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Solid-State NMR, Rotational Resonance
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Fyfe C (1983) Solid State NMR for Chemists. Guelph: CFC Press. Harris RK and Grant DM (1996) Encyclopedia of Nuclear Magnetic Resonance. Chichester: Wiley. Mehring M (1983) High Resolution NMR in Solids, 2nd edn. Berlin: Springer-Verlag. Slichter CP (1983) Principles of Magnetic Resonance, 3rd edn. Springer: Berlin.
Schmidt-Rohr K and Spiess HW (1994) Multidimensional Solid-State NMR and Polymers. London: Academic Press. Traficante DD (ed) Concepts in Magnetic Resonance, An Educational Journal. New York: Wiley.
Solid-State NMR, Rotational Resonance David L Bryce and Roderick E Wasylishen, Dalhousie University, Halifax, Nova Scotia, Canada Copyright © 1999 Academic Press
Introduction One of the primary goals of solid-state NMR spectroscopists has been to develop techniques that yield NMR spectra of solid samples with resolution approaching that observed for samples in isotropic liquids. Rapidly spinning samples about an axis inclined at the magic angle (arccos(1/√3) = 54.7356 °) relative to the applied static magnetic field has been found to be highly effective in this regard. In addition, high-power decoupling of abundant spins (e.g. 1H) eliminates heteronuclear spinspin coupling interactions (direct dipolar and indirect J-coupling) involving the abundant spins when dilute spins are examined. The availability of commercial NMR instrumentation that permits users to apply these two techniques has contributed to spin- NMR becoming a routine method for examining a wide range of solid materials. Finally, cross-polarization (CP) from abundant spins to dilute spins has been important in improving the sensitivity of the dilute-spin NMR experiment. Ironically, it is sometimes desirable to selectively reintroduce interactions which are effectively averaged in the magic-angle-spinning (MAS) experiment. Dipolar coupling, for instance, may be recovered in the form of the direct dipolar coupling constant (RDD) between isolated spin pairs. The value of RDD is of interest due to its simple relationship with the distance separating two spins, r12 (Eqn [1])
MAGNETIC RESONANCE Applications where P0 is the permeability of free space, and Ji are the magnetogyric ratios of the nuclei under consideration. Rotational resonance (RR) is a MAS NMR technique which selectively restores the dipolar interaction between a homonuclear spin pair, thus allowing the determination of the dipolar coupling constant, RDD, and hence, the internuclear distance. Historically, the RR phenomenon was discovered by Andrew and co-workers in a 31P NMR study of phosphorus pentachloride, which consists of PCl 4+ and PCl6− units in the solid state. This group noticed that when the rate of sample spinning matched the difference in resonance frequencies of the nonequivalent phosphorus centres, their peaks broadened and the rate of cross-relaxation was enhanced. It is now known that if the RR condition is satisfied, direct dipolar coupling is restored selectively to a homonuclear spin pair. That is, if the sample spinning rate is adjusted to a frequency, Qr, such that
where n is an integer, generally 13, and Q and Q are the isotropic resonant frequencies of spins 1 and 2 respectively, then the two nuclei are said to be in RR. As a result, dipolar coupling between the nuclei is restored (via the flip-flop term in the dipolar Hamiltonian), and line broadening of the resonances at Q and Q is observed (see Figure 1). Additionally, a rapid oscillatory exchange of Zeeman
SOLID-STATE NMR, ROTATIONAL RESONANCE 2137
of the nucleus. These induced local fields are proportional to the applied field. In frequency units, the Hamiltonian operator which accounts for both the Zeeman interaction and this chemical shielding (CS) interaction is
where
Figure 1 Effect of RR on the 13C NMR line shape of Ph13CH213COOH. Spectra acquired at 4.7 T (50.3 MHz). Top spectrum at n = 1 RR, Qrot = 7207 Hz. Bottom spectrum off RR, Qrot = 10 000 Hz.
magnetization occurs. In fact, it is this exchange of magnetization rather than the line shape which is usually monitored in order to determine the dipolar coupling constant. In order to generate an exchange curve which may be analysed and simulated, one of the two resonances involved must be inverted selectively; the intensity difference of the peaks is then monitored as function of time. Obviously, it is highly desirable to develop techniques capable of recovering weak dipolar coupling constants from high-resolution NMR spectra obtained under MAS conditions. The focus of the present discussion will be to provide an overview of the basic RR scheme. First, the theory of RR will be outlined, followed by a discussion of the most important experimental techniques employed to measure dipolar coupling constants under conditions of RR. Finally, some examples that illustrate the applications and limitation of the techniques will be described.
Theory Restoring the dipolar interaction: a theoretical approach
The most important interaction in NMR results from the application of a large external magnetic field, *0, to the sample. Termed the Zeeman interaction, its effect on the normally degenerate nuclear spin energy levels is to cause them to split. The Zeeman levels are perturbed by local fields generated by the motion of electrons in the vicinity
and Viso is the isotropic chemical shielding constant. The interaction of interest in RR is the dipolar interaction, an orientationally dependent throughspace spinspin coupling, which leads to a perturbation of the CS-perturbed Zeeman energy levels. For a homonuclear two-spin system, the truncated dipolar Hamiltonian operator is given by the following:
Here, Î+ and Î− are the raising and lowering operators and T is the angle between the applied magnetic field and the internuclear vector, r12. The factor containing the raising and lowering operators is sometimes referred to as the flip-flop term. The final interaction that must be considered is the indirect spinspin coupling interaction, which is mediated by the intervening electrons. The indirect spinspin Hamiltonian, J, is often ignored because it is frequently considerably smaller than DD. Up until this point, we have implicitly assumed time independence of the interactions and their corresponding Hamiltonian operators. This assumption is valid for a rigid stationary sample. However, when the sample is spun rapidly, each of the internal Hamiltonians becomes time-dependent. For example, Z,CS becomes time-dependent when there is chemical shielding anisotropy due to the fact that the orientations of the chemical shielding tensors relative to the applied magnetic field change as the sample
2138 SOLID-STATE NMR, ROTATIONAL RESONANCE
rotates. Then Equation [3] becomes
where (Vi,zz − Viiso) is a measure of the orientation dependence of the chemical shielding and [(t) represents the time dependence of the interaction:
Here, C1, C2, S1, and S2 are constants that depend on the nature of the interaction (i.e. CS, dipolar) and Zr is the rotor angular frequency. Summing the CS-perturbed Zeeman Hamiltonian given in Equation [6] with the time-dependent dipolar Hamiltonian gives the total Hamiltonian
components of the dipoledipole coupling which depend on Euler angles defining the crystallite orientation with respect to the rotor frame. They are timeindependent. The spin part of the truncated dipolar Hamiltonian is
To transform the total Hamiltonian into the doubly rotating frame of reference defined by the Zeeman interactions, the propagator is
The Zeeman terms and the Îz terms of the dipolar Hamiltonian are unaffected by this rotation since it is about the z -axis, and so the desired transformation is:
The result of the transformation gives a periodic interaction frame dipolar Hamiltonian,
The parameter [(t) completely describes the time-dependence of a rotating solid. In order to understand some of the essential features of the RR experiment, it is convenient to assume negligible chemical shielding anisotropy. Under these conditions, Z1 and Z2, the CS-perturbed Zeeman angular frequencies, are independent of time. In addition, it is convenient to use the spherical tensor notation to describe the direct dipolar interaction. Thus, the total Hamiltonian is
where Z = Z1 − Z2. If we re-express the rotational resonance condition [2] in angular frequency units as nZr = Z , and if | RDDWr | 1, where Wr = νr−1 is the rotor period, then the time-independent terms vanish and the time average of Equation [14] over one rotor period is
where
Here, Ad(t) represents the spatial dependence of the dipolar Hamiltonian, and dm are Fourier
where m = ± 1, ± 2. The result of this exercise is that at rotational resonance, parts of the flip-flop term do not average to zero and will therefore contribute to the MAS NMR spectrum.
