Analytical theory of γ-encoded double-quantum recoupling sequences in solid-state nuclear magnetic resonance
Introduction
Solid-state NMR is one of the most powerful experimental methods for addressing molecular structural problems, especially for insoluble or poorly crystalline systems. Many applications of solid-state NMR in polycrystalline or disordered materials employ a combination of methods: (i) magic-angle sample spinning [1], [2] for improving resolution and sensitivity, (ii) cross-polarization [3], [4] (CP) to enhance the NMR signals from weakly magnetic nuclear isotopes, and (iii) recoupling techniques [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] to reintroduce informative nuclear spin interactions which are averaged out by MAS.
Recoupling sequences employ radiofrequency pulse sequences which are synchronized with the sample rotation, in order to impede the averaging effect of the magic-angle-spinning on selected nuclear spin interactions. The through-space dipole–dipole coupling is usually the target of recoupling sequences, since this coupling encodes important geometric information through its dependence on the inverse cube of the internuclear distance [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The chemical shift anisotropy [21], or a combination of interactions [1], [22] may also be recoupled. The design of recoupling pulse sequences is facilitated by the use of symmetry theory [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
In most cases, the magnitude of a recoupled spin interaction depends on the relative orientation of the local molecular environment and the sample holder. It is convenient to discuss this dependence using the Euler angle triplet ΩMR = {αMR, βMR, γMR} which defines the relative orientation of a reference frame M which is fixed with respect to the molecular framework, and a reference frame R fixed with respect to the rotating sample holder. The z-axis of frame R is defined to be along the sample rotation axis. In a disordered sample, the three Euler angles ΩMR = {αMR, βMR, γMR} are random, isotropically distributed variables. This paper uses the notation and conventions for Euler angles and reference frames given in Ref. [11].
In general, a recoupling sequence leads to a first-order average Hamiltonian for a recoupled interaction Λ of the following form:where is a spherical tensor operator of rank λ and component index μ, defined with respect to rotations of the resonant nuclear spins. The rank λ and component index μ depend on the nature of the recoupling pulse sequence. For example, double-quantum dipolar recoupling [5], [6], [7], [8], [9], [17], [18] has λ = 2 and μ = 2; zero-quantum homonuclear dipolar recoupling [13], [15] has λ = 2 and μ = 0. Heteronuclear dipolar recoupling of the REDOR type [23], [24] has λ = 1 and μ = 0.
The amplitude of the recoupled interaction is given by the complex number , which depends on the orientation ΩMR. The form of this orientation-dependence also depends on the recoupling sequence. In general, both the magnitude and the phase of the amplitude depends on any combination of the three angles ΩMR = {αMR, βMR, γMR}. However, for an important class of recoupling sequences, the recoupling amplitude has the form:where the phase angle is a function of the single Euler angle γMR, while the magnitude is independent of γMR. Pulse sequences which generate average Hamiltonians obeying Eq. (2) are termed γ-encoded [6]. Examples of γ-encoded recoupling phenomena include rotational resonance [25], [26], homonuclear rotary resonance (HORROR) [6], and a range of symmetry-based recoupling sequences [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], including C7 [7] and POST-C7 [8]. Recoupling sequences which are not γ-encoded include REDOR [23], [24], TEDOR [27], DRAMA [5], RFDR [28], and some supercycled symmetry-based recoupling sequences such as SR26 [17].
The attribute of γ-encoding provides both advantages and disadvantages. In general, the reduced orientation-dependence of γ-encoded pulse sequences leads to stronger oscillations of the NMR signal as a function of the recoupling interval, which allows a more accurate determination of the recoupled interaction magnitude. In the case of double-quantum recoupling, the maximum achievable double-quantum-filtering efficiency in powder samples is significantly higher for γ-encoded sequences than for non-γ-encoded sequences. In addition, γ-encoded sequences allow a finer time-resolution of the recoupling intervals compared to non-γ-encoded sequences [20]. On the other hand, γ-encoded sequences tend to be less robust at long times compared to non-γ-encoded sequences [17].
The orientation-dependence of the recoupling sequence leads to an orientation-dependent NMR signal. Consider an NMR experiment involving a recoupling sequence of duration τ. The powder-average NMR signal is an orientational average of the form:where the form of s(τ,ΩMR) depends on the recoupling sequence, the orientational angles ΩMR, and the experimental protocol, as discussed below. Average Hamiltonian theory often provides a closed analytical expression for the signal from a given molecular orientation, s(τ,ΩMR), under suitable approximations. In some cases, closed analytical expressions are also available for the powder average signal in Eq. (3). Such analytical solutions are very useful for molecular structural studies since they allow a rapid fitting of the experimental data to structural constraints. For example, analytical formulae based on quarter-integer-order Bessel functions were derived by Mueller for the case of the REDOR dephasing curve [29], [30], [31]. Similar functions were derived for the case of the non-γ-encoded supercycled SR26 sequence [17]. These analytical solutions were essential for the structure determination of network solids by double-quantum 29Si NMR, since in that case many thousands of structural models were tested by comparison of theory and experiment [32], [33].
