2)~qKqKJqKqK(~qK),

[5bl

(W%?Khrot)

=

(3/2)21qKpKJqKpK(oqK),

15cl

[54

( W!KqKhrot, = %Kdqxqx@‘qd, ( W8KPKhmt)

=

6%KpKJqKpK@‘A,K

+

OpK),

WI

178

BORZO AND MACIEL

( wKPKhrot~= VQKPKJ,KPK(%K - OPK).

WI

The spectral densities in Eqs. [5] are given by

JOKhK(4 = 1+TZKY2 with g, h = 9, p; and the constants are defined as VQKhK

=

(1~10)ti3/2hfi2~;i%K*

Here ~QK~K is the distance between nucleus IO and nucleus Ih in a configuration K 7 and TBKhKis the correlation time of the radius vector rSKhK. We coifine our discussion of Eq. [ 11 to steady-state experiments performed at some time I = t,, when the time derivatives of all the magnetizations are equal to zero. In addition, the assumption is made throughout our discussion that rf fields are either sufficiently strong to cause complete saturation or so weak as to cause negligible saturation. The discussion will thus be facilitated by the definition of the fractional enhancement of a qK resonance as

Thus, for example, the fractional enhancement of a qK resonance on steady-state saturation of an SK resonance is given, using Eq. [l], by -

(~d(~QK")(~8~~~)

C

+ (TK)-t&L(SK)

(~P)(~*~~~)(~P~~QI~~K(SK)

P#QJ fQdsK)

=

RQK + (TK)-l

for all q # S. Clearly, f&SK) = (0 - M:K)/M:K = - 1. The fractional enhancement of a qL resonance on steady-state saturation of an SK resonance is given by f*&sK)

=

1 pzq

(nP)(uQ~pL)(~P~~Q)fP~(SK)

+

(‘d-lfqdSK)

RQL + (7J1

for all q = 1, 2, 3, . . . , f. In general then, equations describing fractional enhancements in a system of n (Il)n,(12)n,(13)n, * * * (I&,,, units exchanging between two configurations constitute sets of {(t - 1) + t} linear equations, in as many unknowns, for each of the 2t single irradiations possible. It may be remarked that for a system of n (I&&J1 units where -yl = -yZ,the equations we have presented here reduce to those previously discussed by Combrisson et al. (3). It may also be noted that by dropping the exchange terms from Eq. [I], and the subscripts K from Eq. [l] and all other subsidiary equations pertaining to Eq. [l], we are left with equations describing a system of n (Il)nl(12),&3),,3 . . . (It),, units in the absence of exchange. In the extreme-narrowing approximation (0~7-2, G l), and assuming also that uQp = (v*~&,~) = {( Wgp)D(mt,- ( Wijp)Dt,,,~j}, these equations describing a system of n (Il),,(12),,(I& . . * (It),, units in the absence of exchange are entirely consistent with, but more general than, equations previously presented to describe the nuclear Overhauser effect in rigid molecules (7); they are more general in the sense that the individual cases discussed previously are all special cases of the present, more general formalism.

NUCLEAR

OVERHAUSER

EFFECTS

IN EXCHANGING

SYSTEMS

179

DISCUSSION

We discuss four hypothetical examples of various systems to which Eq. [l] could be applied. The first involves a conformational exchange, in which configurations A and B are chemically nonequivalent. In the other three examples, the configurations are chemically equivalent; the first two examples in this set involve conformational exchange, the third chemical exchange. Additional examples are given elsewhere (4). Example 1: Exchange between Isomers This example of the use of Eq. [l] involves a conformational which configurations A and B are chemically nonequivalent:

oo- syn configuration

exchange in

00-anti A

configuration

B

Although this is a hypothetical example chosen to illustrate some of the features of Eq. [l], nuclear Overhauser effects have been observed in similar molecules by Combrisson et al. (3). They discussed their experimental results in terms of a formalism which was adequate for their purposes, but which is less general than the one presented herein. The basic features of Eq. [l] which are brought out in Example 1, however, are all present in the less general equations given by Combrisson et al. (3). If we designate the aldehydic hydrogen (a) as I, (n, = l), a fluorine (f) as IZ (n2 = 3), and a methyl hydrogen (m) as I3 (n3 = 3), then the conformational process depicted above can be formally represented as l/74\ (IlA)dhAh(I3Ah

configuration

= l/TB

A

UIBMI~BMI~B)~~

configuration

B

Equation [l] with q, p = 1, 2, 3 can be used to write a set of six equations for the time derivatives of the aldehydic proton, fluorine, and methyl proton magnetizations in configurations A and B; these six individual equations apply within the limit where r1\ and 78 are sufficiently long compared with the inverse of the separations ( OqA - OqBl -l that two sets of resonances, arising from molecules in configurations A and B, can be observed.2 Likewise, Eqs. [2] and [3] can be used to write three sets of three equations each, describing the relationships among the equilibrium aldehydic proton, fluorine, and methyl proton magnetiza* These six equations can be combined into three expressions for the time derivatives of the aldehydic proton, fluorine, and methyl proton magnetizations in the limit where coalescence of the I,, and be resonances occurs. See also footnote 3.

