One of the most commonly observed dynamic effects in small-molecule NMR is a broadening or doubling of resonances caused by restricted rotation of a tertiary amide bond (2.6) due to its partial double-bond character. This example serves as a useful introduction to the influence on NMR spectra of dynamic chemical exchange within molecules, and to illus- trate this we first consider the classic example of N,N-dimethylacetamide (DMA, 2.7). In this, the two N-methyl groups may be defined as sitting either cis or trans to the carbonyl oxygen and if they were to remain in these positions (ie if there were no rotation about the N-CO bond) we would expect the two groups to experience different chemical environ- ments and thus appear with different chemical shifts. Indeed, the ambient temperature spectrum of DMA displays two N-methyl resonances because rotation around the amide bond is sufficiently slow that the two discrete environments are apparent (Fig. 2.45a). However, at the higher temperature of 420 K only one N-methyl resonance is observed which now arises from six equivalent protons. Under these conditions, amide bond rotation causes the N-methyl groups to exchange their environments rapidly; moreover, this is sufficiently fast that only a single averaged resonance is detected partway between the individual resonance positions (Fig. 2.45c). Between these two extremes the exchange behaviour is said to occur at an intermediate rate and the appearance of the spectrum becomes dominated by line broadening that is sensitive
FIGURE 2.44 Dynamic molecular processes. Examples of intramolecular dynamic exchange equilibria that may be slow enough to produce directly observable effects in NMR spectra.
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40 High-Resolution NMR Techniques in Organic Chemistry
to the exchange rate (Fig. 2.45b). This regime yields more complex spectra that can inform on exchange rate constants and receives further attention below. Notice that the appearance of the C-methyl resonance of DMA does not vary significantly with temperature since its environment remains unchanged following amide bond rotation, although the chemical shifts of all three methyl groups do show a small temperature dependence, a commonly observed effect. To continue, we need to determine what defines a process as being termed slow or fast in the context of NMR spectra, and this is most easily done by again considering exchange between only two discrete sites.
2.6.1.1 Two-Site Exchange: Equal Populations
Let us consider the general case of two-site exchange between two equally populated sites A and B for which the equilib- rium may be written:
A kk B (2.15)
A≡kkB
FIGURE 2.45 The 250-MHz 1H NMR spectra of N,N-dimethylacetamide in DMSO. The spectra show the methyl group behaviour when the amide bond rotation rate is considered to be (a) slow (295 K) (b) intermediate (350 K) and (c) fast (420 K). The box shows the appearance of temperature- dependent water resonance. Methyl resonances B and C are broadened slightly in (a) owing to their mutual, unresolved long-range coupling.
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Introducing High-Resolution NMR Chapter | 2 41
where the forward and reverse first-order rate constants k are necessarily equal. We shall first consider a system which lacks any resolved J coupling; so, we need only consider the resonance frequencies for the two sites A and B (defined in hertz) and the magnitude of their frequency differences ∆AB = |A – B|. An example of this might be the two N-methyl groups of DMA above representing sites A and B that undergo dynamic interconversion. The exchange process will interchange the locations and hence the environment of sites A and B such that the resonance frequency of each will jump whenever an exchange event occurs. In the case of DMA, the methyl groups will move between the cis and trans positions relative to the carbonyl group. The influence of this process on NMR lineshapes as a function of the exchange rate constant k is illustrated in Fig. 2.46. Experimentally, changes in exchange rates are typically induced by changes in sample temperature.
When the exchange rate is very slow (ie when the lifetimes of each state τ are very long, τ = 1/k), the resonances for each site are sharp and clearly resolved yielding discrete lines of equal intensity. At the limit of this so-called slow exchange regime the dynamic exchange process has a negligible influence on the observed NMR spectrum and what is seen is merely
FIGURE 2.46 Two-site exchange for equal populations. Calculated spectra for two nuclei undergoing exchange between two equally populated sites A and B with frequency separation ∆AB of 50 Hz. The arrow indicates the position of exchange-averaged resonance. Linewidths in the absence of exchange were 1 Hz.
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42 High-Resolution NMR Techniques in Organic Chemistry
the discrete resonances of each site. Measurement of exchange rates in this regime require the use of magnetisation transfer experiments, as described in Section 2.6.3.
As the rate progressively increases, each resonance initially begins to broaden, reducing in peak height, and the two lines will then appear to move closer; the exchange process is said to lead to exchange- or dynamic-broadening of resonances.
