^{1}, Marina Carravetta

^{2}, Jacco D. van Beek

^{2,a)}, Wai Cheu Lai

^{2}and Malcolm H. Levitt

^{2,b)}

### Abstract

We present the theoretical principles of supercycled symmetry-based recoupling sequences in solid-state magic-angle-spinning NMR. We discuss the construction procedure of the SR26 pulse sequence, which is a particularly robust sequence for double-quantum homonuclear dipole-dipole recoupling. The supercycle removes destructive higher-order average Hamiltonian terms and renders the sequence robust over long time intervals. We demonstrate applications of the SR26 sequence to double-quantum spectroscopy, homonuclear spin counting, and determination of the relative orientations of chemical shiftanisotropytensors.

This research was supported by the ERSRC (UK) and the NSF (Norway). The authors would like to thank H. Luthman for synthesis of the -all--retinal, A. Sebald for loaning the DAF sample, and O. G. Johannessen for instrumental support.

I. INTRODUCTION

II. THEORY

A. Symmetry-based recoupling

1. Selection rule analysis

2. Chemical shift interference

3. Instrumental phase errors

4. Effective Hamiltonian

B. Modified cycles

1. -shifted cycles

2. Phase-inverted cycles

3. -shifted phase-inverted cycles

C. Supercycles

1. The supercycle

2. The supercycle

3. The supercycle

D. Double-quantum filtering

1. Symmetrical protocol

2. Constant-time protocol

III. NUMERICAL CALCULATIONS

A. Effective Hamiltonian

1. Single cycle

2. The supercycle

3. The supercycle

4. The supercycle

B. Double-quantum trajectories

1. Single cycles

2. Supercycles

IV. METHODS

A. Samples

B. NMR experiments

V. APPLICATIONS

A. Determination of distances

B. Determination of relative CSA orientations

C. Multiple-quantum spin counting

VI. CONCLUSIONS

### Key Topics

- Chemical shifts
- 54.0
- Tensor methods
- 19.0
- Sequence analysis
- 15.0
- Nuclear magnetic resonance
- 14.0
- Anisotropy
- 13.0

## Figures

Construction of the SR26 supercycle. (a) The inversion element is a composite pulse of the form . (b) Two elements are derived from by (i) imposing an overall phase shift of , leading to , or (ii) changing the sign of all phases, followed by an overall phase shift of , leading to . In the case of SR26, the symmetry numbers are and , and . (c) The cycle is constructed by concatenating 13 pairs. The total sequence duration is equal to four rotor periods. (d) The SR26 supercycle is constructed by concatenating with the cycle (derived by changing the sign of all phases), the cycle (derived from by an overall phase shift), and the cycle (derived from by an overall phase shift as well as a change in sign of all phases). The SR26 supercycle has a duration of 16 rotor periods.

Construction of the SR26 supercycle. (a) The inversion element is a composite pulse of the form . (b) Two elements are derived from by (i) imposing an overall phase shift of , leading to , or (ii) changing the sign of all phases, followed by an overall phase shift of , leading to . In the case of SR26, the symmetry numbers are and , and . (c) The cycle is constructed by concatenating 13 pairs. The total sequence duration is equal to four rotor periods. (d) The SR26 supercycle is constructed by concatenating with the cycle (derived by changing the sign of all phases), the cycle (derived from by an overall phase shift), and the cycle (derived from by an overall phase shift as well as a change in sign of all phases). The SR26 supercycle has a duration of 16 rotor periods.

Theoretical double-quantum filtering trajectories for SR26 recoupling. (a) Theoretical trajectories for the symmetric procedure, as given in Eq. (54), for a selection of distances. (b) Theoretical trajectories for the constant time procedure, as given in Eq. (57), for the case . Each curve is normalized against its value for .

Theoretical double-quantum filtering trajectories for SR26 recoupling. (a) Theoretical trajectories for the symmetric procedure, as given in Eq. (54), for a selection of distances. (b) Theoretical trajectories for the constant time procedure, as given in Eq. (57), for the case . Each curve is normalized against its value for .

