Efficient Amplitude-Modulated Pulses for Triple- to Single-Quantum Coherence Conversion in MQMAS NMR

The conversion between multiple- and single-quantum coherences is integral to many nuclear magnetic resonance (NMR) experiments of quadrupolar nuclei. This conversion is relatively inefficient when effected by a single pulse, and many composite pulse schemes have been developed to improve this efficiency. To provide the maximum improvement, such schemes typically require time-consuming experimental optimization. Here, we demonstrate an approach for generating amplitude-modulated pulses to enhance the efficiency of the triple- to single-quantum conversion. The optimization is performed using the SIMPSON and MATLAB packages and results in efficient pulses that can be used without experimental reoptimisation. Most significant signal enhancements are obtained when good estimates of the inherent radio-frequency nutation rate and the magnitude of the quadrupolar coupling are used as input to the optimization, but the pulses appear robust to reasonable variations in either parameter, producing significant enhancements compared to a single-pulse conversion, and also comparable or improved efficiency over other commonly used approaches. In all cases, the ease of implementation of our method is advantageous, particularly for cases with low sensitivity, where the improvement is most needed (e.g., low gyromagnetic ratio or high quadrupolar coupling). Our approach offers the potential to routinely improve the sensitivity of high-resolution NMR spectra of nuclei and systems that would, perhaps, otherwise be deemed “too challenging”.


S1. Procedure for computer-based optimisation of FAM-N pulses
High-throughput computer optimisation was performed using the SIMPSON density matrix simulation program, S1 executed using MATLAB S2 routines. Specified within the optimisation is the external magnetic field strength, the MAS rate, the inherent radiofrequency (rf) nutation rate and the quadrupolar parameters C Q and η Q . The simulation determines the amount of central-transition single-quantum coherences generated from conversion of a unit amount of triple-quantum coherences of the same sign. This process is illustrated below using the example of the conversion of triple-to single-quantum coherences for a single 87 Rb (I = 3/2) species with C Q = 1.2 MHz and η Q = 0, at B 0 = 14.1 T, using rf pulses with ω 1 /2π = 150 kHz at an MAS rate, ω R /2π, of 12.5 kHz.
Step 1: The amount of central-transition single-quantum coherence generated from unit triple-quantum coherence of the same sign is monitored in a series of simulations as the duration of the rf pulse is varied. The point at which maximum conversion is obtained (for one pulse, p1) is highlighted by the red dotted line ( Figure S1). Figure S1. Plot of the amount of single-quantum coherences generated from unit triplequantum coherences for a single 87 Rb (I = 3/2) species with C Q = 1.2 MHz and η Q = 0. The pulse duration resulting in the maximum conversion efficiency is highlighted by the red dotted line 3 Step 2: When the pulse duration resulting in the maximum amount of single-quantum coherences has been determined, a new SIMPSON input file is written by the code, in which a second, oppositely-phased pulse (p2) is added, and the simulation is repeated.
The point at which maximum conversion is now obtained (for two pulses) is highlighted by the red dotted line ( Figure S2).