In this slow–intermediate regime NMR lineshapes are exquisitely sensitive to the exchange rate, which may be related to the observed peak separation ∆O according to
π ( υ ) ( υ )
= √ ∆ − ∆
k √
2
AB 2
O 2
(2.16) The additional line broadening induced as a result of the dynamic exchange ∆υex12 (ie that acting in addition to the natural linewidth in the absence of exchange ∆υ12; see Section 2.4.3) is given by
υ π
∆ ex= k
12 (2.17)
Clearly in this regime the resonances becomes broader as the rate becomes faster, although in principle two separate resonances remain. In reality, it may be that the peaks are sufficiently broad that they may not be readily apparent and may even appear to be missing from a spectrum; this can be an even greater problem in situations of exchange between sites of unequal population, as described below. When peak movement and broadening can be measured, exchange rate constants can be determined either from the above expressions or more commonly nowadays through computer lineshape simulation (see Section 2.6.2).
As the rate further increases, the peaks eventually merge into one and are said to have coalesced. The point at which they do so, when the valley between them just flattens out, is the so-called coalescence point, at which the rate kc will be:
π υ υ
= ∆
√ ≈ ∆
kc 2AB 2.22 AB (2.18)
Thus, if the frequency separation between peaks is known from the spectrum recorded within the slow exchange limit, the exchange rate at coalescence can be determined quite readily. One caveat here is that chemical shifts may also be tem- perature dependent meaning the actual shift differences ∆AB may be altered at coalescence temperature; see discussions on the practicalities of these measurements in Section 2.6.2.
At rates above the coalescence point, the system enters the fast–intermediate regime and only a single merged peak is seen that reflects the intensity sum of the two merged resonances. This peak appears at the midpoint of the two individual resonances since we now observe the average properties of the species undergoing exchange. In this regime the broadening due to exchange is:
υ π υ( )
∆ = ∆
k 2
ex AB
2
12 (2.19)
leading to a reduction in broadening and hence a sharpening of the single resonance as the rate increases. This process is sometimes referred to as exchange- or dynamic-narrowing of resonances. The rate can again be determined from lineshape fitting procedures provided the frequency separation of the resonances in the absence of exchange is known. In the limit of very fast exchange the contribution to the linewidth becomes negligible relative to its natural linewidth and the process becomes too rapid to influence directly the observed NMR spectrum. Although lineshapes can no longer provide informa- tion on exchange rates under these conditions, rapid exchange may still influence spin relaxation rates, allowing exchange rates to be determined through appropriate analysis. However, this finds greatest application in studies of the dynamics and flexibility within macromolecules, and shall not be considered further here.
From the above descriptions, it is apparent that the exchange regimes are determined by the rate of exchange relative to the frequency separation of the exchanging resonances ∆AB (Fig. 2.47). In the slow exchange regime (k << ∆AB), separate resonances for each interchanging site will be observed. In fact, the term slow is somewhat misleading here since the exchange process itself, bond rotation for example, actually occurs rather rapidly. Instead, it is the intervals between the exchange events that become lengthy in this regime, and hence the events infrequent. This in turn means the lifetime of each individual state is also long and its resonance position can be accurately determined. For consistency with common k=π√∆υAB2−∆υO2√2
∆υẵex
∆υẵ
∆υẵex=kπ
kc=π∆υAB√2≈2.22∆υAB
∆υẵex=π∆υAB22k
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Introducing High-Resolution NMR Chapter | 2 43
parlance, but with this in mind, the term slow will continue to be used throughout these discussions to reflect this condi- tion. In the fast-exchange regime (k >> ∆AB), a single resonance is observed representing the averaged properties of each state, which in the case of equally populated sites appears at the midpoint of the two peaks. Here the lifetime of each state is too short for each to be resolved. Between these regimes, within the intermediate-exchange regime (k ∼ ∆AB), exists the coalescence point when the two signals just merge into a single peak. The importance of discussing exchange regimes relative to an appropriate frequency scale cannot be overemphasised, and descriptions of dynamic effects should be referenced to a corresponding NMR timescale, governed by 1/∆. In discussions so far we have referred to events as ranging from slow to fast on the chemical shift timescale since it is chemical shift differences that defined these exchange regimes. Typically, the phrase ‘NMR timescale’ will refer to the chemical shift timescale. Since chemical shift separations typically range from a few hertz up to a few kilohertz, dynamic NMR lineshape studies are most often sensitive to rela- tively slow equilibria occurring on second to millisecond timescales. Processes that are slower than this may be studied through magnetisation transfer techniques, as described in Section 2.6.3. It is equally possible to consider events on a coupling constant timescale where the magnitudes of these provide the frequency reference; see discussions involving scalar coupling below.