In the case of pulse sequences including REDOR [23], [24] and SR26 [17], which are not γ-encoded, a suitable choice of reference frame reduces the triple integral in Eq. (3) to a double integral over the angles βMR and γMR. As shown by Mueller [29], [30], [31], this double integral may be expressed as a closed expression involving quarter-integer-order Bessel functions. The analytical expressions may be evaluated very rapidly, allowing the rapid fitting of molecular structure parameters.
In the case of γ-encoded sequences such as C7 [7], under the specific experimental protocols discussed below, the powder-average double-quantum-filtered NMR signal may be reduced to a single integral over the angle βMR. Although this single integral is superficially simpler than the double integral required for non-γ-encoded sequences, a closed analytical form for the powder average γ-encoded signal has not been available, although an expression involving an infinite Bessel series has been reported [29]. The lack of a closed analytical form has impeded the application of γ-encoded recoupling sequences in iterative molecular structure determination protocols.
In this paper, we provide closed solutions for powder-average γ-encoded NMR signals in terms of Fresnel functions. We compare the Fresnel solutions with numerical simulations, examining the effect of chemical shift anisotropy and rf inhomogeneity in typical experimental regimes. We show that the analytical solutions may be used to extract molecular structure information rapidly and reliably from experimental double-quantum 13C data.
Section snippets
Symmetry-based recoupling
The principles of symmetry-based recoupling sequences have been described in many other places [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and need only be summarized briefly here. There are two major classes of symmetry-based recoupling sequence, denoted and . A pulse sequence with the symmetry or is composed of N elements {ε0ε1… εN−1}, each of which has the same duration τE = nτr/N, where τr is a period of the magic-angle sample rotation, τr
Average Hamiltonian
The average Hamiltonian theory [36] of symmetry-based double-quantum recoupling sequences leads to the following first-order average Hamiltonian, for the case of isolated pairs of spins-1/2, Ij and Ik:where the spherical tensor operators are given by . Here t0 is the time point at which the recoupling sequence is initiated and Jjk is the scalar coupling constant between j and k.
For simplicity, we choose the molecular
Analytical solutions
Special cases of Eq. (29) have been solved by Mueller et al. [29], [30], [31], using Bessel functions [38]. However, the solutions achieved were not in a closed form. Here, we report closed-form solutions in terms of Fresnel functions [39], obtained with the help of Mathematica version 5.2 [40].
After several manual substitutions of variables, Mathematica finds the following solution for Eq. (29):where
Numerical simulations
The analytical solutions were validated by comparisons with accurate numerical simulations of the spin dynamics using realistic parameters. The simulations shown in Fig. 3, Fig. 4 used the spin system parameters defined in Table 1, which are typical for the 13C-labelled retinals used in retinal protein studies. Unless stated otherwise, the simulated magnetic field strength was 9.4 T, and the simulated spinning frequency was 11.000 kHz. The simulated recoupling sequence had the symmetry ,
Experimental results
A sample of [9,10-13C2]-all-E-retinal was obtained as a by-product in the synthesis of [9,10-13C2]-11Z-retinal [44]. The experiments described below were performed on a Varian Infinity + spectrometer, using 30 mg of 10% labeled [9,10-13C2]-all-E-retinal powder packed into a 4mm zirconium oxide rotor, spinning at 10.00 kHz in a magnetic field of 9.39 T (400 MHz proton Larmor frequency).
Longitudinal 13C magnetization was generated by ramped cross-polarization [34] of duration 1.6 ms from the abundant 1
Conclusions
We have derived closed analytical solutions for the powder-average signal amplitudes involved in γ-encoded double-quantum recoupling experiments. These solutions may be evaluated much more rapidly than explicit spin dynamical calculations, which greatly facilitates the determination of dipole–dipole coupling constants from experimental data. The analytical formulae were applied to a dipole coupling estimation for the 13C2 spin pairs in [9,10-13C2]-all-E-retinal. The dipolar coupling estimate
Acknowledgments
This research was supported by the EPSRC (UK). We thank Ole G. Johannessen for experimental support, Marina Carravetta for discussions, and Giulia Mollica for bringing previous NMR applications of Fresnel integrals [41] to our attention.
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