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Some qualitative results of an approximate theoretical treatment of rotational resonance are useful to examine. For n = 1 RR, the splitting of each peak is given by RDD/(2 ), or ∼ 0.35 RDD. For the n = 2 case, the splitting is RDD/4. The splitting decreases as the order of the RR increases. More rigorous treatments also indicate that the splitting decreases as the chemical shielding anisotropy increases. It is important to note that the observed line widths for homonuclear spin systems are not strictly independent of spinning speed; for a spin pair with differing isotropic chemical shifts, the line widths take on a Zr−2 dependence at high spinning speeds. When the RR condition is satisfied, a rapid exchange of Zeeman magnetization occurs in addition to dipolar broadening. We will not present a complete theoretical description of the origins of this exchange, but rather present some of the important approximate results of such a treatment. It is convenient to define
where T is the zero-quantum relaxation time constant, and are the resonant Fourier components associated with the flip-flop term of the dipolar Hamiltonian for RR of order n. To monitor the exchange of magnetization, one plots 〈Îz1 − Îz2〉 as a function of time. In the limit of very fast dephasing where T is relatively short and thus Λ2 0, the decay of magnetization is exponential:
energy level diagram for an isolated homonuclear two-spin system where the two nuclei have resonance frequencies Q and Q is shown in Figure 2. Transitions 1 and 2 correspond to the two isotropic peaks, which would be observed in a MAS NMR spectrum. The difference between the isotropic chemical shifts (or, alternatively, the energies) is, according to the diagram, equivalent to the angular frequency Z∆iso. As shown earlier, rotational resonance occurs when an integer multiple of the spinning frequency is equivalent to Z . In terms of the diagram, it is convenient to think of the mechanical rotation of the sample as supplying the necessary energy for zero-quantum coherence between the two intermediate energy levels. The fact that these two states are linked by mechanical rotation ensures that the dipolar interaction will be recoupled, and that exchange of Zeeman magnetization will occur rapidly. Figure 3 illustrates the exchange experiment, in which one of the transitions is selectively inverted, thus creating a nonequilibrium situation in which spins must relax so that the equilibrium Boltzmann populations are re-established. If we consider the diagram on the left to reflect the excess populations in arbitrary units as determined by the Boltzmann distribution, a selective inversion of transition 1 will result in the population distribution shown on the right. Transition 1 is inverted while the intensity of transition 2 remains unperturbed. Techniques for accomplishing this experimentally will be discussed in the next section. Once the inversion has been carried out, the diagram on the right shows a difference of five population units between the two intermediate energy levels. Rotational resonance provides the zero-quantum coherence necessary for an exchange
In the case of very slow dephasing where T2ZQ is relatively long and Λ2 0, the exchange of magnetization oscillates as it decays:
In practice, the parameters which influence the observed magnetization exchange curve include RDD, T , the magnitude of the principal components of the chemical shielding tensors, the relative orientation of the CS tensors with respect to r12, and the Jcoupling constant. A pictorial representation of rotational resonance and the exchange of Zeeman magnetization
At this point, it is instructive to provide a qualitative picture of the rotational resonance phenomenon. The
Figure 2 Simplified energy level diagram for two spin- nuclei with different isotropic chemical shifts. The two transitions are labelled ‘1’ and ‘2’, and their difference is greatly exaggerated. The energy of the zero-quantum transition is indicated, which corresponds to the mechanical energy supplied at RR. Here, J-coupling is ignored and dipolar coupling is not shown.
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Figure 3 Energy level and population distribution diagrams for two spin- nuclei. The circles indicate the excess population in arbitrary units relative to the least populated level. On the left, an equilibrium Boltzmann-type distribution is represented. Both transitions would show a signal of relative intensity +2. Upon inversion of transition 1 (at right) the populations related to this transition are switched, while the net difference in population for transition 2 is unchanged. Transition 1 would now show an inverted signal with relative intensity −2.
Figure 4 Pulse sequence for carrying out the RR experiment (see text). In the case of the magnetization exchange experiment, CP of the rare spins is followed by a flipback pulse on the rare spin channel, selective inversion of a particular resonance, and a variable delay before acquisition.
informative, and more sensitive to the magnitude of RDD, as will be shown. The exchange experiment
of Zeeman magnetization between these two levels. The zero-quantum relaxation which dampens this exchange is described by the time constant T .
Experimental techniques Pulse sequences and cross-polarization
The basic rotational resonance experiment can be as simple a single-pulse excitation, with the rate of MAS adjusted to satisfy the RR condition. A S/2 pulse followed by acquisition of the free induction decay (FID) will generate a spectrum with significant broadening of the two resonances concerned. Many typical applications involve 13C in the presence of 1H, and benefit from standard CP techniques. Figure 1 shows an example of the line broadening observed for the n = 1 RR condition for the 13C13C spin pair in Ph13CH213COOH, with CP. A typical pulse sequence for carrying out the RR experiment with selective inversion of a particular transition and CP is shown in Figure 4. Note that a flipback pulse is applied to the rare spin channel to store the magnetization along the z axis before carrying out the inversion. The efficiency of CP becomes sensitive to the spinning rate, particularly as Zr increases. One technique which attempts to circumvent this problem is known as variable amplitude cross-polarization (VACP), where the spin-locking pulses vary in amplitude. The goal of the RR experiment is the extraction of the homonuclear dipolar coupling constant, RDD. This can be done by carrying out lineshape simulations. However, in general this is not done because a Zeeman magnetization exchange experiment is more
To generate an exchange curve, one of the two resonances involved must be inverted selectively. By whichever technique a selective inversion is carried out, it is important that the other resonances not be perturbed. The most frequently used inversion techniques in RR experiments are a long, soft pulse or an asynchronous DANTE (delays alternating with nutation for tailored excitation) sequence. In cases where the CS anisotropy at one or both of the sites is comparable to the isotropic chemical shift difference between them, difficulties arise in carrying out the inversion with selectivity. Total sideband suppression pulse sequences combined with their time-reversed counterparts may be used to overcome the difficulties associated with large chemical shift anisotropies. Regardless of what technique is used to establish the initial condition of maximum polarization difference, the next step in the experiment is to allow the exchange of Zeeman magnetization for a variable time, Wm (see Figure 4), before applying a S/2 acquisition pulse. The equilibration of magnetization between the two sets of spins is described by the approximate Equation [17] or [18], depending on the system.
Applications and limitations As mentioned previously, the primary goal of the rotational resonance experiment is to determine the dipolar coupling constant, RDD, from which the internuclear distance, r, may be calculated. Carbon carbon separations as large as 6.8 Å have been successfully determined, which corresponds to measuring a coupling as small as 24 Hz. Occasionally, dihedral angle measurements have also been carried
SOLID-STATE NMR, ROTATIONAL RESONANCE 2141
out using RR. At higher order rotational resonances (i.e. n = 3 or n = 4), where CS anisotropy is more likely to be comparable to Qr, the lineshapes and exchange curves are more sensitive to the orientation of the chemical shielding tensors. In general, when simulations (of either line shapes or exchange curves) are performed, they depend on RDD and, to varying degrees, on the magnitudes of the principal components of the chemical shielding tensors, their orientations with respect to the internuclear vector and with respect to each other, the magnitude of the J-coupling, and the zero-quantum transverse relaxation time constant, T . Lineshape simulations
In order to effectively determine a dipolar coupling constant based on a lineshape simulation, the chemical shielding tensors and their orientations must be known, as well as the J-coupling constant, and T . To determine the principal components of the chemical shielding tensors, a MAS NMR spectrum acquired on a singly-labelled compound in the slowspinning regime may be used to emulate the powder pattern provided the isolated spin approximation is valid. In some cases it is also possible to determine the CS tensor components from a spectrum of the stationary sample. Determining the orientations of the CS tensors is a more involved process, although in some cases careful assumptions and clues from local symmetry may be helpful. In practice, a value for T is usually estimated from the observed line widths off the RR condition.