180

BORZO AND MACIEL

tions in configurations A and B. Assuming all observations are made at some time r = tSSwhen the system is in a steady state achieved on continuous saturation of one of the six resonances, the hfteen equations obtained in this case from Eqs. [ 11, [2], and [3] can be manipulated into a set of thirty expressions for fractional enhancements in terms of relaxation parameters (PA, IZ~~*P*, RqB, nquqBpB) and lifetimes (TA, 78). If, for example, one were to saturate the methyl proton resonance in configuration A at time f = 0, and observe the other resonances in the system at some later time t = t,, when a steady state had been reached, then the fifteen equations obtained from Eqs. [ 11, [2], and [3] would yield expressions forfaA(mA), in terms of relaxation parameters and f,(mA), fadmA), frB(mA), andf,,(mA), lifetimes. Similar sets of five equations each would be obtained for each of the other five single irradiations possible. Within the limit where ?A and 78 are SUffiCieUtly long compared with 1uqA - OqB1-I that coalescence of the IqA and IqB resonances does not occur, three cases of application of the thirty equations can be distinguished: (1) (T&l,

(TB)-l 4 RqA, nq(rqAPA,RqB, nqUqBPB,

(2) (7*)-l,

(T~)-~ - RqA, nq~qAPA,RqB, nquqBpB,

(3) (7*)--l, (T&l

s RqA, nqUqAPA,RqB,

nqUqBpB.

In the first of these cases, the exchange has no direct effect on any of the magnetizations, as the exchange terms can be neglected in the thirty equations. In case (3), the thirty equations reduce to six, as saturation of a resonance in configuration A will have the same effect as saturation of the corresponding resonance in configuration B, and as the fractional enhancements of the remaining unsaturated resonances will be the same in configuration A as in configuration B. The fractional enhancements in this exchange-dominated case are thus weighted average properties, depending on relative populations or lifetimes, as well as on relaxation parameterS.3 For steady-state experiments in the intermediate case when (TA)-1, (78)-l - RqA IZ uqApA, RqB, nqoqBpB the thirty equations apply in full. Thus, in contrast to cask (“1) in which exchange terms drop out, and case (3) in which weighted average properties are observed, information on the exchange time constants, TA and TB, could possibly be obtained from steady-state experiments in case (2). Information on the exchange rates could also be obtained if scalar relaxation of the first kind were the dominant relaxation mechanism, even for cases (1) and (3) above (6). Example 2: Exchange between Chemically Dimethylformamide

Equivalent

Forms of

In this second example, we apply our equations to the conformational exchange in dimethylformamide (DMF), which is due to rotation about the C-N bond: 3 Expressions obtained in the rapid exchange case in the absence of coalescence also apply in the case of coalescence. For example, the expression for&&A) given in the rapid exchange case in the absence of coalescence also describes the fractional enhancement of the methyl proton resonance on steady-state saturation of the aldehydic proton resonance in the case of coalescence.

NUCLEAR

(a)

OVERHAUSER

;Ql3 lb)

“\ NC 0

EFFECTS IN EXCHANGING

-

N\

configuration

CH3 A

(c)

"'A \

\

(a) H\ 0

SYSTEMS ,cH3

181

tb'

J-"\

configuration

*CH3 Cc) B

A limited study of nuclear Overhauser effects in a 2.5% solution of DMF in dimethyl sulfoxide-d, (DMSO-&) was reported by Saunders and Bell in 1971 (8), and will be commented on later. Their experimental results were subsequently discussed by Noggle and Schirmer (9); and relationships between expressions appearing in that discussion and those obtained here have been pointed out by us elsewhere (20). The formalism presented here could be used to describe Example 2 by making the identifications listed below: 11 12 13

aldehydic proton methyl proton in starred methyl group methyl proton in unstarred methyl group

