Encyclopedia of Spectroscopy and Spectrometry || Chemical Exchange Effects in NMR*

Chemical Exchange Effects in NMR

Alex D Bain, McMaster University, Hamilton, ON, Canada

& 1999 Elsevier Ltd. All rights reserved.

This article is reproduced from the previous edition, volume 1,

pp 198–207, & 1999, Elsevier Ltd.

Symbols
B1 radiofrequency field
Fz
total z magnetization
h
Planck’s constant
Ix
perturbation RF field along xffiffiffiffiffiffip i �1
k
rate of exchange
K so
kinetic superoperator matrix
L
Liouville superoperator
Lso
superoperator matrix
Mz
z magnetization of the spin system
R so
Redfield superoperator matrix
T2
spin–spin relaxation time
T1q
T1 in rotating frame
u
x magnetization of spin system
m
y magnetization of spin system
c
magnetogyric ratio
d
chemical shift difference
k
eigenvalue
K
diagonal matrix
/
final state
w
initial state
x1
frequency of RF irradiation
x0
Larmor frequency of the spin.
Figure 1 1H NMR spectra of the two N-methyl groups in

3-dimethylamino-7-methyl-1,2,4-benzotriazine, as a function of

temperature.

Introduction

Chemical exchange, in NMR terms, means that a nucleusmoves among a set of magnetic environments. Thesample is macroscopically at equilibrium, but an indivi-dual nucleus exchanges among a number of sites, so themagnetic properties of the nucleus are modulated by theexchange. Chemical exchange effects were recognizedearly in the development of NMR: at about the sametime (and in the same laboratory) as the scalar coupling.In N,N 0-dimethyl formamide and many other moleculeswith N,N 0-dimethyl groups (Figure 1), the two methylgroups have different chemical shifts if the molecule isrigid. At low temperature this is effectively the case,because of restricted rotation about the C–N bond.However, as the sample is warmed, the two signalsbroaden, coalesce, and eventually start to sharpen up toa single line at the average chemical shift, as shown inFigure 1.

This behaviour is usually interpreted in terms of an‘NMR timescale’. For rotation about the C–N bond,which is slow on this timescale at low temperature, thetwo signals are distinct and relatively sharp. At highertemperature, the rotation is faster and all that one candetect is the average resonance frequency. The broadcoalescence line shape is characteristic of the inter-mediate timescale, and is perhaps the most familiar andobvious manifestation of chemical exchange. This articleconcentrates on intermediate exchange, because a thor-ough understanding of this case clarifies most otheraspects of chemical exchange.

Chemical exchange in all three regimes – slow, inter-mediate and fast – has an effect on the NMR spectrum.

229

230 Chemical Exchange Effects in NMR

Typical values of the parameters mean that processeswith rates in the range of 1–104 s�1 can be studied mosteasily. In other words, this means activation energies forthe reaction are in the range of approximately40–80 kJ mol�1 (10–20 kcal mol�1). This includes con-formational changes in ring structures, restrictedrotations about chemical bonds, ligand rearrangements incoordination complexes, and some intermolecular trans-fer processes. Molecules are dynamic entities, so theeffects of chemical exchange are apparent in many NMRspectra.

The theory of chemical exchange in simple; uncou-pled spin systems have often been couched in terms ofthe Bloch equations, and this is the approach used inmost of the literature. However, it is simpler and easier toconsider the time domain. This time-domain methodallows us to treat all exchanging systems – slow, inter-mediate or fast, coupled or uncoupled – in a consistentand simple way. There is also a simple physical picture ofthe spectroscopic transition probability that helps in thisinterpretation of the theory.