In many cases, not all these parameters are known for the specific spin system under investigation. Therefore, two techniques that may be employed when a lineshape simulation is desirable are (i) a simulation based on known chemical shielding tensors (V), J-coupling constants, and T values, where only RDD is varied; (ii) use of a calibration with respect to similar compounds, where RDD (or r itself) can be extracted analytically from the observed splitting of a resonance. For example, the calibration method (ii) has been employed for a series of 13C-labelled retinals containing vinylic and methyl carbons, shown in Figure 5. The three isotopomers were 13C-labelled at the (10,20), (11,20), and (12,20) positions. It must be emphasized that the required parameters (V, J, T and r) were known independently from X-ray and previous NMR studies. T was estimated using
Figure 5 Structure of the retinal studied using RR, with the labelled carbons indicated. See text for details. (Reprinted with permission of the American Chemical Society from Verdegem PJE, Helmle M, Lugtenburg J and de Groot HJM (1997) (Journal of the American Chemical Society, 119: 169–174).
Equation [19]. The goal of the calibration was to be able to employ a simple, analytic equation relating r to the observed broadening of the vinylic peaks at RR. To accomplish this, the ideal splitting presented above, RDD/(2 ), was plotted against the observed splitting, ∆Z. Simulations showed that ∆Z could be reliably reproduced, independent of the actual shape of the line. The resulting equation,
shows that the approximate theory fits well with experimental results in this case, and allows for a very straightforward determination of r from the observed splitting. The major advantage of using line shape simulations to extract the dipolar coupling constant, in general, is that the spectrometer time involved is less than that for the corresponding magnetization exchange experiment. For molecules similar to the retinal in Figure 5, where r is unknown, the analytical empirical Equation [20] can provide the information after a simple 1D NMR experiment. In spite of the results of the preceding example, the lineshape simulation method has rarely been used in practice, mainly because the RDD values are too small to result in splittings. In such cases, the exchange curve method discussed below is the standard technique for extracting the dipolar coupling constant under RR conditions. Exchange curves and simulations
By far the most common method for deriving structural information under RR conditions is through the analysis and simulation of a magnetization exchange curve. Once a suitable state of polarization difference has been achieved between the two sets of spins, 1 and 2, the delay time, Wm, is varied before applying a S/2 observe pulse and acquiring the spectrum (see Figure 4). Separate NMR experiments
2142 SOLID-STATE NMR, ROTATIONAL RESONANCE
must be performed to generate each point on a magnetization exchange plot. In order to extract the dipolar coupling constant or structural information, the observed magnetization decay must be simulated. Qualitative and relative distance information is more readily available than quantitative information since the exchange curve depends on the same parameters that the RR line shape depends on. Two common procedures for extracting Reff are: (i) comparison of the exchange curve with a series of exchange curves of model compounds for which r is known, and (ii) complete simulation of the exchange curve, where V, J, and T are known (or estimated). The magnetization due to naturally abundant NMR-active spins in the sample must be considered. This is done by subtracting the natural-abundance spectrum from that of the labelled sample. Failure to make such a correction could lead to an overestimation of r and an underestimation of T . A recent example of the application of RR to a structural problem will serve to illustrate its utility as a comparative tool. Figure 6 shows two peptide fragments in different conformations. This compound models the peptide AE1-42, a constituent of the amyloid plaques characteristic of Alzheimers disease. Rotational resonance MAS NMR was used in a qualitative fashion by Costa and co-workers to determine whether the amide conformation in the solid state was cis or trans. From previous
Figure 6 Fragments of the peptide E34-42 showing the cis and trans conformations. Also indicated is the orientation of the carbonyl carbon chemical shielding tensor, with V33 perpendicular to the plane. Note the different orientations of the C–C internuclear vector with respect to the CS tensor components. Reprinted with permission of the American Chemical Society from Costa PR, Kocisko DA, Sun BQ, Lansbury PT Jr and Griffin RG (1997) Journal of the American Chemical Society, 119: 10487–10493)
experiments, model compounds served to give the chemical shielding tensor orientations of the carbonyl carbon. The orientation of the internuclear vector connecting the two labelled carbon atoms with respect to the chemical shielding tensor of the carbonyl carbon is drastically different for the two conformers shown in Figure 6. Note that in the trans conformation, the internuclear vector lies nearly along the V11 component, while in the cis conformation it lies nearly along V22. The dipolar coupling for the two conformations should, however, be nearly identical. Hence, the variable of interest in this experiment is the CS tensor orientation. Experimental Zeeman magnetization exchange curves were generated and matched to simulated curves (Figure 7). It was found that theory matched experiment only when a trans geometry was assumed. The n = 2 RR experiment was used in this case because at higher spinning speed (i.e. n = 1), the orientations of the CS tensors become less influential in determining the course of the magnetization exchange. This example shows that RR experiments can be used for more than simply extracting the dipolar coupling constant and determining an accurate value for r12. In fact, the basic RR technique is probably
Figure 7 Zeeman magnetization exchange plot for the peptide E34-42 fragments shown in Figure 6. The open circles are experimentally determined data points; the solid lines result from simulations assuming a trans geometry; the dotted lines result from simulations assuming a cis geometry. Reproduced with permission of the American Chemical Society from Costa PR, Kocisko DA, Sun BQ, Lansbury PT Jr and Griffin RG (1997) Journal of the American Chemical Society 119: 10487–10493.
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Table 1
A summary of some homonuclear dipolar recoupling techniques.
Name
Acronym
Principle
Rotational resonance
RR
Recouples when the difference in chemical shift frequencies is an integer multiple of the MAS speed.
Dipolar recovery at the magic angle
DRAMA
In its simplest form, a pair of x and –x S pulses separated by a delay, W, results in an observable dipolar broadening (Tycko R and Dabbagh G (1991) Double-quantum filtering in magic-angle-spinning NMR spectroscopy: an approach to spectral simplification and molecular structure determination. Journal of the American Chemical Society 113: 9444–9448).
Simple excitation for the dephasing of the rotational-echo amplitudes
SEDRA
Synchronously applied pulses lead to signal dephasing for dipolar coupled spins (Gullion T and Vega S (1992) A simple magic angle spinning NMR experiment for the dephasing of rotational echoes of dipolar coupled homonuclear spin pairs. Chemical Physics Letters 194: 423–428).
Radio frequency driven dipolar recoupling
RFDR
Rotor-synchronized S-pulses reintroduces flip-flop term (Bennett AE, Ok JH, Griffin RG and Vega S (1992) Chemical shift correlation spectroscopy in rotating solids: radio frequency-driven dipolar recoupling and longitudinal exchange. Journal of Chemical Physics 96: 8624–8627).
Unified spin echo and magic echo
USEME
Spin-echo and magic-echo sequences are applied to recover the dipolar interaction (Fujiwara T, Ramamoorthy A, Nagayama K, Hioka K and Fujito T (1993) Dipolar HOHAHA under MAS conditions for solid-state NMR. Chemical Physics Letters 212: 81–84).
Combines rotation with nutation
CROWN
Dipolar dephasing occurs due to applied RF pulses (Joers JM, Rosanske R, Gullion T and Garbow JR (1994) Detection of dipolar interactions by CROWN NMR. Journal of Magnetic Resonance A106: 123–126).