n, = 1 n, = 3 n3 = 3

In this example, configurations A and B are chemically equivalent, so that l/7, = UT, = UT,,, where 7,, is the first-order lifetime of an (Il)n,(12)n2(13)n, unit in configuration A or B: l/TCS (11‘4)1(12.4)3&*)3 = (11&(12&(13&. l/Tax configuration A configuration B Equation [l] with 4, p = 1, 2, 3 can be used to write a set of six equations for the time derivatives of the 11, 12, and I, magnetizations in configurations A and B. In view of the chemical equivalence of configurations A and B, however, this set of six equations reduces to a set of three expressions for the time derivatives of the aldehydic proton magnetization (designated a), and of the magnetizations of protons in methyl groups which are syn or anti to the aldehydic proton (denoted b and c, respectively). These three individual equations apply within the limit where 7,, is sufficiently long compared with the inverse of the separation IO,, - w, 1-l that two separate methyl proton resonances, arising from protons in methyl groups which are syn or anti to the aldehydic proton, can be observed.4 Assuming all observations are made at some time t = t,, when the system is in a steady state achieved on continuous saturation of one of the three proton resonances, and making use of the relationship@ = i@/3 = M:/3, the three’equations obtained in this case from Eqs. [l] can be manipulated into a set of expressions for the following fractional enhancements: fb(a), fC(a), fa(b), f,(b), fa(c), fb(c). These expressions for fractional enhancements will be in terms of the lifetime, T,,, and of the relaxation parameters RQ, crQp,where 4 In the limit where coalescence of the syn and anti methyl proton resonances occurs, these three equations reduce to two expressions for the time derivatives of the aldehydic and methyl proton magnetizations.

182

BORZO AND MACIEL

R* = 2 Wf + 2[n, - l] We” + c n,[ W= + W#=]

[W

P#9

and CT*== [ wp - wgq,

[6bl

with wp = wp

and

wg= = wg*.

[6cl

Here, q, p = a, b, c, and n, = 1, nb = 3, n, = 3. It may be noted that here the indices denote structural locations (e.g., b designates the syn methyl proton), whereas above they identified specific nuclei in whatever structural positions they happened to occupy in a particular configuration (e.g., starred and unstarred methyl protons). It may also be remarked that these relaxation parameters are like those we defined earlier, except that the subscripts A and B on the relaxation parameters and on the transition probabilities are no longer needed. Within the limit where 7eXis sufficiently long compared with the inverse of the separation 1wb - WC1-l that two separate methyl proton resonances can be observed, three cases of the six expressions for fractional enhancements can be distinguished: (1) (r&-l 4 R*, nq(+qp, (2)

(7ex)-’

- R*, nquqp,

(3) (-r,,)-’ % R*, nqwqp. In the first of these cases, the exchange has no direct effect on any of the magnetizations as the exchange terms simply drop out of the six equations. In this case, the fractional enhancement of a q resonance on steady-state saturation of an s resonance will be given by fq(s) = n,crqsRp - npuqpnsups , RqRP - np~qpnqupq with q, s, p = a, b, c, and n, = 1, nb = 3, n, = 3. For DMF in dilute solution, the two-spin interactions represented in this expression will be intramolecular in nature, and can be evaluated using Eqs. [5]. Assuming that intramolecular amb interactions dominate, then in the extreme-narrowing limit the above equation gives fa(b) = 3aablRa = 0.50 since 3uab = 3( 1/2)y&h ‘
NUCLEAR

OVERHAUSER

EFFECTS IN EXCHANGING

SYSTEMS

183

and Saunders (II) to describe the NOE observed on a single proton magnetization in a set of compounds (including DMF) in CDCl, solution on steady-state saturation of methyl protons (the syn methyl protons for DMF). Reasonable agreement between theory and experiment was demonstrated in that study. The quantity r was interpreted by Bell and Saunders as the distance between the single proton and the point defined by the intersection of the threefold rotation axis of the CHB group with the plane through the three protons of the CHB group. It was thus assumed that the rotational motion of the methyl group will be very fast compared to the relaxation rate of the single proton (Ra for DMF). We return to this point in Example 3, where we show the conditions (within the present formalism) under which such a treatment holds. Cross-correlations, which are not usually found to be of importance in liquids (12), have clearly been neglected in this treatment of the methyl group. [For evidence of the existence in liquids of the effects of cross-correlations due to intramolecular dipole-dipole interactions in a methyl group, see, e.g., Ref. (I2).] In case (3), where exchange dominates, the six equations describing fractional enhancements in this DMF example reduce to uba + uca fb(a)

= f&d

=

fb(c)

= f,(b)

=

Rb + R” + 6ubc ’

[W b’bl

- 1,

and fa(b) = fa(c) = 3(UabR; +“)

.