Before we proceed further we consider briefly why dynamic exchange should lead to the broadening of resonances at all. To begin, we recall that any spin will have a characteristic precession frequency in the transverse plane during the col- lection of its FID (Section 2.2.3). We also note that the observed resonance peak for each site arises from the summed inten- sity of all similar spins in the NMR sample, precessing with a similar frequency. Under conditions of two-site exchange, the change in environment from one site to another leads to the frequency of a spin to jump between two different values whenever an exchange event occurs. If this were to occur so infrequently that a complete FID can be collected before such a jump were to happen, then each spin will exhibit only a single frequency and the exchange process would be too slow to influence the NMR spectrum. Now consider the slow–intermediate exchange situation in which two sites interchange while the FID is being collected. These events are infrequent and random, meaning the jumps will occur at different times for different spins within an ensemble of molecules since they are not correlated events. This will lead to net dephasing of the bulk magnetisation vector arising from the sum of all the spins undergoing frequency jumps, and an associated acceleration of the loss of the net transverse magnetisation beyond that which occurs from natural relaxation processes. As described in Section 2.4.3, faster decay of transverse signal correlates to broader resonance in the transformed NMR spectrum, as is seen in this exchange regime. Still faster exchange rates, that is even shorter lifetimes) give rise to more rapid net magnetisation decay and hence greater resonance broadening. When exchange becomes fast on the shift timescale, the lifetimes of any spin in either of the exchanging sites are so short that little net dephasing of transverse magnetisation can accrue between frequency jumps, meaning the accelerated decay of the NMR response is attenuated and a sharper, averaged resonance is observed, characteristic of the fast-exchange regime.
2.6.1.2 Two-Site Exchange: Unequal Populations
Now consider the result of exchange between two species that exist with an unequal population, a more generalised form of the above equilibrium condition. Continuing with an example of restricted rotation in amide bonds, this is observed for the unsymmetrically substituted tertiary amide 2.8. Rotation about the N-CO bond will change the envi- ronments for all neighbouring groups leading to two different conformational (rotational) forms, commonly known as rotamers, of differing concentrations. Under slow-exchange conditions the 1H spectrum displays the sum of the
FIGURE 2.47 Dynamic exchange regimes. Schematic classification of the dynamic exchange regimes observed in NMR spectra, with approximate rates (k) or lifetimes (τ ) associated with these regimes.
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44 High-Resolution NMR Techniques in Organic Chemistry
spectra for each rotamer so that, for example, the acetyl methyl group and the adjacent methane H2 each give rise to two resonances whose intensities reflect the populations of the interconverting conformers (Fig. 2.48). In this example, it is noteworthy that the carbonyl group exerts a substantial deshielding effect on the adjacent aromatic proton in the major rotamer, giving rise to a shift difference between the rotamers for this proton of a sizeable 2.3 ppm. The fact that these widely separated protons were undergoing exchange was proven by saturation transfer experiments as described in Section 2.6.3, and the assignment of the rotamers was made from NOEs, with all these experiments performed under slow-exchange conditions at 230 K.
In the generic case we may again label the exchanging species as A and B and represent the equilibrium:
A kkA B
B (2.20)
In this case, the rate constants for the forwards (kA) and backwards (kB) processes will differ due to the differing popula- tions such that:
P kA⋅ A =P kB⋅ B (2.21)
where PA and PB represent the fractional populations of species A and B, respectively, with PA + PB = 1. The behaviour of a single, uncoupled resonance as a function of the exchange rate is shown in Fig. 2.49 with kA > kB. The populations of the interchanging species are A:B = 1:2 as reflected in the spectrum at slow exchange. Under slow–intermediate exchange conditions, resonance broadening is apparent and the peaks again tend to move together. However, the extent of broaden- ing is now different for the two resonances, with that of the less populated species showing greater broadening. This is a consequence of its shorter lifetime associated with its higher rate constant. Following from Eq. 2.17, the terms for exchange broadening may be written:
υ π υ
∆ = k ∆ = kπ
A : ex12 A B : ex12 B (2.22)
quantifying the correlation between the exchange rate constant and the extent of resonance broadening.