Double quantum homonuclear rotary resonance
2Q1-HORROR RF field applied at half the rotation frequency in conjunction with RF pulses (Nielsen NC, Bildsøe H, Jakobsen HJ and Levitt MH (1994) Double-quantum homonuclear rotary resonance: efficient dipolar recovery in magic-angle-spinning nuclear magnetic resonance. Journal of Chemical Physics 101(3): 1805–1812).
Melding of spin-locking dipolar recovery at the magic angle
MELODRAMA
Rotor-synchronized 90° phase shifts of the applied spin-locking field (Sun B-Q, Costa PR, Kocisko D, Lansbury PT Jr and Griffin RG (1995) Internuclear distance measurements in solid state nuclear magnetic resonance: Dipolar recoupling via rotor synchronized spin locking. Journal of Chemical Physics 102: 702–707).
Rotational resonance in the R2TR tilted rotating frame
Application of an RF field allows selective recoupling when the chemical shift difference is small (Takegoshi K, Nomura K and Terao T (1995) Rotational resonance in the tilted rotating frame. Chemical Physics Letters 232: 424–428).
Sevenfold symmetric radio- C7 frequency pulse sequence
Seven phase-shifted RF pulse cycles lead to dipolar recoupling (Lee YK, Kurur ND, Helmle M, Johannessen OG, Nielsen NC and Levitt MH (1995) Efficient dipolar recoupling in the NMR of rotating solids. A sevenfold symmetric radiofrequency pulse sequence. Chemical Physics Letters 242: 304–309).
Dipolar recoupling with a windowless multipulse irradiation
DRAWS
Windowless DRAMA sequence (Gregory DM, Wolfe GM, Jarvie TP, Sheils JC and Drobny GP (1996) Double-quantum filtering in magic-angle-spinning NMR spectroscopy applied to DNA oligomers. Molecular Physics 89(6): 1835–1850).
Rotational resonance tickling
R2T
Ramped RF field during the variable delay removes the T dependence (Costa PR, Sun B and Griffin RG (1997) Rotational resonance tickling: accurate internuclear distance measurement in solids. Journal of the American Chemical Society 119: 10821–10830).
Adiabatic passage rotational resonance
APRR
MAS speed varied during CP mixing to achieve more complete polarization transfer (Verel R, Baldus M, Nijman M, van Os JWM and Meier BH (1997) Adiabatic homonuclear polarization transfer in magic-angle-spinning solid-state NMR. Chemical Physics Letters 280: 31–39).
Supercycled POST-C5
SPC-5
Fivefold symmetric pulse sequence leads to homonuclear dipolar recoupling (Hohwy M, Rienstra CM, Jaroniec CP, Griffin RG (1999) Journal of Chemical Physics 110: 7983–7992).
better suited to qualitative distance measurements such as in the example given. It is necessary to make a general comment regarding the influence of molecular motion on the measurement of dipolar coupling constants. In the
context of solid-state NMR, it is not r12 which is directly measured, but rather the dipolar coupling constant. Molecular librations and vibrations will cause a certain degree of averaging of the dipolar interaction and thus RDD. The net result of the
2144 SOLID-STATE NMR, ROTATIONAL RESONANCE
motional averaging of the dipolar coupling is that the calculated distances, r, will be too large. Finally, it is important to recognize that the dipolar coupling constant measured in any NMR experiment also has, in principle, a contribution from the anisotropy in the indirect spinspin coupling, ∆J. That is, only an effective dipolar coupling constant, Reff can be measured where
The last term in equation [21], ∆J/3, is generally ignored. Other homonuclear recoupling methods
Restoring the dipolar coupling between both heteronuclear and homonuclear spin pairs is of great interest. Rotational resonance applies strictly to homonuclear spin pairs, and Table 1 provides a brief overview of some of the other techniques available for recovering the dipolar coupling and extracting Reff for homonuclear spin pair from high-resolution MAS spectra.
Conclusions At present, RR is best suited for use as a qualitative probe into molecular structure rather than a quantitative one. In most experiments which have been done using the basic RR technique, the internuclear distances were known beforehand as a result of other investigations. Further developments of related RR techniques (such as rotational resonance tickling) may prove to be more useful in obtaining quantitative results. Still, the standard RR experiment is an excellent one for confirming distances between homonuclear spin pairs in a proposed structure.
List of symbols Ad(t) = spatial dependence of the dipolar Hamiltonian; B0 = external applied magnetic field; dm = Fourier components of the dipoledipole coupling; h = Planck constant; = Planck constant divided by 2π; = average dipolar Hamiltonian; DD = direct dipolar Hamiltonian operator; J = indirect spinspin coupling Hamiltonian; Z,CS = chemical shielding perturbed Zeeman Hamiltonian operator; Î− = lowering operator; Î+ = raising operator; Îi = spin angular momentum operator for spin i; Îzi = z-component of the spin angular momentum operator for spin i; J = indirect spinspin coupling constant; n = order of the rotational resonance; r12 = distance between spins 1 and 2; internuclear vector; RDD = direct dipolar coupling constant (in Hz);
Reff = observed dipolar coupling constant; t = time; T20 = spin term in the spherical tensor representation of the dipolar Hamiltonian; T = zero-quantum relaxation time constant; U = propagator; Ji = magnetogyric ratio of spin i; ∆J = anisotropy of the indirect spinspin interaction; T = angle between the applied field and the internuclear vector; Λ2 = dephasing parameter; P0 = permeability of free space; Qr = rotor frequency in Hz; Qi, Qiiso = isotropic resonant frequency of nucleus i (in Hz); Qrot = rotor frequency (in Hz); Q1/2 = line width at half-height (in Hz); [(t) = time-dependence of the NMR interactions as a result of sample rotation; Vi = chemical shielding tensor of spin i; Vii = principal component of the chemical shielding tensor (i = 1, 2, 3); Viso = isotropic chemical shielding constant; Wm = variable mixing time; ZB(n) = resonant Fourier components; Zi = CS-perturbed Zeeman angular frequency of spin i (in rad s−1); Zr = rotor frequency (in rad s−1); Z = difference in isotropic angular frequencies of spins 1 and 2. See also: Chemical Exchange Effects in NMR; High Resolution Solid State NMR, 13C; High Resolution Solid State NMR, 1H, 19F; NMR in Anisotropic Systems, Theory; NMR of Solids; NMR Pulse Sequences; NMR Relaxation Rates; Solid State NMR, Methods.
Further reading Andrew ER, Bradbury A, Eades RG and Wynn VT (1963) Nuclear cross-relaxation induced by specimen rotation. Physics Letters 4: 99100. Garbow JR and Gullion T (1995) Measurement of internuclear distances in biological solids by magic-anglespinning 13C NMR. In Beckmann N (ed), Carbon-13 NMR Spectroscopy of Biological Systems , pp. 65115. New York: Academic Press. Griffiths JM and Griffin RG (1993) Nuclear magnetic resonance methods for measuring dipolar couplings in rotating solids. Analytica Chimica Acta 283: 1081 1101. Peersen OB and Smith SO (1993) Rotational resonance NMR of biological membranes. Concepts in Magnetic Resonance 5: 303317. Raleigh DP, Levitt MH and Griffin RG (1988) Rotational resonance in solid-state NMR. Chemical Physics Letters 146: 7176. Smith SO (1993) Magic angle spinning NMR methods for internuclear distance measurements. Current Opinion in Structural Biology 3: 755759. Smith SO (1996) Magic angle spinning NMR as a tool for structural studies of membrane proteins. Magnetic Resonance Review 17: 126. Webb GA, Recent advances in solid-state NMR are reviewed annually in: Nuclear Magnetic Resonance: Specialist Periodical Reports. Cambridge: The Royal Society of Chemistry.