[7cl

Equation [7a] indicates that irradiation of the aldehydic proton resonance will result in nonselective fractional enhancements of the syn and anti methyl proton resonances. From Eq. [7b] we see that in steady-state experiments in this exchange-dominated case, saturation of the anti methyl resonance will result in saturation of the syn methyl resonance, and vice versa. Finally, from Eq. [7c] it is clear that the fractional enhancement of the aldehydic proton resonance will be the same whether the syn or the anti methyl proton resonance is saturated.5 Saunders and Bell (8) also studied a 2.5% solution of DMF in DMSO-d6 at 100 MHz in the temperature range 31 to 90°C. They reported that the fractional enhancement of the aldehydic proton resonance on irradiation of the syn methyl resonance was constant in this temperature range; fa(b) = 0.28 (31-9O”C), while the fractional enhancement of the aldehydic proton resonance on irradiation of the anti methyl protons varied with temperature; fa(c) = 0.03 to 0.28 (31-90°C). 5 It may be noted that in the case of coalescence of the syn and anti methyl proton resonances, Eq. [7a] is the expression obtained for the fractional enhancement of the methyl proton resonance on steady-state saturation of the aldehydic proton resonance, while Eq. [7c] is the expression obtained for the fractional enhancement of the aldehydic proton resonance on steady-state saturation of the methyl proton resonance.

184

BORZO AND MACIEL

They also stated that the methyl proton resonances were still 15.5 Hz apart at 100 MHz and 90°C. They presented a qualitative interpretation of their results. In more quantitative terms, then, we can say that at 90°C and 100 MHz, T,, is sufficiently long compared with 1wb - o,I -’ that two separate methyl proton resonances can be observed, but sufficiently short compared with R*, npqP that the case (3) equations, i.e., Eqs. [7], apply. Indeed, the experimental observation that fa(b) = JQC) (at 90°C and 100 MHz) is the behavior predicted by Eq. [7c] for the exchange-dominated case. If the only relaxation mechanism operable in this DMF system were intramolecular dipolar coupling in which the time dependence was due to isotropic rotational tumbling of molecules, then we could use Eqs. [5] without the’ subscripts K to find the magnitudes of the fractional enhancements which would be expected in this exchange-dominated case. Using Eqs. [5] (without the subscripts K) in conjunction with Eqs. [6] in Eqs. [7a] and [7c], we obtain the results that

Pa’1 and

f,(b) =fa(d =;;zt=;f;,

= 0.50.

ac

[7c'l

We have assumed the extreme-narrowing limit (&z 4 1) in writing these expressions, and have also assumed that the system is characterized by a single rotational correlation time 7,. From Eq. [7c’] it is clear that the maximum positive enhancement possible would be expected for the aldehydic proton in this particular situation. It may be noted that this result is independent of the relative magnitudes of Tab and r acr* from the point of view of the aldehydic proton the molecule thus behaves like a simple two-spin-l/2 system, composed of aldehydic proton spins and of methyl proton spins. However, processes other than intramolecular dipolar coupling could contribute to aldehydic proton relaxation. If this were the case (i.e., if W? = ( W)DN + ( Wotber, etc.), then it is clear from Eqs. [6], [5], and [7c] thatf,(b) andf,(c) would be less than 0.50. The experimental observations mentioned above, thatf,(b) = fa(c) = 0.28 (at 90°C and 100 MHz), thus suggest that other processes do contribute to aldehydic proton relaxation. In order to continue our discussion of the DMF system, expressions which apply in case (2) are required. For steady-state experiments in the intermediate case when (r&l - R*, ~t,c+*pthe six expressions describing fractional enhancements in terms of the lifetime and of the relaxation parameters apply in full. For example, the fractional enhancement of the aldehydic proton on continuous saturation of the syn methyl protons is given in this case by f,(b) =

3uab[RC + (7J1] - 3uac[3vcb - T,, ' Ra[Re + (7e,)-l] - 3(u.E)e( )- ' *

[8a1

The corresponding fractional enhancement of the aldehydic proton on continuous saturation of the anti methyl protons is given by

NUCLEAR

OVERHAUSER

faW =

3@[Rb

EFFECTS

+ (T,J']

IN

EXCHANGING

- 3uab[3ubc -

SYSTEMS

T,,

RYRb + (7&.,)-l] - 3(~.')z(

l )- I *

We consider a case (2) situation in which there are no interactions aldehydic protons and anti methyl protons. In these circumstances, reduces to fa(b) = 3aab/Ra.