As the system moves into the fast–intermediate regime a single resonance is again observed. This now represents a population-weighted average of the exchanging species, meaning the frequency of the averaged peakυABav sits towards that of the more populated species such that:
υABav =PA Aυ +PB Bυ (2.23)
Any further increase in rate constant leads again to greater exchange narrowing of the single resonance, with residual line broadening arising from exchange of:
υ π ( υ )
∆ = ∆
+ P P
k k
ex 4 A B AB
2
A B
12 (2.24)
A≡KBKAB
PA.kA=PB.kB
A:∆υẵex=kAπ B:∆υẵex=kBπ
υABav
υABav=PAυA+PBυB
∆υẵex=4πPAPB∆υAB2kA+kB
FIGURE 2.48 The 1H NMR spectra of the aromatic amide 2.8. The traces show the acetyl methyl and H2 resonances under conditions of slow ex- change at 230 K. From this, the relative populations of rotamers are 5.3:1.
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Introducing High-Resolution NMR Chapter | 2 45
This exchange broadening term can play a very significant role in influencing the linewidths observed for small- molecule ligands experiencing rapid on–off exchange equilibria with a protein receptor. Such dynamic exchange broadening may be indicative of ligand binding and so provides a useful indicator of complex formation with a macromolecular target; studies of protein–ligand binding equilibria are considered in the chapter Protein-Ligand Screening by NMR.
Under conditions of unequal populations, peak broadening in the intermediate exchange regime can mean the point at which the peaks coalesce is difficult, if not impossible, to define and determination of kA and kB is best made through
FIGURE 2.49 Two-site exchange for unequal populations. Calculated spectra for two nuclei undergoing exchange between two unequally populated sites A and B with frequency separation ∆AB of 50 Hz. The population ratio A:B is 1:2 (kA = 2kB) and the spectra are labelled according to the mean rate constant (ẵ(kA + kB) s−1). The arrow indicates the position of exchange-averaged resonance. Linewidths in the absence of exchange were 1 Hz.
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46 High-Resolution NMR Techniques in Organic Chemistry
lineshape simulation. At a practical level, the greater broadening associated with the more dilute species can mean its resonance becomes very difficult to observe in this regime. The presence of chemical exchange may be revealed only by moderate broadening and shift, followed by progressive sharpening, of the major species resonance as sample temperature, and hence the rate constant, is increased. This is illustrated by the exchange spectra with one species in large excess, shown in Fig. 2.50 for A:B = 1:8. Change in the resonance position of the major species B under slow- intermediate exchange and of the averaged peak under fast-intermediate exchange is rather small and only moderate line broadening is observed (compare Fig. 2.46). Moreover, the minor species peak A is barely apparent throughout the intermediate regime and its presence is easily overlooked.
FIGURE 2.50 Two-site exchange for highly unequal populations. Calculated spectra for two nuclei undergoing exchange between two sites of greatly differing populations with frequency separation ∆AB of 50 Hz. The population ratio A:B is 1:8 (kA = 8kB) and the spectra are labelled ac- cording to the mean rate constant (ẵ(kA + kB) s−1). The arrow indicates the position of exchange-averaged resonance. Linewidths in the absence of exchange were 1 Hz.
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Introducing High-Resolution NMR Chapter | 2 47
2.6.1.3 Two-Site Exchange between Scalar Coupled Nuclei
We now extend discussions to include scalar (J)-coupled spins experiencing exchange, and how coupling fine structure may be influenced. Let us first consider exchange between two mutually coupled spins A and B that undergo a symmetri- cal interchange A ⇌ B and share a coupling JAB but have no other coupled partners (Fig. 2.51). This may be an isolated geminal proton pair undergoing mutual exchange of their environments, for example. Under conditions of slow exchange, geminal coupling is apparent and we see the classical roofed coupling structure. As the exchange rate increases the coupling becomes masked by the resonance broadening until at coalescence this is no longer discernible. At the coalescence point the rate constant is given by:
π ( υ ) ( )
= √ ∆ +
k √ 6 J
2
AB 2
AB 2
(2.25) having similarity to Eq. 2.18 for the uncoupled two-spin case. As the system tends to the fast-exchange regime the peaks sharpen as the sites interchange more rapidly. However, the resonance lacks coupling structure, appearing as a broad single resonance until in the fast-exchange regime only a single sharp peak exists. The mutual interconversion of A and B leads to a time-averaged system and as the spins have become equivalent the coupling interaction between them no longer gives rise to an observable splitting and the system has been decoupled by the dynamic process. A real example of such behaviour is seen for diphenyldiazetidinone 2.9, where the nitrogen inversion process leads to the symmetrical interchange of ring CH2 protons, leading to their eventual coalescence and associated decoupling.
k=π√∆υAB2+6JAB2√2
FIGURE 2.51 Two-site exchange for mutually coupled spins. Calculated exchange spectra as a function of the rate constant for two mutually J-coupled spins undergoing symmetrical interconversion (JAB = 12 Hz, ∆AB = 50 Hz).
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