Figure S2.
A second oppositely-phased pulse is added to the SIMPSON input file after p1.
The duration of this pulse is varied and the amount of single-quantum coherences generated is plotted. The new point at which maximum conversion is obtained is highlighted by the red dotted line.
Step 3: The duration of p1 is then increased by one increment and the variation of p2 repeated. If a further increase is observed, this process is repeated, i.e., more increments are added to p1). The point at which maximum conversion efficiency is obtained is again highlighted by the red dotted line, while the dark blue dotted line shows the duration of p1 for which maximum efficiency in the two pulse conversion is obtained ( Figure S3). Figure S3. The duration of p1 is incremented and variation of p2 repeated. If efficiency improves this process is repeated. The total pulse duration resulting in maximum efficiency is shown by the red dotted line, while the dark blue dotted lines shows the duration of p1 when maximum efficiency (of the two pulse sequence) is obtained.
Step 4: The program will continue to increase the length of p1, even if the amount of single-quantum coherences is decreasing, for a fixed number of times, determined by the parameter "countmax" specified in the MATLAB routine. If the maximum signal increases in any one incrementation of p2, countmax is reset to zero. Once countmax is reached, the incrementation of p1 is stopped, and the value of the maximum amount of singlequantum coherences compared to that obtained with the previous increment of p1.
Step 5: For the point where maximum efficiency has been obtained, a new pulse is added (again with opposite phase to the previous one) and steps 3 and 4 are repeated, with a variation in the duration of the N th pulse for a number of increments of the (N -1) th pulse.
( Figure S4). Figure S4. A third (oppositely-phased) pulse is added and the process is repeated.
Step 6: This procedure is repeated until the maximum amount of single-quantum coherence achieved with N pulses is less than that achieved with N -1 pulses. (  6 Figure S6. This procedure is repeated until the maximum amount of single-quantum coherence achieved with N pulses is less than that achieved with N -1 pulses. Step 7: At this point the program terminates. The composite pulse that produced the maximum amount of single-quantum coherence is saved as a text file and as a MATLAB file. If desired, the program can also include the FAM-N pulse directly into a Bruker Topspin 3 pulse sequence for use experimentally. Examples of the MATLAB input and output (text) files for this calculation can be found in the Supporting Information.
The composite pulse obtained in this way is termed here FAM-N. In the example described here, the FAM-N pulse obtained consists of 6 oppositely-phased pulses, with a total duration of 5.63 µs, with components of 1.77, 1.26, 0.81, 0.81, 0.70 and 0.44 µs, respectively, as shown in Figure S7. The optimisation procedure described here does not take into account the effects of varying the lengths of the pulses before the (N -1) th pulse on the overall efficiency of the FAM-N pulse. However, as shown in Table S1, in simulations where the lengths of the first, second, third and fourth pulses of a composite FAM-N pulse, comprising 6 oppositely-phased pulses, were incremented after generation of the optimum pulse following Steps 1-7 above, negligible improvement (a maximum of 0.39%) in the conversion efficiency was observed. Therefore, owing to the additional time cost and very limited benefits of such an additional variation of the pulses, the optimisations reported here omit this process. It should also be noted that the pulse length increments considered here are smaller than the pulse digitisation possible for most spectrometers -for example, the Bruker Avance III spectrometer used in this work has a digitisation limit of 75 ns (twice the value of the pulse length increment during FAM-N optimisation), meaning that any small theoretical gains in signal intensity during the optimisation are likely to be lost during the experimental execution of the pulse.
8 Table S1. The changes to the pulse lengths and signal enhancement (relative to an unmodified pulse) of a FAM-N pulse (N = 6) optimised (for 87 Rb with C Q = 1.2 MHz, ω 1 /2π = 150 kHz) using the method outlined in the text, when pulses before the (N -1) th pulse were reoptimised.

Pulse varied
Change to pulse length / ns (  have been constrained so that each pulse is shorter than the preceding pulse.
In this work, we carried out a procedure conceptually similar to that of Morais et al., but with no constraint on the relative lengths or efficiencies of the individual pulses, and with the pulse durations simply optimised to obtain the overall maximum efficiency. For 87 Rb (9.4 T, C Q = 1 MHz, ω R /2π = 15 kHz MAS, ω 1 /2π = 100 kHz), Table S2 Table 1 of the original work S4 for further details). However, all composite pulses were between 60 and 120% more efficient than single-pulse conversion. using the phase-modulated split-t 1 shifted-echo pulse sequence shown in Figure 1 of the main text, with ω R /2π = 12.5 kHz. In each case, two-dimensional spectra were acquired by averaging 192 transients with a recycle interval of 0.25 s, for 160 increments of 71.1 µs. In all cases, the excitation of triple-quantum coherences was carried out using a pulse duration of 4.75 µs, with ω 1 /2π = 123 kHz. All conversion pulses used with ω 1 /2π = 123 kHz, with the exception of the central-transition selective pulse used in SPAM where ω 1 /2π = 13 kHz. The conversion of triple-to single-quantum coherences was performed using one of the following pulses: (a) single pulse p1 = 1.5 µs (b) FAM-N Generated using C Q = 5.30 MHz, η Q = 0.11 26 oppositely-phased pulses for a total duration of 11.02 µs, with the individual components of 1. 40, 0.54, 0.41, 0.45, 0.40, 0.40, 0.36, 0.40, 0.32, 0.77, 0.72, 0.27, 0.32, 0.32, 0.32,0.32, 0.23, 0.32, 0.36, 0.27,0.32, 0.41, 0.32, 0.41, 0.50, 0.18 µs.