185

Pbl between Eq. [8a] [91

This fractional enhancement (with no direct dependence on r,,) would not vary with temperature in the extreme-narrowing limit, provided that the correlation times associated with the various relaxation mechanisms contributing to the relaxation parameters in Eq. [9] were all the same. This prediction is consistent with the reported measurement (8) of a constant value of fa(b) over the temperature range 31 to 90°C. It would also lead one to expect the constancy in fa(b) to be maintained in other temperature ranges in which the extreme-narrowing approximation is valid. In the same case (2) approximation Eq. [8b] reduces to -3uab[3ubc .fa(C)

=

- (T,.p]

Ra[Rb + (T,~)-~] - ~(cP~)~ *

UW

The neglect of a-c interactions thus results in a fairly complex expression (depending on r,,) for the fractional enhancement of the aldehydic proton on irradiating the anti methyl protons, an expression which might lead one to expect a variation in this fractional enhancement to accompany a change in temperature. For this special case (2) situation, we can also write an alternate expression forf,(c) (entirely equivalent to Eq. [lOa]), this one in terms of fractional enhancements (13):

In view of the measurements of fa(c) and fa(b) which have been reported in the literature (8), Eq. [lob] would lead one to predict that fb(c) would be negative, and directly proportional tofa(c) in the temperature range 31 to 90°C. Of course, this model assumes there are no interactions between aldehydic protons and anti methyl protons, which cannot be strictly correct. However, in view of the P dependence of W2 and WO terms in a dipole-dipole mechanism, we estimate Wzc, WV to be approximately five times smaller than Wgb, Wtb. Example 3: Methyl Rotation Considered as Exchange In Examples 1 and 2, we have considered the protons of a methyl group as members of an (I& unit where nq = 3. Acetaldehyde, 7

\ ,C-c

qq

IP ‘h

h3

for example, would have been considered as a system of n (I,),,(12)n2 units (in the absence of exchange) with I, = the aldehydic proton, n, = 1, and I2 = the

186

BORZO AND MACIEL

methyl proton, nz = 3. In this third example, on the other hand, we view a methyl group as a system of three proton spins undergoing a threefold rotational exchange; and we treat this situation using the extension of the present formalism to three configurations. We then describe the exchange conditions under which the treatment of methyl groups used in Examples 1 and 2 is valid. The equation needed in a three-configuration case is identical to that already presented for the two-configuration case (Eq. [ 11) with the exception that in a threeconfiguration case the inverse lifetimes (r&l become sums of individual configurational rates, kKL (2, 24); i.e.,

[I14 L#K

where K, L = A, B, C, the three chemical configurations. The time derivatives of the macroscopic magnetizations in a configuration K for a three-configuration system will thus be given by dMzK( t) = -[RqK dt

+ 1 kKL](MzK(t) - i@‘) L#K

- c (CJbqKpK) ; i

PfP

(iktgK(t) - ikfgK) + 1 k&ibfzL(t) 1

- Mf),

[Ilb]

L#K

t.

where K, L = A, B, C and q, p = 1, 2, 3, . . . , For acetaldehyde, we thus begin by assuming that the methyl group rotates slowly enough that its rotational rate is of the order of the aldehydic proton relaxation rate, Ra. The three-configuration formalism presented above could then be used to describe the acetaldehyde molecule as a four-spin system undergoing a threefold rotational motion by designating 1, 12 I3 I4 The rotational

methyl proton 1 methyl proton 2 methyl proton 3 aldehydic proton

n, = 1 122= 1

rz3 = 1 n4

exchange process can then be formally

configuration

C

=

1

represented

configuration

as

(1 lB'1 B

In this notation the subscripts 1, 2, 3, 4 represent a particular proton (whatever its position in the molecule) and the subscripts A, B, C represent the three different arrangements of the methyl protons. Here, all three configurations are chemically equivalent, so that k,, = kg,., = kBC = kca = kCA = kAC = krotr where k,t is the

NUCLEAR

OVERHAUSER

EFFECTS

IN

EXCHANGING

187

SYSTEMS

methyl group rotational rate. The constraint on methyl group rotation imposed at the outset is thus that kmt = Ra. Using Eq. [l lb] with 9, p = 1, 2, 3, 4 and K, L = A, B, C and y1 = yZ = y3 = y4, we obtain three sets of four equations each, describing the time derivatives of the 11, Ie, IS, and I4 magnetizations in configurations A, B, and C. In view of the chemical equivalence of the three configurations, however, this set of twelve equations reduces to a set of four equations for the time derivatives of the aldehydic proton magnetization (designated a), and of the magnetizations of the three methyl protons (denoted b, c, and d) as pictured below: methyl

proton

aldehydic proton a L24 _methyl

proton

b

I

--s

d'-

-methyl

proton

Thus, for example, the time derivative of the macroscopic netization would be given by dW(t) dt

c

aldehydic proton mag-

= - (2 w; + W;b + wp + wp + WV + Wgd + Wp}(M;( - { wp

- Wp}(M~(t)

t) - iv;)

- M,b) - { wp - W$c}(MC,(t) - iv:) - {wp

- W$d}(M$(f)

- M,d).