The Bloch Equations Approach

The Bloch equations for the motion of the x and y

magnetizations (usually called the u- and v-mode sig-nals), in the presence of a weak radiofrequency field, B1,are given in the following equation.

du

dtþ u

T2� ðo0 � o1Þu ¼ 0

dudtþ u

T2� ðo0 � o1Þu ¼ gB1Mz

½1�

In this equation, o1 is the frequency of the RF irradia-tion, o0 is the Larmor frequency of the spin, T2 is thespin–spin relaxation time and Mz is the z magnetizationof the spin system. The notation can be simplified bydefining a complex magnetization, M

M ¼ u þ iu ½2�

With this definition, the Bloch equations can bewritten as

dM

dtþ iðo0 � o1ÞM þ

1

T2M ¼ igB1Mz ½3�

In chemical exchange, the two exchanging sites, A andB will have different Larmor frequencies, oA and oB.The exchange will carry magnetization from A to B andvice versa. If we assume equal populations in the twosites, and the rate of exchange to be k, then we can set up

two coupled Bloch equations for the two sites, as

dMA

dtþ iðoA � o1ÞMA þ

1

T2MA � kMB þ kMA

¼ igB1MzAdMB

dtþ iðoB � o1ÞMB þ

1

T2MB � kMA þ kMB

¼ igB1MzB ½4�

The observable NMR signal is the imaginary part ofthe sum of the two steady-state magnetizations, MA andMB. The steady state implies that the time derivativesare zero, and a little further calculation (and neglect ofT2 terms) gives the NMR spectrum of an exchangingsystem as

n ¼ 1

2gB1Mz

� kðoA � oBÞ2

ðoA � o1Þ2ðoB � o1Þ2 þ 4k2½o1 � ðoA þ oBÞ=2�2½5�

This equation can be further extended to systems withunequal populations and more sites, using the sametechniques.

Chemical Exchange in the Time Domain

If we create (by a pulse) the magnetization, MA and MB, attime zero, and then turn off the B1 magnetic field, eqn [4]can be simplified as

d

dt

MA

MB

!¼ �L

MA

MB

!½6�

L ¼idþ 1

T2þ k �k

�k �idþ 1

T2þ k

0BB@

1CCA ½7�

In this equation, the matrix L is given by eqn [7], where wehave made o1¼ (oAþoB)/2 and d¼ (oA�oB)/2:

MAðtÞ

MBðtÞ

!¼ expð�LtÞ

MAð0Þ

MBð0Þ

!½8�

Equation [6] is a set of first-order differential equations,so its formal solution is given by eqn [8], in which exp()means the exponential of the matrix. In practice, wediagonalize the matrix L with a matrix of eigenvectors,U, as in eqn [9] to give a diagonal matrix, L, with theeigenvalues of L down the diagonal:

L ¼ U �1LU ½9�

Figure 2 Decomposition of the coalescence line shape into

individual lines. Top: experimental spectrum from Figure 1;

middle: calculated line shape to match experimental spectrum;

bottom: individual lines calculated as in the Appendix.

Chemical Exchange Effects in NMR 231

Equation [8] becomes eqn [10]. The exponential of adiagonal matrix is again a diagonal matrix with exponen-tials of the diagonal elements, as in eqn [11].

MAðtÞ

MBðtÞ

!¼ Uexpð�LtÞU�1

MAð0Þ

MBð0Þ

!½10�

MAðtÞ

MBðtÞ

!¼ U

e�l1 t 0

0 e�l2 t

!U�1

MAð0Þ

MBð0Þ

!½11�

As was mentioned earlier, the observed signal is theimaginary part of the sum of MA and MB, so eqn [11]predicts that the observed signal will be the sum of twoexponentials, evolving at the frequencies l1 and l2. Thisis the free induction decay (FID). In the limit of noexchange, the two frequencies are simply oA and oB, aswe would expect. When k is non-zero, the mathematicsbecomes slightly more complicated.

If we ignore relaxation and exchange, then L is aHermitian matrix with real eigenvalues and eigenvectors.However, when the exchange is important, the Hermitiancharacter is lost and the eigenvalues and eigenvectorshave both real and imaginary parts. The eigenvalues aregiven by the roots of the characteristic equation:

idþ 1

T2þ k � l �k

�k �idþ 1

T2þ k � l

��

��¼ 0 ½12�

The eigenvalues of eqn [7] are given by the followingequation:

l ¼ 1

T2þ k

� �7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 � d 2p

½13�

These eigenvalues are the (complex) frequencies of thelines in the spectrum, as in eqn [11]: the imaginary partgives the oscillation frequency and the real part gives therate of decay. If kod (slow exchange) then there are twodifferent imaginary frequencies, which become 7d inthe limit of small k (see Figure 2). In fast exchange, whenk exceeds the shift difference, d, the quantity in thesquare root in [13] becomes positive, so the roots arepure real. This means that the spectrum is still two lines,but they are both at the average chemical shift and havedifferent widths. The full expressions for these lineshapes are given in the Appendix.