Fyfe C (1983) Solid State NMR for Chemists. Guelph: CFC Press. Harris RK and Grant DM (1996) Encyclopedia of Nuclear Magnetic Resonance. Chichester: Wiley. Mehring M (1983) High Resolution NMR in Solids, 2nd edn. Berlin: Springer-Verlag. Slichter CP (1983) Principles of Magnetic Resonance, 3rd edn. Springer: Berlin.
Schmidt-Rohr K and Spiess HW (1994) Multidimensional Solid-State NMR and Polymers. London: Academic Press. Traficante DD (ed) Concepts in Magnetic Resonance, An Educational Journal. New York: Wiley.
Solid-State NMR, Rotational Resonance David L Bryce and Roderick E Wasylishen, Dalhousie University, Halifax, Nova Scotia, Canada Copyright © 1999 Academic Press
Introduction One of the primary goals of solid-state NMR spectroscopists has been to develop techniques that yield NMR spectra of solid samples with resolution approaching that observed for samples in isotropic liquids. Rapidly spinning samples about an axis inclined at the magic angle (arccos(1/√3) = 54.7356 °) relative to the applied static magnetic field has been found to be highly effective in this regard. In addition, high-power decoupling of abundant spins (e.g. 1H) eliminates heteronuclear spinspin coupling interactions (direct dipolar and indirect J-coupling) involving the abundant spins when dilute spins are examined. The availability of commercial NMR instrumentation that permits users to apply these two techniques has contributed to spin- NMR becoming a routine method for examining a wide range of solid materials. Finally, cross-polarization (CP) from abundant spins to dilute spins has been important in improving the sensitivity of the dilute-spin NMR experiment. Ironically, it is sometimes desirable to selectively reintroduce interactions which are effectively averaged in the magic-angle-spinning (MAS) experiment. Dipolar coupling, for instance, may be recovered in the form of the direct dipolar coupling constant (RDD) between isolated spin pairs. The value of RDD is of interest due to its simple relationship with the distance separating two spins, r12 (Eqn [1])
MAGNETIC RESONANCE Applications where P0 is the permeability of free space, and Ji are the magnetogyric ratios of the nuclei under consideration. Rotational resonance (RR) is a MAS NMR technique which selectively restores the dipolar interaction between a homonuclear spin pair, thus allowing the determination of the dipolar coupling constant, RDD, and hence, the internuclear distance. Historically, the RR phenomenon was discovered by Andrew and co-workers in a 31P NMR study of phosphorus pentachloride, which consists of PCl 4+ and PCl6− units in the solid state. This group noticed that when the rate of sample spinning matched the difference in resonance frequencies of the nonequivalent phosphorus centres, their peaks broadened and the rate of cross-relaxation was enhanced. It is now known that if the RR condition is satisfied, direct dipolar coupling is restored selectively to a homonuclear spin pair. That is, if the sample spinning rate is adjusted to a frequency, Qr, such that
where n is an integer, generally 13, and Q and Q are the isotropic resonant frequencies of spins 1 and 2 respectively, then the two nuclei are said to be in RR. As a result, dipolar coupling between the nuclei is restored (via the flip-flop term in the dipolar Hamiltonian), and line broadening of the resonances at Q and Q is observed (see Figure 1). Additionally, a rapid oscillatory exchange of Zeeman
SOLID-STATE NMR, ROTATIONAL RESONANCE 2137
of the nucleus. These induced local fields are proportional to the applied field. In frequency units, the Hamiltonian operator which accounts for both the Zeeman interaction and this chemical shielding (CS) interaction is
where
Figure 1 Effect of RR on the 13C NMR line shape of Ph13CH213COOH. Spectra acquired at 4.7 T (50.3 MHz). Top spectrum at n = 1 RR, Qrot = 7207 Hz. Bottom spectrum off RR, Qrot = 10 000 Hz.
magnetization occurs. In fact, it is this exchange of magnetization rather than the line shape which is usually monitored in order to determine the dipolar coupling constant. In order to generate an exchange curve which may be analysed and simulated, one of the two resonances involved must be inverted selectively; the intensity difference of the peaks is then monitored as function of time. Obviously, it is highly desirable to develop techniques capable of recovering weak dipolar coupling constants from high-resolution NMR spectra obtained under MAS conditions. The focus of the present discussion will be to provide an overview of the basic RR scheme. First, the theory of RR will be outlined, followed by a discussion of the most important experimental techniques employed to measure dipolar coupling constants under conditions of RR. Finally, some examples that illustrate the applications and limitation of the techniques will be described.
Theory Restoring the dipolar interaction: a theoretical approach
The most important interaction in NMR results from the application of a large external magnetic field, *0, to the sample. Termed the Zeeman interaction, its effect on the normally degenerate nuclear spin energy levels is to cause them to split. The Zeeman levels are perturbed by local fields generated by the motion of electrons in the vicinity
and Viso is the isotropic chemical shielding constant. The interaction of interest in RR is the dipolar interaction, an orientationally dependent throughspace spinspin coupling, which leads to a perturbation of the CS-perturbed Zeeman energy levels. For a homonuclear two-spin system, the truncated dipolar Hamiltonian operator is given by the following:
Here, Î+ and Î− are the raising and lowering operators and T is the angle between the applied magnetic field and the internuclear vector, r12. The factor containing the raising and lowering operators is sometimes referred to as the flip-flop term. The final interaction that must be considered is the indirect spinspin coupling interaction, which is mediated by the intervening electrons. The indirect spinspin Hamiltonian, J, is often ignored because it is frequently considerably smaller than DD. Up until this point, we have implicitly assumed time independence of the interactions and their corresponding Hamiltonian operators. This assumption is valid for a rigid stationary sample. However, when the sample is spun rapidly, each of the internal Hamiltonians becomes time-dependent. For example, Z,CS becomes time-dependent when there is chemical shielding anisotropy due to the fact that the orientations of the chemical shielding tensors relative to the applied magnetic field change as the sample
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rotates. Then Equation [3] becomes
where (Vi,zz − Viiso) is a measure of the orientation dependence of the chemical shielding and [(t) represents the time dependence of the interaction:
Here, C1, C2, S1, and S2 are constants that depend on the nature of the interaction (i.e. CS, dipolar) and Zr is the rotor angular frequency. Summing the CS-perturbed Zeeman Hamiltonian given in Equation [6] with the time-dependent dipolar Hamiltonian gives the total Hamiltonian
components of the dipoledipole coupling which depend on Euler angles defining the crystallite orientation with respect to the rotor frame. They are timeindependent. The spin part of the truncated dipolar Hamiltonian is
To transform the total Hamiltonian into the doubly rotating frame of reference defined by the Zeeman interactions, the propagator is
The Zeeman terms and the Îz terms of the dipolar Hamiltonian are unaffected by this rotation since it is about the z -axis, and so the desired transformation is:
The result of the transformation gives a periodic interaction frame dipolar Hamiltonian,
The parameter [(t) completely describes the time-dependence of a rotating solid. In order to understand some of the essential features of the RR experiment, it is convenient to assume negligible chemical shielding anisotropy. Under these conditions, Z1 and Z2, the CS-perturbed Zeeman angular frequencies, are independent of time. In addition, it is convenient to use the spherical tensor notation to describe the direct dipolar interaction. Thus, the total Hamiltonian is
where Z = Z1 − Z2. If we re-express the rotational resonance condition [2] in angular frequency units as nZr = Z , and if | RDDWr | 1, where Wr = νr−1 is the rotor period, then the time-independent terms vanish and the time average of Equation [14] over one rotor period is
where
Here, Ad(t) represents the spatial dependence of the dipolar Hamiltonian, and dm are Fourier
where m = ± 1, ± 2. The result of this exercise is that at rotational resonance, parts of the flip-flop term do not average to zero and will therefore contribute to the MAS NMR spectrum.