The four individual equations would apply within the limit where (k&-l is sufficiently long compared with 1wb - w, 1-l, (0, - md 1-l, and (md - wb 1-l that three separate methyl proton resonances can be observed. These four expressions can be combined into two expressions for the time derivatives of the aldehydic (a) and methyl (m) proton magnetizations in the limit where coalescence of the three methyl proton resonances occurs. Within the limit where the rotational exchange of the methyl group is very fast compared to the rates of proton relaxation (i.e., where kmt %=Ra, Rm), two hypothetical cases can thus be distinguished: (1)

hot

e

lob

-

WC (7

1%

-

Od\,

Iwd

-

obl,

(2)

hot

+

1wb

-

%I,

1%

-

odlv

Iad

-

wbl*

In the first of these cases, three methyl proton resonances would be observed, but steady-state saturation of any one of the three methyl resonances would result in saturation of the other two methyl resonances as well, f,(b)

=fdJ)

=.fbk)

=.fd(c)

=.fb(d)

=f,(d)

=

-1;

while the steady-state fractional enhancement of the aldehydic proton would be the same whether the b, c, or d methyl proton resonance were saturated,

f,(b) = fdc) = fat4 =

WZb- Wfjb+ wp - wy + Wry- wp 2W~+W~b+W~b+Wy+W~+W~d+W~d’

Wal

In case (2), where coalescence of the three methyl proton resonances occurs, the fractional enhancement of the aldehydic proton on steady-state,saturation of

188

BORZO AND MACIEL

the single methyl proton resonance would also be described by Eq. [12a]. If we assume that w;b = wt” = w,ad s wy’ I al and wbb

=

WF

=

w,ad

E

warn 0

PI

9

where m refers to some “average” methyl proton, whose concentration times that of the aldehydic proton, then Eq. [12a] reduces to

f,(b) = A(c)= .A@)=f,(m) = 2w~~~~w~,,,w~&,,,, . 1

2

is three

[12bl

0

Thus, the criterion k,, % Ra, Rm, which also makes assumptions [a] and [b] reasonable, emerges as the necessary condition for treating the protons of a rapidly rotating methyl group as members of an (I& unit without explicit consideration of exchange. Example 4: Exchange of Solvent Molecules

Surrounding

a Metal Cation

A more complex phenomenon which could be investigated using the two-configuration formalism presented herein is the NOE observed on the resonance of a solvated spin-l/2 metal nuclide upon irradiation of the solvent resonance(s).6 A simple model for such a system which will be discussed is one in which a diamagnetic metal cation is surrounded by x bound solvent molecules and y unbound solvent molecules, where there is a possibility of exchange between the two types of solvent molecules. We consider a solvent molecule containing z chemically as well as magnetically equivalent nuclear spins (e.g., H,O with z = two proton spins, or NH, with z = three proton spins). Other possibilities, such as proton exchange (for water, for example), or more elaborate models for the exchange of solvent molecules in which there are several solvation shells, could also be treated within the framework of this formalism. The extension of the present formalism to more than two configurations (see, e.g., Eqs. [ll]) allows even more complex models to be constructed. The simple model of metal ion solvation is represented in Fig. la. To include the possibility of chemical exchange between bound and unbound solvent molecules, we arbitrarily select one solvent molecule out of each group for exchange. This distinction is shown in Fig. lb. The two-configuration formalism that has been presented above can be used to write an equation for the time derivative of the macroscopic metal nuclide magnetization in such a system by making the identifications listed below. q = 1,p = 2, 3, 4, 5

11 12 IS

14 16

metal nuclei nuclei nuclei nuclei

ion nucleus in starred solvent molecule in doubly starred solvent molecule in bound unstarred solvent molecules in unbound unstarred solvent molecules

n, = 1 n 2=z

n3 = z n4 = z(x - 1) n5 = z(y - 1)

B Changes in the integrated intensities of the *Wd signal in aqueous solutions of Cd(NO& and of the lmAg signal in aqueous AgNO, upon irradiation at proton frequencies have been observed in this laboratory.