Because of the role of the eigenvectors in eqn [11], thefactor (amplitude) multiplying the complex exponentialis itself complex. The magnitude of the complex ampli-tude gives the intensity of the line and its phase gives thephase of the line (the mixture of absorption and disper-sion). In slow exchange, the two lines have the same real

part, but the imaginary parts have opposite signs, so thephase distortion is opposite. The sum of these distortedline shapes gives the familiar coalescence spectrum, as inFigure 2. In fast exchange, the two lines are both inphase, but one line is negative. This negative line is verybroad, and decreases in absolute intensity as the rateincreases, leaving only the single, positive, in-phase linefor fast exchange.

Therefore, the real and imaginary parts of the fre-quency and the real and imaginary parts of the amplitudeprovide the four parameters that define a line in an NMRspectrum: its intensity, its phase, its position and its width.This gives us a time-domain picture of the chemicalexchange, which can easily be converted to a spectrum byusing the Fourier transform.

Systems with Scalar Coupling

Scalar coupled systems are more complicated, but fun-damentally no different from the uncoupled systemsdescribed in the time domain. If there is scalar couplingin the spectrum, the line shape becomes more complex.For instance, Figure 3 shows the line shape of thediacylpyridine ligand as part of a rhenium complex. Therhenium bonds to the pyridine nitrogen and to onlyone of the carbonyls, lifting the symmetry of protons3 and 5. However, the rhenium can break this latter

Figure 3 Example of chemical exchange in a coupled spin

system. The rhenium can bond to one or other of the carbonyl

oxygen atoms so that the symmetry of the molecule is lifted.

Exchange of the rhenium between carbonyl groups broadens the

signals of protons 3 and 5 (at 8.46 ppm and 7.82 ppm), but leaves

4 (8.38 ppm) unaffected.


coordination and bond to the other carbonyl, whicheffectively interchanges protons 3 and 5 on the pyridinering. During this exchange, they retain their coupling tothe central proton 4, and the shift of 4 does not change.Therefore, proton 4 remains as a sharp triplet even whenprotons 3 and 5 are very broad.

In any strongly coupled spin system, each line inthe spectrum is a mixture of transitions of variousnuclei, which depends on the chemical shifts and cou-pling constants. When a chemical exchange happens, allthe spectral parameters change with it. Therefore, amagnetization (or a coherence) that was associated witha single spectral line in one site may be spread amongseveral lines in the other site. To deal with these com-plexities, we must use the density matrix.

One way of thinking of the density matrix is that it is alist of all the observables of a spin system. For example,the magnetizations of the two exchanging sites form partof the density matrix for that system. At equilibrium, thedensity matrix is just the z magnetization created by thestatic magnetic field. In a pulse Fourier transform NMRexperiment, this z magnetization is flipped into the xy

plane and divided among the individual lines in thespectrum, which then precess around the z axis.

To calculate the spectrum of the system (or any otherobservable), we need to be able to follow the densitymatrix as a function of time. The equation of motion ofthe density matrix (r) is given in eqn [14], where H is theHamiltonian of the spin system.