Projections of the effective Hamiltonian onto two different two-dimensional subspaces, for four different pulse sequences. The numerically evaluated effective Hamiltonian for each molecular orientation is represented by a point. Repetition of the calculation for many molecular orientations generates a cloud. Left column: projections onto real and imaginary double-quantum operators [Eq. (62)]. Right column: projections onto angular momentum operators along the and axes [Eq. (63)]. [(a) and (b)] A single sequence. [(c) and (d)] A supercycle. [(e) and (f)] A supercycle. [(g) and (h)] A supercycle. All simulations were performed using the spin system parameters in Table I, at an external magnetic field of , a spinning frequency of , and a rf nutation frequency of .

Projections of the effective Hamiltonian onto two different two-dimensional subspaces, for four different pulse sequences. The numerically evaluated effective Hamiltonian for each molecular orientation is represented by a point. Repetition of the calculation for many molecular orientations generates a cloud. Left column: projections onto real and imaginary double-quantum operators [Eq. (62)]. Right column: projections onto angular momentum operators along the and axes [Eq. (63)]. [(a) and (b)] A single sequence. [(c) and (d)] A supercycle. [(e) and (f)] A supercycle. [(g) and (h)] A supercycle. All simulations were performed using the spin system parameters in Table I, at an external magnetic field of , a spinning frequency of , and a rf nutation frequency of .

Numerical simulations of symmetric double-quantum filtering trajectory functions as defined in Eq. (51). Unless stated, all simulations use the parameters in Table I, with a magnetic field of , a spinning frequency of , and a rf nutation frequency of . (a) Trajectories without supercycling: (i) ideal trajectory for repetitions of the sequence, omitting all chemical shift terms; (ii) calculated trajectory for but including all parameters in Table I, (iii), as in (ii), but adjusting the rf phases by , as defined in Eq. (A10). (b) Trajectories for supercycles: (iv) calculated trajectory for the supercycle; (v) calculated trajectory for the supercycle; (vi) calculated trajectory for the supercycle; and (vii) result of the analytical formula in Eq. (54).

Numerical simulations of symmetric double-quantum filtering trajectory functions as defined in Eq. (51). Unless stated, all simulations use the parameters in Table I, with a magnetic field of , a spinning frequency of , and a rf nutation frequency of . (a) Trajectories without supercycling: (i) ideal trajectory for repetitions of the sequence, omitting all chemical shift terms; (ii) calculated trajectory for but including all parameters in Table I, (iii), as in (ii), but adjusting the rf phases by , as defined in Eq. (A10). (b) Trajectories for supercycles: (iv) calculated trajectory for the supercycle; (v) calculated trajectory for the supercycle; (vi) calculated trajectory for the supercycle; and (vii) result of the analytical formula in Eq. (54).

Simulated double-quantum filtering efficiency as a function of the phase adjustment parameter [see Eq. (A10)]. Dashed line: simulations of for the sequence, at the point of maximum double-quantum-filtered signal . Solid line: simulations of for the supercycle, at the point of maximum double-quantum-filtered signal . All simulations were performed using the spin system parameters in Table I, at an external magnetic field of , a spinning frequency of , and a rf nutation frequency of .

Simulated double-quantum filtering efficiency as a function of the phase adjustment parameter [see Eq. (A10)]. Dashed line: simulations of for the sequence, at the point of maximum double-quantum-filtered signal . Solid line: simulations of for the supercycle, at the point of maximum double-quantum-filtered signal . All simulations were performed using the spin system parameters in Table I, at an external magnetic field of , a spinning frequency of , and a rf nutation frequency of .

Molecular systems used in the experimental demonstrations. Gray circles indicate labels. (a) -all--retinal, (b) diammonium -fumarate, and (c) -glycine.

Molecular systems used in the experimental demonstrations. Gray circles indicate labels. (a) -all--retinal, (b) diammonium -fumarate, and (c) -glycine.