NUCLEAR

OVERHAUSER

EFFECTS IN EXCHANGING

CD Is**

IS'

0

l/T,

0 m

m

(x-115

(x-1)5

(y-l)5

(Y-1 1s

configuration

189

IS*

Is**

(FJl

SYSTEMS

A

(b)

configuration

6

1. A simple model of solvent exchange in a system containing a solvated metal cation (m). (a) The distinction between x bound solvent molecules (s) and y unbound solvent molecules. (b) Focusing on one solvent molecule within each of the categories shown in part (a), and indicating exchange between two chemically equivalent configurations with lifetimes rA = ra, differing only in the labeling of individual solvent molecules (indicated by one or two asterisks). FIG.

In this example, configurations A and B are obviously chemically equivalent, so that VT, = UT, = l/7,,, where 7,, is the first-order lifetime of a formal structural (11),,(Iz),,(13),,(IJ~~(15)n, unit in configuration A or B. In this model, for each metal nucleus there are x bound and y unbound solvent molecules. The exchange process is then (11A 11 (1ZA 1z (13A 1z (14A ).z(x - 1) (15A 1.?(I/ - 1)

l/T,, =

(1 1B > 1(1 28 1 z (1 38 1P (1 4B 1 .z(.r - 1) (1 5B 1Z(Y - 1).

l/r,,

configuration

A

configuration

B

We also have that lkreX = X/T,, = Y/T”, where 713and T, are the first-order lifetimes of a single solvent molecule in a bound or unbound state, respectively. Equations [l] with q = 1, p = 2, 3, 4, 5 can be used to write a set of two equations for the time derivatives of the I, magnetizations in configurations A and B. We recognize the following chemical equivalences: (a) the (x - 1) bound unstarred solvent molecules and the one bound (singly or doubly) starred solvent molecule, (b) the (y - 1) unbound unstarred solvent molecules and the one un-

190

BORZO

AND

MACIEL

bound (singly or doubly) starred solvent molecule, and (c) configurations A and B. Because of these equivalences, this set of two equations reduces to a single equation for the time derivative of the metal nucleus magnetization, dW(t) T dt

= -R”(Mfy

t) - M,m) -

@b

$

(M,b(t)- M,b) - (TmuYm (M;(t) - M,U), [13] Ys

where b and u refer to nuclei in bound and unbound solvent molecules, respectively. The relaxation parameters in Eq. [13] are given by Eqs. [6], with q = m, p = b, u and with n, = 1, nb = (z)(x), n,, = (z)(y). It may be pointed out that of all the relaxation mechanisms which could contribute to the W values in these relaxation parameters, dipolar coupling and scalar coupling are the only ones which can contribute to W, and W. values; one or both of these relaxation mechanisms must thus be operative if a NOE is to be observed. Referring to Eqs. [6], we can see that the following types of pairwise interactions, which could give rise to a NOE are included in this model: metal ion nucleus-nuclei

in bound solvent molecules

(mmb),

metal ion nucleus*nuclei

in unbound solvent molecules

(m-u).

Metal ion-metal ion interactions, which could in principle give rise to nuclear Overhauser effects in very concentrated solutions, have been neglected in this treatment. For the case where metal nuclide relaxation occurs solely via dipolar coupling between metal ions and bound solvent molecules in which the time dependence of dipolar coupling is due to rotational tumbling, the relaxation parameters in Eq. [13] would be given by Eqs. [6] as R” = 2(W’h,,,

+ (~)(~){(W~~)mrot) + (W’bhrot)l,

urnb= (wi?bhrcd,- (wf;bhrot)9

PM

[14bl

emu= 09 where from Eqs. [5] (without the subscripts K) we have ( K%oot)

= (x)(z)( mbb)Dwot) =

( Wbhm)

=

6Gdmd%

( wtb) Dtrot)

=

&dmb(%

+ -

wbh obh

(x)(z)(3/2)v,bJ,b(o,),

[14cl

[14dl [14el

Here rmb is the distance between a metal nucleus and a bound solvent nucleus, and eb is the correlation time of the radius vector r (from an m nucleus to a b nucleus). Equation [13] applies within the limit where rb and T” are sufficiently long compared with the inverse of the separation 1wb - wU 1-l that two separate solvent resonances, arising from nuclei in bound and unbound solvent molecules, can be observed. It may be noted, however, that there are no exchange terms in Eq. [ 131. This absence of exchange terms is a result of (1) the fact that configura-