ih

2pqq t

r ¼ ½H ; r� ½14�

Some manipulation allows us to reformulate eqn [14] aseqn [15]. Mathematically, this means using superoperatorsin Liouville space L, but the details need not concern ushere. The important point is that the density matrixbecomes a vector of all possible observables of the system –in this case we only deal with the xy magnetizations.Anything does to the system – pulses, free precession,relaxation, exchange, is represented by a matrix:

ih

2pqq t

r ¼ L sor ½15�

If we use frequency units (h/2p¼ 1), then the solution toeqn [15] is given in eqn [16], which is identical to eqn [8]:

rðtÞ ¼ expð�i L sotÞrð0Þ ½16�

Relaxation or chemical exchange can be easily addedin Liouville space, by including a Redfield matrix,Rso (so¼ superoperator), for relaxation, or a kineticmatrix, Kso, to describe exchange. Both relaxation andexchange are described very conveniently by super-operators. The complete equation of motion becomeseqn [17]. Note the similarity to the equations for twouncoupled sites, but now we derived them rigorouslyfrom the equations of motion of the density matrix:

rðtÞ ¼ expð�iL so � R so � K soÞtrð0Þ

¼U expð � LtÞU �1rð0Þ ½17�

The details of what the density matrix is at timezero, what we observe in the receiver and how weconstruct the superoperators for a general spin systemare all beyond the scope of this article. However,the basic idea remains: even the most complex exchangeline shape is a sum of individual lines, with complexfrequencies and complex intensities, as in eqn [17]. Eachof those lines has a transition probability.

Physical Interpretation of the TransitionProbability

The standard expression for the probability of a transi-tion from an initial state, c, to a final state, f, induced


by a perturbation Ix (the RF field along x), is given as

Transition probabilityp f Ixj jch ij j2 ½18�

The matrix element in eqn [18] can be written in a rathermore complicated way, as in eqn [19].

f Ixj jch i

¼ tracef Ixj jfh i c Ixj jfh i

f Ixj jch i c Ixj jch i

!�

0 1

0 0

!( )½19�

The reason for rewriting the equation is that the right-hand side can be interpreted as the dot product of theoperator Ix with the operator |fS/c|. In the Liouvillespace all the operators are vectors. The dot productin Liouville space is defined as the trace of the productof the operators in the original spin space. One importantpoint is that eqn [19] is independent of basis set. Thematrix element in eqn [18] becomes a simple dot productof two Liouville space vectors as in eqn[20].

f Ixj jch i ¼

f Ixj jfh i

c Ixj jfh i

f Ixj jch ic Ixj jch i

0BBBBB@

1CCCCCA�

0

0

1

0

0BBBBB@

1CCCCCA ½20�

A dot product of two vectors gives the projection ofone vector on to the other, so eqn [20] defines the pro-jection of |fS/c| onto Ix. In other words, this is theprojection of the transition onto the total xy magnetiza-tion. The standard transition probability is the square ofthis projection.

This leads to a physical interpretation of the transitionprobability. The spin system at equilibrium is repre-sented by the total z magnetization, Fz. This is the sumof the z magnetizations of the sites, weighted by theequilibrium populations of each site. To distinguish theweighted sum, we give this spin operator a symboldifferent from Iz. A pulse flips this into the xy plane, sothat immediately after the pulse, the spin system isrepresented by Fx. Now each transition receives a share ofthe total x magnetization. Its share is given by the pro-jection of the transition (as an operator) onto the operatorFx. The transitions evolve independently as a function oftime. However, we do not observe the transition directly,but rather the total xy magnetization. The detector issimply a coil of wire, and we measure the xy magneti-zation as a function of time to give us an FID. Fouriertransforming the FID gives us a spectrum. Therefore,each transition contributes to the total signal according toits projection along Ix. The intensity of a transition is theproduct of how much coherence it received from Fx atthe start and how visible it is to the receiver.

Note that the Liouville matrix, iLþRþK may not beHermitian, but it can still be diagonalized. Its eigenvaluesand eigenvectors are not necessarily real, however, andthe inverse of U may not be its complex-conjugatetranspose. If complex numbers are allowed in it, eqn [17]is a general result. Since L is a diagonal matrix we canexpand in terms of the individual eigenvalues, lj . U canbe applied (‘backwards’) to Ix to obtain the followingequation