Pulse sequences for the application of SR26 to the spectroscopy of organic solids. (a) Pulse sequence for the double-quantum filtering of cross-polarized NMR signals. The shaded elements are given a four-step phase cycle to select signals passing through -quantum coherence. The thin rectangles represent pulses. (b) Pulse sequence for the determination of relative CSA tensor orientations. A single rotor period of double-quantum evolution is inserted, interrupted by two strong pulses, and separated by an interval (white rectangles). A series of experiments is performed in which is increased, moving the pulses from the center of the rotational period to the ends of the rotational period .

Pulse sequences for the application of SR26 to the spectroscopy of organic solids. (a) Pulse sequence for the double-quantum filtering of cross-polarized NMR signals. The shaded elements are given a four-step phase cycle to select signals passing through -quantum coherence. The thin rectangles represent pulses. (b) Pulse sequence for the determination of relative CSA tensor orientations. A single rotor period of double-quantum evolution is inserted, interrupted by two strong pulses, and separated by an interval (white rectangles). A series of experiments is performed in which is increased, moving the pulses from the center of the rotational period to the ends of the rotational period .

Double-quantum-filtered signal amplitudes for -all--retinal using the SR26 supercycle. The excitation interval was incremented in steps of a half-supercycle, while the reconversion interval was decremented at the same time to keep the total interval fixed at . The experimental amplitudes (gray curves) are compared with the analytical functions in Eq. (57), adjusting the vertical scale of the analytical functions in each case to obtain the best fit. The dipole-dipole couplings and corresponding internuclear distances are as follows: (i) (, ); (ii) (, ); (iii) (, ); (iv) (, ); and (v) (, ). The bold line (iii) (corresponding to ) is the best fit.

Double-quantum-filtered signal amplitudes for -all--retinal using the SR26 supercycle. The excitation interval was incremented in steps of a half-supercycle, while the reconversion interval was decremented at the same time to keep the total interval fixed at . The experimental amplitudes (gray curves) are compared with the analytical functions in Eq. (57), adjusting the vertical scale of the analytical functions in each case to obtain the best fit. The dipole-dipole couplings and corresponding internuclear distances are as follows: (i) (, ); (ii) (, ); (iii) (, ); (iv) (, ); and (v) (, ). The bold line (iii) (corresponding to ) is the best fit.

Double-quantum evolution trajectories as a function of the separation between the pulses in Fig. 7(b). All simulations and experiments use double-quantum excitation and reconversion intervals of . (a) Experimental data points for diammonium -fumarate (gray squares) and simulated double-quantum trajectory for the parameters given in the first column in Table I. (b) Simulated double-quantum evolution trajectories for the parameters given in the first column in Table I, except for the Euler angles , which are specified in the plot.

Double-quantum evolution trajectories as a function of the separation between the pulses in Fig. 7(b). All simulations and experiments use double-quantum excitation and reconversion intervals of . (a) Experimental data points for diammonium -fumarate (gray squares) and simulated double-quantum trajectory for the parameters given in the first column in Table I. (b) Simulated double-quantum evolution trajectories for the parameters given in the first column in Table I, except for the Euler angles , which are specified in the plot.

Multiple-quantum-filtered signal amplitudes for -glycine using SR26 recoupling sequences of different durations. Left column: multiple-quantum-filtered signal amplitudes, relative to a cross-polarization experiment. Right column: representation of the same data on a logarithmic plot, using gray coloring to represent negative values, which were inverted before taking the logarithm. The error bars in the right-hand column were estimated by analyzing the data from a set of identical experiments (see text). The recoupling durations are the same for the excitation and reconversion sequences and are as follows: (a) 5.4, (b) 10.8 (c) 21.6, and (d) .

Multiple-quantum-filtered signal amplitudes for -glycine using SR26 recoupling sequences of different durations. Left column: multiple-quantum-filtered signal amplitudes, relative to a cross-polarization experiment. Right column: representation of the same data on a logarithmic plot, using gray coloring to represent negative values, which were inverted before taking the logarithm. The error bars in the right-hand column were estimated by analyzing the data from a set of identical experiments (see text). The recoupling durations are the same for the excitation and reconversion sequences and are as follows: (a) 5.4, (b) 10.8 (c) 21.6, and (d) .

## Tables

Spin interaction parameters used in the NMR simulations.

Spin interaction parameters used in the NMR simulations.

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