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Some qualitative results of an approximate theoretical treatment of rotational resonance are useful to examine. For n = 1 RR, the splitting of each peak is given by RDD/(2 ), or ∼ 0.35 RDD. For the n = 2 case, the splitting is RDD/4. The splitting decreases as the order of the RR increases. More rigorous treatments also indicate that the splitting decreases as the chemical shielding anisotropy increases. It is important to note that the observed line widths for homonuclear spin systems are not strictly independent of spinning speed; for a spin pair with differing isotropic chemical shifts, the line widths take on a Zr−2 dependence at high spinning speeds. When the RR condition is satisfied, a rapid exchange of Zeeman magnetization occurs in addition to dipolar broadening. We will not present a complete theoretical description of the origins of this exchange, but rather present some of the important approximate results of such a treatment. It is convenient to define
where T is the zero-quantum relaxation time constant, and are the resonant Fourier components associated with the flip-flop term of the dipolar Hamiltonian for RR of order n. To monitor the exchange of magnetization, one plots 〈Îz1 − Îz2〉 as a function of time. In the limit of very fast dephasing where T is relatively short and thus Λ2 0, the decay of magnetization is exponential:
energy level diagram for an isolated homonuclear two-spin system where the two nuclei have resonance frequencies Q and Q is shown in Figure 2. Transitions 1 and 2 correspond to the two isotropic peaks, which would be observed in a MAS NMR spectrum. The difference between the isotropic chemical shifts (or, alternatively, the energies) is, according to the diagram, equivalent to the angular frequency Z∆iso. As shown earlier, rotational resonance occurs when an integer multiple of the spinning frequency is equivalent to Z . In terms of the diagram, it is convenient to think of the mechanical rotation of the sample as supplying the necessary energy for zero-quantum coherence between the two intermediate energy levels. The fact that these two states are linked by mechanical rotation ensures that the dipolar interaction will be recoupled, and that exchange of Zeeman magnetization will occur rapidly. Figure 3 illustrates the exchange experiment, in which one of the transitions is selectively inverted, thus creating a nonequilibrium situation in which spins must relax so that the equilibrium Boltzmann populations are re-established. If we consider the diagram on the left to reflect the excess populations in arbitrary units as determined by the Boltzmann distribution, a selective inversion of transition 1 will result in the population distribution shown on the right. Transition 1 is inverted while the intensity of transition 2 remains unperturbed. Techniques for accomplishing this experimentally will be discussed in the next section. Once the inversion has been carried out, the diagram on the right shows a difference of five population units between the two intermediate energy levels. Rotational resonance provides the zero-quantum coherence necessary for an exchange
In the case of very slow dephasing where T2ZQ is relatively long and Λ2 0, the exchange of magnetization oscillates as it decays:
In practice, the parameters which influence the observed magnetization exchange curve include RDD, T , the magnitude of the principal components of the chemical shielding tensors, the relative orientation of the CS tensors with respect to r12, and the Jcoupling constant. A pictorial representation of rotational resonance and the exchange of Zeeman magnetization
At this point, it is instructive to provide a qualitative picture of the rotational resonance phenomenon. The
Figure 2 Simplified energy level diagram for two spin- nuclei with different isotropic chemical shifts. The two transitions are labelled ‘1’ and ‘2’, and their difference is greatly exaggerated. The energy of the zero-quantum transition is indicated, which corresponds to the mechanical energy supplied at RR. Here, J-coupling is ignored and dipolar coupling is not shown.
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Figure 3 Energy level and population distribution diagrams for two spin- nuclei. The circles indicate the excess population in arbitrary units relative to the least populated level. On the left, an equilibrium Boltzmann-type distribution is represented. Both transitions would show a signal of relative intensity +2. Upon inversion of transition 1 (at right) the populations related to this transition are switched, while the net difference in population for transition 2 is unchanged. Transition 1 would now show an inverted signal with relative intensity −2.
Figure 4 Pulse sequence for carrying out the RR experiment (see text). In the case of the magnetization exchange experiment, CP of the rare spins is followed by a flipback pulse on the rare spin channel, selective inversion of a particular resonance, and a variable delay before acquisition.
informative, and more sensitive to the magnitude of RDD, as will be shown. The exchange experiment
of Zeeman magnetization between these two levels. The zero-quantum relaxation which dampens this exchange is described by the time constant T .
Experimental techniques Pulse sequences and cross-polarization
The basic rotational resonance experiment can be as simple a single-pulse excitation, with the rate of MAS adjusted to satisfy the RR condition. A S/2 pulse followed by acquisition of the free induction decay (FID) will generate a spectrum with significant broadening of the two resonances concerned. Many typical applications involve 13C in the presence of 1H, and benefit from standard CP techniques. Figure 1 shows an example of the line broadening observed for the n = 1 RR condition for the 13C13C spin pair in Ph13CH213COOH, with CP. A typical pulse sequence for carrying out the RR experiment with selective inversion of a particular transition and CP is shown in Figure 4. Note that a flipback pulse is applied to the rare spin channel to store the magnetization along the z axis before carrying out the inversion. The efficiency of CP becomes sensitive to the spinning rate, particularly as Zr increases. One technique which attempts to circumvent this problem is known as variable amplitude cross-polarization (VACP), where the spin-locking pulses vary in amplitude. The goal of the RR experiment is the extraction of the homonuclear dipolar coupling constant, RDD. This can be done by carrying out lineshape simulations. However, in general this is not done because a Zeeman magnetization exchange experiment is more
To generate an exchange curve, one of the two resonances involved must be inverted selectively. By whichever technique a selective inversion is carried out, it is important that the other resonances not be perturbed. The most frequently used inversion techniques in RR experiments are a long, soft pulse or an asynchronous DANTE (delays alternating with nutation for tailored excitation) sequence. In cases where the CS anisotropy at one or both of the sites is comparable to the isotropic chemical shift difference between them, difficulties arise in carrying out the inversion with selectivity. Total sideband suppression pulse sequences combined with their time-reversed counterparts may be used to overcome the difficulties associated with large chemical shift anisotropies. Regardless of what technique is used to establish the initial condition of maximum polarization difference, the next step in the experiment is to allow the exchange of Zeeman magnetization for a variable time, Wm (see Figure 4), before applying a S/2 acquisition pulse. The equilibration of magnetization between the two sets of spins is described by the approximate Equation [17] or [18], depending on the system.
Applications and limitations As mentioned previously, the primary goal of the rotational resonance experiment is to determine the dipolar coupling constant, RDD, from which the internuclear distance, r, may be calculated. Carbon carbon separations as large as 6.8 Å have been successfully determined, which corresponds to measuring a coupling as small as 24 Hz. Occasionally, dihedral angle measurements have also been carried
SOLID-STATE NMR, ROTATIONAL RESONANCE 2141
out using RR. At higher order rotational resonances (i.e. n = 3 or n = 4), where CS anisotropy is more likely to be comparable to Qr, the lineshapes and exchange curves are more sensitive to the orientation of the chemical shielding tensors. In general, when simulations (of either line shapes or exchange curves) are performed, they depend on RDD and, to varying degrees, on the magnitudes of the principal components of the chemical shielding tensors, their orientations with respect to the internuclear vector and with respect to each other, the magnitude of the J-coupling, and the zero-quantum transverse relaxation time constant, T . Lineshape simulations
In order to effectively determine a dipolar coupling constant based on a lineshape simulation, the chemical shielding tensors and their orientations must be known, as well as the J-coupling constant, and T . To determine the principal components of the chemical shielding tensors, a MAS NMR spectrum acquired on a singly-labelled compound in the slowspinning regime may be used to emulate the powder pattern provided the isolated spin approximation is valid. In some cases it is also possible to determine the CS tensor components from a spectrum of the stationary sample. Determining the orientations of the CS tensors is a more involved process, although in some cases careful assumptions and clues from local symmetry may be helpful. In practice, a value for T is usually estimated from the observed line widths off the RR condition.