NMR signal ¼X

j

ðUIxÞ�j ðU�1FxÞj expð�ilj tÞ ½21�

Each of the terms in the sum is a transition. The NMRsignal is always a sum of decaying sine waves, whose fre-quency and decay rate are given by the imaginary andthe real parts of lj . The intensity is governed by thecoefficients in eqn [21] of this term. This coefficient is theproduct of two terms. The first, (UIx)j , tells us how muchthe coherence overlaps with the receiver. The secondterm, (U�1Fx)j tells us how much the coherence receivedfrom the equilibrium z magnetization. The product ofthese terms is the generalization of the transition prob-ability. If we do not have relaxation, the two terms arecomplex conjugates, and we recover the normal transitionprobability. However, the transition probability for ageneral system may have an imaginary part. If we cling toidea that each term in eqn [21] represents a transition,then an individual transition may be out of phase at t¼ 0.Immediately after the excitation pulse, the total magneti-zation will lie on the y-axis, but the individual componentsmay not. However, our true observable is still only theFID. Whether we want to decompose it into individualtransitions of individual spins depends very much on thesystem we are studying.

Example of Exchange of Coupled Spins

Provided we can construct the matrices, the description ofexchange is quite simple. Again, the details are beyond ourscope, but an example of the Liouvillian (no exchange) fora coupled spin system is given in eqn [22] for an AB spinsystem, with coupling constant J. The eigenvalues of thismatrix are the familiar line positions of an AB system:

ioA i J=2 0 �i J=2

i J=2 ioA �i J=2 0

0 �i J=2 ioB i J=2

�i J=2 0 i J=2 ioB

0BBBBB@

1CCCCCA ½22�

If there is an exchange between A and B given by a rate k,then we set up the two blocks, as we set up the two


exchanging spins in the Bloch equations. Note thatwe have made spin A in one block exchange with spin B inthe other. The full Liouvillian, including exchange,is given in eqn [23], in which dots replace zeroes toemphasize the form of the matrix.

In this particular case, called mutual exchange, thespins simply permute themselves. This matrix can besimplified to half its original size, as in eqn [24], but

ioA � k i J=2 � �i J=2 k � � �

i J=2 ioA � k �i J=2 � � k � �

� �i J=2 ioB � k i J=2 � � k �

�i J=2 � i J=2 ioB � k � � � k

k � � � ioB � k i J=2 � �i J=2

� k � � i J=2 ioB � k �i J=2 �

� � k � � �i J=2 ioA � k i J=2

� � � k �i J=2 � i J=2 ioA � k

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCA

½23�

for non-mutual exchange we must retain eqn [23].

ioA � k i J=2 k �iJ=2

i J=2 ioA � k �iJ=2 k

k �iJ=2 ioB � k iJ=2

�i J=2 k iJ=2 ioB � k

0BBBBB@

1CCCCCA ½24�

Except for its size, this is exactly the same form as inour previous cases. The spectrum can therefore beexpressed as a sum of four lines, as in Figure 4.

Slow Chemical Exchange

The term ‘slow’ in this case means that the exchange rateis much smaller than the frequency differences in thespectrum, so the lines in the spectrum are not sig-nificantly broadened. However, the exchange rate is stillcomparable with the spin–lattice relaxation times in thesystem. In this case, we can measure the rates by doing amodified spin–lattice relaxation experiment, pioneeredby Hoffman and Forsen.

In the absence of exchange (and let us ignore dipolarrelaxation), each z magnetization will relax back toequilibrium at a rate governed by its own T1 as given inthe following equation:

d

dt½MðtÞ �MðNÞ� ¼ � 1

T1½MðtÞ �MðNÞ� ½25�

If we have two sites, A and B, then we can write theanalogous equation for two sites as

d

dt

M AðtÞ �M AðNÞ

M BðtÞ �M BðNÞ

" #

¼

1

TA1

0

01

TB1

0BBB@

1CCCA



" #½26�

If the two sites exchange with rate k during therelaxation, then a spin can relax either through normalspin–lattice relaxation processes or by exchanging withthe other site. Equation [26] becomes

d

dt



" #

¼

1

TA1

� k k

k1

TB1

� k

0BBB@

1CCCA



" #½27�

This equation is very similar to eqns [6] and [7]. Thebasic situation is just as in intermediate exchange, exceptthat now we are dealing with z magnetizations ratherthan xy magnetizations. The frequencies are zero and thematrix now has pure real eigenvalues, but the approach isthe same. In the time domain a relaxation experimentfollows the z magnetizations as a function of time. Asbefore, the time dependence is obtained by diagonalizingthe relaxation–exchange matrix and calculating themagnetizations for each time at which they are sampled.