NUCLEAR

OVERHAUSER

EFFECTS

IN

EXCHANGING

SYSTEMS

191

tions A and B are chemically equivalent in this simple model of chemical exchange between bound and unbound solvent molecules, and (2) the fact that the metal ion does not itself undergo chemical transformation in this model of solvent exchange. Assuming all observation are made at some time t = t,, when the system is in a steady state achieved on continuous saturation of one of the two solvent resonances, and making use of the relationship 1 r& 1 ril = My=-PM; --MM,“, (z)(x) r: (Z)(Y) rs”

Eq. [13] yields

0 = -Rmf,(b)

+ (z)(x)umb 2

- (z)(y)u””

-Y’ ymfu(b)

[W

Ym

for saturation

of the bound solvent resonance, or

0= -Pf,(u)- (z)(x)cF ys ;fb@)

+

(z)(Y)~“”

ys

for saturation of the unbound solvent resonance. For simultaneous the bound and unbound solvent resonances, Eq. 1131 yields fm(bv

U)

=

[(z)(xbmb

+

[15bl

Ym

(z)(Y)~m”I(Ys~Ym)~~m.

saturation of [15cl

As an example of the use of Eqs. [ 151, we consider a situation in which there are no interactions between metal ions and unbound solvent molecules. In such a case, Eqs. [15a] and [15c] would reduce to fm(b) = fm(b,

U) =

{(Z)(X)Um”/Rm}

2

Wal

and Eq. [15b] would reduce to7

If we assume further that metal nuclide relaxation occurs solely via dipolar coupling between metal ions and bound solvent molecules in which the time dependence of dipolar coupling is due to rotational tumbling of m(b), units, then we can use Eqs. [14] in conjunction with Eq. [16a] to write that in the extremenarrowing limit (w%,2 4 l), fm(b)

=

Y&Ym*

The maximum negative enhancement would thus be expected in this particular case of Example 4. In the limit of rapid exchange, where Tb, 7, G 1wb - w, 1-l, Coalescence of the bound and unbound solvent resonances occurs, and a single solvent resonance line centered on a mean frequency w, = P,,ob + P,ou, 7 Note

the similarity

in form

between

Eqs.

[9],

[lob]

and [16a],

[ 16b].

192

BORZO AND MACIEL

where X

pb=2L=*b

+

7,

p,

and

=

XfY

7u

=-7

Tb f

7,

Y X+-Y

will be observed. In this case, Eq. [13] becomes 1

dWY( t> = -z?“(Mgy t) - MF) - dt

(xeb

c

+ ye”)

(M;(t)

- Mg).

[17]

(x + Y)

Equation [17] can be manipulated into an expression for the fractional enhancement of the metal resonance on steady-state saturation of the solvent resonance. It may be noted that this expression for&(s) is identical to the equation for f,(b, u) (Eq. [lk]), obtained in the absence of coalescence, for simultaneous steady-state saturation of the bound and unbound solvent resonances. CONCLUSION

We have considered particular applications of our equations to systems involving conformational exchange or chemical exchange, in which configurations may or may not be equivalent. Information on configurational lifetimes and relaxation rates, which reflect dynamics of molecules or ions in solution, can be obtained in each of the above cases. The expressions should expand the utility of the NOE method in studies of exchanging systems, particularly in the relatively new area of high-resolution NMR of diamagnetic spin-l/2 metals. ACKNOWLEDGMENT This research was supported by the National Science Foundation

under Grant CHE 74-23980.

REFERENCES 1. I. SOLOMON, Phys. Rev. 99, 559 (1955). 2. H. M. MCCONNELL, J. Chem. Phys. 28, 430 (1958). 3. S. COMBRISSON, B. ROQUES, P. RIGNY, AND J. J. BASSELIER, Can. J. Chem. 49, 904 (1971). 4. M. BORZO, Ph.D. thesis, Colorado State University, 1977. 5. J. H. NOGGLE AND R. E. SCHIRMER, “The Nuclear Overhauser Effect,” Academic Press, 1971. 6. A. ABRAGAM, “The Principles London, 1961. 7. Ref. (5), Chap. 3.

New

York/London,

8. J. K. SAUNDERS

AND

R. A. BELL,

of Nuclear

Can.

Magnetism,”

J. Chem.

48,512

J. Gem.

48,

Chap. VIII,

(1971).

9. Ref. (5), Chap. 7, Sect. D. IO. Ref. (4), footnote’(4-7). 11. R. A. BELL AND J. K. SAUNDERS, Can. 12. J. W. HARRELL, JR., J. Mugn. Reson.

1114 (1970).

15, 157 (1974).

13. Ref. (4), Chap. 4, Sect. II. 14. S. FORSBN

AND R. A. HOFFMAN,

.I. Chem.

Phys.

40,

1189 (1964).

Oxford

Univ. Press,