In many cases, not all these parameters are known for the specific spin system under investigation. Therefore, two techniques that may be employed when a lineshape simulation is desirable are (i) a simulation based on known chemical shielding tensors (V), J-coupling constants, and T values, where only RDD is varied; (ii) use of a calibration with respect to similar compounds, where RDD (or r itself) can be extracted analytically from the observed splitting of a resonance. For example, the calibration method (ii) has been employed for a series of 13C-labelled retinals containing vinylic and methyl carbons, shown in Figure 5. The three isotopomers were 13C-labelled at the (10,20), (11,20), and (12,20) positions. It must be emphasized that the required parameters (V, J, T and r) were known independently from X-ray and previous NMR studies. T was estimated using
Figure 5 Structure of the retinal studied using RR, with the labelled carbons indicated. See text for details. (Reprinted with permission of the American Chemical Society from Verdegem PJE, Helmle M, Lugtenburg J and de Groot HJM (1997) (Journal of the American Chemical Society, 119: 169–174).
Equation [19]. The goal of the calibration was to be able to employ a simple, analytic equation relating r to the observed broadening of the vinylic peaks at RR. To accomplish this, the ideal splitting presented above, RDD/(2 ), was plotted against the observed splitting, ∆Z. Simulations showed that ∆Z could be reliably reproduced, independent of the actual shape of the line. The resulting equation,
shows that the approximate theory fits well with experimental results in this case, and allows for a very straightforward determination of r from the observed splitting. The major advantage of using line shape simulations to extract the dipolar coupling constant, in general, is that the spectrometer time involved is less than that for the corresponding magnetization exchange experiment. For molecules similar to the retinal in Figure 5, where r is unknown, the analytical empirical Equation [20] can provide the information after a simple 1D NMR experiment. In spite of the results of the preceding example, the lineshape simulation method has rarely been used in practice, mainly because the RDD values are too small to result in splittings. In such cases, the exchange curve method discussed below is the standard technique for extracting the dipolar coupling constant under RR conditions. Exchange curves and simulations
By far the most common method for deriving structural information under RR conditions is through the analysis and simulation of a magnetization exchange curve. Once a suitable state of polarization difference has been achieved between the two sets of spins, 1 and 2, the delay time, Wm, is varied before applying a S/2 observe pulse and acquiring the spectrum (see Figure 4). Separate NMR experiments
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must be performed to generate each point on a magnetization exchange plot. In order to extract the dipolar coupling constant or structural information, the observed magnetization decay must be simulated. Qualitative and relative distance information is more readily available than quantitative information since the exchange curve depends on the same parameters that the RR line shape depends on. Two common procedures for extracting Reff are: (i) comparison of the exchange curve with a series of exchange curves of model compounds for which r is known, and (ii) complete simulation of the exchange curve, where V, J, and T are known (or estimated). The magnetization due to naturally abundant NMR-active spins in the sample must be considered. This is done by subtracting the natural-abundance spectrum from that of the labelled sample. Failure to make such a correction could lead to an overestimation of r and an underestimation of T . A recent example of the application of RR to a structural problem will serve to illustrate its utility as a comparative tool. Figure 6 shows two peptide fragments in different conformations. This compound models the peptide AE1-42, a constituent of the amyloid plaques characteristic of Alzheimers disease. Rotational resonance MAS NMR was used in a qualitative fashion by Costa and co-workers to determine whether the amide conformation in the solid state was cis or trans. From previous
Figure 6 Fragments of the peptide E34-42 showing the cis and trans conformations. Also indicated is the orientation of the carbonyl carbon chemical shielding tensor, with V33 perpendicular to the plane. Note the different orientations of the C–C internuclear vector with respect to the CS tensor components. Reprinted with permission of the American Chemical Society from Costa PR, Kocisko DA, Sun BQ, Lansbury PT Jr and Griffin RG (1997) Journal of the American Chemical Society, 119: 10487–10493)
experiments, model compounds served to give the chemical shielding tensor orientations of the carbonyl carbon. The orientation of the internuclear vector connecting the two labelled carbon atoms with respect to the chemical shielding tensor of the carbonyl carbon is drastically different for the two conformers shown in Figure 6. Note that in the trans conformation, the internuclear vector lies nearly along the V11 component, while in the cis conformation it lies nearly along V22. The dipolar coupling for the two conformations should, however, be nearly identical. Hence, the variable of interest in this experiment is the CS tensor orientation. Experimental Zeeman magnetization exchange curves were generated and matched to simulated curves (Figure 7). It was found that theory matched experiment only when a trans geometry was assumed. The n = 2 RR experiment was used in this case because at higher spinning speed (i.e. n = 1), the orientations of the CS tensors become less influential in determining the course of the magnetization exchange. This example shows that RR experiments can be used for more than simply extracting the dipolar coupling constant and determining an accurate value for r12. In fact, the basic RR technique is probably
Figure 7 Zeeman magnetization exchange plot for the peptide E34-42 fragments shown in Figure 6. The open circles are experimentally determined data points; the solid lines result from simulations assuming a trans geometry; the dotted lines result from simulations assuming a cis geometry. Reproduced with permission of the American Chemical Society from Costa PR, Kocisko DA, Sun BQ, Lansbury PT Jr and Griffin RG (1997) Journal of the American Chemical Society 119: 10487–10493.
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Table 1
A summary of some homonuclear dipolar recoupling techniques.
Name
Acronym
Principle
Rotational resonance
RR
Recouples when the difference in chemical shift frequencies is an integer multiple of the MAS speed.
Dipolar recovery at the magic angle
DRAMA
In its simplest form, a pair of x and –x S pulses separated by a delay, W, results in an observable dipolar broadening (Tycko R and Dabbagh G (1991) Double-quantum filtering in magic-angle-spinning NMR spectroscopy: an approach to spectral simplification and molecular structure determination. Journal of the American Chemical Society 113: 9444–9448).
Simple excitation for the dephasing of the rotational-echo amplitudes
SEDRA
Synchronously applied pulses lead to signal dephasing for dipolar coupled spins (Gullion T and Vega S (1992) A simple magic angle spinning NMR experiment for the dephasing of rotational echoes of dipolar coupled homonuclear spin pairs. Chemical Physics Letters 194: 423–428).
Radio frequency driven dipolar recoupling
RFDR
Rotor-synchronized S-pulses reintroduces flip-flop term (Bennett AE, Ok JH, Griffin RG and Vega S (1992) Chemical shift correlation spectroscopy in rotating solids: radio frequency-driven dipolar recoupling and longitudinal exchange. Journal of Chemical Physics 96: 8624–8627).
Unified spin echo and magic echo
USEME
Spin-echo and magic-echo sequences are applied to recover the dipolar interaction (Fujiwara T, Ramamoorthy A, Nagayama K, Hioka K and Fujito T (1993) Dipolar HOHAHA under MAS conditions for solid-state NMR. Chemical Physics Letters 212: 81–84).
Combines rotation with nutation
CROWN
Dipolar dephasing occurs due to applied RF pulses (Joers JM, Rosanske R, Gullion T and Garbow JR (1994) Detection of dipolar interactions by CROWN NMR. Journal of Magnetic Resonance A106: 123–126).