There are two main applications of slow chemicalexchange: one is to determine the qualitative mechanismand the other is to measure the rates of the processesas accurately as possible. For the first case, in whichthere is a spectrum in slow exchange, the followingmechanism needs to be established which site isexchanging with which. For this purpose, the

Figure 4 Calculated spectra for mutual exchange in an AB

spin system, as a function of exchange rate. The heavy traces

are the total line shape and the lighter lines show the individual

components. The bottom spectrum shows the typical static AB

spectrum, which broadens and coalesces at higher rates of

exchange.


homonuclear two-dimensional (2D) experiment EXSY(exchange spectroscopy) (the same pulse sequence asNOESY (2D nuclear Overhauser effect spectroscopy),but involving exchange) is by far the best technique touse. If there is no exchange, then the spectrum lies alongthe diagonal and there are no cross-peaks. Exchangebetween sites leads to a pair of symmetrical cross-peaksjoining the diagonal peaks of the same site, so themechanism is very obvious.

The EXSY pulse sequence starts with two p/2pulses separated by the incrementable delay, t1. Thismodulates the z magnetizations, so that the relaxationthat occurs during the mixing time which follows, tm,is frequency labelled. Finally, the z magnetizations aresampled with a third p/2 pulse. Magnetization from a

different site that enters via exchange will have adifferent frequency label. A two-dimensional Fouriertransform then produces the spectrum. However, caremust be taken in choosing the mixing time if there aremultiple exchange processes. If the mixing time is toolong, there is a substantial probability that a spin mayhave exchanged twice in that time, leading to spuriouscross-peaks.

For careful rate measurements, once the mechanism isestablished, it is our opinion that one-dimensional (1D)methods are superior to quantitative 2D ones. Apart fromthe fact that 1D spectra can be integrated more easily, wealso have more control over the experiment. Modernspectrometers can create almost any type of selectiveexcitation, so that the conditions can be controlled at thestart of the relaxation. For two sites, a non-selectiveinversion that inverts both sites equally will mask most ofthe exchange effects and the relaxation will be dominatedby T1. However, if one site is inverted selectively, thenthat site can regain equilibrium by either T1 processes orby exchanging with the other site that was left at equi-librium. The inverted signal will relax at roughly the sumof the exchange and spin–lattice relaxation rate, while thesignal that was unperturbed at the start of the experimentshows a characteristic transient, as in Figure 5. Formultiple sites, a wide range of initial conditions isavailable. Standard non-linear least-squares methodsallow us to fit these curves and derive values for the ratesinvolved.

Fast Exchange

In fast exchange, the spectrum has coalesced, usually to asingle line, so the information in the spectrum is not sorich. For two equally populated sites, for instance, we geta single Lorentzian line with a width proportional to d2/k. The line width (or better, T2) can be measured via aCPMG (Carr–Purcell pulse sequence, Meiboom–Gillmodification) experiment or a T1r measurement. How-ever, if we do not know the frequency difference, d, wecannot obtain the rate. If the exchange rate is not too fast(o104 roughly) modifications of the T2 experiments canhelp. Both experiments have an inherent timescale: inCPMG, it is the timing of the refocussing pulse; in T1r, itis the precession frequency about the spin-locking field.If the exchange is fast with respect to the experiment,we measure a T2 appropriate to the coalesced spectrum.If, however, the exchange is slower than the experimentaltimescale (but still fast with respect to the frequencydifference), the apparent T2 reflects the individual sites.As we change the timescale of the experiment, theapparent T2 changes, and so a value for the rate itself canbe obtained.

Figure 5 Inversion–recovery curves for slow exchange between two sites. Lines A and B are the results from one experiment. They

show the recovery after a non-selective inversion of both sites, showing that the two sites have slightly different T1 values. C and D are

obtained from a different experiment. Line C shows the recovery of a site that has been selectively inverted, and line D shows the

behaviour of the line that was not perturbed in this experiment. The inverted line relaxes faster due to a combination of spin–lattice,

relaxation and exchange, and the unperturbed line shows the characteristic transient.


Conclusions

All of the effects of chemical exchange can be calculatedby following the appropriate magnetizations as a functionof time. For slow exchange, this is the complex couplingof relaxation and exchange that leads to transientbehaviour in modified spin–lattice relaxation experi-ments. For intermediate exchange, the rather complexline shapes can always be decomposed into a sum ofindividual transitions, even though the transitions aredistorted in phase, intensity, position and width by thedynamics of the system. For two uncoupled sites, thiscan be calculated quite easily, but for larger coupledsystems the construction of the matrices may get com-plicated. However, the resulting lines are always gov-erned by a transition probability, provided we extendour definition to allow probabilities with both real andimaginary parts. Working in the time domain is alreadyfamiliar from Fourier transform spectroscopy and itprovides a complete and simple approach to chemicalexchange.

Appendix

Eigenvalues and Eigenvectors for Two-SiteExchange

For two-site equally populated chemical exchange, thetwo magnetizations at time zero are equal, so that eqn[11] can be written as in eqn [28], where the eigenvalues

are complex numbers.

MAðtÞ

MBðtÞ

!¼ U

e l1t 0

0 e l2t

!U�1

1

1

!½28�

Since non c Hermitian matrices are involved, the matrixformed by the eigenvectors will not be unitary, and willhave four independent complex elements. These aresimply call a, b, c and d, so that U is given by:

U ¼a b

c d

!½29�

Regardless of whether U is unitary, its inverse is given byeqn [30], where D is the determinant of eqn [29]:

U �1 ¼ 1

D

d �b

�c a

!½30�

Equation [28] then says that the signal is given by thefollowing, regardless of slow or fast exchange:

Signal ¼ða þ cÞðd � bÞD

expðl1tÞ

þ ðb þ dÞð�c þ aÞD

expðl2tÞ ½31�

The values of the eigenvectors have two forms,depending on whether d4k (slow exchange) or dok

(after coalescence). In the first case, the eigenvalues are


given as

Eigenvalues ¼ � 1

T2þ k

� �7i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid 2 � k 2p

½32�

and a convenient matrix of eigenvectors is given by

k iðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid 2 � k 2p

þ dÞ

�iðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid 2 � k 2p

þ dÞ k

0@

1A ½33�

After coalescence, when the rate is greater than thefrequency difference, the two transitions are both at zerofrequency (i.e. the average chemical shift) but havedifferent widths and intensities. The eigenvalues are purereal, as

Eigenvalues ¼ � 1

T2þ k

� �7


½34�

and the eigenvectors are similar, but reflect the fact thatk2–d2 is now positive:


� id �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2 � d 2p

� id

k k

!½35�

See also: NMR Principles, NMR Pulse Sequences, NMR

Relaxation Rates.

Further Reading

Bain AD (1988) The superspin formalism for pulse NMR. Progress inNuclear Magnetic Resonance Spectroscopy 20: 295--315.

Bain AD and Duns GJ (1996) A unified approach to dynamic NMRbased on a physical interpretation of the transition probability.Canadian Journal of Chemistry 74: 819--824.

Binsch G (1969) A unified theory of exchange effects on nuclearmagnetic resonance lineshapes. Journal of the American ChemicalSociety 91: 1304--1309.

Binsch G and Kessler H (1980) The kinetic and mechanistic evaluationof NMR spectra. Angewandte Chemie, International Edition inEnglish 19: 411--494.

Gutowsky HS and Holm CH (1956) Rate processes and nuclearmagnetic resonance spectra. II. Hindered internal rotation of amides.Journal of Chemical Physics 25: 1228--1234.

Jackman LM and Cotton FA (1975) Dynamic Nuclear MagneticResonance Spectroscopy. New York: Academic Press.

Johnson CS (1965) Chemical rate processes and magnetic resonance.Advances in Magnetic Resonance 1: 33--102.

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Encyclopedia of Spectroscopy and Spectrometry || Chemical Exchange Effects in NMR*