Double quantum homonuclear rotary resonance
2Q1-HORROR RF field applied at half the rotation frequency in conjunction with RF pulses (Nielsen NC, Bildsøe H, Jakobsen HJ and Levitt MH (1994) Double-quantum homonuclear rotary resonance: efficient dipolar recovery in magic-angle-spinning nuclear magnetic resonance. Journal of Chemical Physics 101(3): 1805–1812).
Melding of spin-locking dipolar recovery at the magic angle
MELODRAMA
Rotor-synchronized 90° phase shifts of the applied spin-locking field (Sun B-Q, Costa PR, Kocisko D, Lansbury PT Jr and Griffin RG (1995) Internuclear distance measurements in solid state nuclear magnetic resonance: Dipolar recoupling via rotor synchronized spin locking. Journal of Chemical Physics 102: 702–707).
Rotational resonance in the R2TR tilted rotating frame
Application of an RF field allows selective recoupling when the chemical shift difference is small (Takegoshi K, Nomura K and Terao T (1995) Rotational resonance in the tilted rotating frame. Chemical Physics Letters 232: 424–428).
Sevenfold symmetric radio- C7 frequency pulse sequence
Seven phase-shifted RF pulse cycles lead to dipolar recoupling (Lee YK, Kurur ND, Helmle M, Johannessen OG, Nielsen NC and Levitt MH (1995) Efficient dipolar recoupling in the NMR of rotating solids. A sevenfold symmetric radiofrequency pulse sequence. Chemical Physics Letters 242: 304–309).
Dipolar recoupling with a windowless multipulse irradiation
DRAWS
Windowless DRAMA sequence (Gregory DM, Wolfe GM, Jarvie TP, Sheils JC and Drobny GP (1996) Double-quantum filtering in magic-angle-spinning NMR spectroscopy applied to DNA oligomers. Molecular Physics 89(6): 1835–1850).
Rotational resonance tickling
R2T
Ramped RF field during the variable delay removes the T dependence (Costa PR, Sun B and Griffin RG (1997) Rotational resonance tickling: accurate internuclear distance measurement in solids. Journal of the American Chemical Society 119: 10821–10830).
Adiabatic passage rotational resonance
APRR
MAS speed varied during CP mixing to achieve more complete polarization transfer (Verel R, Baldus M, Nijman M, van Os JWM and Meier BH (1997) Adiabatic homonuclear polarization transfer in magic-angle-spinning solid-state NMR. Chemical Physics Letters 280: 31–39).
Supercycled POST-C5
SPC-5
Fivefold symmetric pulse sequence leads to homonuclear dipolar recoupling (Hohwy M, Rienstra CM, Jaroniec CP, Griffin RG (1999) Journal of Chemical Physics 110: 7983–7992).
better suited to qualitative distance measurements such as in the example given. It is necessary to make a general comment regarding the influence of molecular motion on the measurement of dipolar coupling constants. In the
context of solid-state NMR, it is not r12 which is directly measured, but rather the dipolar coupling constant. Molecular librations and vibrations will cause a certain degree of averaging of the dipolar interaction and thus RDD. The net result of the
2144 SOLID-STATE NMR, ROTATIONAL RESONANCE
motional averaging of the dipolar coupling is that the calculated distances, r, will be too large. Finally, it is important to recognize that the dipolar coupling constant measured in any NMR experiment also has, in principle, a contribution from the anisotropy in the indirect spinspin coupling, ∆J. That is, only an effective dipolar coupling constant, Reff can be measured where
The last term in equation [21], ∆J/3, is generally ignored. Other homonuclear recoupling methods
Restoring the dipolar coupling between both heteronuclear and homonuclear spin pairs is of great interest. Rotational resonance applies strictly to homonuclear spin pairs, and Table 1 provides a brief overview of some of the other techniques available for recovering the dipolar coupling and extracting Reff for homonuclear spin pair from high-resolution MAS spectra.
Conclusions At present, RR is best suited for use as a qualitative probe into molecular structure rather than a quantitative one. In most experiments which have been done using the basic RR technique, the internuclear distances were known beforehand as a result of other investigations. Further developments of related RR techniques (such as rotational resonance tickling) may prove to be more useful in obtaining quantitative results. Still, the standard RR experiment is an excellent one for confirming distances between homonuclear spin pairs in a proposed structure.
List of symbols Ad(t) = spatial dependence of the dipolar Hamiltonian; B0 = external applied magnetic field; dm = Fourier components of the dipoledipole coupling; h = Planck constant; = Planck constant divided by 2π; = average dipolar Hamiltonian; DD = direct dipolar Hamiltonian operator; J = indirect spinspin coupling Hamiltonian; Z,CS = chemical shielding perturbed Zeeman Hamiltonian operator; Î− = lowering operator; Î+ = raising operator; Îi = spin angular momentum operator for spin i; Îzi = z-component of the spin angular momentum operator for spin i; J = indirect spinspin coupling constant; n = order of the rotational resonance; r12 = distance between spins 1 and 2; internuclear vector; RDD = direct dipolar coupling constant (in Hz);
Reff = observed dipolar coupling constant; t = time; T20 = spin term in the spherical tensor representation of the dipolar Hamiltonian; T = zero-quantum relaxation time constant; U = propagator; Ji = magnetogyric ratio of spin i; ∆J = anisotropy of the indirect spinspin interaction; T = angle between the applied field and the internuclear vector; Λ2 = dephasing parameter; P0 = permeability of free space; Qr = rotor frequency in Hz; Qi, Qiiso = isotropic resonant frequency of nucleus i (in Hz); Qrot = rotor frequency (in Hz); Q1/2 = line width at half-height (in Hz); [(t) = time-dependence of the NMR interactions as a result of sample rotation; Vi = chemical shielding tensor of spin i; Vii = principal component of the chemical shielding tensor (i = 1, 2, 3); Viso = isotropic chemical shielding constant; Wm = variable mixing time; ZB(n) = resonant Fourier components; Zi = CS-perturbed Zeeman angular frequency of spin i (in rad s−1); Zr = rotor frequency (in rad s−1); Z = difference in isotropic angular frequencies of spins 1 and 2. See also: Chemical Exchange Effects in NMR; High Resolution Solid State NMR, 13C; High Resolution Solid State NMR, 1H, 19F; NMR in Anisotropic Systems, Theory; NMR of Solids; NMR Pulse Sequences; NMR Relaxation Rates; Solid State NMR, Methods.
Further reading Andrew ER, Bradbury A, Eades RG and Wynn VT (1963) Nuclear cross-relaxation induced by specimen rotation. Physics Letters 4: 99100. Garbow JR and Gullion T (1995) Measurement of internuclear distances in biological solids by magic-anglespinning 13C NMR. In Beckmann N (ed), Carbon-13 NMR Spectroscopy of Biological Systems , pp. 65115. New York: Academic Press. Griffiths JM and Griffin RG (1993) Nuclear magnetic resonance methods for measuring dipolar couplings in rotating solids. Analytica Chimica Acta 283: 1081 1101. Peersen OB and Smith SO (1993) Rotational resonance NMR of biological membranes. Concepts in Magnetic Resonance 5: 303317. Raleigh DP, Levitt MH and Griffin RG (1988) Rotational resonance in solid-state NMR. Chemical Physics Letters 146: 7176. Smith SO (1993) Magic angle spinning NMR methods for internuclear distance measurements. Current Opinion in Structural Biology 3: 755759. Smith SO (1996) Magic angle spinning NMR as a tool for structural studies of membrane proteins. Magnetic Resonance Review 17: 126. Webb GA, Recent advances in solid-state NMR are reviewed annually in: Nuclear Magnetic Resonance: Specialist Periodical Reports. Cambridge: The Royal Society of Chemistry.